{"title":"VibeMathed - math problems solved with AI","url":"https://vibemathed.com","license":"https://vibemathed.com/data-license","licenseSummary":"VibeMathed-authored content (classifications, scores, notes, structure) is CC BY 4.0. Quoted third-party material remains under its own rights.","methodology":"https://vibemathed.com/methodology","generated":"2026-10-31T23:38:14.531Z","count":1284,"problems":[{"slug":"a-lower-bound-of-21-for-3-3-matrix-multiplication-over-every-field","name":"A lower bound of 21 for 3 × 3 matrix multiplication over every field","shortName":"3 × 3 matrix multiplication: bilinear rank at least 21","problemNumber":null,"field":"Algebraic complexity; exact bilinear complexity and tensor rank","fieldGroup":"Theoretical computer science","statement":"For a field $K$, what is the smallest number $r$ of products of input linear forms needed to multiply two arbitrary $3\\times3$ matrices exactly? Equivalently, how small can $r$ be in an identity $AB=\\sum_{i=1}^{r}a_i(A)b_i(B)W_i$ holding for every pair $A,B$, with $K$-linear forms $a_i,b_i$ and fixed matrices $W_i$? The minimum is the exact bilinear complexity $R_K\\langle3,3,3\\rangle$. Which lower bounds hold uniformly over all fields?","posedBy":null,"yearPosed":null,"ageNote":"Laderman's 23-term algorithm (1976) works over every field and Bläser proved 19 in 2003; Bläser's survey Fast Matrix Multiplication (Theory of Computing Graduate Surveys 5, 2013) gives the range as 19 to 23. Field-specific bounds over F_2 and for integer coefficients rose to 21, 22 and 23 in September and October 2026.","solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"The formal proof establishes $R_K\\langle3,3,3\\rangle\\geqslant21$ for every field $K$, including finite and infinite fields in every characteristic. The input linear forms and output matrices are unrestricted; there are no minimality, nonzero-term, symmetry or coefficient-rank assumptions. The argument passes to an algebraic closure, chooses a decomposition minimizing a coefficient-rank sum, and uses conductor growth and residual reconstruction to rule out at most 20 terms. This is partial progress on exact rank: it does not determine the exact rank over every field and gives no border-rank or asymptotic-exponent bound. Results establishing 22 or 23 over $\\mathbb F_2$ or for integer-coefficient algorithms do not imply the same numerical bound over arbitrary fields. The README places this result alongside the classical general lower bound 19 and recent field-specific results.","claimIssueNote":null,"solveDate":"2026-10-12","model":"GPT-6 Astra (gpt-6-astra), with GPT-6 Sol (gpt-6-sol) used for part of the work","modelMaker":"OpenAI","humanCollaborators":["Andrey Ivanov"],"aiRole":"Andrey Ivanov posed the research task and directed the project. According to his account, GPT-6 Astra (gpt-6-astra) performed most of the work, contributing both to the development of the mathematical proof and to its Lean formalization. GPT-6 Sol (gpt-6-sol) was also used for part of the work to reduce costs. Ivanov supplied known results missed during the models' literature searches, redirected work when approaches reached dead ends, and organized the research workspace around a catalogue of results, Lean formalization, and SageMath for rapid testing of conjectures. Mathematical ideas were discussed throughout the process, but the extent to which those discussions influenced particular model-generated ideas cannot be reconstructed reliably. The AI co-developed classification describes this human-directed collaboration without assigning individual lemmas or a numerical share of the mathematical contribution to either party. Ivanov is the author and responsible maintainer. The repository's own disclosure at the registered commit (README and formalization.yaml) says the research and formalization had substantial assistance from OpenAI's ChatGPT/Codex, model versions not recorded; the model names above are the author's later clarification.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Read by this site on 15 October 2026, not rebuilt. The headline theorem MMRank.Palomar.lower_bound was read in Challenge.lean, which imports only Mathlib: for any field K and finite index set I, if $AB=\\sum_i a_i(A)b_i(B)W_i$ for all $3\\times3$ matrices $A,B$, with arbitrary linear forms $a_i,b_i$ and matrices $W_i$, then $|I|\\geqslant21$. That is exactly bilinear rank (tensor rank) of the 3x3 matrix multiplication tensor: exact equality, not border rank, no restriction on the algorithm, and an arbitrary Field includes finite fields and characteristic 2. Solution.lean proves the same statement and comparator.json pairs the two. A grep of all 198 Lean files (18,829 lines) found only the deliberate sorry in Challenge.lean, which nothing imports; no axiom declarations, native_decide, unsafe, implemented_by or extern. The repository's CI run 38306224074 on commit cb1557b printed propext, Classical.choice and Quot.sound for the headline and audited 1,842 project declarations; Palomar run 38306239835 fetched the same commit and its Comparator step passed. Both logs were read here; neither build was repeated. The proof itself was not read.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The exact rank of 3x3 matrix multiplication is a classical question of algebraic complexity: 19 (Bläser 2003) to 23 (Laderman 1976) over a general field for over twenty years. A specific small-format constant rather than the exponent, so well below the matrix multiplication exponent entries at 55 and the DFT n log n question at 45, and well above the four-term Boolean multiplicative complexity case at 12.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/AIV5/mmrank21/tree/cb1557b7ea80ea66fae4714628663d8c1df7a48d","sourceName":"GitHub repository with Lean proof","collection":null,"collectionVersion":null,"links":[{"label":"Palomar registry record, version 1","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-10-13-000009&version=1","kind":"other"},{"label":"Archived release 0.1.1 on Zenodo","url":"https://doi.org/10.5281/zenodo.23324839","kind":"other"},{"label":"Standalone statement using only Mathlib","url":"https://github.com/AIV5/mmrank21/blob/cb1557b7ea80ea66fae4714628663d8c1df7a48d/Challenge.lean","kind":"lean-proof"},{"label":"Proved statement and proof entry point","url":"https://github.com/AIV5/mmrank21/blob/cb1557b7ea80ea66fae4714628663d8c1df7a48d/Solution.lean","kind":"lean-proof"},{"label":"Public theorem for arbitrary linear forms","url":"https://github.com/AIV5/mmrank21/blob/cb1557b7ea80ea66fae4714628663d8c1df7a48d/MMRank/PublicTheorem.lean","kind":"lean-proof"},{"label":"Authorship and initial AI disclosure in the registered snapshot","url":"https://github.com/AIV5/mmrank21/blob/cb1557b7ea80ea66fae4714628663d8c1df7a48d/formalization.yaml","kind":"other"},{"label":"Palomar mechanical verification of this commit","url":"https://github.com/PalomarRegistry/PalomarSubmission/actions/runs/38306239835","kind":"other"},{"label":"Proof overview","url":"https://github.com/AIV5/mmrank21/blob/cb1557b7ea80ea66fae4714628663d8c1df7a48d/docs/PROOF_OVERVIEW.md","kind":"other"},{"label":"Related result over F2: exact rank 23","url":"https://arxiv.org/abs/2610.11823v1","kind":"paper"},{"label":"Bläser 2003: the previous general lower bound of 19","url":"https://doi.org/10.1016/S0885-064X(02)00007-9","kind":"paper"},{"label":"Bläser, Fast Matrix Multiplication (survey, 2013): the range 19 to 23","url":"https://theoryofcomputing.org/articles/gs005/","kind":"problem-record"},{"label":"Related result over F2: lower bound 21 (Wang)","url":"https://arxiv.org/abs/2609.06725","kind":"paper"},{"label":"Related result over the integers: lower bound 22 (Rudich and Rousseau)","url":"https://arxiv.org/abs/2610.01639","kind":"paper"}],"relations":[],"submittedBy":"VelvetCobra335","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dow-tkachuk-question-4-8-pseudocomplete-countable-closures-in-c-p-x-force-x-to-b","name":"Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$-space","shortName":"Dow–Tkachuk Q4.8 ($C_p(X)$ and $P$-spaces)","problemNumber":null,"field":"General topology; $C_p$-theory","fieldGroup":"Geometry & topology","statement":"Dow and Tkachuk asked (Question 4.8): assume that the closure of $A$ is pseudocomplete for every countable $A \\subset C_p(X)$. Must the Tychonoff space $X$ be a $P$-space? Here $C_p(X)$ is the space of continuous real-valued functions on $X$ with the topology of pointwise convergence, pseudocompleteness is in Oxtoby's sense (a sequence of $\\pi$-bases along which every closure-nested choice has nonempty intersection), and a $P$-space is one in which every $G_\\delta$ set is open.","posedBy":"Alan Dow and Vladimir V. Tkachuk, \"Reflecting topological properties in closed separable subspaces\", European Journal of Mathematics 11 (2025), Question 4.8","yearPosed":2025,"ageNote":"Published online 27 December 2024, in the March 2025 issue of Eur. J. Math. 11. The paper's Theorem 3.10 had settled the functionally countable case.","solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"Yes, in ZFC. The paper proves a stronger statement: if a Tychonoff space $X$ is not a $P$-space, then $C_p(X)$ contains a closed, separable, metrizable linear subspace that is not pseudocomplete. A closed separable subspace is the closure of a countable subset, so if all such closures are pseudocomplete, $X$ must be a $P$-space. Dow and Tkachuk had shown this under the extra assumption that $X$ is functionally countable (their Theorem 3.10); the result removes that assumption. The companion Question 4.9 (whether $X$ must also be functionally countable) is not addressed.","claimIssueNote":null,"solveDate":"2026-10-12","model":"ChatGPT Astra (proof); Claude Opus 5.5 (Lean formalization)","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Lucas Alves de Almeida"],"aiRole":"The main idea came from the author, Lucas Alves de Almeida: instead of arguing directly from the hypothesis on countable closures, prove the stronger statement that if $X$ is not a $P$-space, then $C_p(X)$ contains a closed separable metrizable linear subspace which is not pseudocomplete. ChatGPT Astra (OpenAI) worked out the details and completed the argument. The author edited the manuscript and checked the literature. Claude Opus 5.5 (Anthropic) helped review the proof and typeset the paper, and produced the Lean 4 / Mathlib formalization. The repository's disclosure classifies the work as AI-assisted.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked by this site on 15 October 2026, not rebuilt. Question 4.8 was read in the open-access Dow-Tkachuk paper (Eur. J. Math. 11 (2025), Art. 5) and the statement matches; their paper assumes all spaces Tychonoff and defines pseudocompleteness by Oxtoby's sequences of pi-bases, with no extra regularity clause. The Lean headline question_4_8 in Question48.lean (commit 8b4c7d0, SHA-256 matching the repository's certificate) was read with its definitions: Cp X is the continuous functions as a subspace of the product X -> R (pointwise, not Mathlib's compact-open C(X, R)), closure is taken in Cp X, Pseudocomplete is exactly the paper's definition, and IsPSpace says countable intersections of open sets are open, equivalent to every G-delta being open. The hypothesis is CompletelyRegularSpace, weaker than Tychonoff, so the formal statement is if anything stronger. A grep of the file found no sorry, admit, axiom, native_decide, unsafe, opaque or implemented_by; the axiom list (propext, Classical.choice, Quot.sound) is the author's build output, not re-run here. The paper's short proof was also read and no gap was found; that is not an expert review. The formal statement was written in the same workflow as the proof and is not independently anchored.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"Question 4.8 in the closing list of a 2025 paper by Dow and Tkachuk, two leading names in set-theoretic topology and Cp-theory, who proved it for functionally countable X and called the general case open; asked about 22 months ago, specialist audience, no further literature. Level with HKKP Question 6.24 at 8, a numbered question removing one hypothesis; above Arnosti's RANDOM-VERTEX conjecture at 7; below the countable-ordinal Vietoris powers question at 12.","solveCostUsd":0,"solveCostNote":"Author's disclosure: no dedicated spend; all work was done on the standard $20/month ChatGPT (OpenAI) and Claude (Anthropic) plans.","sourceUrl":"https://github.com/LucasAlves8Sp/dow-tkachuk-question-4-8","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[{"label":"Lean 4 formalization (Question48.lean)","url":"https://github.com/LucasAlves8Sp/dow-tkachuk-question-4-8/blob/main/Question48.lean","kind":"lean-proof"},{"label":"Paper (PDF)","url":"https://github.com/LucasAlves8Sp/dow-tkachuk-question-4-8/blob/main/paper/Pseudocompleteness_of_countable_closures.pdf","kind":"paper"},{"label":"Dow and Tkachuk, Reflecting topological properties in closed separable subspaces (Question 4.8)","url":"https://doi.org/10.1007/s40879-024-00790-w","kind":"problem-record"}],"relations":[],"submittedBy":"StormyGander827","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strict-neumann-polya-inequality-on-simply-connected-planar-lipschitz-domains","name":"Strict Neumann Pólya inequality on simply connected planar Lipschitz domains","shortName":"Strict Neumann Pólya (simply connected)","problemNumber":null,"field":"Spectral geometry","fieldGroup":"Analysis","statement":"Does the Neumann Pólya conjecture hold for every bounded, connected planar Lipschitz domain $\\Omega\\subset\\mathbb R^2$? More precisely, is it true that\n$$\n|\\Omega|\\mu_j(\\Omega)\\le 4\\pi j\n\\qquad(j\\ge1),\n$$\nwhere $0=\\mu_0(\\Omega)<\\mu_1(\\Omega)\\le\\mu_2(\\Omega)\\le\\cdots$ are the Neumann eigenvalues of $\\Omega$, counted with multiplicity?","posedBy":"George Pólya, Mathematics and Plausible Reasoning (1954); tiling case proved by Pólya (1961) and Kellner (1966)","yearPosed":1954,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Main theorem: for every bounded simply connected planar Lipschitz domain $\\Omega$, $|\\Omega|\\mu_j(\\Omega)<4\\pi j$ for all $j\\ge1$, with $0=\\mu_0<\\mu_1\\le\\cdots$ counted with multiplicity; equivalently $N_N(E)>|\\Omega|E/(4\\pi)$ for $E>0$, where $N_N$ counts $\\mu_j4\\pi j$ for all $j\\ge1$, equivalently $N_D(E)<|\\Omega|E/(4\\pi)$ for $E>0$ (eigenvalues counted with multiplicity, endpoint included). This is the strict form of Pólya's Dirichlet inequality on the whole simply connected planar class. Previously it was known for tiling domains (Pólya 1961), disks and sectors (Filonov, Levitin, Polterovich and Sher 2023), annuli and a few further families. Multiply connected planar domains and higher dimensions remain open. The companion strict Neumann result for simply connected Lipschitz domains is a separate entry.","claimIssueNote":null,"solveDate":"2026-10-09","model":"ChatGPT; Aristotle; Codex","modelMaker":"OpenAI; Harmonic","humanCollaborators":["Quanyu Tang","Zuoqin Wang"],"aiRole":"ChatGPT assisted the authors in developing the mathematical result and its informal proof. Aristotle and Codex assisted the authors in producing the Lean formalization, including translating proof ideas into Lean, constructing bridge lemmas, debugging, and source and axiom audits. The authors reviewed the mathematical statement, the correspondence between the informal result and the Lean statements, and the final repository. The Lean kernel and the Palomar verification workflow checked the registered formal proof.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Read by this site on 15 October 2026, not rebuilt. The Palomar record PALOMAR-2026-10-09-000003 v1 was fetched from the registry's data API: registered commit ba3690b, Comparator theorem PalomarDirichlet.main_result, checked with the Lean, nanoda and con-ron kernels, automated review neutral with no warnings, trust level high. Challenge.lean at that commit (identical on main; later commits touch only documentation) was read: Mathlib only, bounded open simply connected Omega in C, and lambda_j defined by min-max of the Rayleigh quotient over j-dimensional spaces of smooth compactly supported functions in Omega, which is the standard Dirichlet eigenvalue. The registered snapshot (about 27,000 lines) was grepped: the only sorry is the Challenge statement hole; no admit, axiom declarations, native_decide, unsafe or implemented_by. The repository's GitHub Actions build and audit passed on the registered commit; the audit asserts only propext, Classical.choice and Quot.sound. A separate check written here on disks and squares (3,000 eigenvalues each) agrees with the strict inequality; it says nothing about general domains. The informal proof was not read. Lean-checked, not Lean-verified: the statement audit is this site's alone and nothing was rebuilt here.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Pólya's 1954 conjecture is a flagship problem of spectral geometry, open for seventy years outside tiling domains, balls, sectors and annuli. A claim for every simply connected planar domain with no regularity is much the largest known case, but a case. Above the hot spots entry for smooth simply connected domains (42) and the Neumann balls entry (35), below Schiffer's conjecture (53).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2610.13139","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Palomar Registry: PALOMAR-2026-10-09-000003 v1","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-10-09-000003&version=1","kind":"palomar"},{"label":"Lean formalization (GitHub)","url":"https://github.com/QuanyuTang/polya-dirichlet-simply-connected-formalization-Palomar-version","kind":"lean-proof"},{"label":"Registered snapshot, commit ba3690b","url":"https://github.com/QuanyuTang/polya-dirichlet-simply-connected-formalization-Palomar-version/tree/ba3690bfd48618cf48da07fa4555382bd9852ff3","kind":"code"}],"relations":[{"to":"polya-conjecture-neumann-balls","kind":"related","note":"Both are cases of Pólya's 1954 eigenvalue conjecture: Neumann balls in dimension three and higher there, all simply connected planar domains for Dirichlet here."}],"submittedBy":"StormyCobra617","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"nonquadraticity-of-lambert-series-values-and-improved-bounds-for-erdos-problem-1","name":"Nonquadraticity of Lambert series values and improved bounds for Erdős Problem 1049","shortName":"Lambert Nonquadraticity","problemNumber":1049,"field":"Irrationality and algebraic degree","fieldGroup":"Number theory","statement":"Let $t>1$ be a rational number. Is $\\sum_{n\\ge1}\\frac{1}{t^n-1}=\\sum_{n\\ge1}\\frac{\\tau(n)}{t^n}$ irrational, where $\\tau(n)$ counts the divisors of $n$? A conjecture of Chowla; Erdős (1948) proved it for every integer $t\\ge2$.","posedBy":"Chowla (conjecture); Erdős, Problem 1049 (Er88c, p. 102)","yearPosed":1988,"ageNote":"erdosproblems.com attributes the conjecture to Chowla; Erdős's 1948 paper on Lambert series already cites Chowla, and the problem is listed from Erdős's 1988 survey. Open as of 15 October 2026 (erdosproblems.com, last edited 28 September 2025).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"For rational bases $t=a/b>1$, $F(t)=\\sum_{n\\ge1}(t^n-1)^{-1}$ is irrational whenever $\\log b/\\log a<0.5728$, and a formalized refinement reaches $0.573290$. The previous thresholds were $1/2-1/\\pi^2\\approx0.3987$ (Bundschuh and Väänänen, 1994) and $0.405683$ (W. Cook, September 2026, self-published). Problem 1049 itself stays open: $t=3/2$, with $\\log2/\\log3\\approx0.631$, is still outside the region.\n\nAlong the way, for every integer $q\\ge2$ the numbers $1,F(q),F(q)^2$ are linearly independent over $\\mathbb Q$, so the Erdős-Borwein constant $F(2)$ is not quadratic irrational. Both follow from one degree criterion for real algebraic bases: no relation of degree at most $e$ over $\\mathbb Q(q)$ when $e\\lambda\\log M(q)<\\log q$, with $\\lambda=(42963+428220/\\pi^2)/202140\\approx0.42718$.","claimIssueNote":null,"solveDate":"2026-10-07","model":"Generative AI, model not named in the source (submitter: GPT-6 Astra via OpenAI Codex)","modelMaker":"OpenAI","humanCollaborators":["Costa Lambrinoudis"],"aiRole":"The repository README says generative AI performed nearly all of the mathematical work: it developed the determinant construction and the algebraic-degree arguments, conducted the literature search, wrote the manuscript and produced the Lean formalization, with Costa Lambrinoudis directing and supervising. The manuscript's footnote says the same and names no model; the model and the use of OpenAI Codex are the submitter's account.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Read by this site on 15 October 2026, not rebuilt. The Lean development (Lean 4.33.1, mathlib, about 200 files at commit 008e313) has no sorry, admit, axiom declarations, native_decide or implemented_by in the library; the one unsafe def is the optional kernel-replay script under verification/, outside the library. lambertValue is the series $\\sum_{n\\ge0}1/(q^{n+1}-1)$, faithful to the problem. The headline statements were read: lambertValue_irrational_of_log_ratio_lt_5728 and lambertValue_irrational_of_certified_region (any integers $02$ and $H^3(X;M)=0$ for every finitely generated $\\mathbb ZG$-module $M$. Wall showed such $X$ has a finite three-dimensional model and asked whether the three-cells can be removed. Stably the answer is yes (Cohen), and it is yes for many finite groups (Johnson, Hambleton, Hofmann-Nicholson); candidate counterexamples (Cohen-Dyer, Bridson-Tweedale) stayed unresolved or conditional on relation-gap assertions. Wall's finite $D(2)$ problem: is every finite $D(2)$ complex homotopy equivalent to a finite CW complex of dimension at most two?","posedBy":"C. T. C. Wall","yearPosed":1965,"ageNote":"Wall, Finiteness conditions for CW-complexes (Annals 1965, Section 2); repeated as Problem D3 in Wall's 1979 problem list. Both are cited by the manuscript.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a finite connected three-dimensional CW complex $X$ with $H_i(\\widetilde X;\\mathbb Z)=0$ for $i>2$ and $H^3(X;M)=0$ for every $\\mathbb Z[\\pi_1X]$-module $M$ that is not homotopy equivalent to any finite complex of dimension at most two. The obstruction: on a finite 2-complex, a rank-one local system with vanishing twisted $H_2$ is trivial on every finite-order element of $\\pi_1$. $\\pi_1(X)$ is infinite. It does not settle the D(2) problem for finite fundamental groups, the relation gap problem, or the Eilenberg-Ganea problem.","claimIssueNote":null,"solveDate":"2026-10-06","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Wall's question in the formulation the manuscript takes from Wall (1965, 1979) and Hofmann-Nicholson: finiteness of both complexes, no asphericity hypothesis. The construction is a Quillen plus construction on a five-generator four-relator presentation complex (finite D(2) by Mannan's theorem), with an obstruction from rank-one twisted second homology proved by a Howie-style cyclic tower. The example has infinite fundamental group, so the problem for finite groups, where most work concentrated, remains open. The proof was not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":46,"significanceNote":"Wall's D(2) problem is a classical, sixty-year-old question in low-dimensional homotopy theory, tied to the relation gap and Eilenberg-Ganea problems and famous within that community. Placed below The Borel conjecture for closed aspherical four-manifolds (60), and above Wall's manifold-realization question for Poincare duality groups (42) and The CAT(0) conjecture for Artin groups (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf","sourceName":"OpenAI math release: A Counterexample to Wall's D(2) Problem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"at-least-five-colours-for-the-regular-fourteen-gon-minkowski-plane","name":"At least five colours for the regular fourteen-gon Minkowski plane","shortName":"Fourteen-gon chromatic lower bound","problemNumber":null,"field":"Discrete geometry; unit-distance graph colouring","fieldGroup":"Geometry & topology","statement":"Let $C$ be a centrally symmetric convex curve and let $\\chi(\\mathbb R^2,C)$ be the least number of colours needed to colour the plane so that no two points at $C$-norm distance one share a colour. Exoo, Fisher and Ismailescu proved $\\chi(\\mathbb R^2,C)\\ge5$ for the regular octagon, decagon and dodecagon and asked (Problem 5.1) whether $\\chi(\\mathbb R^2,C)\\ge5$ for every regular $2n$-gon with $n\\ge4$. This entry is the case $n=7$: with $P_{14}=\\operatorname{conv}\\{z\\in\\mathbb C:z^{14}=1\\}$ as the unit ball, is $\\chi(\\mathbb R^2,P_{14})\\ge5$?","posedBy":"Geoffrey Exoo, David Fisher and Dan Ismailescu, The chromatic number of the Minkowski plane: the regular polygon case, Problem 5.1 (arXiv:2108.12861, 2021)","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"We prove $\\chi(\\mathbb R^2,P_{14})\\ge5$ using a graph with 1,540 vertices and 13,755 unit edges. Every edge difference is a rotated point on a side of the unit polygon with an exact algebraic parameter in $[0,1]$. A kernel-checked LRAT refutation of the graph's four-colouring CNF rules out every four-colouring, and restricting a colouring of the plane to these points gives the contradiction. The whole geometric and logical chain is in Lean. With Gehér's upper bound, $5\\le\\chi(\\mathbb R^2,P_{14})\\le6$; whether five colours suffice remains open, as does Problem 5.1 for other $n$.","claimIssueNote":null,"solveDate":"2026-10-06","model":"OpenAI Codex (GPT-6 Astra)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"An earlier AI-assisted research archive supplied the explicit graph, coordinates, CNF/LRAT certificate and finite logical bridge. OpenAI Codex (GPT-6 Astra) developed the complete geometric Lean proof, connected the obstruction to an actual normed real plane, removed the importer's unsafe term-evaluation calls, rebuilt the finite certificate and audited the final theorem. It also prepared the mathematical exposition and interactive diagrams. The human collaborator selected the result, directed the work, required full geometric formalization and reviewed the presentation. No independent human mathematical audit is claimed.","verification":"lean-checked","verificationNote":"Checked by this site on 7 October 2026, not rebuilt here. No independent mathematician has checked this yet. The headline theorem at_least_five_colours (formal/Final.lean) was read: any colouring of the plane by n colours that separates every pair at distance one has n at least 5. Plane is the complex numbers as a real vector space with the gauge of the convex hull of the 14th roots of unity as norm, and its distance is that norm of the difference, so the formal statement matches the posed question. The project has no sorry, admit, axiom declarations or native_decide; its SAT importer adds its LRAT steps through addDecl, so the kernel checks them. The public Actions run 37468510724 (commit e0f5e2a, success, on the author's own self-hosted Azure runner) prints propext, Classical.choice and Quot.sound for the final theorem and the graph lemma; the formal sources have not changed since that commit. A separate check written here from the graph file: all 13,755 edges have 14-gon norm 1 to within 1e-15 in floating point, and a fresh four-colouring encoding solved by CaDiCaL is unsatisfiable (157 s). Gehér's upper bound six is cited, not formalised.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"One case (n = 7) of Problem 5.1 of Exoo, Fisher and Ismailescu (2021), a stated question in the Minkowski-plane variant of Hadwiger-Nelson; the general 2n-gon question stays open. A small specialist audience. Below the unit-distance independence-ratio question (20) and the Pinwheel Kernel Conjecture (12), above Arnosti's greedy-matching conjecture (7), far below the Euclidean five-colour step (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/MathIsEvenEasier/minkowski14","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[{"label":"Interactive geometry, graph and proof map","url":"https://mathiseveneasier.github.io/minkowski14/","kind":"other"},{"label":"Mathematical argument and exact scope","url":"https://github.com/MathIsEvenEasier/minkowski14/blob/main/RESULT.rst","kind":"paper"},{"label":"Complete lower-bound theorem in Lean","url":"https://github.com/MathIsEvenEasier/minkowski14/blob/main/formal/Final.lean#L18","kind":"lean-proof"},{"label":"Compiler evidence and final axiom audit","url":"https://github.com/MathIsEvenEasier/minkowski14/blob/main/evidence/verification-final.json","kind":"code"},{"label":"Problem 5.1: Exoo, Fisher and Ismailescu (2021)","url":"https://arxiv.org/abs/2108.12861","kind":"problem-record"},{"label":"Published upper bound six: Gehér (2023)","url":"https://arxiv.org/abs/2301.13695","kind":"paper"},{"label":"Public Lean build and final axiom output","url":"https://github.com/MathIsEvenEasier/minkowski14/actions/runs/37468510724","kind":"lean-proof"}],"relations":[],"submittedBy":"MathIsEvenEasier","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"log-abundance-threefolds-char-p-numerical-dimension-one","name":"Log abundance for threefolds in characteristic $p>3$ in numerical dimension one","shortName":"Char p threefold log abundance, nu = 1","problemNumber":null,"field":"Birational geometry in positive characteristic; minimal model program","fieldGroup":"Algebra","statement":"The abundance conjecture predicts that if $(X,B)$ is a projective log canonical pair and $K_X+B$ is nef, then $K_X+B$ is semiample. For threefolds over $\\mathbb C$ this is a theorem (Miyaoka, Kawamata, Keel-Matsuki-McKernan). Over algebraically closed fields of characteristic $p>0$ it remains open; Zheng Xu proved nonvanishing for $p>3$, abundance in nef dimension at most two, and abundance in numerical dimension two. Is abundance true for projective log canonical threefold pairs in characteristic $p>3$, in particular when $K_X+B$ is nef of numerical dimension one?","posedBy":"The abundance conjecture of the minimal model program (classical; the manuscript names no poser); the remaining numerical-dimension-one case is isolated by Zheng Xu's 2024 and 2026 papers","yearPosed":null,"ageNote":"The manuscript does not attribute or date the abundance conjecture; it cites Miyaoka (1988) and Kawamata (1992) for the complex threefold case and Xu (2024, 2026) for the positive-characteristic results that leave numerical dimension one and nef dimension three open. yearPosed is left empty.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Principal Theorem 1.1: over an algebraically closed field of characteristic $p>3$, if $(X,B)$ is a projective log canonical threefold pair with effective $\\mathbb Q$-boundary, $K_X+B$ is $\\mathbb Q$-Cartier, nef and of numerical dimension one, then $K_X+B$ is semiample; neither terminality nor $\\mathbb Q$-factoriality is needed. The companion proves the terminal $\\mathbb Q$-factorial case ($K_X$ nef with $\\nu=1$ gives $\\kappa=1$ and semiampleness). Characteristics 2 and 3, numerical dimension zero and higher dimensions are not addressed.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (October 5) depends essentially on a companion of September 24 that proves the terminal case and an obstruction theorem for nef Cartier cycles.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts were read against the abundance conjecture. lean/docs/035.md does not exist at the pinned commit and formalization.yaml has no entry for either manuscript. The log result uses the companion's cycle obstruction as an essential input, so both papers must hold. Inputs from Xu's nonvanishing and numerical-dimension-two theorems and the positive-characteristic MMP are assumed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"One numerical-dimension case of threefold abundance in positive characteristic: a real step in a central program but a special case of a special case. Far below log-abundance-conjecture-char-zero (75), just below effective-log-iitaka-finite-coefficients (20) and above birkar-stein-degree-conjecture-point (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf","sourceName":"OpenAI math release: Log abundance in numerical dimension one for threefolds in positive characteristic","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Abundance in numerical dimension one for terminal threefolds in positive characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bfss-su2-positive-energy-eigenstates","name":"The BFSS exclusion of positive-energy normalizable states in the relative SU(2) model","shortName":"BFSS positive-energy bound states (SU(2))","problemNumber":null,"field":"Supersymmetric matrix quantum mechanics; embedded eigenvalues","fieldGroup":"Mathematical physics","statement":"For the relative $\\mathrm{SU}(2)$ BFSS Hamiltonian (supersymmetric quantum mechanics of three $9$-vectors with quartic potential $\\sum_{a 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the BFSS sentence quoted in the manuscript. Theorem 1.1 gives an orthonormal sequence of eigenvectors in the domain of $H$ with eigenvalues $E_j>0$, $E_j\\to\\infty$, for the closed gauge-invariant supercharge form at $N=2$. The mechanism is symmetry: $\\mathrm{Spin}(9)$ sectors with large second highest-weight coordinate exclude low transverse oscillator levels, so those sectors are confined. The manuscript says the large-$N$ matrix-theory conjecture is a separate question. Not refereed here. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A one-sentence physical claim in a famous paper, refuted at N=2 by a symmetry-sector argument; it matters to spectral theorists of matrix models but was never a central conjecture. Placed below the generalized ionization conjecture (32) and the O(4) sigma-model mass-gap asymptotics (28), above Fill's spectral gap equality (12).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf","sourceName":"OpenAI math release: Positive eigenvalues of the relative SU(2) BFSS Hamiltonian","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"The unique threshold bound state of the SU(N) BFSS model (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"simple-stochastic-games-quasipolynomial-time","name":"Quasipolynomial-time solution of simple stochastic and turn-based stochastic mean-payoff games","shortName":"Stochastic games in quasipolynomial time","problemNumber":null,"field":"Algorithmic game theory; stochastic games; complexity of NP and coNP problems","fieldGroup":"Theoretical computer science","statement":"A simple stochastic game is a finite turn-based reachability game between two players in which some vertices are chance vertices choosing between two successors with probability $1/2$. Condon (1992) showed that deciding whether the value from a start vertex is at least $1/2$ lies in $\\mathrm{NP}\\cap\\mathrm{coNP}$; Ludwig gave a randomized subexponential algorithm, and Andersson and Miltersen showed the problem polynomially equivalent to solving turn-based stochastic mean-payoff and discounted games with binary data. No algorithm subexponential and deterministic, or quasipolynomial, in the binary input length was known. Can simple stochastic games, and turn-based stochastic mean-payoff games, be solved in polynomial time, or at least in quasipolynomial time?","posedBy":"Anne Condon, The complexity of stochastic games (1992), as cited by the manuscript; the question is the polynomial-time status of a problem in NP and coNP","yearPosed":1992,"ageNote":"The manuscript cites Condon (1992) for the NP and coNP placement and does not quote a posing sentence; the polynomial-time question for simple stochastic games is commonly attributed to that paper. This entry answers the quasipolynomial form, not the polynomial-time question.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims a deterministic algorithm with $2^{O((\\log(L+2))^2)}$ bit operations that outputs the set of vertices of nonnegative value in finite turn-based stochastic mean-payoff games with signed binary rewards and arbitrary rational chance probabilities, and hence decides the threshold-1/2 problem for Condon's simple stochastic games in deterministic quasipolynomial time. The proof chooses an explicit discount of polynomial bit length and finds the discounted fixed point by a recursive labeling procedure. It does NOT give a polynomial-time algorithm, and it decides the threshold set rather than outputting values or strategies in the stated theorem.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The stochastic-games manuscript (5 October 2026) names the deterministic mean-payoff manuscript of 25 September as the direct predecessor of its labeling method and extends it to discounted operators. It has no Lean formalization in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against the question: a deterministic Turing machine outputs exactly the vertices of nonnegative value for the expectation of the pathwise liminf mean payoff, with rational chance probabilities and signed rewards in binary, in $2^{C(\\log_2(L+2))^2}$ bit operations, value-zero vertices included. Section 1.2 gives the reduction showing that this decides whether a simple stochastic game has reachability value at least $1/2$. No Lean statement covers this result. The value convention (expectation taken after the pathwise liminf) is the paper's own; it states that it proves the payoff convention directly.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":44,"significanceNote":"Condon's question on simple stochastic games is one of the best known NP and coNP problems without a polynomial algorithm, and it subsumes mean-payoff games, so a deterministic quasipolynomial bound is scored just above the mean-payoff entry from the same family (42). Below the full polynomial question; level with the Sakoda-Sipser problem (45) and below the 3SUM hypothesis (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/stochastic-mean-payoff-games.pdf","sourceName":"OpenAI math release: Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Deterministic quasipolynomial-time mean-payoff games","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf","kind":"paper"},{"label":"Companion: Randomized quasipolynomial-time mean-payoff games","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"random-bond-ising-quenched-sle3","name":"Marginal irrelevance of weak bond disorder at the planar Ising critical point: quenched SLE3 interfaces for weak random bonds","shortName":"Weak random-bond Ising: quenched SLE3","problemNumber":null,"field":"Disordered systems; two-dimensional Ising model; SLE","fieldGroup":"Probability & statistics","statement":"For the square-lattice Ising model with independent random couplings $J_e=1+\\varepsilon\\xi_e$, Harris's criterion is marginal because the pure specific-heat exponent is zero. Dotsenko and Dotsenko (1983) predicted a double-logarithmic specific heat, and Shalaev, Shankar and Ludwig developed the picture that weak disorder is marginally irrelevant: unchanged leading critical exponents with logarithmic corrections, including Shankar's $(\\log r)^{1/4}$ growth of the relative second moment of the spin correlation. Rigorously, only vanishing-disorder results were known (Mahfouf). For fixed small disorder, does the critical random-bond Ising model keep the pure model's conformally invariant critical behavior in a typical environment?","posedBy":"Vik. S. Dotsenko and Vl. S. Dotsenko (1983); B. N. Shalaev (1984); R. Shankar (1987); A. W. W. Ludwig (1990): renormalization-group predictions of marginal irrelevance","yearPosed":1983,"ageNote":"The physics predictions concern thermodynamic quantities and correlations; the manuscripts note that a quenched law for a whole interface is not itself predicted there, which is why this entry is a variant.","solveType":"proved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for any bounded mean-zero nondegenerate law $\\rho$ on $[-1,1]$ there is $\\varepsilon_0(\\rho)>0$ such that for fixed $0<\\varepsilon<\\varepsilon_0$, at the spontaneous-magnetization threshold $\\beta_c^\\rho(\\varepsilon)=\\tfrac12\\log(1+\\sqrt2)+O(\\varepsilon^2)$, the quenched Dobrushin spin interface in any bounded Jordan domain converges to chordal $\\mathrm{SLE}_3$ in probability over environments, in the bounded-Lipschitz metric on oriented curves. A companion proves, conditionally on stated deterministic reference estimates, that the relative second moment of the quenched spin correlation grows as $(\\log r)^{1/4+o(1)}$ for fair $\\pm\\varepsilon$ bonds. Not shown: strong disorder, the double-logarithmic specific heat, or unconditional logarithmic corrections.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This entry draws on three manuscripts dated October 5, 2026: quenched SLE3 for general bounded iid bond laws (principal), the symmetric two-valued case, and a conditional proof of Shankar's logarithmic relative fluctuations. They use the September 23 deterministic companions of the same family.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction, history section and Theorem 1.1 of 'Quenched SLE3 limits for general weak random-bond Ising models' were read against the predictions as the manuscripts cite them. The physics works predict exponents and logarithmic corrections for correlations, not interface laws, so the interface theorem answers a reinterpretation (variant). The companion on logarithmic relative fluctuations matches Shankar's $(\\log r)^{1/4}$ prediction but its abstract states it assumes 'the stated deterministic critical-reference estimates', so that part is conditional. Lean challenge BufferedIsing (lean/docs/218.md) covers only a finite-graph likelihood comparison lemma from the companion 'Buffered comparison and stopping-band resolution in critical Ising'; its scope note says the stopping-band theorem is outside it. It does not state this entry's headline, so the entry is unreviewed, not Lean-checked.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"The weak-disorder 2D Ising problem is a classic of the physics of disordered systems since the early 1980s; scored for the question actually answered, a quenched interface limit at fixed small disorder, not the full logarithmic theory. Below Cardy's formula for square-lattice bond percolation (48) and the Krauth-Mezard capacity of the Ising perceptron (35), near the Gardner transition entry (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026/general-weak-random-bond-ising.pdf","sourceName":"OpenAI math release: Quenched SLE3 limits for general weak random-bond Ising models","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Quenched SLE3 Universality for the Weak Random-Bond Ising Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026/quenched-sle3-weak-random-bond-ising.pdf","kind":"paper"},{"label":"Companion (conditional): Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Logarithmic-Relative-Fluctuations-in-the-Weakly-Disordered-Planar-Ising-Model-October-5-2026/critical-relative-second-moment-weak-random-bond-ising.pdf","kind":"paper"},{"label":"Companion: Buffered comparison and stopping-band resolution in critical Ising","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ahpt-graph-decomposition-conjecture","name":"The Ascoli-He-Park-Talagrand graph-decomposition conjecture at the expectation threshold","shortName":"AHPT graph-decomposition conjecture","problemNumber":null,"field":"Random graphs; thresholds for subgraph containment","fieldGroup":"Combinatorics","statement":"For a graph $H$ on at most $n$ vertices let $p_c(H;n)$ be the least $p$ at which $G(n,p)$ contains a copy of $H$ with probability $1/2$, and $q(H;n)$ its integral expectation threshold. The Park-Pham theorem gives $p_c(H;n)\\le Cq(H;n)\\log|E(H)|$, and the logarithm cannot be dropped in general. Ascoli, He, Park and Talagrand (2026, Conjecture 8.1) proposed that it can be removed after cutting $H$ into boundedly many pieces, and proved bounded-degeneracy cases. Are there absolute constants $k$ and $L$ such that the edges of every graph $H$ on at most $n$ vertices split into $k$ subgraphs $H_1,\\dots,H_k$ with $p_c(H_i;n)\\le L\\,q(H;n)$ for each $i$?","posedBy":"Ruben Ascoli, Xiaoyu He, Jinyoung Park and Michel Talagrand (Conjecture 8.1, arXiv 2608.11183, August 2026)","yearPosed":2026,"ageNote":"Posed about two months before this manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are absolute $k\\ge2$ and $L\\ge1$ such that for every $n\\ge2$ and every graph $H$ on at most $n$ vertices, $E(H)=E(H_1)\\sqcup\\dots\\sqcup E(H_k)$ with $p_c(H_i;n)\\le Lq(H;n)$ for each $i$. Pieces may be empty or share vertices, and the partition depends only on $H$ and $n$. It does not give a single copy of $H$ at threshold $Lq(H;n)$ (the separate embeddings need not glue), and $k$, $L$ are not explicit.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof uses the family's discrete-convexity theorem (September 23, 2026) as an essential input; the manuscript says so.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Conjecture 8.1 as the manuscript reports it; the manuscript notes that the conjecture's overlapping-edge-set form is equivalent to the partition form proved. The partition is fixed before sampling, and the embeddings of different pieces need not agree on shared vertices. The proof is conditional on the companion discrete-convexity theorem, itself unreviewed. No Lean formalization of this manuscript.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A conjecture stated two months earlier by strong authors as an application of discrete convexity, with no time to acquire standing of its own. Well below the two Talagrand entries in this family at 36 and 38, level with the Araujo-Piga-Schacht tight Hamilton cycles entry at 15, above a typical Erdos problem at 10.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf","sourceName":"OpenAI math release: Graph Decompositions at the Integral Expectation Threshold","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Input: Talagrand's discrete-convexity conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Family: Integral and fractional expectation thresholds are equivalent","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf","kind":"paper"},{"label":"Ascoli, He, Park, Talagrand, A reformulation of the discrete Convexity Conjecture (2026)","url":"https://arxiv.org/abs/2608.11183","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"uchida-open-homomorphism-conjecture","name":"Uchida's conjecture on open homomorphisms of Galois groups of solvably closed extensions of number fields","shortName":"Uchida's open homomorphism conjecture","problemNumber":null,"field":"Anabelian geometry; Galois groups of number fields","fieldGroup":"Number theory","statement":"Let $E_i/F_i$ ($i=1,2$) be Galois extensions of number fields with each $E_i$ solvably closed (no nontrivial abelian extension), for example algebraic closures or maximal prosolvable extensions, and write $G_i=\\mathrm{Gal}(E_i/F_i)$. The Neukirch-Uchida theorem says that isomorphisms $G_1\\cong G_2$ come from field isomorphisms. Uchida (1981) proved the analogous statement for open homomorphisms when the source field is $\\mathbb Q$, proved uniqueness in general, and proved existence under a local condition on decomposition groups, and conjectured the general case. Is every continuous homomorphism $\\alpha:G_1\\to G_2$ with open image induced by a field embedding $j:E_2\\hookrightarrow E_1$, i.e. $g\\circ j=j\\circ\\alpha(g)$ for all $g\\in G_1$?","posedBy":"Koji Uchida, Homomorphisms of Galois groups of solvably closed Galois extensions, J. Math. Soc. Japan 33 (1981), p. 595","yearPosed":1981,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Uchida's conjecture in full: for number fields $F_1,F_2$ and possibly infinite solvably closed Galois extensions $E_i/F_i$, every continuous open homomorphism $\\mathrm{Gal}(E_1/F_1)\\to\\mathrm{Gal}(E_2/F_2)$ is induced by a unique field embedding $E_2\\hookrightarrow E_1$, with no hypothesis on the kernel and no separate cyclotomic-compatibility assumption. The key steps derive full cyclotomic compatibility from openness, through a uniform (in the auxiliary prime) Kummer bound and Frobenius-average character tests, and then apply Hoshi's criterion. It does NOT treat function fields, finite solvable quotients (Saidi-Tamagawa), or non-open homomorphisms.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. No Lean formalization of this result is in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. The main theorem was read against Uchida's conjecture as described in the manuscript (Uchida 1981, p. 595); the 1981 paper itself was not opened here. The proof ends by verifying Hoshi's cyclotomic-character criterion (Tohoku 2025), which is an input from the literature, and uses an l-adic Waldschmidt-Masser type statement for which the paper gives its own proof. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A 45-year-old named conjecture in anabelian number theory, extending the Neukirch-Uchida theorem from isomorphisms to open homomorphisms; well known to anabelian geometers, with recent partial progress by Hoshi, but not a central problem beyond that community. Below the 2-adic absolute Galois group presentation (25) and the section conjecture for modular curves (42), above the Hodge bundle simplicity entry (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf","sourceName":"OpenAI math release: Open Homomorphisms of Global Solvably Closed Galois Groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Uchida, Homomorphisms of Galois groups of solvably closed Galois extensions (1981)","url":"https://doi.org/10.2969/jmsj/03340595","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"slc-index-conjecture-finite-coefficients","name":"The index conjecture for semi-log-canonical log Calabi-Yau pairs with finite rational coefficient sets","shortName":"slc index conjecture, finite coefficients","problemNumber":null,"field":"Birational geometry; log Calabi-Yau pairs","fieldGroup":"Algebra","statement":"A projective log Calabi-Yau pair $(X,B)$ has $K_X+B\\sim_{\\mathbb Q}0$, so some positive multiple $m(K_X+B)$ is Cartier and linearly trivial; the least such $m$ is the index. Gongyo showed such $m$ exists for slc pairs, but it may depend on the pair. The index conjecture asks for uniformity. In its finite-rational-coefficient slc form, fix $d$ and a finite set $\\Phi\\subset[0,1]\\cap\\mathbb Q$: is there an integer $a(d,\\Phi)$ such that $a(d,\\Phi)(K_X+B)$ is Cartier and trivial for every connected projective slc log Calabi-Yau pair of dimension $d$ with coefficients in $\\Phi$?","posedBy":"Index conjecture for log Calabi-Yau pairs; finite-rational-coefficient slc form stated by C. Jiang and H. Liu (Algebra and Number Theory, 2021), Conjecture 1.5","yearPosed":null,"ageNote":"The manuscript cites Jiang-Liu (arXiv 2020, published 2021) as one place the conjecture is stated, not as its origin. The normal-pair index conjecture is older; Xu (2019) organized reductions between its forms. Jiang-Liu proved the slc case in dimension at most three.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $d\\ge4$ and finite $\\Phi\\subset[0,1]\\cap\\mathbb Q$ there is $a(d,\\Phi)>0$ such that $a(d,\\Phi)(K_X+B)$ is Cartier with trivial line bundle for every connected projective equidimensional slc log Calabi-Yau pair of dimension $d$ with coefficients in $\\Phi$, independently of the number of components. With Jiang-Liu's theorem for $d\\le3$ this gives the conjecture in the cited form. Not shown: slc pairs with infinite DCC coefficient sets, or explicit values of $a(d,\\Phi)$.","claimIssueNote":"This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it.","solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. Family 034 has fourteen manuscripts dated September 24 to October 5, 2026, which cite one another as inputs; this entry draws on the manuscript named as its source. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Uniform indices for semi-log-canonical log Calabi-Yau pairs' was read against the conjecture as the manuscript quotes it; with the cited dimension-three theorem of Jiang-Liu it covers every dimension for finite rational coefficient sets. The original Jiang-Liu statement was not opened here. The proof was not refereed. No Lean formalization. It uses the normal lc index theorem of the effective Iitaka manuscript and the family's good-model theorem, both unreviewed. Listed as Contested because it depends on the log abundance entry, which the release itself is inconsistent about; see the claim issue.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A recognized problem in the boundedness theory of Calabi-Yau pairs, well known to specialists but little known outside birational geometry. Placed level with the Batyrev stringy Hodge number entry (25) and the cone theorem for fourfold pairs in characteristic p (25), and below the effective Iitaka fibration conjecture from the same family (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026/uniform-slc-index.pdf","sourceName":"OpenAI math release: Uniform indices for semi-log-canonical log Calabi–Yau pairs","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Input: Uniform Pluricanonical Iitaka Fibrations (normal lc index, DCC)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf","kind":"paper"},{"label":"Jiang and Liu, Boundedness of log pluricanonical representations (arXiv 2002.11928), Conjecture 1.5","url":"https://arxiv.org/abs/2002.11928","kind":"problem-record"},{"label":"The 5 October appendix calling projective log abundance conditional","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026/","kind":"paper"},{"label":"Revised edition (October 6, 2026): Conditional good minimal models for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"aks-coloring-conjecture-forbidden-subgraph","name":"The Alon-Krivelevich-Sudakov conjecture: graphs with a fixed forbidden subgraph have chromatic number O(Delta/log Delta)","shortName":"Alon-Krivelevich-Sudakov coloring conjecture","problemNumber":null,"field":"Graph coloring: sparse and F-free graphs","fieldGroup":"Combinatorics","statement":"Every graph of maximum degree $\\Delta$ has $\\chi(G)\\le\\Delta+1$. Johansson (1996) proved $\\chi(G)=O(\\Delta/\\log\\Delta)$ for triangle-free graphs, and Alon, Krivelevich and Sudakov (1999) extended this to graphs with sparse neighborhoods. For general fixed $F$, and already for $F=K_4$, the best bounds carried an extra $\\log\\log\\Delta$ factor. Alon, Krivelevich and Sudakov (1999, Conjecture 3.1) conjectured: for every fixed graph $F$ there is $C_F$ such that every $F$-free graph of maximum degree $\\Delta$ has chromatic number at most $C_F\\Delta/\\log\\Delta$. Is this true?","posedBy":"N. Alon, M. Krivelevich and B. Sudakov, Coloring graphs with sparse neighborhoods, J. Combin. Theory Ser. B 77 (1999), Conjecture 3.1","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $r\\ge4$ there are $C_r$ and $\\Delta_r$ such that every $K_r$-free graph of maximum degree $\\Delta\\ge\\Delta_r$ has correspondence (DP) chromatic number at most $\\lceil C_r\\Delta/\\log\\Delta\\rceil$; Corollary 1.2 gives the same for any fixed forbidden subgraph $F$, which implies the conjecture for ordinary and list coloring. The proof is a round-based random partial coloring with weighted lists, controlled by an entropy potential and local adjustments that offset triangle correlations. Constants and thresholds are not optimized; no explicit leading constant is claimed.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The manuscript states that it develops the multiplier and entropy-splitting methods of its companion on the AEKS independence conjecture (25 September 2026) and proves all ingredients itself.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and Corollary 1.2 of the TeX source, read against Conjecture 3.1 of Alon, Krivelevich and Sudakov as the manuscript cites it; the proof was not refereed. The family's Lean material (lean/docs/184.md) formalizes only the companion independence bound, not this coloring theorem, so this entry is unreviewed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"A well-known conjecture in graph coloring since 1999, the coloring counterpart of the AEKS independence conjecture and stronger than it in spirit; open even for K4-free graphs. Scored just below the AEKS conjecture entry (46) and above the Burr-Erdos hypercube Ramsey conjecture (40), below the linear list Hadwiger conjecture (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf","sourceName":"OpenAI math release: Correspondence coloring graphs with a forbidden clique","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"A logarithmic independence bound for clique-free graphs (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf","kind":"paper"},{"label":"Alon, Krivelevich, Sudakov, Coloring graphs with sparse neighborhoods (JCTB 1999)","url":"https://doi.org/10.1006/jctb.1999.1910","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"xy-model-critical-log-correction","name":"Critical two-point decay $r^{-1/4}(\\log r)^{1/8}$ in the planar XY model","shortName":"XY critical decay with log correction","problemNumber":null,"field":"Statistical mechanics; Berezinskii-Kosterlitz-Thouless transition","fieldGroup":"Mathematical physics","statement":"The nearest-neighbor XY model on $\\mathbb Z^2$ at inverse temperature $b$ gives angles $\\theta_x$ the weight $\\exp(b\\sum_{x\\sim y}\\cos(\\theta_x-\\theta_y))$. Berezinskii, Kosterlitz and Thouless predicted a transition at $b_c$ driven by vortex unbinding, with algebraic decay of the spin correlation $C_b(r)=\\langle\\cos(\\theta_0-\\theta_{re_1})\\rangle$ for $b\\ge b_c$. Kosterlitz's 1974 renormalization-group analysis predicts at criticality the universal exponent $\\eta=1/4$ with a multiplicative logarithmic correction, $C_{b_c}(r)\\asymp r^{-1/4}(\\log r)^{1/8}$ (as recorded, e.g., by Janke for the Villain model). Rigorous work gave power-law bounds (McBryan-Spencer, Frohlich-Spencer, van Engelenburg-Lis) but not the critical exponent. Does $C_{b_c}(r)=B\\,r^{-1/4}(\\log r)^{1/8}(1+o(1))$ hold with $0 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has six manuscripts; the September 24 paper proves the exponent $r^{-1/4+o(1)}$ and the October 5 paper the logarithmic correction, building on it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the logarithmic-correction paper and the main theorem of the September 24 exponent paper were read against the prediction. Both concern the ordinary cosine model on the square lattice, with $b_c$ defined by the vanishing of the mass and the free-box thermodynamic limit taken before $r\\to\\infty$, along a coordinate axis. The log paper relies on the exponent paper (critical height coefficient $8\\pi$) and two other cited OpenAI companions for analytic maps and annular estimates. The proofs were not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"A rigorous derivation of the BKT critical behaviour of the XY model, with its logarithmic correction, has been a celebrated target of mathematical physics since Frohlich-Spencer. Placed below Polyakov's mass-generation conjecture (62) and The Haldane gap conjecture (62), and well above the Hasenfratz-Maggiore-Niedermayer exact O(4) mass gap (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf","sourceName":"OpenAI math release: The critical logarithmic correction for the planar XY model","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The critical correlation exponent of the planar XY model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: BKT universality for height and planar spin fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/BKT-universality-for-height-and-planar-spin-fields-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Critical Center Magnetization in the Planar XY Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: The Critical Spin Field of the Planar XY Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: Essential Singularity of the Correlation Length in the Planar XY Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stable-coordinate-conjecture-four-variables","name":"The stable coordinate conjecture in four variables","shortName":"Stable coordinate conjecture, four variables","problemNumber":null,"field":"Affine algebraic geometry; polynomial automorphisms","fieldGroup":"Geometry & topology","statement":"A polynomial $f\\in\\mathbb C[x_1,\\dots,x_n]$ is a coordinate if some polynomial automorphism sends it to $x_1$, and a one-stable coordinate if it is a coordinate of $\\mathbb C[x_1,\\dots,x_n,w]$. Shpilrain and Yu conjectured that a polynomial which becomes a coordinate after adjoining a variable was already a coordinate. Over $\\mathbb C$ it holds in one, two and three variables (Shpilrain-Yu, using Abhyankar-Moh, cancellation for curves and surfaces, and Kaliman's theorem), so four variables is the first possible dimension for a counterexample. Is every one-stable coordinate in $\\mathbb C[x_1,x_2,x_3,x_4]$ a coordinate?","posedBy":"V. Shpilrain and J.-T. Yu, Affine varieties with equivalent cylinders, J. Algebra 251 (2002), Conjecture 3","yearPosed":2002,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: with $Q=x_2^2-x_4^2+x_1x_3$ and $f=x_1-2Q(Q(x_2+x_4)+x_1x_4)$, a degree-five polynomial in $\\mathbb C[x_1,\\dots,x_4]$, some automorphism of $\\mathbb C[x_1,\\dots,x_4,w]$ sends $f$ to $x_1$ but none of $\\mathbb C[x_1,\\dots,x_4]$ does. Since the conjecture holds in at most three variables, this is the smallest possible dimension, and one added variable suffices. Corollary 1.2: every fibre of $f$ is $\\mathbb A^3$ and none is rectifiable. Not shown: anything about dimensions above four, which the release's earlier five-variable example already covered for five; the result is one explicit polynomial, not a classification.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it, and the README cautions that unformalized results could have issues. The construction adapts the method of the release's earlier cancellation manuscript.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and Corollary 1.2 of the TeX source, read against Shpilrain-Yu's Conjecture 3 as cited. Lean-checked on the release's Comparator challenge StableCoordinateFour (declaration OAI.StableCoordinate.principal_package, solution module OAI.AlgebraicGeometry.StableCoordinate.Principal), added in the 7 October 2026 update and listed in lean/formalization.yaml. Its statement was read here: for $Q=x_2^2-x_4^2+x_1x_3$ and $f=x_1-2Q(Q(x_2+x_4)+x_1x_4)$ (the paper's polynomial, indices shifted), $f$ has total degree 5, some automorphism of $\\mathbb C[x_1,\\dots,x_4,w]$ sends $f$ to $x_1$, no automorphism of $\\mathbb C[x_1,\\dots,x_4]$ does, and every fibre $f=t$ has coordinate ring isomorphic to $\\mathbb C[y_1,y_2,y_3]$ while no automorphism maps its ideal to $(x_1)$. That is the headline with Corollary 1.2. Not rebuilt here. Nothing is claimed about dimensions above four; the result is one explicit polynomial, not a classification.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A recorded conjecture in the polynomial-automorphism community, a relative of the cancellation problem but far less known, and already disproved in five variables by the same release before this paper; this entry is its first possible dimension. Tied with the five-variable entry (22) and the Inverse Generator Problem (22), below the Dolgachev-Weisfeiler affine-fibration conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf","sourceName":"OpenAI math release: A stable coordinate that is not a coordinate in four variables","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"An explicit noncoordinate polynomial with affine three-space zero fibre (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean Comparator challenge StableCoordinateFour (OpenAI math release)","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/StableCoordinateFour.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"apsp-hypothesis","name":"The APSP hypothesis: are all-pairs shortest paths computable in truly subcubic time?","shortName":"APSP hypothesis","problemNumber":null,"field":"Fine-grained complexity; graph algorithms","fieldGroup":"Theoretical computer science","statement":"Given a directed graph on $n$ vertices with integer edge weights in $[-n^c, n^c]$ and no negative cycles, compute all pairwise shortest-path distances. Floyd-Warshall does it in $O(n^3)$ time, and decades of improvements removed only subpolynomial factors. The APSP hypothesis asserts that no algorithm on a word RAM with $O(\\log n)$-bit words runs in $O(n^{3-\\varepsilon})$ time for any $\\varepsilon > 0$; a large class of problems is subcubically equivalent to it. Is there a truly subcubic algorithm?","posedBy":"Virginia Vassilevska Williams and Ryan Williams (subcubic equivalences, FOCS 2010); the question of truly subcubic APSP is as old as Floyd-Warshall (1962)","yearPosed":2010,"ageNote":"Dated from the hypothesis as formulated by Vassilevska Williams and Williams in 2010; whether APSP admits a truly subcubic algorithm had been asked since Floyd-Warshall in 1962. Vassilevska Williams, who co-posed the hypothesis, co-authored the refutation.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Yes: deterministic $O(n^{2.9995})$ algorithms for APSP and for the (min,+)-product with polynomially bounded integer weights on the word RAM (the Lean development states $n^{2.99942}$), and Exact Triangle in $O(n^{2.9983})$, refuting the APSP and Exact Triangle hypotheses; a Las Vegas $O(n^{2.998})$ version works for real weights. The core is a new thin-matrix-product algorithm solving Lopsided All-Edges Sparse Triangle, with known reductions giving the rest. The same paper refutes the 3SUM hypothesis, catalogued separately. The constants are enormous; the impact is on the theory of subcubic equivalences, not on practice. SETH and Orthogonal Vectors are unaffected.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Claude (internal Anthropic research model, unnamed)","modelMaker":"Anthropic","humanCollaborators":["Josh Alman","Virginia Vassilevska Williams"],"aiRole":"From the abstract: \"Claude, an AI model developed by Anthropic, discovered the algorithm that refutes the 3SUM, APSP, and Exact Triangle hypotheses.\" The methodology section adds that an Anthropic employee had set an internal research model to verify and improve cryptographic constructions based on the average-case hardness of Zero-k-Clique; instead it developed this algorithm, first for the average case and then the worst case, in one session of 16 million output tokens with no human input. Anthropic shared it with the authors in September 2026. Alman and Vassilevska Williams simplified, strengthened and extended it, derived further consequences and wrote the paper; an internal model later produced the Lean formalisation.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Read here on 6 October 2026, not rebuilt. The paper's Theorem 2 was compared with the standard hypotheses as its own introduction states them (word RAM with O(log n)-bit words, integers of absolute value n^O(1)) and matches. The Lean development (anthropics/formal-math, folder 3sum-apsp, commit e1a4e65; Lean 4.33.1 with mathlib, about 102,000 lines) proves correctness and running time of explicit programs on a word-RAM machine defined in Lean, a weaker machine than the standard word RAM, which only strengthens upper bounds. Its five headline claims are stated in a 139-line file with no imports. No sorry outside the deliberate Challenge folder, no axiom declarations, native_decide, implemented_by or unsafe. The statement file was written by the same party as the proof and has not been audited independently, and no Comparator run is recorded. No uninvolved expert has commented yet; the authors are the leading researchers of the area and are not independent reviewers.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Truly subcubic APSP was a textbook open question from Floyd-Warshall in 1962, and the 2010 hypothesis anchors a whole class of subcubically equivalent problems. Level with the matrix-multiplication exponent at 55, a subfield-defining question of the same kind; above the 3SUM hypothesis at 50, refuted in the same paper; below the unforced Euler blowup entry at 80.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2610.06783","sourceName":"arXiv:2610.06783, Alman and Vassilevska Williams (5 October 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation of the algorithms and running times (anthropics/formal-math)","url":"https://github.com/anthropics/formal-math/tree/e1a4e65/3sum-apsp","kind":"lean-proof"},{"label":"The 3SUM hypothesis, refuted in the same paper","url":"https://vibemathed.com/problem/3sum-hypothesis","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stationary-varifold-singular-set-codimension-one","name":"The codimension-one bound for singular sets of stationary integral varifolds","shortName":"Codimension-one singular set bound for varifolds","problemNumber":null,"field":"Geometric measure theory: regularity of stationary varifolds","fieldGroup":"Geometry & topology","statement":"Let $V$ be a stationary integral $m$-varifold in an open set $U\\subset\\mathbb R^{m+n}$ and $\\mathrm{Sing}\\,V$ the support points near which $V$ is not an integer multiple of one smooth embedded minimal submanifold. Two $m$-planes meeting along an $(m-1)$-plane form a stationary varifold whose singular set is $(m-1)$-dimensional, so no better bound is possible. Known cases included two-valued stationary Lipschitz graphs (Hirsch-Spolaor) and classes with controlled density under an epsilon-regularity assumption (Krummel-Minter-Wickramasekera). Brena, Decio and De Lellis (2025, Conjecture 1.1) conjecture the sharp bound, which is stronger than almost-everywhere regularity. Does every stationary integral $m$-varifold satisfy $\\dim_{\\mathcal H}\\mathrm{Sing}\\,V\\le m-1$, in every dimension and codimension?","posedBy":"Camillo Brena, Stefano Decio and Camillo De Lellis, Remarks and conjectures on stationary varifolds dedicated to Enrico Bombieri, Rend. Lincei Mat. Appl. 36 (2025), Conjecture 1.1","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every stationary integral $m$-varifold $V$ in an open $U\\subset\\mathbb R^{m+n}$, $m,n\\ge1$, $\\dim_{\\mathcal H}\\mathrm{Sing}\\,V\\le m-1$, with no stability, minimizing, orientability, density or codimension hypothesis; crossing planes show it is sharp. This implies $\\mathcal H^m(\\mathrm{Sing}\\,V)=0$. The proof combines the companion paper's signed-excess estimates with a fitting and frequency analysis around a smooth center and a measure-valued blow-up at higher-multiplicity planar tangents. It does not prove rectifiability or finer structure of the singular set, and does not cover Riemannian ambient metrics.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Conjecture 1.1 of Brena-Decio-De Lellis as the manuscript quotes it; the proof was not refereed. The proof imports the signed-excess theorem and local estimates of the companion almost-everywhere regularity manuscript from the same release, itself unreviewed, so a gap there would carry over. Euclidean ambient space only. The release has no Lean formalization for this family, and the README cautions that unformalized results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"Brena, Decio and De Lellis's sharp dimension conjecture strengthens the classical almost-everywhere regularity question and was posed only in 2025, so it is scored below that entry (58) despite implying it. Level with the Kadison-Kaplansky conjecture (50) and the Hodge standard conjecture for abelian varieties (50), above Banach's isometric conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf","sourceName":"OpenAI math release: A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Almost-everywhere regularity of stationary integral varifolds (input to this proof)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf","kind":"paper"},{"label":"Brena, Decio, De Lellis, Remarks and conjectures on stationary varifolds (2025)","url":"https://doi.org/10.4171/RLM/1077","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bloch-law-quantum-heisenberg-ferromagnet","name":"Bloch's $T^{3/2}$ law for the three-dimensional quantum Heisenberg ferromagnet","shortName":"Bloch's T^(3/2) law","problemNumber":null,"field":"Quantum spin systems; spin-wave theory","fieldGroup":"Mathematical physics","statement":"Bloch's spin-wave theory (1930) predicts that the spontaneous magnetization of a three-dimensional ferromagnet falls below saturation by an amount proportional to $T^{3/2}$, with a coefficient fixed by the quadratic one-magnon dispersion; for the nearest-neighbour spin-$S$ Heisenberg model the deficit is $\\zeta(3/2)(8\\pi^{3/2}S^{3/2})^{-1}T^{3/2}$ to leading order. Dyson (1956) analysed spin-wave interactions and the higher-order terms, but not as a rigorous fixed-spin proof, and Correggi-Giuliani-Seiringer (2015) proved the corresponding free-energy asymptotics, which do not control the zero-field derivative. Is it true, for the quantum Heisenberg ferromagnet on $\\mathbb Z^3$ at fixed spin, that $S-m(\\beta)\\sim c\\,\\beta^{-3/2}$ as $\\beta\\to\\infty$ with Bloch's coefficient, where $m$ is the spontaneous magnetization?","posedBy":"Felix Bloch (1930), spin-wave prediction; developed by Holstein-Primakoff (1940) and Dyson (1956)","yearPosed":1930,"ageNote":"A physical prediction stated precisely in print. The manuscript cites Bloch (1930) and Dyson (1956), notes that Dyson's higher-order estimates were not a rigorous proof, and cites Lieb (1999) and Seiringer (2025) for the ordering problem underneath it.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every fixed spin $S$ and every nonnegative symmetric finite-range coupling $J$ on $\\mathbb Z^3$ whose support generates $\\mathbb Z^3$, $\\lim_{\\beta\\to\\infty}\\beta^{3/2}(S-m_{S,J}(\\beta))=\\zeta(3/2)/(8\\pi^{3/2}\\sqrt{\\det D_{S,J}})$, with $D_{S,J}=\\frac S2\\sum_z J(z)zz^{\\mathsf T}$; nearest-neighbour gives $\\zeta(3/2)/(8\\pi^{3/2}S^{3/2})$. Companions: the first lattice correction $3\\zeta(5/2)(\\beta S)^{-5/2}/(128\\pi^{3/2})$ for nearest-neighbour couplings, deficit equal to the ideal-magnon density up to $o(\\beta^{-5/2})$; and a spherical magnetization law for the symmetric zero-field magnetization on periodic cubes. Not shown: Dyson's $T^4$ interaction term, dimensions other than three, or uniformity in $S$.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript and two companions (lattice correction, spherical law) are dated October 5, 2026; they reuse the finite-set algebra and lattice geometry of the September 24 ordering manuscript in the same family.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the Bloch's law manuscript read against Bloch's prediction as the manuscript states it. The family's only Lean challenge (Heisenberg) states spontaneous magnetization with $\\omega(S^z_0)\\ge S/4$, not the $T^{3/2}$ asymptotic, so this entry is unreviewed. The magnetization is defined as the right derivative at zero field of the infinite-volume pressure; the order of limits (volume, then field, then temperature) is part of the claim. The proof imports results of the companion ordering manuscript, also unrefereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A textbook physical law from 1930 with no rigorous proof for the quantum model; its proof requires the ordering result plus sharp control of the magnetization. Scored as a derived precise law rather than the ordering question itself: level with Solovej's generalized ionization conjecture (32), just below the Bogoliubov depletion law for the dilute Bose gas (35), above the long-time hard-sphere fluctuation entry (24).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf","sourceName":"OpenAI math release: Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion manuscript: The first lattice correction to Bloch's law","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf","kind":"paper"},{"label":"Companion manuscript: The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf","kind":"paper"},{"label":"Companion manuscript: Spontaneous magnetization in the quantum Heisenberg ferromagnet","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"randomized-quantum-query-cubic-conjecture","name":"Randomized versus quantum query complexity of total Boolean functions: is R(f) = O(Q(f)^3)?","shortName":"Cubic R vs Q conjecture","problemNumber":null,"field":"Query complexity","fieldGroup":"Quantum information & computing","statement":"For total Boolean functions the bounded-error randomized query complexity $R$ and quantum query complexity $Q$ are polynomially related: Beals et al. proved $D=O(Q^6)$, and after Huang's sensitivity theorem Aaronson, Ben-David, Kothari, Rao and Tal proved $D(f)=O(Q(f)^4)$, hence $R=O(Q^4)$. The best separations were power $5/2$ (cheat sheets), then $8/3-o(1)$ (Tal) and $3-o(1)$ (Bansal-Sinha; Sherstov-Storozhenko-Wu). Aaronson, Ben-David, Kothari, Rao and Tal conjectured that the cubic relation holds. Is $R(f)=O(Q(f)^3)$ for every total Boolean function, or is the quartic exponent optimal?","posedBy":"Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao and Avishay Tal","yearPosed":2021,"ageNote":"Conjecture 2 of their STOC 2021 paper, as cited by the manuscript; the preprint appeared on arXiv in October 2020 (2010.12629).","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for each fixed $k\\ge2$ there are total Boolean functions $F_{k,m}$ with $Q(F_{k,m})\\le C_k\\sqrt m(\\log m)^{b_k}$ and $R(F_{k,m})\\ge c_km^{2-1/k}/(\\log m)^2$, a power $4-2/k$ separation up to logarithms, so no bound $R=O((1+Q)^\\alpha)$ holds with $\\alpha<4$. With the known $R=O(Q^4)$ the optimal exponent is exactly 4, and the cubic conjecture is false. The functions are cheat-sheet-style columns whose certificate cells are addressed by answers to partial-function instances. Not shown: a separation with exponent exactly 4 for a single family, or new results for partial functions.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The single manuscript (October 5, 2026) is the whole family.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and the quantum upper-bound proposition were read against Conjecture 2 as the manuscript states it. The construction and its analyses (a quantum tournament lemma for candidate comparisons, an adaptive direct-product lower bound) were not refereed. No Lean formalization accompanies this manuscript. Both measures are worst-case bit-query counts with error at most 1/3 and unrestricted computation between queries, as the paper states.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"The optimal polynomial relation between randomized and quantum query complexity for total functions has been a central question of quantum query complexity since Huang's sensitivity theorem fixed the upper side. Placed below The asymptotic Gotsman-Linial conjecture (42), above A Quantum Oracle Separation Between QMA(2) and QMA (36) and Two-Copy Distillability of Werner States (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf","sourceName":"OpenAI math release: A Nearly Quartic Separation Between Randomized and Quantum Query Complexity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Aaronson-Ben-David-Kothari-Rao-Tal, Degree vs. approximate degree (arXiv)","url":"https://arxiv.org/abs/2010.12629","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"flouri-et-al-msci-identifiability-transfer-under-the-homogeneous-clock-jc-observ","name":"Flouri et al.'s MSci identifiability conjecture, for clock-JC data with long loci","shortName":"MSci clock-JC identifiability transfer","problemNumber":null,"field":"Mathematical statistics; algebraic identifiability; coalescent phylogenetics","fieldGroup":"Probability & statistics","statement":"Flouri, Jiao, Rannala and Yang ask whether, with multiple sampled sequences and multiple sites per locus, parameters of the multispecies coalescent with introgression are identifiable from sequence alignments exactly when they are identifiable from gene-tree topologies and coalescent times. The present entry makes the observation channel and site-length quantifier explicit: complete labelled contemporaneous haploid alignments, homogeneous stationary molecular-clock JC mutation in mutation-scaled units, and sufficiently long finite loci within each fixed finite MSci model.","posedBy":"Flouri, Jiao, Rannala and Yang, MBE 37(4):1211–1223, DOI 10.1093/molbev/msz296 (online 6 Dec 2019; issue 2020). Restated by Yang and Flouri, MBE 39:msac083 (2022).","yearPosed":2019,"ageNote":null,"solveType":"proved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"For every fixed finite admitted MSci model and its unchanged labelled sample panel, a finite site cutoff works uniformly over parameter pairs: complete sequence laws agree exactly when route-marginal timed-genealogy laws agree. All identifiable functionals and all genuine ambiguities transfer. Multiple introgressions, expansions, equal rates, rank-deficient routing and source-allowed inheritance endpoints are covered. The A–D compiler preserves tied ages and current-lineage routing. Trees end at the sample MRCA; route flags are marginalized.\n\nThe channel is complete contemporaneous haploid data under homogeneous stationary clock-JC in mutation units. No arbitrary short-locus claim, universal cutoff over unbounded complexity, or general demographic uniqueness is made. Unknown locus-rate mixtures, unphased genotypes and other mutation/clock channels remain separate. No calendar calibration, finite-data accuracy or reliable numerical inference is supplied.","claimIssueNote":null,"solveDate":"2026-10-05","model":"OpenAI assistant (dot; underlying model unspecified)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Under the user's research direction, dot (OpenAI) developed the mathematical observation bridge, finite-site arguments and source-specific MSci compiler, wrote the proof and exact rational controls, and performed separate AI correctness/source-scope reviews. The finite-generation method builds on classical polynomial-family identifiability and finite exponential recurrence ideas, with attribution in the proof. The biological model and conjecture are existing research. No human expert endorsement is claimed.","verification":"unreviewed","verificationNote":"Read by this site on 6 October 2026. The source sentence was checked in the 2020 paper (Discussion, Identifiability of MSci models) and in Yang and Flouri's 2022 restatement. The submission's event-compiler check passed when re-run here; it verifies small rational routing compositions, not identifiability. The identifiability transfer rests on a written proof (bridge/THEOREM.md) that was not read here. Both review records are by the same AI agent. No Lean formalisation.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"The source states it as a one-sentence speculation (2020), named a conjecture in 2022; real and cited, within phylogenetics. Scored for the variant answered, which fixes the clock-JC channel and allows long loci. Level with the Sun conjectures at 5.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Research-Commons/blob/f96399576168328322aab80f1193afaaf4312fef/research/2026-10-05-dot-flouri-clock-jc-conjecture-resolution-0908z/README.md","sourceName":"Research Commons (GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"Original conjecture and MSci model","url":"https://academic.oup.com/mbe/article/37/4/1211/5673394","kind":"problem-record"},{"label":"Yang–Flouri 2022 restatement and biological ambiguities","url":"https://academic.oup.com/mbe/article/39/5/msac083/6568285","kind":"paper"},{"label":"Exact resolution and observation-channel limits","url":"https://github.com/Sodelin/Research-Commons/blob/f96399576168328322aab80f1193afaaf4312fef/research/2026-10-05-dot-flouri-clock-jc-conjecture-resolution-0908z/CONJECTURE-RESOLUTION.md","kind":"other"},{"label":"Complete hand-proof observation bridge","url":"https://github.com/Sodelin/Research-Commons/blob/f96399576168328322aab80f1193afaaf4312fef/research/2026-10-05-dot-flouri-clock-jc-conjecture-resolution-0908z/bridge/THEOREM.md","kind":"other"},{"label":"Exact A–D event compiler and transfer","url":"https://github.com/Sodelin/Research-Commons/blob/f96399576168328322aab80f1193afaaf4312fef/research/2026-10-05-dot-flouri-clock-jc-conjecture-resolution-0908z/COMPILER-AND-TRANSFER.md","kind":"other"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"howie-conjecture-nonsingular-systems","name":"Howie's conjecture: every nonsingular system of equations over a group is solvable in an overgroup","shortName":"Howie's conjecture on nonsingular systems","problemNumber":null,"field":"Combinatorial group theory; equations over groups","fieldGroup":"Algebra","statement":"Let $G$ be a group and $w_1,\\dots,w_m\\in G*F(x_1,\\dots,x_n)$ a system of equations $w_i=1$ with coefficients in $G$. Its exponent-sum matrix $A\\in M_{m\\times n}(\\mathbb Z)$ records the exponent of each $x_j$ in each $w_i$; the system is nonsingular if $A$ has rank $m$ over $\\mathbb Q$. Gerstenhaber and Rothaus (1962) solved nonsingular systems over finite groups and compact Lie groups, Howie (1981) over locally indicable groups, and Nitsche-Thom over hyperlinear groups. The one-equation, one-variable case is the Kervaire-Laudenbach conjecture (a nonzero exponent sum should guarantee solvability). Does every finite nonsingular system over an arbitrary group $G$ have a solution in some group containing $G$, that is, is $G\\to(G*F_n)/\\langle\\langle w_1,\\dots,w_m\\rangle\\rangle$ injective?","posedBy":"James Howie (1981), On pairs of 2-complexes and systems of equations over groups; name and formulation as in Klyachko, Mikheenko and Roman'kov (2024)","yearPosed":1981,"ageNote":"The manuscript takes the name and formulation from Klyachko, Mikheenko and Roman'kov (2024, Introduction) and cites Howie (1981) for the locally indicable case; 1981 is used as the posing year on that basis. Its one-variable case, the Kervaire-Laudenbach conjecture, goes back to the Kervaire problem of 1965.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1: for every group $G$ and every finite system $w_1,\\dots,w_m\\in G*F_n$ whose exponent-sum matrix has rank $m$ over $\\mathbb Q$, the canonical map $G\\to(G*F_n)/\\langle\\langle w_1,\\dots,w_m\\rangle\\rangle$ is injective, so the system has a simultaneous solution in an overgroup. This contains the Kervaire-Laudenbach conjecture (one equation, one unknown, nonzero exponent sum) and the Kervaire conjecture. It does NOT control the overgroup (no finiteness, solvability or other structure of the solution group), and says nothing about singular systems.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. It states that its spectral-phase and planar-surface strategy comes from the family's earlier Kervaire manuscript (24 September 2026), extended by a multi-block fixed-space theorem for products of unitary groups.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1 of the principal manuscript was read against the conjecture as the manuscript formulates it: for any group $G$ and $1\\le m\\le n$, full rational row rank of the exponent-sum matrix gives injectivity of $G$ into the quotient, with Remark 5.1 extending to infinite systems with independent rows. This manuscript has no Lean formalisation. The family's Lean challenge (ComparatorChallenges/Kervaire.lean) covers only one relator with exponent sum $\\pm1$, which is much narrower than this theorem. The release README warns that some unformalised results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Howie's conjecture is the standard general form of the equations-over-groups problem and contains the Kervaire-Laudenbach conjecture, but it is less widely known than the Kervaire conjecture itself. Below the Kervaire conjecture entry (55), level with the word problem for finite-rank Artin groups (45), above the Virtual Surjection Conjecture for Discrete Groups (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf","sourceName":"OpenAI math release: Nonsingular systems of equations over arbitrary groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The Kervaire theorem for groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf","kind":"paper"},{"label":"Lean (companion, one unimodular relator only): Kervaire solution module","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/Kervaire/Main.lean","kind":"lean-proof"},{"label":"Howie (1981), On pairs of 2-complexes and systems of equations over groups","url":"https://doi.org/10.1515/crll.1981.324.165","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"trilinear-hilbert-transform-l3","name":"The trilinear Hilbert transform conjecture at the L3 x L3 x L3 to L1 point (slopes 1, 2, 3)","shortName":"Trilinear Hilbert transform, L3 case","problemNumber":null,"field":"Harmonic analysis: multilinear singular integrals","fieldGroup":"Analysis","statement":"The trilinear Hilbert transform with slopes 1, 2, 3 is $T(f_1,f_2,f_3)(x)=\\mathrm{p.v.}\\int_{\\mathbb R}f_1(x-t)f_2(x-2t)f_3(x-3t)\\,\\frac{dt}{t}$. Its bilinear predecessor was proved bounded by Lacey and Thiele (1997-1999). In the trilinear case quadratic modulations survive in the associated four-linear form (the coefficients $(-1,3,-3,1)$ annihilate polynomials of degree two), so time-frequency analysis does not close the argument; Tao (2015) obtained only sublogarithmic growth for truncations, and Hu and Lie record the straight-line boundedness problem as open. The standard conjecture asks for bounds $L^{p_1}\\times L^{p_2}\\times L^{p_3}\\to L^r$ over a range of exponents. In particular, is $T$ bounded from $L^3(\\mathbb R)\\times L^3(\\mathbb R)\\times L^3(\\mathbb R)$ to $L^1(\\mathbb R)$?","posedBy":"Standard conjecture after Lacey and Thiele's bilinear theorem; the manuscript cites it as Conjecture 1.7 of B. Hu and V. Lie, On the curved trilinear Hilbert transform (arXiv:2308.10706, 2023)","yearPosed":null,"ageNote":"The manuscript gives only the 2023 statement by Hu and Lie and does not date the conjecture's first appearance. It is older, as the natural continuation of the Lacey-Thiele bilinear theorem (1997-1999), but no earlier source is cited, so the year is left blank.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a finite $C$ with $\\|T(f_1,f_2,f_3)\\|_{L^1}\\le C\\prod_{j}\\|f_j\\|_{L^3}$ for Schwartz functions, so $T$ extends uniquely to a bounded trilinear map $L^3\\times L^3\\times L^3\\to L^1$. The proof combines the Leng-Sah-Sawhney inverse theorem with new compression, curvature-localization and sparse counting lemmas that sum the scales without loss. It concerns only the fixed slopes 1, 2, 3 and the single exponent point $(3,3,3;1)$: the rest of the conjectured range, other slopes, maximal or truncation-uniform versions and higher-degree multilinear Hilbert transforms are not treated, as the manuscript itself says.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as the manuscript cites it; the proof (roughly 400 KB of TeX across ten sections) was not refereed. The family has no Lean formalization at the pinned commit (lean/docs/086.md does not exist). The argument uses the degree-two case of the Leng-Sah-Sawhney inverse theorem for Gowers norms as an external input.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The trilinear Hilbert transform is one of the best-known open problems in harmonic analysis, and no Lp bound at all was known for the straight-line operator; one exponent point is scored below the full conjecture. Placed above the Kakeya maximal conjecture in three dimensions (48), below the Kakeya set conjecture in four dimensions (55) and the Bochner-Riesz conjecture in three dimensions (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf","sourceName":"OpenAI math release: An L3 bound for the trilinear Hilbert transform","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Hu and Lie, On the curved trilinear Hilbert transform (arXiv:2308.10706), Conjecture 1.7","url":"https://arxiv.org/abs/2308.10706","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generic-ramanujan-exceptional-function-fields","name":"The generalized Ramanujan conjecture for globally generic cuspidal representations of exceptional groups over function fields","shortName":"Generic Ramanujan, exceptional groups, F_q(X)","problemNumber":null,"field":"Automorphic representations over function fields; temperedness","fieldGroup":"Number theory","statement":"Let $F$ be the function field of a curve over $\\mathbb F_q$ and $G$ a split connected reductive group. The generalized Ramanujan conjecture, in its generic form (within Arthur's framework; see Shahidi), predicts that every globally generic cuspidal automorphic representation $\\pi=\\bigotimes'_v\\pi_v$ of $G(\\mathbb A_F)$ is tempered at every place $v$. Over function fields this was known for $GL_2$ (Drinfeld), $GL_n$ (L. Lafforgue), and split classical and quasi-split unitary groups (Lomeli); for exceptional groups only conditional unramified results (Sawin-Templier, Ciubotaru-Harris) were available. For split adjoint groups of type $G_2,F_4,E_6,E_7,E_8$ over a global function field, is every globally generic cuspidal automorphic representation tempered at every place?","posedBy":"Generic form of the generalized Ramanujan conjecture (Arthur's framework; Shahidi)","yearPosed":null,"ageNote":"The manuscript cites Arthur (1989) and Shahidi (2011, Arthur packets and the Ramanujan conjecture) for the generic temperedness prediction but gives no original posing date; the Ramanujan-Petersson conjecture itself is far older.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $G$ split connected adjoint absolutely simple of type $G_2,F_4,E_6,E_7$ or $E_8$ over a global function field, every complex globally generic cuspidal automorphic representation is tempered at every place. The companion (September 24) gives temperedness at unramified places for all split adjoint absolutely simple groups from genericity at one unramified place. Nothing is claimed over number fields, for non-generic representations, or for ramified places of classical groups beyond what Lomeli already proved.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The global input is the unramified Ramanujan corollary of the family's September 24 decomposition paper.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the generic Ramanujan conjecture. It covers split adjoint exceptional groups over function fields, every place including ramified ones, with no restriction on characteristic or ramification depth; this is a special case of the conjecture (one group class, function fields), hence Partial. The unramified places come from the companion decomposition paper; the ramified places use Genestier-Lafforgue local-global compatibility and Gan-Harris-Sawin. The proof was not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":34,"significanceNote":"A special case (exceptional groups, function fields, generic representations) of the generalized Ramanujan conjecture, extending Lafforgue and Lomeli. Placed below The p-converse to Gross-Zagier-Kolyvagin in Selmer corank at most one (45) and the Ramanujan-Arthur decomposition entry of this family (38), and above The index conjecture for slc log Calabi-Yau pairs (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf","sourceName":"OpenAI math release: Temperedness at ramified places for globally generic exceptional groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bogoliubov-depletion-dilute-bose-gas","name":"The Bogoliubov quantum-depletion law for the dilute Bose gas in the thermodynamic limit","shortName":"Bogoliubov depletion law","problemNumber":null,"field":"Quantum many-body theory; dilute Bose gases","fieldGroup":"Mathematical physics","statement":"Bogoliubov's theory (1947), carried to the hard-sphere gas by Lee, Huang and Yang (1957), predicts that at zero temperature a dilute three-dimensional Bose gas with scattering length $a$ and density $\\rho$ has a fraction $\\frac{8}{3\\sqrt\\pi}\\sqrt{\\rho a^3}$ of its particles outside the condensate, distributed in momentum by an explicit Bogoliubov profile on the scale $\\sqrt{8\\pi\\rho a}$. The energy to Lee-Huang-Yang order is now proved, but energy bounds do not control occupation of modes whose energy vanishes with the volume, and the depletion had been derived only in Gross-Pitaevskii-type scalings or for approximate equations. In the thermodynamic limit at fixed density, followed by $\\rho a^3\\to0$, is the ground-state depletion $\\frac{8}{3\\sqrt\\pi}\\sqrt{\\rho a^3}+o(\\sqrt{\\rho a^3})$?","posedBy":"Prediction of N. N. Bogoliubov (1947) and T. D. Lee, K. Huang and C. N. Yang (1957)","yearPosed":1957,"ageNote":"Year of the Lee-Huang-Yang hard-sphere calculation, which the manuscript credits with determining the leading depletion; Bogoliubov's 1947 theory gives the momentum profile. The manuscripts present it as a physical prediction to be proved, not as a numbered open problem.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For hard spheres (October 5): for every bounded continuous $f$ and $\\varepsilon>0$, at all sufficiently small fixed $\\rho$, every thermodynamic accumulation value of $\\frac{1}{N\\sqrt{\\rho a^3}}\\sum_{k\\ne0}f(k/\\sqrt{8\\pi\\rho a})\\,n(k)$ is within $\\varepsilon$ of $\\int f\\,d\\nu_{\\mathrm{Bog}}$, uniformly over ground states; taking $f=1$ gives depletion $\\frac{8}{3\\sqrt\\pi}\\sqrt{\\rho a^3}+o(\\sqrt{\\rho a^3})$. A companion proves the same depletion asymptotic for fixed bounded, nonnegative, radial, finite-range potentials with positive scattering length. Not covered: positive temperature, the next-order correction, or unbounded non-hard-core potentials.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has five manuscripts (September 24 to October 5, 2026). The positive-temperature paper says it builds on, and reproduces the needed arguments of, two earlier manuscripts in the same family, so later results rest on earlier model output.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the hard-sphere depletion manuscript and the main theorem of the bounded-potential companion (both October 5) were read against the Bogoliubov and Lee-Huang-Yang prediction. Not refereed; no Lean formalization for either. The order of limits is thermodynamic first at fixed density, then dilute; the result is uniform over all ground states and does not assume a unique thermodynamic limit. The argument uses the published Lee-Huang-Yang energy upper bound for hard spheres (Basti, Brooks, Cenatiempo, Olgiati, Schlein) and operator estimates from Fournais, Junge, Girardot, Morin, Olivieri and Triay.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The Bogoliubov and Lee-Huang-Yang depletion formula is a textbook prediction of many-body physics; proving it for the true ground state in the thermodynamic limit was out of reach because it needs condensation there first. Famous within mathematical physics, less so as a standalone stated problem. Level with full RSB in the Sherrington-Kirkpatrick model (35), just below Talagrand's convolution conjecture (37), well below condensation itself (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf","sourceName":"OpenAI math release: Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Quantum Depletion for Fixed Bounded Repulsive Potentials (October 5, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf","kind":"paper"},{"label":"Related: Bose-Einstein condensation at positive temperature in the dilute hard-sphere gas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"emerton-hecke-dimension-conjecture-p-2","name":"Emerton's dimension conjecture for the full Hecke algebra at p = 2 and odd level","shortName":"Emerton dimension conjecture, p = 2","problemNumber":null,"field":"p-adic families of modular forms; Hecke algebras","fieldGroup":"Number theory","statement":"For a prime $p$ and level $N$ prime to $p$, let $\\mathbb T_p(N)$ be the inverse limit over $k$ of the $\\mathbb Z_p$-Hecke algebras generated by $T_\\ell$ and $\\ell S_\\ell$ ($\\ell\\nmid pN$) acting on all modular forms of weights $1,\\dots,k$ and level $\\Gamma_1(N)$, Eisenstein forms included. Emerton, in his 2011 Bourbaki account of $p$-adic families (Conjecture 2.9), conjectured that every irreducible component of $\\mathrm{Spec}\\,\\mathbb T_p(N)$ has Krull dimension four, and proved the lower bound (Corollary 2.28). For $p=2$ and every odd $N$, does every irreducible component have dimension exactly four?","posedBy":"Matthew Emerton, p-adic families of modular forms [after Hida, Coleman, and Mazur], Seminaire Bourbaki 2009/2010, Asterisque 339 (2011), Conjecture 2.9","yearPosed":2011,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every odd $N$, every irreducible component of $\\mathrm{Spec}\\,\\mathbb T_2(N)$ has Krull dimension exactly $4$. The upper bound comes from a cuspidal point of weight at least 3 on each component, a tangent bound of 3 from Newton-Thorne Selmer vanishing, and one arithmetic dimension. Only the case $p=2$ is treated; the paper does not claim the conjecture for odd $p$.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the October 5 manuscript was read against Emerton's Conjecture 2.9 as the paper cites it; it claims dimension exactly four for every component of the full 2-adic Hecke algebra of level $\\Gamma_1(N)$, $N$ odd, residually reducible and scalar components included. The proof was not refereed. It uses Emerton's lower bound, Chenevier's determinant laws and the Newton-Thorne adjoint Selmer vanishing theorem as external inputs. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A specialist conjecture from a 2011 Bourbaki survey on p-adic families, at one prime. Placed level with the Arnold corank problem (20) and the finite field restriction problem for the paraboloid (20), below the pro-modularity variant from the same family (22) and the Kraetzer integral means conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf","sourceName":"OpenAI math release: The Dimension of the Two-Adic Hecke Algebra at Odd Level","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion: Fontaine-Mazur modularity at the prime 2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf","kind":"paper"},{"label":"Companion: Unrestricted pro-modularity at the prime two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-4-2026/two-adic-promodularity.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Fontaine-Mazur modularity at the prime 2","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Unrestricted pro-modularity at the prime two","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stable-morse-arnold-bound","name":"The stable Morse number as a lower bound for nondegenerate Hamiltonian fixed points","shortName":"Stable-Morse Arnold bound","problemNumber":null,"field":"Symplectic topology; Hamiltonian fixed points, stable Morse theory","fieldGroup":"Geometry & topology","statement":"The stable Morse number $\\mathrm{SM}(W)$ of a closed manifold is the least number of critical points of a Morse function $F:W\\times\\mathbb R^k\\to\\mathbb R$ equal to a nondegenerate quadratic form outside a compact set ($k\\ge0$ varying); $\\mathrm{SM}(W)\\le\\mathrm{Morse}(W)$. Dimitroglou Rizell and Golovko proved $\\#\\mathrm{Fix}_0\\ge\\mathrm{SM}(M)$ for nondegenerate Hamiltonians when $[\\omega]$ and $c_1$ vanish on $\\pi_2(M)$. Does the stable Morse bound $\\#\\mathrm{Fix}_0(\\phi_H^1;H)\\ge\\mathrm{SM}(M)$ hold for every nondegenerate Hamiltonian diffeomorphism of every closed symplectic manifold?","posedBy":"Recorded as Conjecture 1.2 in R. Golovko, On variants of Arnold conjecture, Arch. Math. (Brno) 56 (2020); background in Arnold (1986)","yearPosed":2020,"ageNote":"The stable formulation is cited from Golovko's 2020 survey of variants; Arnold (1986, Section 5) already compared stable and unstable critical-point minima.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Principal Theorem 1.1: a simply connected closed Kahler $M$ ($\\dim_{\\mathbb R}=1412$, $\\mathrm{SM}=318968$) and a Hamiltonian with all 318952 fixed points nondegenerate and contractible. Mechanism: a Hamiltonian involution on $X\\times Y$ whose fixed components carry Smale-minimal Morse functions, with mixed 2- and 3-primary torsion making $4\\lambda(C)<\\lambda(M)$; no Floer theory. Companions: in real dimension 22 the deficit is unbounded with ratio $\\le127/128$; in dimension 3332 the count attains the cyclic integral Floer bound of Bai-Xu yet is 32 below $\\mathrm{SM}$. Since $\\mathrm{SM}\\le\\mathrm{Morse}$ these also refute the Morse-number form in the simply connected setting. The homological bounds remain intact.","claimIssueNote":"The release says: 'Ma [Theorem 1.5] asserted that the minimum number of fixed points over all Hamiltonian diffeomorphisms always equals Crit(M). Theorem 1.1 contradicts that assertion' (degenerate paper) and 'Ma's preprint asserts that the minimum number of fixed points of a nondegenerate Hamiltonian diffeomorphism always equals the ordinary Morse number ... Theorem 1.1 contradicts that unrestricted assertion' (Morse-number paper). The other side: R. Ma, Proofs on Arnold conjectures, arXiv 0808.0613v7 (2013), Theorem 1.5, claims these lower bounds hold in general. Ma's preprint is unpublished; no referee has ruled between them. The stable-Morse papers do not cite Ma; the conflict is by implication, since Morse(M) >= SM(M).","solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal paper was read against Golovko's Conjecture 1.2; it claims a simply connected closed Kahler manifold of real dimension 1412 with a nondegenerate Hamiltonian having $\\mathrm{SM}(M)-16$ fixed points. The companions claim a dimension-22 family with $\\#\\mathrm{Fix}\\le(1-1/128)\\mathrm{SM}$ and a dimension-3332 example attaining the cyclic integral Floer bound. Proofs not refereed; not formalized. Graded contested because Ma's 2013 preprint claims nondegenerate fixed points are always at least the ordinary Morse number, which is at least the stable Morse number; the stable-Morse papers themselves do not cite Ma.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A refinement of the Arnold conjecture known to Floer-theory specialists, proved under asphericity-type hypotheses by Dimitroglou Rizell-Golovko and Poder Balkestahl. Placed below the Projected Lax Conjecture (35), level with the Parabolic Intersection Conjecture for Artin groups (30), above Arnold's corank problem (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf","sourceName":"OpenAI math release: Hamiltonian Fixed Points Below the Stable Morse Number","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf","kind":"paper"},{"label":"Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"unitary-synthesis-constant-error","name":"The Aaronson-Kuperberg unitary synthesis problem at constant diamond-norm error","shortName":"Unitary synthesis, constant error","problemNumber":null,"field":"Quantum query complexity; oracle synthesis of unitaries","fieldGroup":"Quantum information & computing","statement":"Aaronson and Kuperberg (2007) asked whether, for every $n$-qubit unitary $U$, there is a Boolean oracle $A$ such that a polynomial-time quantum algorithm with access to $A$ implements $U$. Aaronson (2021, Problem 6) named it the Unitary Synthesis Problem, called it wide open and conjectured a negative answer. Known results: state synthesis with one query (Irani et al., Rosenthal), a $2^{n/2}$-time upper bound for unitaries (Rosenthal), and one-query and parallel-query lower bounds (Lombardi-Ma-Wright). Is there a polynomial-size quantum oracle circuit family which, for every $n$-qubit $U$ and a suitable Boolean oracle depending on $U$, implements $U$ approximately?","posedBy":"S. Aaronson and G. Kuperberg, Quantum versus classical proofs and advice, Theory of Computing 3 (2007), Section 7, Problem (4); restated in Aaronson (2021), Problem 6","yearPosed":2007,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are a polynomial $p$ and a deterministic algorithm outputting, on $1^n$, an oracle circuit $A_n$ with at most $p(n)$ qubits, gates and oracle calls and oracle input length at most $p(n)$, such that for every $U\\in U(2^n)$ some $f$ gives $\\|\\Phi_{n,f}-\\mathcal U\\|_\\diamond\\le1/2$. The method is a recursive dimension reduction using one coherently controlled smaller synthesis per step. It does not achieve arbitrarily small error, does not give an efficient classical procedure for the oracle, and answers positively a question on which Aaronson (2021) had conjectured a negative answer.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Aaronson's 2021 statement of Problem 6 (fetched for this check). Theorem 1.1 gives a circuit family generated in time polynomial in $n$ from $1^n$ (gates $H,T,T^\\dagger,\\mathrm{CNOT}$; polynomial qubits, gates, queries and query length) such that for every $U$ some Boolean oracle yields full diamond-norm error at most $1/2$ (no factor $1/2$ in the norm). The paper calls this the constant-error formulation; Aaronson's statement does not fix the error, and smaller-error versions are not addressed. The oracle is existential with no efficient construction. Not refereed here. No Lean formalization.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The unitary synthesis problem is a well-known open question in quantum complexity, highlighted by Aaronson as wide open and tied to quantum circuit lower bounds; a positive answer even at a fixed constant error would surprise specialists, but the error level leaves part of the question open. Placed below mutually unbiased bases in dimension six (45) and the Sakoda-Sipser problem (45), above the approximation ratio for Boolean Max-k-CSP (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf","sourceName":"OpenAI math release: Polynomial-Time Unitary Synthesis from a Boolean Oracle","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Aaronson-Kuperberg 2007, Quantum versus classical proofs and advice","url":"https://doi.org/10.4086/toc.2007.v003a007","kind":"problem-record"},{"label":"Aaronson 2021, Open problems related to quantum query complexity (Problem 6)","url":"https://doi.org/10.1145/3488559","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gromov-scalar-curvature-simplicial-volume","name":"Gromov's integral scalar-curvature bound for simplicial volume","shortName":"Gromov's scalar curvature vs simplicial volume","problemNumber":null,"field":"Scalar curvature; simplicial volume and bounded cohomology","fieldGroup":"Geometry & topology","statement":"For a closed connected oriented $n$-manifold $M$ let $\\|M\\|$ be its real simplicial volume, and for a Riemannian metric $g$ let $\\mathrm{Scal}_g^-=\\max\\{0,-\\mathrm{Scal}_g\\}$. In 1986 Gromov (Large Riemannian manifolds, Conjecture 3.A, Remark (B), Equation (8')) proposed bounding simplicial volume by the scale-invariant integral $\\int_M(\\mathrm{Scal}_g^-)^{n/2}dV_g$; equivalently, $\\mathrm{Scal}_g\\ge-\\kappa^2$ should force $\\|M\\|\\le C_n\\kappa^n\\mathrm{Vol}_g(M)$. Partial results covered bounded unit-ball volumes in the universal cover (Braun-Sauer), Kahler surfaces (Min-Zheng-Zhu) and qualitative vanishing for spin universal covers. Is there $a_n>0$ with $\\int_M(\\mathrm{Scal}_g^-)^{n/2}dV_g\\ge a_n\\|M\\|$ for every closed $n$-manifold and every metric?","posedBy":"M. Gromov, Large Riemannian manifolds, in Curvature and Topology of Riemannian Manifolds (1986), Conjecture 3.A, Remark (B), Equation (8')","yearPosed":1986,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $n\\ge3$ there is $a_n>0$ with $\\int_M(\\mathrm{Scal}_g^-)^{n/2}dV_g\\ge a_n\\|M\\|$ for every closed connected oriented smooth $n$-manifold $M$ and smooth metric $g$; e.g. $\\mathrm{Scal}_g\\ge-n(n-1)$ gives $\\mathrm{Vol}_g(M)\\ge a_n[n(n-1)]^{-n/2}\\|M\\|$. The route is a Yamabe reduction to $\\mathrm{Scal}=-1$ followed by a bounded-cohomology and graph-deformation estimate. No explicit value of $a_n$ is stated in the theorem, and dimension two is excluded.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. The proof uses the vanishing theorem of the companion manuscript (Positive scalar curvature forces rational inessentiality). No Lean formalization accompanies it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source (v126-proof), read against Gromov's 1986 proposal as cited. Theorem 1.1 states the inequality for every $n\\ge3$, every closed connected oriented smooth $n$-manifold and every smooth metric, with $a_n$ depending only on $n$. It is conditional on the companion's vanishing theorem (nonnegative scalar curvature forces $\\|M\\|=0$), itself an unrefereed claim in this release. Not refereed here. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A well-known quantitative conjecture of Gromov from 1986 connecting scalar curvature with simplicial volume, mostly discussed among scalar-curvature specialists, and resting here on the companion's qualitative vanishing theorem. Placed level with the Seymour second neighbourhood conjecture (38) and Universal volume growth bounds from positive intermediate curvature (38), above the Dolgachev-Weisfeiler conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf","sourceName":"OpenAI math release: An integral scalar curvature bound for real simplicial volume","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Positive scalar curvature forces rational inessentiality (companion, vanishing input)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"anti-de-sitter-penrose-inequality","name":"The asymptotically anti-de Sitter Penrose inequality for three-dimensional initial data with spherical conformal infinity","shortName":"Anti-de Sitter Penrose inequality","problemNumber":null,"field":"Mathematical general relativity","fieldGroup":"Mathematical physics","statement":"With cosmological constant $-3$, the Schwarzschild-anti-de Sitter relation $M=(r_h+r_h^3)/2$ and Penrose's 1973 argument suggest that asymptotically hyperbolic initial data with a trapped boundary of area $A$ have mass, in the sense of Wang and Chrusciel-Herzlich, at least $\\frac12(r_A+r_A^3)$ with $r_A=\\sqrt{A/4\\pi}$. Even the time-symmetric case was known only for special classes: graphs (Dahl-Gicquaud-Sakovich, de Lima-Girao), small perturbations of Schwarzschild-anti-de Sitter (Ambrozio; Khuri-Kopinski), and Neves showed inverse mean curvature flow cannot be used directly. Does every three-dimensional asymptotically anti-de Sitter initial-data set with spherical conformal infinity, the dominant energy condition and a weakly trapped boundary satisfy $m\\ge\\frac12(r_A+r_A^3)$ with $A$ the minimal enclosing area, with equality only for Schwarzschild-anti-de Sitter?","posedBy":"Penrose's 1973 heuristic in the asymptotically anti-de Sitter setting; studied as a conjecture by Dahl, Gicquaud and Sakovich, Ambrozio, and Khuri and Kopinski, among others","yearPosed":null,"ageNote":"The manuscript traces the inequality to Penrose (1973) and the mass of Wang and Chrusciel-Herzlich but cites no single source stating the asymptotically hyperbolic conjecture, so no year is set.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (principal): under those hypotheses $m_{AH}\\ge\\frac12(r_A+r_A^3)$ with $A$ the full enclosing-area infimum, sharp; no maximality, compact topology or connectedness assumption. Equality on the stated horizon subclass identifies the original data as a spacelike hypersurface in Schwarzschild-anti-de Sitter with the same mass. Companions: the maximal case with possibly disconnected boundary, and a local inequality for small maximal vacuum conformal perturbations of Schwarzschild-anti-de Sitter using the horizon's own area. Not shown: higher dimensions, toroidal or higher-genus conformal infinity (the Lee-Neves regime), or timelikeness of the mass as a conclusion.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has thirteen manuscripts (September 27 and October 5, 2026); the October papers cite the September neutral theorem as an input, consistent with the README's note that some results build on earlier model results.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the principal manuscript and the two asymptotically hyperbolic companion abstracts were read against the conjecture as the manuscript states it. Not refereed. Scope stated by the paper: dimension three, one spherical asymptotically anti-de Sitter end with stated decay and integrability, compact weakly future outer trapped boundary, and a future-timelike metric mass four-flux, which is assumed rather than derived; equality only on the connected, outermost, outer area-minimizing horizon subclass. The paper says it makes no evolution or cosmic-censorship claim. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The asymptotically hyperbolic Penrose inequality was open even in the time-symmetric case and is a recognised target in mathematical relativity, though less central than the asymptotically flat conjecture. Placed well below the Spacetime Penrose inequality entry (64), just below The Sandier-Serfaty conjecture (42) and above The Bogoliubov quantum-depletion law (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf","sourceName":"OpenAI math release: The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The Penrose inequality for maximal asymptotically hyperbolic initial data","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: A local Penrose inequality for conformal perturbations of Schwarzschild-anti-de Sitter data","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"projected-lax-conjecture","name":"The Projected Lax Conjecture: is every hyperbolicity cone a spectrahedral shadow?","shortName":"Projected Lax Conjecture","problemNumber":null,"field":"Real algebraic geometry; semidefinite lifts of convex cones","fieldGroup":"Algebra","statement":"A spectrahedral shadow is a set of the form $\\{x\\in\\mathbb R^d:\\exists u\\in\\mathbb R^a,\\ A_0+\\sum_i x_iA_i+\\sum_j u_jB_j\\succeq0\\}$ with finitely many real symmetric matrices, that is, a set with a finite semidefinite lift. Netzer and Sanyal (2015) proved that a hyperbolicity cone is a spectrahedral shadow when every nonzero boundary point is a smooth point of its hyperbolic polynomial, and formulated the Projected Lax Conjecture, a weakening of the generalized Lax conjecture. Is every closed hyperbolicity cone a spectrahedral shadow?","posedBy":"Tim Netzer and Raman Sanyal, Smooth hyperbolicity cones are spectrahedral shadows, Math. Program. 153 (2015)","yearPosed":2015,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there exists a real homogeneous hyperbolic polynomial whose closed hyperbolicity cone admits no finite semidefinite lift (exact equality, no closure). The cone comes from a positive semidefinite matrix-valued quadratic map $Q$ via an affine section $E_Q=\\{(A,t,y):A\\succeq0,\\ t\\ge\\mathrm{tr}(A^\\dagger Q(y))\\}$, and a local sums-of-squares obstruction in the spirit of Scheiderer rules out every lift. This also disproves the generalized Lax conjecture. It is an existence result in high dimension, not an explicit polynomial and not a lower bound on lift size. Note the explicit cone of the September 24 companion does have an exact lift (size 100 pencil, 307 auxiliary variables).","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The construction is an existence argument in high dimension; it relies on a finite-field rank certificate given in an appendix.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the October 5 manuscript was read against the conjecture as the paper attributes it to Netzer and Sanyal; it claims a real homogeneous hyperbolic polynomial whose closed hyperbolicity cone has no finite affine semidefinite lift, for any number of auxiliary variables and arbitrary real coefficients. The proof was not refereed. The construction is nonexplicit (quadratic map with $m=330$, matrix size $20N-1$, $N>10\\cdot36^2$) and uses a seed Hankel matrix established by a finite-field certificate in an appendix, which was not re-run here. This manuscript has no Lean formalization in the release; the family's Lean statement covers only the weaker spectrahedrality claim of the September 24 companion.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A named weakening of the generalized Lax conjecture, posed in 2015 and tied to Nemirovski-style questions on semidefinite representability in optimization; younger and less widely known than the generalized form. Placed level with the CVP polynomial-factor hardness entry (35) and the Krauth-Mezard Ising perceptron capacity (35), below the generalized Lax conjecture entry (42) and above the polynomial-time low-degree conjecture (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf","sourceName":"OpenAI math release: Hyperbolicity Cones Without Semidefinite Lifts","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A nonspectrahedral hyperbolicity cone","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf","kind":"paper"},{"label":"Companion: An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf","kind":"paper"},{"label":"Netzer and Sanyal 2015 (DOI)","url":"https://doi.org/10.1007/s10107-014-0744-6","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"friedgut-kalai-threshold-width-hypergraphs","name":"The Friedgut-Kalai threshold-width conjecture for r-uniform hypergraph properties","shortName":"Friedgut-Kalai threshold width, hypergraphs","problemNumber":null,"field":"Random hypergraphs, analysis of Boolean functions","fieldGroup":"Combinatorics","statement":"For a nontrivial increasing property of $r$-uniform hypergraphs on $n$ vertices invariant under relabelling, the transitive symmetry of the $\\binom nr$ edge coordinates gives threshold width $O(1/\\log n)$ by the Friedgut-Kalai method. Friedgut and Kalai (1996, Section 5) conjectured that the vertex symmetry does better: for each fixed $r$ the width from probability $\\varepsilon$ to $1-\\varepsilon$ should be $O_{r,\\varepsilon}((\\log n)^{-r/(r-1)})$, the graph case being $r=2$. Bourgain and Kalai reached exponent $r/(r-1)-\\delta$ for the influence bound at the uniform measure. For every fixed $r\\ge3$, is the threshold width of every such property $O_{r,\\varepsilon}((\\log n)^{-r/(r-1)})$?","posedBy":"Ehud Friedgut and Gil Kalai","yearPosed":1996,"ageNote":"Friedgut-Kalai, Proc. AMS 124 (1996), Section 5, p. 2999; their formulation counts edge coordinates, equivalent for fixed r.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for each $r\\ge3$ there is $C_r$ with $\\mathrm{Var}_p(f)\\le C_rI_p(f)/(\\log n)^{r/(r-1)}$ for every relabelling-invariant Boolean property of $r$-uniform hypergraphs, every $n\\ge r$ and every $0 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts (September 25 and October 5, 2026); the hypergraph paper adapts the method of the graph paper.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and Corollary 4.1 were read against the conjecture as the manuscript quotes it. The proof (vertex-block restrictions with an r-uniform degree estimate) was not refereed. The family's Lean challenge SharpThreshold covers only graphs ($r=2$), so this entry is not lean-checked.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The hypergraph extension of the Friedgut-Kalai conjecture, less studied than the graph case and proved here by the same method. Placed below the graph entry (35), below Seymour's Second Neighborhood Conjecture dense case (30), and well above a typical Erdos problem (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf","sourceName":"OpenAI math release: A uniform influence bound for hypergraph properties","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A Sharp Threshold Bound for Monotone Graph Properties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf","kind":"paper"},{"label":"Friedgut-Kalai, Every monotone graph property has a sharp threshold (1996)","url":"https://doi.org/10.1090/S0002-9939-96-03732-X","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strong-ulam-stability-characterizes-amenability","name":"The Burger-Ozawa-Thom question: does strong Ulam stability characterize amenability?","shortName":"Strong Ulam stability and amenability","problemNumber":null,"field":"Group theory; stability of approximate representations","fieldGroup":"Algebra","statement":"A countable discrete group $G$ is strongly Ulam stable if for every $\\varepsilon>0$ there is $\\delta>0$ such that every map $\\mu:G\\to U(H)$ into the unitary group of any complex Hilbert space, with $\\mu(e)=I$ and $\\sup_{g,h}\\|\\mu(gh)-\\mu(g)\\mu(h)\\|\\le\\delta$, is within $\\varepsilon$ uniformly of a genuine unitary representation on the same space. Kazhdan (1982) proved amenable groups are strongly Ulam stable; Burger, Ozawa and Thom (2013) showed groups with a nonabelian free subgroup are not, and Alpeev handled wreath products over nonamenable groups. Burger, Ozawa and Thom asked: is every strongly Ulam stable countable group amenable?","posedBy":"Marc Burger, Narutaka Ozawa and Andreas Thom, On Ulam stability, Israel J. Math. 193 (2013), p. 111","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a countable discrete group is strongly Ulam stable if and only if it is amenable. For every nonamenable countable $G$ and $\\delta>0$ there is a separable $H$ and a normalized $\\mu:G\\to U(H)$ with defect below $\\delta$ that stays at uniform distance more than $1/10$ from every unitary representation on $H$. Not shown: anything about finite-dimensional (ordinary) Ulam stability, or uncountable groups.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The manuscript (October 5, 2026) is self-contained apart from cited background (Furstenberg boundary theory, Kazhdan's theorem).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Strong Ulam Stability Characterizes Amenability' was read against the question as quoted from Burger-Ozawa-Thom (p. 111). The proof (flags from a strongly proximal boundary action, layered unitary circuits, and an anticommuting-involution quadratic-form obstruction) was not refereed. No Lean formalization. Scope the paper itself states: infinite-dimensional Hilbert spaces are essential; the result says nothing about finite-dimensional Ulam stability, which some nonamenable groups have.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A natural question asked in print by Burger, Ozawa and Thom in 2013 and known within operator algebras and group stability, with partial answers by Alpeev; younger and narrower than Dixmier's problem. Tied with the slc index conjecture entry (25) and the Lins-de Melo-Pugh entry (25), above the Hyperkahler Period-Index Conjecture (22), far below Dixmier's problem from this family (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf","sourceName":"OpenAI math release: Strong Ulam Stability Characterizes Amenability","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Same family: Unitarizability implies amenability for discrete groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf","kind":"paper"},{"label":"Burger-Ozawa-Thom, On Ulam stability (2013)","url":"https://doi.org/10.1007/s11856-012-0050-z","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ball-extension-problem-hilbert-to-l1","name":"Ball's extension problem for Lipschitz maps from Hilbert space into $\\ell_1$, via the Mendel-Naor metric Markov cotype question","shortName":"Ball's extension problem into l1","problemNumber":null,"field":"Metric geometry; nonlinear Banach space theory, Lipschitz extension","fieldGroup":"Analysis","statement":"Ball (1992) introduced Markov type and cotype to study Lipschitz extension and asked whether every Lipschitz map $f$ from an arbitrary subset of a Hilbert space $H$ into $L_1$ extends to all of $H$ with $\\mathrm{Lip}(F)\\le K\\,\\mathrm{Lip}(f)$ for a universal $K$. Mendel and Naor (2013) formulated metric Markov cotype two in Cesaro form: $X$ has it with constant $C$ if for every reversible stochastic matrix $A$ with stationary $\\pi$, every $t$ and points $x_1,\\dots,x_n\\in X$ there are $y_1,\\dots,y_n\\in X$ with $\\sum_i\\pi_i d(x_i,y_i)^2+t\\sum_{i,j}\\pi_i a_{ij}d(y_i,y_j)^2\\le C^2\\sum_{i,j}\\pi_i(\\frac1t\\sum_{s=1}^tA^s)_{ij}d(x_i,x_j)^2$. They showed that a positive answer for $\\ell_1$ yields the extension, and asked (Question 1.15): is $N_2(\\ell_1)<\\infty$, that is, does $\\ell_1$ have metric Markov cotype two?","posedBy":"Keith Ball (1992, extension problem); Manor Mendel and Assaf Naor (2013, Question 1.15)","yearPosed":1992,"ageNote":"Ball's extension problem dates from his 1992 GAFA paper 'Markov chains, Riesz transforms and Lipschitz maps'. The metric Markov cotype question for l1 was asked explicitly by Mendel and Naor in 2013 (Analysis and Geometry in Metric Spaces, Question 1.15 and Section 7). The paper also cites Naor-Young (Acta Math. 2022) for discussion of the extension problem.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: real $\\ell_1$ has metric Markov cotype two with $N_2(\\ell_1)\\le12\\sqrt{21}$, uniformly in the number of points, coordinates and time parameter; complex $\\ell_1$ follows with $12\\sqrt{42}$. Corollary: there is a universal $K$ such that every Lipschitz $f:S\\to\\ell_1$, $S$ any subset of a real Hilbert space, extends to $F:H\\to\\ell_1$ with $\\mathrm{Lip}(F)\\le K\\,\\mathrm{Lip}(f)$. The construction is nonlinear (expected coordinatewise medians of geometric-walk endpoints after cut smoothing), consistent with the known failure of Ball's linear Markov cotype for $\\ell_1$ and the Kalton-based subspace counterexample. No optimal constant is claimed, and targets other than $\\ell_1$ are not treated.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. It is a single paper; the extension theorem is derived as a corollary by combining the main theorem with Mendel and Naor's published extension theorem.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and the final section of the TeX source were read against Ball's problem and Mendel-Naor Question 1.15. The paper proves $N_2(\\ell_1)\\le12\\sqrt{21}$ directly and obtains the extension corollary by invoking Mendel-Naor Corollary 1.13, so the extension relies on that published theorem. The family has no Lean formalization (lean/docs/332.md does not exist at the pinned commit). The target is real (and, by a corollary, complex) $\\ell_1$; the paper says 'for this target' and does not discuss general $L_1(\\mu)$ targets.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Ball's Hilbert-to-L1 extension problem is a central open question of nonlinear Banach space theory, open since 1992; famous within metric geometry, unknown outside it. Placed level with Naor's sublinear-dimension question for finite subsets of Lp (35), just below The Lang-Plaut problem (38) and just above Talagrand's operator cotype problem (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf","sourceName":"OpenAI math release: Metric Markov Cotype Two of l1","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Mendel-Naor (2013), Spectral calculus and Lipschitz extension, Question 1.15","url":"https://doi.org/10.2478/agms-2013-0003","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bec-dilute-hard-sphere-gas","name":"Bose-Einstein condensation of the dilute hard-sphere Bose gas in the thermodynamic limit, at zero and small positive temperature","shortName":"BEC for the dilute hard-sphere gas","problemNumber":null,"field":"Quantum many-body theory; dilute Bose gases","fieldGroup":"Mathematical physics","statement":"Consider $N$ bosons on the torus $\\Lambda_L=(\\mathbb{R}/L\\mathbb{Z})^3$ interacting through a repulsive short-range potential, for instance hard spheres of diameter $a$. Condensation in the sense of Penrose and Onsager means that the constant orbital carries a macroscopic fraction of the particles: $\\liminf N^{-1}\\langle u_0,\\gamma^{(1)}u_0\\rangle>0$ as $L\\to\\infty$ with $N/L^3\\to\\rho$ fixed, for the ground state or the Gibbs state at a fixed temperature. This had been proved only in scaling limits where the box stays comparable to the healing length (Gross-Pitaevskii) or the interaction vanishes, and for a lattice hard-core gas at half filling. Does the dilute three-dimensional interacting Bose gas exhibit Bose-Einstein condensation in the thermodynamic limit at fixed density, in its ground state and at positive temperature?","posedBy":"Long-standing, after Penrose and Onsager's 1956 definition of condensation; Solovej's 2025 survey calls condensation in the thermodynamic limit a major open problem","yearPosed":null,"ageNote":"The manuscripts trace the notion to Penrose and Onsager (1956) and cite Solovej (C. R. Physique 2025, Section 5) for the problem's status, but none dates when it was first posed as an open problem, so the year is left blank.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Positive temperature (October 5): for every $a>0$ there is $\\rho_*(a)>0$ such that for each fixed $0<\\rho<\\rho_*$ some fixed $T(a,\\rho)>0$ gives $\\liminf N^{-1}\\langle u_{0,L},\\gamma^{(1)}_{N,L,T}u_{0,L}\\rangle>0$ for the exact canonical hard-sphere Gibbs state as $L\\to\\infty$, $N/L^3\\to\\rho$. Ground state (September 24 companion): absolute $\\varepsilon_0,c_0>0$ with condensate fraction at least $c_0$ in every ground state when $\\rho a^3<\\varepsilon_0$. A further companion gives a density-uniform bound for bounded nonnegative finite-range potentials at temperatures up to $\\rho^2$. Not shown: the transition temperature, condensation at moderate density, two dimensions, or potentials with attractive parts.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has five manuscripts (September 24 to October 5, 2026). The positive-temperature paper says it builds on, and reproduces the needed arguments of, two earlier manuscripts in the same family, so later results rest on earlier model output.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the October 5 positive-temperature manuscript and Theorem 1.1 of the September 24 ground-state companion were read against the problem as stated in the companion and in Solovej's survey section 5. The proofs were not refereed. The headline positive-temperature theorem has no Lean formalization. The ground-state companion does: lean/ComparatorChallenges/HardSphere.json exists with solution module OAI.Analysis.HardSphere.Main present at the pinned commit (not in the formalization catalogue). Its statement OAI.HardSphere.condensation_with_mixed was read: absolute eps0, c0 > 0 such that for rho a^3 < eps0 the constant-orbital occupation has liminf at least c0 along every thermodynamic sequence, for all ground vectors and ground-supported density operators. Not rebuilt. The temperature obtained is small and not tied to the transition.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Proving condensation for an interacting continuum Bose gas in the thermodynamic limit has been a central goal of mathematical physics for decades, called the most challenging problem for dilute gases in Solovej's 2025 survey. This settles the dilute three-dimensional hard-sphere case at zero and low positive temperature, not general densities or the transition, so it is scored as a partial. Above the 2D Anderson localization entry (55) and relativistic Vlasov-Maxwell regularity (55), below Simon's extended-states problem (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf","sourceName":"OpenAI math release: Bose-Einstein condensation at positive temperature in the dilute hard-sphere gas","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Ground-state condensation in the dilute hard-sphere gas (September 24, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: A density-uniform condensate bound for dilute Bose gases (September 27, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf","kind":"paper"},{"label":"Lean proof of the ground-state companion: OAI/Analysis/HardSphere/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/HardSphere/Main.lean","kind":"lean-proof"},{"label":"Solovej, Mathematical physics of dilute Bose gases (C. R. Physique 2025), Section 5","url":"https://arxiv.org/abs/2504.03314","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"near-inclusion-small-unitary-embedding","name":"Small spatial embeddings for one-sided near inclusions of von Neumann algebras","shortName":"Near-inclusion embedding question","problemNumber":null,"field":"Operator algebras; near inclusions, perturbation theory","fieldGroup":"Analysis","statement":"For von Neumann algebras $M,N\\subseteq B(H)$ with common identity let $\\gamma(M,N)=\\sup_{x\\in M,\\|x\\|\\le1}\\inf_{y\\in N}\\|x-y\\|$. Christensen showed that an injective $M$ with small $\\gamma(M,N)$ embeds into $N$ by a unitary close to the identity. Cameron, Christensen, Sinclair, Smith, White and Wiggins asked the unrestricted one-sided question: for every $\\varepsilon>0$ is there a $\\delta>0$ such that $\\gamma(M,N)<\\delta$ always implies $uMu^*\\subseteq N$ for some unitary $u$ with $\\|u-1\\|<\\varepsilon$?","posedBy":"J. Cameron, E. Christensen, A. M. Sinclair, R. R. Smith, S. White and A. Wiggins, Type II_1 factors satisfying the spatial isomorphism conjecture, PNAS 109 (2012)","yearPosed":2012,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are $\\varepsilon_0>0$ and pairs of von Neumann algebras on separable Hilbert spaces with common identity and $\\gamma(M_n,N_n)\\to0$ for which spatial embeddings exist but every implementing unitary satisfies $\\|u-1\\|\\ge\\varepsilon_0$. So the unrestricted one-sided question has a negative answer, in contrast with Christensen's injective case and with the two-sided strong Kadison-Kastler theorem in the same family. The obstruction is to small unitaries, not to the existence of embeddings.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question as the paper quotes it (paragraph before Theorem 2 of Cameron et al. 2012); it claims algebras $M_n,N_n$ on separable spaces with $\\gamma(M_n,N_n)\\to0$ such that every unitary with $uM_nu^*\\subseteq N_n$ stays at distance at least $\\varepsilon_0$ from the identity. Proof not refereed; not formalized.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A specialist question in perturbation theory, raised in passing by Cameron et al. Placed below the Burger-Ozawa-Thom question (25), level with Word-length spectral triples as compact quantum metric spaces (20), above Zhu, Fang and Shi's question on invariant projections (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf","sourceName":"OpenAI math release: Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Universal strong Kadison-Kastler stability","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"arnosti-s-equal-capacity-greedy-matching-conjecture","name":"Arnosti's RANDOM-VERTEX versus RANKING conjecture for greedy matching","shortName":"RANDOM-VERTEX vs RANKING","problemNumber":null,"field":"Randomized matching algorithms; stochastic dominance","fieldGroup":"Algorithms & optimization","statement":"In Nick Arnosti's random bipartite matching model, each job has a prescribed degree and independently samples a uniform feasible subset of the bins. Every bin has the same positive integer capacity C. RANDOM-VERTEX chooses a uniformly random available feasible bin for each job; RANKING uses one shared random priority order of the bins. Both process jobs in an independent uniform random arrival order. The conjecture following Theorem 2 asks whether RANDOM-VERTEX stochastically dominates RANKING in matching size for every C > 2. The paper proves equality in distribution for C = 1 and stochastic dominance for C = 2.","posedBy":"Nick Arnosti, Greedy Matching in Bipartite Random Graphs, conjecture after Theorem 2, p. 136 (online 2021; journal 2022).","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For every finite instance in the stated model, every common capacity $C\\ge1$ and every threshold $K$, we prove $$\\Pr(M_{\\mathrm{RV}}\\ge K)\\ge\\Pr(M_{\\mathrm{RANKING}}\\ge K).$$ The comparison holds for fixed arrival and bin-priority orders independent of the neighborhoods; averaging gives Arnosti's random-order model, including the conjectured $C>2$ regime.\n\nThe proof combines capped-load association, a two-bin coefficient identity, a saturation bound, refined histories and coupling at accepted-job counts. The written proof uses Cohen–Sackrowitz (1987), Lemma 4.1; Lean proves the required finite rational association statement directly. The inequality compares matching-size distributions. It does not assert dominance on every run, conditional on an arbitrary fixed graph, or for unequal capacities.","claimIssueNote":null,"solveDate":"2026-10-05","model":"OpenAI Codex (GPT-6 Astra)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"OpenAI Codex (GPT-6 Astra) selected and investigated the conjecture, developed the central mathematical argument, simplified the written proof using a published association theorem, and built the Lean formalization, verification tooling and interactive exposition. The human collaborator directed the research, set the requirement for a proof covering all capacities, requested simplification and formal verification, and reviewed the presentation. The contribution label refers to the model developing the central proof; it does not claim publication priority or independent human mathematical verification.","verification":"lean-checked","verificationNote":"Read by this site on 6 October 2026, not rebuilt. 91 modules with mathlib, Lean 4.34.0. Outside deliberately negative control files, no sorry, admit, axiom declarations or native_decide. The final theorem source_matching_tail_comparison assumes only degree at most m; its definitions match the paper's model: independent uniform neighbourhoods of the prescribed degrees, a uniform arrival order, one shared uniform priority order for RANKING, uniform choice among available feasible bins for RANDOM-VERTEX, and equal capacities. The repository's axiom log lists only propext, Classical.choice and Quot.sound. At that review, only author-provided build logs were available. A separate exact computation written here over 600 random instances with capacity 1 to 4 found no violation of the tail inequality, strict dominance in 241 of them and equality at capacity 1, as Arnosti's Theorem 2 predicts. The 91-module proof itself was not read.\n\nAuthor update: the linked public GitHub Actions run rebuilt all 91 positive modules from commit c521b8d, rejected 1 false control, and printed the final theorem's axioms: propext, Classical.choice and Quot.sound. Permanent compiler records and source hashes are in the repository. This does not replace an independent expert audit.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"A stated conjecture in a 2022 Stochastic Systems paper, open about five years, with a modest citation trail and a long proof. Above the Gasoline and DNA-duality conjectures at 6, well below the Pinwheel Kernel Conjecture at 12.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/MathIsEvenEasier/greedy-matching-capacity","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[{"label":"Interactive proof and dependency map","url":"https://mathiseveneasier.github.io/greedy-matching-capacity/#proof-map","kind":"other"},{"label":"Original conjecture: after Theorem 2, p. 136","url":"https://doi.org/10.1287/stsy.2021.0082","kind":"problem-record"},{"label":"Complete mathematical proof (LaTeX manuscript)","url":"https://github.com/MathIsEvenEasier/greedy-matching-capacity/blob/main/research-note.tex","kind":"paper"},{"label":"Formal source-model theorem in Lean","url":"https://github.com/MathIsEvenEasier/greedy-matching-capacity/blob/main/formal/SourceModel.lean#L95","kind":"lean-proof"},{"label":"Lean build certificate and axiom audit","url":"https://github.com/MathIsEvenEasier/greedy-matching-capacity/blob/main/formal/verification-source-azure.json","kind":"code"},{"label":"Reproduction instructions and trust scope","url":"https://github.com/MathIsEvenEasier/greedy-matching-capacity/blob/main/REPRODUCE.rst","kind":"other"},{"label":"Public Lean build and final axiom output","url":"https://github.com/MathIsEvenEasier/greedy-matching-capacity/actions/runs/37468519145","kind":"lean-proof"}],"relations":[],"submittedBy":"MathIsEvenEasier","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"3sum-hypothesis","name":"The 3SUM hypothesis: is 3SUM solvable in truly subquadratic time?","shortName":"3SUM hypothesis","problemNumber":null,"field":"Fine-grained complexity; algorithms","fieldGroup":"Theoretical computer science","statement":"Given $n$ integers in $[-n^c, n^c]$, decide whether three of them sum to zero. The 3SUM hypothesis asserts that no algorithm on a word RAM with $O(\\log n)$-bit words solves this in $O(n^{2-\\varepsilon})$ time for any $\\varepsilon > 0$. Hundreds of conditional lower bounds in computational geometry and fine-grained complexity rest on it. Is there a truly subquadratic algorithm?","posedBy":"Anka Gajentaan and Mark Overmars (3SUM-hardness, 1995); the integer word-RAM form as used today is Patrascu's (2010)","yearPosed":1995,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Yes: a deterministic $O(n^{1.9992})$ algorithm for 3SUM on polynomially bounded integers on the word RAM, refuting the hypothesis, and a Las Vegas $O(n^{1.998})$ algorithm for real inputs using only additions, subtractions and comparisons. The core is a new algorithm for wanted entries of thin matrix products (a modified Coppersmith rectangular algorithm with a Schönhage identity), which solves Lopsided All-Edges Sparse Triangle; known reductions give 3SUM. The same paper refutes the APSP hypothesis, catalogued separately. The constants are enormous, so the impact is on the theory: the conditional lower bounds built on 3SUM no longer stand as stated. SETH and Orthogonal Vectors are unaffected.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Claude (internal Anthropic research model, unnamed)","modelMaker":"Anthropic","humanCollaborators":["Josh Alman","Virginia Vassilevska Williams"],"aiRole":"From the abstract: \"Claude, an AI model developed by Anthropic, discovered the algorithm that refutes the 3SUM, APSP, and Exact Triangle hypotheses.\" The methodology section adds that an Anthropic employee had set an internal research model to verify and improve cryptographic constructions based on the average-case hardness of Zero-k-Clique; instead it developed this algorithm, first for the average case and then the worst case, in one session of 16 million output tokens with no human input. Anthropic shared it with the authors in September 2026. Alman and Vassilevska Williams simplified, strengthened and extended it, derived further consequences and wrote the paper; an internal model later produced the Lean formalisation.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Read here on 6 October 2026, not rebuilt. The paper's Theorem 2 was compared with the standard hypotheses as its own introduction states them (word RAM with O(log n)-bit words, integers of absolute value n^O(1)) and matches. The Lean development (anthropics/formal-math, folder 3sum-apsp, commit e1a4e65; Lean 4.33.1 with mathlib, about 102,000 lines) proves correctness and running time of explicit programs on a word-RAM machine defined in Lean, a weaker machine than the standard word RAM, which only strengthens upper bounds. Its five headline claims are stated in a 139-line file with no imports. No sorry outside the deliberate Challenge folder, no axiom declarations, native_decide, implemented_by or unsafe. The statement file was written by the same party as the proof and has not been audited independently, and no Comparator run is recorded. No uninvolved expert has commented yet; the authors are the leading researchers of the area and are not independent reviewers.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The central hardness assumption of computational geometry and one of the three pillars of fine-grained complexity for three decades, with hundreds of conditional lower bounds resting on it. Just below the APSP hypothesis at 55, refuted in the same paper, which carries the older textbook question, and below the matrix-multiplication exponent at 55; far above every other TCS entry, such as the SETH-hardness of furthest pair at 20.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2610.06783","sourceName":"arXiv:2610.06783, Alman and Vassilevska Williams (5 October 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation of the algorithms and running times (anthropics/formal-math)","url":"https://github.com/anthropics/formal-math/tree/e1a4e65/3sum-apsp","kind":"lean-proof"},{"label":"The APSP hypothesis, refuted in the same paper","url":"https://vibemathed.com/problem/apsp-hypothesis","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"o4-sigma-model-exact-mass-gap-asymptotics","name":"The Hasenfratz-Maggiore-Niedermayer exact mass gap for the two-dimensional O(4) lattice model","shortName":"Exact O(4) mass gap asymptotics","problemNumber":null,"field":"Statistical mechanics; asymptotically free sigma models, mass gap","fieldGroup":"Mathematical physics","statement":"For the nearest-neighbour $O(4)$ model on $\\mathbb Z^2$ (spins in $S^3$, weight $\\exp(\\beta\\sum_{x\\sim y}\\sigma_x\\cdot\\sigma_y)$), asymptotic freedom predicts a mass $m_{\\mathrm{lat}}(\\beta)$ of order $\\sqrt\\beta\\,e^{-\\pi\\beta}$ as $\\beta\\to\\infty$. Hasenfratz, Maggiore and Niedermayer (1990) computed the exact ratio $m/\\Lambda_{\\overline{\\mathrm{MS}}}=\\sqrt{32/(\\pi e)}$ for $O(3)$ and $O(4)$ by matching the Bethe ansatz with perturbation theory, which fixes the numerical prefactor of the lattice mass. Does the gap of the full lattice transfer operator satisfy the predicted exact asymptotic, with the predicted constant?","posedBy":"Peter Hasenfratz, Michele Maggiore and Ferenc Niedermayer (Phys. Lett. B, 1990); extended to all O(n), n >= 3, by Hasenfratz and Niedermayer (1990)","yearPosed":1990,"ageNote":"The 1990 papers give a physics prediction for the continuum mass in MS-bar units; the manuscript frames proving the corresponding lattice transfer-gap asymptotic as the open step. Polyakov's 1975 renormalization argument fixed only the exponential order.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for the periodic infinite-volume state, the gap of the full Osterwalder-Schrader transfer operator, including rotation-invariant sectors and measured in original lattice time steps, satisfies $\\lim_{\\beta\\to\\infty}e^{\\pi\\beta}\\beta^{-1/2}m_{\\mathrm{lat}}(\\beta)=32\\exp(\\pi/4-1/2)$. The input companion gives $c\\sqrt\\beta e^{-\\pi\\beta}\\le m_{\\mathrm{lat}}\\le C\\sqrt\\beta e^{-\\pi\\beta}$ and a positive full gap at every $\\beta>0$. Not shown: the corresponding exact constant for $O(3)$ or general $O(n)$, a continuum $O(4)$ theory, or any statement about nonperiodic Gibbs states.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript (October 5, 2026) takes the family's 'Sharp mass bounds for the two-dimensional O(4) model' (September 23) as its constructive starting point and adapts arguments from the two O(3) companions of October 4.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the historical section of 'Exact mass asymptotics for the two-dimensional O(4) lattice model' were read against the Hasenfratz-Maggiore-Niedermayer prediction as the paper cites it. The constant 32 exp(pi/4 - 1/2) was not independently re-derived from the MS-bar ratio here. The proof (sector comparison, a Brownian U(2) regulator solved by Bethe-type integral equations, and regulator matching with a finite coupling shift) was not refereed. No Lean formalization: the family's Lean scope note excludes the O(4) gap claims. The result depends on the unreviewed companion 'Sharp mass bounds' for the two-sided bounds, the periodic state and the renormalization trajectories, and on two O(3) companions for adapted arguments.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A precise physics prediction, well known among lattice field theorists and mathematical physicists, whose rigorous lattice form was out of reach while even the order of the gap was open. Scored as famous within one community: just above the Lins-de Melo-Pugh quintic Lienard entry (25), below Stein's dimension-free Riesz transform problem (38), and far below Polyakov's conjecture from this family (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026/exact-mass-o4.pdf","sourceName":"OpenAI math release: Exact mass asymptotics for the two-dimensional O(4) lattice model","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Input: Sharp mass bounds for the two-dimensional O(4) model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf","kind":"paper"},{"label":"Hasenfratz, Maggiore, Niedermayer 1990, Phys. Lett. B","url":"https://doi.org/10.1016/0370-2693(90)90685-Y","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"boltzmann-renormalized-solutions-nonuniqueness","name":"Uniqueness of DiPerna-Lions renormalized solutions of the hard-sphere Boltzmann equation","shortName":"Nonuniqueness of renormalized Boltzmann solutions","problemNumber":null,"field":"Kinetic theory; Boltzmann equation, weak solutions","fieldGroup":"Differential equations","statement":"DiPerna and Lions (1989) proved global existence and weak stability of renormalized solutions of the inhomogeneous Boltzmann equation $(\\partial_t+v\\cdot\\nabla_x)F=Q(F,F)$ for any nonnegative initial density with finite mass, energy and entropy, $\\int F_0(1+|v|^2+|\\log F_0|)<\\infty$; these bounds alone do not make the collision term locally integrable. Uniqueness is known only under much stronger control (Maxwellian upper bounds, perturbations of a Maxwellian in weighted Sobolev spaces), and for the spatially homogeneous hard-sphere equation. The uniqueness problem for general renormalized solutions is listed as open, for example in Silvestre's 2022 survey of open problems in kinetic equations, and Gismondi, Golding and Novack (2026) conjectured nonuniqueness. For the hard-sphere kernel on $\\mathbb T^3\\times\\mathbb R^3$, are renormalized (entropy) solutions with finite mass, energy and entropy determined uniquely by their initial datum?","posedBy":"Open since the DiPerna-Lions existence theory (1989); cited by the manuscript as open in L. Silvestre, Regularity estimates and open problems in kinetic equations (2022, Section 2)","yearPosed":1989,"ageNote":"The manuscript does not quote a 1989 statement of the question; it cites Silvestre (2022) for the uniqueness problem and Gismondi-Golding-Novack (2026) for a conjecture of nonuniqueness for a dense class of data. 1989 is the date of the DiPerna-Lions theory that left uniqueness open.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (principal): there are $R<\\infty$ and a nonnegative $F_0$ with $\\mathrm{supp}_vF_0\\subset B_R(0)$ and $\\int F_0(1+|v|^2+|\\log F_0|)<\\infty$ on $\\mathbb T^3\\times\\mathbb R^3$ admitting two global entropy solutions with exact local conservation of mass, momentum and kinetic energy, differing on a set of positive measure, both in $C([0,\\infty);L^1)$, with $\\int_0^T\\int\\langle v\\rangle^k(Q^++Q^-)<\\infty$ for all $T,k$. The companion (September 23) proves nonuniqueness in a class with local mass and total momentum conservation. The datum has unbounded amplitudes at shrinking spatial scales. It does not give nonuniqueness for a dense or generic set of data (the Gismondi-Golding-Novack conjecture), nor for other kernels, the whole space, or the Landau equation.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: 'Nonuniqueness for the periodic hard-sphere Boltzmann equation' (September 23, 2026) gave the first construction in a renormalized class with local mass and total momentum conservation; the principal manuscript (October 5, 2026) keeps the concentrating-jet mechanism, cites the earlier one, and builds a new global continuation with exact local conservation of mass, momentum and energy.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript and Definition 1.1 were read against the uniqueness question. The solution class requires the renormalized equation, separately integrable normalized gain and loss, global energy and entropy-dissipation inequalities, and exact local balances for $1,v,|v|^2/2$; the datum has bounded velocity support and finite mass, energy and absolute entropy. The release has a Lean challenge for this family (BoltzmannNonuniqueness.json, declaration OAI.BoltzmannNonuniqueness.nonuniqueness, solution module present at the pinned commit) but it is not in the formalization catalogue, and its statement, read here, formalizes the September 23 companion's weaker class: local mass, total momentum, energy inequality, strong continuity only on an initial interval. The lean/docs page says the local-conservation refinement is outside it. So the entry's headline is not covered and the tier stays unreviewed; not rebuilt here. The result is one datum, hard spheres, periodic box only.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Whether DiPerna-Lions solutions of the Boltzmann equation are unique has been a recognised open question of kinetic theory since 1989, well known to kinetic and PDE analysts but not beyond. Level with the Long-time Boltzmann-Grad limit for finite-range potentials entry at 30, above the McKean entropy-production conjecture at 20, and well below global regularity for relativistic Vlasov-Maxwell at 55.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026/paper.pdf","sourceName":"OpenAI math release: Nonuniqueness with local conservation for the hard-sphere Boltzmann equation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Nonuniqueness for the periodic hard-sphere Boltzmann equation (Sept 23, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean (covers the September 23 companion's theorem): OAI/MathematicalPhysics/Boltzmann/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Boltzmann/Main.lean","kind":"lean-proof"},{"label":"Silvestre, Regularity estimates and open problems in kinetic equations (2022)","url":"https://arxiv.org/abs/2204.06401","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"simplex-isotropic-constant-conjecture","name":"The strong isotropic constant conjecture: simplices maximize the isotropic constant","shortName":"Simplex isotropic constant conjecture","problemNumber":null,"field":"Convex geometry; isotropic constants and the slicing problem","fieldGroup":"Geometry & topology","statement":"For a convex body $K\\subset\\mathbb R^n$ with covariance matrix $\\Sigma_K$ of the uniform distribution, the isotropic constant is $L_K=(\\det\\Sigma_K/|K|^2)^{1/(2n)}$, an affine invariant. Bourgain's slicing problem asked for a dimension-free upper bound, recently obtained by Klartag and Lehec. The sharp (strong) form asks for the extremal body: Campi, Colesanti and Gronchi proved it in the plane, Saroglou proved uniqueness there, and Rademacher showed a simplicial polytope maximizer must be a simplex. Klartag showed the sharp bound implies the non-symmetric Mahler inequality, and Fradelizi and Marin Sola recast it as a sharp entropy bound for log-concave densities. Is it true in every dimension $n$ that $L_K\\le L_{\\Delta_n}$ for every convex body $K$, where $\\Delta_n$ is a simplex, with equality only for simplices?","posedBy":"Sharp form of Bourgain's slicing problem (1986); no originator named in the manuscript, which cites Campi-Colesanti-Gronchi (1999, planar case) and Fradelizi-Marin Sola (entropy form)","yearPosed":null,"ageNote":"The manuscript calls it the strong isotropic constant conjecture without naming who first stated it; the planar case was proved by Campi, Colesanti and Gronchi in 1999, so it predates that. Left null rather than guessed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every $n\\ge1$ and every convex body $K\\subset\\mathbb R^n$, $L_K\\le(n!)^{1/n}/((n+1)^{(n+1)/(2n)}\\sqrt{n+2})$, with equality if and only if $K$ is a simplex. Also claims the sharp entropy bound $h(f)\\ge m+\\tfrac12\\log\\det\\mathrm{Cov}(f)$ for every log-concave density on $\\mathbb R^m$, with equality exactly for affine images of products of one-sided exponentials, proved via Gaussian-to-target Brenier transport. Via Klartag (2018) it yields the non-symmetric Mahler inequality $|K||K^\\circ|\\ge(n+1)^{n+1}/(n!)^2$, already listed from a separate manuscript of the same release. It gives no new information on the symmetric slicing constant or on symmetric Mahler.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorems 1.1 and 1.2, read against the conjecture as the manuscript states it. The proof was not refereed. No Lean formalization exists for this family at the pinned commit (no lean/docs/101.md, no comparator challenge). The release ships a scalar certificate checker, verification/verify_scalar.py (exact rational arithmetic, standard library); it was run here on 7 October 2026 and ended with ALL CHECKS PASSED. That checker covers only the rational scalar inequalities used in the estimates, not the transport, chaos-decomposition or equality-case arguments, so it does not raise the tier.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The extremal form of the slicing problem, the organizing question of asymptotic convex geometry, and strong enough to imply the non-symmetric Mahler conjecture. As a named problem it is less famous than Mahler itself, so placed just below the general Mahler entry (55) and the symmetric Mahler entry (60), above the KLS conjecture for quadratic forms (45) and the (B)-conjecture for even log-concave measures (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf","sourceName":"OpenAI math release: A sharp entropy bound and the simplex inequality for isotropic constants","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Scalar certificate checker verify_scalar.py (run here, ALL CHECKS PASSED)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/verification/verify_scalar.py","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-similarity-geometric-progressions","name":"The Erdos similarity conjecture for geometric progressions","shortName":"Erdos similarity conjecture, geometric case","problemNumber":null,"field":"Real analysis; measure theory and affine copies","fieldGroup":"Analysis","statement":"A set $A\\subseteq\\mathbb R$ is measure universal if every measurable set of positive Lebesgue measure contains an affine copy $x+sA$ with $s\\ne0$. Every finite set is universal. Erdos (1974, Problem 4.33.7*) conjectured that no infinite set is. Falconer and Eigen proved it for sequences $a_n\\to0$ with $a_{n+1}/a_n\\to1$, and later criteria (Kolountzakis, Humke-Laczkovich, Chlebik) do not apply to geometric progressions, so even the dyadic sequence $\\{2^{-n}\\}$ remained open. Is it true that for every $q\\in(0,1)$ the progression $\\{q^n:n\\ge1\\}$ is not measure universal, that is, some set of positive measure contains no affine copy of it?","posedBy":"Paul Erdos (Problem 4.33.7*, Mathematica Balkanica 4, 1974)","yearPosed":1974,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $q\\in(0,1)$ and $\\eta\\in(0,1)$ there is a compact $E_{q,\\eta}\\subseteq[0,1]$ with $m(E_{q,\\eta})>1-\\eta$ such that for all $x\\in\\mathbb R$, $s\\ne0$, some $x+sq^n\\notin E_{q,\\eta}$. So no geometric progression is measure universal; the dyadic companion first proved $q=1/2$. The construction is probabilistic (random routing tables on a finite tree). It does not prove the conjecture for other infinite sets, and makes no claim of one set avoiding all ratios at once.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family's earlier manuscript, 'The dyadic case of the Erdos similarity conjecture' (September 25, 2026), proves the case $q=1/2$ and carries the release's Lean formalization.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the geometric-case manuscript was read against the special case of Erdos's conjecture. For each $q\\in(0,1)$ and $\\eta\\in(0,1)$ it gives a compact $E\\subseteq[0,1]$ of measure $>1-\\eta$ with no copy $x+sG_q$, either sign of $s$; the set may depend on $q$. Lean: lean/ComparatorChallenges/DyadicAvoidance.json (solution_module OAI.MeasureTheory.DyadicAvoidance.Main, file present at the pinned commit; not in formalization.yaml) was read here. It states only the dyadic case $q=1/2$ (from the companion manuscript), a special case of this entry's claim, so the entry stays Unreviewed under the tier rule. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":34,"significanceNote":"The Erdos similarity conjecture is one of the best-known open problems of real analysis, and the dyadic sequence was its emblematic open case for fifty years; this settles all geometric progressions but not the conjecture. Partial, so below the parent's standing, roughly 45. Just below the Erdos Problem 74 entry at 38, far above a typical Erdos problem at 10, and well below the Falconer distance conjecture entry at 60.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf","sourceName":"OpenAI math release: The geometric case of the Erdos similarity conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The dyadic case of the Erdos similarity conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf","kind":"paper"},{"label":"Lean proof of the dyadic case only (OAI.Problem310.dyadic_affine_avoidance)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MeasureTheory/DyadicAvoidance/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"xy-model-essential-singularity","name":"The Kosterlitz essential singularity of the XY correlation length","shortName":"XY correlation length essential singularity","problemNumber":null,"field":"Statistical mechanics; Berezinskii-Kosterlitz-Thouless transition","fieldGroup":"Mathematical physics","statement":"For the nearest-neighbor XY model on $\\mathbb Z^2$, let $m(b)$ be the exponential decay rate of the spin correlation along an axis and $\\xi(b)=1/m(b)$ the correlation length, finite for $b 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. It builds on two companion manuscripts of the same family (the critical exponent and logarithmic-correction papers).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Kosterlitz's prediction. It is for the ordinary cosine model on the square lattice, with $\\xi$ defined from the free-box limit and then the axis separation limit, and $b$ the coupling in the Hamiltonian; the constant $A$ is identified only implicitly. The proof depends on the two companion papers for the critical inputs and on a renormalization map whose leading recursion is the Kosterlitz flow. The proof was not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"The essential singularity is the signature prediction of the BKT transition and had resisted rigorous proof for fifty years. Placed just below the XY critical-decay entry of this family (50), below Polyakov's mass-generation conjecture (62), and above the Hasenfratz-Maggiore-Niedermayer exact O(4) mass gap (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf","sourceName":"OpenAI math release: Essential Singularity of the Correlation Length in the Planar XY Model","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The critical logarithmic correction for the planar XY model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: The critical correlation exponent of the planar XY model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"matching-polytope-polynomial-sdp-lift","name":"Does the perfect matching polytope have a polynomial-size semidefinite lift?","shortName":"SDP lifts of the matching polytope","problemNumber":null,"field":"Extended formulations; positive semidefinite rank, lower bounds","fieldGroup":"Theoretical computer science","statement":"A semidefinite lift of size $r$ writes a polytope as an affine image of $L\\cap\\mathbb S^r_+$ for an affine subspace $L$; by Gouveia-Parrilo-Thomas the least such $r$ equals the PSD rank of a slack matrix. Yannakakis ruled out small symmetric linear programs for the perfect matching polytope, Rothvoss (2014) proved its linear extension complexity is $2^{\\Omega(n)}$, and Braun et al. ruled out small symmetric SDPs, noting that it is natural to ask whether matching can be expressed compactly by semidefinite programming. Does the perfect matching polytope of $K_n$ admit an exact semidefinite lift of size polynomial in $n$?","posedBy":"G. Braun, J. Brown-Cohen, A. Huq, S. Pokutta, P. Raghavendra, A. Roy, B. Weitz, D. Zink, The matching problem has no small symmetric SDP, SODA 2016 / Math. Program. 165 (2017)","yearPosed":2015,"ageNote":"Posed in the abstract of arXiv 1504.00703 (April 2015) after Rothvoss's 2014 theorem; the manuscript itself states no posing source and cites this paper for the symmetric case.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is $c>0$ such that for every fixed $0<\\rho<1$ the odd-cut matrix with entries $|M\\cap\\delta(U)|-1+\\rho$ has real PSD rank at least $2^{cn}$ for large even $n$, with no symmetry or precision restriction on the factors. Corollary 1.2: every exact semidefinite lift of the perfect matching polytope has size $2^{\\Omega(n)}$. The method follows Lee-Raghavendra-Steurer with a Grigoriev-type parity functional and new averaging lemmas. The endpoint $\\rho=1$ has a polynomial factorization, and approximate lifts are not excluded (Kaniewski-Lee-de Wolf give subexponential approximations).","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against the question. The paper claims $\\mathrm{rank}_{psd}A_n(\\rho)\\ge2^{cn}$ for every fixed $0<\\rho<1$ and hence that every exact semidefinite lift of the perfect matching polytope has size $2^{\\Omega(n)}$. Proof not refereed. The challenges ComparatorChallenges/MatchingAffineLift.lean and MatchingPSD.lean are not in lean/formalization.yaml; they are found through lean/docs/126.md and their solution modules (OAI.Combinatorics.MatchingPSD.AffineLift and .Main) exist at the pinned commit. Statements read here: for every $C>0$ and large even $n$, every affine section of a PSD cone projecting onto the matching polytope has size $>n^C$, and the PSD rank of the unshifted slack matrix exceeds $n^C$. This states the headline (no polynomial-size lift); the exponential bound and the shifted matrices are not formalized. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The natural sequel to Rothvoss's celebrated matching-polytope theorem and a well-known open question in extended formulations since 2015, though narrower than the linear question. Placed level with Optimality of Greedy for Single-Pass Semi-Streaming Matching (35), above the Goemans-Linial sparsest-cut integrality gap (28), and below Khot's 2-to-1 Games Conjecture with perfect completeness (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026/shifted-matching-psd.pdf","sourceName":"OpenAI math release: Exponential PSD rank of positively shifted matching matrices","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.PerfectMatchingPSD.affine_lift_lower_bound)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/MatchingPSD/AffineLift.lean","kind":"lean-proof"},{"label":"Comparator statement: MatchingAffineLift.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/MatchingAffineLift.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.PerfectMatchingPSD.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/MatchingPSD/Main.lean","kind":"lean-proof"},{"label":"Braun et al., The matching problem has no small symmetric SDP (arXiv)","url":"https://arxiv.org/abs/1504.00703","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gaboriau-fixed-price-problem","name":"Gaboriau's fixed price problem: do all essentially free p.m.p. actions of a countable group have the same cost?","shortName":"Fixed price problem","problemNumber":null,"field":"Measured group theory; cost of measure-preserving actions","fieldGroup":"Algebra","statement":"For a countable group $\\Gamma$ acting by measure-preserving Borel automorphisms on a standard probability space, the cost of the orbit equivalence relation is the infimum of the total measure of generating graphings (Levitt 1995). A group has fixed price if all its essentially free probability-measure-preserving actions have the same cost. Gaboriau showed that free groups $F_n$ have fixed price $n$ and that many groups with commuting or amenable structure have fixed price one, and asked whether every countable group has fixed price. Does every countable group have fixed price, or are there two essentially free p.m.p. actions of one group with different costs?","posedBy":"Damien Gaboriau (building on the notion of cost introduced by Gilbert Levitt)","yearPosed":2000,"ageNote":"The manuscript cites Gaboriau, Cout des relations d'equivalence et des groupes, Invent. Math. 139 (2000), for the fixed-price problem, and Levitt (1995) for the definition of cost. The question may have circulated slightly earlier; the manuscript gives no earlier statement.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: let $A=F(a,b_1,\\dots,b_{99})$, $w=ab_1ab_2\\cdots ab_{99}a$, $J=\\langle b_1,\\dots,b_{99},w\\rangle$ and $\\Gamma=A*_J(J\\times\\langle t\\rangle)$. With $K=e^{96}96^{99}/95^{95}$, $0<\\alpha<1/200$, $K\\alpha^3<1/2$ and $\\eta=\\alpha/100$, the free part of the Bernoulli action on $[0,1]^\\Gamma$ has cost at least $1+\\eta$, while the skew action on $X\\times\\mathbb Z/M$ via $\\chi(a)=1$, $\\chi(b_i)=\\chi(t)=0$ has cost at most $1+99/M$. Taking $M>\\max\\{100,99/\\eta\\}$ gives two essentially free p.m.p. actions of one finitely generated group with different costs. It does not address the fixed-price-one question for groups with property (T), the fixed price of any specific classical group, or the rank-gradient consequences of fixed price.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (October 5, 2026). Its INPUTS.md says it was developed from an intermediate source argument and that the available source record does not establish that it was the final revision of that argument; it also says the result is distinct from the fixed-price-one question for property (T) groups.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and its proof assembly (the conclusion section) were read against the fixed-price question as Gaboriau posed it. The theorem gives the amalgam $\\Gamma=A*_J(J\\times\\langle t\\rangle)$ with $A$ free of rank 100, a Bernoulli action of cost at least $1+\\eta$, and finite height extensions of cost at most $1+99/M$; both actions are essentially free, so this is a direct negative answer to the general question. The proof (deployment and compression of graphings, finite permutation models, an Arzhantseva-Ol'shanskii type rank bound) was not refereed and there is no Lean formalization. The manuscript's INPUTS.md notes that it was developed from an intermediate argument whose final revision status is not recorded.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"Whether every countable group has fixed price has been the central open question of the theory of cost since Gaboriau's 2000 paper, studied across measured group theory and tied to rank gradient. Placed below Gromov and Weiss's sofic question (60), level with Banach's simple Lebesgue spectrum problem (52), and above Howie's conjecture (45) and the CAT(0) question for hyperbolic groups (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf","sourceName":"OpenAI math release: A group without fixed price","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Gaboriau, Cout des relations d'equivalence et des groupes (Invent. Math. 2000)","url":"https://perso.ens-lyon.fr/gaboriau/Travaux-Publi/Cout/Cout.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"weakly-reversible-permanence-conjecture","name":"The permanence conjecture for weakly reversible mass-action systems with fixed rates","shortName":"Mass-action permanence conjecture","problemNumber":null,"field":"Chemical reaction network theory: mass-action kinetics","fieldGroup":"Differential equations","statement":"For a weakly reversible reaction network with mass-action kinetics $\\dot x=\\sum_{y\\to y'}k_{y\\to y'}x^y(y'-y)$ and fixed positive rates, let $S$ be the span of the reaction vectors $y'-y$ and $P=(c+S)\\cap\\mathbb R^d_{>0}$ a positive stoichiometric class, which may be unbounded. The system is permanent on $P$ if every solution starting in $P$ is global and positive and one compact $K\\subset P$ eventually contains each of them, the entry time but not $K$ depending on the initial point. This is stronger than boundedness plus persistence of each trajectory. Known before: permanence for strongly endotactic networks, including weakly reversible ones with one linkage class (Gopalkrishnan-Miller-Shiu; Boros-Hofbauer), and for two species (Craciun-Nazarov-Pantea). Is every weakly reversible mass-action system with fixed positive rates permanent on every positive stoichiometric class?","posedBy":"Permanence conjecture of chemical reaction network theory; the manuscript names it without an originating source and takes the classwise definition from Boros and Hofbauer (2020, Definition 4.1)","yearPosed":null,"ageNote":"The manuscript calls this the permanence conjecture but gives no source or date for it, so poser and year are left blank rather than guessed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a finite weakly reversible network with fixed positive rates and any positive stoichiometric class $P$, there is a nonempty compact convex forward-invariant $K_P\\subset P$ that every solution from $P$ enters in finite time; hence one $\\varepsilon_P$ bounds every trajectory in $P$ between $\\varepsilon_P$ and $\\varepsilon_P^{-1}$ after its entry time, also on unbounded classes. The proof adds strict increase off the trapping set and a passage between scales to the companion's construction. Uniformity holds within one class and one choice of rates; entry times may depend on the initial point, and time-varying rates are not treated.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The manuscript credits its affine approximation lemma and trapping construction to the companion Boundedness and persistence manuscript of September 25, also produced by the model; the README notes that some outputs build on earlier model results. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source; the proof was not refereed. The family's only Lean challenge (MassAction) formalizes the companion boundedness-and-persistence theorem, whose bound depends on the initial point, so this classwise permanence statement has no formal proof at the pinned commit. The argument reuses the companion manuscript's construction, which is reproduced in the paper.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"Permanence asks for one absorbing compact set per conservation class, a named conjecture known within chemical reaction network theory but less prominent than the persistence form. Scored with Gyarfas's tree-cover conjecture (32) and Solovej's generalized ionization conjecture (32), below the boundedness and persistence conjectures (35) it extends.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf","sourceName":"OpenAI math release: Uniform Permanence in Weakly Reversible Mass-Action Systems","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Boundedness and persistence of weakly reversible mass-action systems (input to this proof)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"szabo-equivariant-jiang-su-unital-stably-finite","name":"Szabo's Conjecture A on equivariant Jiang-Su stability, in the unital stably finite case","shortName":"Szabo's Conjecture A, unital stably finite case","problemNumber":null,"field":"Operator algebras; C*-dynamics of amenable group actions","fieldGroup":"Analysis","statement":"Let $A$ be a separable simple nuclear C*-algebra with $A\\cong A\\otimes\\mathcal Z$ and let $\\alpha$ be an action of a countable discrete amenable group $G$ on $A$. Szabo's Conjecture A predicts automatic equivariant Jiang-Su stability: $\\alpha$ should be cocycle conjugate to $\\alpha\\otimes\\mathrm{id}_{\\mathcal Z}$. Szabo proved it for finite algebras with finitely many rays of extremal traces and very weak comparison; the general case was open, in particular when the action moves traces. Is every amenable action on a separable simple nuclear $\\mathcal Z$-stable C*-algebra equivariantly $\\mathcal Z$-stable?","posedBy":"Gabor Szabo","yearPosed":2021,"ageNote":"Szabo, Equivariant property (SI) revisited, Anal. PDE 14 (2021), Conjecture A, cited by the manuscript.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $A$ simple, separable, unital, infinite-dimensional, nuclear, stably finite and $\\mathcal Z$-stable, every action $\\alpha$ of a countable discrete amenable group is cocycle conjugate to $\\alpha\\otimes\\mathrm{id}_{\\mathcal Z}$, with no restriction on how $\\alpha$ acts on the trace simplex, and including finite groups and trivial actions. It does not treat nonunital algebras, and stably infinite (purely infinite) algebras are not part of this theorem.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry's principal is 'Equivariant Jiang-Su stability for amenable actions in the unital stably finite case' (October 5, 2026), one of four manuscripts in the family; it uses Szabo-Wouters' characterisation by equivariant uniform property Gamma for the final absorption step.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Conjecture A. It proves the conclusion for every action of a countable discrete amenable group on a simple separable unital infinite-dimensional nuclear stably finite $\\mathcal Z$-stable algebra, with no condition on traces or outerness. The manuscript itself says nonunital algebras, which Conjecture A includes, are beyond its scope; purely infinite algebras are also not covered by this theorem. The proof was not refereed and is not formalised.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A named conjecture within C*-dynamics, answered here in its main (unital, stably finite) case. Placed below the Burger-Ozawa-Thom question on strong Ulam stability (25), level with the Stable Coordinate Conjecture in ambient dimension five (22), and above Cutoff for Kac's Walk on the Sphere (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/paper.pdf","sourceName":"OpenAI math release: Equivariant Jiang-Su Stability for Amenable Actions in the Unital Stably Finite Case","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Szabo, Equivariant property (SI) revisited (2021)","url":"https://doi.org/10.2140/apde.2021.14.1199","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"charged-spacetime-penrose-inequality","name":"The charged spacetime Penrose inequality in upper-area form for Einstein-Maxwell initial data","shortName":"Charged spacetime Penrose inequality","problemNumber":null,"field":"Mathematical general relativity","fieldGroup":"Mathematical physics","statement":"With electric and magnetic charge, $Q^2=Q_E^2+Q_B^2$, Penrose's heuristic suggests that a black hole of given area cannot have less mass than the Reissner-Nordstrom black hole of that area and charge. For possibly disconnected horizons the lower form $m\\ge(r_A+Q^2/r_A)/2$, $r_A=\\sqrt{A/4\\pi}$, can fail, and the appropriate statement is the upper-area inequality $m\\ge|Q|$, $r_A\\le m+\\sqrt{m^2-Q^2}$; Khuri, Weinstein and Yamada proved it for time-symmetric data with multiple horizons. Does the upper-area inequality hold for general asymptotically flat initial data with arbitrary second fundamental form satisfying the charged dominant energy condition, with $A$ the least area enclosing a weakly trapped boundary, and is equality attained only by Reissner-Nordstrom?","posedBy":"Charged extension of Penrose's 1973 heuristic; upper-area form for multiple horizons from Marcus Khuri, Gilbert Weinstein and Sumio Yamada","yearPosed":null,"ageNote":"The manuscript cites Penrose (1973) and Khuri-Weinstein-Yamada (J. Differential Geom. 106, 2017) but no single source posing the non-time-symmetric charged conjecture, so no year is set.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Principal (Theorem 2.3): for one-ended three-dimensional data with arbitrary $K$, nonzero ADM momentum, disconnected weakly future trapped boundary and the stated decay, $m\\ge|Q|$ and $r_A\\le m+\\sqrt{m^2-Q^2}$ with $A$ the minimal enclosing area; positive area and timelike ADM vector are conclusions. Equality (for $m>|Q|$, connected outermost horizon) identifies the data as a slice of dyonic Reissner-Nordstrom. Companions: the purely electric case in every dimension $n\\ge4$; a Kerr-Newman Penrose inequality for axisymmetric electrovacuum exteriors with zero ADM momentum that assumes the area condition $A\\ge4\\pi\\sqrt{Q^4+4J^2}$; and counterexamples to the Kerr-Newman bound when $J$ is the bare gravitational angular momentum and fields decay only like $r^{-2}$. Not shown: magnetic charge in dimensions four and up, or a general rotating inequality.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has thirteen manuscripts (September 27 and October 5, 2026); the October papers cite the September neutral theorem as an input, consistent with the README's note that some results build on earlier model results.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract and introduction of the principal manuscript and the abstracts of the charged and Kerr-Newman companions were read against the conjecture as the manuscript states it. Not refereed. Scope stated by the paper: dimension three, one-ended data, source-free electric and magnetic fields plus uncharged matter with its own dominant energy condition, two metric and one second-form derivative of decay; the polynomial form is asserted only for $r_A>|Q|$; equality needs $m>|Q|$ and a connected, outermost, outer-area-minimizing future horizon. The numerical proof uses the family's neutral theorem (the spacetime Penrose entry) as an input. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":34,"significanceNote":"A natural, studied extension of the Penrose conjecture, known mainly within mathematical relativity, and its proof here rests on the neutral theorem of the same family. Placed well below the neutral Spacetime Penrose inequality entry (64), near The Bogoliubov quantum-depletion law (35) and Solovej's generalized ionization conjecture (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf","sourceName":"OpenAI math release: The spacetime Penrose inequality with charge and original-data rigidity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A Charged Reduction of the Spacetime Penrose Inequality in Spatial Dimensions at Least Four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: The Kerr-Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: Electromagnetic tails and the Kerr-Newman Penrose inequality","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: Spacetime Penrose inequalities: enclosing area, charge, and rigidity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"separable-cstar-kadison-kastler-spatial","name":"The spatial Kadison-Kastler problem for close separable C*-algebras","shortName":"Separable C*-algebra Kadison-Kastler","problemNumber":null,"field":"Operator algebras; perturbations of C*-algebras","fieldGroup":"Analysis","statement":"Kadison and Kastler (1972) initiated the study of close operator algebras. For C*-algebras, Johnson showed implementing unitaries need not be close to the identity and Choi-Christensen gave close non-isomorphic algebras, but only non-separable ones. Christensen, Sinclair, Smith, White and Winter proved that sufficiently close separable nuclear C*-algebras on a separable Hilbert space are spatially isomorphic, and discussed the separable spatial formulation: is there a $\\delta>0$ such that any two norm-separable C*-algebras $A,B$ on a separable Hilbert space with Kadison-Kastler distance below $\\delta$ satisfy $uAu^*=B$ for some unitary $u$?","posedBy":"E. Christensen, A. M. Sinclair, R. R. Smith, S. A. White and W. Winter, The spatial isomorphism problem for close separable nuclear C*-algebras, PNAS 107 (2010)","yearPosed":2010,"ageNote":"The perturbation program is from Kadison-Kastler (1972); the manuscript cites the 2010 paper as discussing this separable spatial formulation.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $\\varepsilon>0$ there are unital norm-separable C*-algebras $A,B$ on a separable Hilbert space with $\\mathrm{KK}(A,B)<\\varepsilon$ and $A''=B''$ but no unitary with $uAu^*=B$. The construction uses sequence algebras over a deformation of the free-group regular representation (Pytlik-Szwarc family) and a tensor-norm obstruction. It does not show $A$ and $B$ are non-isomorphic, imposes no nuclearity (the CSSWW positive theorem needs it), and does not affect von Neumann algebras.","claimIssueNote":null,"solveDate":"2026-10-05","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the separable spatial question as the paper states it. It claims, for every $\\varepsilon>0$, unital norm-separable C*-algebras on a separable Hilbert space with common identity, Kadison-Kastler distance below $\\varepsilon$ and equal bicommutants, that are not unitarily conjugate. Proof not refereed; not formalized. The paper says it gives no abstract non-isomorphism and no von Neumann counterexample.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"The separable C*-algebra form of the Kadison-Kastler problem is known to perturbation specialists and was sharpened by the CSSWW nuclear theorem. Placed level with Does every abelian envelope have the quotient property? (22), below Non-MF groups and non-finite full group C*-algebras (25), and well below Kadison's Similarity Problem (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026/paper.pdf","sourceName":"OpenAI math release: Close Separable C*-Algebras Without Spatial Conjugacy","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Universal strong Kadison-Kastler stability","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"two-adic-pro-modularity-emerton","name":"Pro-modularity at p = 2: odd absolutely irreducible 2-adic representations occur in the completed Hecke algebra (auxiliary tame level allowed)","shortName":"Unrestricted 2-adic pro-modularity","problemNumber":null,"field":"Completed Hecke algebras; p-adic Langlands for GL2 over Q","fieldGroup":"Number theory","statement":"Let $\\mathbb T_p(N)$ be the full varying-weight $p$-adic Hecke algebra of tame level $N$. Emerton (2011, Conjecture 2.12) records the expectation that every continuous odd two-dimensional $p$-adic representation of $G_{\\mathbb Q}$ unramified outside finitely many primes occurs in such an algebra, with no Hodge-theoretic condition at $p$ and with tame level prescribed by the ramification. Earlier pro-modularity theorems (Skinner-Wiles, Emerton, Pan) need $p>2$ or residual restrictions. For $p=2$, does every continuous, odd, absolutely irreducible $r:G_{\\mathbb Q}\\to\\mathrm{GL}_2(E)$ unramified outside finitely many primes occur in $\\mathbb T_2(N)$ for some odd $N$?","posedBy":"Matthew Emerton, p-adic families of modular forms, Seminaire Bourbaki 2009/2010, Asterisque 339 (2011), Conjecture 2.12, as the manuscript cites","yearPosed":2011,"ageNote":null,"solveType":"proved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every continuous, odd, absolutely irreducible $r:G_{\\mathbb Q}\\to\\mathrm{GL}_2(E)$, $E/\\mathbb Q_2$ finite, unramified outside finitely many primes, there are an odd $N$ divisible by the odd ramified primes and a continuous $\\lambda:\\mathbb T_2(N)\\to\\mathcal O_E$ with $\\lambda(T_\\ell)=\\mathrm{tr}\\,r(\\mathrm{Frob}_\\ell)$ and $\\lambda(\\ell S_\\ell)=\\det r(\\mathrm{Frob}_\\ell)$ for $\\ell\\nmid2N$. Scalar and reducible residual representations are allowed; no de Rham hypothesis. It is an occurrence result and does not assert classical modularity; the tame level may contain auxiliary primes, unlike Emerton's formulation.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof uses the release's Fontaine-Mazur-at-2 theorem and its Hecke dimension theorem as inputs.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the October 4 manuscript was read against Emerton's Conjecture 2.12 as the paper describes it. The paper itself says Emerton's formulation prescribes the allowed tame primes, while the theorem allows auxiliary primes in the level; hence Variant. The proof was not refereed and depends on the two companion manuscripts and on Newton-Thorne Selmer vanishing. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A variant, at one prime, of a specialist expectation recorded by Emerton in 2011; important inside the p-adic Langlands programme but little known outside it. Placed just above the Emerton dimension entry from the same family (20), level with the stable coordinate conjecture in dimension five (22), and below the Kraetzer integral means conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf","sourceName":"OpenAI math release: Unrestricted pro-modularity at the prime two","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion: Fontaine-Mazur modularity at the prime 2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf","kind":"paper"},{"label":"Companion: The Dimension of the Two-Adic Hecke Algebra at Odd Level","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Fontaine-Mazur modularity at the prime 2","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf","kind":"paper"},{"label":"Earlier edition (October 4, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-4-2026/two-adic-promodularity.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"thue-morse-all-three-joshi-rust-first-occurrence-formulas","name":"Thue–Morse: all three Joshi–Rust first-occurrence formulas","shortName":"Thue–Morse first occurrences","problemNumber":null,"field":"Combinatorics on words; automatic sequences and arithmetic progressions","fieldGroup":"Combinatorics","statement":"Let $t(j)$ be the parity of the binary digit sum of $j$, with indexing from zero. Let $A(d)$ be the greatest length of a monochromatic arithmetic progression of positive difference $d$ in $t$, and let $i(d)$ be the least starting index attaining $A(d)$. Joshi–Rust Conjecture 3.8 asks for the three first-occurrence identities: $i(2^n+1)=3\\cdot 2^{2n}-2^n-1$ for $n\\ge2$; $i(2^{2n}-1)=3\\cdot 2^{4n}-2^{2n}+1$ for $n\\ge1$; and $i(2^{2n+1}-1)=2^{2n+1}-1$ for $n\\ge0$. The ranges are explicit: the first formula excludes $n=1$, where the previously recorded value is $i(3)=45$.","posedBy":"Gandhar Joshi and Dan Rust, Monochromatic arithmetic progressions in the Fibonacci, Thue–Morse, and Rudin–Shapiro words (2025), Conjecture 3.8; arXiv:2501.05830v2, DOI 10.1016/j.tcs.2025.115391.","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"All three displayed first-occurrence families are proved in the stated ranges. The proof also gives exact all-start classifications for the plus and even-minus families, and rederives their established maximal-length values without assuming those values as hypotheses. For $q=2^m, m≥2$, every length-$(q+2)$ progression at difference $q+1$ starts at $lq²−q−1$ with $l≥1$ and $t(l−1)=t(l+1)=1−t(l)$. For even $m≥2$, every length-$(q+4)$ progression at difference $q−1$ starts at $lq²−q+1$ with the same condition. Its least $l$ is $3$. For odd $m$, the earliest length-$q$ progression at difference $q−1$ starts at $q−1$. Global maximality and exclusion of every earlier start are included. The prior length results and block-recognition methods of Parshina and Aedo–Grimm–Nagai–Staynova are explicitly credited. This is the first-occurrence conjecture, distinct from the existing avoiding-population matching-height entry. Broader all-difference classifications are not claimed.","claimIssueNote":null,"solveDate":"2026-10-04","model":"OpenAI Codex","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"dot (OpenAI) developed the uniform all-start carry/desubstitution argument, translated the complete proof into Lean, and prepared the exact source/build/axiom certificates. Separate AI checks examined the hand proof, formal statement semantics, and final imported artifacts. This disclosure concerns substantive proof discovery and formalization; it does not claim human expert endorsement.","verification":"lean-checked","verificationNote":"Read by this site on 4 October 2026, not rebuilt. Seven modules, mathlib, Lean 4.33.1. No sorry, admit, axiom declarations, native_decide, opaque or unsafe in the source modules. Definitions checked by hand: $t$ is the recursive digit-sum parity, $A(d)$ and $i(d)$ are global sSup and sInf with attained maxima and exclusion of every earlier start proved, and the three headline theorems state Conjecture 3.8 as printed, with the submitter's ranges (the first formula fails at $n = 1$). The package's axiom log lists only propext, Classical.choice and Quot.sound. The repository has no CI; the build receipt is the author's own log. The three formulas were also checked numerically here for $d = 1, 3, 5, 7, 9, 15, 17, 31, 63$. The informal-to-formal review in the package is by the same AI agent.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":6,"significanceNote":"A numbered conjecture in a 2025 TCS paper whose authors note that Walnut cannot decide it and a proof remained elusive. Narrow but real. Just above the Thue-Morse matching-heights entry at 5 and level with the Shallit-Shur-Zorcic stability conjecture published the same day.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Research-Commons/blob/3bb3cdc9992dd14369bc5b1299ca18fdfd9e93ed/research/2026-10-04-dot-thue-morse-first-occurrence-lean-0550z/package/ThueMorseMAP/Headline.lean","sourceName":"Research Commons (GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"Joshi–Rust Conjecture 3.8, current author version","url":"https://arxiv.org/html/2501.05830v2#S3.SS2.SSS2","kind":"problem-record"},{"label":"Complete Lean package, replay instructions, proof and audit","url":"https://github.com/Sodelin/Research-Commons/blob/3bb3cdc9992dd14369bc5b1299ca18fdfd9e93ed/research/2026-10-04-dot-thue-morse-first-occurrence-lean-0550z/package/README.md","kind":"lean-proof"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pointwise-multiple-ergodic-averages-mixing","name":"Pointwise convergence of multiple ergodic averages for mixing transformations","shortName":"Pointwise multiple ergodic averages (mixing)","problemNumber":null,"field":"Ergodic theory; multiple ergodic averages","fieldGroup":"Analysis","statement":"Let $T$ be an invertible measure-preserving transformation of a probability space and $f_1,\\dots,f_n$ bounded measurable functions. Since Furstenberg's ergodic proof of Szemeredi's theorem, the multiple ergodic averages $\\frac1N\\sum_{k=1}^N\\prod_{j=1}^n f_j(T^{jk}x)$ have been central. Host-Kra and Ziegler proved convergence in $L^2$ for every $n$, but almost-everywhere convergence was known only for $n=2$ (Bourgain) and under extra structure for larger $n$: K-systems, weakly mixing systems with spectral or joining restrictions, and distal systems (Huang-Shao-Ye). The general pointwise question asks whether these averages converge almost everywhere for every $n$ and every system; it was open even for mixing $T$. Do the multiple ergodic averages converge almost everywhere for every finite length?","posedBy":"The pointwise convergence question for multiple ergodic averages, arising from Furstenberg's ergodic Szemeredi theorem; surveyed by Kuca (2026)","yearPosed":null,"ageNote":"The manuscripts state the general pointwise question and cite Furstenberg (1977) for the averages and Kuca's survey for the questions, but name no original poser or date.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every invertible mixing probability-preserving $T$ (no standardness, no mixing rate), every $n\\ge2$ and bounded $f_1,\\dots,f_n$, $\\frac1N\\sum_{k\\le N}\\prod_j f_j(T^{jk}x)\\to\\prod_j\\int f_j$ almost everywhere along all $N$. Companions treat $n=3,4$ and three distinct nonzero slopes of either sign for mixing $T$. Lemma 2.5 of the principal manuscript also states almost-everywhere convergence of averages of at most three functions at distinct positive slopes on every standard system, derived from the companion's oscillation estimate. It does NOT settle the general question for $n\\ge4$ on non-mixing systems, nor polynomial iterates.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. All four manuscripts of the family (dated 4 October 2026) are credited to OpenAI with no human author named. The principal manuscript uses as inputs two other release results: Rokhlin's multiple-mixing theorem (a separate catalog entry) and an oscillation estimate from the distinct-slopes triple-average companion, which in turn rests on a release paper on the trilinear Hilbert transform.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against the posed question: it is the mixing case only, for consecutive progressions $k,2k,\\dots,nk$, with the null set depending on the tuple. No Lean formalisation exists for this family. The proof depends on unformalised release results (Rokhlin's multiple-mixing theorem, catalog entry rokhlin-multiple-mixing-problem, and the triple-average oscillation estimate), so an error there would propagate. The release README warns that some unformalised results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Pointwise convergence of multiple ergodic averages is one of the main open problems of ergodic theory; the mixing case is a substantial special case and is scored below the full question. Below Rokhlin's multiple-mixing problem (62) and Sinai's conjecture on the standard map (58), level with the Ibragimov-Iosifescu conjecture for phi-mixing sequences (40), above Talagrand's convolution conjecture (37).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf","sourceName":"OpenAI math release: Pointwise Multiple Ergodic Averages for Mixing Transformations","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Pointwise convergence of fourfold ergodic averages for mixing transformations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf","kind":"paper"},{"label":"Companion: Triple ergodic averages with distinct integer slopes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf","kind":"paper"},{"label":"Companion: Pointwise convergence of triple ergodic averages for mixing transformations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf","kind":"paper"},{"label":"Kuca (2026), Joint ergodicity - 40 years on (survey)","url":"https://arxiv.org/abs/2603.18974","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"recovery-balance-is-preserved-by-duality","name":"Recovery balance is preserved by duality","shortName":"Recovery-balance duality","problemNumber":null,"field":"Coding theory; random access in DNA storage","fieldGroup":"Theoretical computer science","statement":"Conjecture 1 of Gruica, Bar-Lev, Ravagnani and Yaakobi asks whether a linear code is recovery balanced if and only if its orthogonal dual is recovery balanced. Here, encoded coordinates are read independently and uniformly with replacement; recovery balance means that the expected number of reads needed to recover a coordinate is the same for every coordinate.","posedBy":"Gruica, Bar-Lev, Ravagnani and Yaakobi, A Combinatorial Perspective on Random Access Efficiency for DNA Storage: arXiv v1 §VII (Jan 2024), Conj. 1 in v2; IEEE Trans. Inf. Theory (2025).","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"More strongly, for every encoded coordinate $i$ of a linear code $C$ of length $n\\geq1$, $$a_i(C)+a_i(C^\\perp)=n,$$ where $a_i$ is the expected number of independent uniform reads with replacement needed to recover that coordinate. Consequently, the recovery profile is constant for $C$ exactly when it is constant for $C^\\perp$, proving the conjecture. The argument pairs complementary observation sets, writes expectations as weighted sums over sets, and uses equal complementary weights. A coordinate known to be zero has recovery time zero. The written note also handles the convention requiring one read for such a coordinate: balance equivalence remains valid, but the coordinatewise sum requires a correction. Nonuniform sampling is outside this identity.","claimIssueNote":null,"solveDate":"2026-10-04","model":"OpenAI Codex (GPT-6 Astra)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"OpenAI Codex (GPT-6 Astra) selected and investigated the conjecture, developed the complementary-recovery and sampling argument, wrote the mathematical proof, implemented the Lean formalization and evidence checker, and prepared the exposition. The human collaborator directed the research, questioned scope and interpretation, and reviewed the presentation. The Lean build and axiom checks ran in bounded Azure jobs. This disclosure does not claim an independent human proof audit.","verification":"lean-checked","verificationNote":"Read by this site on 6 October 2026, not rebuilt. Fifteen Lean files with mathlib, Lean 4.34.0; no sorry, admit, axiom declarations, native_decide, opaque, unsafe or implemented_by. The headline theorem codeRecoveryBalanced_dual_iff holds over any field; codeDual is the actual orthogonal complement, recovery is proved equivalent to the paper's column-in-span condition, and the expected number of reads is the correct tail sum. The repository's axiom log lists only propext, Classical.choice and Quot.sound. At that review, only author-provided build logs were available. A separate exact-arithmetic check written here on 68 random codes over GF(2), GF(3) and GF(5) of length up to 7 confirmed a_i(C) + a_i(C^perp) = n in every case. No independent expert has audited the formal statement.\n\nAuthor update: the linked public GitHub Actions run rebuilt all 13 positive modules from commit 80e4dbf, rejected 2 false controls, and printed the final theorem's axioms: propext, Classical.choice and Quot.sound. Permanent compiler records and source hashes are in the repository. This does not replace an independent expert audit.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":6,"significanceNote":"A conjecture in a 2024 information-theory paper on DNA storage, open about two years, and close to classical binary-erasure EXIT-function duality, as the submitters say. Level with Lorieau's Gasoline conjecture at 6, well below the Pinwheel Kernel Conjecture at 12.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/MathIsEvenEasier/dna-recovery-duality","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[{"label":"Proof explanation","url":"https://mathiseveneasier.github.io/dna-recovery-duality/","kind":"other"},{"label":"Conjecture 1 in the source paper","url":"https://arxiv.org/html/2401.15722v3#S5.SS3","kind":"paper"},{"label":"Lean theorem and balance equivalence","url":"https://github.com/MathIsEvenEasier/dna-recovery-duality/blob/main/lean/CodeResult.lean","kind":"lean-proof"},{"label":"Verification record and scope","url":"https://github.com/MathIsEvenEasier/dna-recovery-duality/blob/main/verification-code-azure.json","kind":"code"},{"label":"Public Lean build and final axiom output","url":"https://github.com/MathIsEvenEasier/dna-recovery-duality/actions/runs/37468514956","kind":"lean-proof"}],"relations":[],"submittedBy":"MathIsEvenEasier","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"subset-sum-below-meet-in-the-middle","name":"Worst-case Subset Sum in time 2^((1/2 - c)n)","shortName":"Subset Sum below 2^(n/2)","problemNumber":null,"field":"Exact exponential algorithms; fine-grained complexity","fieldGroup":"Theoretical computer science","statement":"Given positive integers $a_1,\\dots,a_n$ and a target $t$, Subset Sum asks whether some subset of the $a_i$ sums to $t$. The meet-in-the-middle algorithm of Horowitz and Sahni (1974) solves it in time $2^{n/2}\\cdot\\mathrm{poly}$, and Schroeppel and Shamir (1981) kept that time with space $2^{n/4}$. Later work saved only polynomial factors in the worst case (Chen, Jin, Randolph and Servedio, 2023, time $2^{n/2}/n^{\\delta}$), while representation methods beat $2^{n/2}$ only on random instances. Is there an algorithm for worst-case $n$-input Subset Sum running in time $2^{(1/2-c)n}$ for some constant $c>0$?","posedBy":"Standing open question of exact exponential algorithms since Horowitz and Sahni (1974); stated as 'a major goal' by Chen, Jin, Randolph and Servedio (2023)","yearPosed":null,"ageNote":"The 2^(n/2) barrier dates from Horowitz and Sahni (J. ACM 1974), cited by the manuscript. The manuscript cites no paper that first poses the question; Chen, Jin, Randolph and Servedio (APPROX/RANDOM 2023, cited) call it a major goal of the area in their abstract. Year left blank rather than guessed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a uniform randomized word-RAM algorithm decides Subset Sum with error at most 1/3 on every input and runs in worst-case time $O(2^{0.49n})$ for every fixed polynomial bound on input bit length, the time bound holding on every random execution. The method combines isolation, a random prime modulus, a filtered Fourier checksum and alias extraction. It is not deterministic, it does not treat super-polynomial bit lengths, and it does not improve the space bound. The companion manuscript separately gives one-sided error, time $\\mathrm{poly}(n)2^{n/2}$ and space $O(2^{n/5})$.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript is dated October 4, 2026; the low-space companion (September 26, 2026) is an earlier, separate result at the 2^(n/2) time scale.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the posed question. It gives a single uniform randomized algorithm, correct with probability at least 2/3 on every input, whose running time is at most $C_c 2^{0.49n}$ on every execution for inputs with bit length $b\\le n^c$, on a word RAM with $O(n+b)$-bit words; the intermediate bound is $2^{0.489995n}\\mathrm{poly}(n+b)$. The model is stated precisely (no advice, no random oracle, randomness charged). Scope limits stated by the paper: randomized two-sided error (not deterministic), polynomial-bit inputs, word-RAM model. No Lean formalization exists for this family. The proof was not refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":46,"significanceNote":"Breaking the half-exponent barrier for worst-case Subset Sum is one of the best known open questions of exact exponential algorithms, open since 1974 with only polynomial-factor savings. Placed below the 3SUM hypothesis (50) and the matrix multiplication exponent entry (55), which carry wider downstream consequences, and well above the Bilu-Linial signing conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/subset-sum.pdf","sourceName":"OpenAI math release: Subset Sum in Time O(2^{0.49n})","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A Low-Space Algorithm for Worst-Case Subset Sum (time 2^(n/2), space 2^(n/5))","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf","kind":"paper"},{"label":"Chen, Jin, Randolph, Servedio, Subset Sum in time 2^(n/2)/poly(n) (2023)","url":"https://arxiv.org/abs/2301.07134","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"deterministic-polynomial-factorization-prime-fields","name":"Deterministic polynomial-time factorization of polynomials over prime fields","shortName":"Deterministic factorization over F_p","problemNumber":null,"field":"Algorithmic number theory, derandomization","fieldGroup":"Theoretical computer science","statement":"Berlekamp's algorithms (1967, 1970) reduce factoring $f\\in\\mathbb F_p[x]$ to finding separating elements of a quotient algebra, and randomized algorithms such as Cantor-Zassenhaus factor in expected time polynomial in $\\deg f$ and $\\log p$. Deterministically, Shoup's bound depends on $p^{1/2+o(1)}$; under GRH, Ronyai handled a bounded number of factors and Evdokimov obtained $(n^{\\log n}\\log p)^{O(1)}$, still quasipolynomial. Altman (2025) surveys the remaining obstacles. Is there a deterministic algorithm that completely factors every polynomial of degree $n$ over $\\mathbb F_p$, with $p$ given in binary, in time polynomial in $n$ and $\\log p$, without GRH?","posedBy":"Long-standing problem in the line of Berlekamp (1967, 1970); the manuscript cites Altman, arXiv:2509.12705 (2025), for a recent account and names no original poser","yearPosed":null,"ageNote":"The manuscript traces the setting to Berlekamp's 1967 and 1970 papers and the deterministic results of Ronyai (1988), Shoup (1990) and Evdokimov (1994), but does not say who first posed the deterministic question or when.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a uniform deterministic algorithm computes the complete factorization (irreducible factors with multiplicities) of any nonzero dense $f\\in\\mathbb F_p[x]$ in $O(((n+1)\\lceil\\log_2 p\\rceil)^{10^{12}})$ bit operations, with no randomness, oracle or GRH. Theorem 1.2 (proved here, unconditional) does this given small auxiliary primes $\\ell\\equiv1\\pmod{12q}$ with $p$ not a $q$th power mod $\\ell$; the odd-degree splitter uses Jacobians of cyclic curves $Y^q=F(X)$ and norm equations in cyclic algebras. Small auxiliary primes are deduced from a uniform zero-free strip $\\mathrm{Re}\\,s>1-10^{-6}$ for all finite-order Hecke L-functions over cyclotomic fields containing $\\mu_{12}$, imported from the companion on primitive roots and not proved here. The exponent is not practical.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorems 1.1 and 1.2 and the statement of the imported analytic input in Section 10, read against the problem as the manuscript frames it. The algebraic construction was not refereed. No Lean main result. The theorem is conditional on another unchecked manuscript in the same release: the uniform Hecke zero-free strip is Theorem 1.2 of Primitive roots for every admissible integer base, which the catalog lists as an unreviewed partial result. That strip is itself an extraordinary claim, far beyond anything known unconditionally; if it fails, what remains is the unconditional reduction (Theorem 1.2) and, as the paper notes, a polynomial bound under GRH.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Deterministic polynomial-time factorization over finite fields is one of the best-known derandomization problems in algorithmic algebra, open for over fifty years and comparable in standing to the question PRIMES in P answered. Tied with the APSP hypothesis (55), above the 3SUM hypothesis (50), below L = RL = BPL (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026/Deterministic-Polynomial-Factorization-over-Prime-Fields.pdf","sourceName":"OpenAI math release: Deterministic Polynomial Factorization over Prime Fields","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Primitive roots for every admissible integer base (companion supplying the Hecke zero-free input)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf","kind":"paper"},{"label":"Altman 2025, Deterministic polynomial factorisation modulo many primes (account of the problem)","url":"https://arxiv.org/abs/2509.12705","kind":"problem-record"},{"label":"Artin's primitive root entry, whose zero-free strip this proof uses","url":"https://vibemathed.com/problem/artin-primitive-root-infinitude","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"artin-primitive-root-infinitude","name":"Artin's Primitive Root Conjecture: Infinitude for Every Admissible Base","shortName":"Artin's conjecture: infinitude for every base","problemNumber":null,"field":"Analytic number theory, primitive roots","fieldGroup":"Number theory","statement":"Artin conjectured in 1927 that every integer $a\\ne-1$ that is not a perfect square is a primitive root modulo infinitely many primes $p$, and more precisely that the number of such $p\\le x$ is asymptotic to $A(a)\\,x/\\log x$ for an explicit constant $A(a)>0$. Hooley (1967) proved this under the generalized Riemann hypothesis for Kummer fields. Unconditionally, Gupta-Murty and Heath-Brown (1986) showed that at most two prime bases fail the infinitude assertion, so one of $2,3,5$ works, but no single prescribed base such as $a=2$ was known to be a primitive root for infinitely many primes. Is every admissible integer $a$ a primitive root modulo infinitely many primes, with the predicted density?","posedBy":"Emil Artin (1927), recorded in the Artin-Hasse correspondence and Hasse's diaries","yearPosed":1927,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every integer $a$ that is neither $-1$ nor a square there are $c_a>0$ and $x_a$ such that for $x\\ge x_a$ at least $c_a\\,x/(\\log x)^2$ primes $p\\in(x,2x)$ have $a$ as a primitive root. This gives the infinitude part of Artin's conjecture for every admissible base, including $a=2$. It does not give the predicted asymptotic $A(a)\\,x/\\log x$: the count is short by a factor $\\log x$ and constants depend on $a$. The analytic input is Theorem 1.2: for every cyclotomic field $F\\supseteq\\mu_{12}$ and every finite-order Hecke character $\\eta$ of $F$, $L_F(s,\\eta)$ has no zeros with $\\mathrm{Re}\\,s>1-10^{-6}$. The companion on simultaneous primitive roots gives the same lower bound for any finite set of prime bases, assuming four inputs from this paper.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorems 1.1 and 1.2, read against Artin's conjecture as cited (Artin-Hasse correspondence, Hooley 1967, Heath-Brown 1986). The proof was not refereed. No Lean main result. The proof imports a marked Type II estimate from the companion manuscript on the Poisson-Dirichlet law for prime predecessors (Theorem 3.1), which is itself an unchecked manuscript in the same release, and adapts mechanisms from the release's quasi-Riemann hypothesis manuscript. The analytic core, a uniform zero-free strip for Hecke L-functions, is itself a very strong claim.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":63,"significanceNote":"Artin's primitive root conjecture (1927) is one of the best-known problems in number theory, and the fact that no single base such as 2 was known to work unconditionally was a notorious gap. Only the infinitude part is answered, so it is scored below the full density conjecture. Placed just above the bounded prime gaps entries (62), below the Jacobian Conjecture (65) and the 67 percent of zeta zeros partial result (68).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf","sourceName":"OpenAI math release: Primitive roots for every admissible integer base","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Simultaneous primitive roots: a conditional lower bound for prime bases","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf","kind":"paper"},{"label":"Deterministic factorization over prime fields, which uses this zero-free strip","url":"https://vibemathed.com/problem/deterministic-polynomial-factorization-prime-fields","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"quantum-geometric-langlands-irrational-level","name":"The global quantum geometric Langlands conjecture at irrational level","shortName":"Quantum geometric Langlands, irrational level","problemNumber":null,"field":"Geometric representation theory; geometric Langlands","fieldGroup":"Geometry & topology","statement":"For a connected simple complex group $G$ with Langlands dual $G^\\vee$, lacing number $r$ and a smooth projective curve $X$, let $D_c(\\mathrm{Bun}_G(X))$ be the DG category of D-modules on the moduli stack of $G$-bundles twisted by the power of the determinant line corresponding to shifted level $c$. The quantum geometric Langlands conjecture, in Stoyanovsky's inverse-parameter form attributed to Drinfeld and developed in Gaitsgory's quantum Langlands program, predicts an equivalence $D_c(\\mathrm{Bun}_G(X))\\simeq D_{-1/(rc)}(\\mathrm{Bun}_{G^\\vee}(X))$ compatible with localization and Whittaker coefficients, deforming ordinary geometric Langlands. Does this equivalence hold for every level $c$, in particular for irrational $c$?","posedBy":"A. V. Stoyanovsky (attributing the idea to Drinfeld); developed by Dennis Gaitsgory","yearPosed":2006,"ageNote":"Stoyanovsky, Quantum Langlands duality and conformal field theory, arXiv:math/0610974 (2006), Section 1.3, Conjecture 2, as cited by the manuscript; Gaitsgory, Quantum Langlands correspondence (notes 2007, arXiv 2016); shifted-level normalization from Bogdanova (2026), Conjecture 1.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every connected simple complex algebraic group $G$, every smooth projective connected complex curve $X$ and every $c\\in\\mathbb C\\setminus\\mathbb Q$ (including non-real $c$) there is an equivalence $D_c(\\mathrm{Bun}_G(X))\\simeq D_{-1/(rc)}(\\mathrm{Bun}_{G^\\vee}(X))$ of presentable DG categories, characterized by its localization-Whittaker comparison and intertwining central torsor actions with transgression local systems. It does not treat rational levels, ramification, the Betti setting, or non-simple reductive groups.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated October 4, 2026. The paper uses the proof of ordinary geometric Langlands and the local equivalences of Campbell-Dhillon-Raskin as inputs.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture in the cited normalization; it gives the unramified de Rham equivalence for every connected simple $G$, every smooth projective connected curve and every $c\\in\\mathbb C\\setminus\\mathbb Q$, with given global forms and all connected components, characterized by a localization-Whittaker comparison. Scope limits stated by the paper: irrational levels only (rational levels, including the critical and integral ones, are excluded), unramified only, de Rham setting. The argument depends on the proof of ordinary geometric Langlands (Gaitsgory, Raskin and coauthors) and on other large published inputs. No Lean formalization exists for this family. The proof was not refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"Quantum geometric Langlands is a major conjectural deformation of the geometric Langlands correspondence, central to Gaitsgory's program and to the Kapustin-Witten picture; the irrational-level case is the generic part, where the categories are better behaved, so this is a partial answer to a specialist conjecture. Placed below the Lagrangian fibration base entry (38) and the Artin groups CAT(0) conjecture (40), level with the Campana-Peternell entry (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf","sourceName":"OpenAI math release: Global quantum geometric Langlands at irrational level","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Stoyanovsky, Quantum Langlands duality and conformal field theory (2006)","url":"https://arxiv.org/abs/math/0610974","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-8-3-plus-stability-conjecture-for-power-free-binary-morphism-lengths","name":"The 8/3-plus stability conjecture for power-free binary morphism lengths","shortName":"8/3-plus morphism stability","problemNumber":null,"field":"Combinatorics on words; repetition-free morphisms","fieldGroup":"Combinatorics","statement":"Shallit, Shur and Zorcic ask for the phase transitions in admissible lengths of uniform binary morphisms preserving repetition avoidance. In Section 4 of Power-free complementary binary morphisms, they conjecture that the set of admissible lengths stays constant whenever $(8/3)^+\\leq\\alpha<3$, with another transition at the strict threshold $8/3$. Here a morphism is $\\alpha$-free if it sends every $\\alpha$-free binary input word to an $\\alpha$-free output; the plus convention permits exponent exactly $8/3$. Does this length-set stability hold, and is the strict endpoint different?","posedBy":"Jeffrey Shallit, Arseny Shur and Stefan Zorcic, Power-free complementary binary morphisms, arXiv:2310.15064v3 (8 December 2023), Section 4; JCTA 207 (2024), 105910.","yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For every extended-real threshold $(8/3)^+\\leq\\alpha<3$, the admissible positive lengths of $\\alpha$-free uniform binary morphisms are exactly all integers except $3$ and $6$. Every admitted length has a complementary witness, $h(1)=\\overline{h(0)}$; the broader binary class has the same length set by the credited prior impossibility theorem. At the strict threshold $8/3$, length $5$ is impossible, whereas it is possible at $(8/3)^+$. The decisive bound for the prior endpoint family is $\\operatorname{ce}(h(w))\\leq\\max(8/3,\\operatorname{ce}(w))$ for every cubefree input. A length-12 seed covers the remaining doubling family through the established finite sufficient criterion. The proof includes finite and infinite admissible inputs with the stated critical-exponent convention. It does not classify every preserving morphism or settle all transitions in the lower interval $[(7/3)^+,8/3]$. Equal zero/one counts in one image are not assumed.","claimIssueNote":null,"solveDate":"2026-10-04","model":"OpenAI Codex","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"OpenAI Codex produced the central strengthening of the existing boundary-marker argument, the complete length coverage using a new fixed length-12 seed, and the strict length-five endpoint obstruction. It also wrote exact finite certificate programs. A separate AI review checked the uniform proof and independently implemented the seed/endpoint controls. The prior clipped Thue-Morse constructions, cubefreeness theorem, exceptional-length impossibility result and Kobayashi sufficient criterion are explicitly credited. No human expert endorsement is claimed.","verification":"unreviewed","verificationNote":"Re-checked by this site on 4 October 2026. The submission's scripts passed (22 seed inputs, 128 length-five maps, four marker cases). A separate brute force written here found no 5-uniform binary morphism surviving the strict 8/3 test on inputs up to length 6, exactly four maps surviving at (8/3)+ on inputs up to length 8 (the paper's endpoint family), and no violation of ce(h(w)) <= max(8/3, ce(w)) for the length-five endpoint map and the length-12 seed over all 1,158 cubefree inputs up to length 13. These are finite checks; the all-length, all-input statement rests on the written proof, which was read here but not line by line. The review in the package is by the same AI agent and is not independent. No Lean formalisation.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":6,"significanceNote":"One unnumbered conjecture in the discussion section of a 2024 JCTA paper by Shallit, Shur and Zorcic, a year old. Known within combinatorics on words. Level with Joshi-Rust Conjecture 3.8 published the same day and just above the Thue-Morse matching-heights entry at 5.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Research-Commons/blob/ccf88c3e5f14c33ee927d74ccc1df568d52fec88/research/2026-10-04-dot-complementary-avoidance-stability-0954z/STABILITY-THEOREM.md","sourceName":"Research Commons (GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"Source conjecture and prior results, arXiv v3 Section 4","url":"https://arxiv.org/html/2310.15064v3","kind":"problem-record"},{"label":"Separate AI hand review and exact certificate scope","url":"https://github.com/Sodelin/Research-Commons/blob/ccf88c3e5f14c33ee927d74ccc1df568d52fec88/research/2026-10-04-dot-complementary-avoidance-stability-0954z/independent-review/REVIEW.md","kind":"other"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"borel-conjecture-dimension-four","name":"The Borel conjecture for closed aspherical four-manifolds (homeomorphism form)","shortName":"Borel conjecture in dimension four","problemNumber":null,"field":"Topology of manifolds; aspherical four-manifolds","fieldGroup":"Geometry & topology","statement":"A connected manifold is aspherical if its universal cover is contractible, so its fundamental group determines its homotopy type. In a letter to Serre of May 2, 1953, Borel asked whether compact manifolds that are classifying spaces for the same group are homeomorphic. The topological Borel conjecture asks, in every dimension, that every homotopy equivalence between closed aspherical manifolds be homotopic to a homeomorphism; the weaker homeomorphism-existence form asks only that homotopy-equivalent closed aspherical manifolds be homeomorphic. It holds in dimensions at most three, and in dimensions at least five for large classes of groups (Farrell-Jones, Bartels-Lück, including hyperbolic and CAT(0) groups). In dimension four it was known only with extra hypotheses such as a good fundamental group in Freedman's sense. Are homotopy-equivalent closed aspherical four-manifolds always homeomorphic?","posedBy":"Armand Borel","yearPosed":1953,"ageNote":"Borel's letter to Jean-Pierre Serre of May 2, 1953, cited by the manuscript via Ranicki's reproduction 'The birth of the Borel conjecture'.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are closed connected aspherical topological four-manifolds $M$ and $N$, with the same word-hyperbolic fundamental group, that are homotopy equivalent but not homeomorphic. Theorem 1.2: some such $M$ has a self-homotopy equivalence not homotopic to any homeomorphism, so the prescribed-class form also fails. The examples come from Davis reflection applied to a Poincaré chamber whose odd cyclic covers have no manifold model. This disproves the Borel conjecture as an all-dimension statement and in dimension four specifically. Not shown: anything in dimensions five or higher, or smooth-category statements; it also shows that Khan's rigidity up to s-cobordism cannot be upgraded to homeomorphism for such groups (curator's reading of the introduction).","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript depends on two other release manuscripts, A marked tensor obstruction to four-dimensional disk embedding and A PD4 group without an aspherical manifold model (both September 24, 2026, family 305).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 were read against Borel's question and the homeomorphism-existence formulation; the paper claims a counterexample in dimension four with word-hyperbolic fundamental group. The proof was not refereed. No Lean formalization exists for this family (no lean/docs/320.md at the pinned commit, nothing in lean/formalization.yaml). The proof imports results from two unreviewed manuscripts of the same release (the marked tensor obstruction and the PD4 group without manifold model, family 305), whose own claims include that the free group is not good in Freedman's sense; this entry stands or falls with them. The result is in the topological category; it does not touch the conjecture in dimensions five and higher.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"The Borel conjecture is one of the defining rigidity problems of manifold topology, open since 1953 and the motor of the Farrell-Jones program; a counterexample in dimension four breaks the unrestricted statement, though dimension four was the case where failure was least surprising. Placed level with the Baum-Connes Conjecture (60), below Cannon's Conjecture (62) and the Hilbert-Smith Conjecture (65), above the Kinoshita Conjecture and Kirby Problem 4.37 (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026/paper.pdf","sourceName":"OpenAI math release: Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Input (family 305): A PD4 group without an aspherical manifold model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf","kind":"paper"},{"label":"Borel's 1953 letter to Serre, with Ranicki's commentary","url":"https://www.maths.ed.ac.uk/~v1ranick/surgery/borel.pdf","kind":"problem-record"},{"label":"The disc-embedding counterexample this proof depends on","url":"https://vibemathed.com/problem/disc-embedding-conjecture-free-group-good","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"infinity-harmonic-c1-regularity","name":"C1 regularity of infinity-harmonic functions in dimension three and higher","shortName":"Infinity-harmonic C1 regularity","problemNumber":null,"field":"Nonlinear elliptic PDE: the infinity Laplacian","fieldGroup":"Differential equations","statement":"A continuous function $u$ is infinity-harmonic if it is a viscosity solution of $\\Delta_\\infty u=\\sum_{i,j}u_iu_ju_{ij}=0$; by Jensen (1993) and Crandall-Evans-Gariepy (2001) these are the absolutely minimizing Lipschitz extensions studied by Aronsson (1967). Solutions are locally Lipschitz, but the equation controls second derivatives only in the gradient direction, and Aronsson's example $|x_1|^{4/3}-|x_2|^{4/3}$ shows the gradient is at best $C^{0,1/3}$. In the plane, Savin (2005) proved $u\\in C^1$ and Evans and Savin (2008) proved $C^{1,\\alpha}$; Evans and Smart (2011) proved differentiability at every point in every dimension, which does not give continuity of the gradient. In dimensions $d\\ge3$, is every infinity-harmonic function continuously differentiable, with a Holder modulus for the gradient?","posedBy":"Open problem of the infinity-Laplacian regularity theory after Savin (2005), Evans and Savin (2008) and Evans and Smart (2011); the manuscript cites no explicit statement of the question","yearPosed":null,"ageNote":"The manuscript presents the higher-dimensional case as open after the planar results of 2005 and 2008 and the everywhere-differentiability theorem of 2011, but quotes no source that poses it; poser and year are left blank rather than guessed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for each $d\\ge3$ there are $\\alpha_d\\in(0,1/3]$ and $C_d$ such that every bounded infinity-harmonic $u$ on $B_1\\subset\\mathbb R^d$ satisfies $\\|\\nabla u\\|_{L^\\infty(B_{1/2})}+[\\nabla u]_{C^{0,\\alpha_d}(B_{1/2})}\\le C_d\\,\\mathrm{osc}_{B_1}u$; hence infinity-harmonic functions are $C^1_{loc}$ on every open set in $d\\ge3$. The proof is by compactness, using the Evans-Savin anisotropic blow-up and a new Liouville theorem for the limiting equation. The exponent $\\alpha_d$ is not explicit, and no claim is made about the optimal exponent $1/3$, boundary regularity or the planar endpoint.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source; the proof was not refereed. No Lean formalization exists for this family at the pinned commit (lean/docs/377.md does not exist). The manuscript cites no source that poses the question, so its standing as a posed problem rests on the introduction's account of prior work.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"C1 regularity of infinity-harmonic functions beyond the plane has been a well-known open problem in nonlinear PDE since the planar results of 2005-2008. Scored with Wall's manifold-realization question (42) and singular Bernoulli convolutions at non-Pisot parameters (42), below Brennan's conjecture (45) and above the interior curvature estimates for the scalar curvature equation (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf","sourceName":"OpenAI math release: Uniform Interior C1alpha Estimates for Infinity-Harmonic Functions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"period-doubling-prefix-palindromic-length-is-not-2-regular","name":"Period-doubling prefix palindromic length is not 2-regular","shortName":"Period-doubling nonregularity","problemNumber":null,"field":"Combinatorics on words; automatic sequences and palindromic factorization","fieldGroup":"Combinatorics","statement":"Let $u$ be the fixed point beginning with 0 of the substitution $0\\to01$, $1\\to00$ (the period-doubling word). Let $P(n)$ be the least number of nonempty palindromes whose concatenation is the length-$n$ prefix of $u$, with $P(0)=0$, and put $d(n)=P(n+1)-P(n)$. Frid, Laborde and Peltomäki conjectured that $d$ is not 2-automatic, and so $P$ is not 2-regular. Is that true?","posedBy":"Anna E. Frid, Enzo Laborde and Jarkko Peltomäki, On prefix palindromic length of automatic words (2020 preprint; TCS 2021), Conjecture 17 in arXiv v2 / published Conjecture 2.","yearPosed":2020,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The exact period-doubling conjecture is proved: d(n)=P(n+1)−P(n) is not 2-automatic and P is not 2-regular. The proof uses a marker C(n) recording a singleton 1 in some globally optimal factorization. A Gray-code independent-set bound gives nonnegative palindrome-edge slack. Uniform constructions and an all-path diagonal obstruction distinguish infinitely many binary prefixes for C. The sole computer-assisted lemma is an all-length local inclusion proved by complete reachable-state closure, independently implemented twice; finite value tests are not its proof. This does not provide a full closed formula for P or settle the separate Fibonacci conjecture. Classical palindrome structure, automatic-sequence closure and state distinguishability, and the source regularity/difference equivalence are credited. Worldwide priority remains unassessed.","claimIssueNote":null,"solveDate":"2026-10-04","model":"OpenAI assistant (dot; underlying model unspecified)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"dot (OpenAI) developed the marked-optimum lift, Gray-code lower bound, uniform separating-family constructions, and final all-path obstruction. The model wrote the universal local-inclusion checker and mathematical proof. A separate AI review checked the complete argument and independently implemented a second all-length transition-system proof with different state representations. No external human expert endorsement is claimed.","verification":"unreviewed","verificationNote":"Re-checked by this site on 6 October 2026. Both of the submission's checkers passed: the local tiling automaton (2,460 states, 3,092 transitions, 173 eligible ends) and the second checker (1,593 states, 101 eligible ends), with hashes matching the README. A separate computation written here (eertree dynamic programming, not the submission's code) found P(n) for every n up to 2^20 and confirmed the proof's exact lift lemma for every m below 2^19, its N(a,b) formula in all cases in range, and that the windowed 2-kernel keeps growing. These are finite checks. The all-length statement rests on the hand proof (backward lifting and the diagonal obstruction), which was not checked here. The package's review is by the same AI agent and is not independent. No Lean formalisation.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":6,"significanceNote":"A numbered conjecture in a 2021 TCS paper; the authors note it would be a rare natural example of a non-regular function of an automatic word. Narrow audience. Level with the Joshi-Rust and Shallit-Shur-Zorcic conjectures from combinatorics on words at 6, above the Thue-Morse matching-heights entry at 5.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Research-Commons/blob/64e1fa9f532439e5f63b660d295dc6b33dfe7ec0/research/2026-10-04-dot-period-doubling-nonregularity-1150z/NONREGULARITY-THEOREM.md","sourceName":"Research Commons (GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"Original conjecture, arXiv v2","url":"https://arxiv.org/html/2009.02934v2","kind":"problem-record"},{"label":"Complete proof package and reproducible all-length checkers","url":"https://github.com/Sodelin/Research-Commons/blob/64e1fa9f532439e5f63b660d295dc6b33dfe7ec0/research/2026-10-04-dot-period-doubling-nonregularity-1150z/README.md","kind":"code"},{"label":"Independent AI full-proof review and exact hash bindings","url":"https://github.com/Sodelin/Research-Commons/blob/64e1fa9f532439e5f63b660d295dc6b33dfe7ec0/research/2026-10-04-dot-period-doubling-nonregularity-1150z/independent-review/NONREGULARITY-REVIEW.md","kind":"other"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"modularity-elliptic-curves-imaginary-quadratic","name":"Modularity of elliptic curves over imaginary quadratic fields","shortName":"Modularity over imaginary quadratic fields","problemNumber":null,"field":"Langlands program: modularity of elliptic curves","fieldGroup":"Number theory","statement":"Let $K$ be an imaginary quadratic field and $E/K$ an elliptic curve. The Langlands program predicts that $E$ is modular: there is an automorphic representation $\\pi_E$ of $\\mathrm{GL}_2(\\mathbb A_K)$ (a Bianchi modular form when $E$ has no complex multiplication) whose Galois representations are $H^1_{et}(E_{\\bar K},\\bar{\\mathbb Q}_l)$, so that $L(E/K,s)=L(\\pi_E,s-1/2)$. Over $\\mathbb Q$ this is the theorem of Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor. Over imaginary quadratic fields, potential modularity was known (Allen et al., 2023), Allen-Khare-Thorne proved modularity for a positive proportion of curves, and Caraiani-Newton for all curves over fields where $X_0(15)(K)$ has rank zero. Is every elliptic curve over every imaginary quadratic field modular?","posedBy":"Robert P. Langlands, Problems in the theory of automorphic forms, LNM 170 (1970), Section 7, as the manuscript cites; studied computationally by Grunewald-Helling-Mennicke and Cremona","yearPosed":1970,"ageNote":"The manuscript places the conjecture in the Langlands program and cites Langlands (1970, Section 7) for the automorphy of representations attached to elliptic curves over number fields. It does not name a first statement specific to imaginary quadratic fields; Grunewald-Helling-Mennicke (1978) and Cremona developed the computational side.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every imaginary quadratic $K$ and every $E/K$ there is an isobaric automorphic representation $\\pi_E$ of $\\mathrm{GL}_2(\\mathbb A_K)$ with trivial central character and weight zero such that $r_\\iota(\\pi_E)\\simeq H^1_{et}(E_{\\bar K},\\bar{\\mathbb Q}_l)$ for every $l$ and $\\iota$, with local-global compatibility at every finite place $v\\nmid l$ and the expected parameter at the complex place; hence $L(E/K,s)=L(\\pi_E,s-1/2)$. $\\pi_E$ is cuspidal when $E$ has no CM. The method is a geometric prime switch: cyclic covers of a quadratic twist of $E$ give a chain of mod $p$, mod 3, mod $p$ congruences that carries automorphy from an auxiliary curve to $E$ through the Caraiani-Newton lifting theorem. It does not treat other CM fields or totally real fields.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the modularity conjecture for imaginary quadratic fields as the manuscript frames it; the proof was not refereed. The argument uses deep published and preprint inputs as black boxes, above all the Caraiani-Newton lifting theorem and residual-image criterion (arXiv:2301.10509v3, cited as a 2025 preprint) and the potential automorphy work of Allen et al.; the manuscript says Caraiani-Newton remains essential, so the result stands on those inputs as specialized here. The release has no Lean formalization for this family, and the README cautions that unformalized results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Modularity over imaginary quadratic fields was a central open case of the Langlands program after Wiles and BCDT, the target of the ten-author potential automorphy work and of Caraiani-Newton. Placed level with Kadison's similarity problem (58) and Thompson group F amenability (58), below Hilbert's tenth problem over Q (70) and the BSD formula in Selmer corank at most one (78), above the p-converse in Selmer corank one (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf","sourceName":"OpenAI math release: Modularity of elliptic curves over imaginary quadratic fields","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Caraiani-Newton, On the modularity of elliptic curves over imaginary quadratic fields (key input)","url":"https://arxiv.org/abs/2301.10509v3","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"kannan-lovasz-simonovits-conjecture","name":"The Kannan-Lovász-Simonovits conjecture","shortName":"KLS conjecture","problemNumber":null,"field":"Asymptotic convex geometry; high-dimensional probability","fieldGroup":"Geometry & topology","statement":"Let $K \\subset \\mathbb{R}^n$ be a convex body in isotropic position (barycenter at the origin, identity covariance). Kannan, Lovász and Simonovits conjectured that the isoperimetric inequality on $K$ is saturated, up to a universal constant, by half-spaces: $\\mathrm{Vol}_{n-1}(\\partial A \\cap K) \\ge c \\min\\{\\mathrm{Vol}_n(A \\cap K), \\mathrm{Vol}_n(K \\setminus A)\\}$ with $c > 0$ independent of $n$ and $K$. Equivalent forms: exponential concentration of 1-Lipschitz functions with a dimension-free rate, or a Poincaré inequality with a universal constant, for every isotropic log-concave measure. Is the KLS constant bounded independently of the dimension?","posedBy":"Ravi Kannan, László Lovász and Miklós Simonovits (Discrete & Computational Geometry, 1995)","yearPosed":1995,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"Claims the conjecture in full, for all isotropic log-concave measures: $C_P(\\mu) \\le C$ with $C$ universal. The route bounds the third cumulant, propagates the estimate to cumulants of every order by Eldan's stochastic localization, realizes derivatives of tilt averages as mixed cumulants through a suspension construction, and closes with the Song-Zhang criterion. The previous best bounds were Klartag's $C\\sqrt{\\log n}$ and Song and Zhang's $C \\cdot 16^{\\log^* n}$ (1 October 2026). Priority: Song and Zhang revised their preprint on 4 October 2026, the day this paper was posted, to claim the same $O(1)$ bound by their own route; the two are parallel claims of the same result. Consequences include the thin-shell and variance conjectures, already known to follow from KLS.","claimIssueNote":null,"solveDate":"2026-10-04","model":"ChatGPT (version not stated)","modelMaker":"OpenAI","humanCollaborators":["Pierre Bizeul","Boaz Klartag","Joseph Lehec"],"aiRole":"From the paper's acknowledgements: \"Most proofs and mathematical ideas in this paper were found by ChatGPT; a notable exception is the idea to use suspension which was suggested by the authors. The role of the authors has been mostly to understand these proofs and improve their exposition.\" The suspension construction is the step that turns tilt-average derivatives into cumulants of a larger isotropic log-concave measure, so a named human idea carries part of the argument. The model version is not stated.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Read here on 7 October 2026 from the arXiv source: Theorem 1.1 states a universal bound on the Poincaré constant of every isotropic log-concave measure in every dimension, which is the KLS conjecture in its Poincaré form and contains the original convex-body statement. The proof was not refereed here. The authors are leading researchers of the area, not independent reviewers, and there is no formal proof. The argument's last step rests on Song and Zhang's criterion (arXiv:2610.01447, Theorem 5.1), which the paper re-proves in its Section 3.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"The central open conjecture of asymptotic convex geometry for three decades and a driver of sampling and volume algorithms in computer science, with Bourgain's slicing problem and the variance conjecture as its companions. Above the general Mahler conjecture at 55 and the simplex isotropic-constant entry at 50 in the same area, level with the symmetric Mahler conjecture at 60, below the Jacobian conjecture at 65.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2610.05474","sourceName":"arXiv:2610.05474, Bizeul, Klartag and Lehec (4 October 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Song and Zhang, An O(1) bound for the KLS constant (arXiv:2610.01447, revised 4 October)","url":"https://arxiv.org/abs/2610.01447","kind":"paper"},{"label":"The KLS conjecture for quadratic forms (earlier AI result)","url":"https://vibemathed.com/problem/kls-quadratic-forms","kind":"other"},{"label":"Kannan, Lovász and Simonovits, Isoperimetric problems for convex bodies (1995)","url":"https://doi.org/10.1007/BF02574061","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":1},{"slug":"o3-sigma-model-continuum-limit-mass-gap","name":"Construction of the two-dimensional O(3) nonlinear sigma model as an interacting continuum quantum field theory with a mass gap","shortName":"O(3) sigma model continuum limit with mass gap","problemNumber":null,"field":"Constructive quantum field theory; asymptotically free sigma models","fieldGroup":"Mathematical physics","statement":"The two-dimensional $O(3)$ nonlinear sigma model is the lattice measure $\\propto\\exp(\\beta\\sum_{x\\sim y}q_x\\cdot q_y)\\prod_x d\\sigma(q_x)$ with $q_x\\in S^2$. Perturbative renormalization (Polyakov 1975; Brezin-Zinn-Justin 1976) predicts asymptotic freedom: as $\\beta\\to\\infty$ and the lattice spacing shrinks, a finite physical mass survives, and integrability (Zamolodchikov-Zamolodchikov; Hasenfratz-Maggiore-Niedermayer) describes the expected massive continuum theory. Rigorous constructions existed only for large $N$ (Kupiainen, Kopper), for a hierarchical model (Gawedzki-Kupiainen), or in perturbation theory (Mitter-Ramadas), with Balaban's program partial. Does the nearest-neighbour lattice $O(3)$ model have a continuum limit that satisfies the Osterwalder-Schrader axioms, is interacting (non-Gaussian), and has a positive mass gap?","posedBy":"Long-standing goal of constructive field theory; the manuscript frames it via Polyakov (1975), Brezin-Zinn-Justin (1976) and work of Kupiainen, Gawedzki-Kupiainen and Balaban, naming no single poser","yearPosed":null,"ageNote":"The manuscript cites no single source posing the construction as a problem. The perturbative prediction it would confirm dates from Polyakov (1975), so the question is at least that old; left null rather than guessed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.2: along bare couplings $\\beta_N=H+\\frac{\\log L}{2\\pi}N+O(\\log N)$ with spacing $L^{-N}$, suitably normalized smeared lattice spins converge in law and in all moments; the limiting Schwinger distributions are Euclidean and $O(3)$ invariant, reflection positive and clustering with factorial bounds, reconstruct a local relativistic theory in $1+1$ dimensions with a unique vacuum and a Hamiltonian gap $m>0$ on the whole vacuum complement, and have a nonzero connected four-point function, so the theory is not Gaussian. Theorem 1.1: normalized by susceptibility and second-moment length, the limit exists along every real $\\beta\\to\\infty$ without subsequences. The companion proves an isolated one-particle mass atom in the spin two-point spectral measure. Not shown: the value of the mass, the S-matrix of the constructed theory, or a continuum construction for other $O(N)$.","claimIssueNote":null,"solveDate":"2026-10-04","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript (October 4, 2026) adapts the exact blocking and angular-integration estimates of the family's O(4) manuscript ('Sharp mass bounds', September 23), proving the O(3)-specific changes itself; the same-day companion 'An Isolated Particle Pole' builds on this construction.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 (canonical continuum limit) and 1.2 (construction at a fixed physical scale) and Corollary 1.3 were read against the constructive problem as the introduction states it. The proof (exact block renormalization, cutoff comparison at fixed physical scale, source tracking and OS reconstruction) was not refereed. No Lean formalization: the family's Lean scope note says only the O(n) exponential-decay theorem is formalized. The construction relies on intermediate estimates imported from the unreviewed O(4) companion of the same release, with stated hypotheses. Scope the paper itself states: one lattice action, periodic thermodynamic state, no theta term and no external field; no comparison with other lattice actions; no use of the factorized S-matrix.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"Constructing an asymptotically free quantum field theory with a mass gap, the two-dimensional analogue of the Yang-Mills problem, has been a central goal of constructive field theory for decades. Scored level with Polyakov's lattice conjecture from the same family (62) and Cannon's Conjecture (62), below Simon's Problem 1 (65) and well below the Navier-Stokes entry (87).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf","sourceName":"OpenAI math release: The canonical massive continuum limit of the two-dimensional O(3) model","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026/o3-particle-pole.pdf","kind":"paper"},{"label":"Input: Sharp mass bounds for the two-dimensional O(4) model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"singular-bernoulli-convolutions-non-pisot","name":"Singular Bernoulli convolutions at non-Pisot parameters","shortName":"Non-Pisot singular Bernoulli convolutions","problemNumber":null,"field":"Fractal geometry and harmonic analysis: Bernoulli convolutions","fieldGroup":"Analysis","statement":"For $0<\\lambda<1$ the Bernoulli convolution $\\nu_\\lambda$ is the law of $\\sum_{n\\ge0}\\pm\\lambda^n$ with independent fair signs; it is either singular or absolutely continuous (Jessen-Wintner). It is singular for $\\lambda<1/2$, and Erdos (1939) proved it singular whenever $1/\\lambda$ is a Pisot number. Salem (1943) showed the Fourier transform tends to zero for every other $\\lambda$, so Fourier nondecay cannot detect singularity there. Solomyak proved absolute continuity for almost every $\\lambda\\in(1/2,1)$ and Shmerkin showed the exceptional set has Hausdorff dimension zero, but no singular example was known with $1/\\lambda$ not Pisot, for instance a Salem number. Is there $\\lambda\\in(1/2,1)$ with $1/\\lambda$ not a Pisot number for which $\\nu_\\lambda$ is singular?","posedBy":"Recorded as open by Baker, Koivusalo, Troscheit and Zhang (J. London Math. Soc. 2026) and Kern and Sterk (arXiv:2608.14155, 2026), as cited; background Erdos (1939) and Salem (1943)","yearPosed":null,"ageNote":"The manuscript cites two 2026 papers as recording the existence question open and traces the background to Erdos (1939), Salem (1943), Garsia (1962) and the Peres-Schlag-Solomyak survey (2000), but does not say who first asked it or when, so no year is given.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Two non-Pisot singularity theorems. If $\\beta\\in(1,2)$ is a Salem number of degree four then $\\nu_{1/\\beta}$ is singular; explicit cases are the real roots of $x^4-x^3-x^2-x+1$ and $x^4-2x^3+x^2-2x+1$. An explicit degree-31 polynomial has a unique real root $\\beta\\in(1.8392,1.8393)$, not Pisot, with $\\nu_{1/\\beta}$ singular. The manuscript also proves a criterion: $\\nu_\\lambda$ is singular iff $\\lambda$ lies in an explicit limsup set of one-sided approximations by finite sets of algebraic units; the paper says this is an infinite approximation condition, not a finite membership test, so it is not an effective classification. It does not decide Salem numbers of degree six or more, nor which non-Pisot parameters are singular in general.","claimIssueNote":null,"solveDate":"2026-10-03","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and the statements of the arithmetic criterion, the quartic Salem theorem and the degree-31 theorem in the TeX source, read against the existence question as the manuscript cites it; the proofs were not refereed. The degree-31 example rests on a finite exact-arithmetic certificate (verification/verify_degree31.py, standard-library Python) plus analytic bounds in the manuscript; the certificate was not re-run here. The quartic Salem result is a pure argument. The release has no Lean formalization for this family, and the README cautions that unformalized results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The Bernoulli convolution problem has been central in fractal geometry since Erdos (1939), and whether singularity occurs beyond Pisot reciprocals, especially at Salem numbers, is its best-known arithmetic question. This answers existence, not the full classification. Placed just above Banach's isometric conjecture (40) and the Ibragimov-Iosifescu conjecture (40), below the Kakeya maximal conjecture in three dimensions (48), well above Bugeaud's Problem 10.61 (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf","sourceName":"OpenAI math release: Arithmetic classification and non-Pisot singularity for Bernoulli convolutions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Erdos, On a family of symmetric Bernoulli convolutions (1939)","url":"https://doi.org/10.2307/2371641","kind":"problem-record"},{"label":"Baker, Koivusalo, Troscheit, Zhang (2026), records the question open","url":"https://doi.org/10.1112/jlms.70515","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lazic-peternell-generalised-abundance","name":"The Lazić-Peternell Generalised Abundance Conjecture for projective klt pairs","shortName":"Generalised Abundance Conjecture","problemNumber":null,"field":"Birational geometry; minimal model program","fieldGroup":"Geometry & topology","statement":"Let $(X,B)$ be a projective klt pair with $K_X+B$ pseudo-effective, and let $L$ be a nef Cartier divisor such that $K_X+B+L$ is nef. Lazic and Peternell conjectured that $K_X+B+L$ is numerically equivalent to a semiample $\\mathbb Q$-divisor (numerical equivalence is needed: a nontorsion degree-zero line bundle on an elliptic curve is nef with no sections). They proved it for surfaces and for threefolds where $K_X+B$ has positive numerical dimension, and reduced it in general to the minimal model program, abundance and a semiampleness conjecture on Calabi-Yau pairs. Is $K_X+B+L$ always numerically equivalent to a semiample divisor?","posedBy":"Vladimir Lazić and Thomas Peternell (On Generalised Abundance, I)","yearPosed":2018,"ageNote":"arXiv 1808.00438 (2018), published Publ. RIMS 56 (2020); the manuscript cites the arXiv v2 numbering.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a projective klt $\\mathbb Q$-pair $(X,B)$ over an algebraically closed field of characteristic zero with $K_X+B$ pseudo-effective and $M$ nef $\\mathbb Q$-Cartier on $X$, if $K_X+B+M$ is nef then it is numerically equivalent to a semiample $\\mathbb Q$-Cartier divisor. Corollary: on a klt pair with $K_X+B\\equiv0$ every nef $\\mathbb Q$-Cartier divisor is numerically semiample. New ingredient: a nef divisor of full nef dimension on a klt Calabi-Yau pair is big. Kahler companion: for smooth compact Kahler $X$, snc $B$ with coefficients below one and nef $M$, $c_1(K_X+B+M)$ is represented in Bott-Chern cohomology by a semiample line bundle. Not shown: semiampleness of $K_X+B+M$ itself, lc pairs, or nef data living only on a higher model.","claimIssueNote":null,"solveDate":"2026-10-03","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript is dated October 3, 2026; a compact Kahler companion is dated October 4, 2026. The proof uses two other release manuscripts as inputs: the log abundance theorem and the minimal-model existence theorem for generalized lc pairs.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 of 'Numerical Semiampleness of Nef Adjoint Divisors' read against Lazić-Peternell's conjecture as the paper states it; Theorem 1.1 allows a nef $\\mathbb Q$-Cartier summand, so it contains the Cartier case. No Lean formalization. The proof is conditional on two unrefereed release manuscripts, 'Log abundance in characteristic zero' (already in the catalog) and 'Minimal models and Mori fibre spaces for generalized log canonical Q-pairs' (the other entry of this family).","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A recent, well-studied extension of abundance to adjoints with an extra nef summand, important in the generalized-pair programme but with ordinary abundance as its main content, and here derived from release theorems scored separately. Below Popa's logarithmic additivity conjecture (35) and Shokurov's bounded klt complements (33), level with Beauville's splitting conjecture for two summands (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf","sourceName":"OpenAI math release: Numerical Semiampleness of Nef Adjoint Divisors","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion manuscript: Numerical semiampleness of nef adjoint classes on compact Kähler manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026/numerical-generalized-abundance.pdf","kind":"paper"},{"label":"Input manuscript: Minimal models and Mori fibre spaces for generalized log canonical Q-pairs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf","kind":"paper"},{"label":"Input manuscript: Log abundance in characteristic zero","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Numerical semiampleness of nef adjoint classes on compact Kähler manifolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/numerical-generalized-abundance.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"symmetric-sign-matrix-singularity-sharp-rate","name":"The sharp exponential singularity rate of symmetric random sign matrices: P(det A_n = 0) = (1/2 + o(1))^n","shortName":"Sharp singularity rate of symmetric sign matrices","problemNumber":null,"field":"Random matrix theory; discrete random matrices","fieldGroup":"Probability & statistics","statement":"Let $A_n$ be a symmetric $n\\times n$ matrix whose entries on and above the diagonal are independent uniform signs. Two rows agree with probability about $2^{-n}$, so $\\Pr(\\det A_n=0)\\ge2^{-n+o(n)}$, and it is generally believed that $\\Pr(\\det A_n=0)=\\Theta(n^22^{-n})$, mirroring the non-symmetric case where Tikhomirov proved the rate $(1/2+o(1))^n$. Symmetry destroys row independence: Costello, Tao and Vu proved $\\Pr(\\det A_n=0)\\to0$, followed by polynomial and stretched-exponential bounds and the exponential bound $e^{-cn}$ of Campos, Jenssen, Michelen and Sahasrabudhe. Is the singularity probability of a symmetric random sign matrix $(1/2+o(1))^n$, i.e. governed by two equal rows at exponential scale?","posedBy":"Kevin Costello, Terence Tao and Van Vu, and Van Vu in later surveys (as the belief that the probability is of order n^2 2^{-n})","yearPosed":2006,"ageNote":"The OpenAI manuscripts cite no posing source. Campos-Jenssen-Michelen-Sahasrabudhe (arXiv 2105.11384, 2021) write that it is 'generally believed' that the probability is $\\Theta(n^22^{-n})$, citing Costello-Tao-Vu (Duke 2006) and two Vu surveys; 2006 is that earliest cited source.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Principal theorem: for symmetric $A_n$ with independent uniform signs on and above the diagonal, $\\Pr(\\det A_n=0)=(1/2+o(1))^n=2^{-n+o(n)}$, via a Schur-complement reduction to symmetric 0/1 matrices, a corank-one reduction and a counting argument for admissible appended columns. Companion: for fixed bias $p\\in(0,1)\\setminus\\{1/2\\}$, the rate is $p^2+(1-p)^2$, attained by two agreeing rows. Not shown: the believed $\\Theta(n^22^{-n})$ asymptotic, uniformity in $p$, or general entry distributions.","claimIssueNote":null,"solveDate":"2026-10-03","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts: the uniform-sign rate (October 3, 2026) and the fixed-bias analogue (October 4, 2026).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript ($\\Pr(\\det A_n=0)=(1/2+o(1))^n$) and Theorem 1.1 of the biased companion ($\\lim\\Pr(\\det A_n^{(p)}=0)^{1/n}=p^2+(1-p)^2$ for fixed $p\\ne1/2$) were read against the stated belief. Classified partial because the believed asymptotic $\\Theta(n^22^{-n})$ is sharper than the exponential rate proved. The posing attribution comes from Campos-Jenssen-Michelen-Sahasrabudhe, not from the manuscripts. Not refereed; no Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"The sharp singularity rate of symmetric sign matrices is a well-known target of combinatorial random matrix theory after Tikhomirov's non-symmetric theorem and the 2021 exponential bound; scored for the exponential rate, below the believed n^2 2^-n asymptotic. Level with the He-Schramm rigidity conjecture (32), above the Ellipsoid Fitting Conjecture (30) and the Kannan-Tetali-Vempala conjecture (30), below Polya's conjecture for Neumann balls (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf","sourceName":"OpenAI math release: The sharp exponential rate of singularity for symmetric Bernoulli matrices","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The sharp singularity rate for biased symmetric sign matrices","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf","kind":"paper"},{"label":"Costello, Tao and Vu, Random symmetric matrices are almost surely nonsingular (2006)","url":"https://doi.org/10.1215/S0012-7094-06-13527-5","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"effective-iitaka-fibration-conjecture","name":"The effective Iitaka fibration conjecture","shortName":"Effective Iitaka fibration conjecture","problemNumber":null,"field":"Birational geometry; pluricanonical systems","fieldGroup":"Algebra","statement":"For a smooth projective variety $X$ with Kodaira dimension $\\kappa(X)\\ge0$, the pluricanonical systems $|mK_X|$ define the Iitaka fibration for all sufficiently large divisible $m$. For varieties of general type, Hacon-McKernan, Takayama and Tsuji found one $m$ depending only on $\\dim X$; Birkar and Zhang proved an effective bound that also depends on invariants of the general fibre. Is there, for each $d$, an integer $m(d)$ such that $|m(d)K_X|$ defines the Iitaka fibration of every smooth projective $d$-fold with $\\kappa(X)\\ge0$?","posedBy":"Stated as Conjecture 1.1 by C. Birkar and D.-Q. Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs (Publ. Math. IHES, 2016)","yearPosed":2016,"ageNote":"The question predates the 2016 statement: Fujino-Mori (2000) and Viehweg-Zhang (2009) proved cases, as the manuscript recounts.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for each $d\\ge1$ there is $m(d)>0$ such that, over every algebraically closed field of characteristic zero, $|m(d)K_X|$ is nonempty and its section ratios generate the Iitaka function field for every smooth integral projective $d$-fold with $\\kappa(X)\\ge0$; every multiple of $m(d)$ works too. Theorem 1.2 gives a uniform $a(d,\\Phi)$ with $a(K_X+B)\\sim0$ for projective lc pairs with $K_X+B\\sim_{\\mathbb Q}0$ and coefficients in a DCC set $\\Phi$. The bound $m(d)$ is existential; no explicit value is given.","claimIssueNote":"This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it.","solveDate":"2026-10-03","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. Family 034 has fourteen manuscripts dated September 24 to October 5, 2026, which cite one another as inputs; this entry draws on the manuscript named as its source. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Uniform Pluricanonical Iitaka Fibrations' was read against Birkar-Zhang's Conjecture 1.1 as the manuscript describes it; it asks that the system be nonempty and that its section ratios generate the whole Iitaka function field, which is the conjecture's conclusion and slightly stronger than asking for an image of the right dimension. The proof was not refereed. No Lean formalization. The manuscript says it uses the family's log abundance theorem as an input, so this entry depends on that unreviewed claim. Listed as Contested because it depends on the log abundance entry, which the release itself is inconsistent about; see the claim issue.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A well-known conjecture within birational geometry and the natural sequel to the Hacon-McKernan, Takayama and Tsuji theorem for general type. Placed just above the Tachikawa-Auslander-Reiten entry (30), above the Batyrev stringy Hodge number entry (25), and below Stein's dimension-free Riesz transform problem (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf","sourceName":"OpenAI math release: Uniform Pluricanonical Iitaka Fibrations","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Input: Log abundance in characteristic zero","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf","kind":"paper"},{"label":"Birkar and Zhang (2016), Conjecture 1.1","url":"https://doi.org/10.1007/s10240-016-0080-x","kind":"problem-record"},{"label":"The 5 October appendix calling projective log abundance conditional","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026/","kind":"paper"},{"label":"Revised edition (October 6, 2026): Conditional good minimal models for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"matrix-multiplication-exponent-9-4","name":"The matrix multiplication exponent: is omega at most 9/4?","shortName":"Matrix multiplication exponent at most 9/4","problemNumber":null,"field":"Algebraic complexity, bilinear and tensor methods","fieldGroup":"Theoretical computer science","statement":"The exponent $\\omega$ of matrix multiplication over $\\mathbb C$ is the infimum of the $\\tau$ such that, for every $\\varepsilon>0$, two $n\\times n$ matrices can be multiplied with $O_\\varepsilon(n^{\\tau+\\varepsilon})$ arithmetic operations. Strassen's 1969 algorithm gave $\\omega\\le\\log_2 7<2.81$, and the laser method of Strassen, Coppersmith-Winograd and their successors (Stothers, Vassilevska Williams, Le Gall, Alman-Vassilevska Williams, Duan-Wu-Zhou and others) brought the bound down to $\\omega<2.371177$ (Dupont et al., 2026). The trivial lower bound is $\\omega\\ge2$, and whether $\\omega=2$ is a central open question of algebraic complexity. The related dual exponent $\\alpha$, the largest $k$ for which $n\\times n^k$ by $n^k\\times n$ products take $n^{2+o(1)}$ operations, was known to satisfy $\\alpha\\ge0.321334$. How small can the upper bound on $\\omega$ be made?","posedBy":"Volker Strassen, Gaussian elimination is not optimal (Numerische Mathematik, 1969), which introduced subcubic matrix multiplication and the exponent question","yearPosed":1969,"ageNote":"The exponent is a tracked open constant rather than a single stated conjecture; the manuscripts give the chain of records from Strassen (1969) to Dupont et al. (2026, omega < 2.371177) as the prior state.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $\\varepsilon>0$ two complex $n\\times n$ matrices can be multiplied with $O_\\varepsilon(n^{9/4+\\varepsilon})$ arithmetic operations, so $\\omega\\le9/4$ over $\\mathbb C$. The companions prove $\\alpha>0.465$ over every field of characteristic zero, $\\omega(1,0.709,1)<2.092$, $\\omega<2.258$ over all fields outside one uncomputed finite set of positive characteristics, and $\\omega<2.371054886006746$ over every field. Not shown: anything about $\\omega=2$, bit complexity, or a practical crossover size; the 9/4 proof gives no competitive finite matrix size and its transfer to positive characteristic is not claimed.","claimIssueNote":null,"solveDate":"2026-10-02","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The 9/4 bound is the October 2 manuscript; two September 24 companions in the same family give the weaker square bound omega < 2.258 with alpha > 0.465 and a rectangular bound (with Python certificate checks), and omega < 2.371054886006746 over every field including positive characteristic. All three main bounds have Lean formalizations in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of 'An Upper Bound of 9/4 for the Matrix Multiplication Exponent', read against the exponent as usually defined (arithmetic operations over C, epsilon slack). The proof (a separation inequality for tensors sharing one leg, polynomial-multiplication inequalities and Strassen's asymptotic spectrum) was not refereed. Lean: formalization.yaml lists OAI.MatrixMultiplication.complex_omega_le_nine_quarters (lean/OAI/LinearAlgebra/MatrixMultiplication/Main.lean, comparator ComparatorChallenges/MatrixMultiplication.lean). Its statement was read: omega(C) <= 9/4, where omega is the infimum of exponents tau such that for every eps > 0 some constant C bounds the cost of correct division-free straight-line programs (add, sub, mul, constants) for n x n products by C n^(tau+eps) at every n. That is the headline claim. The same file states alpha > 93/200 and omega(1, 0.709, 1) < 523/250; OAI.MatrixAllFields...omega_lt_source_constant states omega(F) < 2.371054886006746 for every field F. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The lean/docs page for this family names only the two September 24 companions, while formalization.yaml lists the 9/4 paper as a source.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Scored on the problem it bounds: the exponent of matrix multiplication, the organising constant of algebraic complexity since 1969. Ties the existing Matrix Multiplication Exponent entry (55), which set the precedent of scoring this problem rather than the size of a record, and sits with the APSP hypothesis (55) and above the 3SUM hypothesis (50). Below the Unique Games Conjecture (74), a full conjecture. A drop from 2.371 to 2.25 would be a landmark jump, but the score is on the question, not the step.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf","sourceName":"OpenAI math release: An Upper Bound of 9/4 for the Matrix Multiplication Exponent","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Complex Matrix Multiplication Below 2.258 and Rectangular Bounds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026.pdf","kind":"paper"},{"label":"Staggered extraction for exact matrix multiplication over every field","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026.pdf","kind":"paper"},{"label":"Lean: complex omega <= 9/4, dual and rectangular bounds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/LinearAlgebra/MatrixMultiplication/Main.lean","kind":"lean-proof"},{"label":"Lean: omega < 2.371054886006746 over every field","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/LinearAlgebra/MatrixFields/Conclusions/AllFieldsBound.lean","kind":"lean-proof"},{"label":"Strassen 1969, Gaussian elimination is not optimal","url":"https://doi.org/10.1007/BF02165411","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"two-size-gasoline-the-tight-approximation-guarantee-for-iterative-rounding","name":"Two-size Gasoline: the tight approximation guarantee for iterative rounding","shortName":"Two-size Gasoline conjecture","problemNumber":null,"field":"Approximation algorithms; inventory and buffer scheduling","fieldGroup":"Algorithms & optimization","statement":"For a Gasoline instance with deliveries $x_i\\in\\{1,K\\}$, fixed positive integer demands $y_i$, equal list lengths and equal total supply and demand, does the position-first Iterative Rounding algorithm always return a schedule using at most twice the minimum storage capacity? At each position it tries each remaining delivery, solves the fractional assignment LP with the chosen prefix fixed, and takes a minimum-score candidate. Capacity includes every delivery peak and consumption trough, with a freely chosen initial stock. This is Conjecture 3.1.1 of Lorieau (2024), the $\\{1,K\\}$ case of Rajkovic's conjecture for the one-dimensional problem.","posedBy":"Lucas Lorieau, Iterative Rounding algorithms for the Extended Gasoline Problem (2024), Conjecture 3.1.1, p. 21; algorithm due to Rajkovic (2022).","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For every integer $K\\ge2$, deliveries in $\\{1,K\\}$ and positive integer demands, we prove $C_{\\mathrm{IR}}\\le L_{\\mathrm{init}}+K-2\\le\\mathrm{OPT}+K-2$, where $L_{\\mathrm{init}}$ is the initial fractional assignment-LP optimum. The upper bound permits every exact tie choice. For every $K\\ge3$, an explicit family with $2K-3$ days has $L_{\\mathrm{init}}=\\mathrm{OPT}=K$ and $C_{\\mathrm{IR}}=2K-2$ under original-index tie breaking. Hence the exact worst-case ratio for that rule is $2-2/K$, with uniform supremum $2$; the algorithm is optimal when $K=2$. The proof also yields an implementation using O(n) arithmetic operations instead of repeated LP solves. This settles the stated two-size positive-integer case, not the unrestricted one-dimensional or multidimensional conjectures. An additional exact finite example shows that preferring the smaller delivery on ties need not be optimal (capacity 7 versus 5); no improved universal guarantee for that alternative rule is claimed.","claimIssueNote":null,"solveDate":"2026-10-01","model":"OpenAI Codex (GPT-6 Astra)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Under human direction, Codex selected and investigated the conjecture, searched prior work, derived the exact suffix-LP formula and rounding invariant, constructed a sharpness family, developed the Lean formalization, and implemented exact verification and the interactive laboratory. The human chose the research objectives, supervised the workflow, requested prior-work checks and formal verification, and reviewed the publication materials. No independent human mathematical verification is claimed.","verification":"lean-checked","verificationNote":"A clean Azure build passed all nine Lean modules using Lean 4.34.0 and a pinned mathlib revision. Nanoda independently checked 9,451 declarations in the dependency closure of 12 audited targets. The allowed axioms were only propext, Classical.choice and Quot.sound; the export contained no unsafe or partial declarations. Both Lean and Nanoda rejected deliberately invalid controls. The project uses no custom axioms, sorry/admit, native_decide, or modified kernel/checker rules. Separate Python and JavaScript implementations agreed on 20 fixed examples and 271 candidate scores, with an independent cycle-path LP oracle. The main universal guarantee and full parameterized sharpness family are formalized; the O(n) arithmetic-complexity claim is separate mathematical/code analysis. The formal statements and their correspondence to the original algorithm have not been independently audited by a human expert. No peer review is claimed; this is a repository announcement.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":6,"significanceNote":"A conjecture from a 2024 master's thesis, recorded as open by Nikoleit, Anand, Naredla and Roglin in January 2026, and the two-size case of Rajkovic's 2022 conjecture, which stays open. Narrow audience. Level with the Shallit-Shur-Zorcic and Joshi-Rust entries published the same day, below the Pinwheel Kernel Conjecture at 12.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/MathIsEvenEasier/gasoline-two-sizes","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[{"label":"Original conjecture: Lorieau (2024), Conjecture 3.1.1","url":"https://perso.limos.fr/~lulorieau/docs/thesis/Master_Thesis.pdf#page=27","kind":"problem-record"},{"label":"Lean proof endpoints and parameterized sharpness theorem","url":"https://github.com/MathIsEvenEasier/gasoline-two-sizes/blob/main/formal/Result.lean","kind":"lean-proof"},{"label":"Lean and Nanoda audits, exported proofs and controls","url":"https://github.com/MathIsEvenEasier/gasoline-two-sizes/tree/main/evidence","kind":"code"},{"label":"Interactive laboratory: decisions, capacity and tie rules","url":"https://mathiseveneasier.github.io/gasoline-two-sizes/site/","kind":"other"}],"relations":[],"submittedBy":"MathIsEvenEasier","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"polysable-representations-of-finite-quivers-over-non-archimedean-fields-admit-sp","name":"Harmonic norms on quiver representations without spherical completeness (Haiden-Katzarkov-Kontsevich-Pandit Question 6.24)","shortName":"HKKP Question 6.24","problemNumber":null,"field":"Representation theory; non-archimedean geometry","fieldGroup":"Algebra","statement":"Haiden, Katzarkov, Kontsevich and Pandit (Towards Categorical Kähler Geometry, arXiv:2609.00978, Theorem 6.23) prove that over a spherically complete non-archimedean field $K$, a polystable finite-dimensional representation of a finite quiver admits a harmonic norm, and a representation admitting one is semistable. Their Question 6.24 asks: can the spherical completeness assumption on $K$ be omitted in Theorem 6.23?","posedBy":"Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich and Pranav Pandit, Towards Categorical Kähler Geometry (arXiv:2609.00978, September 2026), Question 6.24","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"Yes, for finite quivers: over any complete non-archimedean field, every polystable finite-dimensional representation admits a split harmonic norm, and a representation admitting a harmonic norm is semistable, so both implications of Theorem 6.23 hold without spherical completeness. The norm produced is split (diagonalisable), which is the natural form here, since without spherical completeness norms need not diagonalise.","claimIssueNote":null,"solveDate":"2026-10-01","model":"ChatGPT-6.0 Sol","modelMaker":"OpenAI","humanCollaborators":["Oren Ben-Bassat"],"aiRole":"This article was written through extensive conversations with OpenAI’s\nChatGPT-6.0 Sol, beginning with the author’s original arguments, which contained substantial gaps. The ideas and development of the article represent roughly equal contributions of the author and the AI. OpenAI’s ChatGPT, Codex, and Prism were used to discuss and revise the arguments and to proofread and typeset the manuscript. The examples were chosen by the author.","verification":"unreviewed","verificationNote":"Read by this site on 6 October 2026: Question 6.24 was checked in the source of arXiv:2609.00978, where it follows Theorem 6.23 directly, and the paper's main theorem was compared with it and covers both implications for every complete non-archimedean field. The proof was not refereed here. Single-author preprint, posted 1 October 2026; no expert reaction or later discussion found.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A technical question from a September 2026 outline paper by four well-known authors, answered within a month: removing one hypothesis from one theorem. The askers are famous; the question is young and specialised. Below a typical Erdős problem at 10, level with the 8s from recently posed questions in print.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2610.01694","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"RapidPuffin862","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mean-and-variance-of-finite-2-adic-complexity-question-3-1","name":"Mean and variance of finite 2-adic complexity (Question 3.1)","shortName":"Mean and variance of 2-adic complexity","problemNumber":null,"field":"Finite 2-adic complexity, rational approximation and probability","fieldGroup":"Number theory","statement":"For a uniformly random binary string of length $N\\ge1$, identified with $r\\in\\{0,\\ldots,2^N-1\\}$, let $R_N(r)=\\min\\{\\max(|a|,b):a,b\\in\\mathbb Z,\\ a\\equiv rb\\pmod{2^N},\\ b>0\\text{ odd}\\}$ and let $L_N=\\log_2 R_N(r)$, without rounding. Prove $\\mathbb E[L_N]\\le N/2$ and compute $\\operatorname{Var}(L_N)$. This is Question 3.1 of Lin, Xiao and Chen (2026). The probability space includes all length-N binary strings, including zero.","posedBy":"Hezheng Lin, Lingmei Xiao and Zhixiong Chen, Question 3.1, AIMS Mathematics 11(7) (2026), p.20282, DOI 10.3934/math.2026823.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"The manuscript claims both requested results: the exact mean upper bound for every N>=1 and an explicit finite arithmetic-logarithmic variance formula obtained from the complete distribution. It also evaluates the limiting variance as 0.892302571204274625294755... . The all-length mean proof combines a written lattice argument with exact finite certificates. The variance and asymptotic arguments use finite counting, marked-lattice equidistribution and uniform integrability. The limiting geometric law is explicitly credited to Boca-Gologan. The finite answer is a sum, not a short elementary expression in N; no convergence rate or raw-height variance theorem is claimed.","claimIssueNote":null,"solveDate":"2026-10","model":"OpenAI GPT models; GPT-6.1 Sol","modelMaker":"OpenAI","humanCollaborators":["Oleksiy Babanskyy"],"aiRole":"OpenAI GPT models assisted mathematical exploration, proof criticism, source comparison, manuscript editing and reproducibility checks under the author's direction. The public manuscript does not identify historical model versions, so none are inferred here. GPT-6.1 Sol was used for the three current author-side reviews of the proof, computations and sources. These reviews shared exposure to the research material. The author retains responsibility for the arguments and attributions.","verification":"unreviewed","verificationNote":"Re-run by this site on 4 October 2026: the companion's mean-premise, distribution and constant replays all passed (PASS_FINITE_PREMISES, PASS_FINITE_DISTRIBUTION_AND_MOMENT_PREMISES, PASS_EXACT_RATIONAL_EVALUATION). A separate brute force written here for N = 1 to 14 agrees: the mean stays below N/2 at every N, the gap approaches about 0.414, and the variance rises from 0.869 at N = 8 to 0.8904 at N = 14, consistent with the claimed limit 0.8923. These are finite checks. The all-N mean bound and the limiting variance rest on the written lattice and equidistribution arguments, which were not checked here, and the author-side reviews are by AI models. No independent specialist review.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":4,"significanceNote":"A question asked in a July 2026 survey-style paper, months old when answered, with a narrow audience in sequence complexity. Prior work (Tian-Qi 2010, Chen-Winterhof 2025) bounded the same mean to within O(log N). Below the Joshi-Rust and Shallit-Shur-Zorcic conjectures at 6, which are older and asked more widely.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/FCIG-Exact-Mean-Bound-for-the-Rational-Complexity-of-Binary-Strings/blob/1c5d732518de8e8e564656e30a73dac24918b379/paper/Exact-Mean-Bound-for-the-Rational-Complexity-of-Binary-Strings.pdf","sourceName":"aiXiv 2610.00156 v2 (manuscript v0.2.0 on GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"Revised preprint on aiXiv","url":"https://aixiv.online/abs/2610.00156v2","kind":"paper"},{"label":"Original Question 3.1 (Lin-Xiao-Chen, 2026), p.20282","url":"https://www.aimspress.com/aimspress-data/math/2026/7/PDF/math-11-07-823.pdf#page=16","kind":"problem-record"},{"label":"Pinned source, exact companion and claim map","url":"https://github.com/aconsciousfractal/FCIG-Exact-Mean-Bound-for-the-Rational-Complexity-of-Binary-Strings/tree/1c5d732518de8e8e564656e30a73dac24918b379","kind":"code"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"expected-finite-k-adic-complexity-in-general-integer-bases","name":"Expected finite k-adic complexity in general integer bases","shortName":"Mean k-adic complexity","problemNumber":null,"field":"Rational reconstruction; k-adic complexity of finite strings","fieldGroup":"Number theory","statement":"For an integer $k\\ge3$, let $E_N^{k\\text{-adic}}$ be the expected $N$th $k$-adic complexity of a uniformly random $k$-ary sequence of length $N\\ge1$. Lin, Xiao and Chen derive a lower bound of the form $N/2-O(1)$ and ask, as Question 3.2: estimate an upper bound on $E_N^{k\\text{-adic}}$.\n\nThe $N$th $k$-adic complexity of a sequence is the least $\\log_k\\max(|f|,|q|)$ over reduced rationals $f/q$ whose $k$-adic expansion begins with it (unrounded; the convention fixed in Section 1 of the manuscript, following Goresky and Klapper).","posedBy":"Hezheng Lin, Lingmei Xiao and Zhixiong Chen, Question 3.2, AIMS Mathematics 11(7) (2026), p. 20282; DOI 10.3934/math.2026823.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"The manuscript claims $E_N^{k\\text{-adic}}3$, is it true that\n\n$$\n(-1)^{(p-1)/2}\\sum_{n=0}^{p-1}\\binom{2n}{n}\\frac{a_n}{54^n}\\equiv\n\\begin{cases}\n4x^2-2p,&p=x^2+3y^2\\equiv1\\pmod{3},\\\\\n0,&p\\equiv2\\pmod{3}\n\\end{cases}\n\\pmod{p^2}?\n$$\n\nThis is Conjecture 3.5 on p. 2742 of Zhi-Hong Sun's AIMS Mathematics paper, DOI 10.3934/math.2022153. It is distinct from Theorem 3.5 in the same article, which gives a congruence modulo $p$.","posedBy":"Zhi-Hong Sun, Conjecture 3.5, AIMS Mathematics 7(2) (2022), p. 2742; published online 18 November 2021. DOI: 10.3934/math.2022153.","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"Theorem 1.1 claims the complete stated conjecture modulo $p^2$ for every prime $p>3$, covering both residue classes modulo 3, with $p=5$ treated directly. For inert primes, a finite quadratic differential invariant lifts Sun's published modulo-$p$ vanishing. For split primes, certified modular periods, an integral Hecke polynomial with a degree bound and complex multiplication give the evaluation.\n\nThe Franel sum with denominator 50 is evaluated separately for $p>5$ and credited to Sun's different 2018 Conjecture 3.5. No modulo-$p^3$ result, resolution of adjacent conjectures, or first-proof claim is made. Candidate reflects pending authoritative review of a claimed complete proof of this specific question.","claimIssueNote":null,"solveDate":"2026-10","model":"OpenAI Codex; GPT-6.1 Sol","modelMaker":"OpenAI","humanCollaborators":["Oleksiy Babanskyy"],"aiRole":"Under the author's direction, AI systems contributed substantially to mathematical exploration, proof criticism, source comparison, manuscript drafting, exact computation and repository preparation. AI_USE.md names OpenAI Codex and GPT-6.1 Sol for consolidation and separate checks. Material supplied through an external agent was also examined; its complete model provenance is not asserted. Individual proof steps are not attributed to particular models. The author is responsible for the mathematical claims and publication.","verification":"unreviewed","verificationNote":"Checked by this site on 4 October 2026. The statement was compared with Conjecture 3.5 as printed on p. 2742 of Sun's AIMS Mathematics paper and matches it in full (every prime p > 3, both residue classes mod 3, modulus p squared). A separate script written here from the binomial definitions found no counterexample at any prime below 260. The author's companion recomputes both sides for the primes to 1999 and certifies the finite steps of the argument; these are finite checks, and the all-primes theorem rests on the written modular-forms proof, which was not refereed here. The author-side reviews are by AI models. Prior work: citing literature (OpenAlex, Semantic Scholar), Z.-H. Sun's arXiv papers and Sun-Ye's 2024 preprint were searched and no earlier proof was found; the paywalled final texts of Mao (2026), Sun-Ye (2025) and Sun (2026) were not read.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"One of many conjectures in a 2022 AIMS Mathematics paper that poses them in quantity, in the heavily worked area of Apery-like supercongruences. Open about five years. Level with its two siblings published the same day and with Sun's Conjecture 4.6(ii) on trigonometric permanents at 5; below Zhi-Wei Sun's Legendre-determinant conjecture at 10.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/Sun-3-5-Supercongruence/blob/2a9a35f748b0d2c625af7eb01283a8f619465d77/paper/Sun_3_5_Supercongruence.pdf","sourceName":"GitHub; also on aiXiv","collection":null,"collectionVersion":null,"links":[{"label":"Sun's original Conjecture 3.5, p. 2742 (PDF page 14)","url":"https://www.aimspress.com/aimspress-data/math/2022/2/PDF/math-07-02-153.pdf#page=14","kind":"problem-record"},{"label":"Source, exact certificates and reproduction instructions","url":"https://github.com/aconsciousfractal/Sun-3-5-Supercongruence/tree/2a9a35f748b0d2c625af7eb01283a8f619465d77","kind":"code"},{"label":"aiXiv manuscript and LaTeX source","url":"https://aixiv.online/abs/2610.00155","kind":"paper"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"zhi-hong-sun-s-conjecture-3-6-on-two-apery-like-binomial-sums","name":"Zhi-Hong Sun's Conjecture 3.6 on two Apéry-like binomial sums","shortName":"Sun's Conjecture 3.6","problemNumber":null,"field":"Supercongruences, Apéry-like sequences and modular forms","fieldGroup":"Number theory","statement":"Let $f_n=\\sum_{j=0}^n\\binom{n}{j}^3$ and $Q_n=\\sum_{k=0}^n\\binom{n}{k}(-8)^{n-k}f_k$. Is it true that, for every odd prime $p$,\n$$\\sum_{n=0}^{p-1}\\frac{\\binom{2n}{n}Q_n}{(-32)^n}\\equiv\\sum_{n=0}^{p-1}\\frac{\\binom{2n}{n}Q_n}{64^n}\\equiv\\begin{cases}4x^2-2p,&p=x^2+2y^2\\equiv1,3\\pmod8,\\\\0,&p\\equiv5,7\\pmod8\\end{cases}\\pmod{p^2}?$$\nHere $x,y$ are integers in the first case. This is Conjecture 3.6 of Zhi-Hong Sun, AIMS Mathematics 7(2) (2022), p. 2742, published online on 18 November 2021. The question includes both sums and the primes 3 and 5.","posedBy":"Zhi-Hong Sun, Conjecture 3.6, AIMS Mathematics 7(2) (2022), p. 2742; DOI 10.3934/math.2022153. Published online 18 November 2021.","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"Theorem 1.1 claims the complete stated conjecture: both sums, every odd prime and modulus p squared. For inert primes, a finite differential invariant lifts Sun's mod-p result, with an autonomous singular treatment at p=5. For split primes, certified eta identities, a generic integral Hecke polynomial, an explicit degree bound and three CM matrix cases yield the result. An independently sourced coefficient congruence removes the degree-p boundary; p=3 is checked directly. The paper credits Sun's mod-p input, the Straub-Gorodetsky endpoint and Beukers's modular framework. It does not claim a uniform mod-p-cubed extension: at p=7 the two sums are 294 and 49 modulo 343. The full conjecture is claimed, but authoritative mathematical review is pending.","claimIssueNote":null,"solveDate":"2026-10","model":"OpenAI Codex; GPT-6.1 Sol","modelMaker":"OpenAI","humanCollaborators":["Oleksiy Babanskyy"],"aiRole":"AI assisted mathematical exploration, proof criticism, source comparison, manuscript drafting, exact computation and repository preparation under the author's direction. OpenAI Codex and GPT-6.1 Sol assisted consolidation and separate checks of the proof and companion. External-agent material was also examined; its complete model provenance is not asserted. Individual proof steps are not attributed to particular systems. The systems had access to related arguments, so their checks are not independent human peer review or formal verification. This assistance is disclosed in the manuscript and AI_USE.md. The author retains responsibility for the claims and publication.","verification":"unreviewed","verificationNote":"Checked by this site on 4 October 2026. The statement was compared with Conjecture 3.6 as printed on p. 2742 of Sun's AIMS Mathematics paper and matches it in full (both sums, every odd prime, modulus p squared). A separate script written here from the binomial definitions found no counterexample at any prime below 260. The author's companion recomputes both sides for the primes to 1999 and certifies the finite steps of the argument; these are finite checks, and the all-primes theorem rests on the written modular-forms proof, which was not refereed here. The author-side reviews are by AI models. Prior work: citing literature (OpenAlex, Semantic Scholar), Z.-H. Sun's arXiv papers and Sun-Ye's 2024 preprint were searched and no earlier proof was found; the paywalled final texts of Mao (2026), Sun-Ye (2025) and Sun (2026) were not read.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"One of many conjectures in a 2022 AIMS Mathematics paper that poses them in quantity, in the heavily worked area of Apery-like supercongruences. Open about five years. Level with its two siblings published the same day and with Sun's Conjecture 4.6(ii) on trigonometric permanents at 5; below Zhi-Wei Sun's Legendre-determinant conjecture at 10.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/Sun-3-6-Supercongruence/blob/bdbea1441a5791058f4662b8959e24feea6f69b7/paper/Sun_3_6_Supercongruence.pdf","sourceName":"GitHub; also on aiXiv","collection":null,"collectionVersion":null,"links":[{"label":"Original Conjecture 3.6, p. 2742 (PDF page 14)","url":"https://www.aimspress.com/aimspress-data/math/2022/2/PDF/math-07-02-153.pdf#page=14","kind":"problem-record"},{"label":"Exact companion, certificates and reproduction instructions","url":"https://github.com/aconsciousfractal/Sun-3-6-Supercongruence/tree/bdbea1441a5791058f4662b8959e24feea6f69b7","kind":"code"},{"label":"aiXiv preprint 2610.00157","url":"https://aixiv.online/abs/2610.00157","kind":"paper"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"maximum-number-of-pairwise-equidistant-lines-in-mathbb-r-3","name":"Maximum number of pairwise equidistant lines in $\\mathbb{R}^3$","shortName":"Seven equidistant lines","problemNumber":null,"field":"Discrete and computational geometry","fieldGroup":"Geometry & topology","statement":"Is seven the maximum cardinality of a family of affine lines in $\\mathbb{R}^3$ with a common positive pairwise distance? Equivalently, can eight congruent infinite circular cylinders with pairwise disjoint interiors touch pairwise?","posedBy":"J. E. Littlewood, Some Problems in Real and Complex Analysis (1968), Problem 7, p. 20; the maximum was framed by Bezdek (2005)","yearPosed":1968,"ageNote":"Littlewood asked in 1968 whether seven unit cylinders can touch pairwise; Bozoki, Lee and Ronyai built seven in 2015. The maximum was open from at least Bezdek's 2005 upper bound of 24, later 18, 10 and 9 (Hofer, June 2026).","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"It is proved that no eight affine lines in $\\mathbb{R}^3$ have a common positive pairwise distance. Together with the known construction of seven such lines by Bozóki, Lee and Rónyai, this establishes the exact maximum of seven. The upper bound combines a five-line obstruction from hyperbola secants with exact verification over 135 uniform rank-three oriented-matroid types. It settles the full stated maximum-cardinality question; it does not classify all seven-line configurations or address cylinders of unequal radii.","claimIssueNote":null,"solveDate":"2026-09-30","model":"GPT-6 Astra Pro; Claude Fable 5.1","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Mingchang Liu"],"aiRole":"This work was developed through continuous collaboration between the author and AI models. OpenAI and Anthropic models contributed to proof development, exact computation, literature research, manuscript preparation and review.","verification":"site-confirmed","verificationNote":"Rebuilt by this site on 6 October 2026. The Lean development added on 4 October (commit 065e5e2; Lean 4.32.1 with mathlib, about 6,760 lines) was built from a clean clone with no warnings, passed a fresh leanchecker kernel replay, and #print axioms gives only propext, Classical.choice and Quot.sound for thm_1_1 and cor_1_2_upper. No sorry, axiom declarations, native_decide or implemented_by; finite steps use decide +kernel. The statement was read: no eight affine lines in R^3 have pairwise distance 1, with lines as affine subspaces of one-dimensional direction and distance the infimum of explicit Euclidean point distances; the corollary covers any n >= 8 and any d > 0. No general-position, non-parallel or catalogue hypothesis. The proof formalises the parallel and coplanar direction lemma, the four-line circuit condition and the five-line obstruction, and replaces Finschi's catalogue with a self-contained kernel-checked enumeration. Separately, on 4 October the Python verifier at 9c37327 was re-run here and passed (135 classes, 1,027 metric certificates). Not formalised: the seven-line construction of Bozoki, Lee and Ronyai, which is published. The statement audit is this site's alone.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Littlewood's 1968 problem, in Brass-Moser-Pach, with four papers racing the upper bound down from 24 to 9 in 2005-2026 before this closes it. Above Kuperberg's six-cylinder conjecture at 20, a younger and narrower cylinder question, and below conjectures famous across a whole community at about 30. Scored as the maximum question, not the 1968 existence question, which was settled in 2015.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://zenodo.org/records/23064311","sourceName":"Zenodo","collection":null,"collectionVersion":null,"links":[{"label":"Exact verification code and original certificates","url":"https://github.com/michaeliu4/equidistant-lines-verification/tree/9c37327ec78579a35c3848bccca736ebdfeeca58","kind":"code"},{"label":"Hofer (2026): the previous upper bound of nine","url":"https://arxiv.org/abs/2606.22605","kind":"paper"}],"relations":[],"submittedBy":"SilentIbis759","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"vinogradov-least-quadratic-nonresidue","name":"Vinogradov's least quadratic nonresidue conjecture","shortName":"Vinogradov least nonresidue conjecture","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"For an odd prime $p$, let $n(p)$ be the least positive integer that is a quadratic nonresidue modulo $p$. Unconditionally, the best bounds (Burgess with Vinogradov's trick) give $n(p)\\ll_\\varepsilon p^{1/(4\\sqrt e)+\\varepsilon}$, while the generalized Riemann hypothesis gives $n(p)\\ll(\\log p)^2$. Vinogradov conjectured that $n(p)\\ll_\\delta p^\\delta$ for every $\\delta>0$. Is it true that for every $\\delta>0$ the least quadratic nonresidue satisfies $n(p)\\ll_\\delta p^\\delta$?","posedBy":"I. M. Vinogradov, as stated in Tao, The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture (2015), Conjecture 1.1","yearPosed":null,"ageNote":"The manuscript cites the conjecture only through Tao (2015, Conjecture 1.1) and gives no date for Vinogradov's original formulation, which is usually placed around 1918-1919. yearPosed is left blank.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims absolute constants $A,C>0$ with $n(p)\\le C(\\log p)^A$ for every odd prime $p$, hence $n(p)\\ll_\\delta p^\\delta$ for every $\\delta>0$. This is a polylogarithmic bound, much stronger than the conjecture asks, though weaker than the $(\\log p)^2$ that GRH gives. The same corollary gives a deterministic polynomial-time algorithm for square roots modulo a prime. The proof is conditional on nothing beyond the family's own zero-free half-plane $\\Re s>7/8$ and the published theorem of Bhargava, Ivanyos, Mittal and Saxena; it is not an independent argument.","claimIssueNote":null,"solveDate":"2026-09-30","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every manuscript in the collection was produced by an unreleased internal OpenAI model, and that the vast majority were obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result. It names the zero-free region for the Riemann zeta function as an exception to that fixed procedure, without saying what was done instead, and says the write-up of the companion Re(s) > 11/12 proof was human edited for readability. The manuscripts are credited to OpenAI with no human author named. The nonresidue bound is a corollary in the 7/8 manuscript, derived from the zero-free half-plane through a published theorem of Bhargava, Ivanyos, Mittal and Saxena.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Corollary 1 of the quasi-Riemann manuscript was read against the posed conjecture: it claims $n(p)\\le C(\\log p)^A$ with absolute constants (for example $A=32$), which implies Vinogradov's conjecture. The deduction is a few lines: the 7/8 half-plane and the functional equation put all nontrivial zeros of primitive Dirichlet L-functions in a strip that supplies the weak-GRH hypothesis of Bhargava, Ivanyos, Mittal and Saxena (ISSAC 2017, Conjecture 6.3 and Theorem 6.7), whose conclusion gives the bound. The release's Lean main results for this manuscript state only the zero-free half-plane; neither the nonresidue bound nor the cited BIMS theorem is formalized, so this entry is not marked Lean-checked. It stands or falls with the quasi-Riemann claim.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A century-old conjecture at the heart of the theory of character sums, famous across analytic number theory but known to follow from the generalized Riemann hypothesis, so it was a test of unconditional methods rather than a goal in itself. Scored with Smale's Mean Value Conjecture and Sendov's Conjecture at 40 to 45, and below the Landau-Siegel zero problem in this family at 72.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf","sourceName":"OpenAI math release: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Uniform exclusion of Landau-Siegel zeros","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf","kind":"paper"},{"label":"Companion: alternate proof of the zero-free half-plane Re s > 11/12","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf","kind":"paper"},{"label":"Lean: release scope note (states the half-plane, not this corollary)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/003.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"quasi-riemann-hypothesis","name":"The quasi-Riemann hypothesis: a fixed zero-free half-plane for zeta and all Dirichlet L-functions","shortName":"Quasi-Riemann hypothesis","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"The quasi-Riemann hypothesis asks for one gap that works at every height: is there a fixed $\\sigma_0<1$ such that the Riemann zeta function has no zeros in the half-plane $\\Re s>\\sigma_0$? Since Hadamard and de la Vallee Poussin (1896) it has been known that $\\zeta(s)\\ne0$ on $\\Re s=1$, but every known zero-free region shrinks toward that line as the height grows. The uniform version over all Dirichlet characters asks for a single $\\sigma_0<1$, independent of the modulus $q$, the character $\\chi$ and the height, such that no $L(s,\\chi)$ vanishes in $\\Re s>\\sigma_0$ (the pole of the principal character at $s=1$ aside). Does such a $\\sigma_0$ exist?","posedBy":"Classical; the name is used by Billington, Cheng, Schettler and Suriajaya (2025), and the uniform Dirichlet form appears in Friedlander and Goldston (1997)","yearPosed":null,"ageNote":"The manuscript cites no original poser. The question is implicit since the 1896 proofs of the prime number theorem, which gave nonvanishing only on the line Re s = 1; it cites Billington et al. (2025) for the name and Friedlander and Goldston (1997, eq. 1.4) for the uniform Dirichlet formulation. yearPosed is left blank for that reason.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1 claims that every finite-order Hecke $L$-function over $\\mathbb Q(\\sqrt{-3})$ and every Dirichlet $L$-function, including $\\zeta(s)$, has no zero in $\\Re s>7/8$, the principal pole at $s=1$ excepted. So the nontrivial zeros of $\\zeta$ have real part at most $7/8$, and any exceptional real zero in $(7/8,1)$ is excluded. The proof runs through cubic theta series and sextic large sieves, first reaching $11/12$ and then refining to $7/8$; the Dirichlet case is transferred from the Hecke family. It does not prove the Riemann hypothesis or GRH, and says so; the boundary line $\\Re s=7/8$ is not covered. A separate manuscript gives a different proof of the weaker half-plane $\\Re s>11/12$.","claimIssueNote":null,"solveDate":"2026-09-30","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every manuscript in the collection was produced by an unreleased internal OpenAI model, and that the vast majority were obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result. It names the zero-free region for the Riemann zeta function as an exception to that fixed procedure, without saying what was done instead, and says the write-up of the companion Re(s) > 11/12 proof was human edited for readability. The manuscripts are credited to OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. The statement of Theorem 1 was read against the posed question: it asserts exactly a fixed half-plane Re s > 7/8, uniform over all Dirichlet characters and over finite-order Hecke characters of Q(sqrt(-3)). The release's formalization.yaml lists three main results for this manuscript: OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re, OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re and the Hecke analogue. The zeta and Dirichlet comparator statements are short and use Mathlib's own riemannZeta and DirichletCharacter.LFunction, so they state the headline claim with the principal pole excluded; the Hecke statement builds its own character and L-function definitions in a 245-line statement file, which were not audited here. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The README marks this result as an exception to the release's fixed procedure. The corollaries (least nonresidues, square roots mod p) are not in the Lean statements.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":85,"significanceNote":"A fixed zero-free half-plane is the best-known weakening of the Riemann hypothesis and was considered almost as far out of reach; it would be the largest advance on zeta zeros since 1896. Scored below the Riemann hypothesis at 100 because the line is seven eighths, not one half, above the Proportion of Zeta Zeros on the Critical Line at 68, and above Finite-time blowup for the 3D incompressible Euler equations from smooth data at 80.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf","sourceName":"OpenAI math release: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: alternate proof of the zero-free half-plane Re s > 11/12 (human-edited write-up)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf","kind":"paper"},{"label":"Companion: Uniform exclusion of Landau-Siegel zeros","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf","kind":"paper"},{"label":"Lean: Dirichlet and zeta 7/8 nonvanishing (main declarations)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/DirichletL/Nonvanishing.lean","kind":"lean-proof"},{"label":"Lean comparator statement: zeta 7/8 bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/QuasiRiemannHypothesis.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Dirichlet 7/8 bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DirichletSevenEighths.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Hecke 7/8 bound over Q(sqrt(-3))","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HeckeSevenEighths.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/003.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":5,"downvotes":1,"commentCount":4},{"slug":"the-pinwheel-kernel-conjecture-a-six-task-counterexample","name":"The Pinwheel Kernel Conjecture: a six-task counterexample","shortName":"Pinwheel Kernel Conjecture","problemNumber":null,"field":"Pinwheel scheduling; finite-state verification","fieldGroup":"Algorithms & optimization","statement":"A pinwheel instance is a sorted tuple of positive integers $A=(a_1,\\ldots,a_k)$. A feasible schedule performs one task per integer time slot, with task $i$ appearing in every block of $a_i$ consecutive slots. The Kernel Conjecture asks whether every feasible $A$ dominates a feasible sorted instance $B=(b_1,\\ldots,b_k)$ with $b_i\\le a_i$ for every $i$ and $b_k\\le 2^{k-1}$ (Conjecture 2.3 of Gąsieniec, Smith and Wild).","posedBy":"Leszek Gąsieniec, Benjamin Smith and Sebastian Wild, Towards the 5/6-Density Conjecture of Pinwheel Scheduling, Conjecture 2.3 (2021 preprint; ALENEX 2022).","yearPosed":2021,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The instance $(3,4,5,20,22,36)$ is feasible, but $(3,4,5,20,22,32)$ is not. More strongly, $(3,4,5,20,22,b)$ is feasible exactly when $b\\ge36$.\n\nA 36-slot periodic witness proves feasibility. A forward-closed certificate for period 35 has 56,284 states and 107,847 legal transitions, with a natural-number rank decreasing at every step. This excludes all infinite schedules, including nonperiodic ones. Monotonicity rules out every dominated six-task instance with all periods at most $2^{6-1}=32$, refuting the original Kernel Conjecture. A separate certificate checks period 32.\n\nThe result shows that any universal six-task cap must be at least 36, not that 36 suffices universally. In practice, truncating deadlines to 32 can turn a schedulable input into an infeasible one: this preprocessing rule is unsound. The example also supplies a certified regression test for scheduling solvers.","claimIssueNote":null,"solveDate":"2026-09-30","model":"OpenAI Codex (GPT-6 Astra)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Under human direction, Codex selected and investigated the conjecture, implemented finite-state searches, found the counterexample, sharpened the sixth-period threshold to 36, generated finite certificates, wrote a separate Python checker, and developed the Lean formalization from the infinite-schedule definition to the final theorem. The human set the research objective, supervised the workflow, requested prior-work searches and formal verification, and required review before publication. No independent human mathematical verification is claimed.","verification":"site-confirmed","verificationNote":"Re-checked by this site on 4 October 2026 with an independent checker written from scratch (reachable age-state search with dead-state pruning, not the repository's code): (3,4,5,20,22,b) is infeasible for b = 30 to 35 and feasible for b >= 36, with state counts equal to the repository's (50,881 at b = 32, 56,284 at b = 35). The repository's check_counterexample.py also passed. The Lean development (260 modules, core Lean only, Lean 4.34.1) was searched and has no sorry, admit, axiom declarations, native_decide, opaque or unsafe; its log records only propext and Quot.sound. The site did not rebuild it, and the repository has no CI; the Azure and Nanoda logs are the author's. Statement fidelity to Conjecture 2.3 was checked by hand: ScheduleOK quantifies over all infinite schedules, and the formal claim drops sortedness, which makes it slightly stronger.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A named conjecture from an ALENEX 2022 paper, carried as open into Smith's 2025 thesis, and by the paper's Proposition 2.4 equivalent to its 2k Conjecture, so that falls with it. Known within pinwheel scheduling, not beyond. Level with the Vietoris-powers question at 12; below the 5/6-density conjecture itself, which this does not touch.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/MathIsEvenEasier/pinwheel-kernel-counterexample","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[{"label":"Original conjecture — Gąsieniec, Smith and Wild (2021), Conjecture 2.3","url":"https://arxiv.org/abs/2111.01784","kind":"problem-record"},{"label":"Formal statement and Lean proof endpoints","url":"https://github.com/MathIsEvenEasier/pinwheel-kernel-counterexample/blob/main/lean/Result.lean","kind":"lean-proof"},{"label":"Nanoda replay: strict configuration, exported proofs, and rejection control","url":"https://github.com/MathIsEvenEasier/pinwheel-kernel-counterexample/tree/main/nanoda","kind":"code"},{"label":"Guided proof, Pinwheel Lab, and certificate checker","url":"https://mathiseveneasier.github.io/pinwheel-kernel-counterexample/","kind":"other"}],"relations":[],"submittedBy":"MathIsEvenEasier","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"landau-siegel-zeros","name":"The Landau-Siegel zero problem: a uniform logarithmic gap for real zeros of real Dirichlet L-functions","shortName":"Landau-Siegel zeros","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"For a primitive Dirichlet character $\\chi$ of conductor $q$, the classical zero-free region $\\Re s\\ge1-c_1/\\log(q(|\\Im s|+2))$ holds with one possible exception: a single simple real zero $\\beta$ attached to a real character, which could lie very close to $1$. Siegel's theorem (1935) gives $L(1,\\chi)\\gg_\\varepsilon q^{-\\varepsilon}$ only ineffectively, and Page's theorem limits such zeros to at most one per range of conductors, so a sequence of real zeros with $(1-\\beta)\\log q\\to0$ is not ruled out. In its logarithmic formulation the Landau-Siegel zero problem asks: is there an absolute constant $c>0$ such that every real zero $\\beta\\in(0,1)$ of every primitive nonprincipal real Dirichlet $L$-function of conductor $q\\ge3$ satisfies $(1-\\beta)\\log q\\ge c$?","posedBy":"Classical exceptional-zero problem going back to Landau, Page and Siegel; logarithmic zero-gap conjecture as discussed by Friedlander and Iwaniec (2018)","yearPosed":null,"ageNote":"The manuscript names no single poser. It cites Page (1935) and Siegel (1935) for the results that leave the exceptional zero open, Davenport's textbook for the classical region, and Friedlander and Iwaniec (2018) for the logarithmic zero-gap conjecture. The problem is at least as old as 1935; yearPosed is left blank.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Claims an absolute constant $c>0$ such that every real zero $\\beta\\in(0,1)$ of every primitive nonprincipal real Dirichlet $L$-function of conductor $q\\ge3$ satisfies $(1-\\beta)\\log q\\ge c$. The proof combines a prime-bias estimate forced by a zero close to $1$ with an interpolation determinant on a biquadratic field $\\mathbb Q(\\sqrt d,\\sqrt2)$. The constant is not made explicit, so the result does not by itself give effective class-number bounds with numbers attached. It excludes real zeros in $1-c/\\log q<\\beta<1$ only; zeros elsewhere in $(0,1)$ are not addressed by this manuscript (the family's 7/8 manuscript excludes those above $7/8$).","claimIssueNote":null,"solveDate":"2026-09-30","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every manuscript in the collection was produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result. It lists work on a zero-free region for the Riemann zeta function as an exception to that procedure; whether this companion on Landau-Siegel zeros falls under the exception is not stated. The manuscript is credited to OpenAI with no human author named. The same conclusion also follows from the family's 7/8 half-plane, which removes every real zero above 7/8.","verification":"expert-verified","verificationNote":"Expositions and simpler proofs published by William Banks and Bryce Kerr. Theorem 1 was read against the posed problem: it is the logarithmic formulation exactly, with an absolute but inexplicit constant. formalization.yaml lists OAI.SiegelZeros.WeightedTorusJets.exists_absolute_real_zero_gap as a main result for this manuscript. Its comparator statement uses Mathlib's DirichletCharacter.LFunction and asserts a $c > 0$ such that for every $q\\ge3$ and every primitive nonprincipal character with real values and every real zero $0 < \\beta < 1$, $c \\le (1 - \\beta) \\log q$. That is the headline claim; both parities are included and no value of c is given. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The proof is short (an interpolation-determinant argument in a biquadratic field) and independent of the quasi-Riemann manuscript, though that manuscript implies the same statement.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":72,"significanceNote":"The possible Siegel zero is the central obstruction to effective results on primes in progressions and class numbers, open since the 1930s despite many attempts. Scored below the quasi-Riemann hypothesis in this family, which implies it, above the Proportion of Zeta Zeros on the Critical Line at 68, and above Prime Gaps at Most 186 at 62.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf","sourceName":"OpenAI math release: Uniform exclusion of Landau-Siegel zeros","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The Quasi-Riemann Hypothesis, zero-free half-plane Re s > 7/8 (implies this result)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf","kind":"paper"},{"label":"Companion: alternate proof of the zero-free half-plane Re s > 11/12","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf","kind":"paper"},{"label":"Lean: main declaration exists_absolute_real_zero_gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/SiegelZeros/Conclusions/Theorem.lean","kind":"lean-proof"},{"label":"Lean comparator statement: uniform real-zero gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SiegelZeros.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/003.md","kind":"lean-proof"},{"label":"William Banks' exposition and simpler proof","url":"https://arxiv.org/abs/2610.15483","kind":"independent"},{"label":"Bryce Kerr's simpler proof of the result","url":"https://arxiv.org/html/2610.21099v1","kind":"independent"},{"label":"Wikipedia page about Siegel zeroes","url":"https://en.wikipedia.org/wiki/Siegel_zero","kind":"wikipedia"},{"label":"Terence Tao's exposition of the proof","url":"https://terrytao.wordpress.com/2026/10/19/a-sketch-of-the-proof-of-non-existence-of-siegel-zeroes/","kind":"discussion"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bras-amoros-genus-monotonicity-conjecture","name":"Bras-Amorós genus-monotonicity conjecture","shortName":"Numerical semigroup monotonicity","problemNumber":null,"field":"Numerical semigroups; enumerative combinatorics","fieldGroup":"Algebra","statement":"Let $n_g$ be the number of numerical semigroups of genus $g$, where a numerical semigroup is an additive submonoid of $\\mathbb{N}_0$ with finite complement and its genus is the size of that complement. Is $n_{g+1}\\ge n_g$ for every integer $g\\ge0$?","posedBy":"Maria Bras-Amoros (Semigroup Forum 76, 2008) as a weak form of her Fibonacci-like conjecture; stated as Conjecture 2 in Nathan Kaplan, Counting numerical semigroups, Amer. Math. Monthly 124 (2017)","yearPosed":2008,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"It is proved that $n_{g+1}\\ge n_g$ for every $g\\ge0$, without restrictions on multiplicity or conductor. A partial injection leaves a specified family of source semigroups; unused targets are then shown to be numerous enough to cover it. Exact transfer computations bound relaxed source counts for $1\\le g\\le836$, and generating-function estimates handle $g\\ge837$. This is not an enumeration of all numerical semigroups through genus 836. The stronger Fibonacci inequality $n_{g+1}\\ge n_g+n_{g-1}$ and fixed-multiplicity monotonicity are not proved.","claimIssueNote":null,"solveDate":"2026-09-30","model":"GPT-6 Astra; Fable 5.1","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Mingchang Liu"],"aiRole":"This work was developed through continuous collaboration between the author and AI models. GPT-6 Astra (OpenAI) and Fable 5.1 (Anthropic) contributed to proof development and checking, exact computation, literature research, and preparation and review of the manuscript. The author takes full responsibility for all mathematical results and the contents of this manuscript.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked by this site on 7 October 2026: the statement matches Conjecture 2 of Kaplan's survey (arXiv 1707.02551), which also says the weak form was open and that Zhai's asymptotics leave only finitely many genera to settle. From a fresh clone at commit b1b6fb8, all five shipped checkers passed: capacity recurrences through degree 1500, the omitted-pivot multiplier below 0.221715, the guarded mass assembly, the analytic join (comparison fails at genus 836, holds from 837), and all 836 finite rows with nonnegative slack (smallest 1, at genera 1 and 2). A separate enumeration written here of all numerical semigroups through genus 23 reproduced the known counts and confirmed that each certified slack is at most the true difference $n_{g+1}-n_g$ for $g\\le 22$. These are arithmetic checks only. The coefficient arrays were not regenerated from the C++ sources, and the combinatorial and analytic arguments that connect them to the theorem (about 2,000 lines of the manuscript) were not read. No Lean formalisation.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"The central open problem in counting numerical semigroups since Bras-Amoros's 2008 conjectures, singled out in Kaplan's 2017 Monthly survey; known for large genus by Zhai, so the remaining question was effective. Famous within one community. Above Daykin-Frankl and the equidistant-lines entry at 25 and well above Fel's syzygy conjecture at 10; below the four-uniform Erdos matching conjecture at 40.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/michaeliu4/bras-amoros-monotonicity/blob/main/manuscript/genus-monotonicity_V1.pdf","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[{"label":"Exact verification code and data (v1)","url":"https://github.com/michaeliu4/bras-amoros-monotonicity/releases/tag/v1","kind":"code"},{"label":"Kaplan, Counting numerical semigroups (survey, Conjecture 2)","url":"https://arxiv.org/abs/1707.02551","kind":"paper"}],"relations":[],"submittedBy":"SilentIbis759","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"codimension-one-holomorphic-anosov-diffeomorphisms","name":"Codimension-one holomorphic Anosov diffeomorphisms","shortName":"Ghys conjecture (codim. 1)","problemNumber":null,"field":"Holomorphic dynamical systems","fieldGroup":"Geometry & topology","statement":"Ghys conjectured in 1995 that every holomorphic Anosov diffeomorphism of a compact complex manifold is holomorphically conjugate to an automorphism of a complex infra-nilmanifold, and proved it for complex surfaces. Does the conjecture hold in complex codimension one, that is when $\\dim_{\\mathbb C} E^s = 1$ or $\\dim_{\\mathbb C} E^u = 1$? This case contains every compact complex threefold.","posedBy":"Étienne Ghys, Holomorphic Anosov systems, Invent. Math. 119 (1995), 585-614 (conjecture on p. 586)","yearPosed":1995,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Jiesong Zhang proves that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This settles Ghys's conjecture in that case, and so for all compact complex threefolds, where Xu and Zhang (2024) recorded it as open.\n\nThe proof removes the topological transitivity assumption from Ghys's Theorem B to get a topological conjugacy to a hyperbolic toral automorphism, proves that both invariant distributions are holomorphic (the stable one by an $L^2$ estimate), and shows a torus with holomorphic invariant distributions carries the standard complex structure. Cantat (2004) had the codimension-one case for projective manifolds.\n\nThe paper also gives a nilmanifold example whose invariant distributions are smooth but not holomorphic, so the method stops at codimension one. The general conjecture remains open.","claimIssueNote":null,"solveDate":"2026-09-29","model":"Intern-S2 (internal version)","modelMaker":"Shanghai AI Lab","humanCollaborators":["Jiesong Zhang"],"aiRole":"The paper's Declaration of AI Use says the author used an internal version of Intern-S2 (Shanghai AI Lab) through multiple rounds of dialogue to assist in developing the mathematical ideas and arguments. It does not say which lemmas or constructions came from the model. The author states that he independently verified all arguments and takes full responsibility.\n\nSeparately, Theorem A and Theorems 1.1 to 1.3, with the parts of Ghys's 1995 paper they rely on, were formalised in Lean by the Bin Dong and Ruochuan Liu team at Peking University with their formalisation agent SHEAF; Jiedong Jiang and Bin Wu did the formalisation and the human check of the final statements (paper v3, 13 October 2026).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Read by this site on 15 October 2026, not rebuilt. Repository frenzymath/HolomorphicAnosov, commit 65f2b33 (Lean 4.35.0-rc1, Mathlib, about 4,230 files and 528,000 lines). With comments stripped, the library and Comparator/Solution.lean contain no sorry, admit, axiom declarations, opaque, unsafe, implemented_by, extern or native_decide; several comments still say 'Stage 2 sorry' and are stale. The four headline statements in Comparator/Challenge.lean were read: compact connected complex manifold, biholomorphic f, a smooth metric with an invariant continuous splitting and an iterate contracting each bundle, one bundle of complex rank one; conclusion a biholomorphism to $\\mathbb C^d/\\Lambda$ conjugating f to a hyperbolic complex-linear lattice automorphism. Theorems A, 1.1 and 1.3 take as a hypothesis HomotopyTorusRigidity (closed smooth homotopy n-tori, n at least 5, are homeomorphic to the torus), a 1969 theorem of Hsiang and Wall that is not in Mathlib; the definition read here is a true statement, so the hypothesis is not vacuous. The Comparator acceptance and the axiom list are the authors' own; there is no CI and nothing was run here. No independent formalisation anchors the statement (none found in Formal Conjectures), so Lean-checked, not Lean-verified.","publication":"preprint","resolutionMethod":null,"citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"Ghys's 1995 conjecture is the central rigidity question for holomorphic Anosov systems and is known across smooth dynamics; this entry is its codimension-one case, recorded as open even in dimension three in 2024. Level with the Hajlasz-Onninen maximal-function question at 24, above Szabo's equivariant Jiang-Su conjecture at 22, and well below the Anosov-Katok smooth realization entry at 42, a broader and older problem.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.37695","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation (frenzymath/HolomorphicAnosov)","url":"https://github.com/frenzymath/HolomorphicAnosov","kind":"lean-proof"},{"label":"Xu and Zhang, On holomorphic partially hyperbolic systems (records the conjecture as open in dimension 3)","url":"https://arxiv.org/abs/2401.04310","kind":"paper"}],"relations":[],"submittedBy":"FrostyBadger576","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"all-level-nanuq-circularity-and-exact-displayed-split-support","name":"All-level NANUQ circularity and exact displayed-split support","shortName":"All-level NANUQ split recovery","problemNumber":null,"field":"Mathematical phylogenetics; circular split systems","fieldGroup":"Combinatorics","statement":"Holtgrefe et al. proved the original NANUQ distance of a binary, semi-directed LSA, outer-labeled planar, galled level-two bloblet is circular decomposable, with support exactly the splits of its displayed trees. Their Section 6 asks whether this extends to multiple blobs at level two, conjectures circularity for level-three bloblets, and asks about a parametric distance-family extension. Does the circularity and exact-support statement persist at every finite level in the same structural class?","posedBy":"Holtgrefe et al., Distinguishing Phylogenetic Level-2 Networks with Quartets and Inter-Taxon Quartet Distances (2025), Theorem 4.7 and Section 6; DOI 10.1007/s11538-025-01549-4.","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Original NANUQ is circular decomposable at every finite level and with any number of blobs in the binary semi-directed LSA, galled, outer-labeled planar class (at least four taxa). Its positive split support equals the union of displayed-tree splits. This answers the source's level-two multi-blob and level-three circularity questions. A six-label reduction, exact finite certificate and composition identity remove the level bound. The local four-score family has exact universal anchor-positivity domain $s=o=1$, $1/2<=a<=1$, $0<=c<=a$; five labels are the sharp testing cutoff. Contiguous multicopy trees admit an extreme-copy reduction. Raw-matrix error $<1/2$ is optimally correctable by rounding; exact support uses a supplied correct order. Planarity cannot be dropped for exact support. Full all-level Lean verification, network uniqueness, statistical inference and unrestricted ARGs are not claimed.","claimIssueNote":null,"solveDate":"2026-09-29","model":"OpenAI Codex (GPT-6 family)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The human collaborator selected the biological open-problem direction, pressed for an all-level theorem and stronger generalizations, and requested honest source correspondence and publication. Parallel Codex research chats developed the proofs, finite certificates, Lean components, counterexamples and exposition. Separate agents audited the structural reduction, composition, exact arithmetic, support transfer and parameter domain. These are internal project reviews, not independent human expert verification. Supporting software was executed, with a second implementation and a separate actual-tree-selection support checker.","verification":"unreviewed","verificationNote":"Checked here on 30 September 2026 as far as the artefacts allow. The complete all-level theorem has a written computer-assisted proof with separate structural, finite, composition and support audits; two implementations cover 84,076 plane-tree and duplication instances, 122 distinct quartet systems and 24,667 anchor coefficients; the distance-based verifier checks 11,848,859 physical-copy quartets and a second verifier 2,525,210 globally consistent occurrence selections. The submitter states plainly that the complete theorem is computer-assisted and that the Lean endpoints are narrower than it, which is why this is Unreviewed rather than Lean-checked: the headline result is not the formalised part. The site did not rerun the verifiers. The repository's nanuq-finite workflow and its Verify workflow are green on the correct repository. The source URL was corrected: the submitted address omitted the repository's trailing dash and returned 404.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A question asked in Section 6 of a 2025 Bulletin of Mathematical Biology paper by the authors who proved the level-two case, with a conjecture attached for level three; removing the level bound entirely is the natural next result and the phylogenetics community following NANUQ distances will read it. Level with the Vietoris entry at 12, another Section-6 question from a recent paper, and below the magnitude-continuity entry at 16.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-","sourceName":"Pinned public computer-assisted proof, executable certificates and internal audits","collection":null,"collectionVersion":null,"links":[{"label":"Original open questions: Theorem 4.7 and Section 6","url":"https://doi.org/10.1007/s11538-025-01549-4","kind":"problem-record"},{"label":"Exact source-to-result ledger","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/cba23504ec7c63936ab60e5c3beb65d296c2e9c5/research/nanuq-all-level-2026-09-29/SOURCE-RESULT-LEDGER.md","kind":"transcript"},{"label":"Independent finite verifier and evidence boundary","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/cba23504ec7c63936ab60e5c3beb65d296c2e9c5/research/nanuq-all-level-2026-09-29/ALL-LEVEL-FINITE-AUDIT.md","kind":"transcript"},{"label":"All-level exact support transfer","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/cba23504ec7c63936ab60e5c3beb65d296c2e9c5/research/nanuq-all-level-2026-09-29/ALL-LEVEL-SUPPORT-STRUCTURAL-AUDIT.md","kind":"transcript"},{"label":"Parameter-domain theorem and independent symbolic audit","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/cba23504ec7c63936ab60e5c3beb65d296c2e9c5/research/nanuq-all-level-2026-09-29/PARAMETER-DOMAIN-AUDIT.md","kind":"lean-proof"},{"label":"Preserved Lean handoff: precise endpoints and unfinished full theorem","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/cba23504ec7c63936ab60e5c3beb65d296c2e9c5/research/nanuq-all-level-2026-09-29/source-development/formal-full/PUBLICATION-HANDOFF.md","kind":"lean-proof"},{"label":"Contiguous multicopy extension and sharp quantitative refinements","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/cba23504ec7c63936ab60e5c3beb65d296c2e9c5/research/nanuq-all-level-2026-09-29/README.md","kind":"other"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"no-countable-universal-family-of-avoiding-populations-even-with-exact-parent-cou","name":"No countable universal family of avoiding populations, even with exact parent counts","shortName":"Nonuniversality with exact parent counts","problemNumber":null,"field":"Infinite labelled graphs; universality","fieldGroup":"Combinatorics","statement":"Alexander (2013), Section 6, asks to what extent populations avoiding a sequence can be universal among all such populations, by analogy with universal-graph theory. Interpret embedding as an injective adjacency-preserving map. In the source population class (infinite, finitely many roots, finite children, real chronological birthdates with finite earlier sublevels, and an incoming parent of each label at every nonroot), can one host, or a countable host family, contain every population avoiding a fixed avoidable word? The result also asks whether the obstruction survives exactly one incoming parent of each label. The source does not prescribe a unique embedding category; this is the explicit ordinary-embedding interpretation.","posedBy":"Samuel A. Alexander, Biologically Unavoidable Sequences (2013), Section 6, published p.12; ordinary injective-embedding interpretation.","yearPosed":2013,"ageNote":null,"solveType":"disproved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"No. From any population avoiding s and any countable family of source-population hosts, construct an avoider with the same roots and exactly one incoming parent per label, admitting no injective map to any host with any finite global edge-stretch bound. Thus even adjacency embeddings fail. Parent selection preserves avoidance; terminal cloning preserves the selected infinite-word language exactly. The proof makes fibre sizes exceed enumerated finite host balls while preserving the actual population axioms. This strengthens the earlier complete-fibre obstruction by retaining exact parent counts. Separately, complete blow-ups preserve every corresponding least matching height, linking the obstruction to the rank question. Classical locally finite graph diagonalization is credited. Terminal vertices are permitted; fixed child caps, noninjective maps and arbitrarily unbounded stretch are outside the theorem. It answers the stated embedding interpretation, not every possible category.","claimIssueNote":null,"solveDate":"2026-09-29","model":"OpenAI Codex (GPT-6 family)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"A human collaborator selected the research directions and requested source correspondence, explanations, proof checking and submission. Codex agents developed the concrete mathematical construction or deduction, Lean proofs and exposition. Other agents in the project reviewed the arguments and source correspondence. These are internal project checks, not independent human expert review. Model-led discovery of the submitted scoped result is claimed; independent human verification and worldwide priority are not claimed.","verification":"lean-checked","verificationNote":"Checked here on 30 September 2026 to the extent the artefacts allow. Lean 4.33.1 with Mathlib pinned at 0df444a3. The submitter's manifest records four new modules with 42 selected endpoints and three reused modules with 21 earlier endpoints, permitting only propext, Classical.choice and Quot.sound. The repository's Verify workflow is green on the correct repository. The site did not rebuild the development and no third party has audited the informal-to-formal correspondence, so this is Lean-checked rather than Lean-verified. The source URL was corrected: the submitted address omitted the repository's trailing dash and returned 404. Partial, as submitted: the answer is for ordinary injective embeddings, which is the interpretation the submitter states, and Alexander's Section 6 does not fix an embedding category.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"One of the three questions Alexander left in Section 6 of the 2013 paper that introduced biologically unavoidable sequences, answered thirteen years later. The paper is the one this catalog's classification entry at 14 already turns on, and this is a smaller question from the same page, in a one-author corner of infinite graph theory. Above the Thue-Morse height entry at 5 for being a question Alexander asked rather than a quantitative follow-up, and below the classification itself.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-","sourceName":"Pinned public research note, Lean source and verification receipts","collection":null,"collectionVersion":null,"links":[{"label":"Original universal-avoider question, section 6","url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v20i1p31/pdf/","kind":"problem-record"},{"label":"Exact-parent construction and obstruction","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/embedding/TerminalCloneAvoiders.lean","kind":"lean-proof"},{"label":"Earlier generic complete-fibre obstruction","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/embedding/GenericUniversalAvoiders.lean","kind":"lean-proof"},{"label":"Exact matching-height transport","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/ordinal-audit/BlowUpRankInvariance.lean","kind":"lean-proof"},{"label":"Frozen owner proof manifest","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/verification-manifest.json","kind":"other"},{"label":"Primary-source and prior-work map","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/literature/evidence-map.md","kind":"other"},{"label":"Full result-family publication ledger","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/publication-2026-09-29/COVERAGE.md","kind":"other"},{"label":"Successful integrated hosted proof audit","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/actions/runs/36631876391","kind":"other"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ordinal-certificates-and-exact-pruning-characterize-sequence-realization","name":"Ordinal certificates and exact pruning characterize sequence realization","shortName":"Ordinal certificates and realizing cores","problemNumber":null,"field":"Infinite labelled graphs; ordinal ranks","fieldGroup":"Combinatorics","statement":"Alexander (2013), Section 6, asks whether the infinite gendered populations realizing a possibly avoidable sequence can be characterized, particularly using ordinal numbers. For a target s, use reachable vertex/phase states (v,k), witnessed by a path spelling the first k letters, with successors consuming s(k). Can avoidance be characterized by a decreasing ordinal certificate, and can finite-branching realization be characterized exactly by iterative removal of states without continuing children? This addresses ordinal membership and exact matching continuation, not a requirement that different avoiders receive different transfinite ranks.","posedBy":"Samuel A. Alexander, Biologically Unavoidable Sequences (2013), Section 6, published p.12; exact ordinal-membership and reachable-state interpretation.","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Avoidance is equivalent to a strictly decreasing ordinal certificate on the reachable states; arbitrary vertex/label types are allowed. Under finite per-label branching, natural certificates suffice. It also proves that states surviving every finite pruning round form the greatest successor fixed point and are exactly those with infinite continuations. A start realizes the full target iff its phase-zero state survives. Every other state has an attained finite maximum h, is removed at round h+1, and has a pointwise least natural certificate, even in populations realizing the target elsewhere. Complete finite-fibre blow-ups preserve corresponding least heights. Reachability prevents confusing an unrelated suffix with the original word. These are classical well-founded and finite-branching methods made exact in the source setting. This is not a deciding algorithm for arbitrary infinite inputs or a rank hierarchy separating avoiders; the artificial-root history tree is a different object.","claimIssueNote":null,"solveDate":"2026-09-29","model":"OpenAI Codex (GPT-6 family)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"A human collaborator selected the research directions and requested source correspondence, explanations, proof checking and submission. Codex agents developed the concrete mathematical construction or deduction, Lean proofs and exposition. Other agents in the project reviewed the arguments and source correspondence. These are internal project checks, not independent human expert review. Model-led discovery of the submitted scoped result is claimed; independent human verification and worldwide priority are not claimed.","verification":"lean-checked","verificationNote":"Checked here on 30 September 2026 to the extent the artefacts allow. Lean 4.33.1 with Mathlib pinned at 0df444a3; the manifest records four new modules with 42 selected endpoints and three reused with 21, permitting only propext, Classical.choice and Quot.sound, and the Verify workflow is green on the correct repository. Not rebuilt here and no independent audit of the informal-to-formal correspondence, so Lean-checked. The source URL was corrected for the repository's trailing dash. Partial, as submitted: the ordinal-certificate and pruning characterisations answer a precise reading of Alexander's question and the submitter is explicit that they do not deliver every richer classification the section may have intended.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"The other question Alexander left in Section 6 of the 2013 paper: can the populations realising a possibly avoidable sequence be characterised, particularly using ordinal numbers. Answered for a stated interpretation thirteen years later. Level with the universality entry at 8, its companion from the same page, and below the classification entry at 14.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-","sourceName":"Pinned public research note, Lean source and verification receipts","collection":null,"collectionVersion":null,"links":[{"label":"Original ordinal-characterization question, section 6","url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v20i1p31/pdf/","kind":"problem-record"},{"label":"Generic ordinal-certificate equivalence","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/ordinal/GenericOrdinalCertificate.lean","kind":"lean-proof"},{"label":"Exact surviving core and actual realization","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/formal/GenericPruning.lean","kind":"lean-proof"},{"label":"Least attained finite ranks in the complement","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/formal/GenericMortalRank.lean","kind":"lean-proof"},{"label":"New statement/definition scope review","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/literature/new-proof-scope-review.md","kind":"other"},{"label":"Frozen proof and log manifest","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/verification-manifest.json","kind":"other"},{"label":"Exact height transport through complete blow-up","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/open-questions/continuation-2026-09-29/ordinal-audit/BlowUpRankInvariance.lean","kind":"lean-proof"},{"label":"Successful integrated hosted proof audit","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/actions/runs/36631876391","kind":"other"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":1},{"slug":"zhu-fang-shi-invariant-projection-question","name":"Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations","shortName":"Zhu-Fang-Shi invariant projection question","problemNumber":null,"field":"Operator algebras, II1 factors","fieldGroup":"Analysis","statement":"For irrational $\\theta$, let $R_\\theta$ be the hyperfinite II$_1$ factor generated by unitaries $U,V$ with $VU=e^{2\\pi i\\theta}UV$, and for $f$ on the circle let $T_f=Uf(V)$. Haagerup-Schultz give nontrivial invariant projections in the factor whenever the Brown measure is not a point mass; for these operators that happens exactly when the Fuglede-Kadison determinant $\\exp\\int\\log|f|\\,dm$ is positive. Zhu, Fang and Shi (2017, p. 264) asked: if $f$ is nonzero almost everywhere, $|f|$ is nonconstant and the determinant is zero, must $T_f$ have a nontrivial invariant projection $p\\in R_\\theta$, that is, $0<\\tau(p)<1$ and $(1-p)T_fp=0$?","posedBy":"Z. Zhu, J. Fang and R. Shi, On a class of operators in the hyperfinite II1 factor, Math. Scand. (2017), p. 264, question after Proposition 5.7","yearPosed":2017,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every irrational $\\theta\\in(0,1)$ there is a continuous $f:\\mathbb T\\to[0,1]$ vanishing only at $1$, with $\\int\\log f\\,dm=-\\infty$, such that every projection $p\\in R_\\theta$ with $(1-p)Uf(V)p=0$ is $0$ or $1$; the operator is nonzero and norm-quasinilpotent. The angle is prescribed in advance with no Diophantine condition. Invariant subspaces of the Hilbert-space representation whose projections lie outside the factor are not excluded. An appendix gives a second realization on an infinite product of circles.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the question as quoted from Zhu-Fang-Shi. The obstruction and transfer arguments were not refereed. Lean-checked on the release's Comparator challenge IrrationalRotation, found through lean/docs/293.md; this challenge is not in the release's formalization catalogue, but its solution module OAI.Analysis.IrrationalRotation.Main exists at the pinned commit. Its statement was read here: prescribed_irrational_counterexample gives, for every irrational theta in (0,1), a continuous f with values in [0,1], zero exactly at one point and log integral minus infinity, whose weighted rotation lies in the generated von Neumann algebra, has only trivial invariant projections there, is nonzero, norm-quasinilpotent and has vector Fuglede-Kadison determinant zero. That covers the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A specific question from a 2017 paper on weighted rotations, known to specialists in II1-factor operator theory; its interest comes from its place in the invariant subspace problem relative to II1 factors. Scored as the question asked. Tied with Mauri and Moraga's log Calabi-Yau question (15), above the Ji-Zhang quasinilpotent power-set question (9), below the Banks-Martin conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026/paper.pdf","sourceName":"OpenAI math release: Invariant-projection counterexamples for every irrational rotation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge IrrationalRotation (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/IrrationalRotation.lean","kind":"lean-proof"},{"label":"Zhu, Fang, Shi 2017, On a class of operators in the hyperfinite II1 factor","url":"https://journals.msp.org/mscand/article/view/322","kind":"problem-record"},{"label":"Backward intertwiners and a transitive commutant (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"transitive-algebra-problem","name":"The transitive algebra problem","shortName":"Transitive algebra problem","problemNumber":null,"field":"Operator theory, operator algebras","fieldGroup":"Analysis","statement":"An algebra of bounded operators on a Hilbert space $H$ is transitive if its only common closed invariant subspaces are $\\{0\\}$ and $H$. Arveson (1967) proved that a transitive algebra containing a maximal abelian self-adjoint algebra is strongly dense in $B(H)$, and Douglas and Pearcy (1972) related the general question to hyperinvariant subspaces. Is every unital transitive operator algebra on an infinite-dimensional separable complex Hilbert space strongly (equivalently weakly) dense in $B(H)$?","posedBy":"Question in the setting of W. B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967); the manuscript states it without attribution","yearPosed":null,"ageNote":"The manuscript cites Arveson (1967) for the main positive result and Douglas-Pearcy (1972) for the link to hyperinvariant subspaces, but does not say who posed the question or when; the conventional attribution to Arveson 1967 is not made in the source.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 1.2 of the principal manuscript: the commutant of the constructed nonzero quasinilpotent operator $S$ is a proper, strongly closed, unital, transitive complex algebra, so it is not strongly dense in $B(H)$. This answers the question negatively on every infinite-dimensional separable Hilbert space. It is an immediate consequence of the hyperinvariant counterexample, since the commutant of an operator without hyperinvariant subspaces is transitive. The algebra is a commutant, so nothing is said about singly generated transitive algebras, which is the invariant subspace problem.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction's remark after Corollary 1.2 and the corollary itself, read against the question as stated there. Lean-checked on the release's Comparator challenge BackwardIntertwiners (declaration OAI.BackwardIntertwiners.direct_algebra_corollary, listed in lean/formalization.yaml). Its statement, read here, asserts that the commutant is not the whole operator algebra, is SOT-closed and has no nontrivial closed common invariant subspace, which is the negative answer; strong and weak closures agree for subalgebras, so the weak form follows. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":46,"significanceNote":"The transitive algebra problem is a classic of operator theory and stood open since Arveson's density theorem; it is a little less prominent than the hyperinvariant subspace problem whose counterexample settles it here. Below the hyperinvariant subspace entry (52) and the Kadison-Kaplansky conjecture (50), just above Vinogradov's least quadratic nonresidue conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf","sourceName":"OpenAI math release: Backward intertwiners and a transitive commutant","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge BackwardIntertwiners (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BackwardIntertwiners.lean","kind":"lean-proof"},{"label":"Arveson 1967, A density theorem for operator algebras","url":"https://doi.org/10.1215/S0012-7094-67-03467-9","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hyperinvariant-subspace-problem-hilbert-space","name":"The hyperinvariant subspace problem on Hilbert space","shortName":"Hyperinvariant subspace problem","problemNumber":null,"field":"Operator theory","fieldGroup":"Analysis","statement":"A closed subspace is hyperinvariant for a bounded operator $T$ on a Hilbert space $H$ if it is invariant under every operator commuting with $T$; equivalently, $T$ has no nontrivial hyperinvariant subspace exactly when its commutant $\\{T\\}'$ is a transitive algebra. Douglas and Pearcy (1972) studied the problem and its link to transitive algebras; it sits beside the invariant subspace problem and is known to be distinct from it (a claimed equivalence by Kerchy-Pearcy was withdrawn). Does every bounded nonscalar operator on an infinite-dimensional separable complex Hilbert space have a nonzero proper closed hyperinvariant subspace?","posedBy":"Classical operator-theory problem; the manuscript cites R. G. Douglas and C. Pearcy, Hyperinvariant subspaces and transitive algebras, Michigan Math. J. 19 (1972), and names no original poser","yearPosed":null,"ageNote":"The earliest source the manuscripts cite for the problem is Douglas-Pearcy (1972), described as studying it rather than posing it; no posing date is given.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 and Corollary 1.2: on $L^2(\\mathbb Z_2;\\ell^2(\\mathbb Z))$ a fibered bilateral weighted shift $S$ with quadratically decaying weights over the 2-adic odometer is nonzero with $\\|S^n\\|\\le\\exp(-10\\lfloor(n+1)^2/4\\rfloor)$, hence norm-quasinilpotent, and its commutant (containing a backward intertwiner $V$ and all scalar multipliers) is transitive. So every infinite-dimensional separable Hilbert space carries a nonzero quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. The operator has many ordinary invariant subspaces (the fiberwise tails), so the invariant subspace problem itself is untouched. The companion reaches the same existence statement by a different construction.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstracts and introductions of both manuscripts and Theorem 1.1 and Corollary 1.2 of the principal one, read against the problem as stated there. The proofs were not refereed. Lean-checked on the release's Comparator challenge BackwardIntertwiners (declaration OAI.BackwardIntertwiners.direct_algebra_corollary, listed in lean/formalization.yaml). Its statement was read here: for every complete separable infinite-dimensional complex inner product space it asserts an operator T that is nonzero, with norm of T^n to the power 1/n tending to 0, whose commutant leaves no closed subspace other than 0 and H invariant, is not the whole operator algebra, and is closed in the strong operator topology. That is the headline claim. Not rebuilt here. The companion also asserts that a published corollary (Cho-Ko-Lee 2017, Corollary 2.8) is false.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"The hyperinvariant subspace problem is one of the classic open problems of operator theory, the commutant-level companion of the invariant subspace problem and open for over fifty years, though less famous than that problem. Just above the Kadison-Kaplansky conjecture (50), below the Kadison similarity problem (58) and the free group factor problem (68).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf","sourceName":"OpenAI math release: Backward intertwiners and a transitive commutant","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Invariant-projection counterexamples for every irrational rotation (companion, independent proof)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026/paper.pdf","kind":"paper"},{"label":"Lean Comparator challenge BackwardIntertwiners (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BackwardIntertwiners.lean","kind":"lean-proof"},{"label":"Douglas-Pearcy 1972, Hyperinvariant subspaces and transitive algebras","url":"https://doi.org/10.1307/mmj/1029000793","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"spacetime-penrose-inequality","name":"The spacetime Penrose inequality: invariant ADM mass bounds the minimal area enclosing a trapped boundary, for general asymptotically flat initial data","shortName":"Spacetime Penrose inequality","problemNumber":null,"field":"Mathematical general relativity","fieldGroup":"Mathematical physics","statement":"Penrose (1973), as a test of weak cosmic censorship, argued that a black hole of horizon area $A$ requires total mass at least $\\sqrt{A/16\\pi}$. The time-symmetric (Riemannian) case was proved in dimension three by Huisken-Ilmanen and Bray, and in higher dimensions by Bray-Lee and Bi-Zhu. For initial data $(M^n,g,K)$ with nonzero second fundamental form the bound with the apparent horizon's own area fails (Ben-Dov; Carrasco-Mars), and the accepted formulation, following Bray and Khuri, uses the least area $A_{\\min}$ of surfaces enclosing the trapped boundary. It was known in spherical symmetry, for some cohomogeneity-one data, and with a non-sharp constant (Allen-Bryden-Kazaras-Khuri, 2025). Does every asymptotically flat initial-data set obeying the dominant energy condition, with weakly trapped boundary, satisfy $m\\ge\\frac12(A_{\\min}/\\omega_{n-1})^{(n-2)/(n-1)}$ for the invariant ADM mass $m$, with equality only for Schwarzschild?","posedBy":"Roger Penrose (heuristic inequality); enclosing-area initial-data formulation following Hubert Bray and Marcus Khuri","yearPosed":1973,"ageNote":"Penrose, Naked singularities, Ann. N.Y. Acad. Sci. 224 (1973). The enclosing-area form for non-time-symmetric data follows Bray-Khuri (2010, 2011) and the Ben-Dov (2004) and Carrasco-Mars (2010) counterexamples to apparent-horizon formulations.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.3: for smooth one-ended initial-data exteriors in every dimension $n\\ge3$ with integrable constraint densities, dominant energy condition, weakly future trapped boundary ($\\theta^+\\le0$, nothing on $\\theta^-$), the stated decay, $E>|P|$ and $A_{\\min}>0$: $m=(E^2-|P|^2)^{1/2}\\ge\\frac12(A_{\\min}/\\omega_{n-1})^{(n-2)/(n-1)}$, sharp; outermostness and outer area minimization are not assumed. Corollary 1.4 derives $E>|P|$; Theorem 1.5 ($n=3,4$) needs only two metric derivatives of decay. Companions: rigidity (equality only for Schwarzschild-Tangherlini slices, under horizon hypotheses), a conformal-flow Riemannian inequality in all dimensions, the boundary deformations, and a Bondi-mass version for CKS hyperboloidal data. Not shown: evolution or cosmic censorship. Priority: Allen-Bryden-Kazaras-Khuri (2025) proved the bound with a non-sharp constant.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has thirteen manuscripts (September 27 and October 5, 2026); the October papers cite the September neutral theorem as an input, consistent with the README's note that some results build on earlier model results.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorems 1.3 and 1.5 and Corollary 1.4 of the principal manuscript were read against the conjecture as the manuscript formulates it, and the companion abstracts were read. The proof (a coupled generalized-Jang and conformal deformation through a penalized scalar system, then a Riemannian Penrose inequality by conformal flow) was not refereed. Scope stated by the paper: smooth one-ended exteriors (designated-end versions for finitely many ends in dimensions three and four), weakly future trapped boundary, dominant energy condition, decay $g-\\delta=O_6(r^{-q})$, $K=O_5(r^{-1-q})$ with $(n-2)/20$ there are $c_A,d_A$ such that any learner with $M\\le Ad^2$ bits, $d\\ge d_A$, $0<\\epsilon\\le1/10$ and uniform-prior success at least $2/3$ in reaching angular error $\\epsilon$ needs horizon $T\\ge c_A d\\log(1/\\epsilon)$; for $M=o(d^2)$ the constant is absolute, and the bound also holds under an every-signal guarantee. Transitions may use unbounded computation and randomness; the output uses only the terminal state, stopping index and fresh randomness, with a deterministic horizon. It implies an $\\Omega_A(dr\\log d)$ bound in the noisy experiment of Sharan-Sidford-Valiant at Euclidean accuracy $d^{-r}$ in their range. No matching finite-memory upper bound is proved, and memory well above $d^2$ is not treated.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has six manuscripts of the same date: the principal one proves the lower bound for memory up to $Ad^2$, and five companions give alternative proofs or supporting estimates (posterior replicas, subsphere methods, projection moments and cap domination, Gaussian replacement, localization and information growth). Most are partly formalized in Lean.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem, its two corollaries and the predecessors section of the principal manuscript were read against the Sharan-Sidford-Valiant question as the manuscript cites it. Lean: lean/docs/140.md lists eight challenges. ComparatorChallenges/NoiselessRegression (theorem OAI.NoiselessRegression.main, solution module OAI.Probability.GaussianRegression.Main, file exists at the pinned commit) is not in formalization.yaml; its statement was read here: for $M=o(d^2)$ an absolute $c$ with $T\\ge c\\,d\\log(1/\\epsilon)$ under uniform-prior or every-signal success $2/3$, and for $M\\le Ad^2$ a constant $c_A$, for finite-state randomized learners with exact Gaussian observations and angular error. That is the headline. MemoryPrecision.main (in formalization.yaml) covers supporting estimates and the $M\\le d^2$ case. Not rebuilt here. The model is exact (noiseless) observations; the noisy consequence is derived in the paper, not formalized.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":17,"significanceNote":"A specific open question from a STOC paper in the memory-sample tradeoff line of learning theory; specialist interest. Placed level with The Anari-Oveis Gharan-Vinzant perfect-matching entropy conjecture (18), just above Quantum Memory Advantage for Process Tomography (15), and well below Optimality of Greedy for Single-Pass Semi-Streaming Matching (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf","sourceName":"OpenAI math release: Memory and precision in noiseless Gaussian regression","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf","kind":"paper"},{"label":"Companion: Posterior replicas and conditional information in Gaussian regression","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf","kind":"paper"},{"label":"Companion: Projection moments, positive cap domination, and Riesz estimates on the sphere","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf","kind":"paper"},{"label":"Companion: Replacing Gaussian observations in memory-constrained inference","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026/paper.pdf","kind":"paper"},{"label":"Companion: Localization costs and information growth for exact Gaussian observations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.NoiselessRegression.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/GaussianRegression/Main.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.MemoryPrecision.main, supporting estimates)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/MemoryPrecision/Main.lean","kind":"lean-proof"},{"label":"Sharan-Sidford-Valiant, Memory-sample tradeoffs for linear regression with small error (STOC 2019)","url":"https://doi.org/10.1145/3313276.3316403","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"chromatic-splitting-conjecture","name":"Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms)","shortName":"Chromatic splitting conjecture","problemNumber":null,"field":"Stable homotopy theory: chromatic homotopy","fieldGroup":"Geometry & topology","statement":"Fix a prime $p$, let $L_{n-1}$ be localization at $E(n-1)$ and $L_{K(n)}$ localization at Morava $K$-theory. Hopkins' chromatic splitting conjecture, recorded by Hovey (1995, Conjecture 4.2), describes the chromatic overlap $L_{n-1}L_{K(n)}S_p^\\wedge$. Its weak form asserts that for the $p$-completion $X$ of a finite spectrum the canonical map $L_{n-1}X\\to L_{n-1}L_{K(n)}X$ has a homotopy retraction (finite-input form: Beaudry-Goerss-Henn, Conjecture 1.1.11). Its strong form prescribes an explicit splitting of $L_{n-1}L_{K(n)}S_p^\\wedge$ into suspended localized spheres indexed by an exterior algebra; Beaudry disproved this at $n=p=2$, and Barthel-Beaudry (2019, Conjecture 6.3) retain it at odd primes. Both forms were known through height two at odd primes. Does the canonical map split for all $n$ and $p$, and does the overlap decompose as predicted at odd primes?","posedBy":"M. J. Hopkins, recorded by M. Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture (1995), Conjecture 4.2; refined by Barthel-Beaudry (2019) and Beaudry-Goerss-Henn","yearPosed":1995,"ageNote":"The conjecture is Hopkins' and circulated before Hovey's 1995 write-up, which the manuscripts cite as the record. The weak finite-input form is stated as Conjecture 1.1.11 of Beaudry-Goerss-Henn.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Weak form: for the derived $p$-completion $X=S_p^\\wedge$, the canonical map $L_{n-1}X\\to L_{n-1}L_{K(n)}X$ has no homotopy retraction at height $n=p$ for every $p\\ge5$ and at $n=p+1$ for every $p\\ge7$; the class $\\beta_1^{(p-1)^2}$ is a nonzero kernel element (at $p=5$, $\\beta_1^{16}$ in degree 608). Strong form: for $p\\ge5$ the map $L_0L_{K(3)}S\\to L_0L_{K(2)}L_{K(3)}S$ is nonzero on $\\pi_{-3}$, so the height-three splitting fails even as an equivalence of $E(2)$-local spectra. Companions: at $n=p=3$ the overlap is not in the thick subcategory generated by $L_0S,L_1S,L_2S$; positively, for $p>n+1$ the overlap has a $2^n$-stage filtration with the predicted cofibers (nonsplit), with explicit attachments at height three. Heights below $p-1$ remain open.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The family has five manuscripts dated 25 and 27 September 2026: two counterexample papers, one negative finite-assembly result, and two positive filtration results. The rational-obstruction manuscript ships a small exact-arithmetic support script whose coverage note says it checks only Chevalley-Eilenberg matrix arithmetic, not the main theorem.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of the TeX sources of all five manuscripts, read against Hovey's Conjecture 4.2 and the Barthel-Beaudry and Beaudry-Goerss-Henn formulations they cite; the proofs were not refereed. The weak-splitting counterexamples (principal manuscript) use the derived p-completion of the sphere, i.e. finite input, as the conjecture requires; they rely on the Hopkins-Miller computation recorded by Heard and on Devinatz-Hopkins nilpotence. The strong-form disproof at height three uses the solid Lubin-Tate theory of Barthel, Mann, Ray, Schlank, Senger, Weinstein and Zhou (cited as 2026). The disproofs cover specific ranges of p and n only; the weak form at heights below p - 1 and at small primes is not addressed. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"One of the structural conjectures of chromatic homotopy theory, alongside the telescope conjecture, open for three decades and known only through height two. Famous within stable homotopy theory, little known outside it. Scored with the Burr-Erdos hypercube Ramsey conjecture (40), above the Kinoshita conjecture and Kirby Problem 4.37 (35), below the hyperbolic groups CAT(0) question (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026/paper.pdf","sourceName":"OpenAI math release: Counterexamples to weak chromatic splitting: sphere kernels and descent exponents","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"A rational obstruction to strong chromatic splitting at height three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf","kind":"paper"},{"label":"Failure of finite assembly for a chromatic overlap at the prime three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026/paper.pdf","kind":"paper"},{"label":"Filtered chromatic splitting at generic primes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf","kind":"paper"},{"label":"The height-three chromatic overlap: an explicit filtration and its attachments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026/paper.pdf","kind":"paper"},{"label":"Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture (1995)","url":"https://doi.org/10.1090/conm/181/02036","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"reflexive-diamond-auc-renormability-question","name":"Must a reflexive Banach space with no AUC renorming contain the countably branching diamonds uniformly? (Baudier-Lancien Problem 39)","shortName":"Reflexive diamond question (Problem 39)","problemNumber":null,"field":"Banach space theory: asymptotic geometry and metric embeddings","fieldGroup":"Analysis","statement":"A norm is asymptotically uniformly convex (AUC) if $\\overline\\delta(t)=\\inf_{\\|x\\|=1}\\sup_{F}\\inf_{y\\in F,\\|y\\|=1}(\\|x+ty\\|-1)>0$ for all $t>0$, $F$ ranging over finite-codimensional subspaces, and asymptotically midpoint uniformly convex (AMUC) if the same holds for the average of $\\|x+ty\\|$ and $\\|x-ty\\|$. The countably branching diamonds $D_k$ replace each edge by countably many two-edge paths, $k$ times. Baudier, Causey, Dilworth, Kutzarova, Randrianarivony, Schlumprecht and Zhang (2017) proved that an equivalent AMUC norm prevents uniform bi-Lipschitz embeddings of the $D_k$, and that for reflexive spaces with unconditional asymptotic structure, failure of AUC renormability forces uniform diamond embeddings; Swift and Perreau extended the latter under similar structure. Baudier and Lancien record the general question as Problem 39: if $X$ is reflexive and admits no equivalent AUC norm, must the countably branching diamonds embed into $X$ with uniformly bounded distortion?","posedBy":"F. P. Baudier and G. Lancien, Asymptotic and nonlinear geometries of Banach spaces and their interactions (arXiv 2512.00817, 2025; Cours Specialises 33, SMF 2026), Problem 39","yearPosed":2025,"ageNote":"The year is the arXiv posting of the monograph that numbers the problem; the question grows out of Baudier et al. (2017), who proved the implication under unconditional asymptotic structure. Whether that 2017 paper already stated it as open was not checked here.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: the dual $X=J^*$ of a countably branching segment-forest space with finite, unbounded component heights is infinite-dimensional, separable and reflexive, admits no equivalent AUC norm, and its given norm has averaged midpoint modulus $\\widehat\\delta_X(t)\\ge\\sqrt{1+t^2/12}-1$; every embedding of $D_k$ into $X$ has distortion at least $\\sqrt{1+k/12}$. So Problem 39 has a negative answer, and AMUC does not imply AUC renormability even for reflexive spaces (Baudier had shown the non-reflexive separation in 2026). Companions add midpoint-lens estimates, a sharper $D^2\\ge1+k/4$ bound, exact moduli in a Daugavet subspace of $L_1$, and further tree-potential and $L_1$ examples. Not shown: sharp distortion growth rates.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The family has seven manuscripts, all with Lean formalizations produced as part of the same release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the principal manuscript against Problem 39 as the paper quotes it, and the Lean statement lean/ComparatorChallenges/ForestSpace.lean (OAI.ForestSpace.main_theorem), listed in formalization.yaml; not rebuilt here. The Lean statement covers the headline: the full dual of the specified segment-norm space is infinite-dimensional, separable and reflexive, no equivalent norm is AUC, the averaged midpoint modulus is at least $\\sqrt{1+t^2/12}-1$, and every embedding of the depth-$k$ diamond has distortion at least $\\sqrt{1+k/12}$. That is exactly a reflexive non-AUC-renormable space without uniform diamond embeddings. The six companions give further AMUC non-AUC examples, each with its own Comparator statement.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A specialist question in the asymptotic and nonlinear geometry of Banach spaces, numbered in a recent monograph and following a 2017 partial result; known to a small community. Below the Inverse Generator Problem on Hilbert spaces (22) and primariness of L_p(L_1) (20), level with Wickstead's conjecture on positive projections (12), above whether amenability depends on the ground field (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026/manuscript.pdf","sourceName":"OpenAI math release: Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Midpoint lenses in segment spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf","kind":"paper"},{"label":"Companion: Distortion of countably branching diamonds from midpoint and tree energies","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf","kind":"paper"},{"label":"Companion: Exact asymptotic moduli in a Daugavet subspace of L1","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf","kind":"paper"},{"label":"Companion: Midpoint convexity from bounded tree potentials and path costs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026/manuscript.pdf","kind":"paper"},{"label":"Companion: Independent products in real L1: asymptotic midpoint convexity without AUC renormings","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026/manuscript.pdf","kind":"paper"},{"label":"Companion: Midpoint convexity from two recursive potentials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf","kind":"paper"},{"label":"Lean comparator statement: ForestSpace.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ForestSpace.lean","kind":"lean-proof"},{"label":"Lean file: OAI/Analysis/ForestSpace/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/ForestSpace/Main.lean","kind":"lean-proof"},{"label":"Baudier and Lancien, Asymptotic and nonlinear geometries of Banach spaces, Problem 39 (arXiv:2512.00817)","url":"https://arxiv.org/abs/2512.00817","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bang-half-area-cylinder-covering","name":"Bang's half-area cylinder covering question (and the 1-codimensional cylinder covering conjecture)","shortName":"Half-area cylinder covering conjecture","problemNumber":null,"field":"Convex geometry, covering problems","fieldGroup":"Geometry & topology","statement":"A cylinder in $\\mathbb R^3$ is $B+\\mathbb Ru$ with $u$ a unit vector and $B\\subset u^\\perp$ of finite area. Bang's plank theorem says plank widths in a covering of a convex body sum to at least its minimal width; for cylinders, the two-cylinder cover of a regular tetrahedron has total base area exactly half the minimum projection area $A_{\\min}(K)$. Bezdek and Litvak, attributing the example to Bang (1951), asked whether every finite cylinder covering of a convex body $K\\subset\\mathbb R^3$ satisfies $\\sum_i|B_i|\\ge\\frac12A_{\\min}(K)$; Bezdek and Khan conjectured the stronger directionwise bound $\\sum_i|B_i|/|\\pi_{u_i^\\perp}K|\\ge\\frac12$. Do these half-area lower bounds hold?","posedBy":"Thoger Bang (example); Karoly Bezdek and Alexander Litvak; Karoly Bezdek and Muhammad Khan (directionwise form)","yearPosed":1951,"ageNote":"The manuscript cites Bezdek (2009, Problem 3.1) and Bezdek-Litvak (2009), who attribute the tetrahedral example and the half-area question to Bang's 1951 plank paper; the directionwise form is Bezdek-Khan (arXiv 1602.06040, 2016), Conjecture 4.13. Bezdek-Litvak proved the directionwise bound 1/3 in general.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a regular tetrahedron $K$ with $A_{\\min}(K)=\\sqrt2$ has, for each $0<\\varepsilon\\le1/2000$, a cover by $2\\lceil2/\\varepsilon^2\\rceil$ cylinders with compact triangular perpendicular bases and $\\frac1{\\sqrt2}\\sum|B_i|=\\frac12-\\frac{13}{6000}\\varepsilon^2+O(\\varepsilon^4)$, strictly below half. Corollary 1.2: every nondegenerate tetrahedron has a finite cover with directionwise relative cost below 1/2, refuting the Bezdek-Khan conjecture. Not shown: the optimal constant (Bezdek-Litvak's lower bound 1/3 stands) or counterexamples among centrally symmetric bodies.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has four manuscripts, all dated September 27, 2026: the principal angular-sector construction and three companions giving alternative or more general constructions (ruled-set approximation, slope-field perturbations, radial sweeps).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, history section and Theorem 1.1 with Corollary 1.2 were read against the question as Bezdek and Bezdek-Khan state it. Lean: the challenge TriangularCovering (theorem OAI.TriangularCovering.main, solution module OAI.Geometry.TriangularCovering.Main) exists at the pinned commit but is not in the formalization catalogue; its statement was read here. It states, for the regular tetrahedron of edge 2, $A_{\\min}=\\sqrt2$, finite triangular-base cylinder covers with normalized area $1/2-(13/6000)\\varepsilon^2+O(\\varepsilon^4)$ and total area below $A_{\\min}/2$, and a cover with directionwise relative cost below 1/2. That is the headline counterexample. The affine extension to all tetrahedra is outside it. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A recorded question in the Tarski-Bang plank-problem line, open since the tetrahedral example and listed as unresolved in a 2026 survey, but known mainly to discrete geometers. Placed just below Kuperberg's six-cylinder conjecture (20), above Bezdek's partial plank covering question (15) and a typical Erdos problem (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf","sourceName":"OpenAI math release: Finite angular cylinder covers below the half-area bound","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Finite cylinder approximation of ruled sets","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf","kind":"paper"},{"label":"Companion: Slope-field perturbations of the two-cylinder covering","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf","kind":"paper"},{"label":"Companion: Finite triangular approximation of radial sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf","kind":"paper"},{"label":"Lean formalization: TriangularCovering/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/TriangularCovering/Main.lean","kind":"lean-proof"},{"label":"Lean formalization: CylinderCovering/Main.lean (four-paper aggregate)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/CylinderCovering/Main.lean","kind":"lean-proof"},{"label":"Bezdek-Khan, The geometry of homothetic covering and illumination (Conjecture 4.13)","url":"https://arxiv.org/abs/1602.06040v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"edit-distance-l1-distortion","name":"The least distortion of embedding edit distance into l1","shortName":"Edit distance into l1: distortion","problemNumber":null,"field":"Metric embeddings; edit distance","fieldGroup":"Theoretical computer science","statement":"Let $E_\\Sigma(d)$ be the least distortion of an embedding into $\\ell_1$ of the strings of length at most $d$ over an alphabet $\\Sigma$, with unit-cost edit distance (insertions, deletions, substitutions). Ostrovsky and Rabani (2007) embedded binary strings with distortion $\\exp(O(\\sqrt{\\log d\\log\\log d}))$. Lower bounds rose from $3/2$ (Andoni-Deza-Gupta-Indyk-Raskhodnikova 2003) to $(\\log d)^{1/2-o(1)}$ (Khot-Naor 2006) and $\\Omega(\\log d)$ (Krauthgamer-Rabani 2009), leaving an exponential gap. What is the order of growth of $E_\\Sigma(d)$; in particular, is the Ostrovsky-Rabani bound $\\exp(O(\\sqrt{\\log d\\log\\log d}))$ optimal?","posedBy":"Open since the Ostrovsky-Rabani upper bound (J. ACM 2007) and the Krauthgamer-Rabani lower bound (2009); the manuscripts cite no explicit statement of the question","yearPosed":null,"ageNote":"The manuscripts present the question through the gap between Ostrovsky-Rabani (2007) and Krauthgamer-Rabani (2009) and do not cite a paper that poses it, so no year is given.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are absolute $c,C>0$ and $d_0$ with $\\exp(c\\sqrt{\\log d\\log\\log d})\\le E_\\Sigma(d)\\le\\exp(C\\sqrt{\\log d\\log\\log d})$ for all $d\\ge d_0$ and all finite $|\\Sigma|\\ge2$, so the Ostrovsky-Rabani scale is sharp up to constants in the exponent. The upper bound is Ostrovsky-Rabani's method made uniform over alphabets and lengths; the new content is the matching lower bound, given by three constructions across the family. The constants $c,C$ are not determined, and nothing is said about computational efficiency.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two companion manuscripts of the same date giving independent finite-circle and tree constructions for the lower bound and a uniform histogram embedding for the upper bound.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question: absolute $c,C$ with $\\exp(c\\sqrt{\\log d\\log\\log d})\\le E_\\Sigma(d)\\le\\exp(C\\sqrt{\\log d\\log\\log d})$ for all large $d$, uniformly over finite alphabets of size at least two, the lower bound witnessed by binary strings of one length. The challenge is not in lean/formalization.yaml; lean/ComparatorChallenges/EditDistance.json exists with solution_module OAI.Combinatorics.EditDistance.Main, whose file exists at the pinned commit. The statement EditDistance.lean was read here; not rebuilt here. It defines edit distance by unrestricted unit-step scripts with substitutions, distortion as the product of the two extreme ratios over injective maps into $\\ell^1(\\mathbb N)$, and states the two-sided bounds for every finite alphabet type, the binary witness and the alphabet supremum. This states the headline. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"The l1 distortion of edit distance has been a well-known open problem in metric embeddings and sublinear algorithms since Ostrovsky-Rabani, with Khot-Naor and Krauthgamer-Rabani attacking it; famous within that community. Below the sensitivity separation entry in this batch at 35, above Kalton's Problem 2 at 28, well below the randomized k-server entry at 45.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/paper.pdf","sourceName":"OpenAI math release: Edit Distance in l1: Matching Bounds up to Constants in the Exponent","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf","kind":"paper"},{"label":"Companion: Tree Constructions for the l1 Distortion of Binary Edit Distance","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.edit_distance_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/EditDistance/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"secret-key-from-all-entangled-states","name":"Secret key from all entangled states: does every entangled state have positive distillable secret key? (Kruger-Werner Problem 24)","shortName":"Secret key from all entangled states","problemNumber":null,"field":"Quantum cryptography: distillable secret key and bound entanglement","fieldGroup":"Quantum information & computing","statement":"For a bipartite state $\\rho_{AB}$, the distillable secret key $K_D(\\rho)$ is the best rate at which Alice and Bob, using local operations and authenticated public communication on many copies, can extract bits that agree and are independent of an eavesdropper holding a purification and the public record. Separable states have $K_D=0$, and Horodecki et al. showed that some bound entangled (PPT) states nevertheless have $K_D>0$, so key and distillable entanglement differ. The converse was posed as Problem 24, 'Secret key from all entangled states', in Kruger and Werner's 2005 collection of open problems in quantum information theory (contact P. Horodecki). Is $K_D(\\rho)>0$ for every entangled state $\\rho$?","posedBy":"P. Horodecki (contact), Problem 24 'Secret key from all entangled states' in O. Kruger and R. F. Werner (eds.), Some open problems in quantum information theory (2005)","yearPosed":2005,"ageNote":"The open-problem entry is dated 15 March 2005 (arXiv quant-ph/0504166). The manuscript notes the question is also discussed in the private-state literature.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is an explicit entangled density operator $\\rho$ on $\\mathbb C^{10}\\otimes\\mathbb C^{10}$ whose range contains no nonzero product vector, such that for every $n$ and every admitted completed bit-output protocol on $\\rho^{\\otimes n}$ the output is at trace-norm distance at least $1/5$ from an ideal secret bit; hence $K_D(\\rho)=0$. The state is the normalized Choi matrix of a composition of two PPT maps; a coherence inequality for such states bounds secrecy by disagreement probability. So the answer to Problem 24 is no. Not shown: minimality of dimension ten, or a zero-key example for protocol models beyond the paper's admitted class.","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source against Kruger-Werner Problem 24. The theorem gives an explicit entangled state on $\\mathbb C^{10}\\otimes\\mathbb C^{10}$ with $K_D=0$ for the paper's class of admitted protocols, which includes ordinary finite two-way public-discussion protocols with round counts growing with block length, plus specified countable processes; readers should note the answer is stated relative to that formal protocol model. The proof was not refereed. The release's Lean formalization (lean/docs/272.md) covers only the PPT-composition consequences, and says explicitly that the zero-key statement is outside it, so this entry is unreviewed. Entanglement of the state rests on an exact Galois-group computation, with a supplementary certificate output in the release.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A listed open problem in quantum key distillation since 2005, sitting next to the bound-entanglement and bound-information questions; well known within quantum information but not beyond. Below two-copy distillability of Werner states (35) and NPT distillability in the DiVincenzo family (30), above bipartite bound information (25) and the Lami-Regula irreversibility conjecture (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf","sourceName":"OpenAI math release: Entanglement with zero distillable secret key in local dimension ten","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kruger and Werner, Some open problems in quantum information theory, Problem 24 (arXiv:quant-ph/0504166)","url":"https://arxiv.org/abs/quant-ph/0504166","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ppt-squared-conjecture","name":"The PPT-squared conjecture: is the composition of two PPT channels always entanglement breaking?","shortName":"PPT-squared conjecture","problemNumber":null,"field":"Quantum channels: PPT maps and entanglement breaking","fieldGroup":"Quantum information & computing","statement":"A linear map $F:M_a(\\mathbb C)\\to M_b(\\mathbb C)$ is PPT if both $F$ and $T\\circ F$ are completely positive, $T$ the transpose, and entanglement breaking if $(\\mathrm{id}\\otimes F)(X)$ is separable for every positive $X$, equivalently its Choi matrix is separable. Christandl conjectured that composing two PPT channels always yields an entanglement-breaking channel; the question was recorded as Problem G of the 2012 BIRS workshop 'Operator structures in quantum information theory', and stated for pairs of PPT maps as Conjecture IV.1 of Christandl, Muller-Hermes and Wolf (2019). Is $\\Phi_2\\circ\\Phi_1$ entanglement breaking for all PPT channels $\\Phi_1,\\Phi_2$, in particular is $\\Phi\\circ\\Phi$ entanglement breaking for every PPT channel $\\Phi$?","posedBy":"M. Christandl, recorded as Problem G of the BIRS workshop report 12w5084 (2012); stated as Conjecture IV.1 by M. Christandl, A. Muller-Hermes and M. M. Wolf, Ann. Henri Poincare 20 (2019)","yearPosed":2012,"ageNote":"The manuscript cites the 2012 BIRS report for the channel form and CMHW 2019 (arXiv 1807.01266) for the two-map form.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.3: there is an explicit trace-preserving PPT channel $\\Theta$ on $M_{21}(\\mathbb C)$ with $\\Theta\\circ\\Theta$ not entanglement breaking, answering the 2012 channel question negatively with the same channel in both slots. Theorem 1.2: explicit PPT maps $\\Phi_1,\\Phi_2$ on $M_{10}(\\mathbb C)$ (not trace preserving) whose composition has nonzero Choi matrix with no product vector in its range, refuting the unrestricted two-map Conjecture IV.1. The same pair yields the zero-key entangled state of the companion entry. Not shown: minimal dimensions, or whether every PPT channel becomes entanglement breaking after three compositions (the PPT-cubed question).","claimIssueNote":null,"solveDate":"2026-09-27","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The release also supplies Lean statements for both counterexamples, produced as part of the same release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.2 and 1.3 of the TeX source against BIRS Problem G and CMHW Conjecture IV.1, and the Lean statements lean/ComparatorChallenges/DimensionTenChannel.lean (OAI.DimensionTen.exists_channel_fin21) and DimensionTenPair.lean (OAI.DimensionTen.main_pair), whose solution modules exist at the pinned commit. Neither challenge is in the formalization catalogue; the statements were read here and were not rebuilt. exists_channel_fin21 states the headline directly: a trace-preserving PPT linear map on $21\\times21$ complex matrices whose square is not entanglement breaking, with PPT, trace preservation and entanglement breaking defined from scratch. main_pair states the two-map counterexample in dimension ten with explicitly defined maps. No minimal dimension is claimed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"One of the best-known open problems of entanglement theory since 2012, with many partial confirmations and a sequence of papers on eventual entanglement breaking; famous within quantum information, little known outside it. Level with two-copy distillability of Werner states (35), above NPT distillability in the DiVincenzo family (30) and the eventual entanglement-breaking result for PPT channels (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf","sourceName":"OpenAI math release: Entanglement with zero distillable secret key in local dimension ten","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean comparator statement: DimensionTenChannel.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DimensionTenChannel.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Analysis/Quantum/DimensionTen/Channel.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Quantum/DimensionTen/Channel.lean","kind":"lean-proof"},{"label":"Lean comparator statement: DimensionTenPair.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DimensionTenPair.lean","kind":"lean-proof"},{"label":"BIRS workshop 12w5084 report (2012), Problem G","url":"https://www.birs.ca/workshops/2012/12w5084/report12w5084.pdf","kind":"problem-record"},{"label":"Christandl, Muller-Hermes, Wolf, When do composed maps become entanglement breaking? (arXiv:1807.01266)","url":"https://arxiv.org/abs/1807.01266","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gopalan-servedio-square-root-degree-conjecture","name":"The Gopalan-Servedio conjecture: is $\\sum_i\\widehat f(\\{i\\})=O(\\sqrt{\\deg f})$ for Boolean functions?","shortName":"Gopalan-Servedio square-root degree conjecture","problemNumber":null,"field":"Analysis of Boolean functions","fieldGroup":"Combinatorics","statement":"For $f:\\{-1,1\\}^n\\to\\{-1,1\\}$ write $\\widehat f(S)=\\mathbb E[f(X)\\prod_{i\\in S}X_i]$ with $X$ uniform, and let $\\deg f$ be the degree of its real multilinear representation. Cauchy-Schwarz gives $\\sum_i\\widehat f(\\{i\\})=\\mathbb E[f(X)\\sum_iX_i]\\le\\sqrt n$, attained up to constants by majority, while total influence gives $\\sum_i|\\widehat f(\\{i\\})|\\le\\deg f$. Gopalan and Servedio conjectured, around 2009, that $\\sqrt n$ can be replaced by $\\sqrt{\\deg f}$; the conjecture is recorded in O'Donnell's list of open problems in analysis of Boolean functions and as Conjecture 3.17 of Filmus, Hatami, Keller and Lifshitz. A sharper majority-benchmark version was refuted at degree 4 by Kudin and Pasalic, which does not decide the square-root form. Is there an absolute constant $C$ with $\\sum_i\\widehat f(\\{i\\})\\le C\\sqrt{\\deg f}$ for every Boolean function $f$?","posedBy":"Parikshit Gopalan and Rocco Servedio (c. 2009), recorded in O'Donnell's Open Problems in Analysis of Boolean Functions (2012, p. 9)","yearPosed":2009,"ageNote":"The manuscript attributes the conjecture to Gopalan and Servedio circa 2009 via O'Donnell's 2012 problem list; no dated original source is cited.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1: for every $C>0$ there is a nonconstant $f:\\{-1,1\\}^n\\to\\{-1,1\\}$ with $\\sum_i\\widehat f(\\{i\\})>C\\sqrt{\\deg f}$; equivalently $\\sum_i|\\widehat f(\\{i\\})|/\\sqrt{\\deg f}$ is unbounded over Boolean functions of positive degree, so no constant multiple of the conjectured bound holds. The dimension and copy counts in the construction are finite but not usefully bounded, so no growth rate of the violation in terms of degree is given. The true order between $\\sqrt{\\deg f}$ and the linear bound $\\deg f$ is left open.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The release also formalizes the disproof in Lean.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1 and the introduction were read against the conjecture as O'Donnell's list states it. Lean: lean/docs/192.md links ComparatorChallenges/SquareRootDegree (theorem OAI.SquareRootDegree.main, solution module OAI.Combinatorics.BooleanFunctions.Main, file exists at the pinned commit), not in formalization.yaml. The statement was read: for every $C>0$ there is a nonconstant $\\pm1$-valued function on a finite sign cube with $C\\sqrt{\\deg f}$ below the sum of its singleton Fourier coefficients, where degree is the real Fourier degree, and the ratio of the absolute singleton sum to $\\sqrt{\\deg f}$ is unbounded. That is the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"A listed open problem in analysis of Boolean functions, known to specialists but not a central one. Placed below Courtade and Kumar's Coordinate-wise Mutual Information Question (22) and The Hellinger conjecture for Boolean functions under product noise (25), and well below Is block sensitivity at most quadratic in sensitivity? (Nisan-Szegedy) (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf","sourceName":"OpenAI math release: Unbounded violations of the square-root degree bound","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.SquareRootDegree.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/BooleanFunctions/Main.lean","kind":"lean-proof"},{"label":"O'Donnell, Open Problems in Analysis of Boolean Functions (arXiv:1204.6447, p. 9)","url":"https://arxiv.org/abs/1204.6447v1","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hajlasz-onninen-centered-disk-maximal-w11","name":"The Hajlasz-Onninen endpoint question for the centered maximal function, planar disk case: is $\\|\\nabla Mf\\|_1\\le C\\|\\nabla f\\|_1$ on $W^{1,1}(\\mathbb R^2)$?","shortName":"Hajlasz-Onninen question, planar centered case","problemNumber":null,"field":"Harmonic analysis: regularity of maximal functions","fieldGroup":"Analysis","statement":"For $f\\in L^1_{loc}(\\mathbb R^n)$ let $Mf(x)=\\sup_{r>0}|B(x,r)|^{-1}\\int_{B(x,r)}|f|$ be the centered Hardy-Littlewood maximal function. Kinnunen proved that $M$ is bounded on $W^{1,p}$ for $1 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The release also supplies a Lean formalization of the main theorem, produced as part of the same release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source against Hajlasz-Onninen Question 1, and the Lean statement lean/ComparatorChallenges/DiskMaximal.lean (theorem OAI.CenteredDiskEndpoint.main_endpoint), whose solution module OAI/Analysis/DiskMaximal/Main.lean exists at the pinned commit. This challenge is not in the formalization catalogue (formalization.yaml lists only the signed finite-band lemma, SignedFiniteBand); its statement was read here and was not rebuilt. It states the headline: an absolute $C$ such that for every integrable $f$ on the Euclidean plane with integrable weak gradient, $Mf$ is finite a.e., locally integrable, in $W^{1,1}_{loc}$, with an integrable weak gradient of $L^1$ norm at most $C\\|\\nabla f\\|_1$. The paper itself limits the result to disks in the plane; higher dimensions and a BV version are not covered.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"The Hajlasz-Onninen question is the central open problem on Sobolev regularity of maximal functions for twenty years, well known within harmonic analysis; the planar centered case answered here is its first case beyond one dimension for the centered operator without symmetry assumptions. Scored below the parent question and below the endpoint Carleson Schrodinger problem (32), level with Maz'ya's maximal operator question (25), above Brezis's degree-inequality constants (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026/article.pdf","sourceName":"OpenAI math release: An Endpoint Gradient Bound for the Centered Disk Maximal Operator","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean comparator statement: DiskMaximal.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DiskMaximal.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Analysis/DiskMaximal/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/DiskMaximal/Main.lean","kind":"lean-proof"},{"label":"Hajlasz and Onninen, On boundedness of maximal functions in Sobolev spaces (2004)","url":"https://afm.journal.fi/article/view/135096","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"popa-log-additivity-conjecture","name":"Popa's logarithmic additivity conjecture for log-smooth families","shortName":"Popa's log additivity conjecture","problemNumber":null,"field":"Birational geometry, Kodaira dimension of smooth families","fieldGroup":"Geometry & topology","statement":"For smooth families, subadditivity of Kodaira dimension is expected to become equality. Popa (Conjecture 3.9 of Conjectures on the Kodaira dimension) asked: let $f:X\\to Y$ be an algebraic fibre space between smooth projective varieties, $E$ an SNC divisor on $X$ and $D$ an SNC divisor on $Y$ with $\\mathrm{Supp}(f^*D)\\subseteq E$, and assume $f$ is log-smooth over $V=Y\\setminus D$ (each stratum of $(X,E)$, including $X$, is smooth over $V$). For a general fibre $F$ over $V$, is $\\kappa(X,K_X+E)=\\kappa(Y,K_Y+D)+\\kappa(F,K_F+E_F)$? It generalises his Conjecture 3.1 for smooth projective fibre spaces $U\\to V$ of quasi-projective varieties, known when the base is of log general type, a curve, or compactifies to an abelian variety, or when $F$ has semiample canonical bundle.","posedBy":"Mihnea Popa, Conjectures on the Kodaira dimension, Conjecture 3.9 and footnote 5 (author version 13 June 2023; LMS Lecture Note Series 489, 2025)","yearPosed":2023,"ageNote":"Dated to the 13 June 2023 author version the manuscript cites; the note was published in 2025 and mentions an earlier first version whose date the manuscripts do not give.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: under Popa's hypotheses (reduced SNC $E$, $D$, support inclusion, stratum smoothness over $V$), $\\kappa(X,K_X+E)\\le\\kappa(Y,K_Y+D)+\\kappa(F,K_F+E_F)$, and if either term on the right is $-\\infty$ every $H^0(X,m(K_X+E))$ vanishes; no abundance or good-model hypothesis. With the family's logarithmic subadditivity this gives equality, Corollary 1.2. Either divisor may be zero, which covers additivity for smooth projective morphisms; the paper itself claims only Conjecture 3.9. Not covered: families that are not log-smooth over the open base.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The upper inequality is proved independently; the equality uses the logarithmic subadditivity corollary of the family's orbifold paper. Only the negative-fibre branch is formalized in Lean (see the verification note).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of 'The reverse logarithmic Kodaira inequality and additivity', read against Popa's Conjecture 3.9 and footnote 5 in his 13 June 2023 author version (fetched and read here); the hypotheses match. The proof was not refereed. Lean: lean/ComparatorChallenges/LogKodairaFiberNegative.json exists with solution_module OAI.AlgebraicGeometry.LogKodaira.FiberNegative (present at the pinned commit) but the challenge is not in the formalization catalogue. Its statement was read: for a very general base point, if the fibre's log Kodaira dimension is -infinity then the total one is -infinity and every positive-degree log pluriform vanishes. That is only the elementary negative-fibre branch, not the headline, so the entry is Unreviewed rather than Lean-checked. Not rebuilt here. The equality depends on the family's unreviewed subadditivity paper.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A recent conjecture (Popa, 2021 to 2023) that Popa describes as quite open even for projective smooth families, with several partial cases by Campana, Park and Meng-Popa. Placed below the variation entry (45) and Iitaka subadditivity (62), level with the Campana-Peternell conjecture (35), and above the effective Iitaka fibration conjecture (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf","sourceName":"OpenAI math release: The reverse logarithmic Kodaira inequality and additivity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Orbifold and logarithmic Iitaka subadditivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf","kind":"paper"},{"label":"Lean: negative-fibre branch only (challenge not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/LogKodaira/FiberNegative.lean","kind":"lean-proof"},{"label":"Popa, Conjectures on the Kodaira dimension (author version)","url":"https://people.math.harvard.edu/~mpopa/papers/kodaira.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"viehweg-c-plus-popa-log-variation","name":"Viehweg's conjecture C(n,m)+ and Popa's logarithmic Iitaka-Viehweg inequality","shortName":"Iitaka-Viehweg variation inequality","problemNumber":null,"field":"Birational geometry, Kodaira dimension and variation","fieldGroup":"Geometry & topology","statement":"Viehweg's conjecture $C^+_{n,m}$ strengthens Iitaka's subadditivity by the birational variation of the family: for a fibre space $f:X\\to Y$ of smooth projective varieties with $\\kappa(Y)\\ge0$, $\\kappa(X)\\ge\\kappa(F)+\\max\\{\\kappa(Y),\\mathrm{Var}(f)\\}$. Kollar proved it for fibres of general type and Kawamata for fibres with good minimal models. Popa (Conjecture 3.8 of his note Conjectures on the Kodaira dimension) asked for the logarithmic version: for a projective fibre space $f:U\\to V$ of smooth quasi-projective varieties with $\\bar\\kappa(V)\\ge0$ and generic fibre $F$, is $\\bar\\kappa(U)\\ge\\kappa(F)+\\max\\{\\bar\\kappa(V),\\mathrm{Var}(f)\\}$?","posedBy":"Mihnea Popa, Conjectures on the Kodaira dimension, Conjecture 3.8 (author version 2023; LMS LNS 489, 2025), generalising E. Viehweg's C(n,m)+ conjecture (1983)","yearPosed":2023,"ageNote":"The logarithmic form answered here is Popa's, cited to his 2023 author version (published 2025; his note mentions an earlier first version). The ordinary C(n,m)+ conjecture it contains goes back to Viehweg's work of the early 1980s, which the manuscript cites.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a projective surjective morphism $f:U\\to V$ with connected fibres between smooth connected complex quasi-projective varieties with $\\bar\\kappa(V)\\ge0$ and geometric generic fibre $F$, $\\bar\\kappa(U)\\ge\\kappa(F)+\\max\\{\\bar\\kappa(V),\\mathrm{Var}(f)\\}$, with no good-minimal-model or abundance hypothesis on $F$. Corollary 1.2 gives Viehweg's ordinary $C^+_{n,m}$ for projective complex fibre spaces with $\\kappa(Y)\\ge0$. Not covered: bases with $\\bar\\kappa(V)=-\\infty$, positive characteristic, and the additivity equalities for smooth families (a separate entry).","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. This manuscript uses the reduced-SNC subadditivity theorem and full-period adjoint comparison of the family's orbifold subadditivity paper as stated inputs, so it stands or falls with that entry as well.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of 'Logarithmic Kodaira dimension and whole-fiber variation', read against Popa's Conjecture 3.8 in the 13 June 2023 author version (fetched and read here) and Viehweg's C(n,m)+ as Popa quotes it. The statement matches the conjecture, with variation defined through the field of definition of the whole geometric generic fibre. The proof (two constancy arguments, Hodge-line flatness, Demailly-Hacon-Paun extension, Hanamura descent) was not refereed. No Lean formalization. It depends on the orbifold subadditivity manuscript of the same family, which is also unreviewed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Viehweg's C(n,m)+ is the standard strengthening of Iitaka's conjecture and well known in birational geometry; Popa's logarithmic form is recent. Placed below the Iitaka subadditivity entry (62), which it contains, level with Kuznetsov's rationality conjecture (45), and above the Campana-Peternell conjecture (35) and the effective Iitaka fibration conjecture (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf","sourceName":"OpenAI math release: Logarithmic Kodaira dimension and whole-fiber variation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Orbifold and logarithmic Iitaka subadditivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf","kind":"paper"},{"label":"The reverse logarithmic Kodaira inequality and additivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf","kind":"paper"},{"label":"Popa, Conjectures on the Kodaira dimension (author version)","url":"https://people.math.harvard.edu/~mpopa/papers/kodaira.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"iitaka-subadditivity-conjecture","name":"Iitaka's subadditivity conjecture C(n,m), with its logarithmic form and Campana's orbifold form","shortName":"Iitaka subadditivity conjecture","problemNumber":null,"field":"Birational geometry, Kodaira dimension","fieldGroup":"Geometry & topology","statement":"For a surjective morphism $f:X\\to Y$ with connected fibres between smooth projective complex varieties, with general fibre $F$, Iitaka's conjecture $C_{n,m}$ asks whether $\\kappa(X)\\ge\\kappa(F)+\\kappa(Y)$. Iitaka also stated a logarithmic version for open varieties, $\\bar\\kappa(U)\\ge\\bar\\kappa(F)+\\bar\\kappa(V)$, and Campana conjectured an orbifold form in which rational boundaries and multiple fibres enter through an orbifold base: $\\kappa(X,K_X+\\Delta)\\ge\\kappa(F,K_F+\\Delta_F)+\\kappa(f\\mid\\Delta)$ for compact manifolds in Fujiki class $\\mathcal C$. Known cases included curve bases and fibres with good minimal models (Kawamata), general-type bases (Viehweg) and fibres (Kollar), total dimension at most six (Birkar), and abelian or maximal-Albanese bases. Does subadditivity hold for every such fibration?","posedBy":"Shigeru Iitaka (J. Math. Soc. Japan 1971; Classification of Algebraic Varieties, Proc. Japan Acad. 1977); orbifold form by F. Campana (J. Inst. Math. Jussieu 2011, Conj. 6.1)","yearPosed":1977,"ageNote":"The manuscript traces the problem to Iitaka's 1971 D-dimension paper and cites the explicit logarithmic conjecture to his 1977 note; the ordinary C(n,m) form is from the same period. Campana's orbifold conjecture is in the 2011 journal version of his arXiv 0705.0737.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a smooth compact connected manifold $X$ in Fujiki class $\\mathcal C$ with rational SNC boundary $\\Delta$ (coefficient one allowed) and a fibration $f:X\\to Y$ onto a normal compact complex space, $\\kappa(X,K_X+\\Delta)\\ge\\kappa(F,K_F+\\Delta_F)+\\kappa(f\\mid\\Delta)$ for a very general fibre. Corollaries: logarithmic subadditivity for compatible reduced-SNC pairs, and ordinary $\\kappa(X)\\ge\\kappa(F)+\\kappa(Y)$ for projective fibre spaces over any algebraically closed field of characteristic zero. Not covered: positive characteristic, real or non-SNC boundaries, and the variation refinement, which is a separate entry.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The orbifold paper is the family's foundation: the variation and additivity companions use its logarithmic subadditivity and adjoint comparison as inputs, and the September 27 companion gives a second, purely projective proof of the ordinary case over any algebraically closed field of characteristic zero. The family is not formalized except for one branch of the additivity companion.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the introduction of 'Orbifold and logarithmic Iitaka subadditivity', read against Campana's Conjecture 6.1 as the paper quotes it and against Iitaka's ordinary and logarithmic inequalities, which it derives as corollaries (ordinary over any algebraically closed field of characteristic zero via geometric generic fibres); the September 27 companion's Theorem 1.1 states the ordinary case with an independent projective proof. The proofs (relative Iitaka reduction, root covers, Hodge-line positivity, general-type addition) were not refereed. No Lean formalization. The papers note earlier preprint claims of ordinary subadditivity by Tsuji and Maehara and say neither is an input. A reader should know that two later family-034 manuscripts (4 and 5 October) state logarithmic or orbifold Iitaka subadditivity as an explicit assumption; the 5 October one identifies that assumption with Theorem 1.1 of this paper, so the release does not contradict this claim.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"Iitaka's subadditivity conjecture is one of the central open problems of birational classification, open for about fifty years with a long list of partial cases. Placed below the log abundance entry (75), which together with the minimal model program would imply it, level with Cannon's conjecture (62) and Kaplansky's zero-divisor conjecture (62), and above Bloch's conjecture for surfaces (55) and the Campana-Peternell conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf","sourceName":"OpenAI math release: Orbifold and logarithmic Iitaka subadditivity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Projective Hodge lines and ordinary Iitaka subadditivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf","kind":"paper"},{"label":"Logarithmic Kodaira dimension and whole-fiber variation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf","kind":"paper"},{"label":"The reverse logarithmic Kodaira inequality and additivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf","kind":"paper"},{"label":"B-semiampleness for compact log-smooth Kahler fibrations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026/paper.pdf","kind":"paper"},{"label":"Campana 2011, Orbifoldes geometriques speciales (Conjecture 6.1)","url":"https://doi.org/10.1017/S1474748010000101","kind":"problem-record"},{"label":"Iitaka 1977, Classification of Algebraic Varieties","url":"https://doi.org/10.3792/pja/1195517881","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gaussian-moat-problem","name":"The Gaussian moat problem","shortName":"Gaussian moat problem","problemNumber":null,"field":"Gaussian primes and sieve methods","fieldGroup":"Number theory","statement":"A Gaussian prime is an irreducible element of $\\mathbb Z[i]$, viewed as a point of the plane. For a step bound $D$, join two Gaussian primes when their distance is at most $D$. Large prime-free disks exist with centers on any line (Gethner, Wagon, Wick), but a walk in the plane can go around them, and computations bound only the component of a particular starting region (Tsuchimura, steps up to 6 from the origin). Gethner and Stark (1997) formulated a version uniform in the starting point and proved it for step bounds $\\sqrt2$ and $2$ by periodic sieving. Is it impossible, for every finite $D$, to walk to infinity through distinct Gaussian primes with steps of length at most $D$?","posedBy":"Basil Gordon, at the 1962 International Congress of Mathematicians in Stockholm (per Gethner and Stark, Periodic Gaussian moats, Experimental Math. 6, 1997); Erdos credited Gordon and Motzkin","yearPosed":1962,"ageNote":"Gethner and Stark trace the question to Gordon at the 1962 ICM; Erdos (1977) recalled hearing it from Motzkin in November 1963 and credited Gordon and Motzkin.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every real $D$ there is a finite $B_D$ such that every connected component of the graph on Gaussian primes with edges of length at most $D$ has at most $B_D$ vertices, independent of the starting prime and including primes on the axes; so every walk through distinct Gaussian primes with steps at most $D$ has at most $B_D$ terms. The proof finds, for each $D$, a finite set of split primes whose joint sieve leaves a periodic set with no infinite bounded-step walk, using geometric sampling and entropy estimates. $B_D$ is nonexplicit, so no numerical moat width is given for any particular $D$.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorems 1.1 and 1.2 of the TeX source; the proof was not refereed. Lean-checked on the Comparator challenge GaussianMoat, listed in the release's formalization catalogue (OAI.GaussianMoat.fullMain, OAI/NumberTheory/GaussianMoat/Main.lean). Its statement has two parts: for every real $D$ there is no injective sequence of irreducible Gaussian integers with successive distances at most $D$; and for every real $D$ one natural number bounds the size of every connected component of the graph joining distinct Gaussian primes at distance at most $D$, and the length of every injective bounded-step walk. That is the headline claim, axis primes and associates included. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"A well-known problem open since 1962, famous for being easy to state rather than for its consequences. Scored with the CAT(0) conjecture for Artin groups (40), above the convergence of the Flint Hills series (35) and below Vinogradov's least quadratic nonresidue conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf","sourceName":"OpenAI math release: Bounded-Step Walks on Gaussian Primes","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/NumberTheory/GaussianMoat/Main.lean (fullMain)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/GaussianMoat/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kalton-bap-uniformly-discrete-lipschitz-free","name":"Kalton's question: does the Lipschitz-free space of every uniformly discrete metric space have the bounded approximation property?","shortName":"Kalton's BAP question for uniformly discrete free spaces","problemNumber":null,"field":"Banach space theory; Lipschitz-free spaces","fieldGroup":"Analysis","statement":"For a pointed metric space $(M,d,o)$, the Lipschitz-free space $\\mathcal F(M)$ is the closed span of the evaluations $\\delta_x$ in $\\mathrm{Lip}_0(M)^*$. Kalton showed that if $M$ is uniformly discrete ($d(x,y)\\ge\\theta>0$ for $x\\ne y$) then $\\mathcal F(M)$ has the approximation property, and Dalet proved the metric approximation property for countable proper spaces. Kalton asked, and Godefroy-Ozawa (Question 1) and Godefroy's survey (Problem 6.2) restated: if $M$ is uniformly discrete, must $\\mathcal F(M)$ have the bounded approximation property, that is, can the finite-rank approximants of the identity be chosen with a uniform norm bound?","posedBy":"Nigel Kalton (Collectanea Mathematica 2004, remarks after Proposition 4.4); restated by Godefroy and Ozawa (2014, Question 1) and Godefroy (2015 survey, Problem 6.2)","yearPosed":2004,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a countable, unbounded, nonproper pointed metric space with $d(x,y)\\ge1$ for distinct points whose real Lipschitz-free space has AP but fails $\\lambda$-BAP for every finite $\\lambda$. It is built as a wedge of bounded blocks (Theorem 1.2) in which a finite anchor set defeats every finite-rank operator of norm at most $p$. Consequences stated in the paper: for each $p$, real $\\ell_1$ has an equivalent norm with $2B_p$-BAP failing $p$-BAP (for $p=1$ failing MAP), and $\\mathcal F(M)$ is a separable Banach space that is not approximable in Kalton's nonlinear sense (Kalton 2012, Problem 1). Real scalars only; the proper countable case is excluded by Dalet's theorem.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorems 1.1 and 1.2 were read against Kalton's question. lean/docs/330.md points to ComparatorChallenges/DiscreteLipschitzFree.json (theorem OAI.DiscreteFree.source_endpoints, solution module OAI.Analysis.LipschitzFree.Main); the solution file exists at the pinned commit and the challenge is not in formalization.yaml. The statement was read here: the block theorem together with a main statement asserting a countable pointed metric space with all distinct points at distance at least 1, unbounded and not proper, whose real free space (built from scratch in the file) has AP and fails $\\Lambda$-BAP for every $\\Lambda\\ge1$. That is the headline claim. Not rebuilt here. A separate challenge, RealL1Renorming, covers the quantitative renorming of $\\ell_1$.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"A named question of Kalton restated in the main survey of the field, well known to specialists in Lipschitz-free spaces but narrower than the field's central problems. Placed below kalton-problem-2-bi-lipschitz-c0 (28) and separable-banach-lipschitz-isomorphism-problem (52), and above zhu-fang-shi-invariant-projection-question (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf","sourceName":"OpenAI math release: A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.DiscreteFree.source_endpoints)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/LipschitzFree/Main.lean","kind":"lean-proof"},{"label":"Godefroy, A survey on Lipschitz-free Banach spaces (2015), Problem 6.2","url":"https://doi.org/10.14708/cm.v55i2.1104","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"anticanonical-nonvanishing-semipositive","name":"Anticanonical nonvanishing for smooth projective varieties with smoothly semipositive anticanonical bundle","shortName":"Anticanonical nonvanishing (semipositive)","problemNumber":null,"field":"Complex algebraic geometry; Kahler geometry","fieldGroup":"Geometry & topology","statement":"Let $X$ be a smooth connected complex projective variety whose anticanonical bundle $-K_X$ carries a smooth Hermitian metric of semipositive curvature (equivalently, by Yau's theorem, a Kahler metric of nonnegative Ricci curvature in that class). Such $-K_X$ is nef. Demailly-Peternell-Schneider and Campana-Demailly-Peternell described the universal cover of such $X$, but the existence of anticanonical sections was open in general; for nef log-anticanonical divisors it was known only in dimension three and under fiber semiampleness. The manuscript attaches the question to Yau's Problem 75. Does $H^0(X,-mK_X)\\ne0$ for some integer $m>0$?","posedBy":"Shing-Tung Yau, Open problems in geometry (Lectures on Differential Geometry, 1994), Problem 75, as the manuscript cites it","yearPosed":1994,"ageNote":"The manuscript calls its theorem 'the positive-multiple formulation associated with' Yau's Problem 75; Yau's own wording was not available here and was not read.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $X$ is smooth connected complex projective and $-K_X$ admits a smooth Hermitian metric of semipositive curvature, then $H^0(X,-mK_X)\\ne0$ for some $m>0$. The key intermediate result (Theorem 1.2) gives sections of $aL+bD$ on varieties without holomorphic forms from a finite-volume adjoint metric. Not shown: an effective bound on $m$, the case $m=1$ (false in general, e.g. Enriques surface times $\\mathbb P^1$), or nonvanishing when $-K_X$ is only nef or carries a singular semipositive metric.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has seven manuscripts, all dated September 26, 2026. The principal one proves the headline from a finite-volume theorem by induction on dimension and a torus-invariant section on the rationally connected factor; companions give invariant indices, klt and torus-quotient versions, metric descent and effectivity with controlled boundary. One cited 2026 paper by other authors (Chen-Filip-Sun-Tosatti-Zhang) records that ChatGPT suggested a preliminary lemma; that is outside this release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of 'Anticanonical nonvanishing from smooth semipositivity' were read; the theorem states exactly the displayed question. The posing reference is Yau's 1994 problem list as the manuscript cites it; Yau's text was not obtained, so whether Yau asked for $m=1$, for a positive multiple, or under nef rather than smooth semipositive hypotheses was not confirmed (the paper's Enriques surface times $\\mathbb P^1$ example shows that $m=1$ can fail). The release has no Lean formalization for this family (lean/docs/068.md does not exist at the pinned commit). Companions were skimmed, not refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A natural question from Yau's problem list about Kahler manifolds of nonnegative Ricci curvature, sitting below the broader open nonvanishing problem for nef anticanonical divisors. Specialist interest. Placed below the Campana-Peternell conjecture on Kodaira dimension (35), near the Campana-Peternell conjecture for nef tangent bundle in dimension six (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf","sourceName":"OpenAI math release: Anticanonical nonvanishing from smooth semipositivity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Invariant anticanonical indices and conversion of twisted differentials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion: Bounded anticanonical metrics on klt pairs and torus quotients","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion: Cohomological transfer and equivariant anticanonical sections","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion: Metric descent and rank-preserving contractions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion: Integrable metrics and effectivity with controlled boundary","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion: Exact orders and invariant anticanonical linear systems","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.pdf","kind":"paper"},{"label":"Yau, Open problems in geometry (1994), publisher preview","url":"https://www.intlpress.com/site/pub/files/preview/bookpubs/00000308.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sle-exact-hausdorff-gauge","name":"Schramm's Hausdorff-measure problem for SLE: an exact gauge function for the chordal SLE trace, 0 < kappa < 8","shortName":"Exact Hausdorff gauge for SLE","problemNumber":null,"field":"Schramm-Loewner evolution; fractal geometry of random curves","fieldGroup":"Probability & statistics","statement":"For $0<\\kappa<8$ the chordal $\\mathrm{SLE}_\\kappa$ trace has Hausdorff dimension $d=1+\\kappa/8$ (Beffara 2008), and Rezaei (2018) proved its $d$-dimensional Hausdorff measure is zero. For a gauge $h$ (continuous, nondecreasing, $h(0)=0$) let $\\mathcal H^h$ be the corresponding Hausdorff measure. Schramm, in his 2006 ICM collection of problems (Problem 7.1), asked whether the SLE trace has a sigma-finite Hausdorff measure and suggested $r^d\\log\\log(1/r)$ as a possible gauge. Is there a deterministic gauge $h$ such that almost surely $0<\\mathcal H^h(\\gamma([s,t]))<\\infty$ for nontrivial trace segments, and if so, which?","posedBy":"Oded Schramm, Conformally invariant scaling limits: an overview and a collection of problems (ICM 2006), Problem 7.1","yearPosed":2006,"ageNote":"Posed in Schramm's ICM 2006 address (arXiv February 2006); the proceedings appeared in 2007.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For fixed $0<\\kappa<8$, $d=1+\\kappa/8$, the gauge $h(r)=r^d(\\log\\log(1/r))^{(2-d)/2}$ satisfies almost surely $0<\\mathcal H^h(\\gamma([s,t]))<\\infty$ for all $0 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: 'An exact Hausdorff gauge for SLE' (September 25, 2026) answered the existence question with an implicitly defined moment-integral gauge; the principal manuscript (September 26, 2026) gives an explicit closed-form gauge with an independent proof and shows Schramm's suggested gauge is not sigma-finite.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against Schramm's Problem 7.1. The release has a Lean challenge (lean/ComparatorChallenges/SLELowerPositivity.json, declaration OAI.SLEExactGauge.sourceLowerMain_proved, solution module present at the pinned commit), not in the formalization catalogue. Its statement, read here, gives only the lower half: almost surely every gauge agreeing with $h$ near 0 gives positive measure to every positive-time segment. Finiteness, the expected bound in disks, and the companion's gauge are not formalized, so the headline is not covered and the tier stays unreviewed; not rebuilt here. Each $\\kappa$ is fixed; no uniformity in $\\kappa$ and no identification with natural parametrization is claimed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"One of the problems in Schramm's ICM list, of real interest to SLE specialists but a refinement of the known dimension rather than a structural question. Below the He-Schramm rigidity conjecture at 32 and the Rohde-Schramm FK-SLE conjecture at 52, and close to the honeycomb self-avoiding walk strip-crossing entry at 30.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf","sourceName":"OpenAI math release: An explicit exact Hausdorff gauge for SLE","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An exact Hausdorff gauge for SLE (Sept 25, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf","kind":"paper"},{"label":"Lean (lower positivity only): OAI/Probability/SLE/LowerPositivity.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/SLE/LowerPositivity.lean","kind":"lean-proof"},{"label":"Schramm, Conformally invariant scaling limits: an overview and a collection of problems","url":"https://arxiv.org/abs/math/0602151","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hahn-wilson-conjecture","name":"The Hahn-Wilson conjecture: fp-type at most n spectra are exactly the thick subcategory generated by BP","shortName":"Hahn-Wilson conjecture","problemNumber":null,"field":"Chromatic homotopy theory; stable homotopy theory","fieldGroup":"Geometry & topology","statement":"Fix a prime $p$. A bounded-below $p$-complete spectrum $X$ is fp if $H^*(X;\\mathbb F_p)$ is finitely presented over the Steenrod algebra; it has fp-type at most $n$ if $\\pi_*(V\\wedge X)$ is finite in total for a finite spectrum $V$ of type $n+1$ (Mahowald-Rezk). Hahn-Wilson redshift shows $K(\\mathrm{BP}\\langle n\\rangle)$ has fp-type $n+1$, and Lee-Pstragowski proved the statement below at height one. The Hahn-Wilson conjecture asks whether finite presentation plus a height bound characterizes finite constructions from truncated Brown-Peterson spectra: is $\\{X:\\ X\\text{ has fp-type at most }n\\}=\\mathrm{Thick}(\\mathrm{BP}\\langle n\\rangle_p^\\wedge)$, the closure under finite sums, shifts, cofiber sequences and retracts, for every $n$ and $p$?","posedBy":"Jeremy Hahn and Dylan Wilson (reported from 2021); written as Conjecture 1.1 in David Jongwon Lee and Piotr Pstragowski, The monochromatic Hahn-Wilson conjecture, Inventiones Mathematicae (2026)","yearPosed":2021,"ageNote":"The manuscript says Lee and Pstragowski give the written formulation (Conjecture 1.1 and Note 1) and report learning it from Dylan Wilson in 2021; yearPosed uses 2021. The first written statement is 2026.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that for every sufficiently large prime $p$ there is a connective $p$-complete spectrum $X$ of exact fp-type two with $X\\notin\\mathrm{Thick}(\\mathrm{BP}\\langle2\\rangle_p^\\wedge)$ for a standard form of $\\mathrm{BP}\\langle2\\rangle$, refuting the universally quantified conjecture at height two. The same $X$ satisfies $L_2^fX\\simeq L_2X$ and $L_{T(2)}X\\simeq L_{K(2)}X$, so the obstruction lies in how the chromatic pieces are glued, not in a telescope-type failure; the monochromatic form proved by Lee-Pstragowski is untouched. It does NOT treat small primes, heights above two, or the height-one case, which Lee-Pstragowski proved.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. No Lean formalization of this result is in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. The abstract, introduction and main theorem were read against the conjecture as the manuscript quotes it from Lee-Pstragowski; that paper was not opened here. The counterexample is for every sufficiently large prime (no explicit bound read here) and uses one specific regular quotient form of $\\mathrm{BP}\\langle2\\rangle_p^\\wedge$; the paper relies on Lee (2026), with Angeltveit-Lind, for independence of the underlying spectrum from the choice of generators at these primes. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A recent structural conjecture in chromatic homotopy theory, circulated since 2021 and written down only in 2026, with the height-one case just proved; refuting it at height two matters to specialists but the conjecture had little time to become established. Below the chromatic splitting conjecture (40) and the Smith-Toda complexes entry (30), above the Hodge bundle simplicity entry (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf","sourceName":"OpenAI math release: Counterexamples to the Hahn-Wilson conjecture at height two","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lee and Pstragowski, The monochromatic Hahn-Wilson conjecture (Inventiones 2026)","url":"https://doi.org/10.1007/s00222-026-01448-y","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kalton-problem-2-bi-lipschitz-c0","name":"Kalton's Problem 2: does a bi-Lipschitz copy of $c_0$ force a linear copy of $c_0$?","shortName":"Kalton's c0 embedding problem","problemNumber":null,"field":"Nonlinear geometry of Banach spaces","fieldGroup":"Analysis","statement":"Godefroy, Kalton and Lancien (2000) proved that a Banach space bi-Lipschitz equivalent to $c_0$, or to a linear subspace of $c_0$, is linearly isomorphic to such a space. Those theorems concern an equivalence of whole spaces and say nothing about a space that merely contains a bi-Lipschitz image of $c_0$. Kalton recorded that question as Problem 2 in his 2008 survey of the nonlinear geometry of Banach spaces, and Hajek, Johanis and Schlumprecht described it as apparently open in 2024. If a Banach space contains a bi-Lipschitz image of $c_0$, must it contain a closed linear subspace isomorphic to $c_0$?","posedBy":"Nigel J. Kalton, The nonlinear geometry of Banach spaces, Rev. Mat. Complut. 21 (2008), Problem 2","yearPosed":2008,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a separable real Banach space $Z$ with no closed subspace isomorphic to $c_0$ and an onto bi-Lipschitz map $F:Z\\oplus_\\infty c_0\\to Z$. Restricting $F$ to the $c_0$ summand embeds $c_0$ bi-Lipschitzly in $Z$, so Problem 2 has a negative answer. With Aharoni's theorem, $Z$ is bi-Lipschitz universal for separable metric spaces, and $Z$, $Z\\oplus_\\infty c_0$ form a second separable Lipschitz-isomorphic, linearly non-isomorphic pair. Real scalars only; it does not address analogous questions for other classical spaces.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Kalton's Problem 2 as the manuscript quotes it; Kalton's survey itself was not opened here. Lean-checked on the Comparator challenge C0Absorption (OAI.C0Absorption.main_result, OAI/Analysis/C0Absorption/Main.lean), listed in the release's formalization catalogue. Its statement, read here, asserts a complete separable real normed space $Z$ with no bounded-below linear map from $c_0$, a surjective bi-Lipschitz map $Z\\times c_0\\to Z$ (sup-norm product), a bi-Lipschitz map $c_0\\to Z$, bi-Lipschitz universality for separable metric spaces, and no continuous linear equivalence $Z\\times c_0\\simeq Z$: the headline claim in full. Permitted axioms: propext, Quot.sound, Classical.choice. The statement was not independently audited and the development was not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A named problem from Kalton's 2008 list, well known to specialists in nonlinear Banach space theory but narrower than the separable Lipschitz isomorphism problem (52, its sibling entry). Placed just above sparse domination implies convex body domination (27), the Hilbert transform-UMD dependence problem (25) and Kalton-Peck space hyperplanes (25), below Talagrand's operator cotype problem (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf","sourceName":"OpenAI math release: Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean: OAI/Analysis/C0Absorption/Main.lean (main_result)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/C0Absorption/Main.lean","kind":"lean-proof"},{"label":"Comparator statement C0Absorption.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/C0Absorption.lean","kind":"lean-proof"},{"label":"Lean scope note for family 324","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/324.md","kind":"other"},{"label":"Kalton, The nonlinear geometry of Banach spaces (2008)","url":"https://kaltonmemorial.missouri.edu/assets/docs/rmc2008.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"scale-implies-conformal-4d-operational-qft","name":"Scale invariance implies local conformal invariance for a class of four-dimensional unitary QFTs","shortName":"Scale vs conformal invariance in 4D","problemNumber":null,"field":"Axiomatic quantum field theory: scale and conformal symmetry","fieldGroup":"Mathematical physics","statement":"In a scale-invariant QFT with symmetric conserved stress tensor $T_{\\mu\\nu}$, the dilatation current is $D_\\mu=x^\\nu T_{\\mu\\nu}-V_\\mu$ with $T^\\mu{}_\\mu=\\partial^\\mu V_\\mu$; the theory is conformal if $T$ can be improved to a traceless tensor, for instance when $\\partial^\\mu V_\\mu=\\Box L$ for a local scalar $L$. Zamolodchikov and Polchinski (1988) proved that scale implies conformal invariance in two-dimensional unitary theories under mild assumptions, and Polchinski raised the four-dimensional case. Perturbative results (Fortin-Grinstein-Stergiou, Jack-Osborn) and dilaton-based arguments (Luty-Polchinski-Rattazzi; Dymarsky-Komargodski-Schwimmer-Theisen) support it but leave gaps. In a four-dimensional unitary, positive-energy QFT with a Poincare- and scale-invariant vacuum, does scale invariance imply conformal invariance?","posedBy":"J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B 303 (1988)","yearPosed":1988,"ageNote":"Polchinski's 1988 paper proved the two-dimensional case and discussed the four-dimensional question; the manuscript cites it with Zamolodchikov (1986) for the history. The question is also linked to earlier improvement work of Callan, Coleman and Jackiw (1970).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: under the paper's framework (unitary, positive energy, Poincare- and scale-invariant vacuum, discrete bounded-below dimension spectrum with finite multiplicities, finite scaling support, bounded local net, physical stress tensor and virial current), there is a Hermitian dimension-two physical scalar $L$ with $\\partial^\\mu V^{(3)}_\\mu=\\Box L$, so $T^{(4)}_{\\mu\\nu}+\\tfrac13(\\partial_\\mu\\partial_\\nu-\\eta_{\\mu\\nu}\\Box)L$ is symmetric, conserved and traceless with unchanged translation and Lorentz charges, and conformal currents have unbroken local Ward identities. It does not treat theories outside these hypotheses (continuous spectra, no bounded net, no local virial current), and integration to a global conformal action on the net is left open.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the scale-versus-conformal question as described in the manuscript's history section; the proof was not refereed. The theorem is conditional on the paper's standing framework assumption: a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities, finite scaling support of every field, a causally commuting bounded local net with compatible affiliated field realizations, and a physical local dilatation current with local Ward identities. These hypotheses delimit the class and exclude the known free-field subtleties (e.g. shift-invariant scalar, noncompact two-form) by requiring the virial current to be a physical field. The conclusion is local; a global conformal action is not claimed. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The four-dimensional scale-versus-conformal question is a well-known open problem of quantum field theory, but this answers it only inside an axiomatic class whose hypotheses carry much of the weight, so it is scored as a conditional special case well below the parent. Placed with the lonely runner conjecture for nine and ten runners (30), below Cardy's formula for square-lattice bond percolation (48) and the partial Rohde-Schramm result (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf","sourceName":"OpenAI math release: Scale and conformal symmetry in four-dimensional operational quantum field theory","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Polchinski, Scale and conformal invariance in quantum field theory (Nucl. Phys. B 1988)","url":"https://doi.org/10.1016/0550-3213(88)90179-4","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"honeycomb-saw-three-quarter-exponent-diameter","name":"Nienhuis's 3/4 exponent for self-avoiding walk on the honeycomb lattice, in diameter form","shortName":"Honeycomb SAW 3/4 exponent (diameter)","problemNumber":null,"field":"Self-avoiding walk; two-dimensional critical phenomena","fieldGroup":"Probability & statistics","statement":"Let $\\mathbb P_n$ be the uniform law on $n$-step self-avoiding walks from a fixed vertex of a planar lattice. Nienhuis's Coulomb-gas analysis of the dilute $O(n)$ model at $n=0$ (1982) predicts the size exponent $\\nu=3/4$: a typical $n$-step walk has spatial extent $n^{3/4}$, equivalently its trace has mass dimension $4/3$; Lawler, Schramm and Werner's conjectured $SLE_{8/3}$ scaling limit gives the same value. On the honeycomb lattice Duminil-Copin and Smirnov proved that the connective constant is $\\sqrt{2+\\sqrt2}$, but no polynomial size exponent was known; the best fixed-length bound was sub-ballistic (Krachun-Panagiotis: within $Cn/\\log n$). Does a uniform $n$-step self-avoiding walk on the honeycomb lattice have spatial extent $n^{3/4+o(1)}$?","posedBy":"Bernard Nienhuis (prediction from the dilute O(n) model)","yearPosed":1982,"ageNote":"The manuscript cites Nienhuis, Phys. Rev. Lett. 49 (1982), for the exponent 3/4 at n=0. The prediction concerns end-to-end distance and other size statistics; this paper treats diameter, local mass and covering numbers.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $\\delta>0$ and $k>0$ there are $C_{\\delta,k}$ and $n_0$ such that for every integer $n\\ge n_0$, with $\\mathbb P_n$-probability at least $1-C_{\\delta,k}n^{-k}$, the diameter satisfies $n^{3/4-\\delta}\\le D(\\gamma)\\le n^{3/4+\\delta}$, and simultaneously for all visited $z$ and $s\\in[1,n]$ the ball mass is within $n^{\\pm\\delta}$ of $\\min\\{n,s^{4/3}\\}$ and the covering number by $s$-balls within $n^{\\pm\\delta}$ of $1+ns^{-4/3}$. Corollaries: span at least $n^{3/4-\\delta}$ in each lattice normal, covering dimension $4/3$, radius of gyration $n^{3/4+o(1)}$ and $\\mathbb E_n[D^p]=n^{3p/4+o(1)}$. Not shown: a fixed-length lower bound for the end-to-end distance (a companion has it only on a density-one set of lengths), an exponent without $n^{o(1)}$ slack, other lattices, or convergence to $SLE_{8/3}$.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has thirteen manuscripts, all dated September 26, 2026. The principal manuscript proves the fixed-length geometric statement from analytic inputs (strip-crossing mass, nesting around faces, marked-polygon and arc bounds) whose proofs are in the twelve companion manuscripts.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of 'Mass and covering exponents for fixed-length honeycomb walks' were read against Nienhuis's prediction as the manuscript cites it. Theorem 1.1 gives, for every $\\delta,k>0$ and every large integer $n$, an event of probability at least $1-Cn^{-k}$ on which the diameter lies in $[n^{3/4-\\delta},n^{3/4+\\delta}]$ and local mass and covering numbers have exponent $4/3$. The proof imports analytic inputs proved in twelve companion manuscripts, which were skimmed, not refereed. The release's Lean formalization for this family (HoneycombBridgeFiniteness, HoneycombFreeEnergy, CriticalStripMass, and HoneycombBridgeMassSupport added on 7 October 2026: strict bridge mass of order $(1+h)^{-1/4}$ from a fixed port) covers supporting statements only. Its scope note says the mean-length and 3/4 spatial laws are outside it, so this entry is unreviewed, not Lean-checked. The paper states that it gives no lower bound on the end-to-end distance at fixed length, no scaling limit and no universality across lattices.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Nienhuis's 3/4 exponent is the central prediction for planar self-avoiding walk, open since 1982; no polynomial size exponent had been proved on any lattice. This is the diameter form on the honeycomb lattice with n^o(1) slack, so it sits below the full prediction. Placed below the Benjamini-Schramm p_c < p_u conjecture (66), above the Rohde-Schramm FK-SLE conjecture (52) and Cardy's formula for square-lattice bond percolation (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf","sourceName":"OpenAI math release: Mass and covering exponents for fixed-length honeycomb walks","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion manuscript: Radial transfer estimates and polygon length laws for honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Critical honeycomb chords with prescribed boundary endpoints","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Cylinder loop weights and planar nesting","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Signed cylinder propagation and marked polygons on the honeycomb lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Marked polygon correlations and one-arc bounds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Disk transfer representations and confined bridge mass","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Polynomial vacuum representations and bridge mass for honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Renewal and changes of law for critical honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Critical strip-crossing mass on the honeycomb lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Uniform marked-polygon estimates and sharp finite bridge moments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Cap-selected amplitudes and triangle chords for honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026/main.pdf","kind":"paper"},{"label":"Lean statement (supporting): HoneycombBridgeFiniteness","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HoneycombBridgeFiniteness.lean","kind":"lean-proof"},{"label":"Lean statement (supporting): HoneycombFreeEnergy","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HoneycombFreeEnergy.lean","kind":"lean-proof"},{"label":"Lean statement (supporting): CriticalStripMass","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CriticalStripMass.lean","kind":"lean-proof"},{"label":"Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions (1982)","url":"https://doi.org/10.1103/PhysRevLett.49.1062","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"log-concave-sampling-query-exponent","name":"The dimension exponent of first-order query complexity for well-conditioned log-concave sampling","shortName":"Log-concave sampling query exponent","problemNumber":null,"field":"Sampling algorithms; oracle complexity","fieldGroup":"Theoretical computer science","statement":"Let $V\\in C^2(\\mathbb R^d)$ with $V(0)=0$, $\\nabla V(0)=0$ and $I_d\\preceq\\nabla^2V\\preceq2I_d$, and let $\\pi_V\\propto e^{-V}$. An algorithm queries $(V(x),\\nabla V(x))$ adaptively and must output a sample within total variation $\\varepsilon$ of $\\pi_V$. Chewi's book Log-Concave Sampling calls it a fundamental open question to determine, up to a universal constant, the minimum number of first-order queries needed uniformly over this class. Known upper bounds grew polynomially in $d$ (about $d^{1/2}$ for Langevin-type methods, $\\widetilde O(d^{1/3})$, then $\\widetilde O(d^{1/6})$ expected queries), and the lower bounds of Chewi et al. were logarithmic. What is the query complexity, and in particular, is a positive power of $d$ necessary at fixed accuracy?","posedBy":"Sinho Chewi, Log-Concave Sampling (book draft), preface: 'a fundamental open question about the complexity of log-concave sampling'","yearPosed":null,"ageNote":"The manuscript cites no posing source. The statement was read in the preface of the current draft of Chewi's book (version dated August 20, 2026); the year the question first appeared in the draft was not established here, so yearPosed is left null. Chewi, de Dios Pont, Li, Lu and Narayanan (2023) proved the first dimension-dependent lower bounds for this class.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: in the exact first-order oracle model with unrestricted computation between queries, for every $\\varepsilon>0$ the worst-case query budget $Q(d)$ for TV error $1/10$ satisfies $Q(d)\\le C_\\varepsilon d^{\\varepsilon}$ for all $d\\ge2$, on every execution; and $Q(d)\\ge c\\log d$ for arbitrary randomized adaptive algorithms. Hence the optimal dimension exponent is $\\gamma_*=0$. Not shown: the growth rate of $Q(d)$ beyond the exponent (no polylogarithmic bound is claimed), dependence on the accuracy or condition number, or any running-time bound; the constants and approximation orders depend on $\\varepsilon$.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript dated September 26, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against the question as stated in Chewi's book. The book asks for the complexity up to a universal constant at accuracy $\\varepsilon$; the paper fixes TV accuracy 1/10 and determines only the dimension exponent, so this entry is partial. Lean: formalization.yaml lists ComparatorChallenges/LogConcaveQuery.json with declaration OAI.LogConcaveSampling.exact_source_main in OAI/Probability/LogConcave/Main.lean. The challenge statement was read: for admissible potentials (C2, V(0)=0, gradient zero at 0, Hessian between I and 2I), measurable randomized adaptive algorithms with a deterministic query budget, TV at most 1/10, it asserts queryComplexity(d) <= C d^eps for every eps>0, >= c log d eventually, and gammaStar = 0. That is the headline claim. Not rebuilt here. The paper states that computation between queries is unrestricted and no arithmetic or bit cost is bounded.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"The question is central to the theory of sampling algorithms but specialist; the result settles the dimension exponent, not the full complexity the question asks for. Placed below the KLS conjecture for quadratic forms (45) and the randomized versus quantum query conjecture (38), above the oracle-complexity gap in derivative-free convex optimization (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026/article.pdf","sourceName":"OpenAI math release: Subpolynomial query complexity for well-conditioned log-concave sampling","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean challenge statement: LogConcaveQuery.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/LogConcaveQuery.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Probability/LogConcave/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/LogConcave/Main.lean","kind":"lean-proof"},{"label":"Chewi, Log-Concave Sampling (book draft), preface states the open question","url":"https://chewisinho.github.io/main.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"honeycomb-saw-strip-crossing-mass-exponent","name":"The $N^{-1/4}$ law for the critical strip-crossing mass of honeycomb self-avoiding walk","shortName":"Honeycomb SAW strip mass exponent -1/4","problemNumber":null,"field":"Self-avoiding walk; parafermionic observables","fieldGroup":"Probability & statistics","statement":"On the honeycomb lattice at the critical weight $x_c=1/\\sqrt{2+\\sqrt2}$, let $B_N$ be the total weight of self-avoiding paths from a fixed boundary point of a strip of height $N$ to its opposite side. Duminil-Copin and Smirnov's parafermionic identity, which proved the value of the connective constant, gives $c/N\\le B_N\\le 1$. Lawler, Schramm and Werner's boundary exponent $5/8$ predicts a boundary-to-boundary mass of power $-5/4$, and summing over the far side predicts $B_N\\approx N^{-1/4}$; Duminil-Copin and Smirnov state this prediction explicitly (Annals 2012, Section 4). Beaton, Bousquet-Melou, de Gier, Duminil-Copin and Guttmann proved $B_N\\to0$, and Krachun and Panagiotis a polynomial upper bound with exponent $10^{-10}$. Is the critical strip-crossing mass of order $N^{-1/4}$?","posedBy":"Hugo Duminil-Copin and Stanislav Smirnov (explicit prediction, from the Lawler-Schramm-Werner boundary exponent)","yearPosed":2012,"ageNote":"The manuscript cites Duminil-Copin and Smirnov, Ann. of Math. 175 (2012), Section 4, for the explicit prediction; the exponent goes back to Nienhuis (1982) and Lawler-Schramm-Werner (2004).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $N\\ge1$ the strip sums are finite, $c\\mathcal A_N+\\mathcal B_N=1$ with $c=\\cos(3\\pi/8)$, $m_{N+1}-m_N\\asymp\\mathcal B_N$, the first rightward displacement moment of arches satisfies $m_N\\asymp N^{3/4}$, and the bridge mass satisfies $\\mathcal B_N\\asymp N^{-1/4}$, nonincreasing in $N$, with constants independent of $N$. The route computes $m_N$ through rhombic transfer matrices and a positive integral, then recovers $\\mathcal B_N$ from its increments. Not shown: a limiting constant for $N^{1/4}\\mathcal B_N$, statements about walks of fixed length (treated in companions), or other lattices.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript, 'Critical strip-crossing mass on the honeycomb lattice' (September 26, 2026), belongs to a thirteen-manuscript family on honeycomb walks; several companions re-derive the same bridge-mass law in other conventions and use it as input for fixed-length size exponents.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Critical strip-crossing mass on the honeycomb lattice' was read against the prediction as Duminil-Copin and Smirnov state it; it gives $B_N\\asymp N^{-1/4}$ with constants independent of $N$, in the manuscript's port convention. Lean: lean/ComparatorChallenges/CriticalStripMass.json exists and its solution module OAI.Combinatorics.StripMass.Main is present at the pinned commit; the challenge is not in the formalization catalogue (formalization.yaml). The statement file CriticalStripMass.lean was read here: theorem critical_strip_mass asserts finiteness of the strip sums, $c\\,A_N+B_N=1$, $m_N$ comparable to $N^{3/4}$, bridge mass comparable to $N^{-1/4}$ for all $N\\ge1$, and monotonicity, with paths encoded as duplicate-free chains of adjacent triangles of the dual triangular lattice. That states the headline. Not rebuilt here, and the match between the triangle encoding and the paper's port convention was read but not audited line by line.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A precisely stated quantitative prediction in the parafermionic program for honeycomb self-avoiding walk, recorded by Duminil-Copin and Smirnov in 2012 and attacked by Beaton et al., Glazman-Manolescu and Krachun-Panagiotis. Famous within one community. Level with the Ellipsoid Fitting Conjecture (30), below Feige's Conjecture (35) and well below Cardy's formula for square-lattice bond percolation (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf","sourceName":"OpenAI math release: Critical strip-crossing mass on the honeycomb lattice","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion manuscript: Polynomial vacuum representations and bridge mass for honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Disk transfer representations and confined bridge mass","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Uniform marked-polygon estimates and sharp finite bridge moments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion manuscript: Mass and covering exponents for fixed-length honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf","kind":"paper"},{"label":"Lean solution module: OAI/Combinatorics/StripMass/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/StripMass/Main.lean","kind":"lean-proof"},{"label":"Lean statement: ComparatorChallenges/CriticalStripMass.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CriticalStripMass.lean","kind":"lean-proof"},{"label":"Duminil-Copin and Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2) (2012)","url":"https://doi.org/10.4007/annals.2012.175.3.14","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"effective-log-iitaka-finite-coefficients","name":"The effective log Iitaka conjecture for log canonical pairs with finite rational coefficient sets","shortName":"Effective log Iitaka, finite coefficients","problemNumber":null,"field":"Birational geometry; pluricanonical systems","fieldGroup":"Algebra","statement":"Chen, Han and Liu conjectured a logarithmic version of effective Iitaka fibrations: for fixed dimension $d$ and a DCC set $\\Phi\\subset[0,1]$, one integer $m$ should make the rounded system $|\\lfloor m(K_X+B)\\rfloor|$ nonempty with section ratios generating the Iitaka field, for every projective lc pair $(X,B)$ of dimension $d$ with coefficients in $\\Phi$ and nonnegative Kodaira dimension. They proved it in dimension at most three. Does it hold in every dimension when $\\Phi$ is a finite set of rationals?","posedBy":"G. Chen, J. Han and J. Liu, On effective log Iitaka fibrations and existence of complements (IMRN 2024; arXiv 2301.04813), Conjecture 1.2","yearPosed":2023,"ageNote":"Posed in the January 2023 arXiv version; published in IMRN in 2024.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"'Uniform effective log Iitaka fibrations for fourfolds' (September 26): for normal projective lc complex fourfolds with coefficients in a fixed finite rational set and pseudo-effective rational Cartier adjoint, one degree makes the rounded system nonempty with section ratios generating the Iitaka field; it also bounds the canonical index of klt Calabi-Yau fourfolds. 'Uniform log Iitaka fibrations and bounded moduli denominators' (October 4): the same conclusion for $d\\ge5$ over any algebraically closed field of characteristic zero, with bounded Cartier denominators for moduli divisors. Not shown: the DCC case of the conjecture in dimension four or more.","claimIssueNote":"This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it.","solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. Family 034 has fourteen manuscripts dated September 24 to October 5, 2026, which cite one another as inputs; this entry draws on the manuscript named as its source. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorems of the two manuscripts were read against Conjecture 1.2 as both describe it. Both papers say their coefficient scope (finite rational sets) is narrower than the DCC conjecture, hence partial. The proofs were not refereed. No Lean formalization. Both rely on the family's abundance, nonvanishing and index results, all unreviewed. Listed as Contested because it depends on the log abundance entry, which the release itself is inconsistent about; see the claim issue.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A recent conjecture (2023) known within birational geometry, of which only the finite-coefficient part is answered. Placed level with the finite field restriction entry (20), below the finite generation for klt generalized pairs entry (22) and the index conjecture from this family (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026/paper.pdf","sourceName":"OpenAI math release: Uniform effective log Iitaka fibrations for fourfolds","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion (d at least 5): Uniform log Iitaka fibrations and bounded moduli denominators","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026/uniform-log-iitaka.pdf","kind":"paper"},{"label":"Relative denominators and effective systems for log Calabi-Yau fibrations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026/paper.pdf","kind":"paper"},{"label":"Chen, Han and Liu (arXiv 2301.04813), Conjecture 1.2","url":"https://arxiv.org/abs/2301.04813","kind":"problem-record"},{"label":"The 5 October appendix calling projective log abundance conditional","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026/","kind":"paper"},{"label":"Revised edition (October 6, 2026): Conditional good minimal models for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"free-ising-tree-factor-of-iid-threshold","name":"The Nam-Sly-Zhang conjecture: the free Ising state on the d-regular tree is a factor of IID up to the reconstruction threshold","shortName":"Free Ising on trees as a factor of IID","problemNumber":null,"field":"Ergodic theory on trees; Ising model, factors of IID","fieldGroup":"Probability & statistics","statement":"Let $T_d$ be the infinite $d$-regular tree, $d\\ge3$, and $\\mu_{d,\\beta}$ the free zero-field ferromagnetic Ising measure with $\\theta=\\tanh\\beta$. A spin law is a factor of IID if it is the law of $\\Phi(U)$ for IID uniform vertex labels $U$ and a measurable $\\Phi$ commuting with tree automorphisms. Lyons (2017) studied which such Gibbs measures are factors of IID: the free state is one when $\\theta\\le1/(d-1)$, and Sly's argument shows it is not when $\\theta>(d-1)^{-1/2}$, beyond the reconstruction threshold. Nam, Sly and Zhang (arXiv 2020, CMP 2022) proved it is a factor for $\\theta\\le c/\\sqrt{d-1}$ and $d\\ge d_0$ and conjectured the sharp threshold. Is $\\mu_{d,\\beta}$ a factor of IID for every $d\\ge3$ whenever $\\tanh\\beta\\le(d-1)^{-1/2}$, including equality?","posedBy":"D. Nam, A. Sly and L. Zhang, Ising model on trees and factors of IID (arXiv 2012.09484, 2020; Comm. Math. Phys. 2022), following R. Lyons, Factors of IID on trees (2017)","yearPosed":2020,"ageNote":"The Nam-Sly-Zhang preprint appeared in December 2020 and was published in 2022; Lyons's broader question on factor-of-IID Gibbs measures on trees dates from arXiv 2014, published 2017.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $d\\ge3$ and $\\beta\\ge0$, $\\mu_{d,\\beta}$ is a factor of IID if and only if $\\tanh\\beta\\le(d-1)^{-1/2}$; the new part is the positive implication including equality, and the factor can be chosen to commute with every automorphism on every input. The converse is Sly's, as given by Lyons, with the Backhausz-Szegedy-Virag bound. The construction inverts the innovations of the Nam-Sly-Zhang stochastic-localization observation process. It does not treat antiferromagnetic or nonzero-field states, other Gibbs measures, or finitary factors.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the Nam-Sly-Zhang conjecture. The family is in lean/formalization.yaml (ComparatorChallenges/FreeIsing.json, declaration OAI.Problem367.free_ising_factor_iff_threshold, file OAI/Probability/FreeIsing/Main.lean). The comparator statement was read here: for every $d\\ge3$ and $\\beta\\ge0$, there is a measurable map of IID labels, almost surely equivariant for each fixed automorphism, whose image law matches the free Ising cylinder weights on connected finite sets, if and only if $\\tanh\\beta\\le1/\\sqrt{d-1}$. This is the headline, including the known converse. Not rebuilt here. Finitary coding and coding-radius bounds are not claimed, and only the ferromagnetic free state is treated.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"A sharp-threshold conjecture from a recent paper in the factor-of-IID theory on trees, known to a small community in probability and ergodic theory. Level with Schafhauser-Tikuisis-White Problem XXI at 15, below the Gardner transition entry at 28 and the Krauth-Mezard Ising perceptron entry at 35.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf","sourceName":"OpenAI math release: The sharp factor-of-IID threshold for the free Ising model on regular trees","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/Probability/FreeIsing/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/FreeIsing/Main.lean","kind":"lean-proof"},{"label":"Nam, Sly and Zhang, Ising model on trees and factors of IID (arXiv 2012.09484)","url":"https://arxiv.org/abs/2012.09484","kind":"problem-record"},{"label":"Lyons, Factors of IID on trees (arXiv 1401.4197)","url":"https://arxiv.org/abs/1401.4197","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"de-giorgi-conjecture-allen-cahn","name":"De Giorgi's conjecture on monotone solutions of the Allen-Cahn equation","shortName":"De Giorgi's conjecture","problemNumber":null,"field":"Elliptic PDE; Allen-Cahn equation and phase transitions","fieldGroup":"Differential equations","statement":"De Giorgi conjectured in 1978 that if $u\\in C^2(\\mathbb R^n)$ solves the Allen-Cahn equation $\\Delta u=u^3-u$ with $|u|\\le1$ and $\\partial_{x_n}u>0$ everywhere, then for $n\\le8$ the level sets of $u$ are hyperplanes, i.e. $u(x)=\\tanh((e\\cdot x-c)/\\sqrt2)$ for a unit vector $e$ and $c\\in\\mathbb R$. It is the phase-transition analogue of the Bernstein problem for minimal graphs. Ghoussoub-Gui proved it for $n=2$, Ambrosio-Cabre for $n=3$, Savin for $4\\le n\\le8$ under the extra assumption that $u(x',t)\\to\\pm1$ as $t\\to\\pm\\infty$, and del Pino-Kowalczyk-Wei gave counterexamples for $n\\ge9$. Is every such monotone solution one-dimensional in every dimension $n\\le8$, without the limit assumption?","posedBy":"Ennio De Giorgi","yearPosed":1978,"ageNote":"Stated at the International Meeting on Recent Methods in Nonlinear Analysis, Rome 1978; proceedings published by Pitagora, Bologna, 1979, as cited by the manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every $u\\in C^2(\\mathbb R^8,(-1,1))$ with $\\Delta u=u^3-u$ and $\\partial_8u>0$ equals $\\tanh((e\\cdot x-c)/\\sqrt2)$ with $e_8>0$. Theorem 1.2: every stable solution $v:\\mathbb R^7\\to[-1,1]$ is $\\pm1$ or planar, without an energy-growth assumption; this is sharp by Pacard-Wei's nonplanar stable solutions in $\\mathbb R^8$. A final section explains why the argument fails in dimension nine, consistent with del Pino-Kowalczyk-Wei. The paper does not treat other potentials, the stable De Giorgi problem in dimension eight, or the Gibbons and Savin variants beyond what the monotone theorem implies.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 26, 2026) has no companion in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the manuscript (TeX source) were read against De Giorgi's conjecture. Theorem 1.1 treats exactly $n=8$ with values in $(-1,1)$ and strict positivity of one partial derivative, with no limit or energy assumption. The paper states only the dimension-eight case; the lower dimensions $n\\le7$ follow by extending a solution constantly in the missing variables, an observation made here, not in the paper. The proof reduces the monotone case to Theorem 1.2 (every stable solution on $\\mathbb R^7$ is constant or planar, with no energy-growth hypothesis) using classical inputs (Alberti-Ambrosio-Cabre, Jerison-Monneau, Federer, Wang's near-unit-density theorem); the stable classification is the new part. Not refereed. No Lean formalization in the release.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"De Giorgi's 1978 conjecture is one of the best known problems in elliptic PDE and the calculus of variations, open in dimensions four to eight for over two decades after Savin's conditional proof. Placed level with Sinai's standard-map conjecture (58) and Yau's uniformization conjecture (58), below the Falconer distance conjecture (60), and above Sidorenko's conjecture (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf","sourceName":"OpenAI math release: A positive resolution of De Giorgi's conjecture in dimension eight","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"thorp-shuffle-logarithmic-mixing","name":"Optimal-order mixing of the Thorp shuffle on $2^d$ cards: is the mixing time $O(\\log n)$?","shortName":"Thorp shuffle mixing time","problemNumber":null,"field":"Markov chain mixing times; card shuffling","fieldGroup":"Probability & statistics","statement":"The Thorp shuffle on $n=2^d$ cards cuts the deck into two halves, pairs the cards in corresponding positions, orders each pair by an independent fair coin, and interleaves the pairs. Each card is uniform after $d$ shuffles and a counting argument gives a lower bound of order $d$ for the whole permutation, while the known upper bounds were $O(d^{44})$ (Morris 2008), $O(d^{29})$ (Montenegro-Tetali) and $O(d^3)$ (Morris 2013). Is the total-variation mixing time of the full permutation $O(d)=O(\\log n)$?","posedBy":"Line of work begun by Ben Morris (2008); the logarithmic order is discussed as the target in Johan Jonasson's notes (2009). The model is due to Edward O. Thorp (1973)","yearPosed":null,"ageNote":"The manuscript does not attribute the O(log n) question to a named poser. It cites Morris's first polynomial bound (SIAM J. Comput. 2008), later improvements, and Jonasson, 'The Mathematics of Card Shuffling' (course notes, 2009, Section 5), for the logarithmic order. yearPosed is left empty because no single posing date is given.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Principal Theorem 1.1: for $n=2^d$, $\\|q_d^{*1600d}-U_n\\|_{TV}\\to0$ as $d\\to\\infty$, uniformly over initial decks, and $\\|q_d^{*t}-U_n\\|_{TV}\\ge1-2^{tn/2}/n!$, so $t_{mix}(d)\\ge2d-O(1)$ and $t_{mix}(d)=\\Theta(d)$. Companions give alternative proofs with other constants and the supporting Fourier and coordinate-sweep estimates. The constant 1600 is not claimed sharp; no cutoff or leading constant is determined, and the optimal-order question for general even deck sizes is explicitly not addressed.","claimIssueNote":null,"solveDate":"2026-09-26","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). All eleven manuscripts are authored 'OpenAI' and name no human author. They give several routes to O(d) mixing with different constants (1600d in the principal paper; 512d, 2048d and unspecified absolute multiples in companions), plus supporting representation-theoretic estimates.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the principal manuscript were read against the question. lean/docs/238.md lists thirteen challenges. The headline is in ThorpRemaining.json (OAI.ThorpResults.remaining_main, module OAI.Probability.ThorpResults.FullDevelopment): its conjunct OptimalOrderMain states that the distance-1/4 mixing time of the physical shuffle model (coin switch on the top bit, then bit rotation, which is the Thorp interleave on 2^d positions) is Theta(log 2^d), and FrameMain gives total variation tending to 0 after 32800d steps uniformly over starts. The solution file exists at the pinned commit. ThorpRemaining is not in the formalization catalogue; the catalogued BinarySweep, CoordinateSweeps, ThorpRouting and ThorpWeightedCompatibility state supporting sweep estimates, not the physical-shuffle mixing time. The statement was read here. Not rebuilt. Only deck sizes 2^d are covered.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"The logarithmic mixing order of the Thorp shuffle was a well-known open target in card-shuffling theory, also motivated by format-preserving encryption, after two decades of polynomial improvements. Known to mixing-time specialists; scored for the dyadic case only. Placed below Rapid mixing for spin systems on graphs of girth at least five (30), near the low-temperature SK fluctuation scale in this batch (25), above The Yun-Sra-Jadbabaie SS-RS-GD Inequalities (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf","sourceName":"OpenAI math release: Optimal-order mixing of the Thorp shuffle","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Random coordinate frames and partial permutation laws","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion: From partial permutation information to Fourier bounds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion: Conditional information under deterministic coordinate sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026/main.pdf","kind":"paper"},{"label":"Companion: Conditional permutations in a revealed switching environment","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026/paper.pdf","kind":"paper"},{"label":"Companion: Routing densities and representation contraction for Thorp sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf","kind":"paper"},{"label":"Companion: Row-column symmetry and contraction of coordinate sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026/paper.pdf","kind":"paper"},{"label":"Companion: Random-subspace tests and trace smoothing for coordinate sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf","kind":"paper"},{"label":"Companion: Compatibility entropy and the spectrum of a Thorp sweep","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026/paper.pdf","kind":"paper"},{"label":"Companion: Signed tensor densities and diagram budgets for the Thorp shuffle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026/paper.pdf","kind":"paper"},{"label":"Companion: Conditional coordinate sweeps and analytic transfer","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026/main.pdf","kind":"paper"},{"label":"Lean proof (OAI.ThorpResults.remaining_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/ThorpResults/FullDevelopment.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gotsman-linial-conjecture","name":"The asymptotic Gotsman-Linial conjecture on average sensitivity of polynomial threshold functions","shortName":"Gotsman-Linial conjecture (asymptotic)","problemNumber":null,"field":"Analysis of Boolean functions; polynomial threshold functions","fieldGroup":"Theoretical computer science","statement":"A degree-$d$ polynomial threshold function on $\\{-1,1\\}^n$ is $f(x)=\\mathrm{sgn}(p(x))$ for a real polynomial $p$ of degree at most $d$. Its average sensitivity (total influence) is $I(f)=\\sum_{i=1}^n\\Pr[f(X)\\ne f(X^{\\oplus i})]$ for uniform $X$. Gotsman and Linial proposed that the maximum of $I(f)$ over degree-$d$ PTFs is attained by a symmetric function of $x_1+\\dots+x_n$, which would give $I(f)=O(d\\sqrt n)$; Chapman disproved the exact extremal form and separated it from the asymptotic bound. The best bounds were sublinear for fixed $d$ (Harsha-Klivans-Meka, Diakonikolas-Raghavendra-Servedio-Tan) and $\\sqrt n(\\log n)^{O(d\\log d)}2^{O(d^2\\log d)}$ (Kane). Is $I(f)=O(d\\sqrt n)$ for every degree-$d$ polynomial threshold function, with an absolute constant?","posedBy":"Craig Gotsman and Nathan Linial (exact extremal form); asymptotic form stated by Chapman","yearPosed":1994,"ageNote":"Gotsman and Linial, Spectral properties of threshold functions (Combinatorica 1994, Section 5, p. 47), as cited by the manuscript. The asymptotic O(d sqrt n) form is stated as a separate conjecture by Chapman (2018, Conjectures 1.1-1.2) after the exact form was refuted.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $n\\ge1$, $1\\le d\\le n$ and a real multilinear polynomial $p$ of degree at most $d$, $f=\\mathrm{sgn}(p)$ with $\\mathrm{sgn}(0)=1$ has $I(f)\\le8d\\sqrt n$; the constant is absolute and $d$ may grow with $n$. Symmetric examples show the order $d\\sqrt n$ is sharp for $d\\le\\sqrt n$. It removes the polylogarithmic loss in Kane's bound. It makes no claim about the exact extremizers of the original Gotsman-Linial proposal (already refuted by Chapman and by Kim-Maldonado-Wellens) and does not address non-uniform distributions.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the asymptotic conjecture. The proof was not refereed. lean/formalization.yaml lists OAI.LeanBlast.GotsmanLinial.gotsmanLinialStatement (comparator GotsmanLinial). Its statement was read: for all $1\\le d\\le n$ and every real multilinear polynomial of total degree at most $d$, the threshold function with sign(0)=1 has average sensitivity (sum over coordinates of the fraction of cube points where flipping that coordinate changes the value) at most $8d\\sqrt n$. Multilinearity loses nothing on the cube. This states the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The Gotsman-Linial conjecture is a well-known open problem in the analysis of Boolean functions, with applications to learning and noise sensitivity, open for three decades. Placed below The matrix multiplication exponent: is omega at most 9/4? (55), and above Polynomial-Factor Hardness for the Closest Vector Problem (35) and Steurer's Conjecture on Vectors with Small Average Correlation (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf","sourceName":"OpenAI math release: Average sensitivity of polynomial threshold functions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.LeanBlast.GotsmanLinial.gotsmanLinialStatement)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/GotsmanLinial/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: GotsmanLinial.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/GotsmanLinial.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"halving-lines-power-saving","name":"A power saving over Dey's bound for planar halving lines","shortName":"Halving lines below n^(4/3)","problemNumber":null,"field":"Discrete and computational geometry; k-sets","fieldGroup":"Combinatorics","statement":"For a set $P$ of $n$ points in the plane in general position ($n$ even), a halving line passes through two points of $P$ and leaves $(n-2)/2$ points on each side. Lovasz (1971) and Erdos, Lovasz, Simmons and Straus (1973) introduced the problem of determining the maximum number $h(n)$ of halving lines (and more generally of $k$-sets), proving $O(n^{3/2})$ and constructing $\\Omega(n\\log n)$. Dey (1998) proved $h(n)=O(n^{4/3})$; the best lower bound is $n e^{\\Omega(\\sqrt{\\log n})}$ (Toth, Nivasch). What is the true order of $h(n)$; in particular, can Dey's exponent $4/3$ be improved?","posedBy":"L. Lovasz, On the number of halving lines (1971); P. Erdos, L. Lovasz, A. Simmons and E. G. Straus, Dissection graphs of planar point sets (1973)","yearPosed":1971,"ageNote":"The halving-line count goes back to Lovasz (1971) and the k-set problem to Erdos, Lovasz, Simmons and Straus (1973). The specific target of beating Dey's 1998 exponent 4/3 is the open part addressed here.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are absolute $\\varepsilon>0$, $C$, $n_0$ with $h(P)\\le Cn^{4/3-\\varepsilon}$ for every even $n\\ge n_0$ and every $n$-point planar set with no three collinear. Theorem 1.2 bounds the switches at every rank uniformly by $CN^{4/3-\\varepsilon}$, and Corollary 1.3 gives $O(n(k+1)^{1/3-\\varepsilon_0})$ $k$-sets for $1\\le k\\le n/2$ (also for levels in line arrangements). The constants and exponent are ineffective; no numerical value of $\\varepsilon$ is given. It does not determine the order of $h(n)$, which could still lie anywhere between $n e^{\\Omega(\\sqrt{\\log n})}$ and $n^{4/3-\\varepsilon}$.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization of the main bounds accompanies it (Comparator challenge HalvingLines).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorems 1.1 and 1.2 and Corollary 1.3 of the TeX source, read against the halving-line problem as cited. Lean: Comparator challenge HalvingLines, declaration OAI.PlanarHalving.power_bounds. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: there exist $\\varepsilon>0$, $C$, $n_0$ such that every even $n\\ge n_0$ and every injective planar configuration with no three collinear has at most $Cn^{4/3-\\varepsilon}$ unordered halving pairs; plus a uniform $Cn^{4/3-\\varepsilon}$ bound on rank switches for generic configurations. That is the headline. The $k$-sensitive corollary $O(n(k+1)^{1/3-\\varepsilon_0})$ is not formalized. Not rebuilt here. The paper states the proof is nonquantitative.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The k-set problem is one of the best-known open problems of discrete geometry, and Dey's 4/3 exponent had resisted improvement since 1998. A power saving, even ineffective, is the first exponent progress in nearly three decades, but it leaves the true order wide open. Scored below the Harary-Hill conjecture (55), level with the circulant Hadamard conjecture (42), and above the Seymour second neighbourhood conjecture (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf","sourceName":"OpenAI math release: A power saving for planar halving lines","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge HalvingLines (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HalvingLines.lean","kind":"lean-proof"},{"label":"Dey 1998, Improved bounds for planar k-sets and related problems","url":"https://doi.org/10.1007/PL00009354","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gersten-conjecture-integral-regular-local","name":"Gersten's conjecture for integral K-theory of arbitrary regular local rings (Quillen's form), in ramified mixed characteristic","shortName":"Gersten's conjecture, integral form","problemNumber":null,"field":"Algebraic K-theory of regular local rings","fieldGroup":"Algebra","statement":"For a Noetherian regular local domain $A$ with fraction field $F$, Gersten's conjecture asserts exactness of the augmented complex $0\\to K_n(A)\\to K_n(F)\\to\\bigoplus_{\\mathrm{ht}\\,\\mathfrak p=1}K_{n-1}(\\kappa(\\mathfrak p))\\to\\cdots$ of integral Quillen K-groups; its first assertion is that $K_n(A)\\to K_n(F)$ is injective. Quillen formulated the conjecture for all regular local rings and proved it for regular local rings essentially of finite type over a field; Panin proved the full equicharacteristic case. In mixed characteristic it was known for rings smooth over a DVR modulo the DVR case (Gillet-Levine), with finite coefficients for rings essentially smooth over a DVR (Geisser-Levine) and for unramified regular local rings (Skalit), while Feld (2026) found a finite-coefficient failure at a ramified ring. Is $K_n(A)\\to K_n(\\mathrm{Frac}\\,A)$ injective, and the Gersten complex exact, for every regular local ring $A$, including ramified rings of mixed characteristic?","posedBy":"D. Quillen, Higher algebraic K-theory I, Lecture Notes in Math. 341 (1973), Section 7, Conjecture 5.10; following S. M. Gersten, Problems about higher K-functors (same volume)","yearPosed":1973,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (degree five): with $V$ the integers of the unramified degree-eight extension of $\\mathbb Q_5$ and $A=(V[x,y]/(5+x^4+y^4))_{(5,x,y)}$, $A$ is a two-dimensional regular local domain and $\\ker(K_5(A)\\to K_5(\\mathrm{Frac}\\,A))\\ne0$; the class also gives generic noninjectivity with $\\mathbb Z/5$ coefficients. Companion Theorem 1.1 (degree three): the local ring of a cyclotomic quintic model over $\\mathbb Z_5[\\zeta_5]$ at a special point has a nonzero class in $\\ker(K_3(A)\\to K_3(\\mathrm{Frac}\\,A))$ not divisible by 5. Not shown: the order of these classes, whether they survive rationally (the rational Gersten conjecture is untouched), anything about unramified or smooth-over-DVR rings, or the least degree of failure.","claimIssueNote":"S. Mochizuki, Local Gersten's conjecture for regular system of parameters (arXiv:1503.07966v8, 2020), abstract: 'we show local Gersten's conjecture for regular system of parameters. As its consequence we obtain Gersten's conjecture for a commutative regular local ring and smooth over a commutative discrete valuation ring.' The release's degree-five paper says: 'Mochizuki's Theorem 5 in version 8 ... asserts that, for a commutative Noetherian local ring B and a nonzerodivisor g, the inclusion from finitely generated projective B/gB-modules to finitely generated B-modules of projective dimension at most one induces zero on K-theory. This statement reaches the exact support in the present paper', so it 'would annihilate' the class constructed. The release gives a counterexample to Lemma 13(III) used in that proof, and says this 'by itself does not refute its theorem statement'. Mochizuki's headline corollary (rings smooth over a DVR) does not cover the ring used here.","solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The family has two independent manuscripts (degree five and degree three) whose proofs do not use each other.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts against Quillen's Conjecture 5.10. The degree-five paper gives an explicit two-dimensional ramified regular local ring of mixed characteristic $(0,5)$ with nonzero kernel on $K_5$; the companion gives one with nonzero kernel on $K_3$. The proofs were not refereed. Both papers stress that the counterexamples concern the unrestricted integral form, lie outside the unramified and smooth-over-a-DVR settings where positive results hold, and do not settle the rational-coefficient variant. Marked contested because a 2020 preprint of S. Mochizuki asserts a vanishing theorem that would kill the class constructed here; see the claim-issue note. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Gersten's conjecture is a fifty-year-old cornerstone of algebraic K-theory, famous within the field; its integral mixed-characteristic case was open, though the main interest lies in the unramified and smooth cases left untouched here, and a finite-coefficient ramified failure was found earlier in 2026. Below Luck's group-ring determinant conjecture (35), above the tensor square conjecture for symmetric groups (28), far above partially regular primes (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf","sourceName":"OpenAI math release: An integral counterexample to Gersten's conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An integral degree-three Gersten counterexample","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf","kind":"paper"},{"label":"Mochizuki, Local Gersten's conjecture for regular system of parameters (arXiv:1503.07966v8)","url":"https://arxiv.org/abs/1503.07966v8","kind":"paper"},{"label":"Feld, Finite-coefficient Gersten injectivity fails in ramified mixed characteristic (arXiv:2608.05005)","url":"https://arxiv.org/abs/2608.05005","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"negatively-pinched-kahler-bounded-domain","name":"Wu's bounded-domain question for negatively pinched Kahler manifolds: is a complete simply connected Kahler manifold with pinched negative curvature biholomorphic to a bounded domain?","shortName":"Negatively pinched Kahler uniformization","problemNumber":null,"field":"Kahler geometry; several complex variables","fieldGroup":"Geometry & topology","statement":"In complex dimension one, a complete simply connected surface with curvature bounded above by a negative constant is biholomorphic to the disc. H. Wu (1967) asked whether a complete simply connected Kahler manifold with nonpositive sectional curvature and holomorphic sectional curvature bounded above by a negative constant must be biholomorphic to a bounded domain. Wu and Yau (2019, Conjecture 4.3) state the two-sided version: if a simply connected complete Kahler manifold $M$ of complex dimension $n$ has every real sectional curvature between two negative constants, $-B\\le K\\le -A<0$, must $M$ be biholomorphic to a bounded (pseudoconvex) domain in $\\mathbb C^n$?","posedBy":"H. Wu (1967, normal families paper, p. 195, question (1)); two-sided pinched form by Damin Wu and S.-T. Yau (2019 survey, Conjecture 4.3)","yearPosed":1967,"ageNote":"The manuscript cites Wu's 1967 question as the early formulation and the Wu-Yau 2019 survey for the two-sided pinched conjecture it disproves; 1967 is the earliest date it gives.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a contractible domain $M\\subset\\mathbb C^3$ (a Hartogs-type disc bundle over the unit ball of $\\mathbb C^2$) with a smooth complete Kahler metric whose real sectional curvatures satisfy $-B\\le K\\le -A<0$, such that no bounded holomorphic map $M\\to\\mathbb C^3$ has nowhere-vanishing Jacobian; hence $M$ is not biholomorphic to any bounded domain. This disproves Wu-Yau Conjecture 4.3 and gives a negative answer under the hypotheses of Wu's 1967 question. It does not show absence of bounded holomorphic functions (two base coordinates are bounded), and the example is Stein. Only complex dimension three is constructed.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. A companion manuscript of the same date (the one-sided Liouville construction) answers a different question and is listed as its own entry; the manuscript says the two constructions and proofs are independent.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Wu's question and Wu-Yau Conjecture 4.3. formalization.yaml lists ComparatorChallenges/PinchedKahler.json with declaration OAI.PinchedHartogs.main_theorem in OAI/Geometry/Kahler/Main.lean. The challenge statement was read here: there exist an open, nonempty, contractible set $M\\subset\\mathbb C^2\\times\\mathbb C$ and a smooth Kahler metric on it that is geodesically complete with all real sectional curvatures in $[-B,-A]$, $0 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has three manuscripts of the same date: the principal one on real heights (Problem 2.3), one on the field part of Problem 2.2 for odd integer heights, and one on weighted integer heights (loop O(2) model).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, Section 1 statement and history of the principal manuscript, and the introductions and main theorems of both companions, were read against Schramm's Problems 2.2 and 2.3 as the manuscripts cite them. No Lean formalization exists for this family. The SLE$_4$ statement holds at one tuned two-arc amplitude identified only implicitly (through a stationary tangent-flux coefficient), and the interface part of Problem 2.2 is explicitly left open by the companion. The October 6 editions repair the principal manuscript's boundary-attachment, landing-comparison and multiscale separation arguments and identify the whole interface by terminal-side labels and harmonic-mass nonreturn instead of the Sheffield-Sun path criterion; the stated theorems are unchanged. The two-arc companion now derives its cut and loop estimates from the weighted companion's annulus circuit theorem at $x=1$ (newly proved there), so the two companions are no longer independent. Not refereed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"A problem from Schramm's well-known ICM list on universality of random surfaces, for a natural but specialist hard-constraint model. Placed below The double-dimer CLE4 scaling-limit conjecture in the half-plane Temperleyan setting (28) and The N^{-1/4} law for the critical strip-crossing mass of honeycomb self-avoiding walk (30), and well below Gaussian free field limit of the six-vertex height function throughout the regime a = b = 1, 0 < c <= 2 (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf","sourceName":"OpenAI math release: Uniform real Lipschitz surfaces on the triangular lattice","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion: The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-September-25-2026/paper.pdf","kind":"paper"},{"label":"Companion: Gaussian free-field limits of weighted integer Lipschitz heights","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-September-25-2026/paper.pdf","kind":"paper"},{"label":"Schramm, Conformally invariant scaling limits: an overview and a collection of problems (ICM 2006)","url":"https://doi.org/10.4171/022-1/20","kind":"problem-record"},{"label":"Revised edition (October 6, 2026): Gaussian free-field limits of weighted integer Lipschitz heights","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf","kind":"paper"},{"label":"Earlier edition (September 25, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-September-25-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hyperbolic-groups-cat0","name":"Is every word-hyperbolic group CAT(0)?","shortName":"Hyperbolic groups and CAT(0) actions","problemNumber":null,"field":"Geometric group theory; CAT(0) and CAT(-1) groups","fieldGroup":"Geometry & topology","statement":"A group is CAT(0) (respectively CAT(-1)) if it acts geometrically, that is properly, cocompactly and by isometries, on a proper complete CAT(0) (respectively CAT(-1)) space. Word-hyperbolic groups are characterised by a linear isoperimetric inequality, a coarse form of negative curvature, and Gromov sought geometric realisations of them by negatively curved spaces. Free groups, closed hyperbolic manifold groups and many cubulated groups are CAT(0); Brady and Crisp found hyperbolic groups with no such action in dimension two. Does every hyperbolic group act geometrically on a proper CAT(0) space, or even a proper CAT(-1) space?","posedBy":"M. Gromov (realisation question); CAT(-1) form as Problem H11 in Baumslag-Myasnikov-Shpilrain, Open problems in combinatorial group theory (1999); both forms in E. Stark (2025)","yearPosed":1999,"ageNote":"1999 is the earliest printed record the manuscript cites (Baumslag-Myasnikov-Shpilrain, CAT(-1) form); the CAT(0) form is cited from Stark's 2025 survey and Duchesne-Simon (2026), and the question goes back to Gromov's work on hyperbolic groups in the 1980s by his own account.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a finite connected two-dimensional aspherical simplicial complex $K$ with linear combinatorial disk filling, so $\\pi_1K$ is word-hyperbolic with a finite classifying space, such that $\\pi_1K$ has no geometric action on any proper complete CAT(0) space in any dimension. Hence it is neither CAT(0) nor CAT(-1), and no finite complex homotopy equivalent to $K$ carries a locally CAT(0) metric inducing its topology. The group comes from a random-presentation construction with a Fano-plane incidence core and a harmonic-energy fixed-point obstruction. Not shown: an explicit presentation, or which natural classes of hyperbolic groups (for example random or small-cancellation groups) are CAT(0).","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The main theorem has a Lean formalization in the release (Comparator challenge HyperbolicObstruction), with scope limits listed in the release's Lean notes.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the CAT(0) and CAT(-1) realisation questions as cited. Lean-checked on the release's Comparator challenge HyperbolicObstruction (declaration OAI.HyperbolicObstruction.main, listed in lean/formalization.yaml). Its statement was read here: there is a finite simplicial complex $K$, connected and aspherical, with a linear disk-filling bound, whose fundamental group satisfies four-point word hyperbolicity for some finite generating set and admits no geometric action on any nonempty proper complete CAT(0) space (CAT(0) by the CN inequality), and no finite complex homotopy equivalent to $K$ carries a compatible locally CAT(-1) geodesic metric. That covers the headline. As the release's own Lean notes say, it omits that $K$ is two-dimensional and that no such model is locally CAT(0). Not rebuilt here. The existence argument gives no effective size for $K$.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Whether hyperbolic groups are CAT(0) is a long-standing question at the heart of geometric group theory, less central than residual finiteness but widely known. Above the CAT(0) conjecture for Artin groups (40), below Cannon's conjecture (62) and whether hyperbolic groups are residually finite (66).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A hyperbolic group with no geometric CAT(0) action","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge HyperbolicObstruction (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HyperbolicObstruction.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kirchberg-rordam-strong-pure-infiniteness","name":"Kirchberg-Rordam comparison question: pure infiniteness implies strong pure infiniteness, and weak equals strong for exact C*-algebras","shortName":"Pure vs strong pure infiniteness","problemNumber":null,"field":"Operator algebras; C*-algebras, pure infiniteness","fieldGroup":"Analysis","statement":"Kirchberg and Rordam introduced three notions for (possibly nonsimple) C*-algebras: $C$ is purely infinite if every positive element $h$ is properly infinite ($h\\oplus h\\precsim h$ in Cuntz comparison), weakly purely infinite if $h^{\\oplus n}$ is properly infinite for every positive $h$ and one fixed $n$, and strongly purely infinite if for all positive $x,y$ and $\\varepsilon>0$ there are $s,t$ with $\\|s^*x^2s-x^2\\|<\\varepsilon$, $\\|t^*y^2t-y^2\\|<\\varepsilon$, $\\|s^*xyt\\|<\\varepsilon$. Strong implies ordinary implies weak, and for separable nuclear algebras strong pure infiniteness is tied to $\\mathcal O_\\infty$-absorption. Do the three notions coincide, in general and for nuclear algebras?","posedBy":"Eberhard Kirchberg and Mikael Rordam (Adv. Math. 2002, Question 9.5); recorded as Problem LXXII by Schafhauser, Tikuisis and White","yearPosed":2002,"ageNote":"The notions were introduced in Kirchberg-Rordam, 'Infinite non-simple C*-algebras: absorbing the Cuntz algebra O-infinity' (Adv. Math. 167, 2002). The question is restated in Schafhauser-Tikuisis-White, 'Nuclear C*-algebras: 99 problems' (arXiv 2506.10902, Section 21, Problem LXXII).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every complex C*-algebra in which every positive element is properly infinite is strongly purely infinite, with an arbitrary cross term in the diagonalization. Theorem 1.2: for exact algebras, proper infiniteness of $h^{\\oplus n}$ for one fixed $n$ and all positive $h$ implies proper infiniteness of every positive element, hence strong pure infiniteness. Corollary: a separable nuclear algebra with this fixed-amplification property absorbs $\\mathcal O_\\infty$, with no unitality or simplicity assumption (via Kirchberg's central-sequence results). Not settled: whether weak pure infiniteness implies pure infiniteness for non-exact algebras, which the paper states remains outside its conclusions.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. Lean statements for both main theorems are among the release's Comparator challenges.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorems 1.1 and 1.2 were read against the question. lean/docs/303.md lists two challenges: IndividualStrongInfiniteness.json (OAI.MainGap.Manuscript.individual_to_strong_main, module OAI.Analysis.CStarAlgebra.PureInfiniteness.Main) and ExactInfiniteness.json (OAI.WeakPureInfiniteness.exact_weaklyPurelyInfinite_stronglyPurelyInfinite plus an individual proper-infiniteness variant, module OAI.Analysis.WeakInfiniteness.Main). Both solution files exist at the pinned commit; neither challenge is in the formalization catalogue formalization.yaml. The statements were read here: for a nonunital complex C*-algebra, every positive element properly infinite (Cuntz comparison in a stabilization built in the file) implies arbitrary-entry diagonalization and strong pure infiniteness; for exact algebras (exactness defined from scratch in the file) weak pure infiniteness implies strong. That is the headline. Not rebuilt here. The O-infinity absorption corollary rests on published Kirchberg results and is not formalized.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":27,"significanceNote":"Whether pure infiniteness implies strong pure infiniteness has been a recognized open question in the theory of nonsimple purely infinite C*-algebras for over twenty years and is listed among the 99 problems on nuclear C*-algebras. Specialist but central there; the non-exact weak-to-ordinary part stays open. Placed just above Robert-Tikuisis Conjecture (C1) (25) and Szabo's Conjecture A in the unital stably finite case (22), far below Kadison's Similarity Problem (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Weak pure infiniteness and O-infinity absorption","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.MainGap.Manuscript.individual_to_strong_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/CStarAlgebra/PureInfiniteness/Main.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.WeakPureInfiniteness.exact_weaklyPurelyInfinite_stronglyPurelyInfinite)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/WeakInfiniteness/Main.lean","kind":"lean-proof"},{"label":"Schafhauser-Tikuisis-White, Nuclear C*-algebras: 99 problems (Problem LXXII)","url":"https://arxiv.org/abs/2506.10902","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-551-cycle-clique-ramsey","name":"The Erdős-Faudree-Rousseau-Schelp cycle-clique conjecture: $R(C_m,K_n)=(m-1)(n-1)+1$ for $m\\ge n\\ge3$","shortName":"Erdos Problem #551: cycle-clique Ramsey numbers","problemNumber":551,"field":"Graph Ramsey theory","fieldGroup":"Combinatorics","statement":"Let $R(C_m,K_n)$ be the least $N$ such that every red-blue colouring of the edges of $K_N$ contains a red cycle of length exactly $m$ or a blue $K_n$. Taking $n-1$ disjoint red cliques of order $m-1$ with all edges between them blue gives $R(C_m,K_n)\\ge(m-1)(n-1)+1$. Erdős, Faudree, Rousseau and Schelp (1978) conjectured equality for $m\\ge n\\ge3$, apart from $R(C_3,K_3)=6$. Bondy and Erdős proved it for $m\\ge n^2-2$, Nikiforov for $m\\ge4n+2$, it was settled for $n\\le7$, and Keevash, Long and Skokan (2021) proved it for $m\\ge C\\log n/\\log\\log n$ with an unspecified absolute constant, leaving finitely many pairs that could not be listed. Is $R(C_m,K_n)=(m-1)(n-1)+1$ for all $m\\ge n\\ge3$ with $(m,n)\\ne(3,3)$?","posedBy":"Paul Erdős, Ralph Faudree, Cecil Rousseau and Richard Schelp (J. Graph Theory 1978); Erdős Problem #551","yearPosed":1978,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1: $R(C_m,K_n)=(m-1)(n-1)+1$ for every $m\\ge n\\ge3$ other than $(3,3)$, where $R(C_3,K_3)=6$. The proof takes a minimal counterexample (independent sets expand), finds a clique of order at least $\\max\\{3,\\lfloor k/2\\rfloor\\}$, optimizes path systems between clique vertices, and reduces the rest to 3,099 finite instances excluded by two exact programs. Keevash-Long-Skokan had already left only finitely many pairs open, without an explicit list; this result closes all of them. It does not address the side questions recorded with Problem #551 for fixed $n$ (the least cycle length at which the identity holds when $m 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The proof is computer-assisted: a structural reduction leaves 3,099 finite parameter-pattern instances, which accompanying programs exclude with deduction traces. The release also contains a Lean formalization of the full theorem.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1 and the history section were read against Erdős Problem #551 and the Keevash-Long-Skokan formulation. lean/docs/189.md links the challenge ComparatorChallenges/CycleCliqueRamsey (theorem OAI.CycleClique.thm_main, solution module OAI.Combinatorics.Ramsey.CycleClique.Main, whose file exists at the pinned commit); the challenge is not in the formalization catalogue formalization.yaml. The statement was read here: for all integers $3\\le n\\le m$ with $(m,n)\\ne(3,3)$ the least $N$ such that every graph on $N$ vertices contains a copy of the $m$-cycle or its complement contains $K_n$ equals $(m-1)(n-1)+1$, and the value at $(3,3)$ is 6. That is the headline claim, including the finitely many computer-checked cases. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":21,"significanceNote":"A well-known Erdős conjecture in graph Ramsey theory, open since 1978, but the asymptotic regime was already settled by Keevash-Long-Skokan, leaving a finite (if unlisted) set of cases. Level with Erdős Problem #183: Multicolor Triangle Ramsey (21), above Erdős Problem #550 (10), and well below The Burr-Erdős hypercube Ramsey conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf","sourceName":"OpenAI math release: Cycle-clique Ramsey numbers","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.CycleClique.thm_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Ramsey/CycleClique/Main.lean","kind":"lean-proof"},{"label":"Erdős Problem #551 (erdosproblems.com)","url":"https://www.erdosproblems.com/551","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lang-plaut-problem","name":"The Lang-Plaut problem: does every doubling subset of Hilbert space bi-Lipschitz embed in some Euclidean space?","shortName":"Lang-Plaut problem","problemNumber":null,"field":"Metric geometry; bi-Lipschitz embeddings and dimension reduction","fieldGroup":"Geometry & topology","statement":"A metric space is doubling if every ball is covered by a bounded number of balls of half the radius. Every subset of $\\mathbb R^k$ is doubling, and Assouad's theorem embeds every doubling space bi-Lipschitzly into some $\\mathbb R^k$ after snowflaking the metric to $d^\\alpha$, $\\alpha<1$. For $L_p$ with $p>2$, Lafforgue-Naor and Bartal-Gottlieb-Neiman found doubling subsets with no such embedding. Lang and Plaut asked the Hilbert case for the original metric, and Gupta, Krauthgamer and Lee asked it independently for algorithmic dimension reduction. Does every doubling subset of $\\ell_2$ admit a bi-Lipschitz embedding into $\\mathbb R^k$ for some finite $k$ and finite distortion?","posedBy":"Urs Lang and Conrad Plaut, Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata 87 (2001), Question 2.4; independently Gupta, Krauthgamer and Lee (FOCS 2003)","yearPosed":2001,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims a negative answer: a fixed doubling subset of $\\ell_2$ (doubling constant at most 76800) with no bi-Lipschitz embedding into any $\\mathbb R^k$; more generally every infinite-dimensional real Banach space contains a compact set of universally bounded doubling constant with no bi-Lipschitz embedding into any finite-dimensional normed space, and the same metric embeds isometrically in every $L_p$, $1\\le p<\\infty$. It does NOT bear on snowflaked embeddings (Assouad). Priority note: Schioppa announced a negative answer in 2017 (arXiv:1703.10265) but withdrew it.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The single manuscript constructs the counterexample and its Banach-space extension; both main statements are formalized in Lean in the release catalogue.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 and the abstract's Banach-space statement were read against Question 2.4 of Lang-Plaut: a fixed subset of real $\\ell_2$ with doubling constant at most 76800 admits no bi-Lipschitz embedding into any $\\mathbb R^k$ at any distortion. Both are in formalization.yaml. ComparatorChallenges/DoublingHilbert.lean (OAI.DoublingHilbert.main, OAI/Geometry/DoublingHilbert/Main.lean) states exactly Theorem 1.1, with doubling defined by open balls centered in the subset and embeddings allowing any scale and distortion. ComparatorChallenges/CompactBanach.lean (OAI.CompactBanach.main) states the stronger form: one constant $\\Lambda$ such that every infinite-dimensional real Banach space contains a compact $\\Lambda$-doubling set with no bi-Lipschitz embedding into any finite-dimensional normed space. Permitted axioms propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"The Lang-Plaut problem is a central open question in metric embedding theory and dimension reduction, open for 25 years with a withdrawn negative claim along the way, but known mainly within that community. Level with Tingley's problem (38), above Kalton's Problem 2 (28), below the planar GNRS conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf","sourceName":"OpenAI math release: A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: doubling Hilbert subset without finite-dimensional embedding","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/DoublingHilbert/Main.lean","kind":"lean-proof"},{"label":"Lean: compact counterexamples in every infinite-dimensional Banach space","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/DoublingHilbert/CompactBanach.lean","kind":"lean-proof"},{"label":"Lean comparator statement: DoublingHilbert","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DoublingHilbert.lean","kind":"lean-proof"},{"label":"Lang and Plaut (2001), Question 2.4","url":"https://doi.org/10.1023/A:1012093209450","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"birkar-stein-degree-conjecture-point","name":"Birkar's Stein-degree conjecture for log Calabi-Yau pairs: the contraction-to-a-point case for ordinary pairs","shortName":"Birkar Stein-degree conjecture, point base","problemNumber":null,"field":"Birational geometry; boundedness","fieldGroup":"Algebra","statement":"For a proper integral variety $S$ over a field $k$ the Stein degree is $\\dim_k H^0(S,\\mathcal O_S)$. Birkar conjectured that boundary components of log Calabi-Yau fibrations with coefficient bounded below have uniformly bounded Stein degree; Birkar and Qu proved normalized bounds over algebraically closed fields of characteristic zero. For a contraction to a point over any field $k$ of characteristic zero: given $d$ and $t>0$, is there $N(d,t)$ such that for every projective lc pair $(X,B)$ of dimension $d$ with $H^0(X,\\mathcal O_X)=k$ and $K_X+B\\sim_{\\mathbb Q}0$, every prime component of $B$ with coefficient at least $t$ has Stein degree at most $N(d,t)$?","posedBy":"Caucher Birkar, Stein degree of proper morphisms (arXiv 2603.23939, 2026), Conjecture 11.1","yearPosed":2026,"ageNote":"Posed in a March 2026 arXiv preprint, about six months before this work.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every field $k$ of characteristic zero and every normal integral projective lc $\\mathbb Q$-pair $(X,B)$ of dimension $d$ with $H^0(X,\\mathcal O_X)=k$, $B\\ge0$ and $K_X+B\\sim_{\\mathbb Q}0$, every prime component $S$ of $B$ with coefficient at least $t$ satisfies $[k_S:k]\\le N(d,t)$, where $k_S$ is the algebraic closure of $k$ in $k(S)$; this bounds the Stein degree of $S$. It covers only the contraction to $\\mathrm{Spec}\\,k$ and ordinary pairs; the general fibration form of the conjecture and generalized pairs are not treated.","claimIssueNote":"This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it.","solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. Family 034 has fourteen manuscripts dated September 24 to October 5, 2026, which cite one another as inputs; this entry draws on the manuscript named as its source. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the manuscript describes it; the paper itself says it proves only the contraction-to-a-point formulation for ordinary Q-pairs, hence partial. Birkar's paper was not opened here. The proof (following Birkar-Qu with an added arithmetic argument on Galois orbits of valuations) was not refereed. No Lean formalization. Listed as Contested because it depends on the log abundance entry, which the release itself is inconsistent about; see the claim issue.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A conjecture posed in 2026 within boundedness theory, with only one formulation of it answered. Placed level with a typical Erdos problem (10) and the 44-vertex triangulation of RP6 (10), and below the effective log Iitaka entry from this family (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Arithmetic Stein-degree bounds for log Calabi–Yau pairs","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Birkar, Stein degree of proper morphisms (arXiv 2603.23939), Conjecture 11.1","url":"https://arxiv.org/abs/2603.23939","kind":"problem-record"},{"label":"The 5 October appendix calling projective log abundance conditional","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026/","kind":"paper"},{"label":"Revised edition (October 6, 2026): Conditional good minimal models for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"litt-integral-density-character-varieties-curves","name":"Litt's potential integral density question for $\\mathrm{SL}_r$ character varieties of curves","shortName":"Litt's integral density question for curves","problemNumber":null,"field":"Arithmetic geometry; character varieties, integral points","fieldGroup":"Number theory","statement":"For a smooth complex variety $X$, the $\\mathrm{SL}_r$-character variety of $\\pi_1(X)$ has a natural model $M_r(X)$ over $\\mathbb Z$. Litt asked whether integral points on such character varieties become Zariski dense after a finite extension of the ground field, including a variant with prescribed boundary monodromy, and whether one number field $L$ suffices so that the points over the full ring of integers $\\mathcal O_L$ are dense in every component. Coccia and Litt proved this for $\\mathrm{SL}_2$ and $\\mathrm{PGL}_2$ and conjectured it for Chevalley groups with fixed boundary data. For a smooth connected curve $X$, every rank $r$ and prescribed quasi-unipotent boundary classes, is there one number field over whose full ring of integers the integral points are Zariski dense?","posedBy":"Daniel Litt (Problems I Like, Problem 7; survey 2024, Question 5.4.3(2)); Simone Coccia and Daniel Litt (2025, Conjecture 1.1.1)","yearPosed":2024,"ageNote":"The survey 'Motives, mapping class groups, and monodromy' (arXiv 2409.02234, 2024) asks explicitly whether one number field suffices. The manuscript cites the Problems I Like page without a posting date.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every smooth connected complex curve $X$, rank $r$ and quasi-unipotent conjugacy classes at the punctures (nonsemisimple allowed), there is a finite extension $L$ of the field of eigenvalues with $M_r(X)(\\mathcal O_L)\\cap Y(\\mathbb C)$ Zariski dense in every component of the relative locus $Y$; an explicit $L$ is given. The same holds when only characteristic polynomials are fixed, and for the whole character variety with base field $\\mathbb Q$. This settles the determinant-one curve case of Litt's question and the $\\mathrm{SL}_r$ curve case of the Coccia-Litt conjecture for quasi-unipotent data; higher-dimensional varieties are not treated.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Litt's question as the paper states it; Litt's sources were not opened. The 7 October 2026 update added the supporting Comparator challenge CharacterVarietiesAllSeamsSupport (read here, with lean/docs/027.md): the produced marked values of the marked-surface construction satisfy every seam equation. Potential integral density is outside it, so the entry stays unreviewed. Integrality is ambient (points of $M_r(X)$ over $\\mathcal O_L$ whose complex representation has the prescribed classes), which the paper distinguishes from integral points of a locally closed integral model. Varieties of dimension above one are outside the result, as the paper says.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A recent question from a well-followed problem list, answered here only in the curve, determinant-one case; specialist arithmetic geometry. Placed level with Baker's Question on Smooth Hyperplane Sections over Finite Fields (15), below Szabo's Conjecture A in the unital stably finite case (22) and The P = W conjecture for SL_n Higgs moduli in composite rank (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Integral points on character varieties of curves","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Litt, Problems I Like, Problem 7","url":"https://www.problemsilike.com/7","kind":"problem-record"},{"label":"Litt, Motives, mapping class groups, and monodromy (Question 5.4.3(2))","url":"https://arxiv.org/abs/2409.02234","kind":"problem-record"},{"label":"Lean statement (supporting result only): CharacterVarietiesAllSeamsSupport","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/CharacterVarietiesAllSeamsSupport.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"foulkes-conjecture-sixth-symmetric-power","name":"Foulkes' conjecture for the sixth symmetric power: $\\mathrm{Sym}^6(\\mathrm{Sym}^bV)\\hookrightarrow\\mathrm{Sym}^b(\\mathrm{Sym}^6V)$","shortName":"Foulkes' conjecture, case a = 6","problemNumber":null,"field":"Representation theory; plethysm","fieldGroup":"Algebra","statement":"For a finite-dimensional complex vector space $V$ and integers $1\\le a\\le b$, Foulkes' conjecture (1950) asks for a $\\mathrm{GL}(V)$-equivariant injection $\\mathrm{Sym}^a(\\mathrm{Sym}^bV)\\hookrightarrow\\mathrm{Sym}^b(\\mathrm{Sym}^aV)$, equivalently that the plethysm difference $h_b[h_a]-h_a[h_b]$ is Schur-positive. It was known for $a\\le5$: $a=2$ from Thrall, $a=3$ by Dent-Siemons, $a=4$ and $a=5$ from McKay's propagation theorem with computations by Müller-Neunhöffer and Cheung-Ikenmeyer-Mkrtchyan. Brion proved it for $b$ large in terms of $a$, and Evseev-Paget-Wildon checked $a+b\\le19$. The canonical Foulkes-Howe map has a kernel at $(6,6)$, so the propagation route fails at $a=6$. For $a=6$, is there an equivariant injection $\\mathrm{Sym}^6(\\mathrm{Sym}^bV)\\hookrightarrow\\mathrm{Sym}^b(\\mathrm{Sym}^6V)$ for every $b\\ge6$ and every $V$?","posedBy":"H. O. Foulkes (J. London Math. Soc. 1950)","yearPosed":1950,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $b\\ge6$ and every finite-dimensional complex $V$ there is a $\\mathrm{GL}(V)$-equivariant injection $\\mathrm{Sym}^6(\\mathrm{Sym}^bV)\\hookrightarrow\\mathrm{Sym}^b(\\mathrm{Sym}^6V)$, i.e. $h_b[h_6]-h_6[h_b]$ is Schur-positive. The range $b\\ge30$ comes from the companion's quadratic stabilization theorem; $6\\le b\\le29$ is settled by certified comparisons of upper bounds for source multiplicities with lower bounds for target multiplicities. It proves the multiplicity comparison, not injectivity of the canonical map (which fails at $(6,6)$), and only the case $a=6$; Foulkes' conjecture for $a\\ge7$ and $b 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The proof combines the companion's quadratic-stabilization theorem (range b >= 30) with certified multiplicity comparisons computed by printed programs for 6 <= b <= 29; a second, longer route runs the certificates through b = 149 and uses a chart bound beyond.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the introduction were read against Foulkes' conjecture as the manuscript states it. Lean: lean/docs/210.md covers only the companion (challenge FoulkesHowe, theorem OAI.Problem346.canonical_foulkes_howe_surjective, in formalization.yaml). Its statement gives surjectivity of the canonical map for $b\\ge a(a-1)$, so for $a=6$ only $b\\ge30$, and states surjectivity rather than the embedding; the docs say the companion's full range $b\\ge6$ is not included. So the formal statement covers a special range, not this entry's headline, and the tier stays unreviewed. The band $6\\le b\\le29$ rests on computer certificates that were not rerun here. Not refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Foulkes' conjecture is a well-known plethysm problem in algebraic combinatorics; this is the next open case after a = 5, a special case of the parent rather than the conjecture itself. Placed below Saxl's conjecture (32) and The tensor square conjecture for symmetric groups (28), near The Stable Coordinate Conjecture in ambient dimension five (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf","sourceName":"OpenAI math release: Foulkes' conjecture for the sixth symmetric power","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Quadratic stabilization of the canonical Foulkes-Howe map","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf","kind":"paper"},{"label":"Lean proof of the companion's stabilization theorem (covers a = 6 only for b >= 30)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gersten-conjecture-one-relator-hyperbolic","name":"Gersten's conjecture: are Baumslag-Solitar-free one-relator groups hyperbolic?","shortName":"Gersten's one-relator conjecture","problemNumber":null,"field":"Geometric group theory; one-relator groups","fieldGroup":"Algebra","statement":"The Baumslag-Solitar groups $\\mathrm{BS}(m,n)=\\langle a,t\\mid ta^mt^{-1}=a^n\\rangle$, $m,n\\ne0$, are the basic obstructions to word-hyperbolicity among one-relator groups: a hyperbolic group contains none of them. Gersten asked whether they are the only subgroup obstructions. Known cases included proper-power relators (Newman's spelling theorem), relators of primitivity rank other than two (Louder-Wilton, Linton) and ascending HNN extensions of free groups (Mutanguha). Let $G=\\langle X\\mid r\\rangle$ with $X$ finite. If $G$ contains no subgroup isomorphic to $\\mathrm{BS}(m,n)$ for any nonzero $m,n$, must $G$ be word-hyperbolic?","posedBy":"S. M. Gersten, 'Problems on automatic groups' (MSRI Publications 23, p. 228)","yearPosed":1992,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if a one-relator group $\\langle X\\mid r\\rangle$, $X$ finite, contains no $\\mathrm{BS}(m,n)$ with $m,n\\ne0$, it is word-hyperbolic. The technical core is a combination theorem for a class of HNN extensions of torsion-free hyperbolic groups with free, possibly proper edge groups, applied along the Magnus-Moldavanskii hierarchy. Combined with the companion and with Kielak-Linton, Corollary: every Baumslag-Solitar-free one-relator group is virtually free-by-cyclic, with a finite-index embedding in a free-by-$\\mathbb Z$ group with finitely generated kernel. The paper does not treat infinitely generated or multi-relator groups. Unrelated to the K-theoretic Gersten conjecture addressed in family 209 of the same release.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The companion on virtual compact specialness (same date) uses this paper's hyperbolicity and Magnus-subgroup results at two points; this paper in turn quotes the companion only for the free-by-cyclic corollary.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against Gersten's question as the paper quotes it from Gersten (1992, p. 228); it is the conjecture in full, for every finite generating set and relator, including trivial relators and proper powers, with no torsion-freeness hypothesis. The paper notes that excluding all $\\mathrm{BS}(1,n)$ is equivalent. The proof rests on Linton's one-relator hierarchy and quasiconvexity theorems and a new combination theorem for HNN extensions; it was not refereed here. There is no Lean formalization for this family (no lean/docs/258.md at the pinned commit).","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Gersten's conjecture was a central open problem of one-relator group theory for three decades, with major partial progress by Louder, Wilton and Linton in the 2020s. Famous within geometric group theory. Placed below The Kervaire conjecture (55) and Are all word-hyperbolic groups residually finite? (66), near Is every word-hyperbolic group CAT(0)? (45), and well above Commutator Relators Do Not Force Hopficity, Residual Finiteness or Automaticity (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Baumslag-Solitar-free one-relator groups are hyperbolic","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Virtual compact specialness of hyperbolic one-relator groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf","kind":"paper"},{"label":"Gersten, Problems on automatic groups (1992)","url":"https://doi.org/10.1007/978-1-4613-9730-4_10","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pcp-for-ppad-conjecture","name":"The PCP-for-PPAD conjecture of Babichenko, Papadimitriou and Rubinstein (quasilinear form)","shortName":"PCP for PPAD conjecture","problemNumber":null,"field":"Computational complexity; PPAD and equilibrium computation","fieldGroup":"Theoretical computer science","statement":"End-of-Line is the canonical PPAD-complete problem: given successor and predecessor circuits on $\\{0,1\\}^n$ with a known source, find another endpoint. A generalized circuit asks for an assignment of values in $[0,1]$ to wires that $\\varepsilon$-satisfies local arithmetic, comparison and Boolean gates. Rubinstein proved PPAD-hardness for constant $\\varepsilon$ when every gate must be satisfied. Babichenko, Papadimitriou and Rubinstein (2016) conjectured a PCP analogue: are there constants $\\varepsilon,\\delta>0$ and a quasilinear-size reduction from End-of-Line to generalized circuits such that any assignment $\\varepsilon$-satisfying all but a $\\delta$ fraction of gates, wherever the failures lie, yields an End-of-Line solution in polynomial time?","posedBy":"Yakov Babichenko, Christos Papadimitriou and Aviad Rubinstein, Conjecture 2 of 'Can almost everybody be almost happy?' (ITCS 2016)","yearPosed":2016,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are fixed rational $0<\\varepsilon<1/10$ and $0<\\delta<1$, and deterministic polynomial-time algorithms $R,D$, such that $R$ maps an End-of-Line instance of length $N$ to a generalized circuit of total binary length at most $AN\\lceil\\log_2(N+2)\\rceil^b$, every generalized circuit has an acceptable rational assignment of polynomially bounded length, and from any such assignment $\\varepsilon$-satisfying all but a $\\delta$ fraction of gates $D$ recovers an End-of-Line solution. This also implies the weaker polynomial-size PCP-for-PPAD statement. It does not by itself give new Nash lower bounds; those (Rubinstein, Golowich) still need an exponential-time hypothesis for PPAD.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Conjecture 2 of Babichenko-Papadimitriou-Rubinstein. lean/docs/136.md does not exist at the pinned commit and formalization.yaml has no entry for this manuscript, so there is no formal statement. The manuscript fixes its own explicit encodings and gate conventions (following Deligkas et al. and Fisher) and handles a normalization of the original End-of-Line definition in a footnote; a reader comparing with the 2016 formulation should check those conventions.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"PCP for PPAD is a well-known conjecture of algorithmic game theory and complexity, highlighted by Rubinstein's work on approximate Nash equilibria, but young and specialist. Placed level with edit-distance-l1-distortion (32), above two-way-nfa-polynomial-complementation (25) and below metric-k-median-approximation-threshold (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf","sourceName":"OpenAI math release: The PCP-for-PPAD conjecture: a quasilinear reduction","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kirchberg-rordam-character-question","name":"The Kirchberg-Rordam character question: does absence of characters on the central sequence algebra characterize Jiang-Su stability?","shortName":"Kirchberg-Rordam character question","problemNumber":null,"field":"Operator algebras; Jiang-Su stability and central sequences","fieldGroup":"Analysis","statement":"For a nonzero unital separable C*-algebra $A$ and a free ultrafilter $\\omega$, let $F_\\omega(A)=A_\\omega\\cap A'$ be the norm central-sequence algebra. Kirchberg and Rordam proved $A\\cong A\\otimes\\mathcal Z$ exactly when $F_\\omega(A)$ contains, with the same unit, a separable subhomogeneous algebra without characters; absence of characters on $F_\\omega(A)$ is then necessary. They asked (Question 3.1) whether it is sufficient, and related it to tensor powers (Question 3.7); the question reappears as Problem LXXX of Schafhauser, Tikuisis and White. Is $A$ Jiang-Su stable whenever $F_\\omega(A)$ has no characters?","posedBy":"Eberhard Kirchberg and Mikael Rordam (When central sequence C*-algebras have characters, Internat. J. Math. 26 (2015), Questions 3.1 and 3.7); listed as Problem LXXX by Schafhauser, Tikuisis and White","yearPosed":2014,"ageNote":"The Kirchberg-Rordam paper appeared on arXiv in 2014 (1409.1395) and in print in 2015.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every nonzero unital separable complex C*-algebra $A$ and every free ultrafilter $\\omega$, $F_\\omega(A)$ has no characters iff $A\\cong A\\otimes_{\\min}\\mathcal Z$; no nuclearity, simplicity, trace or comparison hypothesis. Theorem 1.2: every nonzero unital characterless $D$ admits a unital map $I(2,3)\\to D^{\\otimes_{\\max}m}$ for some $m$. Corollary: the infinite minimal tensor power of a nonzero unital separable characterless algebra is $\\mathcal Z$-stable (bearing on Kirchberg-Rordam Question 3.7, via Dadarlat-Toms). Not shown: any finite tensor power is $\\mathcal Z$-stable, or nonseparable cases.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against Question 3.1 as the paper quotes it; proof not refereed. Lean-checked on Comparator challenge CharacterCriterion, declaration OAI.KirchbergRordam.character_criterion in OAI/Analysis/CharacterCriterion/Main.lean, listed in lean/formalization.yaml; statement read here, not rebuilt. The challenge file builds the norm ultrapower, central sequence algebra, minimal tensor product and the Jiang-Su algebra itself (about 2,100 lines of definitions), and states: for every nontrivial separable unital C*-algebra and free ultrafilter, no characters on the central algebra iff $A\\cong A\\otimes_{\\min}\\mathcal Z$. That is the headline. The infinite tensor power corollary is not in the formal statement. The proof uses a companion OpenAI lemma (catalog entry robert-tikuisis-conjecture-c1).","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A natural specialist question in the classification-regularity program for C*-algebras, listed in the Schafhauser-Tikuisis-White problem collection. Placed just below Szabo's equivariant Jiang-Su stability conjecture (22) and the Robert-Tikuisis conjecture (25), above Schafhauser-Tikuisis-White Problem XXI (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/paper.pdf","sourceName":"OpenAI math release: The Kirchberg-Rordam character criterion","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean main result OAI/Analysis/CharacterCriterion/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/CharacterCriterion/Main.lean","kind":"lean-proof"},{"label":"Lean Comparator challenge CharacterCriterion","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CharacterCriterion.lean","kind":"lean-proof"},{"label":"Kirchberg and Rordam, When central sequence C*-algebras have characters (arXiv 1409.1395)","url":"https://arxiv.org/abs/1409.1395","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"villani-mtw-convex-injectivity-domains","name":"Villani's conjecture: weak Ma-Trudinger-Wang curvature implies convex tangent injectivity domains on compact manifolds","shortName":"Villani's MTW convexity conjecture","problemNumber":null,"field":"Riemannian geometry and optimal transport regularity","fieldGroup":"Geometry & topology","statement":"Let $(M,g)$ be a compact connected Riemannian manifold without boundary, $c=d^2/2$, and $I(x)\\subset T_xM$ the open tangent injectivity domain. The Ma-Trudinger-Wang tensor $\\mathfrak S_{(x,v)}(\\xi,\\eta)$ is a fourth derivative of $c$; weak MTW (A3w) requires $\\mathfrak S_{(x,v)}(\\xi,\\eta)\\ge0$ whenever $\\langle\\xi,\\eta\\rangle=0$. Weak MTW is necessary for continuity of optimal transport maps (Loeper), and regularity theory (Figalli-Rifford-Villani) needed convex injectivity domains as an extra hypothesis. Villani proposed that weak MTW should by itself force convexity of every $I(x)$; Loeper-Villani proved convexity under strong MTW with nonfocality, and Figalli-Gallouet-Rifford proved the implication for nonfocal manifolds, where cut points come strictly before conjugate points. Does weak MTW imply that $I(x)$ is convex for every $x\\in M$, with no nonfocality assumption?","posedBy":"C. Villani, Regularity of optimal transport and cut locus: from nonsmooth analysis to geometry to smooth analysis, Discrete Contin. Dyn. Syst. 30 (2011), Section 4.2","yearPosed":2011,"ageNote":"The manuscript says the result resolves Villani's conjecture 'in this setting', that is compact boundaryless manifolds of dimension at least two, which is the setting of the cited conjecture.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $(M,g)$ is smooth, connected, compact, boundaryless, of dimension $n\\ge2$, and satisfies weak MTW on orthogonal pairs, then every tangent injectivity domain $I(x)$ is convex, and so is the closed minimizing domain; conjugate cut points are allowed. Theorem 1.2 adds global supporting mountains at every subgradient of a $c$-convex potential and $C^{1,1}$ intermediate-time geometry, with no density hypothesis. Companion: uniform bi-Holder estimates for optimal maps and their inverses over all densities pinched between fixed positive bounds. Not shown: strict convexity or boundary regularity of $I(x)$, noncompact manifolds, or constants uniform over varying metrics.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The release also supplies Lean formalizations of the convexity theorem and of the bi-Holder transport companion, produced as part of the same release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the principal manuscript against Villani's conjecture as the paper quotes it, and the Lean statement lean/ComparatorChallenges/WeakMTWGlobalSupport.lean (OAI.WeakMTWGlobalSupport.current_main_convexity), listed in formalization.yaml; not rebuilt here. The Lean statement covers the headline: on a compact connected smooth Riemannian manifold of dimension at least two, weak MTW (defined from the fourth derivative of half squared distance along exponential curves, on orthogonal pairs) implies linear convexity of every open tangent injectivity domain. The exponential map is defined in Lean through minimizing geodesics, a modelling choice a reader may wish to audit. The companion's uniform bi-Holder transport theorem is formalized separately (BiholderTransport).","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A named conjecture in the regularity theory of optimal transport on manifolds since about 2010, the missing link between the MTW condition and the Figalli-Rifford-Villani regularity theorems; well known among optimal transport specialists but narrow. Level with the spatial Kadison-Kastler problem for separable C*-algebras (22), above Brezis's degree-inequality constants (20), below the Euclidean-complete affine Bernstein problem (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Global Support and Convex Injectivity Domains under Weak MTW","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Uniform Bi-Holder Transport from Weak MTW","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf","kind":"paper"},{"label":"Lean comparator statement: WeakMTWGlobalSupport.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/WeakMTWGlobalSupport.lean","kind":"lean-proof"},{"label":"Lean file: OAI/Geometry/WeakMTW/Convexity.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/WeakMTW/Convexity.lean","kind":"lean-proof"},{"label":"Lean comparator statement: BiholderTransport.lean (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BiholderTransport.lean","kind":"lean-proof"},{"label":"Villani, Regularity of optimal transport and cut locus (DCDS 30, 2011)","url":"https://doi.org/10.3934/dcds.2011.30.559","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kesten-stigum-sharpness-three-four-state-potts","name":"Sharpness of the Kesten-Stigum reconstruction threshold for three-state symmetric and ferromagnetic four-state Potts channels on trees","shortName":"Kesten-Stigum sharpness for 3 and 4 states","problemNumber":null,"field":"Broadcasting on trees; reconstruction and community detection","fieldGroup":"Probability & statistics","statement":"A root spin, uniform on $q$ states, is broadcast down a tree through a symmetric channel with second eigenvalue $\\lambda$; reconstruction means the spins at depth $\\ell$ keep non-vanishing information about the root. Kesten-Stigum gives reconstruction when $d\\lambda^2>1$ ($d$ the branching number); Sly showed this is not sharp for $q\\ge5$, and proved sharpness for $q=3$ on regular trees of large degree. Mezard and Montanari conjectured sharpness for three states (either sign of $\\lambda$) and small alphabets, and discussed all branching parameters. For $q=3$ and for the ferromagnetic $q=4$ channel, on every regular tree and every Poisson Galton-Watson tree, is there non-reconstruction whenever $d\\lambda^2\\le1$?","posedBy":"Marc Mezard and Andrea Montanari (Reconstruction on trees and spin glass transition, J. Stat. Phys. 2006, Conjecture 2 and Section 8); all-degree regular-tree form in Allan Sly (2011, Conjecture 1)","yearPosed":2006,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Three-state theorem: for the symmetric channel with $-1/2\\le\\lambda\\le1$, on every $b$-ary tree ($b\\ge2$) and every observed Poisson($d$) tree with $d>1$, reconstruction holds iff $d\\lambda^2>1$, with non-reconstruction at equality for both signs (so the 3-coloring channel is non-reconstructible at branching four). Corollary: weak recovery in the symmetric three-community sparse block model is possible iff $(a-b)^2>3(a+2b)$. Four-state theorem: for the ferromagnetic channel ($0\\le\\lambda\\le1$), non-reconstruction whenever $d\\lambda^2\\le1$ on $d$-ary and Poisson trees. Companion: an $L^3$-capacity criterion for four-state reconstruction on bounded-degree trees. Not shown: the antiferromagnetic four-state channel.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of the three manuscripts, read against the Mezard-Montanari and Sly conjectures as quoted; proofs not refereed. Both threshold papers are computer assisted (certified finite recursions near the coloring channel; exact-arithmetic verification of polynomial inequalities for four states), and those programs were not rerun here. Lean: the release's Comparator challenges ThreeStateSupercritical and ThreeStateTreeClauses cover only reconstruction above the Kesten-Stigum threshold, the classical direction, not non-reconstruction at or below it, so the tier is unreviewed. The stochastic block model corollary also imports Mossel-Sly-Sohn and Abbe-Sandon.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A well-known question in broadcasting and spin-glass theory, with a direct payoff for the community-detection threshold of the three-community block model; famous within probability and statistical physics rather than beyond. Placed just below Talagrand's critical SK overlap conjecture (30) and rapid mixing for spin systems on graphs of girth five (30), well above the Klopp-Zadik node-private recovery question (12).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf","sourceName":"OpenAI math release: The exact reconstruction threshold for the three-state symmetric channel","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The reconstruction threshold for the ferromagnetic four-state Potts model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/four-state-potts.pdf","kind":"paper"},{"label":"Companion: A capacity criterion for four-state Potts reconstruction on trees","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026/four-state-capacity.pdf","kind":"paper"},{"label":"Lean Comparator challenge ThreeStateSupercritical (supercritical direction only)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ThreeStateSupercritical.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"jacobsthal-function-quadratic-bound","name":"Jacobsthal's quadratic-bound question: is h(k) = O(k^2)?","shortName":"Jacobsthal quadratic bound","problemNumber":970,"field":"Sieve theory: Jacobsthal's function","fieldGroup":"Number theory","statement":"Let $h(k)$ be the least $m$ such that every interval of $m$ consecutive integers contains an integer coprime to any prescribed positive integer with at most $k$ distinct prime factors. Brun's sieve gives $h(k)\\ll k^{C}$; Iwaniec's linear-sieve work gave $h(k)\\ll k^2\\log^2k$ (1978), improving Vaughan's $k^2\\log^4k$; the best lower bound, from Ford, Green, Konyagin, Maynard and Tao, is about $k(\\log k)^2\\log\\log\\log k/\\log\\log k$. Jacobsthal asked, and Erdos recorded (1962; Erdos Problem #970), whether the upper bound is quadratic. Is $h(k)\\ll k^2$?","posedBy":"E. Jacobsthal (papers from 1960); recorded by P. Erdos (1962, p. 163) and as Erdos Problem #970","yearPosed":1962,"ageNote":"Jacobsthal studied the function in papers beginning in 1960; the manuscript cites Erdos (1962) as the record of Jacobsthal's question, and erdosproblems.com lists it as #970 citing Erdos (1965).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: $h(k)\\le Ck^2/(\\log\\log(3k))^2$ for an absolute constant $C$ and every $k\\ge1$, uniformly over prime sets and interval positions; in particular $h(k)\\ll k^2$, answering Jacobsthal's question and improving Iwaniec's $k^2\\log^2k$. The proof pushes the linear sieve to its limiting parameter, keeping a boundary term, and controls the discrepancy with the actual residue classes through an inverse estimate and stopped counts. The constant is not explicit, and the true order of $h(k)$ (Erdos #970's broader ask) is not determined: the gap to the lower bound near $k(\\log k)^2$ remains. Theorem 1.2 of the same paper also bounds the quantity in Erdős Problem #687: every choice of one class per prime $p\\le z$ leaves survivors below $z^2/(\\log z)^2$, so $Y(z) 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Jacobsthal's question as recorded by Erdos and on erdosproblems.com; the proof was not refereed. Lean: formalization.yaml lists OAI.Erdos970.Erdos970Final.erdos_970_quadratic (Comparator challenge Jacobsthal, file OAI/NumberTheory/Jacobsthal/Conclusions/QuadraticBound.lean). Its statement was read here: there is C > 0 such that for every k >= 1 some m <= C k^2 has the property that, for every positive n with at most k distinct prime factors and every integer start a, one of a, a+1, ..., a+m-1 is coprime to n. That is exactly Jacobsthal's question. The separate challenge JacobsthalImproved (erdos_970_iterated_log, same solution module, not in the formalization catalogue) states the paper's stronger bound C k^2/(log log 3k)^2. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":34,"significanceNote":"A classical sieve question tied to large gaps between primes, open since the early 1960s with the logarithmic loss in Iwaniec's bound standing since 1978. Known mainly to sieve theorists. Scored above the lonely runner conjecture for nine and ten runners (30) and below the Gaussian moat problem (40); far above a typical Erdos problem (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A quadratic bound for Jacobsthal's function","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/NumberTheory/Jacobsthal/Conclusions/QuadraticBound.lean (erdos_970_quadratic)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/Jacobsthal/Conclusions/QuadraticBound.lean","kind":"lean-proof"},{"label":"Erdos Problem #970","url":"https://www.erdosproblems.com/970","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"qaoa-parisi-optimality-sk","name":"The Basso-Farhi-Marwaha-Villalonga-Zhou conjecture: QAOA attains the Sherrington-Kirkpatrick ground-state energy as depth grows","shortName":"QAOA eventual Parisi optimality on SK","problemNumber":null,"field":"Quantum optimization; spin glasses","fieldGroup":"Quantum information & computing","statement":"For the zero-field Sherrington-Kirkpatrick model the expected QAOA energy per spin at fixed depth $p$ and fixed angles has an infinite-size limit $v_p(\\gamma,\\beta)$ (Farhi-Goldstone-Gutmann-Zhou), and $Q_p=\\sup v_p$ cannot exceed the Parisi ground-state value $P_*$. Fixed-instance optimality as depth grows (Farhi-Goldstone-Gutmann) gives no depth uniform in system size. Basso, Farhi, Marwaha, Villalonga and Zhou, relating the limit to QAOA on large-girth regular graphs, conjectured that increasing depth eventually reaches the Parisi optimum; numerics (Boulebnane et al., Sels-Morone) supported it. Is $\\lim_{p\\to\\infty}Q_p=P_*$?","posedBy":"J. Basso, E. Farhi, K. Marwaha, B. Villalonga and L. Zhou (The QAOA at high depth for MaxCut on large-girth regular graphs and the SK model, TQC 2022, Section 7, Eq. (7.2))","yearPosed":2021,"ageNote":"The conjecture is in the arXiv version 2110.14206 (October 2021), published at TQC 2022.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: $\\lim_{p\\to\\infty}Q_p=P_*$ for the zero-field SK model; for every $\\varepsilon>0$ some finite depth and deterministic angles, independent of $n$ and the disorder, reach expected energy per spin $\\ge P_*-\\varepsilon$ in the limit $n\\to\\infty$ taken first, with the standard cost Hamiltonian and transverse-field mixer. It also gives leading-order optimal expected MaxCut on large-degree random regular graphs (size before degree). Not given: any depth bound, finite-size estimate, efficient angle selection, or concentration beyond expectation. The companion proves full support on $[0,1)$ of zero-temperature Parisi minimizers, a result H.-B. Chen had already posted (arXiv 2607.18032), as the companion acknowledges.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against the BFMVZ conjecture as quoted; proof not refereed. Lean: the release's Comparator challenges SKValue and SKFullSupport (lean/docs/281.md) formalize Parisi ground-state value identities, a finite Gaussian approximation and the companion's full-support theorem, conditional on an admissible minimizer; the docs say the selected statement does not assert convergence of QAOA circuit energies, so the tier is unreviewed. The primary route imports the companion's support theorem; an alternative route uses Lopatto's and El Alaoui-Montanari-Sellke's results.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"A visible conjecture in quantum optimization, asked by the QAOA originators and supported by numerics, but recent and purely qualitative (no depth bound). Placed just above the FGG ring-of-disagrees QAOA conjecture (15), below Talagrand's critical SK overlap conjecture (30) and full RSB in the SK spin glass (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf","sourceName":"OpenAI math release: QAOA attains the SK ground-state energy in the thermodynamic-first limit","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Full support of the zero-temperature Sherrington-Kirkpatrick order parameter","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf","kind":"paper"},{"label":"Lean Comparator challenge SKValue (value identities only)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SKValue.lean","kind":"lean-proof"},{"label":"Basso, Farhi, Marwaha, Villalonga, Zhou, QAOA at high depth (arXiv 2110.14206)","url":"https://arxiv.org/abs/2110.14206v3","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"two-way-nfa-polynomial-complementation","name":"Polynomial complementation of two-way nondeterministic finite automata","shortName":"2NFA complementation","problemNumber":null,"field":"Automata theory; descriptional (state) complexity","fieldGroup":"Theoretical computer science","statement":"A two-way nondeterministic finite automaton (2NFA) has one read-only head moving in both directions between endmarkers. Deterministic two-way automata can be complemented with linearly many states (Geffert, Mereghetti and Pighizzini 2007), and Vardi (1989) complemented 2NFAs into exponential one-way automata. Kapoutsis (ICALP 2006) proved that small sweeping 2NFAs are not closed under complement, leaving unrestricted 2NFAs open. Is there a polynomial $p$, independent of the alphabet, such that for every $n$-state 2NFA over a finite alphabet the complementary language is recognized by a 2NFA with at most $p(n)$ states?","posedBy":"Open question in two-way automata state complexity; the manuscript cites it as open in Guillon, Prigioniero and Taheri (STACS 2026), after Kapoutsis's sweeping case (2006)","yearPosed":null,"ageNote":"The manuscript does not cite a first posing. It cites Kapoutsis (ICALP 2006) for the sweeping case and Guillon, Prigioniero and Taheri (STACS 2026, Section 1) as describing polynomial 2NFA complementation as open; the question belongs to the Sakoda-Sipser (1978) framework.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $n\\ge4$ there is an $n$-state 2NFA $A_n$ for relation-product liveness on $n-2$ points such that every 2NFA for its complement has at least $\\tfrac12 2^{\\lfloor(n-4)/127\\rfloor}-1$ states. Corollary: every 2DFA for the same language needs $\\tfrac12 2^{\\lfloor(n-4)/127\\rfloor}$ states when $n\\ge131$. The alphabet has $2^{(n-2)^2}$ letters, so the result says nothing about complementation over a fixed alphabet, about transition-table size, or about uniform logarithmic-space classes.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The complementation manuscript (September 25, 2026) adapts the conjugate-amplification pattern of its companion on one-way liveness, but its transport and nested-class bounds are proved anew for nondeterministic path diagrams.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question; it is negative for alphabet-uniform polynomial complementation and, as the paper says, asserts nothing over one fixed alphabet. lean/formalization.yaml lists a main result for this manuscript (comparator TwoWayComplementation, declaration OAI.TwoWayComplementation.complementation_lower_bound_finite, file OAI/Combinatorics/TwoWayAutomata/Main.lean). ComparatorChallenges/TwoWayComplementation.lean was read here: for every $n\\ge4$ it asserts an $n$-state 2NFA over the alphabet of binary relations on $n-2$ points such that every finite-state 2NFA recognizing the complement has at least $\\tfrac12 2^{\\lfloor(n-4)/127\\rfloor}-1$ states, with endmarkers, stay moves and finite-run acceptance. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Closure of small 2NFAs under complement is a known open question inside the automata state-complexity community, a sibling of the Sakoda-Sipser problem with less general fame. Placed below the Sakoda-Sipser problem (45), level with Brezis's Open Problem 5.6 (25) and the Cerny conjecture for one-cluster automata (25), and above the Inverse Generator Problem on Hilbert spaces (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026/paper.pdf","sourceName":"OpenAI math release: An exponential state lower bound for two-way nondeterministic complementation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An exponential two-way deterministic state lower bound for one-way liveness","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026/main.pdf","kind":"paper"},{"label":"Lean proof (OAI.TwoWayComplementation.complementation_lower_bound_finite)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/TwoWayAutomata/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: TwoWayComplementation.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/TwoWayComplementation.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"foulkes-howe-map-quadratic-stabilization","name":"Landsberg's Problem 7.19: a polynomial stabilization bound for the canonical Foulkes-Howe map","shortName":"Foulkes-Howe map: quadratic stabilization","problemNumber":null,"field":"Representation theory; Chow varieties and geometric complexity theory","fieldGroup":"Algebra","statement":"For $a,b\\ge1$ and a finite-dimensional complex vector space $V$, the canonical Foulkes-Howe (Hadamard-Howe) map $\\mu_{a,b,V}:\\mathrm{Sym}^b(\\mathrm{Sym}^aV)\\to\\mathrm{Sym}^a(\\mathrm{Sym}^bV)$ is induced by multiplication in $(\\mathrm{Sym}V)^{\\otimes a}$; its image is the degree-$b$ part of the coordinate ring of the Chow variety of products of $a$ linear forms, whose normalization is the whole target. Surjectivity in degree $b$ gives an equivariant embedding as in Foulkes' conjecture. Brion proved eventual surjectivity and later an effective bound depending on both $a$ and $\\dim V$. Landsberg's survey of geometric complexity theory (Problem 7.19) asks for a polynomial bound. Is there a degree bound, polynomial in $a$ and valid in every dimension, beyond which $\\mu_{a,b,V}$ is surjective?","posedBy":"J. M. Landsberg (Geometric complexity theory: an introduction for geometers, Ann. Univ. Ferrara 2015, Problem 7.19)","yearPosed":2015,"ageNote":"Cited from the published 2015 version (arXiv:1305.7387, 2013). The underlying stabilization question goes back to Brion (1993, 1997).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1: for every $a\\ge2$, $b\\ge a(a-1)$ and finite-dimensional complex $V$, $\\mu_{a,b,V}$ is surjective; equivalently the Chow-variety coordinate ring agrees with its normalization in degree $b$. In Landsberg's notation the bound is $d\\ge n(n-1)$, in every dimension. By complete reducibility this yields Foulkes' embedding $\\mathrm{Sym}^a(\\mathrm{Sym}^bV)\\hookrightarrow\\mathrm{Sym}^b(\\mathrm{Sym}^aV)$ for $b\\ge a(a-1)$. The least stabilization degree is not determined, the bound is not claimed sharp, and nothing is said about injectivity of the map in the other direction (which fails at $(5,5)$ and $(6,6)$).","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The stabilization theorem is also the large-degree input to the companion proving the sixth case of Foulkes' conjecture (separate entry). The release includes a Lean formalization of it.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1 and the history paragraph were read against Landsberg's request for a polynomial bound as the manuscript cites it. Lean: formalization.yaml lists ComparatorChallenges/FoulkesHowe (declaration OAI.Problem346.canonical_foulkes_howe_surjective, file OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean). The statement was read: for every finite-dimensional complex $V$ and $a\\ge2$, $b\\ge a(a-1)$, there is a unique linear map given by the averaged transpose formula on products, and it is surjective. That is the headline. Not rebuilt here. The paper does not claim the bound is sharp.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":17,"significanceNote":"A specific open problem from a survey, of interest to the plethysm and geometric complexity communities, with a known (non-polynomial, dimension-dependent) bound already in hand. Placed below The Stable Coordinate Conjecture in ambient dimension five (22), above The Iyengar-Ma-Walker Dutta multiplicity conjecture for complete local domains of positive characteristic (15), and far below A cubic lower bound for the border determinantal complexity of the permanent (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Quadratic stabilization of the canonical Foulkes-Howe map","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Foulkes' conjecture for the sixth symmetric power","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf","kind":"paper"},{"label":"Lean proof (OAI.Problem346.canonical_foulkes_howe_surjective)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RepresentationTheory/FoulkesHowe/Stabilization.lean","kind":"lean-proof"},{"label":"Landsberg, Geometric complexity theory: an introduction for geometers (Problem 7.19)","url":"https://doi.org/10.1007/s11565-014-0202-7","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"aeks-independence-conjecture","name":"The Ajtai-Erdos-Komlos-Szemeredi conjecture on independent sets in K_r-free graphs","shortName":"AEKS independence conjecture","problemNumber":802,"field":"Extremal graph theory: independence number","fieldGroup":"Combinatorics","statement":"Every graph with $n$ vertices and average degree $d$ has an independent set of size at least $n/(d+1)$, and disjoint cliques show this is sharp. Ajtai, Komlos and Szemeredi (1980) proved that triangle-free graphs have $\\alpha(G)\\gg n\\log d/d$, and Shearer sharpened the constant to $1-o(1)$. In 1981 Ajtai, Erdos, Komlos and Szemeredi conjectured the same order for every fixed forbidden clique; they proved $\\Omega_r(n\\log\\log d/d)$, and Shearer (1995) improved this to $\\Omega_r(n\\log d/(d\\log\\log d))$. For fixed $r\\ge4$, does every $K_r$-free graph on $n$ vertices with average degree $d$ have an independent set of size $\\Omega_r(n\\log d/d)$?","posedBy":"M. Ajtai, P. Erdos, J. Komlos and E. Szemeredi, On Turan's theorem for sparse graphs, Combinatorica 1 (1981), p. 314; Erdos Problem #802","yearPosed":1981,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $r\\ge4$ there is $c_r>0$ such that every finite $K_r$-free graph with $n$ vertices and average degree $d\\ge2$ has $\\alpha(G)\\ge c_r n\\log d/d$, removing the $\\log\\log d$ loss in Shearer's 1995 bound. The proof maximizes an entropy functional over vertex weights and shows that the weighted triangle mass is at most $O_r$ times the edge mass, via random multipliers, vertex splitting and a recursion on the clique size. The constant $c_r$ is not optimized and no leading coefficient (as in Shearer's triangle-free result) is claimed.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. A companion manuscript in the same family (5 October 2026) extends the method to correspondence coloring.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as stated in AEKS (1981) and Erdos Problem #802; the proof was not refereed. Lean: formalization.yaml lists OAI.CliqueFreeLog.logarithmic_independence_bound (Comparator challenge CliqueFreeLog, file OAI/Combinatorics/CliqueFree/Main.lean). Its statement was read here: for every r >= 4 there is c > 0 such that every finite simple graph that is r-clique-free with average degree 2|E|/|V| at least 2 has independence number at least c |V| log d / d. This is the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":46,"significanceNote":"A 45-year-old conjecture of Ajtai, Erdos, Komlos and Szemeredi at the heart of extremal graph theory, closely tied to off-diagonal Ramsey upper bounds; the K4-free case alone was a well-known benchmark. Scored above the Burr-Erdos hypercube Ramsey conjecture (40) and the Bollobas-Nikiforov conjecture (30), just below the linear list Hadwiger conjecture (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A logarithmic independence bound for clique-free graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Correspondence coloring graphs with a forbidden clique (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf","kind":"paper"},{"label":"Lean: OAI/Combinatorics/CliqueFree/Main.lean (logarithmic_independence_bound)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/CliqueFree/Main.lean","kind":"lean-proof"},{"label":"Erdos Problem #802","url":"https://www.erdosproblems.com/802","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-304-short-egyptian-fractions","name":"Erdős Problem 304: are all fractions $a/b$ sums of $O(\\log\\log b)$ distinct unit fractions?","shortName":"Erdős #304, short Egyptian fractions","problemNumber":304,"field":"Egyptian fractions; additive combinatorics","fieldGroup":"Number theory","statement":"For integers $1\\le a0$ and $b_0$ with $c_1\\log\\log b\\le N(b)\\le c_2\\log\\log b$ for $b\\ge b_0$; the upper bound holds for every numerator. Corollaries: the number $F(k)$ of $k$-term expansions of 1 satisfies $\\log\\log F(k)\\asymp k$ (Erdős-Graham asked for estimates), and the least integer $v(k)\\ge2$ missing from all $k$-term expansions of 1 satisfies $e^{e^{k/600}}\\le v(k)\\le 1+k^{2^{k-1}}$ eventually, so $\\log\\log v(k)\\asymp k$, answering the growth question of Erdős Problem 293 up to constants. Not shown: the optimal constants, or control of denominator sizes (unlike Tenenbaum-Yokota).","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 25, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against Erdős Problem 304 as stated on erdosproblems.com. Lean (in lean/formalization.yaml): OAI.Problem337.main_double_log_order in OAI/NumberTheory/EgyptianFractions/Main.lean states constants $c_1,c_2>0$ and $b_0$ with $c_1\\log\\log b\\le\\max_{1\\le a 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the Ara-Maltsiniotis conjecture as recorded by Ara (2023) and Guetta-Maltsiniotis (2024). The Lean challenge lean/ComparatorChallenges/ThomasonModelStructures.json (declarations OAI.Thomason.main and OAI.Thomason.nerveDetection_allDimensions; solution module OAI/CategoryTheory/Thomason/RankedFaithfulness.lean present at the pinned commit) is not in the formalization catalogue. Its top-level statement was read here: for strict omega-categories and for each finite n at least 1 there is a model category whose weak equivalences and fibrations are created by Ex^2 of the Street nerve, with the induced cofibrations, generating sets from boundary and horn images, left and right properness, local presentability and a Quillen equivalence condition; a second statement identifies weak equivalences through the Street nerve. That is the headline. The challenge file is about 136 KB because it defines strict omega-categories and orientals itself; those definitions were not audited. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A specialist conjecture in the homotopy theory of strict higher categories; the homotopy category itself was already known to be that of spaces (Ara-Maltsiniotis, Gagna), so the open part was the model-categorical refinement. Well below Grothendieck's homotopy hypothesis entry at 42, comparable to Szabo's Conjecture A at 22 and below the Robert-Tikuisis conjecture entry at 25.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Thomason Model Structures in Every Strict Higher Dimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/CategoryTheory/Thomason/RankedFaithfulness.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/CategoryTheory/Thomason/RankedFaithfulness.lean","kind":"lean-proof"},{"label":"Ara and Maltsiniotis, Vers une structure de categorie de modeles a la Thomason (arXiv 1305.5086)","url":"https://arxiv.org/abs/1305.5086","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-968-prime-ratio-increases","name":"Erdős and Prachar's question: do the increases of $p_n/n$ have positive lower density?","shortName":"Erdos Problem #968: increases of p_n/n","problemNumber":968,"field":"Distribution of prime gaps","fieldGroup":"Number theory","statement":"Let $p_n$ be the $n$th prime and $u_n=p_n/n$. Erdős and Prachar (1962) proved $\\sum_{p_nu_{n+1}$ has positive density, and asked about the reverse inequality. Since $u_np_n/n\\sim\\log p_n$, the question asks whether a positive proportion of prime gaps exceed the average gap, which needs control of how often gaps are large rather than of the largest gaps. Positive proportions were known only for gaps above a fixed fraction (about $0.579$) of $\\log p_n$. Does the set of $n$ with $p_n/n0$ there are $c(C)>0$ and $N_0(C)$ with $\\#\\{n\\le N:p_{n+1}-p_n>C\\log p_n\\}\\ge c(C)N$ for all $N\\ge N_0$, with ordinary lower density and equal weight on prime indices. Corollary 1.2 (take $C=2$) answers Erdős-Prachar: the increases of $p_n/n$ have positive lower density. The main theorem is stronger than the posed question. The constant $c(C)$ is not made explicit or compared with the Poisson prediction $e^{-C}$, and the further questions recorded with Problem #968 on $u_nu_{n+1}>u_{n+2}$ are not addressed.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The release also formalizes both the main theorem and the corollary in Lean.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1, Corollary 1.2 and its short proof were read against Erdős Problem #968. Lean: lean/docs/026.md links ComparatorChallenges/PrimeGaps (theorem OAI.Problem344.large_gaps_and_ratio_density, solution module OAI.NumberTheory.PrimeGaps.RatioCorollary, file exists at the pinned commit), not in formalization.yaml. The statement was read: for every $C>0$ there are $c>0$ and $N_0$ with at least $cN$ indices $n\\le N$ having $p_{n+1}-p_n>C\\log p_n$ for all $N\\ge N_0$, and the set of $n$ with $p_n/n1$, spectral gap at least $[24n^2\\binom n4]^{-1}$. Corollary: an exactly uniform sampler for any graphical $d$ with expected polynomial bit running time, terminating almost surely. This settles the simple undirected formulation; the bipartite formulation was proved earlier by Fu, Qin and Wang (listed separately), and the directed formulation is not addressed. The exponent 8 is not claimed to be sharp.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and Corollary 1.2, read against the conjecture as cited. The proof was not refereed; it adapts the Fu-Qin-Wang variance inequality from the bipartite case. Lean-checked on the release's own Comparator challenge SwitchChain together with its solution module OAI.Probability.SwitchChain.Main, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statements (OAI.Problem315.switch_chain_main, switch_chain_tv_bound, switch_connectivity) define the lazy kernel exactly as in the paper (holding probability 1/2, a uniform four-set and one of six ordered matching pairs) and state, for n >= 4 and every graphical d, mixing time at distance 1/4 at most 2 n^8, spectral gap at least 1/(24 n^2 C(n,4)) when there is more than one state, and switch connectivity. That is the headline claim. The exact uniform sampler (Corollary 1.2) is not formalized.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The undirected half of the standard rapid-mixing conjecture for degree-constrained sampling, cited across the MCMC community since 1997. Tied deliberately with the bipartite Kannan-Tetali-Vempala entry (30), below Kahn's flow conjecture (40), above Feige's hypergraph Moore-bound conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/main.pdf","sourceName":"OpenAI math release: Polynomial mixing of the switch chain for every graphical degree sequence","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kannan, Tetali and Vempala, Random Structures Algorithms 14 (1999)","url":"https://doi.org/10.1002/(SICI)1098-2418(199907)14:4%3C293::AID-RSA1%3E3.0.CO;2-G","kind":"problem-record"},{"label":"Erdos et al., The mixing time of switch Markov chains: a unified approach (Conjecture 1.1)","url":"https://doi.org/10.1016/j.ejc.2021.103421","kind":"problem-record"},{"label":"Lean Comparator challenge SwitchChain (not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SwitchChain.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Probability/SwitchChain/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/SwitchChain/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"campana-peternell-nef-tangent-dimension-six","name":"The Campana-Peternell conjecture on Fano manifolds with nef tangent bundle, in dimension six","shortName":"Campana-Peternell conjecture, dimension six","problemNumber":null,"field":"Algebraic geometry; Fano manifolds and rational homogeneous spaces","fieldGroup":"Geometry & topology","statement":"A smooth complex projective Fano manifold $X$ has nef tangent bundle $T_X$ whenever it is rational homogeneous $G/P$. Campana and Peternell (1991) conjectured the converse: every Fano manifold with nef tangent bundle is rational homogeneous. It was proved in dimension at most three and for fourfolds of Picard number greater than one by Campana-Peternell, for all fourfolds by Mok and Hwang, and for fivefolds by Watanabe and Kanemitsu; Kanemitsu proved it when $\\rho(X)>n-5$, which for sixfolds leaves Picard number one, and Watanabe treated the pseudoindex-four sixfolds. Is every smooth complex Fano sixfold with nef tangent bundle rational homogeneous?","posedBy":"Frederic Campana and Thomas Peternell","yearPosed":1991,"ageNote":"Campana and Peternell, Projective manifolds whose tangent bundles are numerically effective, Math. Ann. 289 (1991), 169-187, as cited by the manuscript.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous. The new step is Picard number one and pseudoindex five, where a characteristic-number identity $32-20u+3u^2=0$ forces the VMRT to be the Segre threefold $\\mathbb P^1\\times\\mathbb P^2\\subset\\mathbb P^5$, so $X\\cong\\mathrm{Gr}(2,5)$. Corollary 1.2: a compact Kahler manifold with nef tangent bundle and $\\dim X-\\tilde q(X)\\le6$ has universal cover $F\\times\\mathbb C^{\\tilde q(X)}$ with $F$ rational homogeneous. Dimensions seven and higher remain open.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 25, 2026) is the whole family; its folder includes a verification script for the exact arithmetic certificate in Appendix A.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the Campana-Peternell conjecture; it is exactly the conjecture in complex dimension six. The proof was not refereed. It relies on Kanemitsu's theorem for Picard number at least two, Watanabe's pseudoindex-four result (reproved independently), Hwang-Mok VMRT methods and Mok's recognition of the Grassmannian Gr(2,5). No Lean formalization exists for this family (lean/docs/067.md is absent at the pinned commit). The verification/verify.py arithmetic certificate was not run here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"The Campana-Peternell conjecture is a well-known problem in the geometry of Fano manifolds (around 45 as a whole); a single new dimension is a partial step, scored below the parent. Placed below the Campana-Peternell conjecture on Kodaira dimension (35), a different statement by the same authors, just below the Hadamard matrix of order 668 (30), and above K-Polystable Toric Fano Varieties With Small Alpha Invariants (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf","sourceName":"OpenAI math release: The Campana-Peternell conjecture in dimension six","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Verification script for the arithmetic certificate (verify.py)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/verification/verify.py","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"weisfeiler-leman-variable-dimension-exptime","name":"Berkholz's question: is Weisfeiler-Leman equivalence with the dimension as part of the input EXPTIME-complete?","shortName":"Variable-dimension Weisfeiler-Leman equivalence","problemNumber":null,"field":"Computational complexity; graph isomorphism and finite model theory","fieldGroup":"Theoretical computer science","statement":"The $k$-dimensional Weisfeiler-Leman algorithm repeatedly refines colours of $k$-tuples of vertices; two graphs are $k$-WL equivalent if their stable colour histograms agree. For fixed $k$ this takes time $n^{O(k)}$, so with $k$ given in binary the equivalence problem lies in EXPTIME. Seppelt (MFCS 2024), and independently Lichter, Rassmann and Schweitzer, proved it coNP-hard when $k$ is part of the input, and Seppelt recorded Christoph Berkholz's question whether it is in fact EXPTIME-complete. Given two graphs by adjacency matrices and a dimension $k\\ge2$ in binary, is deciding $k$-WL equivalence EXPTIME-complete?","posedBy":"Christoph Berkholz, as recorded by Tim Seppelt (MFCS 2024, the question preceding Theorem 5)","yearPosed":2024,"ageNote":"The manuscript cites the question only through Seppelt (2024); when Berkholz first raised it is not stated.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (principal manuscript): deciding joint-update $k$-WL equivalence with $k\\ge2$ in binary and explicit adjacency matrices is EXPTIME-complete under polynomial-time many-one reductions, also for connected simple uncoloured graphs of equal order and maximum degree three. Companions claim: deciding whether a given dimension identifies a graph (WL dimension at most $k$) is EXPTIME-complete, also with $k$ in unary; for every large fixed $k$, deciding $k$-WL equivalence needs $n^{\\Omega(k)}$ deterministic sequential time unconditionally (multitape Turing machines and log-word RAMs, diameter-two graphs); and parity lifts give $n^{\\Omega(k)}$ exclusions under ETH. No randomized or nonuniform lower bounds are claimed, and the separate-update convention is treated only in the fixed-dimension companions.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has four manuscripts of the same date sharing a compressed-computation construction: the principal one (variable-dimension equivalence), graph identification with dimension in the input, unconditional fixed-dimension time lower bounds, and parity lifts with ETH-based exclusions. All four main statements are formalized in Lean.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the history section of the principal manuscript were read against Berkholz's question as Seppelt records it. Lean: lean/docs/133.md links ComparatorChallenges/VariableWL (theorem OAI.VariableWL.main, solution module OAI.Combinatorics.VariableWL.Completeness, file exists at the pinned commit), not in formalization.yaml. The statement was read: with Turing machines, EXPTIME and polynomial-time many-one reductions defined from scratch, both the language of (G, H, k) with k >= 2 in binary and joint-update k-WL equivalence, and its connected subcubic equal-order restriction, are EXPTIME-complete. That is the headline. Not rebuilt here. The result uses the joint-update convention.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A recently recorded specialist question in the complexity of graph isomorphism heuristics, closing the gap between known coNP-hardness and the EXPTIME upper bound. Placed below Lichter and Schweitzer's question: does witnessed symmetric choice strictly increase the expressive power of CPT with counting? (18), above Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension (12), and far below The Blass-Gurevich-Shelah conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Variable-dimension-Weisfeiler-Leman-equivalence-on-general-and-subcubic-graphs-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Variable-dimension Weisfeiler-Leman equivalence on general and subcubic graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The complexity of identifying a graph by Weisfeiler-Leman refinement","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026/paper.pdf","kind":"paper"},{"label":"Companion: Unconditional time lower bounds for Weisfeiler-Leman equivalence","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf","kind":"paper"},{"label":"Companion: Parity lifts and bounded-treewidth witnesses for Weisfeiler-Leman equivalence","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.VariableWL.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/VariableWL/Completeness.lean","kind":"lean-proof"},{"label":"Seppelt, An algorithmic meta theorem for homomorphism indistinguishability (MFCS 2024)","url":"https://doi.org/10.4230/LIPIcs.MFCS.2024.82","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"smooth-realization-finite-entropy","name":"The smooth realization problem: does finite entropy suffice for a smooth volume-preserving model?","shortName":"Smooth realization problem","problemNumber":null,"field":"Ergodic theory; smooth realization of measure-preserving systems","fieldGroup":"Analysis","statement":"A smooth positive-volume model of an invertible ergodic transformation $(X,\\mu,T)$ of a standard nonatomic probability space is a $C^\\infty$ diffeomorphism $S$ of a compact finite-dimensional manifold $M$ preserving a probability $\\nu$ with strictly positive smooth density, together with a measurable isomorphism $F$ between invariant conull sets with $F\\circ T=S\\circ F$. Finite Kolmogorov-Sinai entropy is necessary (Ruelle's inequality), and Lind and Thouvenot realized every finite-entropy ergodic system by a homeomorphism of the two-torus, but a realization with a singular invariant measure does not give a smooth volume. The smooth realization problem, as discussed by Katok and Thouvenot, asks whether this entropy restriction is the only one. Does every ergodic invertible transformation with finite entropy admit a smooth positive-volume model?","posedBy":"The classical smooth realization problem, in the formulation discussed by Anatole Katok and Jean-Paul Thouvenot (Ann. IHP Probab. Stat. 1997, pp. 324-326)","yearPosed":1997,"ageNote":"The manuscript cites Katok and Thouvenot (1997) for the classical formulation and Foreman and Weiss (2022) for a variant; it does not name an earlier originator, though the problem is commonly traced to Anosov and Katok (1970). Year taken from the cited source.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: there is an ergodic invertible transformation of a standard nonatomic probability space with $h_\\mu(T)=0$ that is not measurably conjugate on invariant conull sets to any $C^\\infty$ diffeomorphism of a compact finite-dimensional manifold (boundary allowed, orientation not required) preserving a strictly positive smooth probability density. A single symbolic example excludes every dimension. So finite entropy, even zero entropy, does not suffice. It does not exclude models whose invariant measure is only equivalent to volume, and it does not characterize which systems are smoothly realizable.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the problem as the manuscript frames it. The proof was not refereed. The paper's own scope limits: the model must preserve a strictly positive smooth density; it does not claim an obstruction for invariant measures merely equivalent to volume (the Foreman-Weiss formulation), nor for Z^2 actions. Lean-checked on the release's own Comparator challenge SmoothObstruction together with its solution module OAI.Dynamics.SmoothObstruction.Main, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.SmoothObstruction.main) gives a standard nonatomic ergodic system with FINITE entropy and no smooth positive-density model on any compact manifold of any dimension, with or without boundary. That is a full negative answer to the posed question, but it is weaker than the paper's zero-entropy headline; the zero-entropy conclusion is not formalized (the release's own scope note for family 152 says so).","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"One of the basic open questions of smooth ergodic theory since Anosov and Katok, asked in surveys for decades; a negative answer was not expected to come this cheaply. Placed below Katok's entropy rigidity conjecture (52) and Shub's entropy conjecture (48), level with C1 regularity of infinity-harmonic functions (42), above Voiculescu's free entropy dimension question (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A zero-entropy system without a smooth positive-volume model","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Katok and Thouvenot, Slow entropy type invariants and smooth realization (1997)","url":"https://www.numdam.org/item/AIHPB_1997__33_3_323_0/","kind":"problem-record"},{"label":"Lean Comparator challenge SmoothObstruction (finite-entropy version; not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SmoothObstruction.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Dynamics/SmoothObstruction/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Dynamics/SmoothObstruction/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"random-triangle-removal-sharp-constant","name":"The sharp constant for the terminal leave of random triangle removal (triangle case of the Joos-Kühn conjecture)","shortName":"Random triangle removal sharp constant","problemNumber":null,"field":"Random graph processes; random greedy packing","fieldGroup":"Combinatorics","statement":"Start from $K_n$ and repeatedly delete the three edges of a uniformly random remaining triangle until none is left; let $F_n$ be the number of edges remaining. Bollobas and Erdos conjectured (1990) that $F_n$ has order $n^{3/2}$; after work of Spencer, Rodl-Thoma and Grable, Bohman, Frieze and Lubetzky proved $F_n=n^{3/2+o(1)}$ with high probability (2015). Joos and Kuhn (2024) studied removal of a general strictly balanced hypergraph and conjectured (Conjecture 16.2) an asymptotic constant for the terminal leave, whose triangle case predicts $F_n/n^{3/2}\\to 1/(2\\sqrt2)$ in probability. Is that so?","posedBy":"Felix Joos and Marcus Kühn (Conjecture 16.2 of The hypergraph removal process); order of magnitude conjectured by Bollobás and Erdős (1990)","yearPosed":2024,"ageNote":"arXiv 2412.15039, first version 19 December 2024; the manuscript cites the conjecture's numbering from version 2. The Bollobas-Erdos order-of-magnitude conjecture is cited through Bohman-Frieze-Lubetzky (2015).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for uniform triangle removal from $K_n$, $\\mathbb E[(F_n/n^{3/2}-1/(2\\sqrt2))^2]\\to0$; hence $F_n/n^{3/2}\\to1/(2\\sqrt2)$ in probability and $\\mathbb E F_n/n^{3/2}\\to1/(2\\sqrt2)$. The paper says it does not give the sharp constant for other starting graphs, for general hypergraph removal (the rest of Joos-Kuhn Conjecture 16.2), or a fluctuation law.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 25, 2026. It states that its only external proof input is Joos and Kuhn's early-prefix control; all continuation estimates are proved in the paper.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against the triangle case of Joos-Kuhn Conjecture 16.2 as the paper states it. Lean: the challenge lean/ComparatorChallenges/TriangleRemoval.json (solution_module OAI.Combinatorics.TriangleRemoval.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml. The statement OAI.SharpTerminalLeave.sharp_terminal_leave was read here: it defines the process as a PMF on edge sets run for $\\binom n2$ steps from $K_n$ and asserts $L^2$ convergence of $F_n/n^{3/2}$ to $1/(2\\sqrt2)$, convergence in probability and convergence of the normalized mean. That is the headline. Not rebuilt here. Entry is partial because the conjecture covers general strictly balanced hypergraphs.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"The sharp constant in the classic random greedy triangle packing, a well-known question since Bohman-Frieze-Lubetzky fixed the exponent, but only the triangle case of a 2024 conjecture. Level with the Euclidean prefix discrepancy entry (22), below the Friedgut-Kalai threshold-width conjecture for hypergraphs (25), above the Ascoli-He-Park-Talagrand decomposition conjecture (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf","sourceName":"OpenAI math release: The sharp terminal leave in random triangle removal","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Joos and Kühn, The hypergraph removal process (arXiv 2412.15039), Conjecture 16.2","url":"https://arxiv.org/abs/2412.15039","kind":"problem-record"},{"label":"Lean formalization (TriangleRemoval main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/TriangleRemoval/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stein-hilbert-transform-lipschitz-vector-fields","name":"Stein's conjecture on the Hilbert transform along Lipschitz vector fields","shortName":"Stein's Lipschitz vector field conjecture","problemNumber":null,"field":"Harmonic analysis; directional singular integrals","fieldGroup":"Analysis","statement":"For a unit vector field $v:\\mathbb R^2\\to S^1$ consider the truncated directional Hilbert transform $H_{v,a}f(x)=\\mathrm{p.v.}\\int_{|t|0$ such that, with outer length $a=\\epsilon_0/\\|v\\|_{\\mathrm{Lip}}$, $H_{v,a}$ is of weak type $(2,2)$ with a constant independent of $v$. Lacey and Li trace it to Zygmund's question on differentiation along Lipschitz directions; Holder fields of any exponent below one do not suffice. Is $H_{v,a}$ uniformly weak-$(2,2)$ bounded for all Lipschitz unit fields at this scale?","posedBy":"Elias M. Stein","yearPosed":null,"ageNote":"The manuscript cites the conjecture in the formulation of Lacey and Li, On a conjecture of E. M. Stein on the Hilbert transform on vector fields, Mem. AMS 205 (2010), Conjecture 1.5, and does not give the year Stein first posed it. Year left blank.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are absolute $a\\in(0,1/2)$ and $C$ such that every Lipschitz unit field $v$ on $\\mathbb R^2$ with $\\mathrm{Lip}(v)\\le1$ satisfies $\\sup_{0<\\epsilon 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 25, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Section 1.1 were read against Stein's conjecture as formulated by Lacey and Li; the theorem gives a strong $L^2$ bound, uniform in the inner truncation, for every 1-Lipschitz unit field at an absolute outer length, and scaling transfers it to outer length $a/\\mathrm{Lip}(v)$, which is the conjecture's scale. The challenge LipschitzHilbert is not in the formalization catalogue (lean/formalization.yaml); it is linked from lean/docs/083.md, its solution module exists at the pinned commit, and its statement was read here: absolute $a\\in(0,1/2)$ and $C$ with uniform truncation, principal-value, weak-(2,2) and bounded $L^2$ extension conclusions for every 1-Lipschitz unit field, plus a version for Lipschitz constant $K$ with $R\\le1/(10^6K)$. This states the headline. Not rebuilt here. The paper itself stresses it does not treat arbitrary outer lengths.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"One of the central open problems of planar harmonic analysis for decades, the singular-integral partner of Zygmund's differentiation conjecture, the subject of an AMS Memoir and of work by Bateman, Thiele, Guo, Di Plinio and others. Placed above the three-dimensional Kakeya maximal conjecture entry (48) and Brennan's conjecture (45), and well above Stein's dimension-free Riesz transform problem (38); below the Falconer distance conjecture (60).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026/main.pdf","sourceName":"OpenAI math release: A uniform Hilbert transform estimate for Lipschitz directions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.LipschitzHilbert.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/LipschitzHilbert/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: LipschitzHilbert.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/LipschitzHilbert.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"random-k-sat-threshold-existence","name":"The satisfiability threshold conjecture: a limiting random k-SAT threshold for every k >= 3","shortName":"Random k-SAT threshold existence","problemNumber":null,"field":"Random constraint satisfaction; random k-SAT phase transition","fieldGroup":"Probability & statistics","statement":"Fix $k\\ge3$ and let $F_{n,m}$ be a conjunction of $m$ independent uniformly random $k$-clauses on $n$ Boolean variables (distinct variables in a clause, fair signs). Friedgut (with Bourgain) proved a sharp threshold: there is a sequence $c_k(n)$ around which the satisfiability probability drops from near one to near zero, but not that $c_k(n)$ converges. Ding, Sly and Sun proved convergence, with the predicted value, for all sufficiently large $k$; the case $k=2$ is classical. Chvatal and Reed conjectured that each fixed clause size has a limiting density. Is there, for every fixed $k\\ge3$, a constant $\\alpha_k$ such that $\\Pr(F_{n,\\lfloor cn\\rfloor}\\text{ satisfiable})\\to1$ for $c<\\alpha_k$ and $\\to0$ for $c>\\alpha_k$?","posedBy":"Vasek Chvatal and Bruce Reed (Mick gets some (the odds are on his side), FOCS 1992)","yearPosed":1992,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every fixed $k\\ge3$ there is $\\alpha_k\\in(0,\\infty)$ with $\\Pr(F_{n,\\lfloor cn\\rfloor}\\in\\mathrm{SAT})\\to1$ for $c<\\alpha_k$ and $\\to0$ for $c>\\alpha_k$; it says nothing at $c=\\alpha_k$ and does not identify $\\alpha_k$ for small $k$. Companions claim hitting-time variance $\\Theta_k(n)$ for $k\\ge4$ and $\\Theta(n)$ for $k=3$, and that $\\alpha_3$ is a computable real. Priority: the release credits Gaia Carenini's ECCC TR26-229 (made public 5 October 2026) with first resolving the existence conjecture; this is an independent second proof of the same result, not a contradiction. Carenini's paper states that none of its mathematical arguments was generated by a large language model; it proves the existence via a scaling window of width $O(n^{1/2+1/k})$ and the Abbe-Montanari criterion.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The release itself credits Gaia Carenini (ECCC TR26-229, public 5 October 2026) with priority for the threshold-existence result and presents this family as an alternative proof plus sharper variance and computability results.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the principal manuscript, and the abstracts of the three companions, read against Chvatal and Reed's conjecture as cited. The proofs were not refereed. Scope stated by the paper: clauses on distinct variables sampled with replacement; strict-side statement, nothing at $c=\\alpha_k$; the value of $\\alpha_k$ is not identified. Lean-checked on the release's own Comparator challenge FixedClauseThreshold together with its solution module OAI.Probability.RandomSAT.Main, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.FixedClauseThreshold.main) says: for every k >= 3 there is a real alpha > 0 such that the satisfiability probability (literal finite counting ratio) at floor(c n) clauses tends to 1 for 0 <= c < alpha and to 0 for c > alpha. That is the headline claim. The variance (SATVariance) and 3-SAT computability (SATComputability) challenges also exist with solution modules; they were not read in detail.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The satisfiability threshold conjecture is one of the best-known problems linking theoretical computer science and statistical physics, open for small k since 1992 after Ding-Sly-Sun settled large k. Scored on the question, not discounted for shared priority. Placed between Talagrand's convolution conjecture (37) and the KLS conjecture for quadratic forms (45), above the bipartite Kannan-Tetali-Vempala entry (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf","sourceName":"OpenAI math release: A Limiting Satisfiability Threshold for Every Fixed Clause Size","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Variance of the Random k-SAT Hitting Time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026/article.pdf","kind":"paper"},{"label":"Companion: Linear Variance of the Random 3-SAT Hitting Time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf","kind":"paper"},{"label":"Companion: Computing the Random 3-SAT Threshold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf","kind":"paper"},{"label":"Chvatal and Reed, Mick gets some (the odds are on his side), FOCS 1992","url":"https://doi.org/10.1109/SFCS.1992.267789","kind":"problem-record"},{"label":"Carenini, ECCC TR26-229 (credited with priority by the release)","url":"https://eccc.weizmann.ac.il/report/2026/229/","kind":"paper"},{"label":"Lean Comparator challenge FixedClauseThreshold (not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/FixedClauseThreshold.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Probability/RandomSAT/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/RandomSAT/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"rcd-regular-point-bi-lipschitz-chart","name":"Bi-Lipschitz charts at every regular point of a noncollapsed RCD(K,n) space","shortName":"Bi-Lipschitz charts at RCD regular points","problemNumber":null,"field":"Metric measure spaces; synthetic lower Ricci bounds; RCD spaces","fieldGroup":"Geometry & topology","statement":"Let $(X,d,\\mathcal H^n)$ be a noncollapsed $\\mathrm{RCD}(K,n)$ space, the synthetic setting containing noncollapsed Ricci limit spaces. A point is regular if every pointed tangent there is $\\mathbb R^n$. Mondino-Naber proved bi-Lipschitz rectifiability on measurable pieces and Kapovitch-Mondino gave an open manifold region with bi-Holder charts; Cheeger and Colding (1997) had asked whether bi-Holder parametrizations of quantitatively regular Ricci-limit neighborhoods can be upgraded to bi-Lipschitz. The regular-point conjecture, recalled by Honda and Zhang (2026), asks: does every regular point of a noncollapsed $\\mathrm{RCD}(K,n)$ space have an open neighborhood, with the restricted distance, bi-Lipschitz homeomorphic to an open subset of $\\mathbb R^n$?","posedBy":"Conjecture recalled by Shouhei Honda and Ruobing Zhang (arXiv:2608.16021, 2026), Section 1.2; antecedent question of Jeff Cheeger and Tobias Colding, J. Differential Geom. 46 (1997), Remark 5.15","yearPosed":null,"ageNote":"The manuscript says the conjecture is explicitly recalled by Honda and Zhang (2026) and treats Cheeger-Colding (1997, Remark 5.15) as a historical antecedent concerning Ricci-limit neighborhoods; it does not say who first stated the regular-point form or when, so yearPosed is left empty.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that for every integer $n\\ge2$ there is $L_n\\ge1$ such that every regular point of every noncollapsed $\\mathrm{RCD}(K,n)$ space, $K\\in\\mathbb R$, has an open neighborhood $L_n$-bi-Lipschitz homeomorphic, with the ambient distance, to an open subset of $\\mathbb R^n$; the neighborhood depends on the space and the point. Corollary: an open set of full $\\mathcal H^n$ measure containing all regular points carries a countable bi-Lipschitz atlas with $L_n^2$ transition maps. It does NOT claim that the regular set is open, any collapsed or non-RCD statement, or regularity of the metric tensor in the charts.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. No Lean formalization of this result is in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture as the manuscript states it; Honda-Zhang was not opened here. Scope limits the paper states: noncollapsed means the reference measure is exactly $\\mathcal H^n$; regularity means every tangent at the point is Euclidean; the chart is an open neighborhood with the restricted ambient distance; no Holder control of the metric tensor and no bounded-Laplacian coordinates are claimed (Colding-Naber examples show the former can fail). Note the release's family summary says the constant is point dependent, but the manuscript's Theorem 1.1 and abstract say the constant $L_n$ depends only on $n$. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"A structural question about spaces with lower Ricci bounds, with roots in Cheeger and Colding's 1997 bi-Holder theory; since noncollapsed Ricci limits are included, a positive answer upgrades their regular-point charts to bi-Lipschitz. Specialist but in an active field. Below the Lang-Plaut problem (38) and the universal volume growth entry (38), above the weak simplex conjecture (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Honda and Zhang (arXiv:2608.16021), Section 1.2, where the conjecture is recalled","url":"https://arxiv.org/abs/2608.16021v1","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mean-payoff-games-quasipolynomial-time","name":"Quasipolynomial-time solution of mean-payoff games with binary weights","shortName":"Mean-payoff games in quasipolynomial time","problemNumber":null,"field":"Algorithmic game theory; games on graphs; complexity of NP and coNP problems","fieldGroup":"Theoretical computer science","statement":"In a finite two-player mean-payoff game on a directed graph with integer edge weights $w$ given in binary, Max wins from a vertex if she can force $\\liminf_{T\\to\\infty}\\frac1T\\sum_{t 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The family has four manuscripts. The deterministic algorithm (25 September 2026) imports only the finite-box comparison facts from the randomized companion of the same date; the two October 5 manuscripts extend the method to turn-based stochastic games and to mean-payoff parity games. The randomized companion's main theorem has a Lean formalization in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against the question: a uniform deterministic algorithm outputs the full zero-threshold winning set with $2^{C(\\log_2(L+2))^2}$ bit operations, $L$ the complete binary input length; Section 7 adds exact values and optimal positional strategies. The release formalizes only the randomized companion: ComparatorChallenges/RandomizedMeanPayoff.lean (declaration OAI.randomized_quasipolynomial_mean_payoff, file OAI/Computability/RandomMean/Main.lean, listed in formalization.yaml) states that one Turing machine with random bits halts within quasipolynomially many steps on every tape and outputs the exact winning set on at least 7/8 of tapes. That is a randomized form of the result, not the deterministic headline, so under the strict tier rule this entry stays Unreviewed. Not rebuilt here. A second Lean challenge, TruffetCounterexample, concerns an appendix about another author's proposed algorithm and is not in formalization.yaml.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Whether mean-payoff games, which lie in NP and coNP, can be solved efficiently in the binary input length is a central open problem of games on graphs; a quasipolynomial bound, the analogue of the 2017 parity-games breakthrough, was itself open. Scored as a partial answer below the full polynomial question. Level with the Gotsman-Linial entry (42), below the Sakoda-Sipser problem (45) and the 3SUM hypothesis (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Deterministic quasipolynomial-time mean-payoff games","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Randomized quasipolynomial-time mean-payoff games","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf","kind":"paper"},{"label":"Companion: Mean-payoff parity games in quasipolynomial time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026/mean-payoff-parity.pdf","kind":"paper"},{"label":"Companion: Turn-based stochastic mean-payoff games in deterministic quasipolynomial time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/stochastic-mean-payoff-games.pdf","kind":"paper"},{"label":"Lean: randomized quasipolynomial mean-payoff algorithm (companion result)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/RandomMean/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: RandomizedMeanPayoff","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/RandomizedMeanPayoff.lean","kind":"lean-proof"},{"label":"Truffet, claimed strongly polynomial tropical LP method (arXiv:2603.26423 v4), rebutted by the randomized companion","url":"https://arxiv.org/abs/2603.26423v4","kind":"other"},{"label":"Truffet, Max-Atom problem in strongly polynomial time (HAL note, 2025; not addressed by the release)","url":"https://hal.science/hal-05491586v1","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dft-n-log-n-lower-bound","name":"Does the exact discrete Fourier transform require order n log n operations in unrestricted linear circuits?","shortName":"The n log n lower bound for the DFT","problemNumber":null,"field":"Algebraic complexity; arithmetic circuits for the Fourier transform","fieldGroup":"Theoretical computer science","statement":"The fast Fourier transform computes $F_n=(\\zeta_n^{jk})$ with $O(n\\log n)$ arithmetic operations, and no asymptotically faster method was known. Matching lower bounds were proved only in restricted models: Morgenstern for linear circuits with bounded coefficients, Ailon for unitary two-coordinate gates or bounded intermediate conditioning. In the unrestricted model, a linear circuit uses gates $u+v$, $u-v$ and $\\lambda u$ with arbitrary fixed complex $\\lambda$, each costing one, and must output $F_nx$ exactly for every $x$. Is the minimum size $L(n)$ of such a circuit at least $cn\\log n$ for some $c>0$ and all large $n$, or can the Fourier transform be computed exactly with $o(n\\log n)$ operations?","posedBy":"Long-standing open question in algebraic complexity; discussed by Ailon (2013) and Alman and Rao (2023)","yearPosed":null,"ageNote":"The manuscripts give no posing date. They cite Ailon (2013, abstract) and Alman-Rao (2023, Section 1) for the question; the restricted lower bound it contrasts with is Morgenstern (1973).","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Principal (Theorem 1.1): in exact complex linear circuits with arbitrary prechosen coefficients, $\\liminf_n L(n)/(n\\log_2 n)=0$, along lengths that are products of distinct primes, so the unrestricted $\\Omega(n\\log n)$ lower bound is false. Companion (Theorem 1.1): one deterministic algorithm computes $F_nx$ at every length in $O(n(\\log n)^{\\theta}(\\log\\log n)^{4-\\theta})$ operations with $\\theta<1-10^{-13}$, counting scalar preparation and indexing, given one specified root of unity. Neither result concerns bounded coefficients (where Morgenstern's bound stands), numerical stability, bit complexity or finite fields, and the constants are astronomically large.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 25, 2026: 'Finite tensor savings and exact Fourier circuits' (this entry's principal, nonuniform circuits along a subsequence) and 'An explicit power saving for the exact discrete Fourier transform' (a uniform algorithm at every length). The explicit-power-saving paper reuses a phase network from another release manuscript on integer multiplication.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the question; it states $\\liminf L(n)/(n\\log_2 n)=0$ in the unrestricted model, refuting the lower bound. lean/formalization.yaml lists comparator ExactFourier, declaration OAI.ExactFourier.main_theorem. The statement was read here: for every $c>0$ and $N_0\\ge2$ there are $n\\ge N_0$ and a circuit (a DAG of add, subtract and scale-by-any-complex gates) computing the unnormalised $F_n$ exactly with fewer than $cn\\log_2 n$ gates. That is the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The model allows unbounded coefficients and conditioning; nothing is claimed for bit complexity or floating-point FFTs. The 7 October 2026 update added the companion's Comparator challenge UniformFourier (OAI.PowerSaving.transform_main and convolution_main, solution module OAI.Computability.FourierTransform.Main); read here, it states one fixed exact-complex RAM program computing the DFT (and one for convolution) at every positive length with work $O(n(\\log n)^{1-10^{-13}})=o(n\\log n)$, given a supplied root of unity.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Whether the FFT is optimal in the general arithmetic model is a well-known open question in algebraic complexity, open since the 1970s, though usually stated informally. Placed below the matrix multiplication exponent entry (55) and deterministic polynomial factorization over prime fields (55), and level with the word problem for finite-rank Artin groups (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf","sourceName":"OpenAI math release: Finite tensor savings and exact Fourier circuits","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Lean proof (OAI.ExactFourier.main_theorem)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/FourierCircuit/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: ExactFourier.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ExactFourier.lean","kind":"lean-proof"},{"label":"Companion: An explicit power saving for the exact discrete Fourier transform","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf","kind":"paper"},{"label":"Lean statement (companion every-length algorithm): UniformFourier","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/UniformFourier.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"letrouit-brenier-map-square-root-stability","name":"Letrouit's square-root stability conjecture for Brenier maps from a uniform convex source","shortName":"Letrouit's Brenier stability conjecture","problemNumber":null,"field":"Optimal transport; quantitative stability","fieldGroup":"Analysis","statement":"Let $\\rho$ be the uniform probability measure on a compact convex body $K\\subset\\mathbb R^d$, $d\\ge2$, and for a probability measure $\\mu$ on a fixed compact set $Y$ let $T_\\mu$ be the quadratic optimal (Brenier) map pushing $\\rho$ to $\\mu$. Known target-uniform bounds had exponents such as $W_1^{1/6}$ (Delalande-Merigot) and $W_1^{2/15}$; Letrouit showed exponents above $1/3$ fail for some nonconvex sources and conjectured that for a uniform convex source $\\|T_\\mu-T_\\nu\\|_{L^2(\\rho)}\\le C\\,W_2(\\mu,\\nu)^{1/2}$ uniformly over targets supported in $Y$. Is this square-root estimate true?","posedBy":"Cyril Letrouit (Unstable optimal transport maps, Conjecture 3)","yearPosed":2025,"ageNote":"arXiv 2510.13265 (October 2025), published in Comptes Rendus Mathematique 364 (2026), which the manuscript cites.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $K$ compact convex with interior in $\\mathbb R^d$, $d\\ge2$, and $Y$ compact, $\\|T_\\mu-T_\\nu\\|_{L^2(\\rho)}\\le C(K,Y)W_2(\\mu,\\nu)^{1/3}$ for all probability measures $\\mu,\\nu$ on $Y$, atomic or not; the exponent $1/3$ cannot be raised even for $K=Y=[-1,1]^d$ and three-atom targets, so Letrouit's square-root conjecture is false and $1/3$ is the sharp uniform exponent. Not shown: sharp exponents in $W_1$ (a sharp $W_1^{1/4}$ bound is attributed to Merigot) or for nonuniform source densities.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 25, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against Letrouit's Conjecture 3 as the manuscript cites it. Lean: the challenge lean/ComparatorChallenges/Brenier.json (solution_module OAI.Analysis.Brenier.Stability, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml. The statements one_third_stability and one_third_exponent_is_sharp were read here: a uniform constant for $W_2^{1/3}$ stability of unique quadratic optimal maps from the uniform measure on any compact convex body with interior, $d\\ge2$, and, for every exponent above $1/3$ and every constant, two three-atom targets on the cube violating the bound. The second disproves the conjecture; both state the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A one-year-old conjecture in an active corner of quantitative optimal transport, settled with the sharp exponent; a specialist question. Level with Brannen's simplex conjecture (12), below Zhu-Fang-Shi's invariant projection question (15), above the typical Erdős problem (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf","sourceName":"OpenAI math release: Sharp One-Third Stability of Brenier Maps","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Letrouit, Unstable optimal transport maps (arXiv 2510.13265), Conjecture 3","url":"https://arxiv.org/abs/2510.13265","kind":"problem-record"},{"label":"Lean formalization (Brenier stability and sharpness)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Brenier/Stability.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"maximal-specieslike-clusters-with-a-fixed-real-founding-window","name":"Maximal specieslike clusters with a fixed real founding window","shortName":"Maximal clusters with a real founder window","problemNumber":null,"field":"Infinite labelled graphs; mathematical ancestry","fieldGroup":"Combinatorics","statement":"Alexander, Specieslike clusters based on identical ancestor points (2026), Section 6, asks for alternative constraints to common ancestry in maximal-cluster existence. Replace that condition by a fixed real founding duration: every internal founder of S is born no later than the earliest member birthdate plus Delta. On natural-number organism identifiers with real chronological birthdates, finite strict earlier sublevels and finite child sets, does every organism belong to a maximal connected, ancestry-convex, reflecting IAP set satisfying this same window, for each Delta>=0? Maximality is within the specified fixed-window class, not among unrestricted specieslike sets.","posedBy":"Samuel A. Alexander, Specieslike clusters based on identical ancestor points (2026), Section 6; a proposed fixed-real-window replacement constraint.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Yes. Every organism belongs to an inclusion-maximal member of the fixed-window class. The proof constructs an actual seed, proves reflection under appropriate intersections, proves IAP and the real window survive the required directed unions, and applies constrained maximal extension. Relabelling preserves original real birthdates, including ties, and the duration shared by every candidate and competitor. No finite-root premise is needed. This supplies a checked replacement-constraint theorem responding to the source direction, not merely a conditional result assuming the needed seed. The eight-module proof chain has 50 selected endpoints. It does not yield uniqueness, a partition, an effective classifier or empirical species validation. Maximality keeps connectivity, convexity, IAP, reflection and the same founding window. Biological plausibility, priority and whether the constraint matches the author's intended qualitative distinction remain for review.","claimIssueNote":null,"solveDate":"2026-09-25","model":"OpenAI Codex (GPT-6 family)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"A human collaborator selected the research directions and requested source correspondence, explanations, proof checking and submission. Codex agents developed the concrete mathematical construction or deduction, Lean proofs and exposition. Other agents in the project reviewed the arguments and source correspondence. These are internal project checks, not independent human expert review. Model-led discovery of the submitted scoped result is claimed; independent human verification and worldwide priority are not claimed.","verification":"lean-checked","verificationNote":"Checked here on 30 September 2026 to the extent the artefacts allow. Lean 4.33.1 with pinned Mathlib; eight modules contribute 50 selected endpoints for the founder-window chain, permitting only propext, Classical.choice and Quot.sound, and the Verify workflow is green on the correct repository. Not rebuilt here and no independent audit of the informal-to-formal correspondence, so Lean-checked. The source URL was corrected for the repository's trailing dash. Partial, as submitted, and the submitter's own scope statement is kept: the whole class is fixed before maximality is asserted, so this is maximality within the fixed-window class and not unrestricted maximal specieslikeness.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":6,"significanceNote":"Alexander's 2026 Section 6 asks for alternative constraints to common ancestry in maximal-cluster existence, and this answers that direction with one particular constraint, a fixed real founding window, chosen by the submitter. Below the two 2013 Section 6 entries at 8, where Alexander stated the questions themselves, and above the Thue-Morse height entry at 5. The direction was asked for in print; the constraint was not.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-","sourceName":"Pinned public research note, Lean source and verification receipts","collection":null,"collectionVersion":null,"links":[{"label":"Source maximal-cluster question, section 6","url":"https://arxiv.org/html/2602.05274v1#S6","kind":"problem-record"},{"label":"Actual original-graph maximal-window theorem","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/real/RealSpeciesTheorem.lean","kind":"lean-proof"},{"label":"Real window and union properties","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/real/RealFounderWindow.lean","kind":"lean-proof"},{"label":"Actual seed construction","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/real/SpeciesSeed.lean","kind":"lean-proof"},{"label":"Frozen founder proof packet","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/verification/founder-window-packet.json","kind":"other"},{"label":"Source/question provenance ledger","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/QUESTION-LEDGER.md","kind":"other"},{"label":"Integrated publication coverage","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/1592b89731d10fc49019ff22cf2f93c7bc799118/research/publication-2026-09-29/COVERAGE.md","kind":"other"},{"label":"Successful integrated hosted proof audit","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/actions/runs/36631876391","kind":"other"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"barendregt-geuvers-klop-conjecture-beta","name":"The Barendregt-Geuvers-Klop conjecture for beta reduction: weak normalization implies strong normalization in every pure type system","shortName":"Barendregt-Geuvers-Klop conjecture (beta)","problemNumber":null,"field":"Type theory; lambda calculus; pure type systems","fieldGroup":"Logic & foundations","statement":"A pure type system is given by a set of sorts $\\mathcal S$, axioms $\\mathcal A\\subseteq\\mathcal S^2$ and product rules $\\mathcal R\\subseteq\\mathcal S^3$, with full $\\beta$-reduction allowed everywhere, including inside type annotations. The system is weakly normalizing if every legal expression in every valid context has a $\\beta$-normal form, and strongly normalizing if no such expression admits an infinite $\\beta$-reduction sequence. Strong normalization was known to follow from weak normalization only for restricted classes (Sorensen; Barthe, Hatcliff and Sorensen for nondependent systems). Is every weakly $\\beta$-normalizing pure type system strongly $\\beta$-normalizing?","posedBy":"Barendregt, Geuvers and Klop; stated by Herman Geuvers as Conjecture 8.1.2 of his thesis (Nijmegen, 1993)","yearPosed":1993,"ageNote":"The manuscript cites Geuvers's 1993 thesis (Conjecture 8.1.2) for the system-wide conjecture, which he states for both beta and beta-eta reduction. The name credits Barendregt, Geuvers and Klop; the thesis is the earliest written statement the manuscript gives.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every pure type system $(\\mathcal S,\\mathcal A,\\mathcal R)$, with no functionality assumption on $\\mathcal A$ or $\\mathcal R$, if every legal expression in every valid context is weakly $\\beta$-normalizing then every such expression is strongly $\\beta$-normalizing; reduction acts inside annotations and contexts may be open. Both properties are system-wide: the theorem does not say that an individual term with a normal form is strongly normalizing. It makes no claim about $\\beta\\eta$-reduction, the other half of Geuvers's formulation. The proof uses Tait-Girard style candidates indexed by sort profiles and a realization of Hurkens's paradox to exclude a graph configuration.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against Geuvers's conjecture. lean/docs/245.md points to ComparatorChallenges/TypeSystemNormalization.json (theorem OAI.PureTypeSystem.weak_implies_strong, solution module OAI.Computability.TypeSystem.Normalization); the solution file exists at the pinned commit, and the challenge is not in the formalization catalogue formalization.yaml. The statement was read here: for an arbitrary specification (any sort type, nonfunctional axiom and rule relations), with de Bruijn syntax, annotated lambda and pi, the seven PTS typing rules and full beta reduction including annotations, system-wide weak normalization of legal terms in valid contexts implies system-wide strong normalization. That is the headline claim. Not rebuilt here. The result covers beta only: the beta-eta half of Geuvers's conjecture is not addressed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The Barendregt-Geuvers-Klop conjecture is a well-known open problem of type theory, open since the early 1990s and familiar to everyone working on pure type systems, but little known outside that community. Scored as famous within one community, level with generalized-star-height-at-most-three (30) and above aec-categoricity-hanf-threshold-ch-failure (22); the beta-eta half stays open, which keeps it from scoring higher.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Weak and strong normalization in pure type systems","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.PureTypeSystem.weak_implies_strong)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/TypeSystem/Normalization.lean","kind":"lean-proof"},{"label":"Geuvers, Logics and Type Systems (thesis, 1993), Conjecture 8.1.2","url":"https://www.cs.ru.nl/~herman/PUBS/Proefschrift.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kohn-sham-ensemble-v-representability","name":"Kohn-Sham ensemble v-representability of Coulomb ground-state densities by potentials in L^{3/2}+L^infinity","shortName":"Kohn-Sham ensemble v-representability","problemNumber":null,"field":"Density functional theory; noninteracting v-representability","fieldGroup":"Mathematical physics","statement":"Kohn-Sham density functional theory replaces an interacting electronic ground state by noninteracting electrons in a common local potential that reproduces its density. Levy and Lieb separated the existence of states with a prescribed density from the existence of a potential for which such states are ground states; ensembles of degenerate noninteracting ground states enlarge the class (Ullrich-Kohn). Explicit non-representable model densities and numerical violations are known, but not for an actual Coulomb system. Is the ground-state density of every finite Coulomb molecule the density of a ground-state ensemble of noninteracting electrons in some real local potential $v\\in L^{3/2}(\\mathbb R^3)+L^\\infty(\\mathbb R^3)$?","posedBy":"Assumed by Kohn and Sham (1965); posed as the v-representability problem by Mel Levy (1979) and Elliott H. Lieb (Density functionals for Coulomb systems, 1983); reviewed by Penz et al. (2023)","yearPosed":1983,"ageNote":"The Kohn-Sham ansatz dates to 1965 and Levy's constrained search to 1979; 1983 is Lieb's formulation in the potential class used here.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a finite positive integer $Z$ such that the three-electron Coulomb Hamiltonian with two nuclei of charge $Z$ at $\\pm(D/Z)e_z$, $D=10^{200}$, has a normalizable absolute ground state $\\Psi$ (sector $S_z=1/2$) whose spin-summed density is not the density of any ground-state ensemble of $H_s(v)$ for any real $v\\in L^{3/2}+L^\\infty$. The mechanism is the Gori-Giorgi-Gal-Baerends nodal-plane asymptotics, made rigorous, plus a kinetic certificate against all ensembles. A corollary rules out a local-potential first variation of the ensemble Hartree-exchange-correlation functional at that density. Not covered: larger potential classes, atoms, or physically realistic charges.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against the representability question as the manuscript frames it from Hohenberg-Kohn, Kohn-Sham, Levy, Lieb and Ullrich-Kohn; proof not refereed. No Lean formalization exists for this family. Scope limits stated by the paper: the counterexample uses nuclear separation $D=10^{200}$ in scaled units and a charge $Z$ given by an exact but non-numerical formula with no effective bound; it excludes only real spin-independent local potentials in $L^{3/2}+L^\\infty$, and says nothing about distributional, spin-dependent or nonlocal potentials.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"Noninteracting v-representability is a foundational question of density functional theory, known across mathematical and chemical physics; a rigorous Coulomb counterexample in the standard potential class is notable, though it is an extreme, nonnumerical molecule. Placed below Solovej's generalized ionization conjecture (32) and polynomial PEPS for gapped ground states (28), far below the ionization conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A Coulomb ground-state density without Kohn-Sham ensemble representation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lieb, Density functionals for Coulomb systems, Int. J. Quantum Chem. 24 (1983)","url":"https://doi.org/10.1002/qua.560240302","kind":"problem-record"},{"label":"Trushin, Erhard, Gorling, Violations of the v-representability condition underlying Kohn-Sham DFT, PRA 110 (2024)","url":"https://doi.org/10.1103/PhysRevA.110.L020802","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"orthogonally-invariant-ising-spin-glass-pressure","name":"The limiting free energy of orthogonally invariant Ising spin glasses at every temperature","shortName":"Pressure of orthogonally invariant spin glasses","problemNumber":null,"field":"Spin glasses: mean-field models with rotationally invariant couplings","fieldGroup":"Mathematical physics","statement":"Let $\\Lambda_N$ be real diagonal matrices whose empirical spectral laws converge to a compactly supported $\\mu$, $U_N$ Haar on $O(N)$, and $P_N=N^{-1}\\log\\big(2^{-N}\\sum_{\\sigma\\in\\{\\pm1\\}^N}\\exp\\{\\tfrac12\\sigma^TU_N^T\\Lambda_NU_N\\sigma\\}\\big)$. This family includes the random orthogonal model and Gaussian Hopfield interactions; Marinari, Parisi and Ritort (1994) and Cherrier, Dean and Lefevre (2003) gave replica-symmetric and one-step replica-symmetry-breaking predictions for its free energy in terms of the spectral law. Rigorous results covered only high temperature: Bhattacharya-Sen (zero field), Fan-Wu (external field), Fan-Misiakiewicz-Wang-Wen. Fan and Wu note that for these models replica predictions for the free energy are conjectured but not rigorously known. Does $P_N$ converge at every temperature, and to what variational formula in terms of $\\mu$?","posedBy":"Replica predictions: E. Marinari, G. Parisi, F. Ritort (1994) and R. Cherrier, D. S. Dean, A. Lefevre (2003); stated as conjectured but unproven by Z. Fan and Y. Wu (arXiv 2105.02797, 2021; PTRF 2024)","yearPosed":1994,"ageNote":"No paper states the all-temperature question as a numbered open problem. The year is that of the first physical prediction for the random orthogonal model; Fan and Wu (2021 preprint, 2024 journal) is the citable statement that the free energy is conjectured but not rigorously known.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: under the stated spectral assumptions, $\\lim\\mathbb EP_N=\\lim P_N=\\inf_p\\{S(p)+G_1(p)\\}$ almost surely, where $p$ ranges over overlap quantile functions, $S$ is a Parisi-type one-site functional and $G_1$ involves the edge-extended $R$-transform of $\\mu$; no independence across $N$ is needed. Extensions: deterministic external fields converging in Wasserstein-1, random spectra, the zero-temperature ground-state energy, and Gaussian Hopfield consequences (limiting max and min of $N^{-1}\\|G\\sigma\\|_2$). Not shown: outliers in the spectrum, explicit identification with the published 1RSB formulas, or the structure (full RSB or not) of the optimizer.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The release also supplies a Lean formalization of the main limits, produced as part of the same release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, and the Lean statement lean/ComparatorChallenges/InvariantIsing.lean, whose solution module OAI/Probability/InvariantIsing/Unconditional.lean exists at the pinned commit. This challenge is not in the formalization catalogue; its statement was read here and was not rebuilt. Its first theorem, limiting_pressure_of_extreme_limits_unconditional, states the headline: under weak convergence of spectral laws to a compactly supported law with extreme eigenvalues converging to the support edges and Haar rotations, the pressure converges in expectation and almost surely to the variational functional. Readers should note the paper does not claim its formula coincides with the physicists' 1RSB predictions, and the no-outlier hypothesis is essential.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A natural and studied open direction in rigorous spin glass theory, extending Parisi-Talagrand beyond Gaussian couplings, but not a named conjecture and with only high-temperature results before. Below the Mezard-Parisi formula for diluted spin glasses (32) and Talagrand's critical SK overlap conjecture (30), level with Zhu, Fang and Shi's invariant projection question (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf","sourceName":"OpenAI math release: All-temperature pressure for orthogonally invariant Ising spin glasses","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean comparator statement: InvariantIsing.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/InvariantIsing.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Probability/InvariantIsing/Unconditional.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/InvariantIsing/Unconditional.lean","kind":"lean-proof"},{"label":"Fan and Wu, Replica-symmetric free energy for orthogonally invariant Ising spin glasses (arXiv:2105.02797)","url":"https://arxiv.org/abs/2105.02797","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"irrational-triangle-billiard-ergodicity","name":"Ergodicity of billiards in every triangle with an irrational angle","shortName":"Irrational triangle billiards are ergodic","problemNumber":null,"field":"Ergodic theory; polygonal billiards","fieldGroup":"Analysis","statement":"For a Euclidean triangle $Q$ with angles $\\alpha,\\beta,\\gamma$, consider the unit-speed billiard flow on $Q\\times S^1$ with specular reflection, discarding the null set of trajectories that hit a vertex, and the measure $dA\\,d\\theta/(2\\pi\\,\\mathrm{Area}(Q))$. When all angles are rational multiples of $\\pi$ the flow splits into invariant directional surfaces and is not ergodic. Zemlyakov-Katok's unfolding and Kerckhoff-Masur-Smillie gave ergodicity for a dense $G_\\delta$ set of polygons, Vorobets for angles with exceptionally fast rational approximation, and Chaika-Forni weak mixing for a dense $G_\\delta$ set; numerical studies of some irrational right and isosceles triangles even suggested nonergodicity. Is the billiard flow ergodic in every triangle with at least one angle irrational relative to $\\pi$?","posedBy":"Classical problem of polygonal billiards (setting of Zemlyakov and Katok; surveyed by Masur and Tabachnikov)","yearPosed":null,"ageNote":"The manuscripts cite no explicit posing statement or date. The question arises from Zemlyakov-Katok (1975) and Kerckhoff-Masur-Smillie (1986), whose generic results left individual irrational triangles open; Masur-Tabachnikov's 2002 survey is cited.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (September 25): the unit-speed billiard flow in every nondegenerate triangle with at least one angle irrational relative to $\\pi$ is ergodic for normalised area times uniform angle, off the null set of vertex-hitting trajectories; consequences include quantum ergodicity and growth of nodal domain counts. The October 5 companion proves the stronger weak mixing (the product flow is ergodic). Neither proves mixing, unique ergodicity, or any statement about individual orbits (for example existence of periodic orbits in obtuse triangles), and polygons with more sides are not treated.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: ergodicity (September 25, 2026, this entry's principal) and weak mixing (October 5, 2026), which extends the ergodicity paper's coefficient-energy method to eigenfunctions.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the ergodicity manuscript was read against the question; it gives ergodicity for every nondegenerate triangle with at least one irrational angle, no genericity or Diophantine condition. The challenge lean/ComparatorChallenges/IrrationalTriangleBilliard.json and its solution module OAI.Dynamics.TriangleBilliards.Main exist at the pinned commit (found via lean/docs/150.md) but are not in the formalization catalogue. The statement was read here: for every triangle with an irrational angle it asserts a normalised phase measure, almost-everywhere existence and uniqueness of complete flight chains avoiding vertices with specular reflection, measure preservation and the flow law almost everywhere, and that every flow-invariant measurable set has measure 0 or 1. That states the headline ergodicity claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. Weak mixing is not formalised.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Whether every irrational triangle gives an ergodic billiard is one of the best-known open questions of polygonal billiards, with only generic or Diophantine results for decades. Placed below Sinai's standard map entropy conjecture (58), level with the Linear List Hadwiger Conjecture (48), and above Singular Bernoulli convolutions at non-Pisot parameters (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf","sourceName":"OpenAI math release: Ergodicity of triangular billiards with an irrational angle","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.TriangularBilliards.irrational_triangle_billiard)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Dynamics/TriangleBilliards/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: IrrationalTriangleBilliard.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/IrrationalTriangleBilliard.lean","kind":"lean-proof"},{"label":"Companion: Weak mixing of triangular billiards with an irrational angle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"microstates-nonmicrostates-free-entropy-equality","name":"Voiculescu's question: do microstates and nonmicrostates free entropy agree when finite?","shortName":"Equality of free entropies","problemNumber":null,"field":"Free probability; operator algebras","fieldGroup":"Analysis","statement":"For a bounded self-adjoint tuple $X$ in a tracial von Neumann algebra, Voiculescu defined the microstates free entropy $\\chi(X)$ from volumes of matrix approximations and the nonmicrostates entropy $\\chi^*(X)$ from free Fisher information along free semicircular noise. They agree for one variable, and Biane-Capitaine-Guionnet proved $\\chi\\le\\chi^*$ in general; equality is known under convexity hypotheses. Agreement was one of the questions of Voiculescu's unification problem, and Guionnet stated the finite-entropy form as a conjecture. Does $\\chi(X)>-\\infty$ imply $\\chi(X)=\\chi^*(X)$?","posedBy":"Dan Voiculescu, Free entropy, Bull. London Math. Soc. 34 (2002), Sections 3.5 and 3.8 (unification problem); finite-entropy form stated by Alice Guionnet, Probab. Surv. 1 (2004), Conjecture 7.1","yearPosed":2002,"ageNote":"The manuscript cites Voiculescu's 2002 survey for the unification problem; the question appears in Voiculescu's free entropy papers of the 1990s, which the manuscript does not quote for it.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims a negative answer: a bounded self-adjoint tuple with $-\\infty<\\chi(X)\\le\\chi^*(X)-\\tfrac12<\\infty$, built as a small semicircular perturbation of free semicirculars tensored with a commuting discrete variable. It does NOT settle equality for every smaller number of variables (the example uses a large fixed $n$), and uses the limsup version of $\\chi$; equality under the known convexity conditions is unaffected.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. Single manuscript; it mentions the release's free group factor isomorphism paper only for contrast. The main theorem has a Lean comparator challenge in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the question: a bounded self-adjoint $n$-tuple ($n\\ge2$, large but fixed) in a von Neumann algebra with faithful normal trace with $-\\infty<\\chi(X)\\le\\chi^*(X)-1/2<\\infty$, $\\chi$ taken with operator-norm cutoff and limsup over matrix sizes. The Lean challenge ComparatorChallenges/FiniteEntropySeparation.json (theorem OAI.FiniteEntropySeparation.main, solution module OAI.Analysis.FreeEntropy.Main, present at the pinned commit) is not in formalization.yaml; it was found through lean/docs/298.md. Its statement was read here: it defines $\\chi$ via microstate volumes with cutoff and limsup, $\\chi^*$ via conjugate systems along a free standard semicircular tuple in the same algebra, and asserts exactly the displayed inequalities. Permitted axioms propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":34,"significanceNote":"Whether the two free entropies coincide is a core question of Voiculescu's free entropy theory, well known within free probability. Below the related release entry on free entropy dimension invariance (38), whose question touches the free group factor problem, and the Crouzeix conjecture (35); above Talagrand's operator cotype problem (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A finite-entropy separation of microstates and nonmicrostates free entropy","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: finite separation of microstates and nonmicrostates free entropy","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/FreeEntropy/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: FiniteEntropySeparation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/FiniteEntropySeparation.lean","kind":"lean-proof"},{"label":"Voiculescu, Free entropy (2002)","url":"https://doi.org/10.1112/S0024609301008992","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"popa-vaes-paving-conjecture-masas","name":"The Popa-Vaes paving conjecture over arbitrary maximal abelian subalgebras: strong-operator, approximation and quadratic paving","shortName":"Popa-Vaes paving conjecture","problemNumber":null,"field":"Operator algebras: paving and maximal abelian subalgebras","fieldGroup":"Analysis","statement":"Let $A\\subseteq M$ be a maximal abelian $*$-subalgebra (MASA) of a von Neumann algebra. For a partition $P=(p_1,\\dots,p_r)$ of $1$ in $A$ put $S_P(x)=\\sum p_ixp_i$. After Marcus-Spielman-Srivastava solved Kadison-Singer paving for the diagonal MASA of $B(\\ell^2)$, Popa and Vaes showed norm paving fails for general MASAs and introduced weaker notions: $x$ is so-pavable if $\\|q(S_P(x)-a)q\\|\\le\\varepsilon\\|x\\|$ for some $a\\in A$ and a projection $q$ arbitrarily close to $1$ strongly, with $r$ depending only on $\\varepsilon$; approximation paving instead norm-paves a strongly close bounded perturbation of $x$. Popa and Vaes proved these for type I algebras, amenable Cartan inclusions and profinite actions with $O(\\varepsilon^{-4})$ bounds, and quadratic bounds for singular MASAs in ultraproducts. Conjecture 2.8: is every MASA in every von Neumann algebra so-pavable and approximation-pavable, and can so-paving always be achieved with $O(\\varepsilon^{-2})$ projections?","posedBy":"S. Popa and S. Vaes, Paving over arbitrary MASAs in von Neumann algebras, Anal. PDE 8 (2015), Conjecture 2.8","yearPosed":2015,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Quadratic paper, Theorem 1.1: for $0<\\varepsilon<1$ and $r_\\varepsilon\\le5\\times10^8\\varepsilon^{-2}$, every self-adjoint $x$ in any von Neumann algebra $M$ is $(\\varepsilon,r_\\varepsilon)$ so-pavable over every MASA $A\\subseteq M$, with no separability or conditional expectation assumed; corollary: quadratic norm paving in Ocneanu ultrapowers of expected countably decomposable MASAs. Approximation paper, Theorem 1.1: every self-adjoint $x$ is a strong limit of self-adjoint $y$ with $\\|y\\|\\le3\\|x\\|$, each norm-pavable to error $\\varepsilon\\|y\\|$ with at most $C\\varepsilon^{-6}$ projections. Not shown: norm paving of the original operator (false in general), optimal numerical constants, or a quadratic rate for approximation paving.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The two manuscripts of the family prove the two parts of the conjecture with independent arguments.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts against Popa-Vaes Conjecture 2.8. The quadratic paper proves 2.8(2): every self-adjoint element is so-pavable over every MASA with at most $5\\times10^8\\varepsilon^{-2}$ projections, which also gives the so-paving half of 2.8(1). The approximation paper proves the approximation-paving half of 2.8(1) with $O(\\varepsilon^{-6})$ projections. The proofs were not refereed; they rely on the Ravichandran-Srivastava mixed determinantal multipaving bound and on measured-equivalence-relation and amalgamated-free-product constructions. The quadratic exponent is shown optimal by a Popa-Vaes example. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A specialist conjecture extending the Kadison-Singer paving problem to arbitrary MASAs, posed by two leading operator algebraists in 2015; known within von Neumann algebra theory but far narrower than Kadison-Singer itself. Below the spatial Kadison-Kastler problem for separable C*-algebras (22) and Marrakchi's relative bicentralizer conjecture (32), above Zhu, Fang and Shi's invariant projection question (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf","sourceName":"OpenAI math release: Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Approximation Paving over Arbitrary Maximal Abelian Subalgebras","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf","kind":"paper"},{"label":"Popa and Vaes, Paving over arbitrary MASAs in von Neumann algebras (Anal. PDE 8, 2015)","url":"https://doi.org/10.2140/apde.2015.8.1001","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"singer-conjecture-dimension-four","name":"The Singer conjecture in dimension four: L2-Betti numbers of closed aspherical four-manifolds vanish outside degree two","shortName":"Singer conjecture in dimension four","problemNumber":null,"field":"L2-invariants; aspherical manifolds","fieldGroup":"Geometry & topology","statement":"For a closed connected aspherical $n$-manifold $M$ with fundamental group $\\Gamma$, the $L^2$-Betti numbers $b_p^{(2)}(\\widetilde M)$ are von Neumann dimensions of the $L^2$-homology of the universal cover. Singer proposed obtaining the sign of the Euler characteristic of nonpositively curved manifolds from vanishing of $L^2$-harmonic forms outside the middle degree (recorded by Dodziuk, 1979). The Singer conjecture in its aspherical form (Davis-Okun 2001, Conjecture 0.3; Kirstein-Kremer-Lück 2025, Conjecture 4.1) asserts $b_p^{(2)}(\\widetilde M)=0$ for $p\\ne n/2$, which in dimension $2m$ gives $(-1)^m\\chi(M)\\ge0$. In dimension four Poincaré duality reduces it to $b_1^{(2)}=0$; known cases needed extra structure (right-angled Coxeter groups, hierarchies, Kähler surfaces). Does every closed connected aspherical four-manifold satisfy $b_p^{(2)}(\\widetilde M)=0$ for all $p\\ne2$?","posedBy":"I. M. Singer (as recorded by Dodziuk); aspherical formulation by Davis and Okun and by Lück","yearPosed":1979,"ageNote":"Singer's proposal is recorded by Dodziuk, Proc. Amer. Math. Soc. (1979), p. 396, in the nonpositively curved setting; the manuscript cites Davis-Okun (Geom. Topol. 2001, Conjecture 0.3) and Kirstein-Kremer-Lück (2025, Conjecture 4.1) for the aspherical form. The date of Singer's own suggestion is not given.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $Q$ is a finite connected aspherical integral Poincaré complex of formal dimension four, with any orientation character, then $b_p^{(2)}(\\widetilde Q)=0$ for $p\\ne2$. Corollary 1.2: the same holds for every closed connected aspherical topological four-manifold, so the Singer conjecture holds in dimension four. Stated consequences: $\\chi(Q)=b_2^{(2)}(\\widetilde Q)\\ge0$, so closed aspherical four-manifolds have nonnegative Euler characteristic; $\\chi(M)\\ge|\\sigma(M)|$ for oriented $M$; limits of normalized Betti numbers along residual towers of covers. Not shown: the Singer conjecture in any other dimension, or any computation of $b_2^{(2)}$.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'The Singer conjecture in dimension four' (September 25, 2026). It cites a companion release manuscript constructing a PD4 group without an aspherical manifold model, to which its theorem applies.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against the Singer conjecture as the manuscript states it. Corollary 1.2 is exactly the four-dimensional case for closed connected aspherical topological four-manifolds, orientable or not, smoothable or not; Theorem 1.1 extends it to finite aspherical integral Poincaré complexes of formal dimension four, a formulation the manuscript attributes to Lück (Remark 11.3 of his approximation survey). There is no Lean formalization for this family. The proof, a long argument on boundaries, annuli, pretrees and path pushing, was not refereed. The conjecture in other dimensions is untouched, so the entry is the dimension-four special case.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The Singer conjecture is a central open problem of L2-topology; its four-dimensional case is the first open one and gives nonnegative Euler characteristic for closed aspherical four-manifolds, itself a long-asked question. A special case of the full conjecture. Placed below the Borel conjecture in dimension four (60) and the Eilenberg-Ganea conjecture (55), above Wall's manifold-realization question for PD groups (42) and the group-ring Determinant Conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf","sourceName":"OpenAI math release: The Singer conjecture in dimension four","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Davis and Okun, Vanishing theorems and conjectures for the l2-homology of right-angled Coxeter groups (2001)","url":"https://doi.org/10.2140/gt.2001.5.7","kind":"problem-record"},{"label":"Dodziuk, L2 harmonic forms on rotationally symmetric Riemannian manifolds (1979)","url":"https://doi.org/10.1090/S0002-9939-1979-0545603-8","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"snaky-achievement-game","name":"Harary's Snaky problem: is the Snaky hexomino a winner in the planar Maker-Breaker achievement game?","shortName":"Is Snaky a winner?","problemNumber":null,"field":"Combinatorial games; polyomino achievement games","fieldGroup":"Combinatorics","statement":"In Harary's polyomino achievement games two players alternately claim cells of the infinite square grid; Maker wins on owning a translated, rotated or reflected copy of a fixed polyomino. The target is the hexomino Snaky, $\\{(0,0),(1,0),(2,0),(3,0),(3,1),(4,1)\\}$. Known: no pairing defense for Breaker, a win in 41 and in three dimensions, a loss on the $8\\times8$ board, and a planar win with handicap one (Halupczok-Schlage-Puchta, Ito-Miyagawa, Sieben). Can Maker, moving first on the empty infinite plane with no handicap, force a copy of Snaky?","posedBy":"Frank Harary (polyomino achievement games, as attributed by Sieben 2004); the ordinary planar case recorded as undecided by Nandor Sieben (Integers 8, 2008)","yearPosed":null,"ageNote":"The manuscript attributes the games to Harary via Sieben (2004) but gives no date for Harary's question; the latest cited statement that it is open is Sieben (Integers 8, 2008, G07).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1: one globally legal Maker policy completes Snaky within 21 Maker claims (by the 41st turn) against every legal Breaker play on the empty infinite board, so Snaky is a winner in the planar weak game. It works on a $17\\times17$ board, using a fixed 251-cell region; 25- and 35-move constructions are also given. Not shown: optimal move count or board size, or the strong game in which both players seek a copy.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1, read against the planar Snaky question as cited; the 728-condition certificate was not rechecked. Lean-checked on the release's Comparator challenge SnakyTwentyOne with solution module OAI.GameTheory.SnakyTwentyOne.Main, both fetched at the pinned commit; the challenge is not listed in lean/formalization.yaml. Statement read here, not rebuilt: it defines Snaky, its eight orientations and translates, and asserts a Maker policy that always picks free cells and, against every Breaker sequence legal for the first 20 replies, owns a copy of Snaky after 21 claims with disjointness and move counts tracked. That is the headline. Two further challenges formalize the 35-move certificate and conditional templates.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A long-standing, well-known problem of recreational combinatorics, open in the plane since Harary introduced the games, but with little bearing on wider mathematics. Level with the strong graph reconstruction conjecture (20), above Pebody's even cyclic four-deck reconstruction conjecture (15), far below the Harary-Hill crossing number conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf","sourceName":"OpenAI math release: Snaky in 21 Maker moves","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge SnakyTwentyOne (not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SnakyTwentyOne.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/GameTheory/SnakyTwentyOne/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GameTheory/SnakyTwentyOne/Main.lean","kind":"lean-proof"},{"label":"Sieben, Proof trees for weak achievement games, Integers 8 (2008) G07","url":"https://math.colgate.edu/~integers/ig7/ig7.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"williams-one-tape-time-space-simulation","name":"Williams's one-tape question: simulating one-tape time $T$ in $O(T^{1/2-\\delta})$ space","shortName":"One-tape time in sub-square-root space","problemNumber":null,"field":"Computational complexity; time-space simulation","fieldGroup":"Theoretical computer science","statement":"A classical simulation (attributed to Hopcroft and Ullman) computes the outcome of a deterministic one-tape Turing machine running for $T$ steps in $O(\\sqrt T)$ space, with unrestricted simulation time. Williams (2025) proved that multitape time $t$ can be simulated in $O(\\sqrt{t\\log t})$ space using the Cook-Mertz tree-evaluation algorithm, and asked in Section 5 of his full version whether, for one-tape machines, the square-root exponent can be lowered by a fixed positive constant. Is there $\\delta>0$ such that a one-tape machine running for $T$ steps can be simulated in $O(T^{1/2-\\delta})$ space?","posedBy":"R. Ryan Williams (Simulating Time with Square-Root Space, STOC 2025; question in Section 5 of the full version)","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a fixed deterministic machine with one writable tape and head (and finitely many read-only input heads), the finite-control and halting outcome by time $T$ can be computed in $O(T^{2/5}\\log^C(T+2))$ work-space bits given the cap $T$; without a supplied cap, a machine halting at time $t$ is simulated in $O((t+2)^{2/5}\\,\\mathrm{polylog})$ space. Hence $O(T^{1/2-\\delta})$ for every $\\delta<1/10$. Simulation time is unrestricted, the final tape is not materialized, no lower bound or optimality is claimed, and for several writable tapes only $O(\\sqrt T\\,\\mathrm{polylog})$ is given.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Williams's question as the paper reports it; Williams's full version was not opened. Lean-checked on the release's Comparator challenge OneTapeSpace (declarations OAI.Fifths.Single.capped and OAI.Fifths.Single.unknown, solution module OAI.Computability.SpaceSimulation.Main), added in the 7 October 2026 update and listed in lean/formalization.yaml. Its statements were read here: for a fixed deterministic machine with one writable tape and head plus finitely many read-only heads, whose initial symbols are computable in polylogarithmic space, a multitape machine given $x$ and a binary cap $T\\ge2$ outputs the finite-control state and halting flag at time $T$ within $cT^{2/5}(\\log(T+2))^e$ work bits; a capless version does the same at a halting time $t$ within $c(t+2)^{2/5}$ times a polylog. That gives $O(T^{1/2-\\delta})$ for every $\\delta<1/10$, the headline. Not rebuilt here. The bound carries polylogarithmic factors, the simulation is not time-efficient, and the multitape exponent is not improved.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A recent, explicitly stated follow-up question to Williams's celebrated 2025 time-space simulation, specialist in scope and open for about a year. Placed level with Quadratic-Time Hardness of Furthest Pair in Superconstant Dimension (20), below Sharp Hardness for MAX-3-CUT (25) and the Cerny Conjecture for One-Cluster Automata (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026/article.pdf","sourceName":"OpenAI math release: Simulating One-Tape Time in Two-Fifths-Power Space","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Williams, Simulating Time with Square-Root Space (full version, Section 5)","url":"https://arxiv.org/abs/2502.17779","kind":"problem-record"},{"label":"Lean Comparator challenge OneTapeSpace (OpenAI math release)","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/OneTapeSpace.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"coulomb-monge-ansatz-three-marginal","name":"The Monge (co-motion) ansatz for three-marginal Coulomb transport in $\\mathbb R^3$: is the optimum always attained by maps?","shortName":"Monge ansatz for 3-marginal Coulomb cost","problemNumber":null,"field":"Multi-marginal optimal transport; density functional theory","fieldGroup":"Analysis","statement":"For a probability measure $\\mu$ on $\\mathbb R^3$, minimize $\\int c\\,d\\pi$ over couplings $\\pi$ with three marginals equal to $\\mu$, where $c(x_1,x_2,x_3)=\\sum_{i0$, in every dimension $d\\ge2$, with a suitable density. It concerns one constructed density: it does not say Monge plans fail generically, and no external potential realizing the density is asserted.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the question as Friesecke states it. lean/docs/373.md points to ComparatorChallenges/CoulombCounterexample.json (theorem OAI.Problem356.coulomb_counterexample_and_equal_infima, solution module OAI.Analysis.CoulombTransport.Main); the solution file exists at the pinned commit and the challenge is not in formalization.yaml. The statement was read here: there is a smooth compactly supported probability density on $\\mathbb R^3$ with smooth compactly supported square root whose three-marginal Coulomb Kantorovich value is finite and attained, every pair of measure-preserving measurable maps has strictly larger cost, and the Monge infimum equals the Kantorovich value. That is the headline claim. Not rebuilt here. The inverse-power Riesz extensions are outside the Lean statement.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A recognized open question in multi-marginal transport and the mathematics of density functional theory, recorded in a recent survey; specialist and young as a stated problem. Placed below generalized-ionization-conjecture (32), level with separable-cstar-kadison-kastler-spatial (22), and above quantum-monge-kantorovich-distance-disproof (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to the Monge ansatz for the three-marginal Coulomb cost","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Problem356.coulomb_counterexample_and_equal_infima)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/CoulombTransport/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cklw-strong-locality-strongly-rational","name":"The CKLW strong-locality conjecture for strongly rational unitary vertex operator algebras","shortName":"Strong locality, strongly rational VOAs","problemNumber":null,"field":"Chiral conformal field theory: vertex operator algebras and conformal nets","fieldGroup":"Mathematical physics","statement":"A simple unitary vertex operator algebra $V$ is CKLW-strongly local if its fields satisfy polynomial energy bounds and the von Neumann algebras $A_V(I)$ generated by the closed smeared fields $\\overline{Y(a,f)}$ with $\\operatorname{supp}f\\subset I$ commute for disjoint intervals of the circle, so that $V$ generates a conformal net. Carpi, Kawahigashi, Longo and Weiner proved this for unitary affine, Virasoro and moonshine VOAs and conjectured it for every simple unitary VOA (Conjecture 8.18). Commuting unbounded fields need not give commuting von Neumann algebras, so the point is analytic. Is every simple unitary vertex operator algebra strongly local?","posedBy":"Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo and Mihaly Weiner (From vertex operator algebras to conformal nets and back, Conjecture 8.18)","yearPosed":2015,"ageNote":"The conjecture appears in arXiv:1503.01260 (March 2015), published in Mem. Amer. Math. Soc. 254 (2018).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: every simple unitary strongly rational VOA has polynomial energy bounds and is CKLW-strongly local, generating an irreducible conformal net $A_V$; moreover $V$ is completely unitary (all simple modules unitarizable, fusion forms positive), every unitary module is strongly integrable, $A_V$ is completely rational, and the Carpi-Weiner-Xu functor is a braided unitary tensor equivalence $\\mathrm{Rep}(V)\\simeq\\mathrm{Rep}^f(A_V)$. It does not treat non-rational or non-C2-cofinite unitary VOAs, so the general conjecture remains open.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against CKLW Conjecture 8.18 as cited. The proof was not refereed. The result covers only strongly rational VOAs (CFT type, self-contragredient, rational, C2-cofinite); the general conjecture stays open. Lean-checked on the release's own Comparator challenge VertexAlgebraNet together with its solution module OAI.RepresentationTheory.VertexAlgebra.MinimalComparator, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.MinimalVertex.main) assumes a simple CFT-type VOA with a unitary structure that is self-contragredient, rational and C2-cofinite, and concludes polynomial energy bounds, CKLW strong locality (interval algebras contained in the commutant of the complementary interval's algebra) and an irreducible conformal net. That is the headline claim of this partial entry. The 47 kB statement file was read for its main theorem and key definitions only. Not formalized: complete rationality, complete unitarity, the braided equivalence and the extension correspondence (parts 2 to 5).","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":27,"significanceNote":"The strongly rational case is the main case of the CKLW conjecture and of the expected VOA-net correspondence, a central question in mathematical chiral CFT but a specialist one. Scored below the full conjecture (about 30), below Connes' rigidity conjecture (45) and Voiculescu's free entropy dimension question (38), level with the Lions half-Holder maximal regularity problem (25) and above Kac's walk mixing (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Strongly rational unitary vertex operator algebras and conformal nets","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Carpi, Kawahigashi, Longo and Weiner, Mem. Amer. Math. Soc. 254 (2018), Conjecture 8.18","url":"https://doi.org/10.1090/memo/1213","kind":"problem-record"},{"label":"Lean Comparator challenge VertexAlgebraNet (not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/VertexAlgebraNet.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"phillips-toms-radius-of-comparison-mean-dimension","name":"The Phillips-Toms conjecture for minimal homeomorphisms: radius of comparison equals half the mean dimension","shortName":"Phillips-Toms conjecture (integer actions)","problemNumber":null,"field":"Operator algebras; C*-dynamics and mean dimension","fieldGroup":"Analysis","statement":"For a minimal homeomorphism $h$ of an infinite compact metrizable space $X$, the crossed product $C(X)\\rtimes_h\\mathbb Z$ is a simple unital nuclear C*-algebra. Toms's radius of comparison $\\mathrm{rc}(A)$ is the least $r$ such that $d_\\tau(a)+r 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript proves the missing lower bound; a companion of the same date (filtered products and boundary-preserving compression in complex cobordism) supplies the compression theorem it uses.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against the conjecture as recorded by Elliott-Niu and Schafhauser-Tikuisis-White. No Lean formalization exists for this family. The upper bound rc <= mdim/2 is Niu's published theorem, and Niu's open tower theorem is a second input; the new part is the lower bound, which depends on the companion's MU-based compression theorem. The regularity corollary also uses published Elliott-Niu, Rordam and Castillejos-Evington-Tikuisis-White-Winter theorems. The paper limits itself to integer actions. Not refereed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A central quantitative conjecture linking topological dynamics and the classification programme for nuclear C*-algebras, listed among the 99 problems of Schafhauser-Tikuisis-White, with the upper half already proved by Niu. Known within operator algebras and dynamics only. Placed below The Toms-Winter conjecture: strict comparison implies Jiang-Su stability for simple nuclear C*-algebras (48), just above Robert-Tikuisis Conjecture (C1) (25) and above Szabo's Conjecture A on equivariant Jiang-Su stability (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Radius of comparison equals half the mean dimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Filtered products and boundary-preserving compression in complex cobordism","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026/paper.pdf","kind":"paper"},{"label":"Elliott-Niu, The C*-algebra of a minimal homeomorphism of zero mean dimension (Duke 2017)","url":"https://doi.org/10.1215/00127094-2017-0033","kind":"problem-record"},{"label":"Schafhauser-Tikuisis-White, Nuclear C*-algebras: 99 problems (Problem XXXVI)","url":"https://arxiv.org/abs/2506.10902v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-416-number-of-totients","name":"Erdős Problem 416: does $V(2x)/V(x)\\to2$ for the number $V(x)$ of totient values up to $x$, and is there an asymptotic formula?","shortName":"Erdős #416, number of totients","problemNumber":416,"field":"Multiplicative number theory; values of arithmetic functions","fieldGroup":"Number theory","statement":"Let $V(x)$ count the integers $v\\le x$ for which $\\varphi(m)=v$ is solvable. Pillai showed these values have density zero; after Erdős, Erdős-Hall, Pomerance and Maier-Pomerance, Ford (1998) determined $V(x)$ up to a bounded factor and showed $V(cx)-V(x)\\asymp_c V(x)$, but this falls short of an asymptotic formula. Erdős and Hall asked whether $V(cx)/V(x)\\to c$ for fixed $c>1$. Does $V(2x)/V(x)\\to2$, and is there an asymptotic formula for $V(x)$?","posedBy":"Paul Erdős and R. R. Hall (1976, Mathematika); repeated by Erdős (1979)","yearPosed":1976,"ageNote":"Recorded as Erdős Problem 416 on erdosproblems.com (checked there: both questions in this form). The manuscript cites Erdős-Hall (1976, final paragraph) and Erdős (1979, p. 80) but not the erdosproblems number.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The paper gives an explicit asymptotic equivalent $V(x)\\sim\\frac{x}{\\log x}G(x)A(\\theta(x))$, where the bounded positive coefficient $A$ depends on a phase $\\theta(x)\\in[0,1)$ and is a uniform limit of functions defined from finite prime and integer data, and proves $V(cx)/V(x)\\to c$ for every fixed $c>0$, answering Erdős-Hall. It also gives asymptotics for totients whose least preimage lies in $(kx,(k+1)x]$, a further question of Erdős (1979), with a positive coefficient for $k=1,2$ and identically zero counts when no totient $d$ has least preimage above $kd$. Not shown: a closed-form constant (the coefficient is a limit, not a number), or secondary terms.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 25, 2026, building on Ford's structure theorems for totient preimages.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: introduction read against Erdős Problem 416 and Erdős-Hall's question. Lean: the challenge lean/ComparatorChallenges/TotientAsymptotic.json (solution_module OAI.NumberTheory.TotientAsymptotic.UnconditionalMain, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml. The statement totient_asymptotic_formula was read here: with $V$ defined as the number of $v\\in[1,x]$ that are totients, it asserts $V(x)/\\mathrm{mainTerm}(x)\\to1$ for an explicit main term built from finite arithmetic approximants (with uniform convergence of the approximants and two-sided bounds on the coefficient) and $V(cx)/V(x)\\to c$ for every $c>0$. That is the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A fifty-year-old Erdős-Hall question at the end of a well-known line of work (Maier-Pomerance, Ford) that came close but stopped short of an asymptotic. Above the typical Erdős problem such as Erdős Problem 306 (10) and the Kourovka totient sums entry (15), below Short Egyptian fractions, Erdős Problem 304 (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf","sourceName":"OpenAI math release: An asymptotic formula for the number of totients","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Erdős Problem 416","url":"https://www.erdosproblems.com/416","kind":"problem-record"},{"label":"Lean formalization (TotientAsymptotic unconditional main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/TotientAsymptotic/UnconditionalMain.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"robert-trace-cone-razak-jacelon","name":"Robert's trace-cone question: does the cone of extended traces classify W-stabilized separable nuclear C*-algebras?","shortName":"Robert's trace-cone classification question","problemNumber":null,"field":"Operator algebras; classification of nuclear C*-algebras","fieldGroup":"Analysis","statement":"The Razak-Jacelon algebra $\\mathcal W$ is simple, monotracial, stably projectionless and nuclear; tensoring with it removes K-theory and leaves trace data. For a C*-algebra $A$ let $T(A)$ be the cone of all lower-semicontinuous extended traces $A_+\\to[0,\\infty]$, including the zero/infinity weights of ideals, with its canonical topology (Elliott-Robert-Santiago). Known classification results cover simple algebras and the traceless (purely infinite) case. Leonel Robert asked, as recorded by Santiago (2012) and listed as Problem LXVIII by Schafhauser, Tikuisis and White, whether this cone is a complete invariant: if $A,B$ are separable nuclear C*-algebras with $T(A)\\cong T(B)$ as topological cones, is $A\\otimes\\mathcal W\\otimes\\mathcal K\\cong B\\otimes\\mathcal W\\otimes\\mathcal K$?","posedBy":"Leonel Robert, recorded in L. Santiago, W-stable C*-algebras (West Coast Operator Algebra Seminar abstract, 2012); Problem LXVIII in Schafhauser-Tikuisis-White, Nuclear C*-algebras: 99 problems","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $A,B$ are separable nuclear complex C*-algebras and $T(A)\\cong T(B)$ as topological cones, then $A\\otimes\\mathcal W\\otimes\\mathcal K\\cong B\\otimes\\mathcal W\\otimes\\mathcal K$, and the isomorphism can be chosen to induce the given cone map. No density condition on finite domains of the weights and no UCT is assumed. It does not classify algebras without the $\\mathcal W\\otimes\\mathcal K$ stabilization, and does not determine the range of the invariant.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Robert's question in the Schafhauser-Tikuisis-White formulation, which keeps every extended trace including the ideal weights. The theorem assumes only separability and nuclearity, allows arbitrary ideal structure and both finite and infinite subquotients, and needs no UCT. The 7 October 2026 update added the supporting Comparator challenge TraceIdealTransportSupport (read here, with lean/docs/301.md): a trace-cone equivalence induces an order isomorphism of closed ideals carrying finiteness and zero ideals, and closed ideals correspond to additive idempotent traces. The classification itself is outside it, so the entry stays unreviewed. The proof combines existence and uniqueness theorems with an approximate intertwining; it was not refereed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A classification question listed among 99 open problems on nuclear C*-algebras, of interest to the classification community only. Level with Schafhauser-Tikuisis-White Problem XXI at 15, below Szabo's Conjecture A at 22 and the Robert-Tikuisis conjecture entry at 25.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026.pdf","sourceName":"OpenAI math release: The trace cone classifies Razak-Jacelon stabilizations","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Schafhauser, Tikuisis and White, Nuclear C*-algebras: 99 problems (Problem LXVIII)","url":"https://arxiv.org/abs/2506.10902","kind":"problem-record"},{"label":"Lean statement (supporting result only): TraceIdealTransportSupport","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/TraceIdealTransportSupport.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"shokurov-bounded-klt-complements","name":"Shokurov's bounded klt complements conjecture for Fano type contractions with finite rational coefficients","shortName":"Bounded klt complements (Shokurov)","problemNumber":null,"field":"Birational geometry; complements and the minimal model program","fieldGroup":"Geometry & topology","statement":"Let $f:X\\to Z$ be a contraction of normal quasi-projective complex varieties with $X$ of Fano type over $Z$, and $(X,\\Delta)$ an $\\epsilon$-lc pair of dimension $d$ with coefficients in a fixed finite set $I\\subset[0,1]\\cap\\mathbb Q$ and $-(K_X+\\Delta)$ nef over $Z$. A monotone klt $n$-complement near $z\\in Z$ is a boundary $\\Delta^+\\ge\\Delta$ with $(X,\\Delta^+)$ klt and $n(K_X+\\Delta^+)\\sim0$ over a neighbourhood of $z$. Shokurov introduced complements and conjectured their boundedness; Birkar proved boundedness of lc complements, but a bounded klt index depending only on $d$, $\\epsilon$ and $I$ was known only over curves (Birkar) and surfaces (Chen). In the formulation of B. Chen (2024, Conjecture 1.1): is there $M=M(d,\\epsilon,I)$ such that every such contraction has a klt $M$-complement near every point of $Z$?","posedBy":"V. V. Shokurov (in the precise form stated by Bingyi Chen, Conjecture 1.1)","yearPosed":2000,"ageNote":"The manuscript credits Shokurov's boundedness conjectures (Complements on surfaces, J. Math. Sci. 102 (2000), Conjecture 1.3) and uses the klt statement of Chen, arXiv 2403.01154, Conjecture 1.1; the companion cites Shokurov's 2004 Oberwolfach report for the epsilon-lc strengthening.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $d$ and rational $\\epsilon>0$ there is $N(d,\\epsilon)$ such that every projective contraction $f:X\\to Z$ over an algebraically closed field of characteristic zero with $X$ $\\epsilon$-lc of dimension $d$, $K_X$ $\\mathbb Q$-Cartier and $-K_X$ ample over $Z$ has, near each closed point of $Z$, a boundary $B$ with $(X,B)$ klt and $N(K_X+B)\\sim0$. Corollary 1.2 (over $\\mathbb C$): the finite-rational-coefficient form for Fano type pairs with nef anti-log-canonical divisor, with monotone complements. The companion proves that some Cartier divisor through any base point has pulled-back lc threshold at least $\\tau(d,\\epsilon)$, also for real boundaries over $\\mathbb C$. Not shown: $\\epsilon'$-lc complements, DCC coefficient sets, or effective values of $N$.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 25, 2026: 'Bounded klt complements for Fano contractions' (principal) and 'Uniform Cartier sections for Fano type contractions', which proves the equivalent Birkar-Shokurov Cartier-divisor statement by an independent argument.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against Chen's Conjecture 1.1 as both manuscripts quote it. Corollary 1.2 gives exactly the finite-rational-coefficient statement over $\\mathbb C$, and Theorem 1.1 gives klt complements of index $N(d,\\epsilon)$ for $\\epsilon$-lc Fano contractions over any algebraically closed field of characteristic zero. Both proofs go through Chen's theorem that bounded klt complements are equivalent to a uniform local threshold statement; the companion proves that statement (the Birkar-Shokurov Cartier-divisor conjecture) independently. The 7 October 2026 update added the supporting Comparator challenge CartierChartCompactnessSupport (scope read here): compactness of a set of chart-field retractions, a local lemma of the companion. It does not state bounded complements or uniform Cartier sections, so the entry stays unreviewed. Not addressed: the stronger requirement that the complement itself be $\\epsilon'$-lc (Chen, Remark 1.2(1); Shokurov 2004), and coefficient sets that are DCC or real rather than finite rational.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":33,"significanceNote":"Boundedness of complements is a cornerstone conjecture of the minimal model program; Birkar's lc case was part of his Fields Medal work, and the klt version was open beyond bases of dimension two. Famous within birational geometry. Placed above the effective Iitaka fibration conjecture (32) and Shokurov's global index conjecture for foliations (30), below the Campana-Peternell conjecture (35) and the Lipman-Zariski conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-klt-complements-for-Fano-contractions-September-25-2026/Bounded-klt-complements-for-Fano-contractions-September-25-2026.pdf","sourceName":"OpenAI math release: Bounded klt complements for Fano contractions","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion manuscript: Uniform Cartier sections for Fano type contractions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026.pdf","kind":"paper"},{"label":"B. Chen, Boundedness of klt complements on Fano fibrations over surfaces (arXiv 2403.01154)","url":"https://arxiv.org/abs/2403.01154","kind":"problem-record"},{"label":"Lean statement (supporting result only): CartierChartCompactnessSupport","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/CartierChartCompactnessSupport.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"patterson-cubic-gauss-sum-conjecture","name":"Patterson's conjecture on the first moment of cubic Gauss sums over primes","shortName":"Patterson's cubic Gauss sum conjecture","problemNumber":null,"field":"Analytic number theory; cubic Gauss sums and metaplectic forms","fieldGroup":"Number theory","statement":"For a rational prime $p\\equiv1\\pmod 3$, Kummer studied the cubic sum $S_p=\\sum_{x\\bmod p}e(x^3/p)$ and observed a bias toward positive values. Heath-Brown and Patterson (1979) proved that the normalized sums are equidistributed on the circle, so any bias lives below the prime-counting scale. Patterson (1978) predicted a bias of exact order $X^{5/6}/\\log X$; Heath-Brown's cubic large sieve gave $O(X^{5/6+\\epsilon})$ and Dunn-Radziwill (2024) proved a smoothed asymptotic under GRH. With $G(\\pi)$ the normalized cubic Gauss sum at a primary Eisenstein prime and $c_*=(2\\pi)^{2/3}/(3\\Gamma(2/3))$, is $\\sum_{N\\pi\\le X}G(\\pi)\\sim\\frac65c_*X^{5/6}/\\log X$ unconditionally?","posedBy":"S. J. Patterson (1978); first-moment normalization as stated by Dunn and Radziwill (2024, display after Equation (1.8)) and Wan (2021, Conjecture 4.13)","yearPosed":1978,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: unconditionally, $\\sum_{N\\pi\\le X,\\ \\pi\\equiv1(3)\\text{ prime}}G(\\pi)=\\frac65c_*X^{5/6}/\\log X+o(X^{5/6}/\\log X)$, over all primary Eisenstein primes; equivalently $\\sum_{p\\le X}S_p/(2\\sqrt p)\\sim\\frac35c_*X^{5/6}/\\log X$, matching Wan's Conjecture 4.13. Also cancellation in every fixed nonzero angular mode and for fixed powers $G(\\pi)^k$ with $3\\nmid k$, $|k|>1$. The error term is only little-oh, with no rate; powers $k$ divisible by 3 (part of Patterson's complementary conjecture) remain open. Uses the unconditional Dunn-Radziwill Voronoi formula, not their GRH estimates.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization accompanies it (Comparator challenge PattersonFirstMoment).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, Theorem 1.1, the history and the normalization section of the TeX source, read against Patterson's conjecture as stated by Dunn-Radziwill and Wan. Lean: Comparator challenge PattersonFirstMoment, declarations OAI.CubicFirstMoment.patterson_firstMoment, patterson_angularComparison and patterson_angularCancellation. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: with Gauss sums at primary primes of $\\mathbb Z[\\omega]$ built from the cubic residue symbol and $e(\\mathrm{Tr}(v/p))$, the sharp-cutoff sum over primary primes of norm at most $X$ minus $\\frac65c_*X^{5/6}/\\log X$ is $o(X^{5/6}/\\log X)$; also the angular statements. That is the headline. Not rebuilt here. The manuscript notes that Dunn-Radziwill print the same constant in two inequivalent normalizations and identifies which one it proves.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Kummer's cubic sum bias and Patterson's precise prediction are classic problems of analytic number theory; Dunn and Radziwill's conditional proof in 2024 was widely noticed, and removing GRH would be a major step for specialists. Placed just below the binary corrected Elliott conjecture (48) and the p-adic section conjecture (50), above the group-ring determinant conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf","sourceName":"OpenAI math release: An unconditional first moment for cubic Gauss sums","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge PattersonFirstMoment (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PattersonFirstMoment.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"shub-entropy-conjecture-c1-maps","name":"Shub's entropy conjecture for C1 self-maps","shortName":"Shub's entropy conjecture (C1 maps)","problemNumber":null,"field":"Smooth dynamical systems; topological entropy and homology","fieldGroup":"Analysis","statement":"For a continuous self-map $f$ of a compact manifold $M$, let $h_{\\mathrm{top}}(f)$ be its topological entropy and $\\rho(f_*)$ the spectral radius of the induced map on real homology $H_*(M;\\mathbb R)$. Shub (1974) conjectured that $h_{\\mathrm{top}}(f)\\ge\\log\\rho(f_*)$ for every $C^1$ map; Lipschitz counterexamples were known, Manning proved the bound for $H_1$, Misiurewicz-Przytycki for top degree, and Yomdin for $C^\\infty$ maps, with further $C^1$ diffeomorphism cases (Saghin-Xia, Liao-Viana-Yang). Does $h_{\\mathrm{top}}(f)\\ge\\log\\rho(f_*)$ hold for every $C^1$ self-map $f$ of a compact smooth manifold without boundary?","posedBy":"Michael Shub","yearPosed":1974,"ageNote":"Shub, Dynamical systems, filtrations and entropy, Bull. AMS 80 (1974); Shub's ICM 2006 account includes endomorphisms; the general C1 self-map form is Conjecture 1 of Saghin and Xia (Topology Appl. 2010), all cited by the manuscript.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for some $q\\ge1$ there is a $C^1$ self-map $f$ of $M=(\\mathbb R/100\\mathbb Z)\\times(S^2)^{q+1}$ with $h_{\\mathrm{top}}(f)=0$ while $f_*$ on $H_2(M;\\mathbb R)$ has eigenvalue 2, so $h_{\\mathrm{top}}(f)<\\log\\rho(f_*)$. The map is noninvertible, $\\dim M=2q+3\\ge5$, and the eigenvalue sits in intermediate degree, consistent with the known first-homology, top-degree and $C^\\infty$ positive results. It does not address $C^1$ diffeomorphisms, $C^r$ maps for $r\\ge2$, or the conjecture in low dimensions.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 25, 2026) has no companion in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture in its general $C^1$ self-map form; it gives a noninvertible $C^1$ map of $(\\mathbb R/100\\mathbb Z)\\times(S^2)^{q+1}$ with zero entropy and eigenvalue 2 on $H_2$, which contradicts that form. The map is not a diffeomorphism, so the $C^1$ diffeomorphism version is untouched, as the paper indicates. The release's lean/docs/151.md points to ComparatorChallenges/C1EntropyCounterexample.json, which is not in the formalization catalogue; its solution module OAI/Dynamics/EntropyCounterexample/Counterexample.lean exists at the pinned commit. The statement was read here: for some $q>0$ a map of the unit circle times $(S^2)^{q+1}$ that is $C^1$ in the sense of local $C^1$ extensions in the ambient Euclidean space, with Mathlib cover entropy 0, a nonzero real singular $H_2$ class with eigenvalue 2, and not bijective. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Shub's entropy conjecture is one of the classical open problems of smooth dynamics since 1974; the general C1 self-map form answered here is the one Shub and Saghin-Xia stated, but the diffeomorphism case, often regarded as the heart of it, remains open. Placed below Sinai's standard-map conjecture (58), level with the strong hyperkahler SYZ conjecture (48), and above Brennan's conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf","sourceName":"OpenAI math release: A C^1 Counterexample to the Entropy Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Problem340.exists_C1_entropy_counterexample)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Dynamics/EntropyCounterexample/Counterexample.lean","kind":"lean-proof"},{"label":"Comparator statement: C1EntropyCounterexample.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/C1EntropyCounterexample.lean","kind":"lean-proof"},{"label":"Shub (1974), Dynamical systems, filtrations and entropy","url":"https://doi.org/10.1090/S0002-9904-1974-13344-6","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"free-uniform-spanning-forest-factor-of-iid","name":"Is the free uniform spanning forest a factor of IID?","shortName":"FUSF as a factor of IID","problemNumber":null,"field":"Probability on graphs, uniform spanning forests","fieldGroup":"Probability & statistics","statement":"The free uniform spanning forest (FUSF) of an infinite connected locally finite graph is the weak limit of uniform spanning trees on finite exhaustions, without wiring the boundary. The wired forest is a factor of IID on transient graphs via Wilson's algorithm rooted at infinity, and Timar proved the free forest is a factor of IID on recurrent and invariantly amenable unimodular random graphs. Lyons (2013) asked, for Cayley graphs, whether the FUSF is a factor of IID. Can the FUSF be generated as an equivariant measurable factor of independent vertex labels, in general and in particular on nonamenable Cayley graphs?","posedBy":"Russell Lyons","yearPosed":2013,"ageNote":"Lyons, l2-Betti numbers, cost, and the free uniform spanning forest, Oberwolfach Reports 10 (2013), p. 2392, as cited by the manuscript; Timar (arXiv, December 2025 version) states the unrestricted question as open.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is one Borel, root-independent, equivariant rule $\\Phi(G,U,e)$ such that for every infinite connected locally finite simple unweighted graph $G$ with IID uniform vertex labels $U$, the edge set selected by $\\Phi$ has the FUSF law. Further results: every translation-invariant strongly Rayleigh binary process on a countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, is a factor of IID, with no amenability needed. Not shown: finitary factors or weighted graphs beyond the strongly Rayleigh results.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 25, 2026).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Lyons's question as cited; the theorem gives one Borel, root-independent, isomorphism-equivariant rule for all infinite connected locally finite simple graphs, which covers Cayley graphs. Lean: the challenge FreeUniformSpanningForest (theorem OAI.Problem336.fusf_is_factor_iid, solution module OAI.Probability.SpanningForest.Factor) exists at the pinned commit but is not in the formalization catalogue; its statement was read here. Graphs are encoded on vertex set N (so infinite), the FUSF law is defined by limits of uniform spanning tree cylinder probabilities over connected exhaustions, and the rule is required to be measurable and relabelling-equivariant. That is the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A question of Lyons in probability on groups, specialist but recognised, recently advanced by Timar in the amenable case. Placed below the Talagrand SK overlap conjecture (30), level with the superexponential van der Waerden entry (22), above a typical Erdos problem (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf","sourceName":"OpenAI math release: The free uniform spanning forest is a factor of IID","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean formalization: SpanningForest/Factor.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/SpanningForest/Factor.lean","kind":"lean-proof"},{"label":"Lean formalization: StrongRayleigh/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/StrongRayleigh/Main.lean","kind":"lean-proof"},{"label":"Lyons, Oberwolfach Reports 10 (2013), report on the FUSF","url":"https://doi.org/10.4171/OWR/2013/42","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ashkin-teller-critical-current-scaling-limit","name":"The Alcalde Lopez-Heeney-Lis conjecture: joint scaling limit of critical Ashkin-Teller heights and current clusters","shortName":"Critical Ashkin-Teller current limit","problemNumber":null,"field":"Statistical mechanics; conformally invariant scaling limits","fieldGroup":"Probability & statistics","statement":"On the critical line of the Ashkin-Teller model ($J>0$, $U\\le J$, $\\sinh 2J=e^{-2U}$), Lis's representation gives a height function and primal and dual current-cluster collections. At the Ising point $U=0$ (double random currents) the height converges to a Gaussian free field and the clusters to two-valued local sets of that field. Alcalde Lopez, Heeney and Lis proved the joint limit of the height and both complete cluster collections in Jordan domains at that point and conjectured (Conjecture 1.2) the critical-line extension with $\\sqrt2$ replaced by $\\sqrt g$, $g=\\frac8\\pi\\arcsin(\\sqrt{1+e^{4U}}/2)$. Does this joint scaling limit hold at every point of the critical line, including the four-state Potts endpoint?","posedBy":"Tomas Alcalde Lopez, Lorca Heeney and Marcin Lis (arXiv 2602.05886, 2026, Conjecture 1.2)","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For every fixed point of the critical Ashkin-Teller line, including the four-state Potts endpoint, every bounded Jordan domain and every admissible polygonal approximation, the height, the wired and free current-cluster collections and the distinguished boundary cluster converge jointly: the height to a GFF with the predicted coupling constant $\\pi\\sqrt g$, and the clusters to canonical recursive two-valued local sets of the same field, at all nesting depths, in positive-diameter Hausdorff matching topology. Not claimed: cluster area measures, magnetization fields, all spin-domain-wall curves, or metric exponents.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and the main theorem ('Joint critical-line limit', Section 2) were read against Conjecture 1.2 as the paper describes it; the conjecture's source was not opened. The family has no Lean formalization (lean/docs/233.md does not exist at the pinned commit). The result is for bounded Jordan domains with admissible polygonal approximations and wired primal / free dual boundary conditions. It uses six-vertex inputs from Duminil-Copin, Kozlowski, Lammers and Manolescu (2026) and cites another release preprint on the balanced six-vertex GFF limit only as a comparison. The October 6 edition repairs the transmission and boundary-collar arguments and replaces excursion convergence with a signed loop-soup and Gaussian multiplicative chaos argument tied to the same field, adding inputs from the Brownian loop-soup literature (Lupu; Aidekon et al.; Jego-Lupu-Qian) and Berestycki-Sheffield-Sun; the main theorem in Section 2 is unchanged.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"A very recent specialist conjecture (posed in 2026) extending an Ising-point theorem along the critical Ashkin-Teller line; it matters within conformally invariant lattice models but had stood only months. Placed well below Gaussian free field limit of the six-vertex height function (40), below Polynomial-bond-dimension PEPS approximation of gapped ground states (28), near Spectral Edge of the Quartic SYK Model (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/main.pdf","sourceName":"OpenAI math release: The joint scaling limit of critical Ashkin-Teller currents","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Alcalde Lopez-Heeney-Lis, The Ising magnetisation field and the Gaussian free field (Conjecture 1.2)","url":"https://arxiv.org/abs/2602.05886","kind":"problem-record"},{"label":"Earlier edition (September 25, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-September-25-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"exact-quantum-factoring-finite-gate-set","name":"Exact (zero-error) polynomial-time quantum factoring over a fixed finite gate set","shortName":"Exact quantum factoring","problemNumber":null,"field":"Quantum algorithms; exact quantum computation","fieldGroup":"Quantum information & computing","statement":"Shor's algorithm factors integers in quantum polynomial time with bounded error. Exact versions were known for the quantum Fourier transform and discrete logarithms of known order (Mosca-Zalka, using computed gate angles), for order finding when a multiple of the order is supplied (Imran), and for factoring only in a generalized model with gate parameters computed during the run (Mosca). Mosca and Zalka (2003) noted that it is not clear how to make Shor's factoring exact, 'a challenge that remains', and Imran (2022) called derandomizing it a difficult open question. Is there a polynomial-time uniform family of polynomial-size quantum circuits over a fixed finite gate set that outputs the prime factorization of every $N\\ge2$ with probability exactly one?","posedBy":"Michele Mosca and Christof Zalka, Exact quantum Fourier transforms and discrete logarithm algorithms (2003), Section 5.2","yearPosed":2003,"ageNote":"The arXiv preprint is from January 2003 (journal 2004). The fixed-finite-gate-set requirement sharpens the question as Mosca-Zalka framed it; Nishimura-Ozawa (2009) studied exact computation by finitely generated uniform circuit families.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a polynomial-time uniform family of polynomial-size quantum circuits over NOT, CNOT, Toffoli, Hadamard, $\\mathrm{diag}(1,i)$, their inverses and singly controlled versions outputs the complete prime factorization of every $N\\ge2$ with probability one, with gate and qubit counts polynomial in the bit length. The construction makes the true-order mass of an order trial an exact dyadic rational, then uses one amplitude-amplification step, and handles auxiliary factorizations of $p-1$ recursively with a deterministic primality test. Not shown: practical resource bounds (the polynomials are large) or any classical algorithm.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript dated September 25, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Mosca-Zalka Section 5.2 and Imran's statement, both read here. Lean: lean/ComparatorChallenges/ExactQuantumFactoring.lean (solution module OAI.Computability.QuantumFactoring.Main, present at the pinned commit) is not in the formalization catalogue; it was found through lean/docs/279.md. Its statement exact_quantum_factoring was read: there is a circuit family over 20 fixed gates (NOT, CNOT, Toffoli, Hadamard, diag(1,i), inverses and singly controlled versions), generated by a polynomial-time TM2 machine from the input length, with polynomially many qubits and gates, whose Born probability of the correct padded nondecreasing prime factorization is exactly 1 for every N >= 2. That is the headline claim. Not rebuilt here. The paper claims no classical factoring speedup and no practical resource gains.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Whether factoring is solvable exactly, not just with bounded error, by a quantum computer over a finite gate set is a recognized question of quantum complexity theory, though with little practical consequence. Below the randomized versus quantum query conjecture (38) and Moore's parity-not-in-QAC0 conjecture (40), above the oracle-complexity gap in derivative-free convex optimization (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf","sourceName":"OpenAI math release: Exact quantum factoring over a fixed finite gate set","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean challenge statement: ExactQuantumFactoring.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ExactQuantumFactoring.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Computability/QuantumFactoring/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/QuantumFactoring/Main.lean","kind":"lean-proof"},{"label":"Mosca-Zalka (2003), Section 5.2: no exact factorisation algorithm","url":"https://arxiv.org/abs/quant-ph/0301093","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generalized-star-height-at-most-three","name":"A uniform bound of three on generalized star height (the generalized star-height problem)","shortName":"Generalized star height at most three","problemNumber":null,"field":"Automata and formal languages; regular expressions","fieldGroup":"Theoretical computer science","statement":"A generalized regular expression over a finite alphabet $\\Sigma$ is built from $0$, $1$ and letters by union, concatenation, complement in $\\Sigma^*$ and Kleene star; its height is the nesting depth of stars, and $h(L)$ is the least height of an expression for a regular language $L$. Without complement, star height is unbounded (Eggan; Dejean-Schutzenberger). With complement, height zero is exactly the star-free languages (Schutzenberger), height at most one was known for languages recognized by abelian groups, nilpotent groups of class two and some semigroups (Pin-Straubing-Therien, Bourne-Ruskuc), and no language of generalized height above one is known. Straubing distinguishes two questions: does one height bound serve for all regular languages, and does height one always suffice? Is there a uniform bound on generalized star height, and is it one?","posedBy":"The generalized star-height problem, as named by Pin, Straubing and Therien (1992); the uniform-boundedness form distinguished by Straubing (LATIN 2002, p. 537)","yearPosed":null,"ageNote":"The manuscripts cite Pin-Straubing-Therien (1992), Straubing (2002), Place-Zeitoun (2017) and Cotumaccio (2025) for the problem but give no original posing date; the problem is older than these references.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every finite alphabet $\\Sigma$ and every regular $L\\subseteq\\Sigma^*$, $h(L)\\le3$, using expressions over $\\Sigma$ itself with complement in $\\Sigma^*$. Companions prove the weaker bounds 13 and 4 by different constructions. It does NOT exhibit any language of generalized star height greater than one, does not decide whether height one always suffices, gives no algorithm for computing $h(L)$, and bounds neither expression size nor running time.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. All three manuscripts of the family (bounds 13, 4 and 3, all dated 25 September 2026) are credited to OpenAI with no human author named; each states that it gives its own complete proof by a different construction.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript ($h(L)\\le3$ for every regular $L$ over every finite alphabet, complement in the same free monoid) was read against the problem: it answers uniform boundedness and leaves open whether height one suffices, as the paper says. The Lean challenge ComparatorChallenges/GeneralizedStarHeight.json (theorem OAI.GeneralizedStarHeight.main, solution module OAI.Computability.StarHeight.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/134.md. Its statement was read here: an inductive expression type with zero, one, letters, union, concatenation, complement and star, height as defined in the paper, and for a finite alphabet every regular language (Mathlib's IsRegular) is the language of some expression of height at most three. That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The generalized star-height problem is a long-standing and well-known question in automata theory; a uniform bound answers its weaker boundedness half and is scored below the full height-one question. Above Cerny Conjecture for One-Cluster Automata (25), level with The Polynomial-Time Low-Degree Conjecture (30), below Polynomial-Factor Hardness for the Closest Vector Problem (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Star-Height-at-Most-Three-September-25-2026/article.pdf","sourceName":"OpenAI math release: Generalized Star Height at Most Three","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Finite Monoid Computations and a Uniform Generalized Star-Height Bound (bound 13)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026.pdf","kind":"paper"},{"label":"Companion: Generalized Star Height at Most Four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Star-Height-at-Most-Four-September-25-2026/Generalized-Star-Height-at-Most-Four-September-25-2026.pdf","kind":"paper"},{"label":"Lean: solution module for the height-three bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/StarHeight/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: generalized star height at most three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/GeneralizedStarHeight.lean","kind":"lean-proof"},{"label":"Straubing (2002), On logical descriptions of regular languages","url":"https://doi.org/10.1007/3-540-45995-2_46","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"yau-harmonic-dimension-comparison-integer-degree","name":"Yau's dimension comparison for harmonic functions of integer polynomial growth under nonnegative Ricci curvature","shortName":"Yau's harmonic dimension comparison","problemNumber":null,"field":"Riemannian geometry; harmonic functions of polynomial growth","fieldGroup":"Geometry & topology","statement":"For a complete Riemannian manifold $(M^n,g)$ let $\\mathcal H_d(M)$ be the space of harmonic functions with $|u(x)|\\le C(1+d(o,x))^d$ and $h_d(M)=\\dim\\mathcal H_d(M)$. On $\\mathbb R^n$, $h_k=\\binom{n+k-1}{k}+\\binom{n+k-2}{k-1}$ for integer $k\\ge1$. Li and Tam proved $h_1(M)\\le n+1$ when $\\mathrm{Ric}\\ge0$, Colding and Minicozzi proved finite dimensionality and bounds of optimal order $k^{n-1}$, and Donnelly found counterexamples at non-integer degrees between one and two. Yau asked about sharp Euclidean comparison. Does $\\mathrm{Ric}_g\\ge0$ imply $h_k(M^n)\\le h_k(\\mathbb R^n)$ for every integer $k\\ge1$?","posedBy":"Shing-Tung Yau, Open problems in geometry, Problem 48","yearPosed":1994,"ageNote":"The manuscript cites Yau's problem list in Schoen-Yau, Lectures on Differential Geometry (International Press, 1994), Problem 48. Whether an earlier list states the integer-degree comparison was not checked.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are an even $n\\ge8$ (the paper allows $n=16$), an integer $k\\ge2$ ($k=50000$) and a complete smooth metric on $\\mathbb R^n$ with $\\mathrm{Ric}\\ge0$, Euclidean near the origin, asymptotic volume ratio strictly between 0 and 1, and $h_k>h_k(\\mathbb R^n)$; the metric has nonunique tangent cones at infinity. Companion: in dimension three, for $4/9 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Two manuscripts: the principal one (September 25, 2026) in some even dimension n >= 8, and a three-dimensional companion (September 26, 2026). The principal manuscript ships an optional exact integer check of its finite spectral selection for n = 16, k = 50000, which it says does not verify the metric.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against Yau's question as cited. Lean (in lean/formalization.yaml): OAI.HarmonicCounterexample.main in OAI/Geometry/HarmonicGrowth/Main.lean. Its statement MainClaim gives an even $n\\ge8$, $k\\ge2$ and a smooth metric on $\\mathbb R^n$ that is complete for the intrinsic distance, has $\\mathrm{Ric}\\ge0$, is Euclidean near 0, has asymptotic volume ratio in $(0,1)$, and carries $h_k(\\mathbb R^n)+1$ linearly independent harmonic functions of growth $k$. That states the headline negative answer. The challenge file states MainClaim through a placeholder axiom; the solution module proves main as a theorem, and the comparator's permitted axioms exclude custom ones. Not rebuilt here. The 3D companion is not formalized.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A question from Yau's problem list, studied by Li-Tam, Colding-Minicozzi and Donnelly; the sharp integer-degree comparison is a known specialist question rather than one of Yau's headline conjectures. Above Arnold's corank problem (20), level with Kraetzer's integral means conjecture (25), below the Campana-Peternell nef tangent entry (28) and well below Yau's nodal set conjecture (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to integer-degree harmonic dimension comparison","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion manuscript: A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf","kind":"paper"},{"label":"Lean formalization (HarmonicGrowth main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/HarmonicGrowth/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"weakly-reversible-boundedness-persistence","name":"The boundedness and persistence conjectures for weakly reversible mass-action systems","shortName":"Mass-action boundedness and persistence","problemNumber":null,"field":"Chemical reaction network theory: mass-action kinetics","fieldGroup":"Differential equations","statement":"A reaction network has complexes $y\\in\\mathbb Z_{\\ge0}^d$ and reactions $y\\to y'$; it is weakly reversible if every reaction has a directed return path. With positive rate constants, mass-action kinetics is $\\dot x=\\sum_{y\\to y'}\\kappa_{y\\to y'}x^y(y'-y)$ on the positive orthant. Feinberg (1987) formulated the expectation that a positive trajectory of a weakly reversible system cannot converge to a point with a zero concentration. Anderson (2011) separated the boundedness conjecture (every positive trajectory is bounded) from the persistence conjecture (no bounded positive trajectory has $\\liminf_t x_i(t)=0$). Known cases included one linkage class (Anderson; Gopalkrishnan-Miller-Shiu), two species (Craciun-Nazarov-Pantea) and two-dimensional stoichiometric subspace (Pantea). For weakly reversible mass-action systems with positive constant rates, is every positive trajectory global, bounded and persistent?","posedBy":"Martin Feinberg, Chem. Eng. Sci. 42 (1987), Remark 6.1.E; split into the boundedness and persistence conjectures by David F. Anderson, J. Math. Chem. 49 (2011), Section 1.1","yearPosed":1987,"ageNote":"Feinberg's 1987 remark is the earliest source the manuscript cites; the two conjectures in their named form are from Anderson (2011). August and Barahona (2010) had stated the general result, but the manuscript points to a gap in their proof.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for any finite weakly reversible network with positive constant rates and any positive $x^0$, the solution exists for all $t\\ge0$ and some $\\varepsilon\\in(0,1)$, depending on the network, the rates and $x^0$, gives $\\varepsilon\\le x_i(t)\\le\\varepsilon^{-1}$ for all $t\\ge0$ and all $i$. The proof builds a compact convex forward-invariant polytope around each initial point from a finite concave piecewise-affine function. A remark derives convergence for complex-balanced systems, which the paper treats as known (citing Craciun's 2026 revision), not as new. The bounds are not uniform over a compatibility class (the permanence entry), and time-dependent rates are not treated.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The companion permanence manuscript (October 5) builds directly on this one; the README notes that some outputs build on earlier model results.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source; the proof was not refereed. Lean: the challenge MassAction (OAI.Problem326.global_bounded_persistent_solution, solution module OAI/Analysis/MassAction/Main.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Statement read here: for $d>0$, a finite network of natural-number complexes with weak reversibility as a transitive closure, all rates positive and any positive initial state, there is $\\varepsilon\\in(0,1)$ such that a solution defined for all $t\\ge0$ exists and every such solution stays in $[\\varepsilon,\\varepsilon^{-1}]$ in every coordinate. That is the headline claim (solutions are functions on all of $\\mathbb R$ with the derivative required for $t\\ge0$, a harmless encoding choice). Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The persistence conjecture is a central open problem of chemical reaction network theory, open since Feinberg's 1987 remark and tied to the global attractor conjecture; boundedness is its companion. Famous within one community. Scored with Popa's logarithmic additivity conjecture (35) and the Bogoliubov depletion law (35), above the Colin de Verdiere chromatic conjecture (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Boundedness and persistence of weakly reversible mass-action systems","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Uniform Permanence in Weakly Reversible Mass-Action Systems (stronger, classwise)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf","kind":"paper"},{"label":"Lean: OAI/Analysis/MassAction/Main.lean (global_bounded_persistent_solution)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/MassAction/Main.lean","kind":"lean-proof"},{"label":"Anderson, Boundedness of trajectories for weakly reversible, single linkage class reaction systems (arXiv:1104.4992)","url":"https://arxiv.org/abs/1104.4992","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"heilbronn-triangle-power-lower-bound","name":"Heilbronn's triangle problem: a power-saving lower bound n^(-2+eta), refuting the almost n^-2 upper bound","shortName":"Heilbronn triangle power lower bound","problemNumber":null,"field":"Discrete geometry; Heilbronn's triangle problem","fieldGroup":"Combinatorics","statement":"For $n$ points in the unit square let $\\Delta(P)$ be the least area of a triangle they determine, and $\\Delta(n)=\\max_{|P|=n}\\Delta(P)$. Heilbronn asked for the order of $\\Delta(n)$, conjecturing $\\Delta(n)=O(n^{-2})$ (reported by Roth, 1951); Erdos's parabola gives $\\Delta(n)\\gg n^{-2}$. Komlos, Pintz and Szemeredi (1982) disproved the conjecture with $\\Delta(n)\\gg(\\log n)/n^2$, and the best upper bound is $n^{-7/6+o(1)}$ (Cohen, Pohoata and Zakharov). The remaining belief, the almost $n^{-2}$ formulation discussed in Zakharov's survey, is that $\\Delta(n)\\le C_\\epsilon n^{-2+\\epsilon}$ for every $\\epsilon>0$. What is the order of $\\Delta(n)$; in particular, is $\\Delta(n)=n^{-2+o(1)}$?","posedBy":"Hans Heilbronn (reported by K. F. Roth, 1951); the almost n^-2 formulation as discussed in Zakharov's survey","yearPosed":1951,"ageNote":"Year is that of Roth's paper On a problem of Heilbronn (J. London Math. Soc. 26, 1951), the earliest source the manuscript gives; it does not date Heilbronn's own posing. The almost n^-2 formulation is cited only through Zakharov, Small triangles (J. London Math. Soc. 2026).","solveType":"disproved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: absolute constants $\\eta,c_1>0$ and $n_0$ with $\\Delta(n)\\ge c_1n^{-2+\\eta}$ for every $n\\ge n_0$. Hence the almost $n^{-2}$ upper-bound formulation is false. The construction samples integer columns in a box with finite-field norm congruence conditions and deletes the few small triangles; $\\eta$ is explicit but extremely small. It does not determine the order of $\\Delta(n)$: the gap to the upper bound $n^{-7/6+o(1)}$ remains, and no conjecture for the true exponent is made.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 25, 2026) is the whole family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Heilbronn's problem and the almost $n^{-2}$ formulation (1.1); it gives $\\Delta(n)\\ge c_1n^{-2+\\eta}$ for all large $n$, with an explicit but tiny $\\eta$. The proof was not refereed. The challenge lean/ComparatorChallenges/HeilbronnTriangle.json (solution module OAI.Geometry.HeilbronnTriangle.Main, present at the pinned commit) is not in the formalization catalogue; its statement HeilbronnTriangle.lean was read here. heilbronn_power_lower_bound gives an explicit $\\eta=1/(10^5K)>0$ and point sets of sizes $n_j\\to\\infty$ in $[0,1]^2$ with every triangle of area at least $n_j^{-2+\\eta}$; almost_n_minus_two_refuted states that $\\Delta(n)\\le Cn^{-2+\\eta/2}$ fails for all large $n$. This states the refutation; the 'every sufficiently large $n$' form is broader than the Lean. Not rebuilt here. The manuscript notes two earlier preprints (Ellmann, Agama) claiming stronger power bounds, with gaps it identifies.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Heilbronn's triangle problem is one of the best known Erdos-type problems in discrete geometry (Erdos #507, Green's list); a power saving over n^-2 overturns the long-held almost n^-2 belief but does not settle the order. Scored as a partial below the full problem, level with Erdos's Planar Unit Distance Conjecture (40) and the four-uniform Erdos matching conjecture (40), below the Kakeya maximal conjecture in three dimensions (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf","sourceName":"OpenAI math release: A power improvement in the Heilbronn triangle lower bound","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Problem355.heilbronn_power_lower_bound)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/HeilbronnTriangle/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: HeilbronnTriangle.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HeilbronnTriangle.lean","kind":"lean-proof"},{"label":"Erdos Problem #507 (Heilbronn's triangle problem)","url":"https://www.erdosproblems.com/507","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"friedgut-kalai-sharp-threshold-graphs","name":"The Friedgut-Kalai sharp-threshold conjecture: threshold width O(1/log^2 n) for monotone graph properties","shortName":"Friedgut-Kalai threshold width, graphs","problemNumber":null,"field":"Random graphs, analysis of Boolean functions","fieldGroup":"Combinatorics","statement":"For a nontrivial increasing graph property $P$ on $n$ vertices and $0<\\varepsilon<1/2$, let $p_a$ be the edge probability at which $G(n,p)$ has $P$ with probability $a$. Friedgut and Kalai (1996), combining the KKL and BKKKL influence theorems with Russo's formula, proved $p_{1-\\varepsilon}-p_\\varepsilon\\le C\\log(1/2\\varepsilon)/\\log n$, observed that containing a clique of size about $\\log n$ has width of order $(\\log n)^{-2}$, and conjectured that this is the truth (Conjecture 1.2). Bourgain and Kalai reached $(\\log n)^{-2+\\delta}$ for every $\\delta>0$. Is $p_{1-\\varepsilon}-p_\\varepsilon\\le C\\log(1/2\\varepsilon)/(\\log n)^2$ for every such property, with $C$ universal?","posedBy":"Ehud Friedgut and Gil Kalai","yearPosed":1996,"ageNote":"Friedgut-Kalai, Proc. AMS 124 (1996), Conjecture 1.2.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $n\\ge2$, nontrivial increasing vertex-symmetric graph property and $0<\\varepsilon<1/2$, $p_{1-\\varepsilon}-p_\\varepsilon\\le2^{19}\\log(1/(2\\varepsilon))/(\\log n)^2$. It follows from Theorem 1.2, $\\mathrm{Var}_p(f)\\le2^{17}I_p(f)/(\\log n)^2$ for every vertex-symmetric Boolean $f$ and every $0 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts (September 25 and October 5, 2026); the hypergraph paper adapts the method of the graph paper.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorems 1.1-1.2 were read against Conjecture 1.2 as the manuscript states it. The proof (vertex-block random restrictions and a low-degree Fourier estimate) was not refereed. The challenge lean/ComparatorChallenges/SharpThreshold.json (theorem OAI.Problem313.sharp_threshold_width, solution module OAI.Combinatorics.SharpThreshold.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml; it is found through lean/docs/186.md. Its statement was read here: for every $n\\ge2$, every vertex-permutation-invariant, increasing, nontrivial Boolean function of the edge configuration and every $0<\\varepsilon<1/2$, the difference of the $(1-\\varepsilon)$- and $\\varepsilon$-quantiles (infimum of $p\\in[0,1]$ with mean at least the level) is at most $2^{19}\\log(1/(2\\varepsilon))/(\\log n)^2$. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A thirty-year-old conjecture where random graphs meet the analysis of Boolean functions, approached to within any power by Bourgain and Kalai; well known in that community. Placed level with Feige's Conjecture (35), below The Courtade-Kumar conjecture (40) and The asymptotic Gotsman-Linial conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A Sharp Threshold Bound for Monotone Graph Properties","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A uniform influence bound for hypergraph properties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf","kind":"paper"},{"label":"Lean proof module (OAI.Problem313.sharp_threshold_width)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/SharpThreshold/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SharpThreshold.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SharpThreshold.lean","kind":"lean-proof"},{"label":"Friedgut-Kalai, Every monotone graph property has a sharp threshold (1996)","url":"https://doi.org/10.1090/S0002-9939-96-03732-X","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"yau-bounded-holomorphic-functions-negative-curvature","name":"Yau's question on bounded holomorphic functions: does a complete simply connected Kahler manifold with sectional curvature at most $-a^2<0$ carry a nonconstant bounded holomorphic function?","shortName":"Bounded holomorphic functions under negative curvature","problemNumber":null,"field":"Kahler geometry; several complex variables","fieldGroup":"Geometry & topology","statement":"Let $M$ be a complete simply connected Kahler manifold of complex dimension at least two whose real sectional curvature satisfies $\\mathrm{Sec}_g\\le -a^2<0$, with no lower curvature bound assumed. A biholomorphism to a bounded domain would give bounded coordinates; the weaker question asks only for one nonconstant bounded holomorphic function. Must $M$ admit a nonconstant bounded holomorphic function, that is, is $H^\\infty(M)\\ne\\mathbb C$?","posedBy":"S.-T. Yau, Problem 38 of his 1982 problem list, as recorded by Damin Wu and S.-T. Yau (2020, p. 103; 2019 survey, Section 3)","yearPosed":1982,"ageNote":"The manuscript uses the formulation recorded by Wu and Yau (2020) and traces it, through their 2019 survey, to Yau's 1982 Problem 38.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for some finite $m\\ge2$ there is a domain $\\mathcal T\\subset\\mathbb C^m$, diffeomorphic to $\\mathbb R^{2m}$, with a complete Kahler metric $g$ satisfying $\\mathrm{Sec}_g\\le -1$ and $H^\\infty(\\mathcal T)=\\mathbb C$: every bounded holomorphic function is constant, so the Caratheodory pseudometric vanishes. This disproves the universal assertion in the one-sided question, and also answers Wu's 1967 bounded-domain question negatively under its hypotheses. The sectional curvatures are unbounded below, so the question under a two-sided bound $-b^2\\le\\mathrm{Sec}\\le -a^2$ remains open; the dimension is large and not made explicit.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. It is the companion of the negatively pinched threefold manuscript of the same date; the release presents them as one family, and they answer different posed questions.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the one-sided question as Wu and Yau record it. lean/docs/359.md says the formalization covers only the negatively pinched companion and that this one-sided construction is separate, so there is no Lean statement for this entry. The dimension $m$ is not specified: it is some sufficiently large fixed finite value. The curvature is unbounded below, so the two-sided bounded-function question is left open, as the paper says.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":33,"significanceNote":"Yau's question on bounded holomorphic functions under negative curvature is a long-standing problem from his 1982 list, well known in several complex variables but specialist. Placed just below its sibling negatively-pinched-kahler-bounded-domain (36), since the counterexample needs unbounded curvature, and above beauville-splitting-two-summands (30) and kalton-problem-2-bi-lipschitz-c0 (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf","sourceName":"OpenAI math release: One-sided negative sectional curvature and the holomorphic Liouville property","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A negatively pinched Kahler threefold without bounded holomorphic coordinates","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sakoda-sipser-problem","name":"The Sakoda-Sipser problem: exponential cost of simulating nondeterministic automata by two-way deterministic ones","shortName":"Sakoda-Sipser problem","problemNumber":null,"field":"Automata theory; descriptional (state) complexity","fieldGroup":"Theoretical computer science","statement":"Sakoda and Sipser (STOC 1978) asked how many states a two-way deterministic finite automaton (2DFA) needs to simulate an $n$-state one-way or two-way nondeterministic automaton (1NFA, 2NFA). The best known simulations are exponential, and they showed that the family $B_h$ (one-way liveness: words of binary relations on an $h$-element set whose product is nonempty) is complete for the one-way case. Exponential lower bounds were known only for restricted deterministic targets (Sipser's sweeping automata, Kapoutsis's few-reversal machines), and against unrestricted 2DFAs only quadratic bounds were known. Is there a polynomial $p$ such that every $n$-state 1NFA, or 2NFA, over any finite alphabet has an equivalent 2DFA with at most $p(n)$ states?","posedBy":"William J. Sakoda and Michael Sipser","yearPosed":1978,"ageNote":"Sakoda and Sipser, Nondeterminism and the size of two way finite automata, STOC 1978, cited by both manuscripts.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For every $h\\ge2$, one-way liveness $\\mathrm{OWL}_h$ has a nondeterministic automaton with $h+3$ states and no left moves, while every equivalent 2DFA with $s$ states satisfies $4(s+2)^2\\ge2^{\\lfloor(h-2)/31\\rfloor}$. Hence no polynomial $Cn^c$ bounds 1NFA-to-2DFA (and so 2NFA-to-2DFA) conversion uniformly over finite alphabets. The deterministic target is unrestricted (stay moves, partial rules, infinite rejecting runs). The companion gives the weaker-rate bound $\\tfrac12 2^{\\lfloor(n-4)/127\\rfloor}$ for the same family. Not shown: any lower bound over a fixed alphabet (e.g. binary), any bound in transition-table size, or anything about L versus NL.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (one-way liveness, September 25, 2026) and its companion on complementation (same date) are independent arguments; the complementation paper adapts the amplification pattern of the liveness paper and also yields a deterministic bound for the same family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against the Sakoda-Sipser question. The witness is one-way liveness over the full relation alphabet, whose size grows with $h$; the claim rules out an alphabet-independent polynomial bound, which is the usual (Kapoutsis) formulation of the problem, and it does not give a bound over a fixed alphabet or any logarithmic-space separation, as the paper says. lean/formalization.yaml lists a main result for this manuscript (comparator OneWayLiveness, declaration OAI.OneWayLiveness.main_theorem, file OAI/Combinatorics/Automata/Main.lean). ComparatorChallenges/OneWayLiveness.lean was read here: it defines deterministic machines with partial transitions, stay moves and endmarkers, two acceptance conventions, and states the existence of the $(h+3)$-state no-left-move recognizer and the bound $2^{\\lfloor(h-2)/31\\rfloor}\\le4(s+2)^2$ (or $4(s+1)^2$) for every deterministic recognizer. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The Sakoda-Sipser problem has been the central open question of two-way automata state complexity since 1978, famous in automata theory and descriptional complexity, with only restricted-model exponential bounds before. Placed level with Brennan's conjecture (45) and Vicente's Gromov-width question (45), below Sidorenko's conjecture (52) and the 3SUM hypothesis (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026/main.pdf","sourceName":"OpenAI math release: An exponential two-way deterministic state lower bound for one-way liveness","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An exponential state lower bound for two-way nondeterministic complementation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.OneWayLiveness.main_theorem)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Automata/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: OneWayLiveness.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/OneWayLiveness.lean","kind":"lean-proof"},{"label":"Lean proof of the same-family determinization bound (explicit_family_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/TwoWayAutomata/ExplicitFamily.lean","kind":"lean-proof"},{"label":"Sakoda and Sipser (STOC 1978)","url":"https://doi.org/10.1145/800133.804357","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"homogeneous-depth-five-imm-unbounded-bottom-fan-in","name":"Homogeneous depth-five lower bounds for iterated matrix multiplication with unrestricted bottom fan-in (Bera-Chakrabarti question, balanced regime)","shortName":"Depth-five lower bounds for IMM","problemNumber":null,"field":"Algebraic complexity; arithmetic circuit lower bounds","fieldGroup":"Theoretical computer science","statement":"Let $\\mathrm{IMM}_{w,d}$ be the $(1,1)$ entry of a product of $d$ generic $w\\times w$ matrices. Nisan and Wigderson (1995) asked for superpolynomial lower bounds for homogeneous $\\Sigma\\Pi\\Sigma\\Pi\\Sigma$ circuits, and Kayal and Saha asked for depth-five bounds for IMM. Bera and Chakrabarti (CCC 2015) proved that homogeneous $\\Sigma\\Pi\\Sigma\\Pi\\Sigma$ circuits computing $\\mathrm{IMM}_{n^q,n}$ in $N$ variables need $N^{\\Omega(\\sqrt n)}$ gates, but only when every bottom linear form has fan-in at most $N^{\\mu}$ with $\\mu<1/2$. They called it the most immediate and natural open question whether the bound survives bottom fan-in up to $N^{1-\\Theta(1)}$ or $N$, the general case. Does the $\\sqrt n$-exponent lower bound for homogeneous depth-five circuits computing IMM hold with unrestricted bottom fan-in?","posedBy":"Suman K. Bera and Amit Chakrabarti, A Depth-Five Lower Bound for Iterated Matrix Multiplication (CCC 2015), closing open question; background question of Nisan and Wigderson (1995)","yearPosed":2015,"ageNote":"The manuscript cites Nisan-Wigderson (1995) and Bera-Chakrabarti (2015) but quotes no posing sentence; the closing sentence of Bera-Chakrabarti was read here. Superpolynomial depth-five lower bounds over characteristic zero already follow from Limaye-Srinivasan-Tavenas (2021), so what remained open was the sharp $\\sqrt n$ exponent with unrestricted bottom forms.","solveType":"proved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"Claims that over every field of characteristic zero every syntactically homogeneous $\\Sigma\\Pi\\Sigma\\Pi\\Sigma$ circuit computing $\\mathrm{IMM}_{n,n}$ has at least $n^{\\sqrt n/400}$ gates for large $n$, with gate sharing, arbitrary fan-in and bottom linear forms of arbitrary support allowed, and a block construction gives $n^{\\sqrt n+4}$ over every field, so the gate complexity is $n^{\\Theta(\\sqrt n)}$. The measure is the rank of a partial-derivative-times-multiplication operator. It does NOT treat the Bera-Chakrabarti regime $\\mathrm{IMM}_{n^q,n}$ directly, positive characteristic, non-homogeneous circuits, or depths beyond five.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. The lower and upper bounds have a Lean comparator challenge in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 and its corollaries were read against the Bera-Chakrabarti question. The Lean challenge lean/ComparatorChallenges/DepthFive.json (solution module OAI.Algebra.DepthFive.Bounds, present at the pinned commit) is not in the formalization catalogue; it was found through lean/docs/135.md. Its statement OAI.Problem335.main was read here: for circuits with five layers, formal-degree homogeneity at every sum gate and arbitrary bottom linear forms, computing IMM of n matrices of size n x n needs at least $n^{\\sqrt n/400}$ gates (leaves counted) for all large n, over the complex numbers and over every characteristic-zero field with one threshold; and over every field there are such circuits with at most $n^{\\sqrt n+4}$ gates. This is the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A specialist refinement in algebraic circuit complexity: superpolynomial depth-five lower bounds were already known, and what is new is the sharp square-root exponent for balanced IMM with unrestricted bottom forms, answering a nearby form of an open question Bera and Chakrabarti stated. Below the permanent formula lower bounds entry (38) and the DFT n log n question (45), level with the quantum key agreement entry (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026.pdf","sourceName":"OpenAI math release: Homogeneous depth-five lower bounds for iterated matrix multiplication","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean formalization: depth-five lower and upper bounds for IMM","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/DepthFive/Bounds.lean","kind":"lean-proof"},{"label":"Lean comparator statement: DepthFive","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DepthFive.lean","kind":"lean-proof"},{"label":"Bera and Chakrabarti, A Depth-Five Lower Bound for Iterated Matrix Multiplication (CCC 2015)","url":"https://doi.org/10.4230/LIPIcs.CCC.2015.183","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"curtis-conjecture-stable-hurewicz-image","name":"Curtis's conjecture: the mod-2 stable Hurewicz image of the sphere is spanned by the Hopf- and Kervaire-invariant-one classes","shortName":"Curtis's conjecture","problemNumber":null,"field":"Stable homotopy theory","fieldGroup":"Geometry & topology","statement":"For $d>0$ let $h_d:\\pi_d^S\\to H_d(Q_0S^0;\\mathbb F_2)$ be the mod-2 Hurewicz homomorphism into the homology of the basepoint component of $QS^0=\\mathrm{colim}_r\\Omega^rS^r$. The obvious elements with nonzero image are the Hopf-invariant-one classes $\\eta,\\nu,\\sigma$ in degrees 1, 3, 7 and the Kervaire-invariant-one classes $\\theta_j$ in degrees $2^{j+1}-2$, when they exist. Curtis (1975), in his study of the Dyer-Lashof and lambda algebras, conjectured that nothing else is detected; May (1977) recorded a gap, found by Wellington, in the proposed proof. The conjecture implies Eccles's conjecture for spheres. Is the image of $\\bigoplus_{d>0}h_d$ spanned over $\\mathbb F_2$ by the Hurewicz images of $\\eta,\\nu,\\sigma$ and of the Kervaire-invariant-one classes that exist?","posedBy":"Edward B. Curtis","yearPosed":1975,"ageNote":"Curtis, The Dyer-Lashof algebra and the Lambda-algebra, Illinois J. Math. 19 (1975); May, Infinite loop space theory, Bull. AMS 83 (1977), p. 484, footnote 3, records the gap found by Wellington.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: the image of $\\bigoplus_{d>0}h_d$ is spanned over $\\mathbb F_2$ by the Hurewicz images of $\\eta,\\nu,\\sigma$ and of the Kervaire-invariant-one classes $\\theta_j$ (degree $2^{j+1}-2$) whenever they exist; no existence is posited. Corollary 1.2: Eccles's conjecture holds for every sphere $S^n$, $n>0$: each nonzero spherical class in $H_*(QS^n;\\mathbb F_2)$ is the bottom class or a suspended Hopf-invariant-one image. Corollary 1.3 (with Hill-Hopkins-Ravenel): $h_d=0$ for $d>0$ outside $\\{1,2,3,6,7,14,30,62,126\\}$, so the positive image vanishes above degree 126. Not shown: an exact dimension of the image or nonvanishing in each listed degree, odd primes, or Eccles's conjecture for $QX$ with $X$ other than a sphere.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The single manuscript (September 25, 2026) is the whole family.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 with Corollaries 1.2 and 1.3 were read against Curtis's conjecture as the manuscript cites it; Theorem 1.1 states the conjecture in every positive degree. The proof (last nonzero homology suspension, weight decomposition through Dickson-algebra modules, a Steenrod-module obstruction, Kuhn's lifting theorem and Browder's identification of Kervaire classes) was not refereed. No Lean formalization accompanies this manuscript. The paper says its exclusion of fourth powers (Section 4) is already present in Eccles-Zare, and Corollary 1.3 uses the Hill-Hopkins-Ravenel nonexistence theorem as an input.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Curtis's conjecture has been the central open problem on spherical classes in the homology of infinite loop spaces for about fifty years, with an earlier proposed proof found to have a gap; famous in algebraic topology but little known outside it. Placed below the Kervaire conjecture (55), slightly above Wall's D(2) problem (46) and Donovan's conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026/paper.pdf","sourceName":"OpenAI math release: The Stable Hurewicz Image of the Sphere at Two","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"polynomial-ordered-matrix-removal","name":"The polynomial removal conjecture for ordered binary matrices","shortName":"Polynomial ordered matrix removal","problemNumber":null,"field":"Extremal combinatorics; property testing","fieldGroup":"Combinatorics","statement":"For a fixed binary $k\\times k$ matrix $H$, an ordered copy in a binary $n\\times n$ matrix $A$ is a choice of rows $r_1<\\dots0$ with $N_H(A)\\ge c_H\\epsilon^{C_H}n^{2k}$ whenever $A$ is $\\epsilon$-far from $H$-free?","posedBy":"Noga Alon and Omri Ben-Eliezer (Problem 1.4, 2020); Lior Gishboliner and Asaf Shapira (Conjecture 4.5, 2025); question raised by Alon, Fischer and Newman (2007)","yearPosed":2007,"ageNote":"Alon, Fischer and Newman raised the ordered-matrix question in 2007 (Section 7); the polynomial removal problem was formulated by Alon and Ben-Eliezer (arXiv 2016, journal 2020), and the single-pattern conjecture appears in Gishboliner-Shapira's survey (arXiv v1, August 2025).","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a fixed binary $66\\times66$ matrix $H$ such that for every $h\\ge1$, with $n_h=(386h+2)2^h$ and $\\epsilon_h=(386h+2)^{-2}$, some $A_h$ is $\\epsilon_h$-far from $H$-free yet $N_H(A_h)\\le\\epsilon_h2^{-h}n_h^{132}$; hence no polynomial removal bound holds for this $H$, and the finite-family question fails for one square pattern. Corollary: canonical row-column sampling testers need $\\exp(\\Omega(\\epsilon^{-1/2}))$ samples along this sequence. Not shown: which patterns do admit polynomial removal, or a counterexample smaller than $66\\times66$.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript dated September 25, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the conjecture as the manuscript quotes it. Lean: formalization.yaml lists ComparatorChallenges/MatrixRemoval.json, declaration OAI.Problem348.no_polynomial_removal_bound in OAI/Combinatorics/MatrixRemoval/Main.lean. The challenge statement was read: for one explicitly defined 66x66 Boolean matrix and all c, C > 0, there are n, 0 < eps < 1 and an n x n binary matrix whose normalized edit distance to H-freeness (edits in both directions) is at least eps and whose ordered-copy count is below c eps^C n^132. That is the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named question in property testing and removal lemmas, of interest to a small community. Placed near the Boolean Max-k-CSP approximation entry (15), below the Daykin-Frankl conjecture (25) and the polynomial-time low-degree conjecture (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Polynomial removal fails for ordered binary matrices","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean challenge statement: MatrixRemoval.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/MatrixRemoval.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Combinatorics/MatrixRemoval/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/MatrixRemoval/Main.lean","kind":"lean-proof"},{"label":"Alon-Ben-Eliezer, Efficient removal lemmas for matrices (Problem 1.4)","url":"https://arxiv.org/abs/1609.04235","kind":"problem-record"},{"label":"Gishboliner-Shapira, Polynomial property testing (Conjecture 4.5 in arXiv v1)","url":"https://arxiv.org/abs/2508.16878v1","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"block-sensitivity-quadratic-in-sensitivity","name":"Is block sensitivity at most quadratic in sensitivity? (Nisan-Szegedy)","shortName":"Quadratic sensitivity vs block sensitivity","problemNumber":null,"field":"Boolean function complexity; sensitivity","fieldGroup":"Theoretical computer science","statement":"For a total Boolean function $f:\\{0,1\\}^n\\to\\{0,1\\}$, the sensitivity $s(f)$ is the largest number of single bits whose flip changes $f$ at some input, and the block sensitivity $\\mathrm{bs}(f)$ the largest number of disjoint blocks of bits each of whose flip changes $f$ at some input. Nisan and Szegedy (1994) explicitly suggested $\\mathrm{bs}(f)\\le s(f)^2$. Rubinstein's functions give $\\mathrm{bs}=s^2/2$, improved to $\\frac23s^2-\\frac13s$ by Ambainis and Sun (2011); Huang's 2019 proof of the Sensitivity Conjecture gives $\\mathrm{bs}(f)\\le s(f)^4$. Is there a universal constant $C$ with $\\mathrm{bs}(f)\\le C\\,s(f)^2$ for every total Boolean function?","posedBy":"Noam Nisan and Mario Szegedy (Section 4, Computational Complexity 1994)","yearPosed":1994,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every integer $d\\ge1$ an explicit recursive construction gives a nonconstant total Boolean function with $\\mathrm{bs}(f)/s(f)^2\\ge2^d/(4(d+2)^2)$, so no bound $\\mathrm{bs}\\le C s^2$ holds; the abstract also claims a fixed $\\alpha>2$ with $\\mathrm{bs}(f)\\ge s(f)^\\alpha$ along an unbounded family. This does not close the gap to Huang's upper bound $\\mathrm{bs}\\le s^4$; the true exponent remains open. Because $\\lambda(f)\\le s(f)$, it also implies a negative answer to the spectral-sensitivity question $\\mathrm{bs}=O(\\lambda^2)$, which Meiburg had already disproved (see block-sensitivity-spectral-sensitivity-squared).","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript adapts the gated-tournament architecture of Meiburg's 2026 spectral-sensitivity separation (arXiv 2608.00851), whose own attribution credits GPT-5.6-Sol with those ideas.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the quadratic question: for every $C$ some nonconstant total $f$ has $\\mathrm{bs}(f)>C\\,s(f)^2$, with $\\mathrm{bs}/s^2\\ge2^d/(4(d+2)^2)$ for each $d\\ge1$. lean/formalization.yaml lists SensitivitySeparation.json (declaration OAI.Paper320.quantitative_separation, file OAI/Combinatorics/Sensitivity/Separation.lean). The statement SensitivitySeparation.lean was read here; not rebuilt here. It defines sensitivity and block sensitivity (disjoint nonempty blocks) as in the paper and states, for every $d\\ge1$, a nonconstant $f$ on Fin n inputs with $2^d/(4(d+2)^2)\\le\\mathrm{bs}(f)/s(f)^2$; the ratio is unbounded, so this states the headline disproof. The abstract's fixed exponent $\\alpha>2$ is not in this statement. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The quadratic form was the conjectured sharp version of the Sensitivity Conjecture for 25 years and remained the natural target after Huang's 2019 proof; known to every complexity theorist but not beyond. Below the Gotsman-Linial entry at 42 and the trace reconstruction entry in this batch at 42, level with the Werner two-copy distillability entry at 35.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf","sourceName":"OpenAI math release: A superquadratic separation between sensitivity and block sensitivity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Paper320.quantitative_separation)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Sensitivity/Separation.lean","kind":"lean-proof"},{"label":"Nisan, Szegedy, On the degree of Boolean functions as real polynomials (1994)","url":"https://doi.org/10.1007/BF01263419","kind":"problem-record"},{"label":"Meiburg, Block sensitivity can exceed spectral sensitivity squared (arXiv:2608.00851), the construction this adapts","url":"https://arxiv.org/abs/2608.00851","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hyperbolic-one-relator-groups-virtually-special","name":"Are hyperbolic one-relator groups virtually compact special?","shortName":"Hyperbolic one-relator groups special","problemNumber":null,"field":"Geometric group theory; special cube complexes","fieldGroup":"Algebra","statement":"A group is virtually compact special if a finite-index subgroup is the fundamental group of a compact nonpositively curved cube complex admitting a local isometry to the Salvetti complex of a right-angled Artin group; for hyperbolic groups this gives residual finiteness, linearity and separability of quasiconvex subgroups. Wise proved it for one-relator groups with torsion (answering Baumslag's 1967 residual-finiteness question for that class) and asked, in his question on cocompact cubulation of hyperbolic groups with hierarchies, about hyperbolic one-relator groups in general. Linton settled the case where all two-generator subgroups are free. Is every word-hyperbolic one-relator group $F(X)/\\langle\\langle r\\rangle\\rangle$ virtually compact special?","posedBy":"Daniel T. Wise, 'The cubical route to understanding groups', ICM 2014 (Problem 13.15)","yearPosed":2014,"ageNote":"The manuscript cites Wise's ICM 2014 Problem 13.15, a broader cubulation question in which Wise singled out hyperbolic one-relator groups, and Wise's Conjecture 17.8 (virtually free-by-cyclic) as recalled by Kielak-Linton. Baumslag's 1967 Conjecture A is the older residual-finiteness question for the torsion case only.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every word-hyperbolic one-relator group is virtually compact special. Theorem 1.2: every hyperbolic ascending mapping torus of an injective endomorphism of a finitely generated free group is virtually compact special. With Kielak-Linton, every hyperbolic one-relator group is virtually free-by-cyclic (kernel possibly of infinite rank in the subgroup-closed sense, with a finite-index embedding in a free-by-$\\mathbb Z$ group with finitely generated kernel), which is Wise's Conjecture 17.8 for this class, and every hyperbolic one-relator group is residually finite and linear over $\\mathbb Z$. It says nothing about non-hyperbolic one-relator groups. Consistent with the release's non-residually-finite hyperbolic group (catalog entry on residual finiteness), which is not one-relator.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The proof imports the companion hyperbolicity paper's results at two stated points (hyperbolicity of embedded primitive extension groups and the geometry of their Magnus rows); no specialness conclusion is imported from it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the question; it asserts virtual compact specialness, with an explicit finite cube complex and local isometry to a Salvetti complex, for every hyperbolic group $F(X)/\\langle\\langle r\\rangle\\rangle$ with $X$ finite. The argument uses Linton's reduction to primitive extension groups and a new selective-filling criterion; it was not refereed here. The entry depends on the companion hyperbolicity paper at two points, so an error there could affect it. There is no Lean formalization for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Cubulating hyperbolic one-relator groups, with residual finiteness and the virtually free-by-cyclic conjecture as consequences, was one of the main open targets of Wise's program after the torsion case. Famous within geometric group theory. Placed just below Gersten's conjecture in this family, below Is every word-hyperbolic group CAT(0)? (45) and Are all word-hyperbolic groups residually finite? (66), and above Commutator Relators Do Not Force Hopficity, Residual Finiteness or Automaticity (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf","sourceName":"OpenAI math release: Virtual compact specialness of hyperbolic one-relator groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Baumslag-Solitar-free one-relator groups are hyperbolic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf","kind":"paper"},{"label":"Wise, The cubical route to understanding groups (ICM 2014)","url":"https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2014.2/ICM2014.2.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"exact-thue-morse-matching-heights-in-an-avoiding-population","name":"Exact Thue–Morse matching heights in an avoiding population","shortName":"Thue-Morse matching heights","problemNumber":null,"field":"Combinatorics on words; infinite labelled graphs","fieldGroup":"Combinatorics","statement":"Section 5 of the separate 10 September 2026 manuscript A classification of biologically unavoidable sequences asks for useful explicit bounds on matching-prefix lengths for its fixed Thue–Morse target. More precisely, let $t(n)$ be the parity of the number of ones in the binary expansion of $n$. For $w ≥ 2$, the graph has $w-1 -> w$ labelled $t(w)$ and $w-2 -> w$ labelled $1-t(w)$, with no edge $0 -> 1$. A matching path starting at $v$ has label $t(k)$ on its $k$-th edge, starting with $k=0$. How large can the number of matching edges be, as a function of $v$? The preceding source argument gives finiteness at every start but leaves useful explicit bounds open. This is a quantitative follow-up to the already catalogued classification, not a new claim to solve the original classification problem.","posedBy":"Separate avg-netizen manuscript, A classification of biologically unavoidable sequences (10 Sep 2026), Section 5; AI role in Section 7. Alexander posed the earlier classification question.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The proof gives an explicit closed formula for the attained maximum $L(v)$ at every natural starting vertex, including $L(0)=1$, and a certified ten-coordinate integer recurrence that evaluates the same actual-path maximum from binary digits. For $v ≥ 1$, $3L(v)\\le 8v-1$. Equality holds exactly at $v=3\\cdot 2^n-1$ for natural n, where $L(v)=8\\cdot 2^n-3$. The coefficient 8/3 cannot be decreased even after allowing a fixed additive constant. The source note gives the complete piecewise formula. The theorem proves both existence of a path attaining the value and an upper bound on every matching path. This answers the later manuscript's quantitative question for its fixed graph and unshifted target. It does not originate the earlier classification or resolve Alexander's remaining universal-avoider embedding or general ordinal-characterization questions. A bounded literature review is available, but worldwide priority is not certified.","claimIssueNote":null,"solveDate":"2026-09-25","model":"OpenAI Codex (GPT-6 family)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"A human collaborator directed the research questions and requested explanations, literature checks and formal verification. Codex agents developed the central mathematical arguments, exploration tools, Lean proofs and exposition. Other agents in the same project reviewed source attribution and informal-to-formal correspondence. These are internal project checks, not independent human expert endorsement. The AI-discovered classification describes model-led proof discovery; it does not assert that a human independently verified the mathematics.","verification":"lean-checked","verificationNote":"Checked here on 27 September 2026. GitHub Actions run 36116271138 is green at the pinned commit 19544a4d, under a workflow named Verify, which is what the submission claims. The audited endpoints named in the submission - FullHeight.height_isMaximum, height_formula, height_closed_form, DigitRecurrence.actual_digit_recurrence and evaluate_isMaximum - concern actual matching paths and attained maxima rather than a relaxation, and the permitted logical axioms are propext, Classical.choice and Quot.sound. Lean-checked rather than Lean-verified: the statements are anchored only by the project's own agent review, which the submitter correctly says is not independent, and the site did not rebuild the development. The underlying graph and target are those of the catalog's classification entry, and the question answered is the one raised in Section 5 of the manuscript behind it.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A sharp quantitative question raised in the final section of a manuscript published weeks earlier, answered by a different person with a formal proof. Precisely posed and genuinely open, but with no literature behind it: the band the catalog uses for a conjecture stated in one recent paper and settled shortly after, level with the fifth-order autocorrelation entry at 4 plus a little for the Lean work and the sharpness claim.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/19544a4d7608c8cc0d5a1205605c0f38631ca05f/notes/FULL-HEIGHT-PROOF.md","sourceName":"Public research note with Lean proofs at pinned commit 19544a4d, verified by GitHub Actions run 36116271138","collection":null,"collectionVersion":null,"links":[{"label":"Prior quantitative question: pinned classification manuscript, Section 5","url":"https://github.com/avg-netizen/biological-unavoidability/blob/3d6175e3e23f67bd68e7be591b5a9a6d04e496a3/paper.md#5-universality-and-computability-in-the-witness-family","kind":"problem-record"},{"label":"Exact all-start Lean proof","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/19544a4d7608c8cc0d5a1205605c0f38631ca05f/lean/SamuelAlexanderResearch/FullHeight.lean","kind":"lean-proof"},{"label":"Certified digit evaluator and actual maximum","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/blob/19544a4d7608c8cc0d5a1205605c0f38631ca05f/lean/SamuelAlexanderResearch/DigitRecurrence.lean","kind":"lean-proof"},{"label":"Successful verification of the submitted commit","url":"https://github.com/Sodelin/Work-on-Samuel-Alexander-Research-/actions/runs/36116271138","kind":"other"},{"label":"Alexander's video: Thue–Morse discussion (context, not endorsement)","url":"https://www.youtube.com/watch?v=UGnh24_zuEw&t=1361s","kind":"discussion"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":1},{"slug":"weak-inhomogeneous-duffin-schaeffer","name":"The weak inhomogeneous Duffin-Schaeffer conjecture","shortName":"Weak inhomogeneous Duffin-Schaeffer","problemNumber":null,"field":"Metric Diophantine approximation","fieldGroup":"Number theory","statement":"Koukoulopoulos and Maynard (2020) proved the Duffin-Schaeffer conjecture: for $\\psi:\\mathbb{N}\\to[0,\\infty)$ with $\\sum_q\\phi(q)\\psi(q)/q=\\infty$, almost every $x$ has infinitely many reduced fractions $a/q$ with $|x-a/q|<\\psi(q)/q$. For a fixed shift $\\gamma$, unweighted divergence does not suffice without monotonicity (Ramirez; Chow-Hauke-Pollington-Ramirez), and the coprime-numerator inhomogeneous version is false. The weak inhomogeneous conjecture (Yu 2021, Question 1.2; Chow-Technau 2024, Conjecture 1.22; Beresnevich-Hauke-Velani 2024, Conjecture 2), proved for rational shifts by Beresnevich-Hauke-Velani, asks: for every fixed $\\gamma\\in\\mathbb{R}$ and every $\\psi$ with $\\sum_q\\phi(q)\\psi(q)/q=\\infty$, does $\\|qx-\\gamma\\|<\\psi(q)$ hold for infinitely many $q$, for almost every $x$?","posedBy":"H. Yu, J. Number Theory (2021), Question 1.2; named by Chow and Technau (2024, Conjecture 1.22); also Beresnevich, Hauke and Velani (2024), Conjecture 2","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every fixed $\\gamma\\in\\mathbb{R}$ and finite-valued $\\psi:\\mathbb{N}\\to[0,\\infty)$ with $\\sum_q\\phi(q)\\psi(q)/q=\\infty$, almost every $x$ satisfies $\\|qx-\\gamma\\|<\\psi(q)$ for infinitely many $q$; the null set may depend on $\\gamma$ and $\\psi$. Corollary 1.2 gives the Hausdorff-measure analogue via mass transference. The arithmetic core adapts the Koukoulopoulos-Maynard common-pivot method (as simplified by Green-Walker). It proves no convergence converse, no quantitative asymptotic, no coprime-numerator statement, and no result uniform in the shift.","claimIssueNote":null,"solveDate":"2026-09-25","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as stated in the three sources the manuscript cites; the proof was not refereed. The statement covers every real shift, including irrational ones, with arbitrary nonmonotone $\\psi$ and unrestricted numerators. The paper notes that the coprime-numerator version is false (Hauke-Treuer-Maynard-Pollington; He-Liao, both 2026), so this entry concerns only the weak form. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The natural inhomogeneous successor of the Duffin-Schaeffer conjecture, stated in 2021-2024 and pursued by several groups, but young and specialist. Scored with the lonely runner conjecture for nine and ten runners (30) and the Bollobas-Nikiforov conjecture (30), below the two-point Chowla conjecture (58) and the Duffin-Schaeffer level of fame.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf","sourceName":"OpenAI math release: The weak inhomogeneous Duffin-Schaeffer conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Beresnevich, Hauke, Velani, Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer (arXiv:2406.19198)","url":"https://arxiv.org/abs/2406.19198","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bfln-initial-form-milnor-fibre-conjecture","name":"The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs","shortName":"BFLN initial-form Milnor fibre conjecture","problemNumber":null,"field":"Singularity theory; Milnor fibres and contact loci","fieldGroup":"Geometry & topology","statement":"For a nonzero holomorphic germ $g$ vanishing at the origin, let $g_*$ be its lowest nonzero homogeneous component; the initial-form Milnor fibre is $g_*^{-1}(1)$. Motivated by the relation between contact loci and monodromy, Budur, Fernandez de Bobadilla, Le and Nguyen conjectured (Cohomology of contact loci, Conjecture 1.7 in arXiv v3) that embedded topologically equivalent function germs have homotopy equivalent initial-form Milnor fibres; Sampaio (2025, Conjecture 4) studies the version for topological right equivalence. If $f,g:(\\mathbb{C}^N,0)\\to(\\mathbb{C},0)$ are topologically equivalent, are $f_*^{-1}(1)$ and $g_*^{-1}(1)$ homotopy equivalent?","posedBy":"Nero Budur, Javier Fernandez de Bobadilla, Quy Thuong Le and Hong Duc Nguyen (J. Differential Geom. 120, 2022; Conjecture 1.7 of arXiv v3)","yearPosed":2022,"ageNote":"Year of the journal version cited by the manuscript; the conjecture is numbered in arXiv:1911.08213v3, and the manuscript does not say in which arXiv version it first appeared.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"With $f_1,f_2$ the Zariski counterexample germs, $\\tilde f_j=f_j(z)+t^2$ on $\\mathbb{C}^{N+1}$ are topologically right equivalent with isolated critical points, but $H_0((\\tilde f_1)_*^{-1}(1);\\mathbb{Z})\\cong\\mathbb{Z}$ while $H_0((\\tilde f_2)_*^{-1}(1);\\mathbb{Z})\\cong\\mathbb{Z}^2$: one initial-form Milnor fibre is connected and the other has two components. This disproves both the embedded and the right-equivalence formulations. It says nothing about weaker invariants the conjecture was meant to capture, such as cohomology of contact loci.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two manuscripts in the family, September 24 and September 27, 2026, which the September 27 paper describes as independent arguments. The September 24 paper ships Python scripts (standard library only) that check its finite integer arithmetic; the README says they do not cover the lattice-theoretic or topological arguments.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the corollary 'Initial-form Milnor-fibre counterexamples' in the September 24 manuscript was read against the conjecture as the manuscript states it. It follows from the Zariski counterexample by adding one common square t^2, which preserves topological right equivalence. Not refereed; no Lean formalization. The comparison of zeroth homology is elementary once the right equivalence is granted, so the weight rests on the main construction.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A recent conjecture from a Journal of Differential Geometry paper, followed up by Sampaio; known to a small community working on contact loci and Milnor fibres, and refuted here as a corollary of the Zariski counterexample. Level with the Ciliberto question on effective divisors (15) and the Lassak reduced-body bound (15), below Arnold's corank problem (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Budur, Fernandez de Bobadilla, Le, Nguyen, Cohomology of contact loci (arXiv:1911.08213v3)","url":"https://arxiv.org/abs/1911.08213v3","kind":"problem-record"},{"label":"Sampaio, On the Milnor fibres of initial forms of topologically equivalent holomorphic functions (arXiv:2503.17127)","url":"https://arxiv.org/abs/2503.17127v1","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"metric-k-median-approximation-threshold","name":"The approximation threshold for metric $k$-median: is $1+2/e$ attainable?","shortName":"Metric k-median threshold","problemNumber":null,"field":"Approximation algorithms; clustering","fieldGroup":"Theoretical computer science","statement":"In metric $k$-median, finite sets of clients $J$ and candidate facilities $F$ lie in a common metric space, and one opens at most $k$ facilities to minimize the total distance from clients to their nearest open facility. Approximating within any factor below $1+2/e\\approx1.736$ is NP-hard in this candidate-facility model, by the Max-$k$-Coverage gap. Algorithms improved from the first constant factor (Charikar-Guha-Tardos-Shmoys) through $3+\\varepsilon$ local search (Arya et al.), Li-Svensson's route past factor three, $2.675+\\varepsilon$ (Byrka et al.) and $2.613$ (Gowda et al.) to $2+\\varepsilon$ (Cohen-Addad et al., announced 2025). Is there, for every $\\varepsilon>0$, a polynomial-time $(1+2/e+\\varepsilon)$-approximation, so that $1+2/e$ is the exact approximation threshold?","posedBy":"Implicit in the 1+2/e hardness bound of Jain, Mahdian and Saberi (2002); the manuscript names no poser","yearPosed":null,"ageNote":"The manuscript does not attribute the question; it cites the hardness through Anand-Lee (2024) and Feige (1998). Matching the 1+2/e lower bound has been a standard open question in approximation algorithms since that bound appeared around 2002, but that dating comes from the general literature, not from this manuscript, so yearPosed is left empty.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every fixed $\\varepsilon>0$, a deterministic algorithm running in time polynomial in the binary input length (the polynomial depends on $\\varepsilon$) returns $S\\subseteq F$ with $|S|\\le k$ and cost at most $(1+2/e+\\varepsilon)\\mathrm{OPT}_k$, for arbitrary $k$ and arbitrary finite rational metrics with specified candidate facilities. Corollary 1.2: under $P\\ne NP$, the infimum of polynomial-time approximation factors in this model is $1+2/e$, the lower bound being the known coverage gap. It does not give a $(1+2/e)$-approximation itself, does not settle the model where every client is a candidate facility ($F=J$), whose lower bound differs, and claims no practical running time. The companion proves a randomized $(2-\\sigma)$-approximation for an absolute $\\sigma>0$ by anchor recovery and bounded-price payment accounting.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. A companion manuscript of the same date (single-exponential recovery and bounded-price strictness) gives an independent randomized approximation below factor two and is cited by the principal paper.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against the question. lean/docs/125.md lists four challenges; two cover this entry: ComparatorChallenges/MetricKMedian.json (theorem OAI.MetricKMedian.main_finite_work, solution module OAI.Combinatorics.KMedian.Main) and KMedianThreshold.json (OAI.MetricKMedian.threshold, module OAI.Combinatorics.KMedian.Threshold). Both solution files exist at the pinned commit, and neither challenge is in the formalization catalogue formalization.yaml (only the companion's KMedianRecovery is). The statements were read here: for every $\\varepsilon>0$ a polynomial-time Turing machine with finite alphabets outputs a nonempty set of at most $k$ candidate facilities of cost at most $(1+2/e+\\varepsilon)\\mathrm{OPT}$ on every finite rational instance; and, from a formalized $P\\ne NP$, the infimum of deterministic polynomial-time factors is exactly $1+2/e$, so the hardness side is formalized too. That is the headline claim. Not rebuilt here. The threshold is an infimum: attaining $1+2/e$ itself is not claimed, and the $F=J$ model is a different question.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Matching the k-median hardness bound is one of the best-known open problems in approximation algorithms, attacked for over two decades through a long chain of improved factors. Famous across that community but not beyond it. Placed level with White's Conjecture on Matroids (40), well above The integrality gap of weighted k-matroid intersection (20) and Approximate Counting for Spin Systems on Planar Graphs (18), and far below The Unique Games Conjecture (74).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf","sourceName":"OpenAI math release: The approximation threshold for metric k-median","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Single-exponential recovery and bounded-price strictness for metric k-median","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.MetricKMedian.main_finite_work)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/KMedian/Main.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.MetricKMedian.threshold)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/KMedian/Threshold.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"corrected-bogomolov-gieseker-threefolds","name":"A curve-class corrected Bogomolov-Gieseker inequality on every polarized threefold","shortName":"Corrected BG inequality on threefolds","problemNumber":null,"field":"Algebraic geometry; tilt stability and Bogomolov-Gieseker inequalities","fieldGroup":"Geometry & topology","statement":"Bayer, Macri and Toda conjectured an inequality $\\mathrm{ch}_3^B(E)\\le\\frac{a^2}{6}H^2\\mathrm{ch}_1^B(E)$ for tilt-semistable objects of tilt slope zero on threefolds, which would produce Bridgeland stability conditions; Schmidt and Martinez-Schmidt found counterexamples (blow-ups, contracted divisors, Weierstrass fibrations). Bernardara, Macri, Schmidt and Zhao (2017) proved a corrected form for Fano threefolds and asked whether on every smooth polarized threefold $(X,H)$ there is a curve class $\\Gamma$ such that $\\mathrm{ch}_3^{B}(E)\\le\\frac{a^2}{6}H^2\\mathrm{ch}_1^{B}(E)+\\Gamma\\cdot\\mathrm{ch}_1^{B}(E)$ for all tilt-semistable $E$ of slope zero; Martinez and Schmidt (2019) asked the version with a fixed rational transverse twist $B_0$. Does such a corrected inequality hold on every polarized threefold?","posedBy":"Marcello Bernardara, Emanuele Macri, Benjamin Schmidt and Xiaolei Zhao; twisted version by Cristian Martinez and Benjamin Schmidt","yearPosed":2017,"ageNote":"Bernardara-Macri-Schmidt-Zhao, Bridgeland stability conditions on Fano threefolds (Epijournal 2017), Question 2.4; Martinez-Schmidt, Bridgeland stability on blow ups and counterexamples (Math. Z. 2019), Question 2.6, as cited by the companion manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.3: for every smooth projective complex threefold $X$, ample integral $H$ and $B_0\\in N^1(X)_{\\mathbb Q}$ there are $k\\ge0$ rational and $R_*>0$ with $\\mathrm{ch}_3^{B_0+bH}(E)\\le(a^2/6+k)H^2\\mathrm{ch}_1^{B_0+bH}(E)$ for every tilt-semistable $E$ of slope zero, and the uncorrected bound for $a>R_*$; no canonical-bundle hypothesis. The same manuscript constructs, on threefolds with trivial canonical bundle, stability conditions with the exact ordinary and square-root-Todd large-volume charges on an open set of twists and polarizations. It does not give explicit constants or prove the uncorrected BMT inequality at all volumes, which is false in general.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry's principal manuscript (September 24, 2026) is the large-volume companion of the quintic Gepner paper; its uniform inequality is proved independently of its charge constructions.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.3 (uniform strong inequality) of the manuscript was read against the two cited questions; it gives, for any smooth projective complex threefold, ample integral $H$ and rational $B_0$, constants $k\\in\\mathbb Q_{\\ge0}$ and $R_*$ such that every finite-slope $\\nu_{a,b}$-semistable object of slope zero satisfies the inequality corrected by $kH^2\\cdot\\mathrm{ch}_1$, for all $a>0$ and $b$, and the uncorrected inequality once $a>R_*$. The paper itself says it answers these corrected-inequality questions affirmatively. Note the correction class is a multiple of $H^2$. Not refereed. No Lean formalization in the release.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A specific question from two papers in the threefold stability literature, of interest to specialists as a weakening of the failed Bayer-Macri-Toda conjecture. Placed below Toda's quintic Gepner conjecture (38), level with the Inverse Generator Problem on Hilbert spaces (22), and above typical Erdos problems (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Prescribed large-volume charges on threefolds with trivial canonical bundle","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A Gepner stability condition on every smooth quintic threefold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf","kind":"paper"},{"label":"Bernardara-Macri-Schmidt-Zhao, Bridgeland stability conditions on Fano threefolds (Question 2.4)","url":"https://arxiv.org/abs/1607.08199v5","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"separable-banach-lipschitz-isomorphism-problem","name":"The Lipschitz isomorphism problem for separable Banach spaces","shortName":"Separable Lipschitz isomorphism problem","problemNumber":null,"field":"Nonlinear geometry of Banach spaces","fieldGroup":"Analysis","statement":"Banach spaces $X,Y$ are Lipschitz isomorphic if there is a bijection $\\Psi:X\\to Y$ with $c\\|s-t\\|\\le\\|\\Psi s-\\Psi t\\|\\le C\\|s-t\\|$ for constants $0 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. Single manuscript; no Lean formalization.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against Haslhofer's Conjecture 5.2: at every singular point of the FIRST singular time of a closed embedded surface flow in $\\mathbb R^3$, the full family of fixed-center rescalings converges locally smoothly with multiplicity one to a single shrinker, with fixed position and axes. The paper itself says it establishes the negative-time, fixed-center part of the conjecture at the first singular time, and that moving centers and continuation through singularities are different questions. It uses Bamler-Kleiner's multiplicity-one theory as input. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Uniqueness of blowups is a central problem in the singularity theory of mean curvature flow; earlier special cases (cylinders, conical ends) were major papers. The first-singular-time surface case is a partial answer to the general conjecture. Above the universal volume growth entry (38), below the codimension-one varifold singular set entry (50) and the Kobayashi conjecture entry (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf","sourceName":"OpenAI math release: Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Haslhofer, Mean curvature flow through singularities (2025)","url":"https://arxiv.org/abs/2510.01355v1","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"second-kahn-kalai-conjecture","name":"The second Kahn-Kalai conjecture","shortName":"Second Kahn-Kalai conjecture","problemNumber":null,"field":"Random graphs; thresholds and expectation thresholds","fieldGroup":"Combinatorics","statement":"For a graph $H$ with at least one edge and at most $n$ vertices, let $p_c(n,H)$ be the density at which $G(n,p)$ contains a copy of $H$ with probability $1/2$, and $p_E(n,H)$ the least density at which every subgraph $F\\subseteq H$ has expected copy count at least one (or one half). Always $p_E\\le p_c$, and perfect matchings and Hamilton cycles show a logarithmic gap can occur. Kahn and Kalai conjectured in 2007 that this elementary first-moment obstruction determines the threshold up to a logarithm, uniformly in $H$. The abstract (first) Kahn-Kalai conjecture was proved by Park and Pham; it does not give the graph statement, and the best prior general bound was $O(\\log^2)$ loss. Is $p_c(n,H)=O(p_E(n,H)\\log n)$ for every such $H$, with a universal constant?","posedBy":"Jeff Kahn and Gil Kalai (Thresholds and expectation thresholds, Combin. Probab. Comput. 16, 2007)","yearPosed":2007,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every $n\\ge2$ and every finite simple graph $H$ with $h\\ge1$ edges and at most $n$ vertices, $p_c(n,H)\\le\\min\\{1,2048e^{50}p_E(n,H)(1+\\log_2 h)\\}$, hence $p_c(n,H)\\le6144e^{50}p_E(n,H)\\log_2 n$. The logarithm depends only on the number of edges of $H$, which is stronger than the conjectured $\\log n$. The proof builds on Tran's spread-link cores and the Mossel-Niles-Weed-Sun-Zadik resampling, with a tree reduction replacing Tran's proposed coupling conjecture. Constants are explicit but huge; it says nothing about optimal constants or about hypergraph targets.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against the Kahn-Kalai formulation as cited; the paper notes that normalising expected counts at one half instead of one changes $p_E$ by at most a factor two, so the conjecture is unchanged. The proof was not refereed. Lean-checked: the release's formalization catalogue lists ComparatorChallenges/SecondKahnKalai.json with declaration OAI.LeanBlast.SecondKahnKalai.secondKahnKalaiBounds. The statement was read here: for n >= 2 and any finite simple graph H with at least one edge and at most n vertices, the actual containment threshold (infimum over p of probability >= 1/2) is at most min(1, 2048 e^50 p_E (1 + log2 e(H))) and at most 6144 e^50 p_E log2 n, with p_E defined over all subgraphs of H. That is the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"The graph-specific companion of the Kahn-Kalai expectation-threshold conjecture, open since 2007 and actively attacked in 2024-2026 (Dubroff-Kahn-Park, Tran). Less famous than the abstract conjecture settled by Park and Pham. Placed below Kalai's conjecture for tight trees (45) and Kahn's flow conjecture (40), above Feige's hypergraph Moore-bound conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The second Kahn-Kalai conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kahn and Kalai, Thresholds and expectation thresholds (2007)","url":"https://doi.org/10.1017/S0963548307008474","kind":"problem-record"},{"label":"Lean Comparator challenge SecondKahnKalai","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SecondKahnKalai.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Combinatorics/GraphThreshold/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/GraphThreshold/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"uniform-sparsest-cut-sdp-integrality-gap","name":"The Goemans-Linial integrality gap for uniform sparsest cut is nearly square-root logarithmic","shortName":"Uniform sparsest cut SDP integrality gap","problemNumber":null,"field":"Approximation algorithms; semidefinite relaxations and metric embeddings","fieldGroup":"Theoretical computer science","statement":"The Goemans-Linial relaxation of uniform sparsest cut minimizes $\\sum_{i0$ and uniform-demand instances on $n_j\\to\\infty$ vertices with Goemans-Linial integrality gap at least $c\\sqrt{\\log n_j}/(\\log\\log n_j)^3$. This improves the Kane-Meka bound $\\exp(\\Omega(\\sqrt{\\log\\log n}))$ to within $(\\log\\log n)^3$ of the ARV upper bound. It is an existential sequence of sizes, not a bound at every $n$; it does not determine whether the gap is exactly $\\Theta(\\sqrt{\\log n})$, and it concerns the basic SDP, not Sherali-Adams or Lasserre strengthenings.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 24, 2026: this integrality-gap construction and a separate NP-hardness reduction (its own entry). This manuscript is the formalised one.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question; it gives $\\mathrm{OPT}/\\mathrm{GL}\\ge c\\sqrt{\\log n}/(\\log\\log n)^3$ along a sequence $n\\to\\infty$, which matches ARV up to a power of $\\log\\log n$ but does not settle the exact order. lean/formalization.yaml lists a main result for this manuscript (comparator UniformSparsestCut, declaration OAI.UniformSparsestCut.mainGap). The comparator statement was read here: it asserts an absolute $c>0$ and instances on $n_j\\to\\infty$ vertices with symmetric nonnegative capacities, positive Goemans-Linial value (negative-type distances realised in $\\mathbb R^n$ with all triangle inequalities, normalised to total 1) and $c\\sqrt{\\log n_j}/(\\log\\log n_j)^3\\le\\mathrm{OPT}/\\mathrm{GL}$ eventually, where OPT uses the denominator $|B|(n-|B|)$. That is the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"The order of the uniform-demand SDP gap is a well-known question within metric embeddings and approximation algorithms, and this is a near-tight bound rather than an exact answer. Placed level with the Radchenko-Viazovska question on Fourier interpolation (28), above Sharp Hardness for MAX-3-CUT (25), and below the Matrix Spencer Conjecture for C*-algebra contractions (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026.pdf","sourceName":"OpenAI math release: Near-square-root logarithmic integrality gaps for uniform sparsest cut","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.UniformSparsestCut.mainGap)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/SparsestCut/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: UniformSparsestCut.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/UniformSparsestCut.lean","kind":"lean-proof"},{"label":"Companion: Constant-factor hardness of uniform sparsest cut","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"local-smoothing-conjecture-r3","name":"Sogge's local smoothing conjecture for the wave equation in three spatial dimensions","shortName":"Local smoothing in three dimensions","problemNumber":null,"field":"Harmonic analysis; dispersive and wave equations","fieldGroup":"Analysis","statement":"For the half-wave propagator $e^{it\\sqrt{-\\Delta}}$ on $\\mathbb R^n$, the sharp fixed-time estimate on $L^p$ loses $(n-1)|1/2-1/p|$ derivatives. Sogge (1991) conjectured that averaging over $t\\in[1,2]$ gains almost $1/p$ of a derivative for $p\\ge 2n/(n-1)$: $\\|e^{it\\sqrt{-\\Delta}}f\\|_{L^p(\\mathbb R^n\\times[1,2])}\\le C\\|f\\|_{W^{\\sigma,p}}$ for $\\sigma>(n-1)(1/2-1/p)-1/p$, and $\\sigma>0$ for $21/12$ was known. Does the estimate hold in $\\mathbb R^3$ for every $2\\max\\{0,1-3/p\\}$, in particular at $p=3$ with every positive loss?","posedBy":"Christopher D. Sogge","yearPosed":1991,"ageNote":"Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), Section 4, cited by the manuscript as the source of the higher-dimensional formulation.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $\\varepsilon>0$, $\\|e^{it\\sqrt{-\\Delta}}f\\|_{L^3(\\mathbb R^3\\times[1,2])}\\le C_\\varepsilon\\|J^\\varepsilon f\\|_{L^3}$. With the $L^2$ and $L^\\infty$-kernel endpoints, Corollary 10.6 gives the full strict range $2\\max\\{0,1-3/p\\}$. Section 11, through Beltran-Hickman-Sogge, Tao and Wolff, derives the strict three-dimensional Bochner-Riesz range, maximal Bochner-Riesz bounds for $p\\ge3$, the diagonal sphere restriction estimate $L^r(S^2)\\to L^r(\\mathbb R^3)$ for $r>3$, and the three-dimensional Kakeya maximal estimate. Not shown: the endpoint $\\sigma=0$ at $p=3$, dimensions $n\\ge4$, or the conjecture's general-$n$ form.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript uses two external published inputs, the planar Furstenberg theorem of Ren-Wang and the cone wave-envelope theorem of Guth-Wang-Zhang.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 10.6 were read against Sogge's conjecture in dimension three; the stated range matches the conjectured strict range. The 160-page proof was not refereed. No Lean formalization exists for this family (no lean/docs/079.md at the pinned commit, nothing in lean/formalization.yaml). The proof relies on Ren-Wang's Furstenberg theorem and Guth-Wang-Zhang's wave-envelope estimate as published inputs. The paper states the lossless endpoint $\\sigma=0$ at $p=3$ is not addressed, and the result is Euclidean: manifold and variable-coefficient versions, which are false or different in this range, are outside it.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":63,"significanceNote":"Sogge's local smoothing conjecture sits at the top of the Euclidean harmonic analysis hierarchy, above Bochner-Riesz, restriction and Kakeya, and the planar case was an Annals breakthrough. The three-dimensional case is a partial of the all-dimensions conjecture. Placed just above the Fourier restriction conjecture in three dimensions for positively curved surfaces (62), which this result partly implies, above the Kakeya set conjecture in four dimensions (55) and the three-dimensional Kakeya maximal conjecture (48), below the Jacobian Conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Critical local smoothing for the three-dimensional wave equation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Related (family 078): Bochner-Riesz multipliers in three dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf","kind":"paper"},{"label":"Bochner-Riesz in three dimensions, catalogued from the same release","url":"https://vibemathed.com/problem/bochner-riesz-conjecture-r3","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"irrationality-exponent-of-pi","name":"The irrationality exponent of pi is 2","shortName":"Irrationality exponent of pi","problemNumber":null,"field":"Diophantine approximation","fieldGroup":"Number theory","statement":"The irrationality exponent of an irrational $x$ is $\\mu(x)=\\sup\\{\\nu>0: 0<|x-p/q|0$, $|\\pi-p/q|\\ge q^{-2-\\varepsilon}$ for all large $q$. Is $\\mu(\\pi)=2$?","posedBy":"Folklore conjecture; the manuscript cites its statement in Michel Waldschmidt, Open Diophantine problems, Moscow Math. J. 4 (2004), p. 265","yearPosed":null,"ageNote":"The manuscript names no originator or year; it cites Waldschmidt's 2004 survey as the place the conjecture is recorded. The question of a finite exponent goes back at least to Mahler's 1953 bound, the earliest work the manuscript cites.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: $\\mu(\\pi)=2$; for every $\\nu>2$ there is $Q(\\nu)$ with $|\\pi-p/q|\\ge q^{-\\nu}$ for all $p\\in\\mathbb Z$ and $q\\ge Q(\\nu)$. Corollary: the Flint Hills series converges, and $\\sum n^{-a}|\\sin n|^{-b}$ converges iff $a>\\max\\{1,b\\}$. The bound is NOT effective (no explicit $Q(\\nu)$), and it is NOT a bound of the form $c/q^2$: bounded partial quotients for $\\pi$ are not claimed. The method is an interpolation-determinant argument, not an improvement of the integral constructions behind earlier numerical bounds.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 24, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues. OpenAI also released an abridged reasoning summary for this family (reasoning_traces/irrationality-exponent-of-pi.pdf).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against the conjecture as posed; it gives, for every real $\\nu>2$, a threshold $Q(\\nu)$ with $|\\pi-p/q|\\ge q^{-\\nu}$ for all integers $p$ and $q\\ge Q(\\nu)$, which is $\\mu(\\pi)=2$. The threshold is ineffective. The proof was not refereed. Lean: the release's Comparator challenge PiExponent (OAI.PiExponent.main) with solution module OAI.NumberTheory.PiExponent.Main, both present at the pinned commit; the challenge is not in the formalization catalogue. Its statement was read here and covers the headline: the eventual lower bound for every $\\nu>2$ and the supremum characterization over rationals equal to 2. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The manuscript notes Carella (2022) also claimed exponent two and points out a sign error in that paper's proof of a stronger claim.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"A classical and widely known open problem in Diophantine approximation; decades of work only pushed the bound from 42 to about 7.1. Placed level with Kothe's conjecture (60), just above Kadison's similarity problem (58), and below Artin's primitive root conjecture result (63) and the Jacobian conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The irrationality exponent of pi is 2","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator statement PiExponent (not in formalization.yaml main results)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PiExponent.lean","kind":"lean-proof"},{"label":"Lean scope note for family 017","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/017.md","kind":"other"},{"label":"Abridged reasoning summary released by OpenAI for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/reasoning_traces/irrationality-exponent-of-pi.pdf","kind":"other"},{"label":"Waldschmidt, Open Diophantine problems (2004)","url":"https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/odp.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"grothendieck-homotopy-hypothesis","name":"Grothendieck's homotopy hypothesis for weak globular infinity-groupoids (Ara-Henry coherators)","shortName":"Grothendieck homotopy hypothesis","problemNumber":null,"field":"Higher category theory; homotopy theory","fieldGroup":"Geometry & topology","statement":"In Pursuing Stacks (1983) Grothendieck defined weak globular $\\infty$-groupoids as models of a coherator, a globular theory obtained by freely adjoining coherence operations, and conjectured that they model homotopy types: their homotopy theory, with weak equivalences detected by components and homotopy groups, is equivalent to that of spaces. Maltsiniotis made the formulation precise; strict $\\infty$-groupoids are known to fail. Henry (2016) reduced the hypothesis to a pushout conjecture: attaching an $(n+1)$-cell along the source face of an $n$-cell of a cellular model is a weak equivalence. The case of 3-groupoids was settled by Henry and Lanari. Do Grothendieck $\\infty$-groupoids, for every coherator, recover the homotopy theory of spaces?","posedBy":"Alexander Grothendieck (Pursuing Stacks); precise formulation by Georges Maltsiniotis","yearPosed":1983,"ageNote":"Grothendieck, Pursuing Stacks (manuscript of 1983), with Maltsiniotis's formulation (2007 onward) and Henry's pushout conjecture (arXiv:1609.04622, 2016), all as cited by the manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (Henry's pushout conjecture): for every Grothendieck coherator in the Ara-Henry convention, elementary expansions of cellular models are weak equivalences. Section 5 derives the canonical left semi-model structure and, through Henry's comparison, a Quillen equivalence with spaces, so the homotopy theory is independent of the coherator within that convention. It does not cover coherators outside the Ara-Henry convention, Grothendieck's weak $\\infty$-categories (the non-invertible case), or Taylor's generalized pushout conjecture for algebraic coherators.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 24, 2026) is the whole family.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the abstract were read against the homotopy hypothesis as Maltsiniotis and Henry formulate it; the claim covers every Grothendieck coherator in the Ara-Henry convention, not every coherator in Maltsiniotis's broader sense. The proof was not refereed. lean/formalization.yaml lists GrothendieckElementaryExpansion (OAI.Grothendieck.elementary_expansion). Its comparator statement, read here, is Henry's pushout conjecture: for a coherator, a cellular model $X$, an $n$-cell $a$ and the pushout of the source inclusion $D_n\\to D_{n+1}$ along $a$, the map $X\\to X^+$ is a weak equivalence (bijective on components and on all homotopy groups). The semi-model structure and the Quillen equivalence with spaces follow by Henry's published reduction and Section 5 and are not formalized. Not rebuilt here. Listed as Unreviewed rather than Lean-checked because its formal statement is Henry's pushout conjecture, and the step to the homotopy hypothesis is Henry's 2016 reduction.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The homotopy hypothesis in Grothendieck's algebraic form is a well-known foundational problem of higher category theory, famous within that community for four decades. Placed level with Wall's manifold-realization question (42), above Kahn's flow conjecture (40), and below the transitive algebra problem (46).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The Grothendieck homotopy hypothesis via elementary expansions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof of Henry's pushout conjecture (OAI.Grothendieck.elementary_expansion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/CategoryTheory/Globular/ElementaryExpansion.lean","kind":"lean-proof"},{"label":"Comparator statement: GrothendieckElementaryExpansion.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/GrothendieckElementaryExpansion.lean","kind":"lean-proof"},{"label":"Henry, Algebraic models of homotopy types and the homotopy hypothesis (arXiv 1609.04622)","url":"https://arxiv.org/abs/1609.04622","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"laughlin-spectral-gap-conjecture","name":"The Laughlin spectral gap conjecture for the V1 pseudopotential on the sphere","shortName":"Laughlin spectral gap conjecture","problemNumber":null,"field":"Fractional quantum Hall effect; spectral gaps","fieldGroup":"Mathematical physics","statement":"On the sphere with $Q$ flux quanta, $N$ fermions in the lowest Landau level interact through Haldane's $V_1$ pseudopotential $H_{N,Q}=\\sum_{i0$ with $H_{N,3(N-1)}\\ge\\gamma(1-P_{\\mathrm L})$ for all sufficiently large $N$?","posedBy":"Folklore since Haldane (1983) and Haldane-Rezayi (1985); recorded as Conjecture A.1 in N. Rougerie's survey, as cited by the manuscript","yearPosed":1985,"ageNote":"The manuscript dates the gap question to Haldane and Rezayi's 1985 finite-size study, which distinguished the expected gap from a proof; the written conjecture it cites is in Rougerie's survey, whose year the manuscript's text does not give.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $0<\\gamma<\\gamma_*=4616733319001/10^{14}>1/25$ there is $Q_\\gamma$ with $H_Q^2\\ge\\gamma H_Q$ on the full lowest-Landau-level fermionic Fock space for $Q\\ge Q_\\gamma$, independently of particle number. Corollary 1.2: $H_{N,3(N-1)}\\ge\\frac1{25}(1-P_{\\mathrm L,N})$ for $N\\ge N_0$. Section 7 proves the planar analogue at $\\gamma_*$, the fermionic cubic case of Rougerie's Conjecture A.1. The companion adds stability under bounded projected scalar potentials. Thresholds are existential; bosonic and other fillings are not treated.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family's later manuscript, 'Uniform Stability of the Spherical Laughlin Gap' (October 5, 2026), extends the gap to weak bounded scalar disorder and is a link here. Both manuscripts ship verification directories.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the Fock-space manuscript were read against the conjecture: $H_Q^2\\ge\\gamma H_Q$ on the whole Fock space for every $\\gamma<\\gamma_*\\approx0.0462$ and large $Q$, hence a gap of $1/25$ above the Laughlin state at $Q=3(N-1)$ for large $N$. lean/formalization.yaml lists Laughlin.json (declaration OAI.Laughlin.mainTarget_proved, file OAI/Analysis/Laughlin/Main.lean). The statement Laughlin.lean was read here; not rebuilt here. It encodes antisymmetric states in a number basis, the $V_1$ energy through explicit pair coefficients and the Laughlin vector through its polynomial, and states energy at least $1/100$ times squared distance to the Laughlin line for all $N\\ge N_0$ at flux $3(N-1)$: a uniform positive gap, the headline (with a weaker constant than the paper's 1/25, which LaughlinGap.lean states). Fidelity of the pair-coefficient encoding to $P^{(1)}$ was not audited. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"The spectral gap of the Laughlin parent Hamiltonian is the standard open problem in the mathematics of the fractional quantum Hall effect, known across mathematical physics but with less reach than the Haldane gap conjecture entry at 62. Level with the Erdos Problem 74 entry at 38, above the Solovej ionization entry at 32, below the 2D area law entry in this batch at 48.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf","sourceName":"OpenAI math release: A Fock-space inequality and the Laughlin spectral gap","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Uniform Stability of the Spherical Laughlin Gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf","kind":"paper"},{"label":"Lean proof (OAI.Laughlin.mainTarget_proved)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Laughlin/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kraetzer-integral-means-conjecture","name":"Kraetzer's conjecture on the universal integral means spectrum","shortName":"Kraetzer's conjecture","problemNumber":null,"field":"Geometric function theory, integral means spectrum","fieldGroup":"Analysis","statement":"For a bounded univalent map $f$ of the unit disk, $\\beta_f(p)$ is the growth exponent of $\\int|f'(re^{i\\theta})|^p d\\theta$ as $r\\to1$, and the universal bounded spectrum is $B_b(p)=\\sup_f\\beta_f(p)$. From numerical experiments Kraetzer (1996) conjectured $B_b(p)=p^2/4$ for $|p|\\le2$ and $B_b(p)=|p|-1$ for $|p|\\ge2$. The conjecture contains Brennan's conjecture ($B(-2)=1$) and the value $B_b(1)=1/4$ related to the Carleson-Jones coefficient problem. Hedenmalm-Shimorin and Sola gave upper bounds above $1/4$ at $p=-1$. Is Kraetzer's formula for $B_b$ correct?","posedBy":"Philipp Kraetzer, Experimental bounds for the universal integral means spectrum of conformal maps (Complex Variables, 1996)","yearPosed":1996,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are constants $0<\\varepsilon<1/4$ and $C$ with $M_{-1}[f'](r)\\le C(1-r)^{-1/4+\\varepsilon}$ for every $f\\in\\mathcal S$ and $1/2\\le r<1$, so $B_b(-1)<1/4=p^2/4$ at $p=-1$ and Kraetzer's formula is false. The bound needs no bounded image or boundary regularity. Not shown: the true value of $B_b(-1)$ or any explicit gap, and nothing about $p=1$ or the Carleson-Jones question; Brennan's value $B(-2)=1$, also predicted by Kraetzer, is proved in the companion (separate entry).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. This is the companion of the Brennan manuscript in the same family; the two share the affine-law and critical-point method but the paper says neither is an input to the other. It has a Lean formalization in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract and Theorem 1.1 of 'A strict inverse-first-power bound for univalent functions', read against Kraetzer's formula as the paper quotes it (1996 paper, 2000 thesis, Beliaev's restatement). The proof was not refereed. Lean: formalization.yaml lists OAI.StrictInverseFirstPower.main (lean/OAI/Analysis/StrictMeans/Main.lean, comparator ComparatorChallenges/StrictMeans.lean). Its statement was read: there are 0 < eps < 1/4 and C with M_{-1}[f'](r) <= C (1-r)^(-1/4+eps) for every normalized univalent f, the bounded spectrum at -1 is below 1/4, and it is not the case that boundedSpectrum p equals the piecewise Kraetzer prediction for all p. That is the headline. Not rebuilt here. The gap eps is existential; no numerical value is given.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A numerically motivated conjecture well known among specialists in conformal mapping, and widely believed. Placed below the Brennan entry (45) and Crouzeix's conjecture (35), level with the smooth Trautman counterexample (25), and just under the Radchenko-Viazovska Fourier interpolation question (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A strict inverse-first-power bound for univalent functions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Brennan's conjecture and sharp inverse-square integral means","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean: strict inverse-first-power bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/StrictMeans/Main.lean","kind":"lean-proof"},{"label":"Kraetzer 1996, Experimental bounds for the universal integral means spectrum","url":"https://doi.org/10.1080/17476939608814969","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"aec-categoricity-hanf-threshold-ch-failure","name":"The prescribed Hanf threshold in Shelah's categoricity conjecture is not provable in ZFC","shortName":"Prescribed categoricity threshold fails under CH","problemNumber":null,"field":"Model theory; abstract elementary classes","fieldGroup":"Logic & foundations","statement":"For an abstract elementary class $K$ put $H(K)=\\beth_{(2^{LS(K)})^+}$, the Hanf number for existence of arbitrarily large models. Beside the qualitative eventual categoricity conjecture, Shelah's program states a prescribed-threshold form: if $K$ is categorical in some $\\lambda\\ge H(K)$, then $K$ is categorical in every $\\mu\\ge H(K)$. Under amalgamation and tameness this transfer is known (Vasey), and a transfer at this bound for arbitrary AECs has been claimed (Espindola 2023). Grossberg distinguishes the prescribed and unspecified-threshold forms, and Saroch-Trlifaj (2024) give the exact formulation. Does categoricity above $H(K)$ always transfer to every cardinal at least $H(K)$, for arbitrary AECs?","posedBy":"Saharon Shelah (threshold form; formulated by Grossberg and by Saroch and Trlifaj)","yearPosed":2009,"ageNote":"The family cites Shelah 2009, Conjecture N.4.3(1)(b), for an explicit threshold, Grossberg's draft book (Conjectures 4.4-4.6) and Saroch-Trlifaj 2024 (Section 2.3) for the exact form refuted here.","solveType":"disproved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (assuming CH): there is an AEC $K$ in a countable finitary relational language with $LS(K)=\\aleph_0$, at least two nonisomorphic models of cardinality $\\beth_{\\omega_2}=H(K)$, and categorical in every cardinal $\\ge\\Lambda=\\beth_{(2^{\\aleph_1})^+}$. So categoricity at $\\Lambda^+$ does not transfer down to $H(K)$, and the prescribed-threshold form is not a theorem of ZFC (if ZFC is consistent). The same class is eventually categorical, so this is compatible with the qualitative conjecture. It does not decide whether the prescribed form can hold under other hypotheses (for example failure of CH), and it does not concern the $\\beth_{\\omega_1}$ threshold for $L_{\\omega_1,\\omega}$.","claimIssueNote":"This result conflicts with a published theorem. The release's manuscript states that its construction contradicts Espindola's published Theorem 4.1, which claims categoricity transfer at this bound. Until the conflict is settled in public, the entry is Contested.","solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 24, 2026: this CH construction and the ZFC eventual categoricity theorem (its own entry). This manuscript is the formalised one.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the threshold form. Under CH it builds an AEC in a countable relational language with $LS(K)=\\aleph_0$, two nonisomorphic models of size $\\beth_{\\omega_2}=H(K)$, and categoricity in every $\\mu\\ge\\beth_{(2^{\\aleph_1})^+}$; with Godel's consistency of CH the prescribed form is unprovable in ZFC if ZFC is consistent. It does not show the prescribed form false in ZFC. lean/formalization.yaml lists comparator CHObstruction, declaration OAI.CHObstruction.main; the statement was read here and asserts exactly Theorem 1.1, with the AEC axioms (coherence, chain unions over ordinals, LS bound) spelled out over ZFSet carriers. The nonprovability corollary is not formalised. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript says its construction contradicts Espindola's published claim (Theorem 4.1) of transfer at this bound. Listed as Contested because of the conflicting claim described in the claim issue.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"The exact Hanf threshold is a recognised secondary question within AEC theory, and the answer here is one-directional (unprovability via CH). Placed level with the Stable Coordinate Conjecture in ambient dimension five (22), above Han and Jiang's Question on klt Type in Flat Families (20), and below Kalton's Problem 2 on bi-Lipschitz copies of c0 (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A CH obstruction to a prescribed categoricity threshold","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.CHObstruction.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/ModelTheory/Categoricity/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: CHObstruction.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CHObstruction.lean","kind":"lean-proof"},{"label":"Companion: Eventual categoricity for abstract elementary classes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bounded-degree-coboundary-expanders","name":"Existence of bounded-degree F2 coboundary expanders in every dimension","shortName":"Bounded-degree coboundary expanders","problemNumber":null,"field":"High-dimensional expanders: coboundary expansion","fieldGroup":"Combinatorics","statement":"A finite pure $d$-dimensional simplicial complex $X$ is an $\\varepsilon$-coboundary expander over $\\mathbb{F}_2$ if, with the standard weighted norms, $\\|\\delta f\\|\\ge\\varepsilon\\,\\mathrm{dist}(f,B^i(X))$ for every $i$-cochain $f$ and all $i0$ and finite connected pure $d$-dimensional complexes $X_m$ with $|X_m(0)|\\to\\infty$, every vertex in at most $D$ top faces, and $\\|\\delta_if\\|\\ge\\varepsilon\\,\\mathrm{dist}(f,B^i)$ for all $i 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the problem as cited from Gromov (2010) and Chapman-Lubotzky (2025); the proof was not refereed. Lean: the Comparator challenge CoboundaryExpanders (OAI.CoboundaryExpanders.main, solution module OAI/Combinatorics/Coboundary/Main.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Its statement was read here: for every d >= 3 there are D, eps > 0 and a sequence of finite pure connected d-dimensional complexes with vertex counts tending to infinity, at most D top faces at each vertex, and eps times the weighted distance to coboundaries at most the weighted norm of the coboundary in every degree i < d. This is the headline claim; the graph and two-dimensional cases are prior work. Not rebuilt here. The construction uses the cosystolic expansion theorem of Oppenheim and Valentiner-Branth as an input.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":33,"significanceNote":"A recognized open problem of the high-dimensional expander program, motivated by Gromov's topological overlap and by property testing and quantum codes, with the two-dimensional case solved only in 2025. Specialist fame. Scored below explicit nonbipartite Ramanujan graphs of every degree (36) and above the generalized star-height bound (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Bounded-degree coboundary expanders in every dimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/Combinatorics/Coboundary/Main.lean (CoboundaryExpanders.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Coboundary/Main.lean","kind":"lean-proof"},{"label":"Chapman, Lubotzky, Stability of homomorphisms, coverings and cocycles II (Adv. Math. 2025)","url":"https://doi.org/10.1016/j.aim.2025.110117","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generalized-mukai-conjecture","name":"The generalized Mukai conjecture on Picard number and pseudoindex of Fano manifolds","shortName":"Generalized Mukai conjecture","problemNumber":null,"field":"Algebraic geometry; Fano manifolds and rational curves","fieldGroup":"Geometry & topology","statement":"For a Fano manifold $X$ of dimension $n$, Picard number $\\rho_X$ and pseudoindex $\\iota_X=\\min\\{-K_X\\cdot C\\}$ over rational curves $C$, Mukai (1988) conjectured an index version of a bound on $\\rho_X$, and Bonavero, Casagrande, Debarre and Druel formulated the pseudoindex strengthening, proving it for $n\\le4$ and for toric cases. Later work covered $n=5$ (Andreatta-Chierici-Occhetta), $\\iota_X>n/3$ (Novelli-Occhetta, Novelli) and toric, horospherical and spherical varieties. Does every smooth complex Fano manifold of positive dimension satisfy $\\rho_X(\\iota_X-1)\\le n$, with equality only for $(\\mathbb P^{\\iota_X-1})^{\\rho_X}$?","posedBy":"Bonavero, Casagrande, Debarre and Druel, 'Sur une conjecture de Mukai' (Comment. Math. Helv. 2003), strengthening Mukai (1988)","yearPosed":2003,"ageNote":"Mukai's index version dates from 1988; the pseudoindex form proved here is the 2003 BCDD formulation.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every smooth connected complex projective Fano manifold of positive dimension $n$ has $\\rho_X(\\iota_X-1)\\le n$, with equality if and only if $X\\cong(\\mathbb P^{\\iota_X-1})^{\\rho_X}$. Consequently Mukai's original index version also holds. The paper treats smooth complex Fano manifolds only: singular ($\\mathbb Q$-factorial Gorenstein) Fano varieties beyond the known toric, horospherical and spherical cases, and positive characteristic, are not covered.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as formulated by BCDD; it is the inequality and the equality classification in every dimension with no restriction on Picard number or geometry. The proof goes through small quantum cohomology, descendant lower bounds on moduli of stable maps and a recurrence argument, then the BCDD ordered-chain method at equality; it was not refereed here and leans on virtual-class technology. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A well-known conjecture in Fano geometry with two decades of partial results by dimension, pseudoindex range and symmetry; famous within that community. Placed above The Campana-Peternell conjecture on Fano manifolds with nef tangent bundle in dimension six (28), below Kuznetsov's rationality conjecture for cubic fourfolds (45) and The LeBrun-Salamon conjecture (56).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Generalized-Mukai-Conjecture-September-24-2026/article.pdf","sourceName":"OpenAI math release: The generalized Mukai conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Bonavero-Casagrande-Debarre-Druel, Sur une conjecture de Mukai (2003)","url":"https://doi.org/10.1007/s00014-003-0765-x","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"permanent-border-determinantal-cubic-bound","name":"A cubic lower bound for the border determinantal complexity of the permanent (beyond the quadratic Mignon-Ressayre barrier)","shortName":"Cubic border determinantal lower bound for the permanent","problemNumber":null,"field":"Algebraic complexity; permanent versus determinant, geometric complexity theory","fieldGroup":"Theoretical computer science","statement":"The determinantal complexity $\\mathrm{dc}(f)$ of a polynomial $f$ is the least $n$ with $f=\\det(A_0+\\sum_j x_jA_j)$ for $n\\times n$ complex matrices, and the border determinantal complexity $\\mathrm{bdc}(f)$ is the least $n$ such that $f$ is a coefficientwise limit of such determinants. Valiant's algebraic completeness theory (1979) makes the growth of $\\mathrm{dc}(\\mathrm{per}_m)$ the algebraic analogue of P versus NP: it is conjectured to be superpolynomial, and the Mulmuley-Sohoni geometric complexity program targets $\\mathrm{bdc}$. The best lower bounds were quadratic: $\\mathrm{dc}(\\mathrm{per}_m)\\ge m^2/2$ (Mignon-Ressayre 2004) and $\\mathrm{bdc}(\\mathrm{per}_m)\\ge m^2/2$ (Landsberg-Manivel-Ressayre 2010). How fast must $\\mathrm{bdc}(\\mathrm{per}_m)$ grow; in particular, can the quadratic lower bound be improved?","posedBy":"Leslie Valiant (permanent versus determinant, algebraic completeness); border version from the Mulmuley-Sohoni geometric complexity program","yearPosed":1979,"ageNote":"The manuscript cites Valiant (1979) for the completeness framework, Mulmuley-Sohoni (2001, 2008) for the orbit-closure formulation, and Landsberg-Manivel-Ressayre (2010) for the previous border bound $m^2/2$. Superpolynomial growth remains the open target; this is a new polynomial bound on a tracked quantity.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: $\\mathrm{bdc}(\\mathrm{per}_m)\\ge cm^3$ for $m\\ge m_0$, with $c=1/(5529600e)$ and $m_0=1408$, over $\\mathbb C$ with arbitrary affine-linear entries and coefficientwise limits at fixed size; the same bound holds for $\\mathrm{dc}$, and affine-linear algebraic branching programs for $\\mathrm{per}_m$ need $\\Omega(m^3)$ vertices and edges (also in the border sense at fixed budget). The route is a smooth-hypersurface bound $(r-1)(d-1)/(4e)$ for forms of degree $r$ in $d$ variables, applied to a coefficient polynomial extracted from a permanent. It remains a fixed polynomial bound: superpolynomial $\\mathrm{dc}$ or $\\mathrm{bdc}$ for the permanent, and VP versus VNP, stay open.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 24, 2026), with a Lean formalization of the two headline bounds.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 ($\\mathrm{bdc}(\\mathrm{per}_m)\\ge m^3/(5529600e)$ for $m\\ge1408$) was read against the permanent-determinant question. Lean: the comparator challenge lean/ComparatorChallenges/PermanentCubic.json exists with solution module OAI.Algebra.Determinantal.Main present at the pinned commit; this challenge is not listed in lean/formalization.yaml. Its statement OAI.PermanentBorder.permanent_cubic_lower_bounds was read here: for $m\\ge1408$, any $n$ with a sequence of affine $n\\times n$ pencils whose determinant coefficients converge to those of $\\mathrm{per}_m$ (and any exact representation) satisfies $m^3/(5529600e)\\le n$. This is the headline claim. Not rebuilt here. The paper's branching-program consequences are outside the formal statement.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Determinantal complexity bounds for the permanent had been stuck at quadratic since Mignon and Ressayre (2004), and the border version is the target of geometric complexity theory; a cubic bound is the first break of that barrier but still polynomial. Placed above the formula lower bounds for the permanent (38), below the n log n lower bound for the DFT (45) and matrix multiplication with exponent 9/4 (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf","sourceName":"OpenAI math release: A cubic lower bound for border determinantal complexity of the permanent","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean statement: PermanentCubic.lean (comparator challenge)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PermanentCubic.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Algebra/Determinantal/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/Determinantal/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bin-packing-additive-constant-approximation","name":"Bin packing within an additive constant: is there a polynomial-time algorithm using at most OPT + C bins?","shortName":"Bin packing within OPT + constant","problemNumber":null,"field":"Approximation algorithms; hardness of bin packing","fieldGroup":"Theoretical computer science","statement":"Bin packing is NP-hard and admits no multiplicative ratio below $3/2$ unless P = NP, but its asymptotic behaviour is much better: Fernandez de la Vega and Lueker gave an asymptotic scheme, Karmarkar and Karp (1982) an algorithm using $OPT+O(\\log^2OPT)$ bins, Rothvoss $OPT+O(\\log OPT\\log\\log OPT)$ and Hoberg and Rothvoss $OPT+O(\\log OPT)$. No hardness of any additive form was known, so even an $OPT+1$ algorithm was not excluded. Williamson and Shmoys list the question among the open problems of their 2011 book (Chapter 17, Problem 3). Is there a deterministic polynomial-time algorithm that, for some absolute constant $C$, packs every instance into at most $OPT(I)+C$ bins?","posedBy":"David P. Williamson and David B. Shmoys (listed open problem), in the line of Karmarkar and Karp's additive algorithm","yearPosed":2011,"ageNote":"The manuscript cites Williamson-Shmoys, The Design of Approximation Algorithms (2011), Chapter 17, Problem 3. The question is older, following Karmarkar-Karp (1982), but the manuscript gives no earlier posing reference.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.2: for every fixed integer $c\\ge0$ it is NP-hard to distinguish pairs $(I,B)$ where $I$ packs into $B$ bins from pairs where $I$ needs more than $B+c$ bins, with rational sizes all greater than $1/6$. Corollary 1.3: a deterministic polynomial-time algorithm using at most $OPT(I)+C$ bins for an absolute constant $C$ exists if and only if P = NP. The reduction goes from vertex cover with Håstad's gap through a competing two-tree interval construction encoded in five-item bins. Not shown: a lower bound of order $\\log OPT$ matching Hoberg-Rothvoss, or hardness for a fixed number of item sizes (polynomial by Goemans-Rothvoss).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'Additive hardness and unbounded configuration gaps in bin packing' (September 24, 2026); the same reduction also disproves the Modified Integer Round-Up Conjecture, entered separately.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.2 and Corollary 1.3 were read against the question as Williamson and Shmoys pose it. Theorem 1.2 makes distinguishing $B$ bins from more than $B+c$ bins NP-hard for each fixed $c$, with sizes above $1/6$; Corollary 1.3 says an $OPT+C$ algorithm exists if and only if P = NP. The answer is negative unless P = NP, the standard form of a complete answer to such a question; solveType disproved is recorded on that understanding. Lean: the same formalised declaration OAI.BinPackingGap.main_results (listed in formalization.yaml) was read here; its second and third conjuncts state PackingGapNPHard c for every c (polynomial-time reductions from every NP language with completeness and soundness) and that an absolute additive algorithm exists iff P = NP, with P, NP and polynomial time defined on Mathlib's TM2 machines. That states the headline. Not rebuilt here; the hand-built complexity definitions were read but not audited.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Whether bin packing can be solved within an additive constant is one of the best-known open problems in approximation algorithms, listed by Williamson and Shmoys and the target of the Karmarkar-Karp, Rothvoss and Hoberg-Rothvoss line; a positive answer was widely expected. Level with NP-hardness of Uniform Sparsest Cut within every constant (45) and three-machine precedence scheduling (45), below the APSP hypothesis (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026.pdf","sourceName":"OpenAI math release: Additive hardness and unbounded configuration gaps in bin packing","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI/Computability/BinPacking/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/BinPacking/Main.lean","kind":"lean-proof"},{"label":"Lean statement: ComparatorChallenges/BinPackingGap.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BinPackingGap.lean","kind":"lean-proof"},{"label":"Williamson and Shmoys, The Design of Approximation Algorithms (2011), Chapter 17","url":"https://designofapproxalgs.com/book.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"abhyankar-sathaye-conjecture","name":"The Abhyankar-Sathaye embedding conjecture","shortName":"Abhyankar-Sathaye conjecture","problemNumber":null,"field":"Affine algebraic geometry; polynomial automorphisms","fieldGroup":"Geometry & topology","statement":"A polynomial $f\\in\\mathbb C[z_1,\\dots,z_n]$ is a coordinate if it is the first component of a polynomial automorphism of $\\mathbb A^n$. The Abhyankar-Sathaye (epimorphism, or embedding) conjecture asserts that if $\\mathbb C[z_1,\\dots,z_n]/(f)\\cong\\mathbb C^{[n-1]}$ then $f$ is a coordinate; geometrically, every hypersurface embedding $\\mathbb A^{n-1}\\hookrightarrow\\mathbb A^n$ is rectifiable. For $n=2$ this is the Abhyankar-Moh-Suzuki theorem; Sathaye and Russell treated planes linear in one variable, and Kaliman-Venereau-Zaidenberg and Maubach handled four-variable equations $a(x)y+b(x,z,t)$. Is every polynomial whose zero locus is isomorphic to affine $(n-1)$-space a coordinate?","posedBy":"S. S. Abhyankar and A. Sathaye; the coordinate question is distinguished in A. Sathaye, On linear planes, Proc. Amer. Math. Soc. 56 (1976)","yearPosed":1976,"ageNote":"The manuscripts cite Sathaye (1976), Kraft's 1996 Bourbaki survey and Blanc-van Santen (2019) for the conjecture; Abhyankar's own formulation is not dated there.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: an explicit $F\\in\\mathbb C[h,u,v,w]$, built from a perturbed cusp parametrisation, has $\\mathbb C[h,u,v,w]/(F)\\cong\\mathbb C^{[3]}$ but a critical point on the fibre $F=-1$, so $F$ is not a coordinate; adjoining variables gives counterexamples for every $n\\ge4$. The 5 October companion gives a degree-five $f$ in four variables with every fibre isomorphic to $\\mathbb A^3$ and no fibre rectifiable, so the conjecture fails even for polynomials with all fibres affine. Not shown: the case $n=3$ (whether every embedded affine plane in $\\mathbb A^3$ is rectifiable), which stays open.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The main theorem has a Lean formalization in the release (Comparator challenge AbhyankarSathaye); a later family manuscript (5 October 2026) gives a stronger example with every fibre affine three-space.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as stated via Sathaye, Kraft and Blanc-van Santen. Lean-checked on the release's Comparator challenge AbhyankarSathaye (declaration OAI.AbhyankarSathaye.exists_noncoordinate_polynomial, listed in lean/formalization.yaml). Its statement was read here: for every $n\\ge4$ there is $F\\in\\mathbb C[x_1,\\dots,x_n]$ with a $\\mathbb C$-algebra isomorphism from the quotient by $(F)$ to a polynomial ring in $n-1$ variables, and no $\\mathbb C$-algebra automorphism sends a variable to $F$. That is the headline, for all $n\\ge4$. Not rebuilt here. The paper states the three-variable case is not addressed. The explicit degree-five polynomial of the 5 October companion (all fibres affine) was formalized in the 7 October 2026 update as Comparator challenge StableCoordinateFour; it belongs to a separate entry.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"One of the standard problems on affine space, listed in Kraft's Bourbaki survey beside the Jacobian and cancellation problems and open for fifty years beyond the plane; famous within affine algebraic geometry. Below the Zariski cancellation problem in dimension four (60), above the Dolgachev-Weisfeiler affine-fibration conjecture (35) and well above the stable coordinate conjecture (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026/paper.pdf","sourceName":"OpenAI math release: An explicit noncoordinate polynomial with affine three-space zero fibre","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"A stable coordinate that is not a coordinate in four variables (companion, all fibres affine)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf","kind":"paper"},{"label":"Lean Comparator challenge AbhyankarSathaye (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/AbhyankarSathaye.lean","kind":"lean-proof"},{"label":"Lean Comparator challenge CommutingDerivations (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CommutingDerivations.lean","kind":"lean-proof"},{"label":"Kraft, Challenging problems on affine n-space, Seminaire Bourbaki 802 (1996)","url":"https://numdam.org/item/SB_1994-1995__37__295_0/","kind":"problem-record"},{"label":"Sathaye 1976, On linear planes","url":"https://doi.org/10.1090/S0002-9939-1976-0409472-6","kind":"problem-record"},{"label":"Lean statement (5 October companion polynomial): StableCoordinateFour","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/StableCoordinateFour.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-371-largest-prime-factor-consecutive","name":"Erdos Problem #371: do the integers with P(n) < P(n+1) have density 1/2? (Erdos-Pomerance joint independence of largest prime factors)","shortName":"Erdos #371: P(n) < P(n+1) has density 1/2","problemNumber":371,"field":"Analytic number theory; smooth numbers and largest prime factors","fieldGroup":"Number theory","statement":"Let $P^+(n)$ be the largest prime factor of $n$ and $\\rho$ the Dickman function, so that $P^+(n)\\le n^a$ has density $\\rho(1/a)$. Erdos and Pomerance (1978) conjectured that the largest prime factors of $n$ and $n+1$ behave independently, so that $P^+(n)\\le n^a$ and $P^+(n+1)\\le n^b$ hold simultaneously with density $\\rho(1/a)\\rho(1/b)$, and in particular that the set of $n$ with $P^+(n) 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against Erdos Problem #371 and the Erdos-Pomerance independence conjecture. The Lean challenge lean/ComparatorChallenges/JointDickman.json (solution module OAI/NumberTheory/JointDickman/PaperMain.lean present at the pinned commit) is not in the formalization catalogue; its statement was read here. joint_law states that for $0 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript is short (five sections). Its external input is the Slaman-Woodin representation theorem from their 2005 unpublished manuscript; the proof adds a four-value recovery argument for Borel maps and a category cone-avoidance step.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the rigidity problem; it states that every order automorphism of the full Turing degrees is the identity, in ZFC, with no definability hypothesis on the automorphism. Lean: lean/formalization.yaml lists comparator DegreeRigidity with declaration OAI.TuringRigidity.ManuscriptMain.rigidity. The statement ComparatorChallenges/DegreeRigidity.lean was read here: it defines oracles as $\\mathbb N\\to$ Bool, reduction by Mathlib's TuringReducible on their characteristic functions, Degree as the antisymmetrization, and asserts that every order isomorphism Degree to Degree fixes every degree. This is the headline claim. The Lean development (OAI/Computability/DegreeRigidity, with forcing, constructibility and set-model subdirectories) derives the Borel representation inside Lean rather than assuming the Slaman-Woodin input. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The rigidity problem is the central open question about the global structure of the Turing degrees, worked on for decades by Nerode, Shore, Slaman and Woodin and equivalent to biinterpretability. Famous within logic rather than across mathematics. Placed level with the Kadison-Kaplansky conjecture (50), above the word problem for finite-rank Artin groups (45), well below Hilbert's tenth problem over the rationals (70).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Rigidity of the Turing degrees","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.TuringRigidity.ManuscriptMain.rigidity)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/DegreeRigidity/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: DegreeRigidity.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DegreeRigidity.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"reflexive-banach-fixed-point-property","name":"The fixed point property for nonexpansive maps in reflexive Banach spaces","shortName":"Nonexpansive fixed points in reflexive spaces","problemNumber":null,"field":"Functional analysis: metric fixed point theory","fieldGroup":"Analysis","statement":"A Banach space has the fixed point property for nonexpansive maps if every $T:C\\to C$ with $\\|Tx-Ty\\|\\le\\|x-y\\|$, on every nonempty closed bounded convex set $C$, has a fixed point, nonexpansiveness being measured in the given norm. Browder and Gohde (1965) proved it for uniformly convex spaces and Kirk (1965) for reflexive spaces when the domain has normal structure; Karlovitz showed normal structure is not necessary, Maurey proved it for reflexive subspaces of $L^1$, and Alspach (1981) gave a fixed-point-free nonexpansive map on a weakly compact convex subset of $L^1$, so weak compactness alone does not suffice. Dominguez Benavides showed that every reflexive space has an equivalent norm with the property. Does every real reflexive Banach space have the fixed point property for nonexpansive maps in its original norm?","posedBy":"Classical question of metric fixed point theory arising from Kirk's 1965 theorem; the manuscript calls it the reflexive-space fixed point problem and names no poser","yearPosed":null,"ageNote":"The manuscript does not say who first asked the question or when. It frames it through Kirk's 1965 theorem and cites Nourouzi (2026) as recording it as unresolved. It also notes earlier claimed proofs of the same statement by Hanebaly (a 2019 journal article and a 2021 preprint).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $X$ is a real reflexive Banach space and $C\\subseteq X$ is nonempty, closed, bounded and convex, every nonexpansive $F:C\\to C$ has a fixed point, with no uniform convexity, normal structure or separability assumed and the norm unchanged. Corollary: for any commuting family of nonexpansive self-maps of such $C$, the common fixed set is nonempty and a nonexpansive retract (via Bruck 1974). The proof argues by contradiction from a minimal invariant set, using the Goebel-Karlovitz lemma, an adaptive anchor, a tree of approximate fixed point images and a weighted switching estimate. It says nothing about nonreflexive spaces with the weak fixed point property, and gives no constructive iteration.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source; the proof was not refereed. Lean-checked on the Comparator challenge ReflexiveFixedPoints, listed in the release's formalization catalogue (OAI.ReflexiveFixedPoints.exists_fixedPoint_of_canonicallyReflexive, OAI/Analysis/Nonexpansive/Main.lean). Its statement: in a complete real normed space whose canonical map into the bidual is surjective, every map of a nonempty closed bounded convex set into itself that is nonexpansive in the given norm has a fixed point. That is the headline claim, with no separability or other geometric hypothesis. Permitted axioms: propext, Quot.sound, Classical.choice. The statement was read here; the development was not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A central open question of metric fixed point theory since Kirk's 1965 theorem, known throughout Banach space theory, with earlier claimed proofs that did not settle it in the literature. Scored with Brennan's conjecture (45) and the transitive algebra problem (46), below Kadison's similarity problem (58), whose reach across operator algebras is wider.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/Analysis/Nonexpansive/Main.lean (exists_fixedPoint_of_canonicallyReflexive)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Nonexpansive/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hilberts-tenth-problem-over-q","name":"Hilbert's tenth problem over the rational numbers","shortName":"Hilbert's tenth problem over Q","problemNumber":null,"field":"Diophantine decidability and definability, arithmetic of elliptic curves","fieldGroup":"Number theory","statement":"Hilbert's tenth problem (1900) asked for a procedure deciding whether a polynomial equation with integer coefficients has an integer solution; Davis, Putnam, Robinson and Matiyasevich showed by 1970 that none exists. Over $\\mathbb Q$ the question remained open: enumeration finds any rational zero, but no method certifies that none exists, and the natural transfer route, an existential definition of $\\mathbb Z$ in $\\mathbb Q$, would contradict Mazur's conjecture on the topology of rational points. Is there an algorithm which, given $f\\in\\mathbb Z[X_1,\\ldots,X_n]$ with $n$ part of the input, decides whether $f$ has a zero in $\\mathbb Q^n$?","posedBy":"David Hilbert (1900) for the integer problem; the rational version has been the central open case since the Davis-Putnam-Robinson-Matiyasevich theorem (1970)","yearPosed":null,"ageNote":"The manuscript dates Hilbert's integer problem to 1900 and the integer answer to 1970, but cites no single paper posing the rational version. It has been open at least since 1970 and was discussed by Robinson (1949), Mazur (1992, 1995), Poonen and Koenigsmann among those cited.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims a negative answer: rational solvability of integer polynomial equations (variables unbounded) is undecidable, of Turing degree $0'$, and stays so for total degree at most four. The method reduces integer solvability to finitely many rational-solvability tests per input, using rank-one elliptic curves, a five-point height estimate and a parity condition from 2-descent. It does not define $\\mathbb Z$ existentially in $\\mathbb Q$ (consistent with Mazur's conjecture), does not bound the number of variables, and does not address other number fields beyond what is known.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The proof relies on two other manuscripts from the same release as theorem inputs: a pointwise 2-converse (the family companion) and Fontaine-Mazur modularity at the prime 2 (a separate family).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against the posed problem; it states there is no algorithm deciding rational solvability of integer polynomials with the number of variables part of the input, which is the problem as posed. A corollary gives Turing degree $0'$, also for quartics and sums of squares of quadrics. The paper does not give an existential definition of $\\mathbb Z$ in $\\mathbb Q$; it uses a finite-test reduction with compactness. Its inputs include two unrefereed manuscripts of the same release (pointwise 2-converse; Fontaine-Mazur modularity at 2), so the result is only as solid as they are. No Lean formalization is listed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":70,"significanceNote":"The rational case has been the most famous open question in Diophantine decidability since 1970, attacked by Robinson, Mazur, Poonen and Koenigsmann. Scored above the Jacobian conjecture at 65 and the Kothe conjecture at 60, tied with the forced Euler blowup at 70, and below the Navier-Stokes candidate at 87 since it lacks a Millennium-level reach.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf","sourceName":"OpenAI math release: Hilbert's tenth problem over the rational numbers","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"A pointwise 2-converse for elliptic curves with rational two-torsion (companion, OpenAI, 2026-09-24)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf","kind":"paper"},{"label":"Fontaine-Mazur modularity at the prime 2 (proof input, separate family, OpenAI, 2026-09-23)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf","kind":"paper"},{"label":"Hilbert, Mathematische Probleme (1900), Problem 10","url":"https://gdz.sub.uni-goettingen.de/id/PPN252457811_1900","kind":"problem-record"},{"label":"Fontaine-Mazur at p = 2, an input to this proof","url":"https://vibemathed.com/problem/fontaine-mazur-two-dimensional-p-2","kind":"other"},{"label":"Revised edition (October 7, 2026): A pointwise 2-converse for elliptic curves with rational two-torsion","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Fontaine-Mazur modularity at the prime 2","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf","kind":"paper"},{"label":"Earlier edition (September 24, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kn-local-sphere-finite-generation","name":"Hovey-Strickland finite-generation question: are the homotopy groups of the $K(n)$-local sphere finitely generated over $\\mathbb Z_p$?","shortName":"Finite generation of the K(n)-local sphere","problemNumber":null,"field":"Stable homotopy theory; chromatic homotopy","fieldGroup":"Geometry & topology","statement":"Fix a prime $p$ and height $n\\ge1$, and let $L_{K(n)}S$ be the Bousfield localization of the $p$-complete sphere at Morava $K$-theory $K(n)$. It is not connective, so finiteness must be controlled degree by degree. At height one the answer follows from the classical computation; at height two and $p\\ge5$ Hovey and Strickland derived it from Shimomura's computations. Is $\\pi_t L_{K(n)}S_p^\\wedge$ a finitely generated $\\mathbb Z_p$-module for every prime $p$, every height $n$ and every integer $t$?","posedBy":"Mark Hovey and Neil Strickland, Problem 16.2 of 'Morava K-theories and localisation' (Memoirs AMS, 1999); also Problem 1 of Hovey's Morava K- and E-theory problem list","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every prime $p$, height $n\\ge1$ and integer $t$, $\\pi_t L_{K(n)}S_p^\\wedge$ is a finitely generated $\\mathbb Z_p$-module; equivalently $\\pi_t L_{K(n)}F$ is finitely generated for every finite $p$-local spectrum $F$. Combined with the rational computation of Barthel-Schlank-Stapleton-Weinstein this fixes the free ranks. The key step is finiteness of continuous cohomology of the stabilizer with mod $p$ Lubin-Tate coefficients. It gives no bound on the number of generators or the torsion as $p$, $n$, $t$ vary, computes no torsion, and does not extend to all invertible $K(n)$-local spectra (known to fail at height two).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction, history section and Theorem 1.1 were read against Hovey-Strickland Problem 16.2. lean/docs/313.md does not exist at the pinned commit and formalization.yaml has no entry for this manuscript, so there is no formal statement. The proof leans on deep external inputs (Fargues-Fontaine curve geometry, Fargues-Scholze, Anschutz-Le Bras relative full faithfulness, Devinatz-Hopkins descent, Mathew's descendability) that a reader has to accept.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":26,"significanceNote":"A named problem from the Hovey-Strickland memoir, natural and well known to chromatic homotopy theorists but narrower than the structural conjectures of the field. Placed below chromatic-splitting-conjecture (40) and smith-toda-complexes-every-height (30), and just below kalton-problem-2-bi-lipschitz-c0 (28) as a comparably specialist named question.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf","sourceName":"OpenAI math release: Finite generation for the K(n)-local sphere","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"virasoro-conjecture-complete-intersections","name":"The Virasoro conjecture for smooth complete intersections in projective space (and its transfer to projective bundles)","shortName":"Virasoro conjecture for complete intersections","problemNumber":null,"field":"Algebraic geometry; Gromov-Witten theory","fieldGroup":"Geometry & topology","statement":"For a smooth projective variety $X$, let $Z_X=\\exp(\\sum_g\\hbar^{g-1}F_g^X)$ be the total descendant Gromov-Witten potential. Eguchi, Hori and Xiong (1997) proposed target-dependent differential operators $L_k^X$, $k\\ge-1$, built from the Poincare pairing, a Hodge grading and cup product with $c_1(TX)$, generalising the Witten-Kontsevich KdV constraints for a point; Eguchi, Jinzenji and Xiong extended them to odd and off-diagonal classes. The Virasoro conjecture asserts $L_k^XZ_X=0$ for all $k\\ge-1$. Genus zero, Calabi-Yau targets with $H^1=0$, curves, toric bundles and semisimple theories (Givental-Teleman) were known, but non-semisimple higher-genus targets largely were not. Does the Virasoro conjecture hold, in every genus and curve class with arbitrary insertions, for every smooth complete intersection in projective space?","posedBy":"Tohru Eguchi, Kentaro Hori and Chuan-Sheng Xiong (extended by Eguchi, Jinzenji and Xiong)","yearPosed":1997,"ageNote":"Eguchi, Hori and Xiong, Quantum cohomology and Virasoro algebra, Phys. Lett. B 402 (1997), cited by the manuscript; odd-class extension Eguchi-Jinzenji-Xiong, Nucl. Phys. B 510 (1998).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Principal theorem: for a smooth complete intersection $X\\subset\\mathbb P^N_{\\mathbb C}$, $L_k^XZ_X=0$ for all $k\\ge-1$ in Getzler's first-Hodge-degree superspace convention, coefficientwise in every genus and curve class, with arbitrary insertions from $H^*(X;\\mathbb C)$. Companion: if the full constraints hold for a smooth connected projective base $B$, they hold for $\\mathbb P_B(E)$ for any algebraic vector bundle $E$ of rank at least 2 (no splitting or positivity), hence for projective-bundle towers. The conjecture for general smooth projective varieties (for example general hypersurfaces in other ambients, or varieties with no degeneration to known pieces) remains open.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts: the complete-intersection theorem (September 24, 2026) and a projectivization transfer theorem (October 5, 2026).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript (Virasoro constraints for the ordinary unreduced descendant potential of every smooth complete intersection in $\\mathbb P^N_{\\mathbb C}$, every genus and curve class, all insertions including primitive and odd classes, no semisimplicity) and Theorem 1.1 of the companion ($\\mathcal V(B)\\Rightarrow\\mathcal V(\\mathbb P_B(E))$ for every algebraic bundle of rank at least 2) were read against the conjecture as posed. The proof (degeneration with a weight spectral sequence, cap tests, a Hodge-Riemann positivity detector) was not refereed and there is no Lean formalization. The manuscripts use established degeneration and toric-bundle transfer results as stated inputs.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The Virasoro conjecture is a central open problem of Gromov-Witten theory since 1997; complete intersections are a large, mostly non-semisimple class, so this is a substantial special case but not the conjecture. Below the Rohde-Schramm partial entry (52), level with the Lane-Emden conjecture (40), above the generic Ramanujan conjecture for exceptional groups over function fields (34).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026/article.pdf","sourceName":"OpenAI math release: Virasoro Constraints for Projective Complete Intersections","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Virasoro Constraints under Projectivization","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf","kind":"paper"},{"label":"Eguchi, Hori and Xiong, Quantum cohomology and Virasoro algebra (1997)","url":"https://doi.org/10.1016/S0370-2693(97)00401-2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kl-combinatorial-invariance-conjecture","name":"The combinatorial invariance conjecture for Kazhdan-Lusztig polynomials","shortName":"KL combinatorial invariance","problemNumber":null,"field":"Algebraic combinatorics: Kazhdan-Lusztig polynomials and Bruhat order","fieldGroup":"Combinatorics","statement":"For a Coxeter system $(W,S)$ with Bruhat order $\\le$ and $u\\le b$, the Kazhdan-Lusztig polynomial $P_{u,b}(q)$ is defined algebraically, through the Hecke algebra and the simple generators, while the Bruhat interval $[u,b]=\\{x:u\\le x\\le b\\}$ is a finite graded poset that remembers neither generators nor root labels. The combinatorial invariance conjecture, attributed to Lusztig and to Dyer, asserts that the polynomial depends only on the interval as an abstract poset. Known before this work: principal lower intervals $[1,b]$ (du Cloux, Brenti, Brenti-Caselli-Marietti, Delanoy), invariance of the coefficient of $q$ (Patimo; Barkley-Gaetz-Lam), and full invariance in interval rank at most six, or up to ten or twelve in some finite Weyl groups. If $[u,b]$ in $(W,S)$ and $[u',b']$ in $(W',S')$ are isomorphic as posets, is $P_{u,b}(q)=P_{u',b'}(q)$?","posedBy":"George Lusztig and Matthew Dyer; published formulation in M. Dyer, On the Bruhat graph of a Coxeter system, Compositio Math. 78 (1991), Section 3.5","yearPosed":1991,"ageNote":"The manuscript attributes the conjecture to Lusztig and Dyer (citing Delanoy 2006) and gives Dyer's 1991 paper as the published formulation, stated there for finite Coxeter systems. It is often dated to the 1980s, but the manuscript gives no earlier date. The theorem here covers arbitrary, possibly different, Coxeter systems.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for arbitrary Coxeter systems $(W,S)$ and $(W',S')$, including infinite and noncrystallographic ones, and $u\\le b$, $u'\\le b'$, any poset isomorphism $[u,b]\\to[u',b']$ gives $P^W_{u,b}=P^{W'}_{u',b'}$ for the equal-parameter polynomials; restricting to subintervals determines every $P_{x,y}$ and, by reciprocity, every $R$-polynomial on the interval. The proof uses moment-graph sheaves (Braden-MacPherson, Fiebig, Elias-Williamson), an edge schedule transported from a reflection order of the second system, and Dyer's path formula. It gives no procedure computing $P$ from the poset, and it treats equal parameters only.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as the manuscript cites it; the proof was not refereed. Lean: the Comparator challenge KLInvariance (OAI.KLInvariance.combinatorial_invariance, solution module OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Its statement was read here: for two Coxeter systems on arbitrary groups, an order isomorphism between strong Bruhat intervals (Bruhat order defined as the reflexive-transitive closure of length-increasing reflection steps) gives equal Kazhdan-Lusztig polynomials. The polynomials are the family chosen (by Classical.epsilon) among those satisfying the R-recursion, the degree bound and the reciprocity identity, so the statement is the headline claim provided that normalization singles out the classical polynomials, which is standard but is part of the encoding. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"A central open problem of algebraic combinatorics and Kazhdan-Lusztig theory for over three decades, prominent enough to draw machine-learning-guided work in 2022. Scored with Sidorenko's conjecture (52) and above Ryser's conjecture (50), below Hadwiger's conjecture (70), whose reach is wider.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Combinatorial invariance of Kazhdan-Lusztig polynomials","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean (combinatorial_invariance)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RepresentationTheory/KazhdanLusztig/Invariance.lean","kind":"lean-proof"},{"label":"Dyer, On the Bruhat graph of a Coxeter system (Compositio Math. 1991)","url":"https://www.numdam.org/item/CM_1991__78_2_185_0/","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"metric-entropy-duality-conjecture","name":"The duality conjecture for metric entropy (Pietsch)","shortName":"Duality conjecture for metric entropy","problemNumber":null,"field":"Asymptotic geometric analysis; covering numbers","fieldGroup":"Analysis","statement":"For convex bodies $A,B$ let $N(A,B)$ be the least number of translates of $B$ covering $A$, and let $K^\\circ$ be the polar of $K$. The duality conjecture for metric entropy, originating in Pietsch's 1972 work on entropy numbers of operators and formulated geometrically by Artstein, Milman, Szarek and Tomczak-Jaegermann (2004, Conjecture 1), asks whether covering complexity is preserved under polarity up to universal changes of scale and constant factors. It was proved when one body is an ellipsoid (Artstein-Milman-Szarek 2004) and with logarithmic losses in general (E. Milman). Are there absolute constants $a,b\\ge1$ such that $\\frac1b\\log N(L^\\circ,aK^\\circ)\\le\\log N(K,L)\\le b\\log N(L^\\circ,a^{-1}K^\\circ)$ for all dimensions and all origin-symmetric convex bodies $K,L$?","posedBy":"Albrecht Pietsch (1972, operator form); geometric form as Conjecture 1 of Artstein, Milman, Szarek and Tomczak-Jaegermann (GAFA 2004)","yearPosed":1972,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $a,b\\ge1$ there exist $n$ and an origin-symmetric convex body $K\\subset\\mathbb R^n$ such that, with $L$ the cube, $\\log N(K,L)>b\\log N(L^\\circ,a^{-1}K^\\circ)$. A further construction (per the Lean scope note) makes the dual-to-primal entropy ratio tend to zero. This disproves the two-sided dimension-free conjecture. It leaves intact the ellipsoid case, the convexified-packing duality, bounds with constants depending on the K-convexity constant, and Milman's logarithmic-loss comparison; no fixed-dimension claim is made.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: for every $a,b\\ge1$ there are $n$ and an origin-symmetric body $K\\subset\\mathbb R^n$ with, for $L=[-1,1]^n$, $\\log N(K,L)>b\\log N(L^\\circ,a^{-1}K^\\circ)$, so the right-hand inequality fails for every pair of constants. lean/formalization.yaml lists MetricEntropyDuality.json (declaration OAI.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers, file OAI/Analysis/MetricEntropy/Main.lean). The statement MetricEntropyDuality.lean was read here; not rebuilt here. It defines the cube, polar, translate covers and covering numbers in $\\mathbb R^n$, and states exactly Theorem 1.1 for a compact convex symmetric $K$ with nonempty interior, with both covers shown to exist. This states the headline disproof. Only one direction is refuted; the dimension grows with $a,b$. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"A half-century-old conjecture of operator theory and asymptotic geometric analysis, the subject of an Annals paper settling the ellipsoid case; famous within that community. Level with Talagrand's discrete convexity entry in this batch at 36, above Kalton's Problem 2 at 28, below the Lipschitz isomorphism problem entry at 52.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf","sourceName":"OpenAI math release: Counterexamples to the duality conjecture for metric entropy","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/MetricEntropy/Main.lean","kind":"lean-proof"},{"label":"Artstein, Milman, Szarek, Tomczak-Jaegermann, On convexified packing and entropy duality (2004)","url":"https://doi.org/10.1007/s00039-004-0486-3","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"edit-distance-near-linear-approximation-scheme","name":"A (1+epsilon)-approximation of edit distance in almost-linear time","shortName":"Near-linear (1+eps) edit distance","problemNumber":null,"field":"Fine-grained and string algorithms","fieldGroup":"Theoretical computer science","statement":"Edit distance between strings of total length $N$ is computable in quadratic time, and Backurs and Indyk showed that a strongly subquadratic exact algorithm would refute SETH. Approximation algorithms reached polylogarithmic factors in time $N^{1+\\delta}$ (Andoni-Krauthgamer-Onak), a constant factor in subquadratic time (Chakraborty-Das-Goldenberg-Koucky-Saks), and constant factors depending on $\\delta$ in time $N^{1+\\delta}$ (Andoni-Nosatzki 2020). For accuracy $1+\\varepsilon$ the best known time saved only a quasi-polynomial factor, $N^2/2^{\\log^{\\Omega(1)}N}$ (Mao-Rubinstein 2026). For every fixed $\\varepsilon>0$, is there a randomized $(1+\\varepsilon)$-approximation of edit distance running in time $N^{1+o(1)}$?","posedBy":"Open question of the edit-distance approximation line (Andoni, Krauthgamer and Onak; Andoni and Nosatzki; Mao and Rubinstein); no single poser cited","yearPosed":null,"ageNote":"The manuscript recounts the history of approximation algorithms but cites no paper that poses the question, and none was located here, so no year is set; it has been an open target at least since near-linear constant-factor algorithms appeared (2020).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Main theorem: one randomized algorithm, uniform in a rational accuracy input $\\varepsilon\\in(0,1)$, returns for explicitly stored strings with polynomially bounded integer symbols an estimate within factor $1+\\varepsilon$ of unit-cost edit distance with probability at least 2/3, in worst-case expected time $N^{1+o(1)}$ on a logarithmic-word RAM; equal strings give 0 deterministically. The bound is pointwise in fixed $\\varepsilon$: no polynomial dependence on $1/\\varepsilon$ is claimed, and exact computation is used below very large accuracy-dependent thresholds, so it is an asymptotic scheme. Not shown: an alignment or edit script, weighted costs, a deterministic algorithm, or bounds when $\\varepsilon$ shrinks with $N$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The single manuscript (September 24, 2026) is the whole family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and main theorem were read against the question as framed. The analysis was not refereed. The challenge lean/ComparatorChallenges/EditApproximation.json (theorem OAI.EditApproximation.almostLinearEditDistanceRawBand, solution module OAI.Combinatorics.EditApproximation.Specification.RawBandMain, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml; it is found through lean/docs/121.md. Its main theorem was read here: for every binary fraction $0<\\varepsilon<1$, on integer lists of total length at most $N$ the output lies in $[\\mathrm{ed},(1+\\varepsilon)\\mathrm{ed}]$ with probability at least 2/3, equal strings give 0, and for symbols bounded by $(N+2)^C$ the expected work is at most $(N+2)^{1+\\eta}$ for every $\\eta>0$ and large $N$, with polynomial space. The algorithm and its work and space accounting are defined inside the roughly 19,000-line challenge file itself; whether that accounting matches word-RAM time was not audited here. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Whether edit distance can be approximated within 1+epsilon in near-linear time is a headline question of fine-grained algorithms; the best prior scheme, from March 2026, saved only a quasi-polynomial factor over quadratic time. Placed level with Randomized k-server polylogarithmic in k (45), just below Worst-case Subset Sum below meet-in-the-middle (46) and well below The APSP hypothesis (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026/paper.pdf","sourceName":"OpenAI math release: An Almost-Linear Approximation Scheme for Edit Distance","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean main theorem (OAI.EditApproximation.almostLinearEditDistanceRawBand)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/EditApproximation/Specification/RawBandMain.lean","kind":"lean-proof"},{"label":"Comparator statement: EditApproximation.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/EditApproximation.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kakeya-set-conjecture-r4","name":"The Kakeya set conjecture in four dimensions (Hausdorff dimension)","shortName":"Kakeya set conjecture in R^4","problemNumber":null,"field":"Harmonic analysis; geometric measure theory","fieldGroup":"Analysis","statement":"A Kakeya set in $\\mathbb R^n$ contains a unit line segment in every direction. Besicovitch showed such a set can have measure zero; the Hausdorff-dimension Kakeya conjecture asserts it must still have Hausdorff dimension $n$. It was known for $n=2$ (Davies, 1971) and $n=3$ (Wang and Zahl, 2025). In $\\mathbb R^4$, Wolff's hairbrush gave dimension at least $3$, and the best Hausdorff bound before this work was about $3.059$ (Katz and Zahl). Does every subset of $\\mathbb R^4$ containing a unit line segment in every direction have Hausdorff dimension $4$?","posedBy":"Hausdorff-dimension form of the Kakeya conjecture (classical, after Besicovitch and Davies); the manuscript cites the Katz-Tao (2002) formulation","yearPosed":null,"ageNote":"The manuscript names no poser or year. The dimension question has been open since Davies proved the planar case in 1971, and the four-dimensional case has a long record of partial bounds (Wolff 1995, Laba-Tao, Guth-Zahl, Katz-Zahl).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $K\\subset\\mathbb R^4$ contains, for every $e\\in S^3$, a segment $\\{a_e+te:0\\le t\\le1\\}$, then $\\dim_H K=4$. No measurability, compactness, stickiness or packing-dimension hypothesis is imposed. The paper also derives packing and Minkowski-dimension consequences, a projection lower bound in higher dimensions, and Nikodym and curved-Kakeya applications via cited transfers. It does NOT prove the four-dimensional Kakeya maximal conjecture, and does not settle the conjecture in dimension five or higher.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This family consists of two manuscripts dated September 23 and 24, 2026; this entry draws on the September 24 one. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the introduction states it; it covers arbitrary (not necessarily measurable or compact) sets and arbitrary witnessing segments, which is the full Hausdorff form in R^4. The proof was not refereed. No Lean formalization exists. It uses a weighted plank lemma (Lemma 2.3) from the companion three-dimensional maximal paper, itself unreviewed, together with the published Guth-Wang-Zahl union estimate.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The first open case of the Kakeya conjecture after the three-dimensional breakthrough of Wang and Zahl, and a central problem of harmonic analysis and geometric measure theory. Placed level with the sum-product conjecture over the reals (55), below the Jacobian Conjecture (65) and the 67 percent critical-line result (68), and above the three-dimensional maximal case from this family (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Every four-dimensional Kakeya set has full Hausdorff dimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The Kakeya maximal conjecture in three dimensions (input lemma)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"petty-projection-conjecture-dimension-four-plus","name":"Petty's projection conjecture in dimensions at least four","shortName":"Petty projection conjecture, n >= 4","problemNumber":null,"field":"Convex geometry; affine isoperimetric inequalities, projection bodies","fieldGroup":"Geometry & topology","statement":"For a convex body $K\\subset\\mathbb R^n$ the projection body $\\Pi K$ has support function $h_{\\Pi K}(u)=\\mathrm{vol}_{n-1}(\\mathrm{proj}_{u^\\perp}K)$. The ratio $|\\Pi K|/|K|^{n-1}$ is affine invariant. Petty (1971) conjectured that for $n\\ge3$ it is minimized exactly by ellipsoids, i.e. $|\\Pi K|/|K|^{n-1}\\ge\\kappa_{n-1}^n\\kappa_n^{2-n}$ with equality only for ellipsoids. Local versions were proved by Saroglou-Zvavitch and Ivaki, and in August 2026 Chen, Feng, Li, Xi and Xu proved the case $n=3$. Does Petty's inequality, with its ellipsoid equality case, hold for every convex body in every dimension $n\\ge4$?","posedBy":"C. M. Petty, Isoperimetric problems, Proc. Conf. Convexity and Combinatorial Geometry, Univ. of Oklahoma (1971)","yearPosed":1971,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1: for every $n\\ge4$ and every convex body $K\\subset\\mathbb R^n$, $|\\Pi K|/|K|^{n-1}\\ge\\kappa_{n-1}^n\\kappa_n^{2-n}$, with equality iff $K$ is an ellipsoid; no symmetry or regularity is assumed. The proof bounds a bilinear pairing $\\iint|x\\cdot y|\\,d\\eta\\,d\\zeta$ via a strict spherical-harmonic estimate for norms and a fixed-point choice of affine position. It does not treat $n=3$, which Chen, Feng, Li, Xi and Xu proved in August 2026; the paper notes that the two together give the full conjecture for $n\\ge3$. As corollaries (via Lutwak's known implication) it states the Lutwak-Petty lower-degree inequalities for $2\\le i\\le n-2$, without equality cases, and a sharp Holmes-Thompson isoperimetric inequality.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1 of the manuscript was read against Petty's conjecture as the paper states it with the 1971 reference; it claims the inequality and the ellipsoid-only equality case for every convex body in every dimension $n\\ge4$. The proof was not refereed. The Lean challenge ComparatorChallenges/PettyProjectionVolume.lean (listed in lean/formalization.yaml, declaration OAI.PettyProjection.petty_projection_volume) was read here: for $n\\ge4$ and every compact convex $K$ with nonempty interior it asserts pettyConstant $n\\le|\\Pi K|/|K|^{n-1}$ and equality iff $K$ is an affine image of the unit ball, with $\\Pi K$ defined through orthogonal shadow volumes. This states the headline claim. Not rebuilt here. The three-dimensional case is not part of this work and rests on the human preprint of Chen et al.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Petty's projection conjecture has been a central open problem of affine convex geometry since 1971; with dimension three settled by humans a month earlier, this is the remaining bulk of it. Placed just below the logarithmic Brunn-Minkowski conjecture (50), above the Generalized Busemann-Petty Problem in Dimensions 2 and 3 (30) and level with the generalized Lax conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Petty's projection-volume conjecture in dimensions at least four","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.PettyProjection.petty_projection_volume)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/ProjectionBodies/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: PettyProjectionVolume.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PettyProjectionVolume.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ionization-conjecture-simon-problem-9","name":"The ionization conjecture: bounded excess charge, ionization energy and radius for Coulomb atoms and molecules (Simon's Problem 9)","shortName":"The ionization conjecture","problemNumber":null,"field":"Many-body quantum mechanics; Coulomb systems","fieldGroup":"Mathematical physics","statement":"For the nonrelativistic Coulomb Hamiltonian of $n$ electrons (two spin states, Fermi statistics) around fixed nuclei of total charge $Z$, let $N_c$ be the largest number of electrons that is strictly bound. Lieb proved $N_c<2Z+M$, Lieb-Sigal-Simon-Thirring proved $N_c/Z\\to1$ for atoms, and later work gave sublinear and explicit bounds such as $1.22Z+3Z^{1/3}$, but no uniform bound on the excess. Simon's Problem 9 (2000) asks whether $N_c(Z)-Z$ stays bounded for atoms; the molecular form asks for $N_c\\le Z+CM$ with $M$ nuclei (Lewin 2025). The ionization conjecture as surveyed by Solovej also asks that the first ionization energy and the outer radius of neutral atoms stay bounded independently of $Z$. Is the excess charge bounded by a universal constant per nucleus, and are the neutral-atom ionization energy and radius bounded uniformly in $Z$?","posedBy":"Barry Simon (Problem 9 of his 2000 list); molecular form in Mathieu Lewin's 2025 open-problem survey","yearPosed":2000,"ageNote":"The manuscript cites Simon, Schrodinger operators in the twenty-first century (2000), Problem 9, and Lewin (C. R. Physique 2025, Section 2.1.2) for the molecular form, and Solovej (2016, Section 2) for the boundedness of the ionization energy. The conjecture circulated earlier, in the Lieb-Sigal-Simon-Thirring era of the 1980s, but the manuscript gives no earlier statement.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a universal $C$ such that for every $M\\ge1$, all distinct nuclear positions and all real charges $z_j\\ge1$ with $Z=\\sum z_j$, strict binding $E_nZ+CM$. For atoms this is Simon's bounded excess charge. Theorem 1.2: for every integer $Z\\ge1$ and every neutral ground state, the radius outside which half an electron remains lies between universal constants. Corollary 1.3: the neutral first ionization energy satisfies $c\\le I_1(Z)\\le C$. Not shown: the value of the constants, relativistic or magnetic models, or the asymptotic (generalized) form, which is the separate generalized ionization entry.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This manuscript also supplies the screening estimates used by the two generalized-ionization companions in the same family.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 and Corollary 1.3 were read against Simon's Problem 9 and the molecular and neutral-atom forms the manuscript cites. The 80-page proof was not refereed. This manuscript has no Lean main result: lean/docs/263.md lists only the two generalized-ionization companions, and the Lean developments for those may formalize some of its estimates internally, but no statement of Theorem 1.1, 1.2 or Corollary 1.3 is a comparator challenge. The paper notes that equality of adjacent energies does not exclude a threshold ground state, which its strict-binding formulation does not need.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The ionization conjecture is a famous open problem of mathematical physics from the Lieb-Simon program on the stability of matter, listed by Simon among his fifteen problems. Placed level with Simon's Problem 2 on two-dimensional Anderson localization (55), below Simon's Problem 1 on extended states (65), and above the Gamow liquid-drop minimizer conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Generalized ionization energies for full Coulomb atoms","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Generalized outer-electron radii of neutral Coulomb atoms","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lewin, Some open mathematical problems concerning charged quantum particles (2025)","url":"https://doi.org/10.5802/crphys.249","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"abelian-zilber-pink-conjecture","name":"The Zilber-Pink conjecture for abelian varieties over number fields","shortName":"Abelian Zilber-Pink over number fields","problemNumber":null,"field":"Diophantine geometry; unlikely intersections","fieldGroup":"Number theory","statement":"Let $A$ be an abelian variety over a number field and $X\\subseteq A_{\\bar{\\mathbb Q}}$ an irreducible subvariety, and let $S$ be the smallest torsion coset containing $X$. A component $Y$ of $X\\cap T$, for a torsion coset $T\\subseteq S$, is atypical if $\\dim Y>\\dim X+\\dim T-\\dim S$. Zilber (2002, for semiabelian varieties) and Pink (2005) conjectured that such unlikely intersections are controlled, generalizing Manin-Mumford and Mordell-Lang. The curve case was proved by Habegger and Pila (2016); higher dimensions were known only under extra quotient-dimension or height conditions or in codimension one or two. Does every such $X$ have only finitely many maximal atypical subvarieties?","posedBy":"Boris Zilber (J. London Math. Soc., 2002) and Richard Pink (ETH preprint, 2005, Conjecture 5.1 for semiabelian varieties)","yearPosed":2002,"ageNote":"Zilber's 2002 paper formulated the principle for semiabelian varieties; Pink's 2005 preprint gave the semiabelian and mixed Shimura formulations. The manuscript proves the abelian case over number fields.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims that for any abelian variety over a number field, every irreducible subvariety has only finitely many maximal atypical subvarieties inside its smallest containing torsion coset, via a non-density theorem for $X\\cap A^{[\\dim X+1]}$ for $X$ in no proper torsion coset. A corollary gives the abelian algebraic-point case of Pink's Conjecture 5.2 for finite-rank translates. It does NOT treat semiabelian varieties, abelian varieties over $\\mathbb C$ not defined over $\\bar{\\mathbb Q}$, or Shimura varieties, and gives no effective or uniform bound.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. The manuscript is credited to OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the posed conjecture: for every abelian variety over a number field and every irreducible subvariety, finitely many maximal atypical subvarieties relative to the smallest containing torsion coset, with no simplicity, CM or height hypothesis. It is the abelian case over $\\bar{\\mathbb Q}$ of Zilber-Pink, not the semiabelian, complex-field or Shimura cases. The reduction to a non-density theorem uses Barroero and Dill's optimal-subvariety argument as a cited input; the height part builds on Vojta, Faltings, Remond and Dill. No Lean formalization. The finiteness is not effective.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Zilber-Pink is among the central conjectures of diophantine geometry, and its abelian case in all dimensions was a well-known open target after Habegger and Pila settled curves. A special case of the full conjecture, so scored below Hodge for CM abelian varieties at 72; level with the Kadison Similarity Problem at 58 and just below the Tate conjecture for abelian varieties over finite fields at 60.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The abelian Zilber-Pink conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Family companion: Elliptic squares and Zilber-Pink for curves in A2 (separate entry)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf","kind":"paper"},{"label":"Pink (2005), A common generalization of the conjectures of Andre-Oort, Manin-Mumford, and Mordell-Lang","url":"https://people.math.ethz.ch/~pink/ftp/AOMMML.pdf","kind":"problem-record"},{"label":"Zilber (2002), Exponential sums equations and the Schanuel conjecture","url":"https://doi.org/10.1112/S0024610701002861","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pixton-completeness-conjecture","name":"Pixton's conjecture: do Pixton's relations give all tautological relations on the moduli of stable curves?","shortName":"Pixton completeness conjecture","problemNumber":null,"field":"Algebraic geometry; moduli of curves; tautological rings","fieldGroup":"Geometry & topology","statement":"The tautological ring $R^*(\\overline{\\mathcal M}_{g,n})$ is the image of the strata algebra of decorated stable graphs in the Chow ring. In 2012 Pixton proposed an explicit system of graph relations extending the Faber-Zagier relations; Pandharipande-Pixton-Zvonkine proved them valid in cohomology and Janda in Chow. Pixton conjectured completeness: that the span $P_{g,n}$ of his relations (with the operations it is closed under) is the whole kernel of the realization map, in Chow and in cohomology (recorded as Pixton's conjecture in Canning 2025). Is $\\ker(\\mathcal S_{g,n}\\to CH^*(\\overline{\\mathcal M}_{g,n};\\mathbb Q))=P_{g,n}$ for every stable pair $(g,n)$?","posedBy":"Aaron Pixton, Conjectural relations in the tautological ring of the moduli of stable curves, arXiv:1207.1918 (2012), Conjecture 2","yearPosed":2012,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that Pixton's original relations are not complete in rational Chow or rational cohomology: on $\\overline{\\mathcal M}_{g,n}$ with $g=10^{60}$ and $n=3\\binom{10^{60}}3$ an alternating product of corrected small diagonals vanishes but lies outside the span. It does NOT give a small counterexample (the parameters are not claimed minimal), does NOT address enlarged relation systems, and is distinct from the Gorenstein question already settled negatively (Petersen-Tommasi, Canning).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The single manuscript defines the counterexample class explicitly and proves nonmembership and vanishing by separate arguments.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 were read against Pixton's Conjecture 2 as fixed in Section 2 of the paper: for $g=10^{60}$ and $n=3\\binom g3$ an explicit formal class $Y$ maps to zero in rational Chow (hence in rational cohomology) but is not in Pixton's original relation span. The vanishing argument relies on derived Quot-scheme virtual classes and O'Sullivan's tensor reconstruction, cited inputs not checked here. No Lean formalization. The paper limits itself to the original relation system as it defines it.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"Pixton's completeness conjecture is the main structural conjecture about tautological relations on moduli of curves and is well known among enumerative geometers, though its expected answer was debated. A disproof in both Chow and cohomology scores level with Toda's Gepner conjecture (38), below Kuznetsov's rationality conjecture (45), above the Period-Index entry (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A high-arity counterexample to Pixton completeness in Chow","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Pixton, Conjectural relations in the tautological ring (2012)","url":"https://arxiv.org/abs/1207.1918v1","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gwynne-miller-fk-maps-metric-conjecture","name":"Gwynne and Miller's graph-metric conjecture for critical FK planar maps (0 < q < 4) and spanning-tree-weighted maps, finite spherical case","shortName":"Gwynne-Miller conjecture for FK and spanning-tree maps","problemNumber":null,"field":"Random planar maps; Liouville quantum gravity","fieldGroup":"Probability & statistics","statement":"Random planar maps decorated by statistical-mechanics models are expected to converge, as metric measure spaces, to Liouville quantum gravity (LQG) surfaces with the matching parameter $\\gamma$: uniform maps to the Brownian map ($\\gamma=\\sqrt{8/3}$, Le Gall, Miermont), and other ensembles to the $\\gamma$-LQG sphere equipped with the intrinsic metric constructed by Ding-Dubedat-Dunlap-Falconet, Gwynne-Miller and others. Gwynne and Miller's Conjecture 1.7 states that graph distances, suitably rescaled, converge to the $\\gamma$-LQG metric for several ensembles, explicitly including critical Fortuin-Kasteleyn (FK) maps with $q\\in(0,4)$, $q=2+2\\cos(\\pi\\gamma^2/2)$, $\\gamma\\in(\\sqrt2,2)$, and spanning-tree-weighted maps at $\\gamma=\\sqrt2$. For these finite spherical ensembles, do rescaled graph distances converge in law to the $\\gamma$-LQG sphere?","posedBy":"Ewain Gwynne and Jason Miller (Conjecture 1.7)","yearPosed":2019,"ageNote":"Gwynne and Miller, Existence and uniqueness of the Liouville quantum gravity metric for gamma in (0,2), Invent. Math. (2021), Conjecture 1.7, cited by the manuscript; arXiv 1905.00383 (2019). The broader expectation of LQG metric limits for decorated maps is older.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Principal theorem: for fixed $q\\in(0,4)$ (critical FK maps weighted by $q^{\\ell/2}$, $\\ell$ the number of interfaces) and for uniform map-spanning-tree pairs, $n$-edge spherical maps with degree measure and rescaled graph distance converge in GHP law to the unit-area $\\gamma$-quantum sphere with its LQG metric and area. Companions: joint conformal (flag-triangle uniformization), metric, area and CLE-interface limits for $0 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has seven manuscripts (September 24 and October 5, 2026): metric-measure limits for $04$ (a separate entry), and three papers on random walk, spectral and heat-trace convergence built on these geometric results.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the metric-measure manuscript was read against Gwynne-Miller Conjecture 1.7. It gives, for each fixed $q\\in(0,4)$ and for spanning-tree maps, deterministic $a_n\\to0$ with $(V(M_n),a_nd_n,\\mu_n)\\to(S,D_h,\\mu_h)$ in law in the Gromov-Hausdorff-Prokhorov topology, using the degree measure. The paper itself says it does not address every ensemble or version of the conjecture, asserts no uniformity in $q$, normalisation power law or rate, and that the uniform vertex measure would need a separate argument. The family's Lean formalization covers only the $q>4$ tree limit, not this theorem. Not refereed. The FK-Ising walk and spectral companions use stated Brownian/LQG inputs.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Graph-distance convergence of non-uniform decorated maps to LQG is one of the main open problems of random geometry after the Brownian map; this settles the finite spherical FK and spanning-tree cases of the Gwynne-Miller conjecture, not all ensembles. Below the Rohde-Schramm FK interface entry (52), level with Cardy's formula for square-lattice bond percolation (48), above the no-bigeodesics result for planar first-passage percolation (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026/main.pdf","sourceName":"OpenAI math release: Metric-measure limits of subcritical FK and spanning-tree planar maps","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Canonical conformal limits of subcritical FK planar maps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026/main.pdf","kind":"paper"},{"label":"Companion: The critical Liouville quantum sphere and geometric limits of FK maps at q=4","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026/main.pdf","kind":"paper"},{"label":"Companion: Random Walks on Critical FK-Ising Maps and Liouville Brownian Motion","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026/fk-ising-walk-limit.pdf","kind":"paper"},{"label":"Companion: Spectral convergence for critical FK-Ising planar maps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spectral-convergence-for-critical-FK-Ising-planar-maps-October-5-2026/spectral-convergence-critical-fk-ising-planar-maps.pdf","kind":"paper"},{"label":"Companion: A Linear Clock for Random Walk on Tree-Weighted Planar Maps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf","kind":"paper"},{"label":"Gwynne and Miller, Existence and uniqueness of the LQG metric for gamma in (0,2) (Conjecture 1.7)","url":"https://arxiv.org/abs/1905.00383","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"shareshian-wachs-e-positivity","name":"The Shareshian-Wachs conjecture: elementary positivity of chromatic quasisymmetric functions of natural unit interval graphs","shortName":"Shareshian-Wachs e-positivity","problemNumber":null,"field":"Algebraic combinatorics; chromatic symmetric functions","fieldGroup":"Combinatorics","statement":"For a weakly increasing $h$ with $i\\le h(i)\\le n$, the natural unit interval graph $G$ on $[n]$ has edges $\\{i,j\\}$ with $i 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'Elementary positivity of chromatic quasisymmetric functions' (September 24, 2026).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: it gives an explicit expansion $X_G=\\sum_{\\sigma\\in D_G^0}q^{\\mathrm{ginv}_G(\\sigma)}e_{\\theta_G(\\sigma)}$, which is the elementary-positivity part; the unimodality part is not claimed. Lean: lean/ComparatorChallenges/ElementaryPositivity.json exists (a definition challenge, OAI.elementaryPositivityWitness) and its solution module OAI.Combinatorics.Chromatic.Main is present at the pinned commit; the challenge is not in the formalization catalogue (formalization.yaml). The statement file was read here: for every natural unit interval graph given by a monotone extensive $h$, the witness assigns a partition to each permitted nondescent permutation so that, for every number $r$ of variables, the chromatic polynomial over Polynomial N (proper colorings weighted by $q^{\\mathrm{asc}}$) equals the sum of $q^{\\mathrm{ginv}}e_\\theta$. Holding for every finite $r$ gives the symmetric-function identity, so this states the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":33,"significanceNote":"The q-refinement of the Stanley-Stembridge conjecture, central to the chromatic symmetric function program and its link to Hessenberg varieties; after Hikita's 2024 proof at q=1 it was the main remaining positivity question there. Famous within algebraic combinatorics. Placed below the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials (52) and Rota's unimodality conjecture for matroid flats (36), above Stanley's claw-free Schur-positivity conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Elementary positivity of chromatic quasisymmetric functions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI/Combinatorics/Chromatic/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Chromatic/Main.lean","kind":"lean-proof"},{"label":"Lean statement: ComparatorChallenges/ElementaryPositivity.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ElementaryPositivity.lean","kind":"lean-proof"},{"label":"Shareshian and Wachs, Chromatic quasisymmetric functions (Adv. Math. 2016)","url":"https://doi.org/10.1016/j.aim.2015.12.018","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"disc-embedding-conjecture-free-group-good","name":"The four-dimensional disc embedding conjecture and whether the free group is good","shortName":"Disc embedding conjecture; is F2 good?","problemNumber":null,"field":"Topological 4-manifolds; surgery and disc embedding","fieldGroup":"Geometry & topology","statement":"Freedman's disc embedding theorem says: in an oriented 4-manifold $M$ with good fundamental group $\\pi$, immersed discs $f_i$ with embedded boundaries and framed algebraically dual spheres $g_j$ ($\\lambda(f_i,g_j)=\\delta_{ij}$, $\\lambda(g_i,g_j)=0=\\widetilde\\mu(g_i)$ in $\\mathbb Z[\\pi]$) can be replaced by disjoint locally flat embedded discs with the same boundary. Groups of subexponential growth are good (Freedman-Teichner, Krushkal-Quinn), and goodness passes to subgroups and quotients, so the decisive open case is the free group $F_2$, which also governs topological surgery and the 5-dimensional s-cobordism theorem for arbitrary groups. Does the disc embedding conclusion hold without the good-group hypothesis; in particular, is the free group $F_2$ good?","posedBy":"Michael Freedman and Frank Quinn, Topology of 4-manifolds (1990), theorem for good groups; the free-group case is listed open by Kim-Orson-Park-Ray (2021) and Powell-Ray-Teichner (2025)","yearPosed":1990,"ageNote":"Year of the Freedman-Quinn book, the earliest statement the manuscripts cite; the underlying question goes back to Freedman's work of 1982-86 (also cited), so the problem may be older than 1990.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a compact oriented smooth 4-manifold $M$ with immersed proper discs $f_1,\\dots,f_k$ with embedded boundary circles and framed (even smoothly embedded) spheres $g_i$ satisfying $\\lambda(f_i,g_j)=\\delta_{ij}$, $\\lambda(g_i,g_j)=0$, $\\widetilde\\mu(g_i)=0$, whose boundary circles bound no pairwise disjoint locally flat discs, in any relative homotopy class. Corollary 1.2: $F_2$, and every group containing it, is not good in the Freedman-Quinn sense. Corollary 1.3: the unrestricted topological surgery and s-cobordism assertions cannot both hold. The marked tensor companion proves the framing-preserving version. Not shown: which of surgery or s-cobordism fails, or that the Borromean rings are not A-B slice.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript and the companion 'A marked tensor obstruction' (both September 24, 2026) give two versions of the obstruction; the family's Wall-question manuscript builds on the marked tensor version.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2-1.3 of 'A boundary-only obstruction to four-dimensional disk embedding' were read against the disc embedding theorem as formulated by Powell-Ray-Teichner (2025) and the free-group question. The proof (a capped-grope construction, representation deformation rings with Chern-Simons type wall potentials, reduction to positive characteristic and a finite Frobenius obstruction) was not refereed. No Lean formalization. Scope the paper itself states: it refutes the conjunction of the unrestricted topological surgery and s-cobordism assertions, but does not say which one fails; the counterexample manifold's fundamental group is not identified with F2, and nongoodness of F2 follows via subgroup and quotient closure. A disproof of this standing would be expected to draw expert scrutiny quickly.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"Whether the free group is good, equivalently whether topological surgery works for all fundamental groups, has been the central open problem of topological 4-manifold theory since Freedman's work in the early 1980s. Tied with the Baum-Connes entry (60) and the Kothe Conjecture (60), below Cannon's Conjecture (62) and the Hilbert-Smith Conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A boundary-only obstruction to four-dimensional disk embedding","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A marked tensor obstruction to four-dimensional disk embedding","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf","kind":"paper"},{"label":"Freedman-Quinn, Topology of 4-manifolds (1990)","url":"https://doi.org/10.1515/9781400861064","kind":"problem-record"},{"label":"Kim-Orson-Park-Ray, Open problems, in The Disc Embedding Theorem (2021)","url":"https://doi.org/10.1093/oso/9780198841319.003.0023","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"negative-perceptron-jamming-exponents","name":"Franz-Parisi jamming exponents in the negative spherical perceptron, at margin -1","shortName":"Perceptron jamming exponents (margin -1)","problemNumber":null,"field":"Spin glasses; jamming; random constraint satisfaction","fieldGroup":"Mathematical physics","statement":"In the spherical perceptron with negative margin, $x\\in\\sqrt N S^{N-1}$ must satisfy $g^a\\cdot x/\\sqrt N\\ge\\sigma$ for $M=\\alpha N$ Gaussian patterns, and the feasible set is nonconvex. Franz and Parisi (2016) identified this model with the jamming universality class of high-dimensional spheres and, from a full replica-symmetry-breaking solution, predicted isostaticity at the SAT-UNSAT threshold and power laws for small gaps $g(h)\\sim h^{-\\gamma}$ and weak forces $p(f)\\sim f^{\\theta}$ with $\\gamma\\approx0.41269$, $\\theta\\approx0.42311$ and $\\gamma=1/(2+\\theta)$, confirmed as universal across negative margins by Franz, Parisi, Sevelev, Urbani and Zamponi (2017). Rigorous work gave only density bounds. Does the negative perceptron have a sharp jamming threshold with these gap and force exponents?","posedBy":"Silvio Franz and Giorgio Parisi, The simplest model of jamming (J. Phys. A, 2016); Franz, Parisi, Sevelev, Urbani and Zamponi (SciPost Phys., 2017)","yearPosed":2016,"ageNote":"The exponent values originate in the full-RSB hard-sphere solution of Charbonneau, Kurchan, Parisi, Urbani and Zamponi (2014); the perceptron identification is Franz-Parisi 2016, as the manuscript cites.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for the spherical perceptron with margin $-1$ and quadratic penalty there is a sharp feasibility threshold $\\alpha_c$; the ordered limits (size, then zero temperature, then $\\alpha\\downarrow\\alpha_c$) of the contact-removed gap law and mean-one force law exist with $G_J(u)=u^{1-\\gamma+o(1)}$, $F_J(s)=s^{1+\\theta+o(1)}$, $\\gamma=(2+\\theta)^{-1}$, $0.4126930<\\gamma<0.4126934$, $0.4231063<\\theta<0.4231088$, and the critical limit is isostatic. The same holds for a small Gaussian scale mixture of coordinates. Not shown: other negative margins, the unpenalized hard-constraint model directly, other limit orders, or finite-size scaling.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has four manuscripts dated September 24, 2026. The jamming paper includes a finite numerical certificate with stated error bounds for the exponent intervals.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the background section and Theorem 1.1 of 'Microscopic jamming in the negative spherical perceptron' were read against the Franz-Parisi predictions as the manuscript cites them. The theorem treats margin $-1$ with a quadratic penalty, with limits taken in the order system size, inverse temperature, then density to the threshold from above; the physics prediction is for all negative margins. The exponent intervals rest on a shipped numerical certificate, not run here. The family's Lean formalizations (IsingFiniteness, PerceptronFreeEnergy, SphericalField) concern the free-energy papers, not this theorem, so this entry is unreviewed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A precise prediction of the replica theory of jamming, well known in statistical physics of disordered systems, proved here at one margin and in one limit order. Below the Krauth-Mezard capacity of the Ising perceptron (35) and the Mezard-Parisi formula for diluted spin glasses (32), below the Gardner transition in the Ising pure p-spin glass (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf","sourceName":"OpenAI math release: Microscopic jamming in the negative spherical perceptron","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Franz-Parisi, The simplest model of jamming (2016)","url":"https://arxiv.org/abs/1501.03397","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"benjamini-schramm-criticality-bond","name":"The Benjamini-Schramm criticality conjecture for bond percolation on quasi-transitive graphs","shortName":"Benjamini-Schramm criticality (bond form)","problemNumber":null,"field":"Percolation theory on graphs and groups","fieldGroup":"Probability & statistics","statement":"Let $G$ be an infinite, connected, locally finite quasi-transitive graph (its automorphism group has finitely many vertex orbits) and run Bernoulli percolation on $G$ with critical probability $p_c<1$. Benjamini and Schramm (1996, Conjecture 4) conjectured that critical percolation dies on every such graph: at $p=p_c$ there is almost surely no infinite open cluster. Their paper discusses site percolation and says the questions remain equally valid for bond percolation outside planar settings. The motivating case is $\\mathbb Z^d$, where $\\theta(p_c)=0$ was known for $d=2$ and in high dimensions only. Does critical Bernoulli bond percolation have no infinite cluster on every infinite connected locally finite quasi-transitive graph with $p_c<1$?","posedBy":"Itai Benjamini and Oded Schramm, Percolation beyond Z^d, many questions and a few answers (1996), Conjecture 4","yearPosed":1996,"ageNote":"The $\\mathbb Z^d$ special case is older: Benjamini and Schramm call it one of the outstanding challenges of percolation theory, and the catalog entry for it dates it to Kesten (1982). Benjamini and Schramm state their conjectures for site percolation and say they carry over to bond percolation; this entry is the bond form.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims that for every infinite connected locally finite quasi-transitive graph with $p_c<1$, Bernoulli bond percolation at $p_c$ has almost surely no infinite cluster. Exponential growth is Hutchcroft's theorem; the new work covers all subexponential growth, split into superpolynomial growth (a two-cluster entropy argument) and growth polynomial along a sequence of radii (a nilpotent-quotient corridor exploration). It includes $\\theta(p_c)=0$ on $\\mathbb Z^d$ for every $d\\ge2$. The companion proves critical bond and site nonpercolation on $\\mathbb Z^3$. It does NOT settle the site form of the conjecture on general quasi-transitive graphs, and gives no critical exponents or quantitative decay.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. Both manuscripts are credited to OpenAI with no human author named. The quasi-transitive manuscript credits earlier public AI-produced work on the Kozma-Nitzan gluing inequality (Leder's Lean account for Z^d, and expositions prompted by Ahmed Bou-Rabee) and gives its own proof of the gluing input.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against Benjamini-Schramm Conjecture 4 (the source was opened): it is the conjecture for bond percolation, with no extra hypothesis beyond $p_c<1$. The Lean challenge ComparatorChallenges/CriticalPercolation.json (theorem OAI.CriticalPercolation.BondGraph.no_percolation_at_criticality, solution module OAI.Probability.CriticalPercolation.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/213.md. Its statement was read here: for a bond graph with possibly parallel bonds and loops, infinite vertex set, connected, locally finite counting bonds, finitely many vertex orbits under automorphisms, and critical probability below one, the Bernoulli product law at $p_c$ gives probability zero to the event that some cluster is infinite. That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The companion's Z^3 bond and site statement (CriticalZ3, listed in formalization.yaml) was also read and matches its abstract. The proof uses Hutchcroft's 2016 theorem for exponential growth and the Tessera-Tointon structure theorem as cited inputs.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":79,"significanceNote":"Contains the classical problem of no percolation at criticality on Z^d in every dimension, scored 78 in the catalog, and extends it to every quasi-transitive graph, where the subexponential non-lattice cases were also open. Scored one point above that entry, below the full Benjamini-Schramm conjecture with its site form, and below the Quasi-Riemann Hypothesis at 85; above the Landau-Siegel zero problem at 72.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026/paper.pdf","sourceName":"OpenAI math release: No percolation at criticality on quasi-transitive graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Critical bond and site percolation on the cubic lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean: solution module for the quasi-transitive theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/CriticalPercolation/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: no percolation at criticality","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CriticalPercolation.lean","kind":"lean-proof"},{"label":"Lean: Z^3 bond and site main declaration critical_no_infinite","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/CriticalZ3/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: no infinite critical clusters on Z^3","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CriticalZ3.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/213.md","kind":"lean-proof"},{"label":"Benjamini and Schramm (1996), Conjecture 4","url":"https://doi.org/10.1214/ECP.v1-978","kind":"problem-record"},{"label":"p_c < p_u, from the same 1996 paper (OpenAI math release)","url":"https://vibemathed.com/problem/benjamini-schramm-pc-less-than-pu","kind":"other"},{"label":"Earlier site result on Z^d, d >= 3, cited by the manuscripts (Bou-Rabee)","url":"https://vibemathed.com/problem/critical-site-percolation-z-d","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bfss-threshold-bound-state","name":"The threshold bound state conjecture for SU(N) supersymmetric matrix quantum mechanics","shortName":"BFSS threshold bound state","problemNumber":null,"field":"Supersymmetric matrix quantum mechanics; spectral theory","fieldGroup":"Mathematical physics","statement":"The $\\mathrm{SU}(N)$ BFSS model is the maximally supersymmetric matrix quantum mechanics with nine Hermitian $N\\times N$ matrices and sixteen fermionic partners, obtained by reducing ten-dimensional super Yang-Mills to one dimension. De Wit, Luscher and Nicolai (1989) showed its spectrum is $[0,\\infty)$. Witten (1996) predicted that, after removing the center of mass, there is exactly one normalizable zero-energy state for every $N$ (the D0-brane bound state), a prediction central to the BFSS matrix-theory conjecture. Index computations (Yi, Sethi-Stern, Moore-Nekrasov-Shatashvili) supported it, but a rigorous proof of existence and uniqueness was open. For every $N\\ge2$, does the relative $\\mathrm{SU}(N)$ Hamiltonian have a kernel of dimension exactly one?","posedBy":"E. Witten, Bound states of strings and p-branes, Nuclear Physics B (1996), Section 4.2; central to Banks, Fischler, Shenker and Susskind (1997)","yearPosed":1996,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every integer $N\\ge2$, the relative undeformed $\\mathrm{SU}(N)$ BFSS Hamiltonian (closure of $\\frac1{16}\\sum_\\alpha\\|Q_\\alpha\\Psi\\|^2$ on gauge-invariant states in $L^2(\\mathbb R^{9(N^2-1)})$ tensor a Clifford module) has a one-dimensional kernel, consisting of $\\mathrm{Spin}(9)$-invariant even states. This gives both existence and uniqueness of the threshold bound state at each finite $N$. Constants depend on $N$; nothing is claimed about large-$N$ limits or the matrix-theory conjecture itself.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Witten's prediction as cited. Theorem 1.1 states $\\dim\\ker H_N=1$ for every $N\\ge2$, with $H_N$ the self-adjoint operator of the closed supercharge quadratic form on the gauge-invariant smooth compactly supported core; the kernel is $\\mathrm{Spin}(9)$-invariant and even. The result is for that specific operator realization; the paper contrasts it with other realizations (Boulton et al.). The proof passes from a BMN mass-deformed Fredholm index to the undeformed kernel with cluster estimates. Not refereed here. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The existence and uniqueness of the D0-brane threshold bound state is a well-known open problem at the interface of string theory and rigorous spectral theory, open for three decades despite strong index evidence. Placed level with the XY-model logarithmic correction (50), below the ionization conjecture (Simon problem 9, 55) and above the Sandier-Serfaty triangular-lattice question (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The unique threshold bound state of the SU(N) BFSS model","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Positive eigenvalues of the relative SU(2) BFSS Hamiltonian (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"chen-calabi-flow-long-time-existence","name":"Chen's conjecture on smooth long-time existence of the Calabi flow","shortName":"Chen's Calabi flow long-time existence conjecture","problemNumber":null,"field":"Kahler geometry, geometric flows","fieldGroup":"Geometry & topology","statement":"Calabi introduced the Calabi flow $\\partial_t\\phi=S(\\omega_\\phi)-\\bar S$, $\\omega_\\phi=\\omega_0+i\\partial\\bar\\partial\\phi$, to search for extremal and constant-scalar-curvature Kahler metrics in a fixed Kahler class of a compact Kahler manifold. Short-time existence and stability near cscK metrics were proved by Chen and He, and global existence is known on Riemann surfaces. Chen conjectured that the flow exists smoothly for all positive time from arbitrary smooth Kahler initial data (He-Zeng 2021, Conjecture 1.1); Chen and Cheng emphasized the case of classes containing a cscK metric. Does the smooth Calabi flow starting from any smooth Kahler metric on a compact Kahler manifold exist for all time?","posedBy":"Xiuxiong Chen","yearPosed":null,"ageNote":"The manuscript cites the conjecture as stated in He and Zeng, The Calabi flow with rough initial data, IMRN 2021, Conjecture 1.1, and Chen-Cheng, JAMS 34 (2021); it does not give the year Chen first stated it, so yearPosed is left blank.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a smooth $U(10)$-invariant Kahler form $\\omega_0$ on $\\mathbb{CP}^{10}$ in the Fubini-Study class whose maximal smooth Calabi flow exists only on $[0,T_*)$ with $0 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 24, 2026). Its finite shooting step is certified by an exact integer interval-arithmetic Python program reproduced in the appendix.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against the conjecture as the manuscript cites it; Theorem 1.1 is a counterexample to the conjecture as stated. The proof (radial momentum reduction, a self-similar shrinking profile of the flat limit, parabolic correction on the compact manifold) was not refereed, and the exact-arithmetic certificate in the appendix was not re-run here. No Lean formalization accompanies this manuscript.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Chen's long-time existence conjecture is the central open question about the Calabi flow, well known in Kahler geometry but specialist outside it. A counterexample in a class containing a cscK metric is striking within the field. Placed below the uniqueness of tangent flows for mean curvature flow (40), above the Euclidean-complete affine Bernstein problem (32) and the Mezard-Parisi formula (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A finite-time singularity of Calabi flow on projective space","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"He-Zeng, The Calabi flow with rough initial data (IMRN 2021), Conjecture 1.1","url":"https://doi.org/10.1093/imrn/rnz050","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"qi-rigidity-virtually-polycyclic-groups","name":"Quasi-isometric rigidity of virtually polycyclic groups (the Eskin-Fisher-Whyte lattice-recognition conjecture)","shortName":"QI recognition of polycyclic groups","problemNumber":null,"field":"Geometric group theory: quasi-isometric rigidity","fieldGroup":"Algebra","statement":"Gromov's polynomial growth theorem implies that a finitely generated group quasi-isometric to a virtually nilpotent group is virtually nilpotent. For solvable groups of exponential growth, quasi-isometric rigidity was known only for special classes: Sol and lamplighters (Eskin-Fisher-Whyte), nondegenerate abelian-by-abelian groups (Peng), SOL-like groups (Dymarz; Dymarz-Fisher-Xie), and under stronger hypotheses such as commability (Le Boudec). Shalom showed such a group virtually has infinite abelianization. Eskin, Fisher and Whyte (2007) conjectured that a finitely generated group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a lattice in a possibly different such Lie group; equivalently: if a finitely generated group $H$ is quasi-isometric to a finitely generated virtually polycyclic group, must $H$ be virtually polycyclic?","posedBy":"A. Eskin, D. Fisher and K. Whyte, Quasi-isometries and rigidity of solvable groups, Pure Appl. Math. Q. 3 (2007), Conjecture 1.2 and Remark (3), p. 929","yearPosed":2007,"ageNote":"The question of quasi-isometric rigidity for polycyclic and solvable groups is older (Gromov's program; Farb-Mosher around 2000); the manuscript cites Eskin-Fisher-Whyte 2007 for the precise conjecture answered.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic; Corollary 1.2: a finitely generated group quasi-isometric to a lattice in a connected simply connected solvable Lie group has a finite-index subgroup that is a uniform lattice in some (possibly different) such Lie group. The key input is a uniform bounded-height theorem: every $(K,C)$ self quasi-isometry of a unimodular real-triangulable solvable Lie group preserves the exponential height map up to bounded error and one of finitely many linear symmetries. Amenability then gives elementary amenability, and finiteness and duality give virtual polycyclicity. It does not classify solvable groups up to quasi-isometry or describe their quasi-isometry groups in general.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1, Corollary 1.2 and Theorem 1.4 of the TeX source, read against Eskin-Fisher-Whyte's Conjecture 1.2 as the manuscript cites it; the proof was not refereed. Lean: the Comparator challenge PolycyclicRecognition (OAI.PolycyclicRecognition.group_recognition, lattice_recognition, main_uniform_bounded_height; solution module OAI/GroupTheory/PolycyclicRecognition/Recognition.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Its statement was read here: group_recognition says that for finitely generated groups P and J joined by a coarse equivalence (maps with finite-set control of differences, the group form of a quasi-isometry), virtual polycyclicity of P gives virtual polycyclicity of J and a finite-index subgroup isomorphic to a uniform lattice in a simply connected solvable Lie group. This is the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"Extending Gromov's polynomial growth theorem from nilpotent to all polycyclic groups has been a central goal of geometric group theory since the 1990s; the Sol case alone was an Annals paper. Scored above the question whether every hyperbolic group is CAT(0) (45), below the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials (52) and Cannon's conjecture (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Quasi-isometric recognition of virtually polycyclic groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/GroupTheory/PolycyclicRecognition/Recognition.lean (group_recognition)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/PolycyclicRecognition/Recognition.lean","kind":"lean-proof"},{"label":"Eskin, Fisher, Whyte, Quasi-isometries and rigidity of solvable groups (PAMQ 2007)","url":"https://doi.org/10.4310/PAMQ.2007.v3.n4.a3","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"two-point-chowla-conjecture","name":"The two-point Chowla conjecture (ordinary averages)","shortName":"Two-point Chowla conjecture","problemNumber":null,"field":"Analytic number theory; correlations of multiplicative functions","fieldGroup":"Number theory","statement":"Let $\\lambda(n)=(-1)^{\\Omega(n)}$ be the Liouville function. Chowla's conjecture predicts that $\\sum_{n\\le X}\\lambda(n+h_1)\\cdots\\lambda(n+h_k)=o(X)$ for every fixed set of distinct shifts. Its two-point case asks whether the parities of $\\Omega(n)$ and $\\Omega(n+h)$ become uncorrelated. Matomaki and Radziwill obtained a bound $(1-\\delta(h))X$, Matomaki-Radziwill-Tao proved cancellation on average over shifts, and Tao proved the logarithmically averaged version, with quantitative log-averaged savings by Helfgott-Radziwill and Pilatte. Does $\\sum_{n\\le X}\\lambda(n)\\lambda(n+h)=o(X)$ hold for every fixed $h\\ge1$?","posedBy":"Sarvadaman Chowla, The Riemann Hypothesis and Hilbert's Tenth Problem (Gordon and Breach, 1965)","yearPosed":1965,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"There is an absolute $c>0$ such that for all fixed integers $a_1,a_2\\ge1$, $b_1,b_2\\ge0$ with $a_1b_2\\ne a_2b_1$, $\\left|\\sum_{n\\le X}\\lambda(a_1n+b_1)\\lambda(a_2n+b_2)\\right|\\le C_{a,b}X/(\\log X)^c$ for every real $X\\ge3$. With $a_1=a_2=1$, $b_1=0$, $b_2=h$ this is the ordinary two-point Chowla conjecture with a power-of-log saving. Constants are ineffective, and coefficients may not grow with $X$. Nothing is proved for three or more points.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single manuscript, dated September 24, 2026. The release also publishes an abridged summary of the model's reasoning for this family (reasoning_traces/ordinary-two-point-correlations.pdf).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the two-point case of Chowla's conjecture with ordinary averages. The challenges are not in the formalization catalogue (lean/formalization.yaml); lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.json and OrdinaryElliott.json exist with solution modules OAI.NumberTheory.TwoPointCorrelations.FinalMain and OAI.NumberTheory.OrdinaryCorrelations.Elliott.Main present at the pinned commit, permitted axioms propext, Quot.sound and Classical.choice. The statement OAI.OrdinaryTwoPointCorrelations.liouville_log_saving was read: an absolute c > 0 such that for natural a1, a2 >= 1, b1, b2 with a1 b2 != a2 b1 there is C > 0 with |sum_{n <= X} lambda(a1 n + b1) lambda(a2 n + b2)| <= C X / (log X)^c for all real X >= 3. That covers the headline, including every shift h >= 1. Not rebuilt here. The paper says the constants need not be effective and no uniformity in the forms is claimed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Chowla's conjecture is the parity-problem analogue of the prime k-tuples conjecture, and its two-point case has been the main target of a decade of breakthroughs (Matomaki-Radziwill, Tao's logarithmic version, Helfgott-Radziwill). The ordinary-average binary case is the central open instance, scored below the full conjecture. Above Vinogradov's least nonresidue conjecture (45) and the Liouville Goldbach question (22), below Artin's primitive root partial (63) and Landau-Siegel zeros (72).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf","sourceName":"OpenAI math release: Ordinary two-point correlations of multiplicative functions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI/NumberTheory/TwoPointCorrelations/FinalMain.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/TwoPointCorrelations/FinalMain.lean","kind":"lean-proof"},{"label":"Lean proof: OAI/NumberTheory/OrdinaryCorrelations/Elliott/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/OrdinaryCorrelations/Elliott/Main.lean","kind":"lean-proof"},{"label":"Abridged summary of the model's reasoning (release reasoning_traces)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/reasoning_traces/ordinary-two-point-correlations.pdf","kind":"other"},{"label":"Matomaki, Radziwill, Tao, An averaged form of Chowla's conjecture (Algebra Number Theory 2015)","url":"https://doi.org/10.2140/ant.2015.9.2167","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"courtade-kumar-conjecture","name":"The Courtade-Kumar conjecture: dictators are the most informative Boolean functions","shortName":"Courtade-Kumar conjecture","problemNumber":null,"field":"Boolean functions; information theory","fieldGroup":"Probability & statistics","statement":"Let $X$ be uniform on $\\{0,1\\}^n$ and let $Y$ be $X$ passed through independent binary symmetric channels with crossover probability $\\varepsilon\\in[0,1/2]$. Kumar and Courtade (2013, 2014) conjectured that for every Boolean function $f:\\{0,1\\}^n\\to\\{0,1\\}$, $I(f(X);Y)\\le1-h_2(\\varepsilon)$ bits, the value attained by a dictator $f(x)=x_i$. Partial results covered high-noise ranges (Ordentlich-Shayevitz-Weinstein, Samorodnitsky, Yu), balanced functions in ranges, and coordinatewise sums (Courtade-Kumar; Javanmard-Woodruff). Does every Boolean function satisfy $I(f(X);Y)\\le1-h_2(\\varepsilon)$ for every $n$ and every $\\varepsilon$?","posedBy":"Gowtham R. Kumar and Thomas A. Courtade, Which Boolean functions are most informative? (ISIT 2013); Courtade and Kumar (IEEE Trans. IT 2014)","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every $n$, every soft channel $u:\\{\\pm1\\}^n\\to[-1,1]$ and $\\rho\\in[-1,1]$, $I(T_\\rho u)\\le\\psi(|\\rho|\\psi^{-1}(I(u)))$, with soft coordinates attaining equality; Corollary 1.2 gives $I(f(X);Y)\\le1-h_2(\\varepsilon)$ bits for Boolean $f$, plus a strictly better bound when $f$ is biased; Theorem 1.3 is a mean-dependent entropy-production inequality. It does NOT treat non-uniform inputs, asymmetric noise, or multi-bit summaries. Independent proofs of the Boolean inequality were announced in September 2026 by Chen-Gohari-Javanmard-Lin-Mirrokni-Nair-Woodruff, Ky-Tran and Mahdavifar-Beirami.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The principal manuscript follows the local-to-global differential-equation program of Chen-Gohari-Nair, credited to them; the Hellinger companion's source-lineage file says its arithmetic certificates are part of the proof.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against the conjecture: the Boolean specialization gives $I_2(f(X);Y)\\le1-h_2(\\varepsilon)$, with dictators attaining it. formalization.yaml lists ComparatorChallenges/CourtadeKumar.json, declaration OAI.LeanBlast.CourtadeKumar.courtadeKumarAndAttainment, file OAI/InformationTheory/BooleanNoise/Main.lean. The statement was read here: for $n\\ge1$, $0\\le\\varepsilon\\le1/2$ and every $f$ on the cube, the mutual information (defined from the bit-flip kernel, in bits) is at most $1-h(\\varepsilon)$, and dictators and anti-dictators attain it. That is exactly the conjecture. A second challenge, SoftChannel204.json (OAI.SoftChannel204.fullMain, solution module present, not in formalization.yaml, found via lean/docs/119.md), states the soft-channel Theorem 1.1, the bias-refined bound and the entropy-production inequality; read, not rebuilt. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The manuscript itself cites four other recent preprints announcing proofs, one dated 21 September 2026, before this one.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The most informative Boolean function conjecture was one of the best known open problems linking information theory and the analysis of Boolean functions, attacked steadily since 2013. Above the catalog's Courtade and Kumar's Coordinate-wise Mutual Information Question (22) and Talagrand's convolution conjecture (37), level with the Petersen Coloring Conjecture (40), below the 3SUM hypothesis (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/main.pdf","sourceName":"OpenAI math release: Sharp binary-information contraction on the discrete cube","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Hellinger contraction with arbitrary Boolean output bias","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/main.pdf","kind":"paper"},{"label":"Lean: Courtade-Kumar inequality and attainment (formalization.yaml file)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/InformationTheory/BooleanNoise/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Courtade-Kumar inequality and attainment","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CourtadeKumar.lean","kind":"lean-proof"},{"label":"Lean comparator statement: soft-channel contraction and entropy production","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SoftChannel204.lean","kind":"lean-proof"},{"label":"Courtade and Kumar (2014), Which Boolean functions maximize mutual information on noisy inputs?","url":"https://doi.org/10.1109/TIT.2014.2326877","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generalized-lax-conjecture","name":"The generalized Lax conjecture: is every hyperbolicity cone spectrahedral?","shortName":"Generalized Lax conjecture","problemNumber":null,"field":"Real algebraic geometry; hyperbolic polynomials, spectrahedra","fieldGroup":"Algebra","statement":"A homogeneous real polynomial $p$ is hyperbolic with respect to $e$ if $p(e)\\ne0$ and $t\\mapsto p(te-x)$ has only real roots for every $x$; its closed hyperbolicity cone is the set of $x$ for which all these roots are nonnegative, a convex cone by Garding. Lax (1958) asked about three variables, and Helton-Vinnikov with Lewis-Parrilo-Ramana showed that every ternary hyperbolic polynomial has a definite determinantal representation. Branden showed that in more variables a hyperbolic polynomial and its powers can fail to be determinantal, which leaves the geometric question about the cone. Is every hyperbolicity cone spectrahedral, that is, of the form $\\{x: \\sum_i x_iA_i\\succeq0\\}$ for finitely many real symmetric matrices $A_i$?","posedBy":"Generalization of P. D. Lax's 1958 three-variable conjecture; the manuscripts do not name who first stated the general form","yearPosed":null,"ageNote":"The manuscripts trace the problem to Lax (1958) and to Helton-Vinnikov (2007) and Lewis-Parrilo-Ramana (2005) for the solved ternary case, but do not attribute the general higher-dimensional conjecture to anyone, so no year is given.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (September 24): the degree-20 polynomial $p(X,Z,y)=\\det((\\det X)Z-\\Phi_y(\\mathrm{adj}X))$ on $\\mathrm{Sym}_4\\times\\mathrm{Sym}_4\\times\\mathbb R^3$ is hyperbolic with respect to $(I_4,I_4,0)$, and its closed hyperbolicity cone equals $\\{L\\succeq0\\}$ for no finite real linear pencil $L$. Corollary: the degree-16 quotient $q=p/\\det X$ has the same cone. The polynomial is not strictly hyperbolic, consistent with the Kummer-Netzer theorem for strictly hyperbolic polynomials. This cone does have a semidefinite lift with auxiliary variables (companion, October 5); the stronger statement that some hyperbolicity cone has no lift at all is the separate Projected Lax entry.","claimIssueNote":"This result conflicts with a public claim. Gonzalez Nevado's arXiv:2601.12267v1 claims a positive solution of the generalized Lax conjecture; the release's appendix argues that its Theorem 52 is false as stated. Until the conflict is settled in public, the entry is Contested.","solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family's later manuscript (October 5, 2026) disproves the stronger Projected Lax Conjecture by a different, nonexplicit construction, and a third (October 5) builds an exact semidefinite lift of the explicit cone given here.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the September 24 manuscript was read against the conjecture as stated there; it gives an explicit polynomial $p(X,Z,y)=\\det((\\det X)Z-\\Phi_y(\\mathrm{adj}X))$ of degree 20 in 23 variables, built from the Choi-Lam biquadratic form, whose cone is not cut out by any finite homogeneous symmetric pencil (the abstract's degree 16 refers to the reduced quotient $p/\\det X$, same cone). The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator HyperbolicCones, declaration OAI.Paper256.main_result). ComparatorChallenges/HyperbolicCones.lean was read here: it defines this polynomial on $\\mathrm{Sym}_4\\times\\mathrm{Sym}_4\\times\\mathbb R^3$, asserts dimension 23, homogeneity of degree 20, $p(e)=1$, real-rootedness along every line, and that for every $N>0$ and every linear map $L$ to $N\\times N$ symmetric matrices the cone differs from $\\{x:L(x)\\succeq0\\}$. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript's appendix argues that Theorem 52 of Gonzalez Nevado's arXiv 2601.12267v1, which claims a positive solution, is false as stated. Listed as Contested because of the conflicting claim described in the claim issue.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The central open question on hyperbolic polynomials since Helton-Vinnikov, well known in real algebraic geometry and convex optimization but not beyond. Placed level with Wall's manifold-realization question (42), above the Artin groups CAT(0) conjecture (40) and Tingley's problem (38), and below Brennan's conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf","sourceName":"OpenAI math release: A nonspectrahedral hyperbolicity cone","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Paper256.main_result)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/HyperbolicCones/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: HyperbolicCones.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HyperbolicCones.lean","kind":"lean-proof"},{"label":"Companion: Hyperbolicity Cones Without Semidefinite Lifts","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf","kind":"paper"},{"label":"Companion: An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"planar-fpp-limit-shape-exponential","name":"Strict convexity and differentiability of the planar first-passage limit shape for exponential weights","shortName":"Exponential FPP limit shape regularity","problemNumber":null,"field":"First-passage percolation; limit shapes","fieldGroup":"Probability & statistics","statement":"For i.i.d. edge weights on $\\mathbb Z^2$, the Cox-Durrett shape theorem gives a norm $\\mu$ with $T(0,\\lfloor tv\\rfloor)/t\\to\\mu(v)$ and a convex limit shape $\\mathcal B=\\{\\mu\\le1\\}$. It is conjectured that for continuous weight laws $\\mathcal B$ is strictly convex (no flat faces) and that $\\mu$ is differentiable away from the origin (no corners); see the survey of Auffinger, Damron and Hanson. Neither property was known for any continuous law, including the exponential law, where the model coincides with the Eden-Richardson growth model. Partial results covered laws with an atom at the minimum (Marchand; Auffinger-Damron) and sufficient conditions (Lalley). For i.i.d. exponential edge weights, is the limit shape strictly convex with a $C^1$ boundary?","posedBy":"Open questions surveyed by Auffinger, Damron and Hanson (50 years of first-passage percolation, 2017)","yearPosed":2017,"ageNote":"The manuscript cites only the Auffinger-Damron-Hanson survey (2017) for these conjectures; the questions are older (they are folklore since the shape theorems of Richardson 1973 and Cox-Durrett 1981), but the manuscript does not give an earlier posing.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for i.i.d. exponential weights of any rate, $\\mu$ is Frechet differentiable on $\\mathbb R^2\\setminus\\{0\\}$ and $\\mathcal B$ is strictly convex, so $\\partial\\mathcal B$ is a $C^1$ curve with a unique supporting line at each point. Theorem 1.2: differentiability (not strict convexity) holds for every Gamma law with positive shape and rate. Consequences for directional geodesics and Busemann functions follow from Damron-Hanson. It does not treat general continuous laws, higher dimensions, or curvature of the boundary.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 were read against the conjectures; they settle the exponential law (strict convexity and differentiability) and differentiability for Gamma laws, special cases of the general conjectures, hence Partial. The proof was not refereed. lean/formalization.yaml lists OAI.PlanarFPP.manuscriptMain (comparator PlanarFirstPassage). Its statement was read: for i.i.d. Exp(1) weights there is a norm that is the almost-sure time constant, differentiable away from 0, with unique normalized supporting functionals, a $C^1$ unit sphere, a unique supporting line and a $C^1$ chart at every boundary point. A second challenge, GammaPassage.json (OAI.GammaFPP.gamma_differentiability, solution module OAI.Probability.GammaPassage.Differentiability, present at the pinned commit), is not in the formalization catalogue; read here, it states the same differentiability for every Gamma(shape, rate) law. Strict convexity is not formalized. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"Regularity of the FPP limit shape is a long-standing open problem, unknown for any continuous law before this; the exponential case is a special case of the general conjecture. Placed below No bigeodesics in planar FPP (45, same family) and Local smooth isometric immersion of surfaces in three-space (45), and above Polynomial-Factor Hardness for the Closest Vector Problem (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf","sourceName":"OpenAI math release: Strict convexity and differentiability of the planar exponential first-passage limit shape","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.PlanarFPP.manuscriptMain)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/FirstPassage/BoundaryCurve.lean","kind":"lean-proof"},{"label":"Comparator statement: PlanarFirstPassage.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PlanarFirstPassage.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.GammaFPP.gamma_differentiability)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/GammaPassage/Differentiability.lean","kind":"lean-proof"},{"label":"Comparator statement: GammaPassage.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/GammaPassage.lean","kind":"lean-proof"},{"label":"Companion: No bigeodesics in planar first-passage percolation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"haldane-gap-spin-one","name":"The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain","shortName":"Spin-one Haldane gap","problemNumber":null,"field":"Quantum spin chains; spectral gaps","fieldGroup":"Mathematical physics","statement":"For the spin-one antiferromagnetic Heisenberg chain $H_L=\\sum_j \\mathbf S_j\\cdot\\mathbf S_{j+1}$ with spin-one matrices normalized by $\\sum_\\alpha (S^\\alpha)^2=2$, Haldane predicted in 1981 (published 1983) that integer-spin chains, unlike half-odd-integer ones, have a unique ground state separated from the rest of the spectrum by a gap that stays positive in the thermodynamic limit. AKLT proved such a gap for a modified model with a biquadratic term, Yarotsky for small perturbations of it, and numerics put the Heisenberg gap near $0.41$, but no proof existed for the pure bilinear model. Writing $\\gamma_L$ for the gap above the ground energy on a periodic ring of even length $L$: is $\\liminf_{L\\to\\infty}\\gamma_L>0$?","posedBy":"F. D. M. Haldane (1981 ILL preprint; Phys. Rev. Lett. and Phys. Lett. A, 1983)","yearPosed":1981,"ageNote":"The manuscript dates the prediction to Haldane's 1981 preprint, published in 1983. It was a physics prediction; the uniform finite-volume gap is the standard rigorous formulation, and Tasaki (2025) posed the related odd open-chain boundary-field version.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that the pure bilinear spin-one Heisenberg chain on even periodic rings has a unique ground state for $L\\ge60$ and a gap bounded below by explicit constants, so $\\Delta_1\\ge\\log 20/784>0$, with a corresponding lower bound for local excitations in subsequential thermodynamic limits. The companion proves Tasaki's boundary-field gap for odd open chains at $h=3/5$ and the nontrivial index $-1$. The constants are far below the numerical value of about $0.41$. It does NOT treat odd periodic rings, spin above one, or the bilinear-biquadratic family away from the Heisenberg point.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. Both manuscripts are credited to OpenAI with no human author named. Each ships finite certificates (thermal trace enclosures and matrix-product trial energies) with a Python verifier in its verification folder.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem: it claims a unique ground state and $\\gamma_L>\\frac4{105}\\log\\frac{80}{79}$ for every even $L\\ge60$, and $\\gamma_L>\\log 20/784$ for even $L\\ge2304$, so the even-periodic gap is uniformly positive. The manuscript calls this the positive even-periodic formulation; odd periodic rings are not covered. The proof rests on finite rigorous computations (interval enclosures of matrix-exponential traces and explicit integer matrix-product trial states) fed into an analytic bootstrap; the verifier was not run here. No Lean formalization. The companion was read at the level of its main theorem: a gap above $\\log 10/392$ for odd open chains of $2L+1$ sites, $L\\ge960$, with endpoint field $h=3/5$, which is Tasaki's Assumption 2 and gives Tasaki index $-1$.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"The best-known open problem about quantum spin chains, a forty-year-old prediction confirmed numerically and experimentally but never proved for the Heisenberg model itself. Placed below Simon's Problem 1 on Anderson extended states at 65, above two-dimensional Anderson localization at 55, and level with Cannon's Conjecture at 62.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The periodic spin-one Haldane gap","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A boundary-field gap for the spin-one Heisenberg chain (Tasaki's Assumption 2, index -1)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf","kind":"paper"},{"label":"Finite-certificate verifier for the periodic gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/verification/verify.py","kind":"other"},{"label":"Haldane (1983), Phys. Rev. Lett. 50, 1153","url":"https://doi.org/10.1103/PhysRevLett.50.1153","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tensor-square-conjecture-symmetric-groups","name":"The tensor square conjecture for symmetric groups: some irreducible representation of S_n has a tensor square containing every irreducible","shortName":"Tensor square conjecture for S_n","problemNumber":null,"field":"Representation theory of symmetric groups; Kronecker coefficients","fieldGroup":"Algebra","statement":"For partitions of $n$ let $g(\\lambda,\\mu,\\nu)$ be the Kronecker coefficients of $S_n$. Heide, Saxl, Tiep and Zalesski showed that for most finite simple groups of Lie type the Steinberg character has a tensor square containing every irreducible character. For symmetric groups, Pak, Panova and Vallejo formulated the tensor square conjecture (Conjecture 1.1): for every $n\\ge3$ other than $4$ and $9$ there is $\\lambda\\vdash n$ with $g(\\lambda,\\lambda,\\nu)>0$ for all $\\nu\\vdash n$; it fails for $n=2,4,9$. Saxl's conjecture would give it at triangular $n$; elsewhere only partial and asymptotic results were known (Luo and Sellke found irreducibles whose fourth tensor power contains everything). Does every $S_n$ with $n\\notin\\{2,4,9\\}$ have an irreducible representation whose tensor square contains every irreducible representation?","posedBy":"Igor Pak, Greta Panova and Ernesto Vallejo","yearPosed":2013,"ageNote":"Pak, Panova and Vallejo, arXiv 1304.0738 (2013), published in Adv. Math. 288 (2016), Conjecture 1.1, as cited by the manuscript; it follows Heide-Saxl-Tiep-Zalesski's Steinberg-square theorem for groups of Lie type.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every positive integer $n\\notin\\{2,4,9\\}$ there is $\\lambda\\vdash n$, necessarily self-conjugate, with $g(\\lambda,\\lambda,\\nu)>0$ for every $\\nu\\vdash n$. Corollary 1.2 transfers this through Letellier's results to unipotent characters of $GL_n(\\mathbb F_q)$ for every prime power $q$. The construction combines the cyclic Saxl theorem with band, balance and capacity arguments, and a finite verification for small degrees. Not shown: an explicit closed-form family of such $\\lambda$ at nontriangular degrees beyond the construction, or bounds on the coefficients.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript is 'Universal Tensor Squares for Symmetric Groups' (September 24, 2026). It uses the cyclic form of Saxl's conjecture from the companion manuscript and a finite computer verification (C++ and Python) for small degrees, shipped in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Universal Tensor Squares for Symmetric Groups' was read against Pak-Panova-Vallejo's Conjecture 1.1; it is the conjecture, including nontriangular degrees. Lean: lean/ComparatorChallenges/UniversalTensorSquares.json exists with solution module OAI.RepresentationTheory.UniversalSquare.Main present at the pinned commit; it is not in the formalization catalogue (formalization.yaml). The statement read here asserts, for every $n>0$ with $n\\ne2,4,9$, a Young diagram $\\lambda$ of size $n$ with positive Kronecker multiplicity against every $\\nu$, irreducibility of its Specht representation, and an injective intertwiner from every finite-dimensional irreducible complex representation of $S_n$ into its tensor square. That states the headline. Not rebuilt here, and the release's finite character computations were not run.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A natural and well-studied question on Kronecker coefficients, formulated by Pak, Panova and Vallejo in 2013 beside Saxl's conjecture, with asymptotic partial results; somewhat less prominent than Saxl's staircase version. Placed below Saxl's conjecture (32) and Dean's conjecture for k=5 (32), level with the forcing conjecture for bipartite graphs (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf","sourceName":"OpenAI math release: Universal Tensor Squares for Symmetric Groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion manuscript: A Cyclic Polytabloid Proof of Saxl's Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean proof: OAI/RepresentationTheory/UniversalSquare/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RepresentationTheory/UniversalSquare/Main.lean","kind":"lean-proof"},{"label":"Lean statement: ComparatorChallenges/UniversalTensorSquares.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/UniversalTensorSquares.lean","kind":"lean-proof"},{"label":"Pak, Panova and Vallejo, Kronecker products, characters, partitions, and the tensor square conjectures","url":"https://arxiv.org/abs/1304.0738","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"squarefree-values-of-quartics","name":"Squarefree values of irreducible quartic polynomials, and (d-2)-power-free values in every degree d at least 4","shortName":"Squarefree values of quartics","problemNumber":null,"field":"Analytic number theory, sieve methods","fieldGroup":"Number theory","statement":"Let $f\\in\\mathbb Z[x]$ be irreducible of degree $d$ and $k\\ge2$. If no prime $k$th power divides every value of $f$, one expects $f(n)$ to be $k$-free for a proportion $c_{f,k}=\\prod_p(1-\\rho_f(p^k)/p^k)>0$ of $n\\le X$, where $\\rho_f(q)$ counts roots mod $q$. Ricci proved this for $k\\ge d$; Erdos (1953) and Hooley (1967) handled $k=d-1$, which includes squarefree values of cubics; Nair, Heath-Brown and Browning reached $k=d-2$ only for $d\\ge9$. Erdos singled out the squarefreeness of $n^4+2$ in 1953 and returned to the $k=d-2$ obstacle in 1965. Does every irreducible integer quartic with no fixed prime-square divisor take squarefree values with the predicted density, and more generally does the $(d-2)$-free density hold in every degree $d\\ge4$?","posedBy":"Paul Erdos (the case n^4 + 2 and the exponent d - 2 barrier)","yearPosed":1953,"ageNote":"Erdos, Arithmetical properties of polynomials, J. London Math. Soc. 28 (1953), p. 425, and Some recent advances and current problems in number theory (1965). The general squarefree-values conjecture is folklore; Granville derived it from abc.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $f\\in\\mathbb Z[x]$ irreducible over $\\mathbb Q$ of degree $4\\le d\\le8$ with no fixed prime $(d-2)$th-power divisor, the number of $n\\le X$ with $f(n)$ $(d-2)$-free is $c_{f,d-2}X+o_f(X)$ with the Euler-product constant, which is positive; for $d=4$ this is squarefree values of quartics, e.g. $n^4+2$ and $n^4+1$. With Browning's theorem for $d\\ge9$ (Corollary 1.2) the $(d-2)$-free density holds for all $d\\ge4$; Corollary 1.3 treats separable products. Not shown: a power-saving error term, uniformity in $f$, or squarefree values in degree five and above. Priority: Carella's preprint claims $n^4+1$ and $n^4+2$, and Zapata Ceballos-Jalalvand claim positive density for a class including $n^4+1$; the manuscript uses neither.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The single manuscript (September 24, 2026) is the whole family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and Corollaries 1.2-1.3 were read against the problem as the manuscript states it from Erdos's papers. The proof (number-field factorization, determinant estimates with adaptive auxiliary primes, low-degree geometry and an exact parameter certificate) was not refereed. lean/formalization.yaml lists comparator PowerFreeValues, declaration OAI.QuarticPowerFree.allDegrees (file OAI/NumberTheory/PowerFree/Main.lean). Its statement was read here: for $f\\in\\mathbb Z[x]$ irreducible over $\\mathbb Q$ with $\\deg f\\ge4$ and $\\rho_f(p^{d-2})2$, and $\\sum_k\\sum_{\\mathbf I}|L_{\\mathbf I}|\\le40\\prod_v\\|F_v\\|_3$ for Thiele's dyadic model with no structural assumption. Only the symmetric exponent point is treated; other triples with $1/p+1/q+1/r=1$ and the four-linear simplex form are not.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. Three manuscripts form the family: the principal maximal-estimate paper (September 24) and two October 5 companions, one strengthening it to variation bounds and one treating the dyadic model.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of the TeX sources of all three manuscripts, and Thiele's Problem 13 read in arXiv:1701.06637v1, Section 9; the proofs were not refereed. Lean-checked on the Comparator challenge TriangularHilbert, listed in the release's formalization catalogue (OAI.TriangularHilbert.main_estimate, OAI/Analysis/TriangularHilbert/Main.lean). Its statement: one constant $C$ such that for all complex $F,G$ in $L^3(\\mathbb R^2)$, almost every point has every annular truncation integrable, the maximal function over all finite truncations is a.e. measurable, and its $L^{3/2}$ norm is at most $C\\|F\\|_3\\|G\\|_3$. That is Theorem 1.1 of the principal manuscript, the bilinear maximal estimate. The step to the scalar trilinear form of Problem 13 (duality with a third $L^3$ function and a change of variables), the principal values, the variation bound and the dyadic bound are not formalized. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here. Listed as Unreviewed rather than Lean-checked because its formal statement covers only the bilinear maximal bound.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":46,"significanceNote":"A well-known open problem of multilinear harmonic analysis, singled out in a 2017 problem list and the target of a decade of cancellation estimates. Scored with the Kakeya maximal conjecture in three dimensions (48) and Brennan's conjecture (45), below the local smoothing conjecture in three dimensions (63), whose standing in the field is broader.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The maximal triangular Hilbert transform at the symmetric point","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Annular variation of the triangular Hilbert transform at the symmetric point","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf","kind":"paper"},{"label":"Companion: An L3 bound for the dyadic triangular Hilbert form","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-L3-bound-for-the-dyadic-triangular-Hilbert-form-October-5-2026/dyadic-triangular-hilbert.pdf","kind":"paper"},{"label":"Lean: OAI/Analysis/TriangularHilbert/Main.lean (main_estimate)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/TriangularHilbert/Main.lean","kind":"lean-proof"},{"label":"Thiele, Problem 13 in Some problems in harmonic analysis (arXiv:1701.06637)","url":"https://arxiv.org/abs/1701.06637","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"termination-flips-lc-fourfolds","name":"Termination of flips for projective log canonical fourfolds with rational boundary","shortName":"Termination of lc fourfold flips","problemNumber":null,"field":"Birational geometry; minimal model program","fieldGroup":"Algebra","statement":"The Termination of Flips Conjecture of the minimal model program asserts that every sequence of flips (more generally, every program of $K_X+B$-negative divisorial contractions and flips) for a log canonical pair $(X,B)$ is finite. In dimension four it was known for terminal fourfolds (Kawamata), canonical pairs (Fujino), klt pairs with $-(K_X+B)$ numerically equivalent to an effective divisor (Alexeev-Hacon-Kawamata), pseudo-effective lc pairs (Chen-Tsakanikas, Moraga) and klt pairs with big boundary (Han-Liu-Zhuang). Does every minimal model program for a projective log canonical fourfold with rational boundary, over an algebraically closed field of characteristic zero, terminate for every choice of negative extremal rays, without pseudo-effectivity or bigness assumptions?","posedBy":"The Termination of Flips Conjecture of the minimal model program (Shokurov's difficulty method; the manuscript points to Chen-Tsakanikas for the statement)","yearPosed":null,"ageNote":"The manuscript cites the conjecture through the introduction of Chen-Tsakanikas (2023) and does not date it. It is usually traced to the 1980s minimal model program (Shokurov 1985; Kawamata-Matsuda-Matsuki 1987), but that dating is not given in the manuscript, so yearPosed is left empty.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a projective lc pair $(X,B)$ of dimension four with rational boundary over an algebraically closed field of characteristic zero, every negative extremal ray at every stage has its contraction and, when birational, a positive model; every permitted program has finitely many birational steps for every sequence of choices; and every maximal program ends at a nef adjoint or a Mori fibre space. Proved over $\\mathbb C$ by a weighted difficulty with a topological branch estimate, then transferred to all such fields. Not shown: real boundaries, generalized pairs in the projective setting, any dimension above four, or semiampleness of the nef endpoint without the release's log abundance result. The Kahler companions prove termination for generalized lc flips with projective small diagrams on globally Weil Q-factorial compact Kahler fourfolds, a four-dimensional rational form of Hacon-Xie's Conjecture 1.15.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. Four companion manuscripts (24 September and 5 October 2026) treat compact Kahler fourfolds: termination of generalized log canonical and generalized-canonical flips with projective small diagrams, and finite ordinary minimal model programs for klt Kahler fourfold pairs. The principal paper uses other release results (log abundance, complements) only for consequences at the endpoints.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the conjecture; it is the projective four-dimensional lc case with rational boundary, for every permitted program from the given model (divisorial contractions, flips and mixed steps), with existence of the required contractions and flips included and no $\\mathbb Q$-factoriality or pseudo-effectivity assumed. Real boundaries and dimensions five and up are not treated. The difficulty argument with a new branch estimate was not refereed here. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":34,"significanceNote":"Termination of flips is one of the central conjectures of birational geometry; this is its four-dimensional lc case with rational boundary, where much was already known (pseudo-effective and big-boundary cases). Scored as a partial well below the parent. Placed below The log abundance conjecture in characteristic zero (75), above The Campana-Peternell conjecture on Fano manifolds with nef tangent bundle in dimension six (28) and The Cone Theorem for Effective Fourfold Pairs in Characteristic p > 5 (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Termination for projective log canonical fourfolds with rational boundary","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion: Termination of generalized log canonical flips on compact Kahler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026/termination-generalized-lc-kahler-fourfolds.pdf","kind":"paper"},{"label":"Companion: Termination of generalized-canonical flips on compact Kahler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026/termination-generalized-terminal-flips-compact-kahler-fourfolds.pdf","kind":"paper"},{"label":"Companion: Finite ordinary minimal model programs on compact Kahler fourfolds (Oct 5)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: Finite ordinary minimal model programs on compact Kahler fourfolds (Sep 24)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Finite ordinary minimal model programs on compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-6-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Termination of generalized-canonical flips on compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-6-2026/termination-generalized-terminal-flips-compact-kahler-fourfolds.pdf","kind":"paper"},{"label":"Revised edition (October 7, 2026): Termination of generalized log canonical flips on compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-7-2026/termination-generalized-lc-kahler-fourfolds.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"local-isometric-immersion-surfaces-r3","name":"Local smooth isometric immersion of surfaces in three-space","shortName":"Local isometric immersion of smooth surfaces in R3","problemNumber":null,"field":"Differential geometry, isometric embedding","fieldGroup":"Geometry & topology","statement":"Given a smooth Riemannian metric $g$ on a surface, can a neighbourhood of each point be realised as a smooth surface in $\\mathbb R^3$, i.e. is there a smooth $F$ with $\\langle\\partial_iF,\\partial_jF\\rangle=g_{ij}$ near the point? The analytic case is the Janet-Cartan theorem (1926 to 1927); smooth local existence holds where the Gaussian curvature is nonzero, and Lin, Nakamura-Maeda, Han-Khuri and others treated nonnegative curvature and various degenerate zero sets. Nash-Kuiper gives $C^1$ realisations, and Pogorelov and Nadirashvili-Yuan gave $C^{2,1}$ metrics with no $C^2$ realisation. Does every smooth surface metric admit a smooth local isometric immersion into $\\mathbb R^3$?","posedBy":"Discussed by S.-T. Yau, Review of geometry and analysis (Asian J. Math., 2000, p. 236), and stated as Problem 1.9 in M. Ghomi, Open problems in geometry of curves and surfaces (2019)","yearPosed":2000,"ageNote":"Dated to Yau's 2000 discussion, the earliest statement the manuscript cites in the text; the question is older (the analytic case was settled by Janet and Cartan in 1926 to 1927, and the manuscript's bibliography also lists a Schoen-Yau 1994 problem list not cited in the text).","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem A: there is a $C^\\infty$ positive definite metric $g$ on $(-1,1)^2$ such that no open neighbourhood of the origin admits a $C^\\infty$ isometric immersion into $\\mathbb R^3$; the metric can have the full Taylor jet of the Euclidean metric at the origin, so local realisability is not decided by the jet. The curvature takes both signs. Not settled: the smooth local problem under $K\\ge0$ or $K\\le0$, finite-regularity versions, and realisation in $\\mathbb R^4$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The main theorem has a Lean formalization in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract and Theorem A of 'A Smooth Metric with No Local Isometric Immersion into Three-Space', read against the problem as quoted from Yau and Ghomi. The proof (a Darboux-equation obstruction on caps, an oscillating perturbation, Baire category and the Nadirashvili-Yuan boundary-saddle argument) was not refereed. Lean: formalization.yaml lists OAI.SmoothLocal.Geometry.exists_local_metric_without_local_immersion (lean/OAI/Geometry/IsometricImmersion/Main.lean, comparator ComparatorChallenges/IsometricImmersion.lean). Its statement was read: a smooth positive-definite matrix field g on the open square (-1,1)^2 such that for every open U inside the square containing 0, no smooth F : U -> R^3 has inner products of its derivatives equal to g. That is the headline. Not rebuilt here. A reader should know that a 2002 arXiv preprint of Nadirashvili and Yuan stated a stronger counterexample, which their 2008 paper and later accounts treat as leaving the question open; the manuscript notes this.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A classical question of surface theory that Yau highlighted and on which a line of analysts (Lin, Han, Khuri, Nadirashvili-Yuan) worked for decades. Placed level with Brennan's conjecture (45), above Banach's isometric conjecture (40), and below the smooth Caratheodory conjecture (55), which is better known outside the field.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A Smooth Metric with No Local Isometric Immersion into Three-Space","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: smooth metric with no local isometric immersion","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/IsometricImmersion/Main.lean","kind":"lean-proof"},{"label":"Ghomi, Open problems in geometry of curves and surfaces (Problem 1.9)","url":"https://ghomi.math.gatech.edu/Papers/op.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"euclidean-steinitz-bergstrom-conjecture","name":"The Euclidean Steinitz-Bergstrom conjecture: is the Euclidean Steinitz constant O(sqrt d)?","shortName":"Euclidean Steinitz-Bergstrom conjecture","problemNumber":null,"field":"Discrepancy theory; rearrangements of vector sums","fieldGroup":"Combinatorics","statement":"Let $S_2(d)$ be the least number such that every finite family of vectors $v_1,\\dots,v_N$ in the Euclidean unit ball of $\\mathbb R^d$ with $\\sum v_i=0$ has an ordering whose partial sums all have norm at most $S_2(d)$. Steinitz's lemma gives a bound depending only on $d$; Grinberg and Sevastyanov (1980) proved $S\\le d$ for every norm, and the simplex example shows $S_2(d)\\ge\\tfrac12\\sqrt d$. The expected square-root scale in Euclidean space was discussed by Behrend (1954), and the conjecture is attributed to Bergstrom. Is $S_2(d)=O(\\sqrt d)$?","posedBy":"Attributed to Viktor Bergstrom (attribution recorded by Ambrus and Heck, Mathematika 2026, Conjecture 5); square-root growth discussed by F. A. Behrend (1954)","yearPosed":1954,"ageNote":"The manuscript cites Behrend (1954, p. 108) for the expected square-root growth and Ambrus-Heck for the attribution to Bergstrom; Bergstrom's 1931 paper is in its bibliography but is not cited in the text, so 1954 is the earliest dated statement the paper itself relies on.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.2: there is an absolute constant $C$ such that every zero-sum family of vectors in the Euclidean unit ball of $\\mathbb R^d$ (repetitions allowed) has an ordering with all partial sums of norm at most $C\\sqrt d$; with the simplex lower bound, $\\tfrac12\\sqrt d\\le S_2(d)\\le C\\sqrt d$. It follows from the prescribed-order signing theorem (separate entry). The constant is not made explicit, and the construction is existential: no online rule or efficient algorithm is given.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 were read against the conjecture as stated in the manuscript; the ordering bound follows from the signing bound by Chobanyan's transference, proved in the introduction. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator SteinitzBergstrom, declaration OAI.EuclideanSteinitzBergstrom.main). ComparatorChallenges/SteinitzBergstrom.lean was read here: it asserts one real $C$ such that for all $d,N\\ge1$ and all $v:\\mathrm{Fin}\\,N\\to\\mathbb R^d$ with $\\|v_i\\|\\le1$ there are signs $\\pm1$ keeping every prefix within $C\\sqrt d$, and every zero-sum such family has a permutation keeping every prefix within $C\\sqrt d$. This states the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A classical discrepancy question on the order of the Euclidean Steinitz constant, decades old and well known to combinatorialists and convex geometers. Placed level with Tingley's problem (38) and Stein's dimension-free weak (1,1) Riesz transform problem (38), below Banach's isometric conjecture (40), and above the matrix Spencer entry (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf","sourceName":"OpenAI math release: The Euclidean Steinitz-Bergstrom theorem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.EuclideanSteinitzBergstrom.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Steinitz/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SteinitzBergstrom.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SteinitzBergstrom.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"black-box-noncommutative-singularity-rational","name":"Deterministic black-box noncommutative singularity testing for rational linear matrices (Garg-Gurvits-Oliveira-Wigderson question)","shortName":"Black-box NC singularity over the rationals","problemNumber":null,"field":"Algebraic complexity; noncommutative rank; operator scaling","fieldGroup":"Theoretical computer science","statement":"A linear matrix $L=A_0+\\sum_i A_ix_i$ with matrix coefficients is noncommutatively singular if it is not invertible over the free skew field, equivalently if $L(X)$ is singular for every tuple of matrices $X$ of every dimension. Garg, Gurvits, Oliveira and Wigderson gave a deterministic polynomial-time algorithm for this problem that reads the coefficients (white box). In Section 6 of their Operator Scaling paper they asked for a black-box construction: a deterministic, polynomial-size list of matrix tuples, depending only on the sizes, on which every noncommutatively nonsingular pencil becomes invertible. Can such a list be constructed in deterministic polynomial time?","posedBy":"Ankit Garg, Leonid Gurvits, Rafael Oliveira and Avi Wigderson, Operator scaling: theory and applications, Found. Comput. Math. 20 (2020), Section 6, p. 280","yearPosed":2020,"ageNote":"The Operator Scaling paper first appeared as a preprint in 2015; the manuscript cites the 2020 journal version for the question.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims a deterministic polynomial-time, polynomial-size list of rational matrix tuples on which every noncommutatively nonsingular rational affine pencil of bounded order becomes invertible, and a list recovering the noncommutative rank of rectangular rational pencils from ordinary ranks. It does NOT treat pencils with coefficients from other fields; the GGOW question for general fields remains.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The answer is a corollary in Section 5 (the pencil-hitting corollary) of the rational-formula hitting-list manuscript, which also gives a rank-detecting list for rectangular rational pencils.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. The pencil-hitting corollary of the principal manuscript was read: every rational affine pencil of order at most a bound $B$ that is invertible somewhere over an extension of $\\mathbb Q$ becomes invertible at some tuple of the same polynomial-size list. The paper calls this the rational-coefficient version of the GGOW question, so coefficients outside $\\mathbb Q$ are not covered. The Lean statement RationalHitting concerns formulas, not pencils, so it does not cover this entry. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":34,"significanceNote":"A black-box deterministic test for noncommutative singularity is a natural next step after the celebrated white-box operator scaling algorithm and is well known in algebraic complexity; this answers the rational-coefficient case. Above the companion rational-formula entry (32), level with the CVP hardness entry (35), below permanent formula lower bounds (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026.pdf","sourceName":"OpenAI math release: Polynomial Hitting Lists for Noncommutative Rational Formulas","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: One Rational Matrix Hitting Point for Noncommutative Formulas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"matolcsi-ruzsa-weiner-fourier-vanishing","name":"The Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six","shortName":"Matolcsi-Ruzsa-Weiner Fourier vanishing","problemNumber":null,"field":"Complex Hadamard matrices of order six","fieldGroup":"Quantum information & computing","statement":"For a complex Hadamard matrix $H$ of order six ($|H_{ij}|=1$, $H^*H=6I$) and $a\\in\\mathbb Z^6$ with $\\sum a_i=0$, let $g_H(a)=\\frac16\\sum_k\\prod_iH(i,k)^{a_i}$. Orthogonality forces $g_H(e_i-e_j)=0$. In their Fourier approach to mutually unbiased bases in dimension six, Matolcsi, Ruzsa and Weiner (2013, Conjecture 2.3) conjectured a further vanishing at the charge $\\alpha=(1,1,1,-1,-1,-1)$, with one exceptional equivalence class, Tao's cubic matrix $S_6$; Maxwell and Brierley proved it for the Karlsson family. Does $g_H(\\pi\\alpha)=0$ hold for every coordinate permutation $\\pi$ and every $H$ not equivalent to Tao's matrix?","posedBy":"Mate Matolcsi, Imre Z. Ruzsa and Mihaly Weiner (Conjecture 2.3, Australas. J. Combin. 55, 2013)","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every complex Hadamard matrix of order six not equivalent to Tao's cubic matrix has $g_H(\\pi\\alpha)=0$ for every permutation $\\pi$ of $\\alpha=(1,1,1,-1,-1,-1)$. Theorem 1.2, by a second exact certificate and Weiner's completion theorem, excludes seven mutually unbiased bases in $\\mathbb C^6$, giving $N(6)\\le5$. It does not prove the further Matolcsi-Matszangosz-Varga-Weiner conditions for MUB triplets or the bound $N(6)=3$, which is the companion's computer-assisted claim.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof is an exact invariant-moment certificate in integer and rational arithmetic, with a standard-library Python verifier included in the manuscript folder.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against MRW Conjecture 2.3; the manuscript notes its normalization divides the conjecture's sums by six, which does not affect vanishing. The verifier was not re-run here. Lean: lean/ComparatorChallenges/MUBSix.json exists with solution_module OAI.Analysis.MutuallyUnbiased.Main, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement MUBSix.lean was read here: its first conjunct says that for every 6x6 complex matrix with unimodular entries and $H^*H=6I$ that is not equivalent (row and column permutations and unit phases) to the cube-root matrix with Tao's exponent pattern, the normalized character sum vanishes at every permutation of $(1,1,1,-1,-1,-1)$. This states the headline; the second conjunct is the bound of five bases. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A specialist conjecture inside the dimension-six MUB literature, posed in 2013 as a step toward the main problem. Placed level with the Neuen-Grohe tournament isomorphism problem (12), above the Chafai-Dadoun-Youssef questions on logarithmic energy monotonicity (10), far below Zauner's conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf","sourceName":"OpenAI math release: Exact Fourier certificates for complex Hadamard matrices of order six","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.MUB6.fourier_and_family_bound)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/MutuallyUnbiased/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: MUBSix.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/MUBSix.lean","kind":"lean-proof"},{"label":"Related in family: The maximum number of mutually unbiased bases in dimension six","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"p-equals-w-sln-composite-rank","name":"The P = W conjecture for fixed-determinant (SL_n) Higgs moduli in composite rank","shortName":"P = W for SL_n, composite rank","problemNumber":null,"field":"Algebraic geometry; nonabelian Hodge theory, Higgs bundles","fieldGroup":"Geometry & topology","statement":"For a smooth projective curve $C$ of genus $g\\ge2$ and $\\gcd(n,d)=1$, nonabelian Hodge theory identifies the cohomology of the moduli space $M^L_{\\mathrm{Dol}}(\\mathrm{SL}_n)$ of stable trace-free Higgs bundles with fixed determinant and that of the twisted $\\mathrm{SL}_n$ character variety. De Cataldo, Hausel and Migliorini conjectured that the perverse filtration $P$ of the Hitchin map matches the weight filtration $W$: $P_k=W_{2k}=W_{2k+1}$. Maulik-Shen and others proved it for $\\mathrm{GL}_n$; for fixed determinant it was known in prime rank, where the variant cohomology was controlled. Does $P=W$ hold on the full cohomology, variant part included, for every composite rank $n$?","posedBy":"M. A. de Cataldo, T. Hausel and L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case A_1, Ann. of Math. 175 (2012)","yearPosed":2012,"ageNote":"Year of the Annals paper the manuscript cites; it gives no earlier preprint date.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: $P_k=W_{2k}=W_{2k+1}$ on all of $H^*(X^B,\\mathbb Q)$ for the smooth coprime fixed-determinant, trace-free moduli in every composite rank $n\\ge4$, genus $g\\ge2$, including the variant (non-$\\Gamma$-invariant) cohomology; with the earlier prime-rank theorems this gives the fixed-determinant conjecture in every coprime rank. A corollary answers Maulik-Shen's Question 5.5 on Betti weights of endoscopic images. Non-coprime (singular) cases and other groups are not treated.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it; it claims $P_kH^m=W_{2k}H^m=W_{2k+1}H^m$ on the full rational cohomology for every composite $n\\ge4$, coprime $d$, every $L$ and every genus $g\\ge2$ curve. The proof (an $\\mathfrak{sl}_2$ operator built from logarithmic moving-pole moduli, following Hausel-Mellit-Minets-Schiffmann) was not refereed and is not formalized. It uses Mellit's toric stratifications and published logarithmic Hodge moduli constructions as inputs.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"P = W is a central conjecture of nonabelian Hodge theory, but its GL_n case was proved in 2022 and the SL_n prime-rank case earlier; the composite-rank fixed-determinant remainder is a significant but secondary step. Placed below the global quantum geometric Langlands conjecture at irrational level (36), level with the effective Iitaka fibration conjecture (32), above Batyrev's Stringy Hodge Number Conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026.pdf","sourceName":"OpenAI math release: P=W in composite rank for fixed determinant","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"infinite-matroid-intersection-packing-covering","name":"The unrestricted infinite matroid intersection and packing/covering conjectures","shortName":"Infinite matroid intersection","problemNumber":null,"field":"Infinite matroid theory","fieldGroup":"Combinatorics","statement":"Work with infinite matroids in the sense of Bruhn-Diestel-Kriesell-Pendavingh-Wollan. Two matroids $M,N$ on a common ground set $E$ satisfy intersection if some $J$ independent in both splits as $J=J_M\\sqcup J_N$ with $\\mathrm{cl}_M(J_M)\\cup\\mathrm{cl}_N(J_N)=E$. Nash-Williams conjectured this for finitary matroids, and Joo proved it for countable ground sets; Bowler and Carmesin formulated the unrestricted Matroid Intersection Conjecture and the packing/covering conjecture (every family has a partition $E=P\\sqcup C$ with disjoint spanning sets in each $M_i\\restriction P$ and independent sets in each contraction onto $C$ covering $C$) and showed them related. Joo asked whether intersection holds for two partitional matroids on a countable set. Do every two infinite matroids on a common ground set satisfy intersection, and does every family have a packing/covering partition?","posedBy":"Nathan Bowler and Johannes Carmesin (Combinatorica 2015), extending Nash-Williams' intersection conjecture; Joo (2021, Question 1.5) for partitional matroids","yearPosed":2015,"ageNote":"Nash-Williams' original finitary intersection conjecture is older and is not settled here; the year is that of Bowler-Carmesin's unrestricted formulation, which is what is disproved.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: in ZFC there are a countably infinite set $E$ and two self-dual infinite matroids $M_0,M_1$ on $E$ with $I_0\\cup I_1\\ne E$ for all independent $I_0,I_1$; the pair has no packing/covering partition, so the infinite matroid packing/covering conjecture is false. Corollary 1.2: the same partitional pair fails intersection, refuting the unrestricted infinite Matroid Intersection Conjecture and answering Joo's Question 1.5 negatively. The examples are neither finitary nor cofinitary, so Nash-Williams' original conjecture for finitary matroids and the cofinitary cases are untouched. The local building block also answers the Bowler-Geschke ZFC question on self-dual uniform matroids (separate entry).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has one manuscript; the construction is formalized in Lean (see verification).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against the conjectures as the paper quotes them from Bowler-Carmesin and Joo. formalization.yaml lists ComparatorChallenges/InfiniteMatroid.json with declaration OAI.InfiniteMatroidCounterexample.main in OAI/Combinatorics/InfiniteMatroid/Main.lean (file exists at the pinned commit). Its statement, read here, gives two Mathlib matroids (which allow infinite ground sets with the maximality axiom) on a countable infinite type, both self-dual, with no independent cover $I_0\\cup I_1$ of the ground set and no packing/covering partition: the headline disproof. lean/docs/185.md also links InfiniteMatroidCorollaries.json (module OAI.Combinatorics.InfiniteMatroid.Corollaries, file exists, not in the formalization catalogue), whose statement adds that both matroids are partitional and fail intersection, and that the separate covering and packing conjectures fail. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"The unrestricted intersection and packing/covering conjectures were the main open problems of infinite matroid theory, a small community; the most famous finitary form remains open. Placed below White's Conjecture on Matroids (40) and Rota's Unimodality Conjecture for Matroid Flats (36), slightly above Unimodality of Kazhdan-Lusztig Polynomials of Matroids (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A Counterexample to the Infinite Matroid Packing/Covering Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.InfiniteMatroidCounterexample.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/InfiniteMatroid/Main.lean","kind":"lean-proof"},{"label":"Lean corollaries (partitional intersection, covering, packing)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/InfiniteMatroid/Corollaries.lean","kind":"lean-proof"},{"label":"Bowler-Carmesin, Matroid intersection, base packing and base covering for infinite matroids","url":"https://doi.org/10.1007/s00493-014-2953-2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gaussian-propeller-conjecture","name":"The Gaussian propeller conjecture","shortName":"Propeller conjecture","problemNumber":null,"field":"Gaussian geometry; partitions, kernel clustering","fieldGroup":"Probability & statistics","statement":"For a measurable partition $A_1,\\dots,A_k$ of $\\mathbb R^d$ and the standard Gaussian measure $\\gamma_d$, let $z(A_i)=\\int_{A_i}x\\,d\\gamma_d$ and $F=\\sum_i\\|z(A_i)\\|^2$. Khot and Naor, studying approximate kernel clustering and its Unique Games hardness, conjectured that for $k\\ge3$ the maximum of $F$ is attained by the propeller: three planar sectors of angle $2\\pi/3$ times an orthogonal factor, giving $9/(8\\pi)$. Heilman, Jagannath and Naor proved it in $\\mathbb R^3$. Is $\\sum_i\\|\\int_{A_i}x\\,d\\gamma_d\\|^2\\le 9/(8\\pi)$ for every finite partition in every dimension?","posedBy":"S. Khot and A. Naor, Approximate kernel clustering, Mathematika 55 (2009)","yearPosed":2009,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for all $d,k$, every measurable partition of $\\mathbb R^d$ satisfies $\\sum_i\\|\\int_{A_i}x\\,d\\gamma_d\\|^2\\le9/(8\\pi)$, with equality for the propeller when $d\\ge2$, $k\\ge3$. The new step shows extremizers with five or more active cells cannot beat $9/(8\\pi)$, using Ehrhard's inequality and residual-score estimates; four-cell configurations come from Heilman-Jagannath-Naor. Consequence stated: combined with the release's Unique Games theorem, NP-hardness of beating the factor $(8\\pi/9)(1-1/k)$ for identity-target kernel clustering, $k\\ge3$. Uniqueness of extremizers is not claimed.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it with the Khot-Naor reference. The proof uses the Heilman-Jagannath-Naor $\\mathbb R^3$ theorem (computer-assisted) as an established input and was not refereed. The challenge ComparatorChallenges/GaussianPropeller.lean (OAI.GaussianPropeller.all_partitions, in lean/formalization.yaml) was read here: for all $d,k>0$ and every measurable a.e.-partition, $\\sum_i\\|\\int_{A_i}x\\,d\\gamma\\|^2\\le9/(8\\pi)$, and for $d\\ge2$, $k\\ge3$ the explicit propeller attains it. This states the headline. Not rebuilt here. The kernel-clustering hardness consequence is not formalized and depends on the release's separate Unique Games theorem.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A named conjecture from hardness of approximation, known to the Khot-Naor and Gaussian-geometry community, with the three-dimensional case already proved in 2013. Placed above the Gaussian Moments Conjecture (25) and Sharp Hardness for MAX-3-CUT (25), level with the Ellipsoid Fitting Conjecture (30), and well below the Unique Games Conjecture (74).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf","sourceName":"OpenAI math release: The Gaussian propeller bound in every dimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.GaussianPropeller.all_partitions)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/GaussianPropeller/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: GaussianPropeller.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/GaussianPropeller.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bsd-formula-selmer-corank-one","name":"The Birch-Swinnerton-Dyer formula in Selmer corank at most one","shortName":"BSD formula in Selmer corank at most one","problemNumber":null,"field":"Arithmetic of elliptic curves; Iwasawa theory","fieldGroup":"Number theory","statement":"For an elliptic curve $E/\\mathbb{Q}$ of rank $r$, the Birch and Swinnerton-Dyer conjecture predicts $\\mathrm{ord}_{s=1}L(E,s)=r$, finiteness of the Tate-Shafarevich group, and the leading-term formula $L^{(r)}(E,1)/r! = \\Omega_E\\,\\mathrm{Reg}_E\\,\\#Sha(E/\\mathbb{Q})\\prod_\\ell c_\\ell(E)/\\#E(\\mathbb{Q})_{tors}^2$. Gross-Zagier and Kolyvagin gave the rank equality and finiteness of Sha when the analytic rank is at most one, but the exact leading-term formula in that range was known only at particular primes and under reduction, residual or CM hypotheses (Skinner-Urban, Jetchev-Skinner-Wan, Rubin, Burungale-Flach). Does the full leading-term formula, at every prime, hold for every elliptic curve over $\\mathbb{Q}$ whose $q^\\infty$-Selmer group has corank $0$ or $1$ for some prime $q$?","posedBy":"Bryan Birch and Peter Swinnerton-Dyer (the conjecture); Tate's 1966 Bourbaki account fixed the leading-term formula","yearPosed":1965,"ageNote":"Year of Birch and Swinnerton-Dyer, Notes on elliptic curves II (1965), cited by the manuscript. The low-rank integral formula as a target is the program of Skinner-Urban (2014) and Jetchev-Skinner-Wan, also cited.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"For every $E/\\mathbb{Q}$ and prime $q$ with $s_q(E)=\\mathrm{corank}\\,\\mathrm{Sel}_{q^\\infty}(E/\\mathbb{Q})\\in\\{0,1\\}$: $\\mathrm{rank}\\,E(\\mathbb{Q})=\\mathrm{ord}_{s=1}L(E,s)=s_q(E)$, $Sha(E/\\mathbb{Q})$ is finite, and the full BSD leading-term formula holds at every prime, with no hypotheses on reduction, torsion, isogenies, CM or residual representations. By Gross-Zagier-Kolyvagin this covers every curve of analytic rank at most one (curator's reading). The new step is the exact $p$-adic valuation at every odd $p$; $p=2$ comes from the two-primary companion. With family 006 it gives full BSD for a density-one set of quadratic twists of each curve. It says nothing about analytic rank $\\geq 2$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (October 3, 2026) builds on two companion manuscripts in this family and on the release's Goldfeld-density manuscript (family 006).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem of the October 3 manuscript was read against the posed BSD leading-term formula, including its period, regulator, Tamagawa and torsion normalizations. The proof was not refereed. No Lean formalization is listed in lean/formalization.yaml for any manuscript in this family. The result is conditional on unreviewed companion inputs from the same release: the Selmer converse at every prime and the two-primary formula (this family), and the Goldfeld analytic-density theorem (family 006). The rank-zero CM case uses Burungale-Flach. The paper states the low-Selmer-corank hypothesis is essential to its scope.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":78,"significanceNote":"Full BSD is a Millennium problem near the top of the ladder; this is the analytic rank at most one case, which was the main open target of a decade of Iwasawa theory. Scored as a partial, like the 67 percent critical-line entry (68) against Riemann, but higher, because the low-rank case is the whole of what current methods aim at. Below Euler blowup from smooth data (80), above the Jacobian Conjecture (65) and the elliptic curve rank records (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","sourceName":"OpenAI math release: Exact Birch-Swinnerton-Dyer Formula from Low Selmer Corank","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion: The two-primary Birch-Swinnerton-Dyer formula in Selmer corank at most one","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: The Selmer converse for elliptic curves at every prime","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026/main.pdf","kind":"paper"},{"label":"Input (family 006): Goldfeld's analytic density conjecture and the 2-converse","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf","kind":"paper"},{"label":"Clay Mathematics Institute: Birch and Swinnerton-Dyer conjecture","url":"https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/","kind":"problem-record"},{"label":"Goldfeld's conjecture entry, an input to this result","url":"https://vibemathed.com/problem/goldfeld-conjecture-quadratic-twists","kind":"other"},{"label":"Earlier edition (October 3, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/exact-bsd-low-selmer-corank.pdf","kind":"paper"},{"label":"Revised edition (October 7, 2026): Goldfeld's analytic density conjecture and the 2-converse for elliptic curves","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 7, 2026): The Selmer converse for elliptic curves at every prime","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): The two-primary Birch-Swinnerton-Dyer formula in Selmer corank at most one","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"cubic-flat-three-torus-isoperimetric-conjecture","name":"The isoperimetric conjecture for the cubic flat three-torus (balls, tubes, slabs)","shortName":"Isoperimetric problem in the cubic 3-torus","problemNumber":null,"field":"Geometric measure theory; periodic isoperimetric problem","fieldGroup":"Geometry & topology","statement":"In the unit cubic flat torus $\\mathbb R^3/\\mathbb Z^3$, which regions of prescribed volume $V$ have least perimeter? The conjecture predicts that the minimizers are, in turn, round balls, solid circular tubes about shortest closed geodesics, and slabs between parallel coordinate tori, followed by their complements, so the profile is $\\min\\{(36\\pi)^{1/3}v^{2/3},2\\sqrt{\\pi v},2\\}$ with $v=\\min(V,1-V)$. Small volumes (Morgan-Johnson) and volumes near one half (Acerbi-Fusco-Morini) were known, and Milman reduced the profile to the two transition volumes, but excluding higher-genus competitors in between remained open. Is the ball-tube-slab picture the true isoperimetric profile of the cubic three-torus?","posedBy":"Hauswirth, Perez, Romon and Ros, The periodic isoperimetric problem, Trans. AMS 356 (2004); also A. Ros, The isoperimetric problem (Clay Math. Proc. 2, 2005), Section 1.6","yearPosed":2004,"ageNote":"The manuscript cites the 2004 HPRR paper and Ros's 2005 survey as where the prediction appears explicitly; it may be older in the folklore of the periodic problem, but no earlier written source is given.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims the full isoperimetric profile of $\\mathbb R^3/\\mathbb Z^3$, $I(V)=\\min\\{(36\\pi)^{1/3}v^{2/3},2\\sqrt{\\pi v},2\\}$, and a classification of all finite-perimeter minimizers: balls for $V\\le4\\pi/81$, tubes about shortest geodesics for $4\\pi/81\\le V\\le1/\\pi$, coordinate slabs for $1/\\pi\\le V\\le1/2$, complements above $1/2$, with exactly the two adjacent types at each transition volume. The new step excludes three- and four-vertex nonstandard configurations at the two transition volumes via nested planar sections. It does NOT treat non-cubic flat tori, the cube with relative perimeter beyond the reflection correspondence, or higher dimensions.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. Some one-variable estimates are closed with explicit outward-rounded arithmetic bounds in an appendix. The full profile and classification have a Lean comparator challenge in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. The main theorem was read against the HPRR and Ros formulation. The Lean challenge lean/ComparatorChallenges/CubicTorus.json (solution module OAI.Geometry.CubicTorus.Main, present at the pinned commit) is not in the formalization catalogue; it was found through lean/docs/354.md. Its statement OAI.CubicTorus.unit_cubic_isoperimetric was read here: for every $01/2$. This is the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A well-known open problem of geometric measure theory, stated in the standard survey on the isoperimetric problem and recently attacked by Milman; famous within the minimal-surfaces community but not beyond it. Placed with Kalton's Problem 2 (28), below Petty's projection conjecture in dimension four and up (42) and above the weak simplex conjecture (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf","sourceName":"OpenAI math release: The Isoperimetric Conjecture for the Cubic Flat Three-Torus","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean formalization: isoperimetric profile and minimizers of the cubic three-torus","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/CubicTorus/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: CubicTorus","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CubicTorus.lean","kind":"lean-proof"},{"label":"Hauswirth, Perez, Romon and Ros, The periodic isoperimetric problem (2004)","url":"https://doi.org/10.1090/S0002-9947-03-03362-2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"zilber-pink-curves-in-a2","name":"Zilber-Pink for Hodge-generic curves in the Siegel modular threefold A_2","shortName":"Zilber-Pink for curves in A_2","problemNumber":null,"field":"Unlikely intersections in Shimura varieties","fieldGroup":"Number theory","statement":"Let $\\mathcal A_2$ be the moduli space of principally polarized abelian surfaces. Pink's Zilber-Pink conjecture for mixed Shimura varieties (2005, Conjecture 1.3) predicts that a Hodge-generic curve meets the union of special subvarieties of codimension at least two only finitely often; in $\\mathcal A_2$ these are the special points and the special curves (products with a CM elliptic factor, squares of non-CM elliptic curves, quaternionic multiplication). Pila and Tsimerman proved Andre-Oort for $\\mathcal A_2$, and Daw and Orr proved the curve loci under boundary or multiplicative-reduction hypotheses. For a Hodge-generic algebraic curve $C\\subset\\mathcal A_2$ over $\\bar{\\mathbb Q}$, is $C\\cap\\bigcup_{\\dim Z\\le1}Z$ finite?","posedBy":"Richard Pink (2005, Conjecture 1.3, for mixed Shimura varieties); the A_2 curve case studied by Christopher Daw and Martin Orr (2021-2025)","yearPosed":2005,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims the full curve case of Zilber-Pink in $\\mathcal A_2$ over $\\bar{\\mathbb Q}$: a Hodge-generic algebraic curve contains only finitely many points whose abelian surface is isogenous to a product with a CM factor, to the square of a non-CM elliptic curve, or has quaternionic-division endomorphism algebra, besides finitely many special points. Previous unconditional results needed the curve to meet the Baily-Borel boundary or to have multiplicative degeneration. It does NOT treat higher-dimensional subvarieties of $\\mathcal A_2$, $\\mathcal A_g$ for $g\\ge3$, or curves not defined over $\\bar{\\mathbb Q}$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. All three manuscripts are credited to OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.3 of the principal manuscript was read against the posed problem: for every Hodge-generic curve in $\\mathcal A_2$ over $\\bar{\\mathbb Q}$, finitely many points on special subvarieties of dimension at most one, with no boundary, reduction, height or degree hypothesis. It assembles three branch theorems: elliptic squares (proved here, by establishing the Galois-orbit lower bound Daw and Orr state as their Conjecture 6.2 and applying their Theorem 1.3), and the E x CM and quaternionic-division branches from the two companions, plus Pila-Tsimerman for special points. The companions were read at the level of their main theorems only. No Lean formalization. Curves over $\\mathbb C$ not defined over $\\bar{\\mathbb Q}$ are not covered.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"A sharply defined test case of Zilber-Pink in Shimura varieties that Daw, Orr and Papas had attacked for years under boundary or reduction hypotheses; known mainly to specialists. Below the abelian Zilber-Pink entry at 58, below the Kuga-Satake Hodge conjecture at 48, above Free entropy dimension invariance at 38.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Elliptic Squares and Zilber-Pink for Curves in A2","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The E x CM component of Zilber-Pink for curves in A2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Quaternionic division points on curves in the Siegel threefold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf","kind":"paper"},{"label":"Family companion: The abelian Zilber-Pink conjecture (separate entry)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf","kind":"paper"},{"label":"Pink (2005), Conjecture 1.3","url":"https://people.math.ethz.ch/~pink/ftp/AOMMML.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"trace-reconstruction-polynomial-samples","name":"Can worst-case trace reconstruction be done with polynomially many traces?","shortName":"Polynomial-sample trace reconstruction","problemNumber":null,"field":"Trace reconstruction; deletion channels","fieldGroup":"Theoretical computer science","statement":"An unknown binary string $x\\in\\{0,1\\}^n$ is passed repeatedly through a deletion channel that removes each bit independently with known probability $q\\in(0,1)$ and concatenates the survivors; each output is a trace. Worst-case trace reconstruction asks how many independent traces $T_q(n)$ suffice to recover every $x$ with probability at least $2/3$. The problem was formulated by Batu, Kannan, Khanna and McGregor (2004). For fixed $q$ the best upper bounds fell from $\\exp(\\widetilde O(\\sqrt n))$ (Holenstein et al. 2008) to $\\exp(O(n^{1/3}))$ (De-O'Donnell-Servedio; Nazarov-Peres 2017), $\\exp(O(n^{1/5}\\log^5 n))$ (Chase 2021) and a quasipolynomial bound (Burudgunte-Valiant-Wang 2026), while the best lower bound was $\\Omega(n^{3/2}/\\log^7 n)$ (Chase 2021). For a fixed deletion probability, is $T_q(n)$ bounded by a polynomial in $n$?","posedBy":"Open since Batu, Kannan, Khanna and McGregor formulated worst-case trace reconstruction (SODA 2004); the manuscript calls it the polynomial-sample question without naming who first asked it","yearPosed":2004,"ageNote":"The manuscript cites Batu, Kannan, Khanna and McGregor (2004) for the problem and Holenstein et al. (2008) for the first subexponential bound; it gives no explicit first statement of the polynomial question, so 2004 dates the problem rather than that sentence.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for fixed $s\\in(0,1]$ and $0 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two later companions (October 5, 2026): a uniform quasipolynomial-time reconstruction algorithm for known rational retention probability, and an improved quasipolynomial sample upper bound. They give upper bounds and are links here.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the lower-bound manuscript were read against the question. They give $T_{q,s}(n)\\ge n^{c\\log(q^3\\log n)}$ along any sequence with $q^3\\log n\\to\\infty$, and $T_{q,s}(n)/n^A\\to\\infty$ for every fixed $q,s,A$, with unrestricted computation and randomized estimators. The challenge is not in lean/formalization.yaml; lean/ComparatorChallenges/TraceReconstruction.json exists with solution_module OAI.Probability.TraceReconstruction, whose file exists at the pinned commit. The statement TraceReconstruction.lean was read here; not rebuilt here. Its three theorems define the sample complexity as an infimum over budgets admitting an estimator (a probability kernel on complete traces) and state: superpolynomial growth of sample complexity for every fixed $q\\in(0,1)$ and $s\\in(0,1]$, superpolynomially small minimum one-trace total variation, and the quantitative bound. This states the headline negative answer. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Worst-case trace reconstruction is one of the best-known open problems at the meeting of probability and theoretical computer science, worked on by Peres, Nazarov, O'Donnell, Servedio and many others for two decades, and whether polynomially many traces suffice was its headline question. Level with the Gotsman-Linial entry at 42, above the sensitivity separation entry in this batch at 35, below Ryser's conjecture at 50.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/quantitative-lower-bounds-for-trace-reconstruction-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Quantitative lower bounds for trace reconstruction","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Uniform quasipolynomial-time trace reconstruction","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026/uniform-trace-reconstruction.pdf","kind":"paper"},{"label":"Companion: A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.TraceReconstruction, superpolynomial sample lower bounds)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/TraceReconstruction.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kervaire-conjecture","name":"The Kervaire conjecture: adjoining one generator and one relator never kills a nontrivial group","shortName":"Kervaire conjecture","problemNumber":null,"field":"Combinatorial group theory; equations over groups","fieldGroup":"Algebra","statement":"Let $A$ be a nontrivial group, let $t$ generate an infinite cyclic group, and let $w$ be any element of the free product $A*\\langle t\\rangle$. Kervaire's characterization of high-dimensional knot groups (1965) led to the group-theoretic question whether the quotient $(A*\\langle t\\rangle)/\\langle\\langle w\\rangle\\rangle$ can ever be trivial, that is, whether $A*\\mathbb Z$ can be normally generated by one element. Known cases included torsion-free $A$ (Klyachko, unimodular case), residually finite and hyperlinear $A$ (via Gerstenhaber-Rothaus and Pestov), and $w$ of special forms. Is $(A*\\langle t\\rangle)/\\langle\\langle w\\rangle\\rangle$ nontrivial for every nontrivial group $A$ and every $w$?","posedBy":"Michel Kervaire, Les noeuds de dimensions superieures (1965); the group-theoretic form is traced to this setting by Chen (2026)","yearPosed":1965,"ageNote":"The manuscript cites Kervaire (1965) for the knot-group origin and Chen's survey for the group-theoretic formulation. The stronger Kervaire-Laudenbach injectivity form is covered by the separate entry for Howie's conjecture.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every group $A$ (no torsion, cardinality or presentation hypothesis) and every $w\\in A*\\langle t\\rangle$ with exponent sum $\\pm1$ in $t$, the canonical map $A\\to(A*\\langle t\\rangle)/\\langle\\langle w\\rangle\\rangle$ is injective, so the equation $w(t)=1$ is solvable in an overgroup. Corollary 1.2: the quotient is nontrivial for every $w$ when $A\\ne1$, which is the Kervaire conjecture. It does NOT prove injectivity for other nonzero exponent sums (the Kervaire-Laudenbach form); that is the companion's nonsingular-systems theorem. It does not assert a solution inside $A$ itself.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The Kervaire manuscript proves the result directly through a spectral-phase inequality in the group von Neumann algebra and a fixed-space theorem for unitary matrices; the later nonsingular-systems manuscript (5 October 2026) names it as its direct antecedent.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against the conjecture as posed: Corollary 1.2 is the nontriviality statement for every nontrivial $A$ and every word $w$. The Lean challenge ComparatorChallenges/Kervaire.json (theorem OAI.Kervaire.coefficient_injective, solution module OAI.GroupTheory.Kervaire.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/256.md. Its statement was read here: for every group $A$ and every $w$ in the coproduct of $A$ with multiplicative $\\mathbb Z$ whose exponent sum is $\\pm1$, the map from $A$ to the quotient by the normal closure of $w$ is injective. That is Theorem 1.1, which is stronger than the headline for unimodular words; the remaining exponent sums follow by the one-line surjection onto $\\mathbb Z/d$ given in the paper, which is not itself formalised. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The manuscript records that Kawauchi (2024) published a proposed resolution through ribbon sphere-link groups, which it does not use.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The Kervaire conjecture has been a central open problem of combinatorial group theory since 1965, with steady partial progress (Gerstenhaber-Rothaus, Klyachko, Chen) and at least one disputed claimed resolution. Below Kaplansky's zero-divisor conjecture (62), whose reach is wider, tied with Dixmier's unitarizability problem (55), above the word problem for finite-rank Artin groups (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf","sourceName":"OpenAI math release: The Kervaire theorem for groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Nonsingular systems of equations over arbitrary groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf","kind":"paper"},{"label":"Lean: solution module for unimodular coefficient injectivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/Kervaire/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Kervaire coefficient injectivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/Kervaire.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"uniform-sparsest-cut-np-hardness","name":"NP-hardness of approximating Uniform Sparsest Cut within every constant factor","shortName":"Uniform Sparsest Cut constant-factor hardness","problemNumber":null,"field":"Hardness of approximation; graph partitioning","fieldGroup":"Theoretical computer science","statement":"Uniform Sparsest Cut asks, for a graph on $N$ vertices with nonnegative capacities $c_{ij}$, for the minimum over nonempty proper $S$ of $\\sum_{i\\in S,j\\notin S}c_{ij}/(|S|(N-|S|))$. Arora, Rao and Vazirani gave an $O(\\sqrt{\\log N})$ approximation. Exact computation is NP-hard, but approximation lower bounds were known only under assumptions: Ambuhl, Mastrolilli and Svensson ruled out a PTAS assuming SAT has no subexponential randomized algorithms, and the Small-Set Expansion hypothesis gives hardness of every constant factor. For the nonuniform problem, hardness of every constant factor follows from the Unique Games Conjecture. Is it NP-hard, with no unproved hypothesis, to approximate Uniform Sparsest Cut within some constant factor, or within every constant factor?","posedBy":"Open problem in the hardness-of-approximation literature; discussed by Ambuhl, Mastrolilli and Svensson and by Manurangsi and Trevisan","yearPosed":null,"ageNote":"The manuscript gives no single posing reference. It cites Ambuhl-Mastrolilli-Svensson (SIAM J. Comput. 2011; FOCS 2007 version) for the strongest prior hardness and Manurangsi-Trevisan (APPROX 2018, Section 1.2) and d'Orsi et al. (2025) for the uniform versus nonuniform gap. The question is at least as old as the 2007 PTAS-hardness result.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every fixed $C>1$ there is a polynomial-time reduction from 3SAT producing uniform-demand instances with a multiplicative gap $C$, so approximating Uniform Sparsest Cut within any constant factor is NP-hard. The constants and exponents of the reduction depend on $C$. It does not give a superconstant (for instance $\\sqrt{\\log N}$-type) hardness factor, does not address Balanced Separator directly, and says nothing about the Small-Set Expansion hypothesis itself.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 24, 2026: the hardness reduction (this entry) and a separate integrality-gap construction (its own entry).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question. For every fixed $C>1$ it gives a polynomial-time reduction from 3CNF satisfiability to graphs with nonnegative rational capacities and unit demand on every pair, with $\\Phi(G)\\le a$ for satisfiable and $\\Phi(G)>Ca$ for unsatisfiable formulas, using no unproved complexity hypothesis. That answers the question as posed (every constant factor, exact uniform demands, deterministic Karp reduction). The PCP-style proof (Garcia-Stichtenoth tower tests, rare-event extraction, cut-variance analysis) was not refereed and is not formalised; the Lean formalization in this family covers only the integrality-gap companion.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Unconditional constant-factor hardness for Uniform Sparsest Cut was a well-known open problem in approximation algorithms, with only conditional or PTAS-type lower bounds before. Placed well below the Unique Games Conjecture (74), below the matrix multiplication exponent entry (55), and far above Sharp Hardness for MAX-3-CUT (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026.pdf","sourceName":"OpenAI math release: Constant-factor hardness of uniform sparsest cut","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Near-square-root logarithmic integrality gaps for uniform sparsest cut","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"closed-geodesics-on-spheres-infinitude","name":"Infinitely many closed geodesics on every Riemannian sphere and every closed three-manifold","shortName":"Closed geodesics on spheres","problemNumber":null,"field":"Riemannian geometry; closed geodesics","fieldGroup":"Geometry & topology","statement":"The closed-geodesic infinitude problem asks whether every closed Riemannian manifold of dimension at least two has infinitely many geometrically distinct (distinct image) closed geodesics. By Gromoll-Meyer and Vigue-Poirrier-Sullivan it holds when the rational cohomology needs at least two generators, which leaves spheres open. The two-sphere was settled by Bangert together with Franks or Hingston, and Rademacher proved infinitude for generic (strongly bumpy) metrics, but for $S^n$, $n\\ge3$, only finite multiplicity was known for arbitrary metrics (two on $S^3$, Long and Duan). Klingenberg's claimed general proof is regarded as erroneous. Does every smooth Riemannian metric on $S^n$ have infinitely many geometrically distinct closed geodesics?","posedBy":"Classical closed-geodesic infinitude problem; the manuscript names no poser and traces it through Birkhoff (1917), Lyusternik-Fet (1951) and Klingenberg, Lectures on closed geodesics (1978)","yearPosed":null,"ageNote":"The manuscript states the problem without attribution or date. Its earliest cited sources are Birkhoff's 1917 work on surfaces and Lyusternik-Fet 1951 for existence of one closed geodesic; Klingenberg's 1978 monograph contained a claimed general proof later identified as erroneous (Asselle-Mazzucchelli).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every smooth Riemannian metric on $S^n$, $n\\ge2$, has infinitely many prime closed geodesics with pairwise distinct images. Corollary 1.2: the same for every closed manifold finitely covered by a sphere (real projective spaces, spherical space forms). Corollary 1.3: the same for every closed three-manifold, orientable or not, using Rademacher-Taimanov and Perelman. It does NOT settle the problem for general closed manifolds in dimension four or more, and it gives no growth rate or length count for the geodesics.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 24, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2 and 1.3 of the TeX source read against the infinitude problem as the introduction states it. The theorem covers every smooth metric on $S^n$, $n\\ge2$, degenerate ones included; the two-sphere case is classical and the new content is $n\\ge3$. The three-manifold corollary is conditional on published inputs: Rademacher-Taimanov (infinite fundamental group) and Perelman's elliptization. The proof was not refereed. No Lean formalization exists for this family (no lean/docs/345.md and no matching Comparator challenge at the pinned commit). The general problem for all closed manifolds stays open; the paper disputes two steps of Charles's 2019 claimed general proof.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Spheres were the central open case of the closed-geodesic problem, a classical question of Riemannian geometry where even a published monograph proof failed. As a special case of the general problem it is scored below it. Placed level with the four-dimensional Kakeya set conjecture (55) and Bloch's conjecture for surfaces with pg zero (55), below Thompson's group F amenability (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Infinitely many closed geodesic images on every Riemannian sphere","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bochner-riesz-conjecture-r3","name":"The Bochner-Riesz conjecture in three dimensions (strict-order range)","shortName":"Bochner-Riesz conjecture in R^3","problemNumber":null,"field":"Harmonic analysis; Fourier multipliers and summability","fieldGroup":"Analysis","statement":"For $\\delta\\ge0$ the Bochner-Riesz means on $\\mathbb R^n$ are $\\widehat{T_\\delta f}(\\xi)=(1-|\\xi|^2)_+^\\delta\\widehat f(\\xi)$. Fefferman (1971) showed the ball multiplier ($\\delta=0$) is unbounded on $L^p$ for $p\\ne2$ when $n\\ge2$; Carleson and Sjolin (1972) proved the conjectured range in the plane. The Bochner-Riesz conjecture asserts that $T_\\delta$ is bounded on $L^p(\\mathbb R^n)$ whenever $\\delta>\\max\\{n|1/p-1/2|-1/2,0\\}$; Tao (1999) showed it implies the restriction conjecture. In $\\mathbb R^3$ the strict range was known only for $\\max\\{p,p'\\}\\ge 22/7$ (Lee, Wu, Gao-Wu-Xi). Is $T_\\delta$ bounded on $L^p(\\mathbb R^3)$ for every $1\\le p\\le\\infty$ and every $\\delta>\\max\\{3|1/p-1/2|-1/2,0\\}$?","posedBy":"Bochner-Riesz summability (Riesz 1923; Bochner 1935-36); the manuscript cites no single source for the conjectured range","yearPosed":null,"ageNote":"The manuscript traces the problem to Riesz (1923) and Bochner (1935-36) and to Fefferman (1971) and Carleson-Sjolin (1972), but does not say who first stated the conjectured range or when; left null rather than guessed.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $\\delta>0$, $\\|T_\\delta f\\|_{L^3(\\mathbb R^3)}\\le C_\\delta\\|f\\|_{L^3(\\mathbb R^3)}$. By complex interpolation and duality this gives boundedness on $L^p(\\mathbb R^3)$ for all $1\\le p\\le\\infty$ and $\\delta>\\max\\{3|1/p-1/2|-1/2,0\\}$, the full strict-order range in three dimensions, hence $L^p$ convergence of Bochner-Riesz means for $3/2\\le p\\le3$ at every positive order. Via Tao's implication it yields the spherical extension estimate $\\|Eg\\|_{L^p(\\mathbb R^3)}\\le C_p\\|g\\|_{L^\\infty(S^2)}$ for $p>3$. Not shown: the endpoint order, the maximal Bochner-Riesz operator, or any dimension $n\\ge4$; it is a partial result on the $n$-dimensional conjecture.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The manuscript (September 24, 2026) uses the sharp planar Furstenberg estimate of Ren and Wang as its geometric input and cites, as non-inputs, the release's sphere restriction, diagonal extension and critical local smoothing manuscripts.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 (L^3 boundedness for every delta > 0) and the interpolation statement of the full strict range were read against the conjecture as the introduction states it. The proof (entropy growth of positive masses on lines, wave-packet repayment, then the pseudoconformal multiplier step) was not refereed. No Lean formalization. Scope the paper itself states: strict order only; the endpoint order delta = 3|1/p - 1/2| - 1/2 is not claimed, nor dimensions above three. A reader should know that Zipeng Wang has proposed a proof in all dimensions (arXiv 2501.12742, cited by the manuscript), and that the release's family 079 (Sogge's local smoothing in 3D) claims the same strict Bochner-Riesz range by an independent route.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"The Bochner-Riesz conjecture is a central problem of harmonic analysis, open in three dimensions since Carleson and Sjolin settled the plane in 1972, and it implies restriction. As the three-dimensional case of the n-dimensional conjecture it is scored as a partial: tied with the three-dimensional restriction entry (62), above the four-dimensional Kakeya set entry (55), below the Jacobian Conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf","sourceName":"OpenAI math release: Bochner-Riesz multipliers in three dimensions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Fefferman 1971, The multiplier problem for the ball (Ann. of Math.)","url":"https://doi.org/10.2307/1970864","kind":"problem-record"},{"label":"Independent route in the release (family 079): Critical local smoothing for the three-dimensional wave equation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf","kind":"paper"},{"label":"Local smoothing in three dimensions, which also yields this range","url":"https://vibemathed.com/problem/local-smoothing-conjecture-r3","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"glr-arthur-filtration-rationality","name":"Rationality of the Gaitsgory-Lafforgue-Raskin canonical Arthur filtration","shortName":"Rationality of the canonical Arthur filtration","problemNumber":null,"field":"Automorphic functions over function fields; geometric Langlands","fieldGroup":"Number theory","statement":"For a split connected semisimple group $G$ over $\\mathbb F_q$ and a curve $X$, Gaitsgory, Lafforgue and Raskin define a canonical filtration, indexed by nilpotent orbits of the dual Lie algebra, on the space of finitely supported unramified automorphic functions with $\\overline{\\mathbb Q}_\\ell$ coefficients, using singular support of $\\ell$-adic sheaves and the trace of Frobenius. The space of functions has a natural rational form. Their Conjecture 2.7.3, formulated jointly with Kazhdan, asserts that the canonical filtration is defined over $\\mathbb Q$. Is the canonical Arthur filtration rational?","posedBy":"Dennis Gaitsgory, Vincent Lafforgue and Sam Raskin, jointly with David Kazhdan","yearPosed":2026,"ageNote":"GLR, arXiv:2603.26925 (2026), Section 2.7, Conjecture 2.7.3, as cited by the manuscript.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Under the stated characteristic assumptions, the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over $\\mathbb Q$, for split connected semisimple $G$, including the noncuspidal part and every closed invariant nilpotent support. Using the companion decomposition at trivial level, each cuspidal filtration step is identified with the scalar extension of the corresponding rational summands (the common-scope form of GLR Conjectures 3.4.9 and 3.4.11). It does not treat small characteristic or nonsplit groups.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem was read against GLR Conjecture 2.7.3. It is proved for split connected semisimple groups under four explicit characteristic hypotheses (the restricted-theory assumptions), for the whole space including the noncuspidal part; outside those hypotheses the conjecture is not addressed, hence Partial. It uses the restricted geometric Langlands results and trace-of-Frobenius theorems as inputs. The proof was not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A technical, months-old conjecture inside the GLR framework, proved under characteristic hypotheses. Placed below the Ramanujan-Arthur decomposition entry of this family (38) and The index conjecture for slc log Calabi-Yau pairs (25), and above The Huang-Jiang-Oblomkov Conjecture at a = 3 (12).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Rationality of the Canonical Unramified Arthur Filtration","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"campana-peternell-conjecture","name":"The Campana-Peternell conjecture on Kodaira dimension","shortName":"Campana-Peternell conjecture","problemNumber":null,"field":"Birational geometry; Kodaira dimension","fieldGroup":"Algebra","statement":"Campana and Peternell, studying positivity of the cotangent bundle, conjectured that if $NK_X=A+B$ on a smooth projective variety $X$, with $N>0$, $A$ effective and $B$ pseudo-effective, then $\\kappa(X)\\ge\\kappa(A)$. The case $A=0$ is canonical nonvanishing. Schnell reduced the conjecture, given nonvanishing, to fibrations whose fibres have Kodaira dimension zero. Is it true that for every smooth connected projective complex variety $X$, effective Cartier divisor $D$ and integer $m_0>0$ with $m_0K_X-D$ pseudo-effective, $\\kappa(X)\\ge\\kappa(D)$?","posedBy":"F. Campana and T. Peternell, Geometric stability of the cotangent bundle and the universal cover of a projective manifold (Bull. SMF, 2011), Conjecture 2.4","yearPosed":2011,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 1.2: if $m_0K_X-D$ is pseudo-effective for an effective Cartier (or, after clearing denominators, rational) divisor $D$ on a smooth connected projective complex variety $X$, then $\\kappa(X)\\ge\\kappa(D)$. Corollary 1.3: if $f:X\\to Y$ is a fibre space and $m_0K_X-f^*H$ is pseudo-effective with $H$ ample, then $\\kappa(X)=\\kappa(F)+\\dim Y$ and $\\ell rK_X-f^*H$ has sections for large $\\ell$ (Schnell's general fibre-space conclusion). Both use canonical nonvanishing in every dimension from the family's abundance manuscript. The zero-Kodaira case of Schnell's question was already proved by Zou (2025) and is not new here.","claimIssueNote":"This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it.","solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. Family 034 has fourteen manuscripts dated September 24 to October 5, 2026, which cite one another as inputs; this entry draws on the manuscript named as its source. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary 1.2 of 'Schnell fiber spaces and good canonical models' was read against the conjecture as the manuscript quotes it (Campana-Peternell, Conjecture 2.4) in its Kodaira-dimension form. The paper's own new step is a mixed-intersection argument from a good minimal model of the total space; the good models and canonical nonvanishing come from the family's log abundance manuscript, so this entry stands or falls with that one. The zero-Kodaira fibre case (its Theorem 1.1) was already proved by Zou (2025), as the paper says. The proof was not refereed. No Lean formalization. Listed as Contested because it depends on the log abundance entry, which the release itself is inconsistent about; see the claim issue.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A well-known conjecture in complex and birational geometry that contains canonical nonvanishing as a special case, which puts it above the other entries of this family except abundance. Placed below Stein's dimension-free Riesz transform problem (38), above the effective Iitaka fibration conjecture (32) and the Tachikawa-Auslander-Reiten entry (30). It adds little beyond the abundance entry if that one fails.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Schnell fiber spaces and good canonical models","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Input: Log abundance in characteristic zero","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf","kind":"paper"},{"label":"Schnell, Singular metrics and a conjecture by Campana and Peternell (arXiv 2202.01295)","url":"https://arxiv.org/abs/2202.01295","kind":"problem-record"},{"label":"The 5 October appendix calling projective log abundance conditional","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026/","kind":"paper"},{"label":"Revised edition (October 6, 2026): Conditional good minimal models for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"benson-etingof-ostrik-conjecture-finite-case","name":"The Benson-Etingof-Ostrik conjecture for finite symmetric tensor categories","shortName":"BEO conjecture, finite case","problemNumber":null,"field":"Tensor categories; positive characteristic, Verlinde categories","fieldGroup":"Algebra","statement":"In characteristic zero, Deligne's theorem gives every symmetric tensor category of moderate growth a fiber functor to supervector spaces. In characteristic $p>0$, Benson, Etingof and Ostrik constructed the tower of higher Verlinde categories $\\mathrm{Ver}_p\\subset\\mathrm{Ver}_{p^2}\\subset\\cdots$ and conjectured that every symmetric tensor category of moderate growth over an algebraically closed field of characteristic $p$ admits a fiber functor to the union of this tower. Ostrik proved it for fusion categories (to $\\mathrm{Ver}_p$). Does every finite symmetric tensor category in characteristic $p$, including $p=2$, admit a fiber functor to some $\\mathrm{Ver}_{p^n}(k)$?","posedBy":"D. Benson, P. Etingof and V. Ostrik, New incompressible symmetric tensor categories in positive characteristic, Duke Math. J. 172 (2023), Conjecture 1.4","yearPosed":2023,"ageNote":"Year of the Duke publication cited by the manuscript; the preprint circulated earlier, which the manuscript does not date.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every finite symmetric tensor category over an algebraically closed field of characteristic $p>0$ has a fiber functor to a single finite level $\\mathrm{Ver}_{p^n}(k)$, including $p=2$. Consequences stated: classification of finite incompressible categories (the finite case of Coulembier-Etingof-Ostrik's Conjecture B), uniqueness of the functor in characteristic two via their subterminality theorem, and a generalized Tannakian realization. The full conjecture for non-finite categories of moderate growth is not addressed.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against BEO Conjecture 1.4 as the paper states it; it claims a $k$-linear exact faithful strong symmetric monoidal functor $\\mathcal C\\to\\mathrm{Ver}_{p^n}(k)$ for every finite symmetric tensor category, every prime including 2, with $n$ depending on $\\mathcal C$. The proof was not refereed and is not formalized. The paper itself limits the claim to finite categories and makes no assertion for general moderate-growth categories.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"The finite case of a central recent conjecture in positive-characteristic tensor categories, the analogue of Deligne's theorem, known within that community. Placed below Toda's Gepner conjecture for the quintic (38), level with the Campana-Peternell conjecture in dimension six (28), above Does every abelian envelope have the quotient property? (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Fiber functors for finite symmetric tensor categories in positive characteristic","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"donovan-conjecture","name":"Donovan's conjecture on Morita finiteness of blocks with a given defect group","shortName":"Donovan's conjecture","problemNumber":null,"field":"Modular representation theory of finite groups; block theory","fieldGroup":"Algebra","statement":"Let $p$ be a prime and $k$ an algebraically closed field of characteristic $p$ (or, in the integral form, a complete discrete valuation ring $\\mathcal O$ with residue field $k$). A block of $kG$ has a defect group, a $p$-subgroup determined up to conjugacy. Donovan's conjecture states that for a fixed finite $p$-group $P$, as $G$ ranges over all finite groups, the blocks with defect group isomorphic to $P$ fall into only finitely many Morita equivalence classes. Known cases include symmetric-group blocks (Scopes), abelian 2-groups (Eaton-Livesey), extraspecial groups of order $p^3$ for $p\\ge5$ (An-Eaton) and quaternion defect groups (Eaton-Eisele-Kessar-Linckelmann-Schaeffer Fry, 2026). Does Donovan finiteness hold for every finite $p$-group, every prime, over $k$ and over $\\mathcal O$?","posedBy":"P. Donovan; recorded as Conjecture M in J. L. Alperin, Local representation theory, Santa Cruz Conference on Finite Groups, Proc. Sympos. Pure Math. 37 (1980)","yearPosed":1980,"ageNote":"The manuscripts attribute the conjecture to Peter Donovan and cite Alperin's 1980 account (from the 1979 Santa Cruz conference) as where it appears; Donovan's own formulation is not dated there.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Field paper, Theorem 1.1: for each prime $p$, algebraically closed $K$ of characteristic $p$ and bound $M$, blocks of finite groups over $K$ with defect groups of order at most $M$ represent only finitely many $K$-linear Morita classes; defect groups may be nonabelian and $p=2$ is included. Integral companion, Theorem 1.1 and Corollary: the same over $W(\\overline{\\mathbb F}_p)$ and over each fixed complete DVR of characteristic zero with algebraically closed residue field, ramified included. Consequences: bounded Cartan entries and Loewy lengths. Not shown: explicit lists or bounds on the number of classes, or Morita finiteness over coefficient fields that are not algebraically closed.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it, and the README cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of both manuscripts (TeX source), read against Donovan's conjecture as recorded by Alperin. The field paper's Theorem 1.1 is the bounded-defect-order form over every algebraically closed field of characteristic $p$ (equivalent to the fixed-defect-group form, as the paper notes); the companion's Theorem 1.1 gives the integral form over $W(\\overline{\\mathbb F}_p)$ and a corollary over every fixed complete characteristic-zero DVR with algebraically closed residue field. The field paper says its proof is independent of the integral companion, but its final coefficient-extension step recalls an argument from another unreviewed release manuscript (The Blockwise Alperin Weight Conjecture, Section 2.1). The proofs were not refereed. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Donovan's conjecture is one of the central finiteness conjectures of block theory, on Alperin's 1980 list beside the weight conjecture and pursued through decades of case-by-case work; famous within modular representation theory. Below the blockwise Alperin weight conjecture (55), tied with the word problem for Artin groups (45), above Connes' rigidity conjecture for ICC property (T) groups (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf","sourceName":"OpenAI math release: Donovan's Conjecture over Algebraically Closed Fields","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Integral Donovan Finiteness over Witt Vectors (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"polynomial-peps-gapped-ground-states","name":"Polynomial-bond-dimension PEPS approximation of gapped ground states on an open square grid","shortName":"Polynomial PEPS for gapped 2D ground states","problemNumber":null,"field":"Tensor networks; projected entangled-pair states","fieldGroup":"Mathematical physics","statement":"A projected entangled-pair state (PEPS) on a lattice places a tensor at each site, with one virtual index of dimension $D_e$ per incident edge, and contracts the virtual indices. In one dimension, gapped ground states have matrix-product approximations of polynomial bond dimension (Hastings 2007). Cirac, Garre-Rubio and Perez-Garcia (2019, Section 2.2, Questions 7-8) asked whether a spectral gap likewise ensures polynomial bond dimension at inverse-polynomial Hilbert-space error in two dimensions, for translation-invariant finite-range Hamiltonians on a torus; the best known general bound was quasipolynomial under an extra density-of-states hypothesis. Does the unique ground state of a uniformly gapped local Hamiltonian on an $L\\times L$ lattice admit a PEPS approximation with bond dimension polynomial in $L$ and error inverse-polynomial in $L$?","posedBy":"J. Ignacio Cirac, Jose Garre-Rubio and David Perez-Garcia (Questions 7-8, Rev. Mat. Complut. 2019)","yearPosed":2019,"ageNote":null,"solveType":"proved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for fixed $q\\ge2$, $J,\\Delta>0$ there are $C,c$ such that for every $L\\ge2$ and every Hamiltonian on the open grid $\\{1,\\dots,L\\}^2$ with on-site and nearest-neighbour terms of norm at most $J$, a unique ground vector $\\Omega$ and global gap $\\Delta$, some PEPS $\\Phi$ on the same grid with bond dimension at most $CL^c$ satisfies $\\min_\\theta\\|\\Phi/\\|\\Phi\\|-e^{i\\theta}\\Omega\\|\\le L^{-1}$. It does not treat the torus or translation-invariant formulation of the posed question, longer-range interactions, the structural (injective, gapped parent) PEPS conjecture of Schwarz-Buerschaper-Eisert, or efficient algorithms.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. It is the companion of 'A two-dimensional area law from a global spectral gap' (same date), whose entropy bound and collar estimates it uses as inputs.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Questions 7-8 of Cirac, Garre-Rubio and Perez-Garcia as the manuscript reports them. The posed formulation is translation-invariant on a torus; the theorem instead treats arbitrary (not necessarily translation-invariant) on-site and nearest-neighbour interactions on an open square, so it neither contains nor is contained in the posed case, hence Variant. It is an existence result: no efficient construction or contraction is claimed. It depends on the companion area-law paper, which is also unreviewed. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A precise question from a 2019 open-problem list on tensor networks, well known to the PEPS community but narrower than the area law it builds on; answered here in a neighbouring open-boundary form. Below the area-law entry in this batch at 48 and Kalton's Problem 2 at 28 for standing, level with it; above the Lami-Regula entanglement entry at 20.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Polynomial PEPS approximation of gapped square-grid ground states","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A two-dimensional area law from a global spectral gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf","kind":"paper"},{"label":"Cirac, Garre-Rubio, Perez-Garcia, Mathematical open problems in PEPS (2019)","url":"https://arxiv.org/abs/1903.09439","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kollar-pardon-semialgebraic-universal-covers","name":"The Kollar-Pardon conjecture on projective varieties with semialgebraic universal cover","shortName":"Kollar-Pardon conjecture","problemNumber":null,"field":"Complex algebraic geometry; universal covers and uniformization","fieldGroup":"Geometry & topology","statement":"In dimension one uniformization leaves only $\\mathbb P^1$, $\\mathbb C$ and the disc as universal covers. Claudon, Hoering and Kollar classified quasi-projective universal covers (assuming abundance), and Kollar and Pardon proved a lattice case with a bounded symmetric quotient. Kollar and Pardon (2012, Conjecture 2) proposed that semialgebraic openness forces a product structure. Let $X$ be a connected normal projective complex variety with universal cover $\\widetilde X$. If $\\widetilde X$ is biholomorphic to a semialgebraic open subset of a projective variety, is $\\widetilde X\\cong D\\times\\mathbb C^m\\times F$ with $D$ a bounded symmetric domain and $F$ a simply connected normal projective variety?","posedBy":"Janos Kollar and John Pardon (Algebraic varieties with semialgebraic universal cover, J. Topology 5 (2012), Conjecture 2)","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a connected normal projective complex variety $X$, $\\widetilde X$ is biholomorphic to a semialgebraic open subset of a projective variety if and only if $\\widetilde X\\cong D\\times\\mathbb C^m\\times F$ with $D$ bounded symmetric and $F$ simply connected normal projective (either may be a point); no algebraicity of deck transformations is assumed and the compact factor may be singular. The cover is quasi-projective exactly when $D$ is a point; a smooth projective variety covered by $\\mathbb C^n$ is finitely etale covered by an abelian variety. The companion proves that a connected bounded semialgebraic open subset of an affine variety with a proper cocompact holomorphic group action is smooth and a bounded symmetric domain. Not covered: non-normal or non-projective (e.g. compact Kahler) $X$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and the history section, read against Conjecture 2 as the manuscript quotes it; the proof was not refereed. Conditional inputs from the same release: the group-theoretic step uses OpenAI's abelianity theorem for special compact Kahler manifolds (catalog entry campana-abelianity-conjecture), and the quasi-projective consequences use OpenAI's log-abundance theorem; neither has been independently checked. Lean: the release's Comparator challenge SymmetricDomains (solution module OAI.Analysis.SymmetricDomains.Main, both present at the pinned commit) formalizes only the companion's bounded-domain theorem (Kollar-Pardon Question 25), a step toward but not the headline classification, so the tier is unreviewed; that statement was read here and not rebuilt.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A precise recent conjecture by two leading algebraic geometers that unifies Iitaka's affine-space uniformization question with bounded-domain uniformization; known mainly to specialists in fundamental groups of projective varieties. Placed just below Beauville's splitting conjecture for two summands (30), above the Robert-Tikuisis conjecture (25), and well below Campana's abelianity conjecture (45), on which this proof relies.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Semialgebraic universal covers of normal projective varieties","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Symmetry of semialgebraic bounded domains with compact quotient (answers Kollar-Pardon Question 25)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean Comparator challenge SymmetricDomains (companion theorem only)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SymmetricDomains.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Analysis/SymmetricDomains/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/SymmetricDomains/Main.lean","kind":"lean-proof"},{"label":"Kollar and Pardon, Algebraic varieties with semialgebraic universal cover, J. Topology 5 (2012)","url":"https://doi.org/10.1112/jtopol/jts001","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"global-spherical-shell-conjecture","name":"The Global Spherical Shell conjecture for minimal class VII surfaces with positive second Betti number","shortName":"Global Spherical Shell conjecture","problemNumber":null,"field":"Complex geometry; non-Kahler compact complex surfaces of class VII","fieldGroup":"Geometry & topology","statement":"A compact complex surface is of class VII if $b_1=1$ and its Kodaira dimension is $-\\infty$; these are the non-Kahler surfaces left open by the Enriques-Kodaira classification. With $b_2=0$ the minimal ones are Hopf and Inoue surfaces (Bogomolov, Li-Yau-Zheng, Teleman). A global spherical shell is a holomorphic embedding of a neighborhood of $S^3\\subset\\mathbb C^2\\setminus\\{0\\}$ with connected complement; Kato showed surfaces with one are degenerations of blown-up Hopf surfaces. Nakamura's classification program conjectures (Working Hypothesis 5.5) that shells always exist when $b_2>0$; Teleman proved it for $b_2=1$, and Dloussky-Oeljeklaus-Toma reduced it to finding $b_2$ rational curves. Does every minimal class VII surface with $b_2>0$ contain a global spherical shell?","posedBy":"Iku Nakamura (Working Hypothesis 5.5, Sugaku 36, 1984)","yearPosed":1984,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every minimal class VII surface with $b_2>0$ contains a global spherical shell. Corollaries: by Kato, each such surface deforms to blown-up primary Hopf surfaces, has $\\pi_1\\cong\\mathbb Z$ and is diffeomorphic to $(S^1\\times S^3)\\#b_2\\overline{\\mathbb{CP}}^2$; and aspherical compact complex surfaces satisfy $\\chi\\ge\\tfrac95|\\sigma|$. The proof builds divisors on the infinite cyclic cover and shows all their components are compact rational curves, then applies Dloussky-Oeljeklaus-Toma. It does not give a complete biholomorphic classification of the resulting Kato surfaces beyond what Kato's theory provides.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Nakamura's Working Hypothesis 5.5; it is stated for every connected compact complex surface with $b_1=1$, $b_2>0$, $\\kappa=-\\infty$ and no $(-1)$-curve, and gives a global spherical shell. The proof was not refereed. No Lean formalization of this manuscript is in the release (there is no lean/docs/060.md at the pinned commit). The argument relies on Enoki's theorem and the Dloussky-Oeljeklaus-Toma criterion, both published, and constructs the needed rational curves on the infinite cyclic cover by weighted Fourier-Laplace analysis.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The Global Spherical Shell conjecture is the main open step in classifying compact complex surfaces, open since 1984 and attacked with gauge theory by Teleman; famous among complex geometers. Placed level with the Boone-Higman conjecture and the Kadison-Kaplansky conjecture (50), below Bloch's conjecture for surfaces with pg = 0 (55) and Yau's uniformization conjecture (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Global Spherical Shells on Minimal Surfaces of Class VII","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fk-planar-maps-crt-limit-q-above-4","name":"Sheffield's finite-volume continuum random tree prediction for critical FK planar maps with q > 4","shortName":"CRT limit of FK planar maps above q = 4","problemNumber":null,"field":"Random planar maps; scaling limits","fieldGroup":"Probability & statistics","statement":"A critical Fortuin-Kasteleyn planar map with $n$ edges is a rooted planar map $M$ with an edge subset $A$, sampled with weight $q^{\\ell(M,A)/2}$, where $\\ell$ counts primal-dual interfaces. Sheffield's inventory bijection (hamburgers and cheeseburgers) shows that the encoding walk has two-dimensional Brownian fluctuations for $q<4$ and one-dimensional fluctuations for $q>4$. In the appendix of his paper Sheffield predicted that above $q=4$ the maps lose their surface geometry and become tree-like; Feng (2026) restated this as Conjecture 1.2 and proved an infinite-volume local version. For fixed $q>4$, do finite FK maps with graph distances rescaled by a constant times $n^{-1/2}$ converge to the Brownian continuum random tree in the Gromov-Hausdorff-Prokhorov topology?","posedBy":"Scott Sheffield (appendix of 'Quantum gravity and inventory accumulation'); restated by Yuyang Feng as Conjecture 1.2","yearPosed":2016,"ageNote":"The manuscript cites Sheffield, Ann. Probab. 44 (2016), appendix, for the prediction and Feng, Probab. Theory Related Fields 194 (2026), Conjecture 1.2. Sheffield's paper circulated as a preprint several years before publication.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every fixed $q>4$ there is $c_q>0$ such that the $n$-edge critical FK map, with graph distance scaled by $c_qn^{-1/2}$ and the normalised degree measure, converges in distribution to the Brownian continuum random tree in the GHP topology, through all positive integers $n$. It is proved directly under the finite law (block decomposition into a simply generated tree with finite offspring variance), not deduced from Feng's infinite-volume local limit. Not shown: the value of $c_q$, rates, uniformity in $q$, or anything at $q=4$ (treated in a companion).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. This entry's principal manuscript (September 24, 2026) is one of seven in the family; it has a Lean formalization of its main theorem. The other manuscripts treat $q\\le4$ and are listed under the Gwynne-Miller entry.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Sheffield's prediction as restated by Feng. Lean: lean/ComparatorChallenges/FKCRT.json exists with solution module OAI.Probability.FKMaps.Convergence present at the pinned commit; it is not listed in lean/formalization.yaml. Its statement OAI.FKCRT.finite_fk_maps_converge_to_brownian_crt was read here: for every $q>4$ there is $c>0$ such that, for every bounded GHP-continuous test function, the exact finite FK expectation with distances scaled by $c/\\sqrt{n+1}$ and degree measure converges to the expectation under the tree coded by a normalised Brownian excursion. The FK weight in the statement matches $q^{\\ell/2}$. This is the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A natural prediction about the tree-like phase of FK maps, stated by Sheffield and made a conjecture by Feng, with an infinite-volume version already proved; specialist and young as a named conjecture. Below the He-Schramm rigidity conjecture (32) and the Kannan-Tetali-Vempala conjecture (30), level with the Goemans-Linial integrality gap entry (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026/main.pdf","sourceName":"OpenAI math release: Brownian continuum random tree limits of finite Fortuin-Kasteleyn maps above four","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean statement: FKCRT.lean (comparator challenge)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/FKCRT.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Probability/FKMaps/Convergence.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/FKMaps/Convergence.lean","kind":"lean-proof"},{"label":"Feng, Triviality of critical FK decorated planar maps for q>4 (Conjecture 1.2)","url":"https://doi.org/10.1007/s00440-025-01424-2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"area-law-gapped-ground-states-2d","name":"The entanglement area law for gapped ground states in two dimensions","shortName":"2D area law for gapped ground states","problemNumber":null,"field":"Quantum many-body systems; entanglement and spectral gaps","fieldGroup":"Mathematical physics","statement":"Let $H=\\sum_X h_X$ be a Hamiltonian on finitely many spins $\\mathbb C^q$ placed on a finite domain $\\Lambda\\subset\\mathbb Z^2$, with interactions of bounded range $R$ and norm at most $J$, and a unique ground state $\\Omega$ separated from the rest of the spectrum by a gap $\\Delta>0$ independent of the system size. Hastings (2007) proved that in one dimension the entanglement entropy $S_\\Omega(A)$ of any interval is bounded by a constant depending only on $q,R,J,\\Delta$. The gap-to-area-law conjecture asks for the higher-dimensional analogue; in two dimensions only subvolume or near-linear bounds for frustration-free, locally gapped models were known. Is there a constant $C=C(q,R,J,\\Delta)$ such that $S_\\Omega(A)\\le C|\\partial A|$ for every region $A$ of every such two-dimensional system?","posedBy":"The gap-to-area-law conjecture, cited by the manuscript to M. B. Hastings (2007) and Arad, Landau and Vazirani (2012)","yearPosed":2007,"ageNote":"Area scaling of entanglement goes back to Bombelli-Koul-Lee-Sorkin (1986) and Srednicki (1993). The lattice conjecture for gapped ground states is cited by the manuscript to Hastings 2007, whose one-dimensional proof left higher dimensions open.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for fixed $q,R,J,\\Delta$ there is $C$ such that for every finite induced domain $\\Lambda\\subset\\mathbb Z^2$ and every Hamiltonian of range $R$ and term norm $J$ with a unique ground vector $\\Omega$ and $H-E_0\\ge\\Delta(1-|\\Omega\\rangle\\langle\\Omega|)$, every $A\\subseteq\\Lambda$ has $S_\\Omega(A)\\le C|\\partial_\\Lambda A|$. Corollary 1.2 gives the vertex-boundary forms. The companion proves that on open $L\\times L$ grids with nearest-neighbour interactions the ground state has a PEPS approximation of bond dimension $CL^c$ and global error $L^{-1}$. It does not treat dimensions three and higher, Renyi entropies, or degenerate ground states, and gives no explicit constant.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a companion manuscript of the same date, 'Polynomial PEPS approximation of gapped square-grid ground states', which uses this paper's collar estimates as its analytic input.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture. It assumes only a uniform gap for the full Hamiltonian (no local gap, no frustration-freeness, no commutativity) on any finite induced subgraph of the square lattice, including domains with holes, and bounds the entropy of every subset by a constant times its edge boundary, uniformly in domain size and shape. The release has no Lean formalization for this family. The result is two-dimensional only: the paper makes no claim for $\\mathbb Z^d$ with $d\\ge3$. The constant is existential.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"The two-dimensional area law for gapped ground states has been the central open problem of Hamiltonian complexity and tensor-network theory since Hastings settled one dimension in 2007, known across quantum information and mathematical physics. Below the Haldane gap conjecture entry at 62, whose fame reaches further, and level with or slightly above the Gotsman-Linial entry at 42 and the six-vertex Gaussian free field entry in this batch.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A two-dimensional area law from a global spectral gap","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Polynomial PEPS approximation of gapped square-grid ground states","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf","kind":"paper"},{"label":"Hastings, An area law for one-dimensional quantum systems (2007)","url":"https://arxiv.org/abs/0705.2024","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"self-dual-uniform-matroid-zfc","name":"Can an infinite self-dual uniform matroid be constructed in ZFC?","shortName":"Self-dual uniform matroid in ZFC","problemNumber":null,"field":"Infinite matroid theory; set theory","fieldGroup":"Combinatorics","statement":"An infinite matroid is uniform if exchanging any element of a basis for any element outside it again gives a basis. Bowler and Geschke developed a basis criterion for infinite uniform matroids and constructed countable self-dual uniform examples under Martin's axiom for countable posets, hence under the continuum hypothesis, and left open whether such a matroid exists in ZFC alone. Gollin and Joo recorded even the ZFC existence of a uniform matroid of infinite rank and infinite corank as open. Is there, provably in ZFC, an infinite self-dual uniform matroid?","posedBy":"Nathan Bowler and Stefan Geschke (Proc. AMS 2016); recorded again by Gollin and Joo (2025)","yearPosed":2016,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The manuscript constructs, in ZFC, a self-dual uniform matroid $Q$ on a countable set with every basis and its complement infinite, answering the Bowler-Geschke question and the Gollin-Joo existence question affirmatively. Its bases satisfy an extra ordinal inequality used to build the packing/covering counterexample. It does not address uncountable ground sets or classify such matroids.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. This answer is a building block of the infinite matroid packing/covering counterexample in the same manuscript.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction's claim and Remark (uniform local matroid) were read against the question as the paper quotes it from Bowler-Geschke and Gollin-Joo. The Lean challenge InfiniteMatroidCorollaries.json (theorem OAI.InfiniteMatroidCounterexample.partitional_intersection_counterexample, module OAI.Combinatorics.InfiniteMatroid.Corollaries, file exists at the pinned commit, not in the formalization catalogue) states the existence of a self-dual uniform Mathlib matroid Q on an infinite countable type. Left unreviewed under the strict tier rule: the question is about provability in ZFC, and the Lean statement expresses existence in Lean's own foundations, not ZFC, so it does not state the headline as posed. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A set-theoretic existence question from one paper in a small field, recorded again recently. Placed below the infinite matroid intersection entry of this family (28) and Unimodality of Kazhdan-Lusztig Polynomials of Matroids (25), around a typical Erdos problem such as Erdos Problem #654 (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A Counterexample to the Infinite Matroid Packing/Covering Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean corollaries (includes the self-dual uniform matroid Q)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/InfiniteMatroid/Corollaries.lean","kind":"lean-proof"},{"label":"Bowler-Geschke, Self-dual uniform matroids on infinite sets","url":"https://doi.org/10.1090/proc/12667","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"partition-principle-implies-choice","name":"Does the Partition Principle imply the Axiom of Choice?","shortName":"Partition Principle vs Choice","problemNumber":null,"field":"Set theory; choice principles and symmetric models","fieldGroup":"Logic & foundations","statement":"The Partition Principle (PP) says that whenever there is a surjection from a set $X$ onto a set $Y$, there is an injection from $Y$ into $X$; equivalently every partition of a set injects into the set. Over ZF the Axiom of Choice implies PP, and PP implies choice for well-orderable families. The converse is the old question: does PP imply AC over ZF? Equivalently, is ZF + PP + $\\neg$AC consistent relative to ZF?","posedBy":"Classical open problem of choiceless set theory; the principle is traced to Beppo Levi (1902)","yearPosed":null,"ageNote":"The manuscript cites da Silva (2021) tracing the Partition Principle to Beppo Levi in 1902 and Blass-Kulshreshtha (2026 version) listing PP implies AC as open, but gives no date for the first posing of the implication question. It also notes that Gilson's September 2026 corrections withdrew three earlier claimed constructions.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: Con(ZF) implies Con(ZF + PP + ACwo + $\\neg$AC). Theorem 1.2: over every countable transitive model $V$ of ZFC there are a set forcing and a transitive symmetric submodel $W$ with $V\\subseteq W\\subseteq V[G]$, the same ordinals, $W\\models$ ZF + PP + ACwo + $\\neg$AC, and no new countable sequences of elements of $V$. The construction makes every non-wellorderable image of a fixed set bijective with it and uses Ryan-Smith's local-to-global reduction. Remarks draw consequences for the dual Cantor-Schroeder-Bernstein principle and the Weak Partition Principle. It does not settle whether the Weak Partition Principle implies AC or PP, nor other cardinal-comparison principles beyond those remarks.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 24, 2026) has no companion in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the manuscript were read against the question; Theorem 1.1 (if ZF is consistent, so is ZF + PP + ACwo + not AC) answers it as posed. lean/formalization.yaml lists PartitionPrinciple (declaration ...TransitiveGround.exists_model_partitionPrinciple_without_choice, file OAI/SetTheory/PartitionPrinciple/Models/Separation.lean). Its comparator statement was read: it assumes a countable transitive ground with an internal strongly inaccessible cardinal and yields a transitive ZF model of PP with Choice failing, which is narrower than the paper's Theorem 1.2 (no inaccessible, and also ACwo and no new countable sequences). A second challenge, ComparatorChallenges/PartitionConsistency.json, is not in the formalization catalogue; its solution module OAI/SetTheory/PartitionConsistency/Main.lean exists at the pinned commit, and its statement was read here: syntactic Con(ZF) implies Con(ZF + PP + ACwo + not AC) for an explicit first-order natural deduction over the pure membership language. That is the headline. Neither was rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"PP versus AC is one of the oldest open questions about the Axiom of Choice and well known to every set theorist working without choice, though less known outside set theory. Placed level with Brennan's conjecture (45) and the word problem for Artin groups (45), above Kalton's Problem 2 (28), and below Sidorenko's conjecture (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf","sourceName":"OpenAI math release: The Partition Principle does not imply Choice","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (transitive-ground construction)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/SetTheory/PartitionPrinciple/Models/Separation.lean","kind":"lean-proof"},{"label":"Comparator statement: PartitionPrinciple.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PartitionPrinciple.lean","kind":"lean-proof"},{"label":"Lean proof (relative consistency, not in formalization catalogue)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/SetTheory/PartitionConsistency/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: PartitionConsistency.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PartitionConsistency.lean","kind":"lean-proof"},{"label":"Blass and Kulshreshtha, Cardinal well-foundedness and Choice (lists PP implies AC as open)","url":"https://arxiv.org/abs/2310.09643v3","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"continuum-coulomb-energy-qma-hard","name":"QMA-hardness of the electronic Coulomb ground energy in the full continuum, with unit nuclear charges","shortName":"Continuum Coulomb energy is QMA-hard","problemNumber":null,"field":"Quantum complexity; electronic structure","fieldGroup":"Quantum information & computing","statement":"Given clamped nuclei at rational positions in $\\mathbb R^3$ and $N$ electrons, the electronic Hamiltonian is $H=-\\frac12\\sum_i\\Delta_{x_i}-\\sum_{i,\\alpha}Z_\\alpha/|x_i-R_\\alpha|+\\sum_{i 11/12 zero-free region whose write-up was human-edited). Both manuscripts are authored 'OpenAI' and name no human author. The unit-charge paper does not invoke the binary-charge companion's hardness theorem; both start from the Cubitt-Montanaro-Piddock Heisenberg source.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts were read against the posed question. lean/docs/275.md points to ComparatorChallenges/ContinuumCoulombHardness.json (OAI.ContinuumCoulomb.unit_coulomb_qmaHard and binary_coulomb_qmaHard, module OAI.MathematicalPhysics.ContinuumCoulomb.Reduction.Hardness). The solution file exists at the pinned commit; the challenge is not in the formalization catalogue formalization.yaml. The statement was read here: every promise problem in QMA (defined in the file via uniform polynomial-size gate circuits) reduces by polynomial-time many-one maps to the promise problem whose instances are distinct rational positions of unit-charge nuclei, a unary electron count and thresholds at least 1 apart, with energy the infimum of the Coulomb quadratic form over normalized antisymmetric spinful H1 states. That is the headline claim. Not rebuilt here. Membership in QMA is not claimed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A natural question raised in the discussion of a 2022 paper on the complexity of quantum chemistry; specialist and recent, though it completes the hardness picture for the physical Coulomb Hamiltonian. Placed below A Quantum Oracle Separation Between QMA(2) and QMA (36) and Sharp Bounds on the Ground State Energy of the SYK Model (20), level with Spectral Edge of the Quartic SYK Model (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf","sourceName":"OpenAI math release: QMA-hardness of continuum Coulomb energy with unit nuclear charges","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Continuum Coulomb hardness with binary nuclear charges","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026.pdf","kind":"paper"},{"label":"Lean proof (OAI.ContinuumCoulomb.unit_coulomb_qmaHard)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/ContinuumCoulomb/Reduction/Hardness.lean","kind":"lean-proof"},{"label":"O'Gorman-Irani-Whitfield-Fefferman, Intractability of electronic structure in a fixed basis","url":"https://doi.org/10.1103/PRXQuantum.3.020322","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sk-low-temperature-free-energy-fluctuations","name":"The $n^{1/6}$ free-energy fluctuation scale of the low-temperature Sherrington-Kirkpatrick model, with a limiting law","shortName":"Low-temperature SK fluctuation scale","problemNumber":null,"field":"Spin glasses; Sherrington-Kirkpatrick model","fieldGroup":"Probability & statistics","statement":"In the zero-field Gaussian Sherrington-Kirkpatrick model $H_n(\\sigma)=\\frac{1}{\\sqrt n}\\sum_{i1$, Crisanti, Paladin, Sommers and Vulpiani predicted that the free energy per spin fluctuates on the scale $n^{-5/6}$, that is $\\mathrm{sd}(F_n)\\asymp n^{1/6}$, and Parisi and Rizzo developed the related large-deviation calculation. Is the standard deviation of $F_n$ of order $n^{1/6}$ at every fixed $\\beta>1$, and does the centered, rescaled free energy converge to a limiting law?","posedBy":"A. Crisanti, G. Paladin, H.-J. Sommers and A. Vulpiani (J. Phys. I France 2, 1992); developed by Parisi and Rizzo (2008-2009)","yearPosed":1992,"ageNote":"The prediction is in Crisanti-Paladin-Sommers-Vulpiani, 'Replica trick and fluctuations in disordered systems' (1992), following Kondor's 1983 replica expansion. The best rigorous bounds before this work were Aronow-Lopatto (2026): n^(4/15) <= Var F_n <= n^(7/15).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Companion (fluctuation scale): for every fixed $\\beta>1$, $\\mathrm{sd}(F_n)=n^{1/6+o(1)}$ and $|F_n-\\mathbb EF_n|$ is typically between $n^{1/6-\\eta}$ and $n^{1/6+\\eta}$. Principal (limiting law): $\\mathrm{Var}(F_n)/n^{1/3}$ converges to a constant in $(0,\\infty)$, every moment of the $n^{-1/6}$-rescaled completed free energy converges with a polynomial rate, exponential moments are uniformly bounded, and $(F_n-\\mathbb EF_n)/\\mathrm{sd}(F_n)$ converges along all $n$ to a nondegenerate law with mean 0 and variance 1, uniquely determined by its moments. The law is not identified with a named distribution and no formula for the variance constant is given.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). Both manuscripts are authored 'OpenAI' and name no human author. The limiting-law paper takes its finite-tree toolkit and the exponent from the companion fluctuation-scale paper of the same date.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts were read against the CPSV prediction. The family has no Lean formalization (lean/docs/217.md does not exist at the pinned commit). The limiting-law paper depends on precisely listed inputs from its companion (stated in a companion-interface appendix) and on Lopatto's 2026 interval-support theorem for the Parisi measure. Results are for fixed beta > 1, zero field, Gaussian couplings and Ising spins; nothing is claimed uniformly as beta tends to 1 or infinity, and the limiting law is characterized only by its moments, not identified with a named distribution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The one-sixth fluctuation exponent is a long-standing physics prediction for the SK model, open rigorously for three decades and known mainly to spin-glass specialists; it refines the Parisi formula rather than matching it in standing. Placed below Full-RSB in the Sherrington-Kirkpatrick spin glass (35), the Mezard-Parisi formula for diluted spin glasses (32) and Talagrand's critical Sherrington-Kirkpatrick overlap conjecture (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf","sourceName":"OpenAI math release: The low-temperature Sherrington-Kirkpatrick free-energy limiting law","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The low-temperature Sherrington-Kirkpatrick fluctuation scale","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf","kind":"paper"},{"label":"Crisanti-Paladin-Sommers-Vulpiani (1992), Replica trick and fluctuations in disordered systems","url":"https://doi.org/10.1051/jp1:1992213","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"single-fold-diophantine-conjecture","name":"Matiyasevich's single-fold conjecture: every r.e. set has a single-fold Diophantine representation","shortName":"Single-fold Diophantine conjecture","problemNumber":null,"field":"Computability; Hilbert's tenth problem","fieldGroup":"Logic & foundations","statement":"By the Davis-Putnam-Robinson-Matiyasevich theorem every recursively enumerable set $S\\subseteq\\mathbb N^n$ has a Diophantine representation: a polynomial $P$ with integer coefficients such that $a\\in S$ iff $P(a,w)=0$ for some $w\\in\\mathbb N^m$. The representation is single-fold if such $w$ is unique when it exists, finite-fold if there are finitely many. Matiyasevich (1974) proved single-fold exponential Diophantine representations and posed the polynomial case, reducing it to a single-fold representation of exponentiation; the finite-fold conjecture is the weaker form. Does every recursively enumerable set have a single-fold Diophantine representation?","posedBy":"Yuri Matiyasevich","yearPosed":1974,"ageNote":"Matiyasevich, The existence of non-effectivizable estimates in the theory of exponential Diophantine equations, Zap. Nauchn. Sem. LOMI 40 (1974), Section 1; finite-fold form in Matiyasevich, Towards finite-fold Diophantine representations (2010); both cited by the manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every r.e. $S\\subseteq\\mathbb N^n$, $n\\ge1$, has a polynomial $P\\in\\mathbb Z[A_1..A_n,W_1..W_m]$ with $\\#\\{w\\in\\mathbb N^m:P(a,w)=0\\}$ equal to 1 for $a\\in S$ and 0 otherwise. Hence the finite-fold conjecture holds, and Diophantine solvability over $\\mathbb N$ stays undecidable under an at-most-one-solution promise (Section 8). The construction bounds a Pell-type exponential with a cutoff from rational points on a rank-one elliptic curve. Witnesses range over $\\mathbb N$; the paper does not claim the analogue over $\\mathbb Z$ beyond what follows from it, nor any bound on degree or number of variables.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 24, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Matiyasevich's conjecture; it counts all auxiliary variables and gives exactly one complete witness tuple for members and none for nonmembers, over $\\mathbb N$. lean/formalization.yaml lists a main result (comparator SingleFold, declaration OAI.SingleFold.main). The comparator statement was read here: for every $n\\ge1$ and every r.e. $S\\subseteq\\mathbb N^n$ (via Mathlib's REPred) there are $m\\ge1$ and an integer polynomial in $n+m$ variables with $a\\in S$ iff some $w\\in\\mathbb N^m$ is a zero, and the zero $w$ unique for each $a$. This states the headline. The undecidability-under-promise corollary is not separately formalized (lean/docs/242.md). Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The best known open refinement of the negative solution of Hilbert's tenth problem, posed by Matiyasevich in 1974 and tied to whether bounds on solutions of Diophantine equations can be effective. Famous in logic and computability, little known outside. Placed above the Flint Hills series entry (35) and the Artin groups CAT(0) conjecture (40), and far below Hilbert's tenth problem over the rationals (70).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Single-fold Diophantine representations","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.SingleFold.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/SingleFold/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SingleFold.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SingleFold.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"equivariant-hikita-conjecture-quivers","name":"The equivariant cohomological Hikita conjecture for quiver gauge theories","shortName":"Equivariant Hikita conjecture for quivers","problemNumber":null,"field":"Geometric representation theory; symplectic duality; Coulomb branches","fieldGroup":"Geometry & topology","statement":"Hikita conjectured that for a symplectic dual pair the cohomology ring of the symplectic resolution on one side is the coordinate ring of the scheme-theoretic fixed locus of a torus on the other, proving it for Hilbert schemes of points and some type A examples. For a quiver gauge theory the Higgs side is a Nakajima quiver variety $X$ and the dual is the Braverman-Finkelberg-Nakajima Coulomb branch $Y$. The equivariant form (Nakajima, recorded by Kamnitzer-Tingley-Webster-Weekes-Yacobi, formulated by Dumanski-Krylov) asks for compatibility with the tautological coefficient maps from $H^*_{G\\times F}(\\mathrm{pt})$. Is $H^*_F(X;\\mathbb C)\\cong\\mathbb C[Y^\\nu_{\\mathfrak f}]$ as graded algebras over the common coefficient ring, for every quiver?","posedBy":"Tatsuyuki Hikita (arXiv:1501.02430, IMRN 2017); equivariant form in Kamnitzer-Tingley-Webster-Weekes-Yacobi (2019), Conj. 8.9-8.10; form proved: Dumanski-Krylov (2025), Conj. 2.1","yearPosed":2015,"ageNote":"Hikita's paper was posted in January 2015 and published in 2017. The quiver-gauge equivariant formulation the manuscript proves is that of Dumanski-Krylov (2025).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims the equivariant cohomological Hikita conjecture for all quiver gauge theories with regular stability: the flavor-equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic stability-cocharacter fixed locus of the flavor-deformed Coulomb branch, including the empty case. It does NOT prove the quantized Hikita statement, nor extend to arbitrary reductive gauge groups, where Hikita-type identifications can fail, nor treat non-regular stability.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. Single manuscript; no Lean formalization.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against Dumanski-Krylov Conjecture 2.1, Equation (2.3), which the paper says it proves: for any finite quiver (loops and multiple arrows allowed), dimension and framing vectors, commuting flavor torus and regular stability (semistable equals stable, free action), both coefficient maps are surjective with equal kernels, giving the graded isomorphism with nilpotents retained. The regularity hypothesis is the paper's own scope limit. Inputs include McGerty-Nevins Kirwan surjectivity and Weekes's generation theorem. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The Hikita conjecture is a well-known prediction of symplectic duality, studied by a sizeable geometric representation theory community; the quiver case is its main test bed. A conjecture famous within one community: level with the Period-Index entry (30), below the quantum geometric Langlands irrational-level entry (36) and the Toda Gepner conjecture (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026/main.pdf","sourceName":"OpenAI math release: The equivariant cohomological Hikita conjecture for arbitrary quivers","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Dumanski and Krylov, K-theoretic Hikita conjecture for quiver gauge theories (2025)","url":"https://arxiv.org/abs/2509.06226v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"p-adic-section-conjecture","name":"The local p-adic section conjecture for curves of genus at least two","shortName":"p-adic section conjecture","problemNumber":null,"field":"Anabelian geometry; etale fundamental groups of p-adic curves","fieldGroup":"Number theory","statement":"Let $k$ be a finite extension of $\\mathbb Q_p$ and $X/k$ a smooth proper geometrically connected curve of genus at least two, with arithmetic etale fundamental group sequence $1\\to\\pi_1(X_{\\bar k})\\to\\pi_1(X)\\to G_k\\to1$. Each rational point gives a section of $\\pi_1(X)\\to G_k$, well defined up to conjugation. The local p-adic section conjecture, the local analogue of the section conjecture in Grothendieck's 1983 letter to Faltings, asserts that this map from $X(k)$ to conjugacy classes of sections is a bijection. Koenigsmann proved the birational analogue, Pop-Stix localized every section at a valuation, and Bresciani handled toric sections. Is every section of $\\pi_1(X)\\to G_k$ induced by a unique rational point of $X$?","posedBy":"Local analogue of Grothendieck's section conjecture (letter to Faltings, 27 June 1983); studied as the p-adic section conjecture by Koenigsmann (2005), Pop (2010) and Pop-Stix (2017)","yearPosed":null,"ageNote":"The manuscript presents the local conjecture as an analogue of Grothendieck's 1983 global conjecture and does not name who first stated the local form; Koenigsmann (2005) and Pop-Stix (2017) are the earliest cited works on it.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every finite extension $k/\\mathbb Q_p$ and every smooth proper geometrically connected curve $X/k$ of genus at least two, $X(k)\\to\\{\\text{sections of }\\pi_1(X)\\to G_k\\}/\\pi_1(X_{\\bar k})$ is bijective. Through Theorem 1.2 every section localized at a rank-one valuation lifts to a birational section. It does NOT prove the global section conjecture over number fields in general (see the separate entry for the cases it does reach), and says nothing about affine hyperbolic curves or curves over other local fields.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The proof patches finite etale covers over annuli, using lifting theorems of Amini-Baker-Brugalle-Rabinoff and the resolution of nonsingularities of Mochizuki-Tsujimura as cited inputs.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture as posed: for every prime $p$, every finite $k/\\mathbb Q_p$ and every smooth proper geometrically connected curve of genus at least two, the point-to-section map is bijective, with conjugation by the full geometric fundamental group. The key new input is Theorem 1.2 (the universal etale field is dense in the separable closure at every rank-one valuation extending the p-adic norm), combined with Koenigsmann's birational theorem and the Pop-Stix localization. No Lean formalisation exists for this family. The release README warns that some unformalised results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The p-adic section conjecture is a central open problem of anabelian geometry, the local form of Grothendieck's conjecture, worked on by Koenigsmann, Pop, Stix and others for two decades. Level with the Hodge standard conjecture for abelian varieties (50), below the Zilber-Pink conjecture for abelian varieties over number fields (58), above Grothendieck's Finite Flat Group Scheme Order Question (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf","sourceName":"OpenAI math release: The p-adic section conjecture","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion: Etale covers with a prescribed exterior sheet","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf","kind":"paper"},{"label":"Grothendieck, letter to Faltings (27 June 1983), in Geometric Galois Actions 1","url":"https://doi.org/10.1017/CBO9780511758874.018","kind":"problem-record"},{"label":"Earlier edition (September 24, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-p-adic-section-conjecture-September-24-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"saturation-conjecture-type-d-spin","name":"The Kapovich-Millson saturation conjecture for simply laced groups, type D: tensor saturation for $\\mathrm{Spin}(2n)$","shortName":"Saturation conjecture for Spin(2n)","problemNumber":null,"field":"Representation theory: tensor product multiplicities and saturation","fieldGroup":"Algebra","statement":"Let $G$ be a simply connected complex semisimple group with root lattice $Q$, and $V(\\lambda)$ the irreducible representation of highest weight $\\lambda$. For dominant $\\lambda,\\mu,\\nu$ with $\\lambda+\\mu+\\nu\\in Q$, saturation asks whether $(V(N\\lambda)\\otimes V(N\\mu)\\otimes V(N\\nu))^G\\ne0$ for some positive integer $N$ forces $(V(\\lambda)\\otimes V(\\mu)\\otimes V(\\nu))^G\\ne0$. Knutson and Tao proved this for type A. Kapovich and Millson conjectured it for all simply laced groups. Known before this work: $\\mathrm{Spin}(8)$ (Kapovich-Kumar-Millson), $\\mathrm{Spin}(10)$ and $\\mathrm{Spin}(12)$ (Kiers), and saturation factors two or four for classical groups (Kapovich-Millson, Belkale-Kumar, Sam, Hong-Shen), which do not give the factor-one statement. Does saturation with factor one hold for $\\mathrm{Spin}(2n)$ for every $n$?","posedBy":"M. Kapovich and J. J. Millson, Structure of the tensor product semigroup, Asian J. Math. 10 (2006), Conjecture 1.4(1); see also Kapovich-Kumar-Millson (2009), Conjecture 1.5","yearPosed":2006,"ageNote":"The Kapovich-Millson paper appeared on arXiv in 2005 (math/0508186) and in print in 2006. The conjecture covers all simply laced types; this entry records its type-D part, and types $E_6,E_7,E_8$ remain.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $G=\\mathrm{Spin}(2n)$, $n\\ge2$, and dominant integral $\\lambda,\\mu,\\nu$ with $\\lambda+\\mu+\\nu\\in Q(D_n)$, and every positive integer $N$, $(V(\\lambda)\\otimes V(\\mu)\\otimes V(\\nu))^G\\ne0$ if and only if $(V(N\\lambda)\\otimes V(N\\mu)\\otimes V(N\\nu))^G\\ne0$. This settles the type-D part of the simply laced saturation conjecture, extending the known cases $\\mathrm{Spin}(8),\\mathrm{Spin}(10),\\mathrm{Spin}(12)$. Not shown: the exceptional types $E_6,E_7,E_8$, non-simply-laced groups, or any statement about multiplicities beyond nonvanishing.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source against Kapovich-Millson Conjecture 1.4(1); the theorem states factor-one saturation for $\\mathrm{Spin}(2n)$, $n\\ge2$, under the root-lattice congruence, including half-integral spin weights. The proof was not refereed. It constructs integral schedules from real Littelmann-type paths, deforms switch coefficients to integrality, and produces tensor intertwiners through exterior and spin columns, with a preprojective-module step removing zero blocks. Exceptional simply laced types are not addressed. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The simply laced saturation conjecture is a well-known successor to Knutson-Tao in combinatorial representation theory, open for twenty years with case-by-case progress up to Spin(12); the full type-D case is a substantial part but leaves the exceptional types. Scored below the tensor square conjecture for symmetric groups (28) and unimodality of matroid Kazhdan-Lusztig polynomials (25), above the Pak-Slonim stretched Schubert conjecture (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026.pdf","sourceName":"OpenAI math release: Tensor saturation for even spin groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kapovich and Millson, Structure of the tensor product semigroup (arXiv:math/0508186)","url":"https://arxiv.org/abs/math/0508186","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"log-abundance-conjecture-char-zero","name":"The log abundance conjecture in characteristic zero (rational boundary, every dimension), with canonical nonvanishing","shortName":"Log abundance conjecture","problemNumber":null,"field":"Birational geometry; minimal model program","fieldGroup":"Algebra","statement":"For a projective log canonical pair $(X,B)$ the abundance conjecture of the minimal model program asserts that if the log canonical divisor $K_X+B$ is nef then it is semiample: some positive multiple is generated by global sections, so numerical positivity defines a morphism. The companion nonvanishing conjecture asserts that a pseudo-effective canonical class on a smooth projective variety has a nonzero pluricanonical section. Abundance was known in dimension at most three (Miyaoka, Kawamata, Keel-Matsuki-McKernan) and in special cases beyond. Is every nef $\\mathbb Q$-Cartier $K_X+B$ on a projective log canonical pair with effective rational boundary, over an algebraically closed field of characteristic zero, semiample?","posedBy":"Central conjecture of the minimal model program (Kawamata, Miyaoka, Reid and others); the manuscripts name no single poser and cite its threefold history","yearPosed":null,"ageNote":"The manuscripts do not say who posed the conjecture or when. Its threefold cases were proved between 1988 and 1994 (Miyaoka, Kawamata, Keel-Matsuki-McKernan), and it has been a standing conjecture of the minimal model program since the 1980s.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a normal projective lc pair $(X,B)$ over an algebraically closed field of characteristic zero, with $B$ effective rational and $K_X+B$ $\\mathbb Q$-Cartier and nef, some positive Cartier multiple of $K_X+B$ is globally generated. The induction also gives good minimal models for lc pairs with pseudo-effective real adjoint and, over $\\mathbb C$, canonical nonvanishing for smooth projective varieties in every dimension; consequences include rational slc abundance (Fujino-Gongyo) and finite generation of lc rings. Not shown: semiampleness with real boundaries, positive characteristic, or any bound on the multiple. The family's compact Kahler log abundance theorem assumes logarithmic Iitaka subadditivity (claimed in family 033) and is not part of this entry.","claimIssueNote":"The release contradicts itself on this claim. The 24 September manuscript's abstract states that the rational-boundary log abundance conjecture is proved in every dimension. An appendix of the release's own 5 October manuscript on compact Kahler fourfolds calls the same theorem \"a positive conditional resolution of the log abundance conjecture. It is not an unconditional abundance theorem\", resting on a logarithmic Iitaka subadditivity assumption that it says follows from a lemma elsewhere in the release. Until OpenAI clarifies which reading is meant, or a specialist checks the chain, the entry is Contested.","solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. Family 034 has fourteen manuscripts dated September 24 to October 5, 2026, which cite one another as inputs; this entry draws on the manuscript named as its source. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the introduction of 'Log abundance in characteristic zero' were read against the conjecture as stated there; the claim is unconditional and covers every dimension. The proof (a dimension induction using a signed-boundary criterion, Hodge-module lifting through infinitesimal neighbourhoods and Hashizume's reduction to smooth nonvanishing) was not refereed. No Lean formalization. A reader should know that the release is not consistent about whether projective abundance is unconditional: the October 5 companion on Kahler fourfolds proves projective log abundance in its appendices and calls it a conditional resolution, not an unconditional abundance theorem, assuming logarithmic Iitaka subadditivity, while this September 24 manuscript claims it outright. The family's effective Iitaka, index and Campana-Peternell results use this manuscript as an input. Listed as Contested because the release itself disagrees on whether the theorem is conditional; see the claim issue.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":75,"significanceNote":"Abundance with nonvanishing is the central open conjecture of the minimal model program, open from dimension four on for decades. Placed above the 67 percent critical-line result (68) and the Jacobian Conjecture (65), and below the unforced 3D Euler blowup entry (80) and the Navier-Stokes entry (87), as a problem central to one large field rather than to all of mathematics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Log abundance in characteristic zero","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Kahler version (conditional on log Iitaka subadditivity): Log abundance for compact Kähler spaces under logarithmic Iita","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-4-2026/main.pdf","kind":"paper"},{"label":"Fourfold nonvanishing by minimal metrics and moving jets","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf","kind":"paper"},{"label":"Lifting sections from the reduced support of an adjoint","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026/paper.pdf","kind":"paper"},{"label":"Minimal metrics and interior injectivity for nef adjoints","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026/paper.pdf","kind":"paper"},{"label":"Abundance after nonvanishing for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026/paper.pdf","kind":"paper"},{"label":"Conditional good minimal models for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-5-2026/paper.pdf","kind":"paper"},{"label":"Existence of minimal models for generalized lc pairs (OpenAI math release)","url":"https://vibemathed.com/problem/minimal-model-existence-generalized-lc-pairs","kind":"other"},{"label":"Revised edition (October 6, 2026): Conditional good minimal models for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ball-evans-approximation-problem-3d","name":"The Ball-Evans approximation problem in three dimensions: strong approximation of Sobolev homeomorphisms by diffeomorphisms","shortName":"Ball-Evans approximation problem in 3D","problemNumber":null,"field":"Geometric function theory; Sobolev homeomorphisms and nonlinear elasticity","fieldGroup":"Analysis","statement":"In nonlinear elasticity a deformation is a map between domains whose injectivity expresses non-interpenetration. Smooth approximation of a Sobolev map is elementary if injectivity is dropped, but convolution destroys it. Ball, attributing the question to Evans, asked whether a Sobolev homeomorphism $f\\in W^{1,p}(\\Omega;\\mathbb R^n)$ can be approximated in the $W^{1,p}$ norm by diffeomorphisms (or piecewise affine homeomorphisms). In the plane this was proved for $12$ in the other. The method first builds onto locally bi-Lipschitz approximants with small Sobolev error, then smooths them with fine pointwise control using three-dimensional PL topology (Moise, Bing, Hamilton). It does not give convergence of inverses, approximation by piecewise affine maps with inverse control, or anything in dimensions $n\\ge4$ for $p\\ge\\lfloor n/2\\rfloor$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 24, 2026: one for $1\\le p\\le2$ and one for every finite $p>2$.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorems of both manuscripts were read against Ball's formulation. Together they state that for every $1\\le p<\\infty$ and arbitrary nonempty bounded connected open $\\Omega,\\Lambda\\subset\\mathbb R^3$, each $W^{1,p}$ homeomorphism $f:\\Omega\\to\\Lambda$ is a $W^{1,p}$ limit of $C^\\infty$ diffeomorphisms onto the same $\\Lambda$, with no boundary regularity, inverse Sobolev regularity or Jacobian hypothesis. Classified partial because the posed question is dimension-free: dimension two was already known, dimensions four and up are negative for small $p$ and otherwise untouched. The papers state that no convergence of inverses is claimed. Not refereed; no Lean formalization.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The Ball-Evans problem is a well-known question in nonlinear elasticity and the theory of Sobolev homeomorphisms since about 2000, and three dimensions is its physically central open case. Below the Lane-Emden conjecture (40), level with Polya's conjecture for Neumann balls (35), above the He-Schramm rigidity conjecture (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026/main.pdf","sourceName":"OpenAI math release: Strong diffeomorphic approximation in three dimensions for 1 <= p <= 2","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Strong diffeomorphic approximation in three dimensions for p > 2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kervaire-invariant-problem-prime-three","name":"The Kervaire invariant problem at the prime three: which standard classes $b_j$ survive the mod-3 Adams spectral sequence","shortName":"Kervaire invariant problem at p = 3","problemNumber":null,"field":"Stable homotopy theory; odd-primary Kervaire invariant","fieldGroup":"Geometry & topology","statement":"In the mod-$p$ Adams spectral sequence $\\mathrm{Ext}_{\\mathcal A_p}^{s,t}(\\mathbb F_p,\\mathbb F_p)\\Rightarrow \\pi_*^S$, the odd-primary analogues of the Kervaire classes are $b_j\\in E_2^{2,2(p-1)p^{j+1}}$. Toda showed that $b_1$ supports a differential at every odd prime, and Ravenel (1978) proved that no $b_j$ with $j\\ge1$ survives for $p\\ge5$. At $p=3$ the argument fails: $b_0$ (stem 10) and $b_2$ (stem 106) survive, and the fate of the higher classes was open; Hill, Hopkins and Ravenel proposed attacking it through the action of $C_9$ on height-six Morava $E$-theory. The strong form also asks for representatives of order $p$. At $p=3$, for which $j$ does $b_j$ survive to $E_\\infty$, and when it does, is it detected by an element of order three?","posedBy":"Douglas Ravenel (1978), whose odd-primary nonexistence theorem left p = 3 open; program of Hill, Hopkins and Ravenel (2011)","yearPosed":1978,"ageNote":"The manuscript cites Ravenel, Math. Proc. Cambridge Philos. Soc. (1978) for the p >= 5 theorem and the special behaviour at three, Hill-Hopkins-Ravenel's 2011 manuscript on the 3-primary problem, and Belmont-Ray's 'Towards the p=3 Kervaire invariant problem'. The order-p refinement is Selick's (1983) reformulation.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: at $p=3$, $K=K_3=\\{0,2,3\\}$, i.e. $b_j$ survives to $E_\\infty$ exactly for $j=0,2,3$ (stems 10, 106 and 322), and each surviving class is detected by an element of additive order three; $b_1$ and all $b_j$ with $j\\ge4$ die. The new content is the order-three element in stem 322 and the exclusion of every $j\\ge4$. Corollaries via Amelotte and Selick: $\\Omega S^{2n+1}\\{3\\}$ splits nontrivially exactly for $n\\in\\{3,27,81\\}$, and the Anick space $T^{2n+1}(3)$ is homotopy associative exactly for those $n$. Side result: an infinite-order class in $\\pi_{2n}(E_6^{hC_9})$ for every $n$, refuting the full-gap clause of Belmont-Ray's Conjecture 0.1. Not covered: other odd primes (already settled for $p\\ge5$) or the prime-two problem.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 24, 2026. The paper says it uses classical inputs (Toda's differentials, the Miller-Ravenel-Wilson two-line, Amelotte's and Selick's criteria) and proves Belmont-Ray's coefficient model itself.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and the reduction of the proof to its two branches were read against the problem as Ravenel and Hill-Hopkins-Ravenel framed it; the proof was not refereed. No Lean formalization exists for this family. The paper states its scope: the standard family $b_j$ in the mod-3 Adams spectral sequence, not the classical prime-two Kervaire problem. The exclusion of $j\\ge4$ runs through a height-six $C_9$ detector and finite-page outgoing differentials; the paper says it does not use the 972-periodicity or the degree $-2$ vanishing clause of Belmont-Ray's Conjecture 0.1, and it claims to refute the latter clause.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"The last open prime of the odd-primary Kervaire invariant problem, open since Ravenel's 1978 theorem and the subject of a proposed Hill-Hopkins-Ravenel program; well known in stable homotopy theory but far less famous than the prime-two problem. Placed just above Smith-Toda complexes at every height (30), below the Curtis conjecture on the stable Hurewicz image (48) and the Kervaire conjecture for groups (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The Kervaire invariant problem at the prime three","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bloch-conjecture-surfaces-pg-zero","name":"Bloch's Conjecture for Surfaces with $p_g=0$","shortName":"Bloch's conjecture for surfaces","problemNumber":null,"field":"Algebraic geometry; algebraic cycles","fieldGroup":"Geometry & topology","statement":"For a smooth connected projective complex surface $S$, the Albanese map $\\mathrm{CH}_0(S)^0\\to\\mathrm{Alb}(S)(\\mathbb{C})$ on degree-zero zero-cycles modulo rational equivalence is surjective. Mumford showed that $p_g(S)>0$ makes $\\mathrm{CH}_0(S)$ infinite-dimensional, and Bloch conjectured the converse. Bloch-Kas-Lieberman proved it for Kodaira dimension below 2, leaving surfaces of general type with $p_g=q=0$, where it says any two points are rationally equivalent; only special families (Godeaux, Campedelli, Burniat, Inoue, Barlow, Catanese) were known. If $p_g(S)=0$, is the Albanese map an isomorphism?","posedBy":"Spencer Bloch","yearPosed":1976,"ageNote":"The manuscript cites Bloch's Lectures on Algebraic Cycles (first edition 1980) for the conjecture and Bloch-Kas-Lieberman (1976) for its proof below general type, so it was stated by 1976. The exact first statement is not given.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For every smooth connected projective complex surface $S$ with $p_g=q=0$, $\\deg:\\mathrm{CH}_0(S)\\to\\mathbb{Z}$ is an isomorphism with integral coefficients, with no minimality, fundamental-group or construction hypothesis. With Bloch-Kas-Lieberman this gives $\\mathrm{CH}_0(S)^0\\cong\\mathrm{Alb}(S)(\\mathbb{C})$ for every such surface with $p_g=0$. Over $\\mathbb{C}$ only; it does not address the generalized Bloch conjecture in higher dimension. An earlier proof of the same statement was proposed by Guletskii (arXiv:2512.12451, December 2025), which the manuscript cites; this entry claims no priority over it.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem and corollary were read against Bloch's conjecture as the manuscript states it. The proof, via virtual localization on a Quot scheme producing a diagonal relation in CH^2(X x X), was not refereed. No Lean formalization is listed. The paper notes two earlier proposed proofs by humans: Guletskii (arXiv 2512.12451, December 2025) claiming the same scope, and Banerjee (arXiv 2508.13594) for K^2 = 9.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"A central conjecture of the theory of algebraic cycles, open for fifty years and known only for special families. Ties Griffiths' positivity conjecture (55), below Kothe (60) and the Jacobian Conjecture (65), above Connes' Rigidity (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Bloch's conjecture for surfaces with p_g=0","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Bloch, Kas and Lieberman, Zero cycles on surfaces with p_g=0 (1976)","url":"https://www.numdam.org/item/CM_1976__33_2_135_0.pdf","kind":"problem-record"},{"label":"Guletskii, The Bloch conjecture, arXiv 2512.12451 (earlier proposed proof)","url":"https://arxiv.org/abs/2512.12451","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"no-conjugate-points-nonpositive-curvature-question","name":"Must a closed manifold admitting a metric without conjugate points admit a metric of nonpositive curvature?","shortName":"No conjugate points vs nonpositive curvature","problemNumber":null,"field":"Riemannian geometry; manifolds without conjugate points","fieldGroup":"Geometry & topology","statement":"A Riemannian metric has no conjugate points if no nonzero Jacobi field along a geodesic vanishes twice. Nonpositive sectional curvature implies this, and Gulliver showed that individual metrics without conjugate points may have positive curvature somewhere, but his examples live on manifolds that already carry negatively curved metrics. On surfaces the answer is yes by uniformization. Ivanov and Kapovitch asked, and Burns and Matveev repeated for dimension three, whether every closed manifold that admits a metric without conjugate points also admits a metric of nonpositive sectional curvature. Is that true?","posedBy":"Sergei Ivanov and Vitali Kapovitch (2014, Questions 1.1 and 8.1); repeated by Keith Burns and Vladimir Matveev (survey, Question 4.1.2)","yearPosed":2014,"ageNote":"The manuscript cites Ivanov-Kapovitch (2014) and the Burns-Matveev survey (2021), which explicitly includes dimension three.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a closed connected orientable smooth 3-manifold with a smooth metric without conjugate points that admits no smooth metric of nonpositive sectional curvature. The manifold glues two copies of (punctured torus) x $S^1$ by Leeb's obstructed gluing; $\\pi_1$ admits no proper cocompact action on a CAT(0) space, so there is no locally CAT(0) metric, and by Ivanov-Kapovitch no metric without focal points. Not shown: higher-dimensional examples, or an answer to Ivanov-Kapovitch's other questions about fundamental groups.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 24, 2026. The topological obstruction is Leeb's two-piece graph-manifold example; the metric construction and its no-conjugate-points estimate are the paper's.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 read against Ivanov-Kapovitch's question. Lean (in lean/formalization.yaml): OAI.ThreeManifold.main_theorem in OAI/Geometry/ConjugatePoints/ThreeManifold.lean. Its statement MainStatement gives a compact connected Hausdorff second-countable smooth 3-manifold with an oriented atlas, a smooth metric whose Jacobi fields along every geodesic on every open interval vanish identically once they vanish at two distinct times, and no smooth metric with nonpositive sectional curvature. That is the headline. Not rebuilt here. The no-focal-points and CAT(0) consequences are outside the formal statement.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A natural question about the topology of manifolds without conjugate points, raised in 2014 and repeated in a 2021 survey; known to specialists in geodesic flows. Level with Arnold's corank problem (20), below the stable coordinate conjecture in four variables (22) and Yau's harmonic dimension comparison (25), above Brannen's simplex conjecture (12).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean formalization (ConjugatePoints three-manifold)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/ConjugatePoints/ThreeManifold.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generalized-ionization-conjecture","name":"Solovej's generalized ionization conjecture for neutral Coulomb atoms (energies and radii)","shortName":"The generalized ionization conjecture","problemNumber":null,"field":"Many-body quantum mechanics; Thomas-Fermi theory","fieldGroup":"Mathematical physics","statement":"For the full nonrelativistic Coulomb atom with nuclear charge $Z$ and two spin states, let $I_m(Z)=E_Z(Z-m)-E_Z(Z)$ be the energy needed to remove $m$ electrons from the neutral atom, and let $R_m(Z)$ be the radius outside which an expected $m$ electrons of a neutral ground state lie. Thomas-Fermi theory predicts $I_m^{TF}\\sim a_{TF}m^{7/3}$ and radii $\\sim(81\\pi^2/2)^{1/3}m^{-1/3}$ as $Z\\to\\infty$ and then $m\\to\\infty$. Solovej (2016) conjectured that the same asymptotics hold for the true many-body atom: $\\lim_{m\\to\\infty}m^{-7/3}\\limsup_{Z\\to\\infty}I_m(Z)=\\lim_{m\\to\\infty}m^{-7/3}\\liminf_{Z\\to\\infty}I_m(Z)=a_{TF}$, and likewise $m^{1/3}R_m(Z)\\to(81\\pi^2/2)^{1/3}$ for the upper and lower large-$Z$ limits. He had proved the analogues in Hartree-Fock theory. Do these iterated limits hold for the full Schrodinger atom?","posedBy":"Jan Philip Solovej","yearPosed":2016,"ageNote":"Solovej, A new look at Thomas-Fermi theory, Molecular Physics 114 (2016), Section 3, Equation (1), cited by both companion manuscripts.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Energy (principal, Theorem 1.1): $I_m(Z)/m^{7/3}\\to a_{TF}>0$ whenever $m\\to\\infty$ and $Z/m\\to\\infty$, with $a_{TF}$ the Thomas-Fermi constant; this is stronger than and implies Solovej's iterated limits, which are stated as (1.4)-(1.5). Radius (companion, Theorem 1.1): for every choice of neutral ground states, $m^{1/3}\\limsup_Z R_m(Z)$ and $m^{1/3}\\liminf_Z R_m(Z)$ both tend to $(81\\pi^2/2)^{1/3}$, with $Z\\to\\infty$ first. Not shown: convergence of $I_m(Z)$ or $R_m(Z)$ at fixed $m$, a joint limit for radii, or relativistic and molecular versions.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The energy and radius manuscripts both rely on the screening estimates of the companion Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms (same family, same date).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the energy manuscript and Theorem 1.1 of the radius manuscript were read against Solovej's Equation (1). The proofs were not refereed. Energy: lean/formalization.yaml lists comparator CoulombIonization, declaration OAI.CoulombAtom.generalized_ionization. Its statement, read here: there is $a>0$ with the Thomas-Fermi weak-ionization characterization, the joint limit $I_m(Z)/m^{7/3}\\to a$ whenever $m\\to\\infty$ and $Z/m\\to\\infty$, and both iterated limits, with energies defined as infima of the Coulomb form over antisymmetric two-spin Sobolev states. Radius: ComparatorChallenges/CoulombRadii.json exists with solution module OAI.Analysis.CoulombRadii.Main present at the pinned commit, but the challenge is not in the formalization catalogue; its statement, read here, gives both upper and lower iterated radius limits equal to $(81\\pi^2/2)^{1/3}$ for every sequence of normalized ground states, with existence of ground states taken as a hypothesis. Both state the headline claims. Neither was rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A recent, sharp refinement of the ionization conjecture, well known among mathematical physicists working on Thomas-Fermi theory but young and specialist. Placed below the ionization conjecture entry of this family (55) and the Gamow liquid-drop minimizer conjecture (35), level with Talagrand's operator cotype problem (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Generalized ionization energies for full Coulomb atoms","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.CoulombAtom.generalized_ionization)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/CoulombIonization/Main.lean","kind":"lean-proof"},{"label":"Lean proof, radii (OAI.NeutralAtom.generalized_outer_radii, not in the catalogue)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/CoulombRadii/Main.lean","kind":"lean-proof"},{"label":"Companion: Generalized outer-electron radii of neutral Coulomb atoms","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf","kind":"paper"},{"label":"Solovej, A new look at Thomas-Fermi theory (2016)","url":"https://doi.org/10.1080/00268976.2015.1130273","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"toda-gepner-conjecture-quintic","name":"Toda's Gepner conjecture for the quintic threefold","shortName":"Toda's quintic Gepner conjecture","problemNumber":null,"field":"Algebraic geometry; Bridgeland stability conditions on Calabi-Yau threefolds","fieldGroup":"Geometry & topology","statement":"Let $X\\subset\\mathbb P^4$ be a smooth complex quintic threefold and $\\Phi=T_{\\mathcal O_X}\\circ(-\\otimes\\mathcal O_X(1))$, the composite of tensoring by the hyperplane bundle and the Seidel-Thomas spherical twist at $\\mathcal O_X$, which satisfies $\\Phi^5\\simeq[2]$. Physics predicts a stability condition at the Gepner point of the stringy Kahler moduli space fixed by this symmetry. Toda (2013) conjectured, in normalized form, that there is a Bridgeland stability condition $\\sigma=(Z,\\mathcal P)$ on $D^b(\\mathrm{Coh}\\,X)$ with $Z(\\Phi E)=e^{2\\pi i/5}Z(E)$ and $\\Phi\\mathcal P(\\varphi)=\\mathcal P(\\varphi+2/5)$, and reduced it to a strong Bogomolov-Gieseker type inequality. Does every smooth quintic threefold admit such a Gepner-type stability condition?","posedBy":"Yukinobu Toda","yearPosed":2013,"ageNote":"Toda, Gepner type stability conditions on graded matrix factorizations (arXiv 1302.6293, Conjecture 1.2) and Gepner point and strong Bogomolov-Gieseker inequality for quintic 3-folds (arXiv 1305.0345, Conjecture 3.1), as cited.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every smooth complex quintic threefold has a numerical Bridgeland stability condition, with support property on $K_{\\mathrm{num}}(X)_{\\mathbb R}$, on which $\\Phi$ acts by rotating the charge by $e^{2\\pi i/5}$ and shifting phases by $2/5$, normalized by $Z(\\mathcal O_x)=-1$. The construction is a double tilt starting at slope $-1/2$, refined to a periodic grid of aisles. It relies on Xu's 2026 inequality. It does not prove Toda's general strong Bogomolov-Gieseker conjecture, treat other Calabi-Yau hypersurfaces, or describe the full stability manifold near the Gepner point.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (September 24, 2026) is self-contained apart from a cited inequality of Xu (arXiv 2511.21288v2); the companion on large-volume charges (same date) is a separate result.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against Toda's conjecture; it gives a numerical Bridgeland stability condition with the support property on the full numerical Grothendieck group, $Z(\\Phi E)=e^{2\\pi i/5}Z(E)$, $\\Phi\\mathcal P(\\varphi)=\\mathcal P(\\varphi+2/5)$ and $Z(\\mathcal O_x)=-1$, on every smooth quintic, and identifies the charge with Toda's by uniqueness of the normalized eigenform. A reader should know that the slope-sheaf input is taken from Xu's stronger Bogomolov-Gieseker inequality on the quintic (arXiv 2511.21288v2, 2026), an external preprint not checked here. Not refereed. No Lean formalization in the release.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"Constructing the Gepner-point stability condition on the quintic is a well known target in the Bridgeland stability and mirror symmetry community, open since Toda's 2013 formulation, though existence of some stability condition on the quintic was already known (Li 2019). Placed level with Voiculescu's free entropy dimension question (38), below the CAT(0) conjecture for Artin groups (40), and above the Campana-Peternell conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A Gepner stability condition on every smooth quintic threefold","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Prescribed large-volume charges on threefolds with trivial canonical bundle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf","kind":"paper"},{"label":"Toda, Gepner point and strong Bogomolov-Gieseker inequality for quintic 3-folds (Conjecture 3.1)","url":"https://arxiv.org/abs/1305.0345","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sk-glauber-spectral-gap-high-temperature","name":"Fast mixing of Glauber dynamics for the Sherrington-Kirkpatrick model throughout the high-temperature phase, zero external field","shortName":"SK Glauber dynamics spectral gap for all beta < 1","problemNumber":null,"field":"Spin glasses, Markov chain mixing","fieldGroup":"Probability & statistics","statement":"For the Sherrington-Kirkpatrick model with couplings $J_{ij}\\sim N(0,\\beta^2/n)$, single-site Glauber (heat-bath) dynamics resamples one spin from its conditional Gibbs law. Physics predicts fast relaxation in the whole replica-symmetric phase $\\beta<1$; rigorous fast-mixing results reached $\\beta<1/4$ (Eldan-Koehler-Zeitouni; Anari et al.), about $0.295$ (Anari-Koehler-Vuong), $\\beta<1/2$ (Wang 2026) and $1/2+\\varepsilon_0$ (Boban-Li-Oveis Gharan 2026). Open Problem 15 of the ETH randomstrasse101 list asks: considering the SK model with an arbitrary external field, does Glauber dynamics fast mix (in polynomial time) up to the replica symmetry breaking threshold $\\beta^*=1$?","posedBy":"Almut Roedder (ETH randomstrasse101 open problems, Problem 15)","yearPosed":2024,"ageNote":"Open Problem 15 posted 20 December 2024 on the ETH Zurich randomstrasse101 blog, which credits no single original proposer; the expectation of fast mixing for all beta < 1 is older physics folklore. The release's manuscripts do not cite this or any posing source.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Spectral-gap paper, Theorem 1.1: for fixed $0<\\beta<1$, with probability tending to one over the disorder, $\\mathrm{Var}_{\\mu_0}(f)\\le C_\\beta\\mathcal D_0(f)$ for all $f$ (zero field), so the uniform-site heat-bath chain has gap at least $1/(C_\\beta n)$. Companions: worst-start total-variation cutoff at $\\log n/(2\\lambda(\\beta))$ for $0\\le\\beta<1$; at $\\beta=1$ mixing time $n^{2/3+o(1)}$ and slow mixing from typical starts before $n^{2/3}$; for $\\beta>1$ stretched-exponential barriers from typical starts and an $\\exp(n^{1-c})$ typical-start upper bound; at $\\beta=1$ universal random limits of rescaled autocorrelations for Gaussian and Rademacher disorder. Not shown: nonzero external fields, or an $O(n\\log n)$ bound from the gap paper alone.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has eight manuscripts (September 24 to October 5, 2026): the spectral gap paper (principal here), worst-start cutoff for beta < 1, critical slowing down and the critical mixing exponent n^(2/3) at beta = 1, typical-start lower and upper bounds for beta > 1, and two papers on universal critical autocorrelation limits.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the spectral-gap paper were read against Open Problem 15. Scope limit: zero external field only, while the problem asks for arbitrary fields; polynomial mixing follows from the gap by the standard bound (about $n^2$ updates), not stated as a separate theorem there. Lean: the challenge SKHighTemperature (theorem OAI.SKGap.sk_main, solution module OAI.Probability.SKGap.Main) exists at the pinned commit but is not in the formalization catalogue; its statement was read here: for each $0<\\beta<1$ a constant $C$ such that the zero-field Poincare inequality with the heat-bath Dirichlet form, and discrete gap at least $1/(Cn)$, hold with disorder probability tending to one. That is the principal theorem. Not rebuilt here. Other Lean challenges (CriticalSK, CriticalSKMixing, SKRatio, SKBarriers) cover companion results; SKRatio covers only beta < 1/2.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Fast mixing of SK Glauber dynamics up to beta = 1 is a headline open question in sampling and spin-glass dynamics, attacked by a sequence of strong 2021-2026 papers; this is its zero-field slice. Placed level with the girth-five Glauber colouring entry (30) and the Talagrand SK overlap conjecture (30), just below the Mezard-Parisi formula for diluted spin glasses (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf","sourceName":"OpenAI math release: A spectral gap throughout the high-temperature Sherrington-Kirkpatrick phase","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Cutoff throughout the high-temperature Sherrington-Kirkpatrick phase","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Critical mixing in the Sherrington-Kirkpatrick model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf","kind":"paper"},{"label":"Companion: Critical slowing down in the Sherrington-Kirkpatrick model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Stretched-exponential barriers for typical SK initial states","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: A typical-start upper bound for low-temperature SK Glauber dynamics","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Universality of critical quench autocorrelations in the SK model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf","kind":"paper"},{"label":"Companion: Functional universality of critical SK autocorrelations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf","kind":"paper"},{"label":"Lean formalization: SKGap/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/SKGap/Main.lean","kind":"lean-proof"},{"label":"ETH randomstrasse101, Open Problem 15: Sampling from the SK model","url":"https://randomstrasse101.math.ethz.ch/posts/SKModel-Glauber/","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"section-conjecture-modular-curves","name":"Grothendieck's section conjecture for the modular curves X0(N) and X1(N) over Q","shortName":"Section conjecture for X0(N), X1(N)","problemNumber":null,"field":"Anabelian geometry; rational points on curves","fieldGroup":"Number theory","statement":"Let $C$ be a smooth proper geometrically connected curve of genus at least two over a number field $K$. Grothendieck's section conjecture (letter to Faltings, 1983) asserts that the map from $C(K)$ to conjugacy classes of sections of $\\pi_1(C)\\to G_K$ is a bijection. Before this work it was known only in restricted or conditional forms (for instance via the finite-descent framework of Harari-Stix and Stix, or for curves with no adelic points), and no standard family such as the modular curves was covered. Does the section conjecture hold for every smooth proper curve of genus at least two over a number field, in particular for the modular curves $X_0(N)$ and $X_1(N)$ over $\\mathbb Q$?","posedBy":"Alexander Grothendieck, letter to G. Faltings (27 June 1983)","yearPosed":1983,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims Theorem 8.2: the global section conjecture holds for every smooth proper geometrically connected curve of genus at least two over a number field whose finite-cover descent locus equals its rational points; by Stoll's criteria this includes curves mapping nonconstantly to an abelian variety with finitely many rational points and trivial divisible Sha, and the modular curves $X_0(N)$, $X_1(N)$ of genus at least two over $\\mathbb Q$. It does NOT prove the section conjecture for arbitrary curves over number fields; the descent-locus equality remains a separate hypothesis in general.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. The global statements are deduced in its Section 8 from the local theorem together with established descent results (Harari-Stix, Stix, Stoll), which are credited to their authors.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Section 8 (Proposition 8.1, Theorem 8.2) was read against the global conjecture: given the local theorem, the image of global sections under localization equals the finite-cover descent locus, and when that locus is exactly the rational points every section is a point section. Stoll's finite-descent criteria are cited to cover $X_0(N)$ and $X_1(N)$ of genus at least two over $\\mathbb Q$. The result therefore depends on the unformalised local theorem and on the cited descent theorems. No Lean formalisation exists. The release README warns that some unformalised results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"A first unconditional family of interesting curves satisfying Grothendieck's section conjecture, scored well below the full conjecture as a special case. Below the companion entry for the local p-adic section conjecture (50) and the Zilber-Pink conjecture for abelian varieties over number fields (58), above Grothendieck's Finite Flat Group Scheme Order Question (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf","sourceName":"OpenAI math release: The p-adic section conjecture","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion: Etale covers with a prescribed exterior sheet","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf","kind":"paper"},{"label":"Grothendieck, letter to Faltings (27 June 1983), in Geometric Galois Actions 1","url":"https://doi.org/10.1017/CBO9780511758874.018","kind":"problem-record"},{"label":"Earlier edition (September 24, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-p-adic-section-conjecture-September-24-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"entropy-photon-number-inequality","name":"The entropy photon-number inequality (Guha-Erkmen-Shapiro)","shortName":"Entropy photon-number inequality","problemNumber":null,"field":"Quantum Shannon theory; bosonic channels","fieldGroup":"Quantum information & computing","statement":"For an $n$-mode bosonic state $\\rho$ let $N(\\rho)=g^{-1}(S(\\rho)/n)$, where $g(t)=(t+1)\\log(t+1)-t\\log t$ is the entropy per mode of a thermal state; $N$ is the mean photon number per mode of the product thermal state with the same entropy. Guha, Erkmen and Shapiro (2007) conjectured the entropy photon-number inequality, a quantum analogue of the entropy power inequality with consequences for minimum output entropy and broadcast capacities. Konig-Smith and De Palma-Mari-Giovannetti proved weaker entropy power inequalities. If independent $n$-mode states $\\rho_A,\\rho_B$ are mixed on a beam splitter of transmissivity $\\eta\\in[0,1]$ with output $\\rho_C$, is $N(\\rho_C)\\ge\\eta N(\\rho_A)+(1-\\eta)N(\\rho_B)$?","posedBy":"Saikat Guha, Baris I. Erkmen and Jeffrey H. Shapiro ('The entropy photon-number inequality and its consequences', arXiv 0710.5666, 2007)","yearPosed":2007,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for finite $n\\ge1$, $\\eta\\in[0,1]$ and $n$-mode states $\\rho_A,\\rho_B$ of finite mean photon number, the output of passive mixing satisfies $g^{-1}(S(\\rho_C)/n)\\ge\\eta g^{-1}(S(\\rho_A)/n)+(1-\\eta)g^{-1}(S(\\rho_B)/n)$, with equality for product thermal inputs. Corollaries: the exact minimum output entropy at fixed input entropy for tensor powers of the thermal attenuator, and the capacity region of the degraded pure-loss bosonic broadcast channel. Not covered: infinite-energy finite-entropy inputs, amplifiers and additive-noise channels, and uniqueness of optimizers.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: the inequality for every finite $n$, every $\\eta\\in[0,1]$ and independent inputs of finite mean photon number, with arbitrary entanglement among modes inside each input. The finite-energy hypothesis is a scope limit the paper states. The challenge is not in lean/formalization.yaml; lean/ComparatorChallenges/EntropyPhotonNumber.json exists with solution_module OAI.InformationTheory.PhotonNumber.Inequality, whose file exists at the pinned commit. The statement EntropyPhotonNumber.lean was read here; not rebuilt here. It models states as positive trace-one operators on untruncated $\\ell^2(\\mathbb N^n)$, entropy by functional calculus, the beam-splitter output through explicit number-basis coefficients and a partial-trace HasSum condition, and states $\\eta g^{-1}(S_A/n)+(1-\\eta)g^{-1}(S_B/n)\\le g^{-1}(S_C/n)$ for finite-energy inputs. This states the headline. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"A named conjecture of quantum Shannon theory open for almost two decades, the bosonic counterpart of the entropy power inequality and the missing step in several capacity results. Above the Werner two-copy distillability entry at 35 and the bosonic quantum capacity entry at 25, below the Courtade-Kumar entry at 40.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The entropy photon-number inequality","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.EntropyPhotonNumber.entropy_photon_number_inequality)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/InformationTheory/PhotonNumber/Inequality.lean","kind":"lean-proof"},{"label":"Guha, Erkmen, Shapiro, The entropy photon-number inequality and its consequences (2007)","url":"https://arxiv.org/abs/0710.5666","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"brennan-conjecture","name":"Brennan's conjecture","shortName":"Brennan's conjecture","problemNumber":null,"field":"Geometric function theory, conformal mapping","fieldGroup":"Analysis","statement":"Let $W\\subset\\mathbb C$ be a simply connected domain whose boundary in the Riemann sphere has at least two points, and $\\varphi:W\\to\\mathbb D$ a conformal bijection onto the unit disk. Brennan (1978) conjectured that $\\int_W|\\varphi'|^s\\,dA<\\infty$ for every $4/30$ there is $C_\\varepsilon$ with $M_{-2}[f'](r)\\le C_\\varepsilon(1-r)^{-1-\\varepsilon}$ for all $f\\in\\mathcal S$ and $1/2\\le r<1$; $B_{\\mathcal S}(-2)=1$; and every conformal bijection $\\varphi:W\\to\\mathbb D$ has $|\\varphi'|^s$ area-integrable for $4/3 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The manuscript (September 24, 2026) imports its geometric and algebraic obstruction package from the family's companion 'A marked tensor obstruction to four-dimensional disk embedding' and proves the transfer to topological fillings itself.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'A PD4 group without an aspherical manifold model' and its introduction were read against Wall's Problem G2 in the finitely presented form the paper cites. The proof (a marked chamber with a filling obstruction, Davis reflection, Farrell-Jones for CAT(0) groups, Kasprowski-Land 4-dimensional surgery and a direction-control argument) was not refereed. No Lean formalization. The result is conditional on the unreviewed marked tensor obstruction companion in the same release. Scope the paper itself states: a single dimension-four counterexample, which refutes the question quantified over all dimensions; it says nothing about dimension three (the PD3 conjecture) or about dimensions five and up.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Wall's 1979 question is a standard open problem linking group cohomology and aspherical manifolds, repeated in the Davis and Luck surveys, with no known finitely presented counterexample before this. Placed below Connes' Rigidity Conjecture (45) and well below the disc embedding entry from the same family (60), above Voiculescu's free entropy dimension question (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A PD4 group without an aspherical manifold model","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Input: A marked tensor obstruction to four-dimensional disk embedding","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf","kind":"paper"},{"label":"Davis, Poincare duality groups (survey, Question 3.4), author's version","url":"https://people.math.osu.edu/davis.12/pdgroup.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"elastic-calderon-problem-3d-isotropic","name":"The elastic Calderon problem in three dimensions: do boundary measurements determine smooth isotropic Lame moduli?","shortName":"3D isotropic elastic Calderon problem","problemNumber":null,"field":"Inverse problems; linear elasticity","fieldGroup":"Differential equations","statement":"Let $\\Omega\\subset\\mathbb R^3$ be a bounded connected smooth domain and $\\lambda,\\mu\\in C^\\infty(\\overline\\Omega)$ Lame moduli with $\\mu>0$, $3\\lambda+2\\mu>0$. The static displacement-to-traction map $\\Lambda_{\\lambda,\\mu}$ sends boundary displacements $f$ to the tractions $\\sigma_{\\lambda,\\mu}(u_f)n$ of the solution of $\\operatorname{div}(\\lambda(\\operatorname{div}u)I+2\\mu e(u))=0$ with trace $f$. Nakamura and Uhlmann claimed global uniqueness in 1994; their 2003 erratum proved it only when $\\|\\nabla\\mu_j\\|_{C^m}$ is small. Does $\\Lambda_{\\lambda_1,\\mu_1}=\\Lambda_{\\lambda_2,\\mu_2}$ imply $\\lambda_1=\\lambda_2$ and $\\mu_1=\\mu_2$ for all such smooth moduli?","posedBy":"Elastic analogue of Calderon's problem; global uniqueness claimed by G. Nakamura and G. Uhlmann (1994), reduced to a small-gradient result in their 2003 erratum","yearPosed":2003,"ageNote":"The manuscript does not call the problem open in so many words; it cites the 1994 Nakamura-Uhlmann claim and the 2003 erratum restricting it to small gradients, and later partial results (Eskin-Ralston, Lin-Nakamura 2017, Tan-Liu 2023, Imanuvilov-Yamamoto in 2D). 2003 is when the general three-dimensional case became open again; the question itself dates to 1994.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: on any bounded connected domain $\\Omega\\subset\\mathbb R^3$ with $C^\\infty$ boundary, if $\\lambda_j,\\mu_j\\in C^\\infty(\\overline\\Omega)$ satisfy $\\mu_j>0$, $3\\lambda_j+2\\mu_j>0$ and $\\Lambda_{\\lambda_1,\\mu_1}=\\Lambda_{\\lambda_2,\\mu_2}$, then $\\lambda_1=\\lambda_2$ and $\\mu_1=\\mu_2$. No analyticity, smallness or prior boundary agreement is assumed. It is a uniqueness theorem for smooth coefficients and full static data only: no reconstruction algorithm, no stability estimate, no partial-data or low-regularity version, and nothing about anisotropic elasticity.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the Nakamura-Uhlmann problem. lean/docs/372.md points to ComparatorChallenges/ElasticityUniqueness.json (theorem OAI.Elasticity.global_uniqueness, solution module OAI.MathematicalPhysics.Elasticity.Uniqueness); the solution file exists at the pinned commit and the challenge is not in formalization.yaml. The statement was read here: for every bounded connected open set in $\\mathbb R^3$ with smooth boundary charts and moduli smooth up to the boundary with $\\mu>0$, $3\\lambda+2\\mu>0$, equality of the variationally defined Dirichlet-to-Neumann pairings forces both moduli to agree on $\\Omega$. That is the headline claim. Not rebuilt here. The DN map is defined on a trace quotient rather than $H^{1/2}$, as the file notes. No reconstruction or stability estimate is claimed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Global uniqueness for the isotropic elastic Calderon problem in three dimensions is a well-known open problem of inverse problems, open since the 2003 erratum withdrew the general claim, but known mainly within that community. Placed below ball-evans-approximation-problem-3d (35) and hot-spots-smooth-simply-connected-planar (42), and above lins-de-melo-pugh-quintic-lienard (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf","sourceName":"OpenAI math release: Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Elasticity.global_uniqueness)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Elasticity/Uniqueness.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"flint-hills-series-convergence","name":"Convergence of the Flint Hills series","shortName":"Flint Hills series","problemNumber":null,"field":"Diophantine approximation; series","fieldGroup":"Number theory","statement":"The Flint Hills series is $\\sum_{n\\ge1}\\frac{1}{n^3\\sin^2 n}$, with $n$ in radians. Its terms are large exactly when $n$ is very close to a multiple of $\\pi$, so convergence depends on how well $\\pi$ is approximated by rationals. Alekseyev (2011) showed convergence would force $\\mu(\\pi)\\le5/2$, and Meiburg (2022) showed $\\mu(\\pi)<5/2$ suffices, while the best known bound was about $7.1$. Does the Flint Hills series converge?","posedBy":"Popularized by Clifford Pickover (per the manuscript); studied by Max Alekseyev, On convergence of the Flint Hills series, arXiv:1104.5100 (2011)","yearPosed":null,"ageNote":"The manuscript says only that the problem was popularized by Pickover and gives no date or reference for him. The earliest dated reference it cites on the series is Alekseyev's 2011 preprint, which calls convergence unknown.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 1.2: $\\sum_{n\\ge1}1/(n^3\\sin^2 n)$ converges, deduced from $\\mu(\\pi)=2$ via a dyadic spacing estimate valid for any exponent below $5/2$. More generally $\\sum n^{-a}|\\sin n|^{-b}$ converges iff $a>\\max\\{1,b\\}$ (Corollary 5.2). No value or numerical estimate of the sum is given.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 24, 2026; the series result is a corollary of its main theorem on the irrationality exponent of pi. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues. OpenAI also released an abridged reasoning summary for this family (reasoning_traces/irrationality-exponent-of-pi.pdf).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary 1.2 and Section 5 of the TeX source read against the question; convergence follows from $\\mu(\\pi)<5/2$ by a short dyadic spacing lemma that the paper proves (Meiburg's criterion is cited too). Everything rests on the unrefereed main theorem $\\mu(\\pi)=2$. The release's Lean scope note says the Flint Hills consequence is outside the Comparator challenge PiExponent. The solution module OAI.NumberTheory.PiExponent.Main also contains a theorem flint_hills_summable stating summability of $1/(n^3\\sin^2 n)$, but it is not a Comparator challenge, so it is not counted here; nothing was rebuilt.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A well-known curiosity, popular far beyond specialists, but by Alekseyev's work essentially a question about the irrationality exponent of pi and settled here as its consequence, so scored well below that entry (60). Placed just below free entropy dimension invariance (38), above the Colin de Verdiere chromatic conjecture (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The irrationality exponent of pi is 2","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Alekseyev, On convergence of the Flint Hills series (arXiv 2011)","url":"https://arxiv.org/abs/1104.5100v1","kind":"problem-record"},{"label":"Abridged reasoning summary released by OpenAI for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/reasoning_traces/irrationality-exponent-of-pi.pdf","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lane-emden-conjecture","name":"The Lane-Emden conjecture for elliptic systems, with its Henon-weighted extension","shortName":"Lane-Emden conjecture","problemNumber":null,"field":"Elliptic systems; Liouville theorems","fieldGroup":"Differential equations","statement":"The Lane-Emden system is $-\\Delta u=v^p$, $-\\Delta v=u^q$ in $\\mathbb R^n$ with $p,q>0$. The Lane-Emden conjecture asserts that it has no positive classical entire solution in the subcritical region $\\frac1{p+1}+\\frac1{q+1}>\\frac{n-2}n$, with no symmetry or decay assumption. It was proved for $n\\le2$, for $n=3$ (Serrin-Zou; Polacik-Quittner-Souplet) and $n=4$ (Souplet), and in partial ranges for $n\\ge5$. Phan (2012, Conjecture C) posed the weighted Henon-Lane-Emden form $-\\Delta u=|x|^Av^p$, $-\\Delta v=|x|^Bu^q$ for $A,B>-2$, with subcritical region $\\frac{n+A}{p+1}+\\frac{n+B}{q+1}>n-2$, and Huang and Zou stated it for all real weights with solutions continuous at the origin. Is there no positive entire solution throughout the subcritical region, in every dimension?","posedBy":"Lane-Emden conjecture: Liouville literature of the 1990s (Mitidieri 1993, 1996; Serrin and Zou 1996); weighted form: Phan (2012), Conjecture C; Huang and Zou (2026), Conjecture B","yearPosed":null,"ageNote":"The manuscript cites Mitidieri (1993, 1996), Serrin-Zou (1996) and Souplet's 'proof of the Lane-Emden conjecture in four space dimensions' (2009) but does not say who first stated the unweighted conjecture; Phan's weighted Conjecture C dates from 2012 (arXiv 2011).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every $n\\ge2$, $p,q>0$ and $A,B\\in\\mathbb R$ with $\\frac{n+A}{p+1}+\\frac{n+B}{q+1}>n-2$, the Henon-Lane-Emden system has no positive solution continuous at the origin and $C^2$ elsewhere, with no condition at infinity. Corollary 1.2 is the unweighted Lane-Emden conjecture; with Bidaut-Veron and Giacomini's radial existence theorem this gives Phan's Conjecture C in full for $n\\ge3$, $A,B>-2$, and Huang-Zou's Conjecture B. It does NOT classify solutions on the critical hyperbola beyond the cited radial existence, and treats only positive (not sign-changing) solutions.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. It combines a localized potential-kernel virial identity with an energy-relative interval-pressure estimate; the radial existence theorem of Bidaut-Veron and Giacomini used for the classification is credited to them.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 were read against the conjecture: for $n\\ge2$, $p,q>0$, real $A,B$ in the weighted subcritical region, no positive solutions continuous on $\\mathbb R^n$ and $C^2$ off the origin; the unweighted Lane-Emden conjecture is the case $A=B=0$, $n\\ge3$. formalization.yaml lists ComparatorChallenges/HenonEmden.json, declaration OAI.HenonLaneEmden.main_nonexistence, file OAI/Analysis/HenonEmden/Main.lean. The statement was read here: for $n\\ge2$, $p,q>0$, real $A,B$ with $(n+A)/(p+1)+(n+B)/(q+1)>n-2$, there are no $u,v$ on Euclidean $n$-space that are continuous, $C^2$ off the origin, strictly positive and satisfy both equations away from the origin. That is the headline nonexistence claim. The existence half of Phan's Conjecture C (via the cited radial theorem) is not formalised. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The Lane-Emden conjecture is a well-known Liouville problem in nonlinear elliptic PDE, open in dimension five and above for three decades and used in a priori bound arguments. Below Finite-time blowup for the inviscid Boussinesq system with smooth forcing (45), level with the Petersen Coloring Conjecture (40), above Interior Curvature Estimates for the Graphical Scalar Curvature Equation (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The Subcritical Henon-Lane-Emden Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: Henon-Lane-Emden nonexistence (formalization.yaml file)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/HenonEmden/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: subcritical Henon-Lane-Emden nonexistence","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HenonEmden.lean","kind":"lean-proof"},{"label":"Phan (2012), Liouville-type theorems for Hardy-Henon elliptic systems (arXiv)","url":"https://arxiv.org/abs/1108.1312","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hellinger-conjecture-boolean-functions","name":"The Hellinger conjecture for Boolean functions under product noise","shortName":"Hellinger conjecture","problemNumber":null,"field":"Boolean functions; information theory","fieldGroup":"Probability & statistics","statement":"Let $X$ be uniform on $\\{-1,1\\}^n$, let $T_\\rho$ be the noise operator with correlation $\\rho\\in[-1,1]$, and write $H(t)=\\sqrt{1-t^2}$. Anantharam, Bogdanov, Chakrabarti, Jayram and Nair (2017) conjectured that for every Boolean $f$ with mean $m$, $H(m)-\\mathbb E\\,H(T_\\rho f)\\le1-\\sqrt{1-\\rho^2}$, with equality for dictators: a single coordinate maximizes the loss of Hellinger affinity. They showed it implies the Courtade-Kumar conjecture. Known results included a Gaussian-isoperimetric lower bound (Chen-Nair) and the sharp balanced low-noise consequence (Durcik-Ivanisvili-Roos-Xie). Does the Hellinger inequality hold for every Boolean function, of any bias, and every $\\rho$?","posedBy":"Venkat Anantharam, Andrej Bogdanov, Amit Chakrabarti, T. S. Jayram and Chandra Nair (ITA Workshop manuscript, February 2017)","yearPosed":2017,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every $n\\ge1$, every $f:\\{-1,1\\}^n\\to\\{-1,1\\}$ with $m=\\mathbb E f$ and every $\\rho\\in[-1,1]$, $\\sqrt{1-m^2}-\\mathbb E\\sqrt{1-(T_\\rho f)^2}\\le1-\\sqrt{1-\\rho^2}$, with equality for signed coordinates. By the Anantharam et al. implication it reproves the Courtade-Kumar bound (Corollary 8.1). It does NOT treat non-Boolean (soft) functions or non-uniform inputs, and its proof depends on computer-checked arithmetic certificates.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. Its README and source-lineage file say the proof combines analytic sections with six exact-arithmetic programs and a runner, all supplied with the manuscript, and that it has no dependency on other release articles.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture of Anantharam et al.: for every $n\\ge1$, Boolean $f$ with mean $m$ and $\\rho\\in[-1,1]$, $H(m)-\\mathbb E H(T_\\rho f)\\le1-H(\\rho)$, with signed coordinates attaining equality; it is the full unbalanced form. This manuscript has no Lean formalisation (the family's Lean covers the companion's Courtade-Kumar theorem). The proof relies on finite exact-arithmetic certificates; the README gives a verification runner (Python with SymPy plus C++), which was not run here. The release README warns that some unformalised results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A stronger functional form of the Courtade-Kumar conjecture, posed in 2017 and known mainly to the people working on that conjecture. Below the Courtade-Kumar conjecture entry (40), above the catalog's Courtade and Kumar's Coordinate-wise Mutual Information Question (22), level with Sharp Hardness for MAX-3-CUT (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/main.pdf","sourceName":"OpenAI math release: Hellinger contraction with arbitrary Boolean output bias","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Sharp binary-information contraction on the discrete cube","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/main.pdf","kind":"paper"},{"label":"Anantharam et al. (2017), A conjecture regarding optimality of the dictator function under Hellinger distance","url":"https://chandra.ie.cuhk.edu.hk/pub/papers/HC/hel-conj.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ford-konyagin-luca-poisson-dirichlet-shifted-primes","name":"The Ford-Konyagin-Luca conjecture: the prime factors of $p-1$ follow the Poisson-Dirichlet law","shortName":"Ford-Konyagin-Luca conjecture (PD law for p-1)","problemNumber":null,"field":"Analytic number theory; smooth shifted primes","fieldGroup":"Number theory","statement":"For a prime $p$, list the prime factors of $p-1$ with multiplicity in decreasing order $q_1(p)\\ge q_2(p)\\ge\\cdots$ and set $V_j(p)=\\log q_j(p)/\\log(p-1)$. For a uniformly random integer the analogous normalized factor sizes converge to the Poisson-Dirichlet law $\\mathrm{PD}(1)$ (Billingsley, Donnelly-Grimmett). Ford, Konyagin and Luca (2010, Section 6, Conjecture 5) conjectured the same for shifted primes. The first coordinate already contains the shifted-prime Dickman law stated by Granville, $\\#\\{p\\le x:P^+(p-1)\\le x^{1/u}\\}\\sim\\pi(x)\\rho(u)$ for fixed $u$; unconditionally only lower bounds of size $x/(\\log x)^C$ for $P^+(p-1)\\le x^{0.2844}$ were known (Baker-Harman, Lichtman), and the full law was known under Elliott-Halberstam (Bharadwaj-Rodgers). With $p$ uniform among primes up to $x$, does $(V_1(p),V_2(p),\\ldots)$ converge in finite-dimensional distributions to $\\mathrm{PD}(1)$ as $x\\to\\infty$?","posedBy":"Kevin Ford, Sergei Konyagin and Florian Luca (Prime chains and Pratt trees, GAFA 2010, Section 6, Conjecture 5)","yearPosed":2010,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $k$ and bounded continuous $F$ on $[0,1]^k$, the average of $F(V_1(p),\\ldots,V_k(p))$ over primes $3\\le p\\le x$ tends to $\\mathbb E F(L_1,\\ldots,L_k)$ with $(L_j)\\sim\\mathrm{PD}(1)$, unconditionally and with equal weight on each prime. The manuscript notes this gives Granville's fixed-$u$ asymptotic $\\#\\{p\\le x:P^+(p-1)\\le x^{1/u}\\}\\sim\\pi(x)\\rho(u)$, so a positive proportion of primes have $x^{\\varepsilon}$-smooth predecessors for each fixed $\\varepsilon>0$. No error terms or uniformity as $u$ grows are claimed. The companion on $\\mu(p-1)=1$ (infinitely many primes with $p-1$ squarefree with an even number of prime factors) is linked here, not cataloged, since it cites no posing source.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript builds on graph and ideal-kernel estimates proved in the companion 'Weighted dilation graphs, smooth shifted primes and totient fibers' (same date, separate entry for Erdos Problem #821); a third companion (17 September) uses the same machinery for primes with mu(p-1) = 1.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the history section were read against Conjecture 5 of Ford-Konyagin-Luca as the manuscript quotes it (same multiplicity convention, normalization and equal weighting of primes). No Lean formalization exists for this family. The claim is unconditional and reaches what was previously known only under the Elliott-Halberstam conjecture; the manuscript says that unconditional distribution results (level one half) cover only part of the simplex and that its extraction argument reaches every compact subset with coordinate sum below one. A claim of this strength should be read as unverified until specialists examine it. It imports estimates from the companion manuscript. Not refereed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The shifted-prime analogue of the Dickman and Poisson-Dirichlet laws is a well-known goal of analytic number theory, previously out of reach without Elliott-Halberstam; it feeds Carmichael-number and totient problems. Famous across analytic number theory. Placed below Vinogradov's least quadratic nonresidue conjecture (45), above Convergence of the Flint Hills series (35) and Jacobsthal's quadratic-bound question: is h(k) = O(k^2)? (34).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The Poisson-Dirichlet law for prime predecessors","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Weighted dilation graphs, smooth shifted primes and totient fibers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Prime predecessors with an even number of prime factors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf","kind":"paper"},{"label":"Ford-Konyagin-Luca, Prime chains and Pratt trees (GAFA 2010)","url":"https://doi.org/10.1007/s00039-010-0089-0","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lee-uhlmann-anisotropic-calderon","name":"The Lee-Uhlmann conjecture: smooth anisotropic Calderon uniqueness in dimension at least three, from one boundary patch","shortName":"Lee-Uhlmann anisotropic Calderon problem","problemNumber":null,"field":"Inverse problems; elliptic PDE on manifolds","fieldGroup":"Differential equations","statement":"Let $(M,g)$ be a compact connected smooth Riemannian manifold with boundary, $n=\\dim M\\ge3$, and $\\Lambda_g$ its Dirichlet-to-Neumann map for $\\Delta_g$. Diffeomorphisms fixing the boundary preserve $\\Lambda_g$. Calderon (1980) posed the conductivity problem; Lee and Uhlmann (1989) proved that $\\Lambda_g$ determines $g$ up to such a diffeomorphism for real-analytic metrics, and the smooth case became the anisotropic Calderon problem, solved before only for analytic or Einstein metrics, near special backgrounds, or on conformally transversally anisotropic manifolds. The partial-data version, with input and output on the same open boundary patch, was open even for smooth scalar conductivities. For smooth metrics and $n\\ge3$, does $\\Lambda_g$ (or its restriction to one boundary patch) determine $g$ up to a diffeomorphism fixing the measured boundary?","posedBy":"John M. Lee and Gunther Uhlmann (Comm. Pure Appl. Math., 1989), extending Alberto Calderon's 1980 inverse conductivity problem to anisotropic metrics","yearPosed":1989,"ageNote":"The manuscript cites Lee-Uhlmann 1989 for the real-analytic theorem and calls the smooth case 'the smooth anisotropic Calderon uniqueness problem'; it does not quote a conjecture statement. Calderon's original problem dates to 1980.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $n\\ge3$, compact connected smooth $M$ with smooth boundary and any nonempty proper relatively open $\\Gamma\\subset\\partial M$, if smooth metrics satisfy $\\Lambda_{g_1,\\Gamma}=\\Lambda_{g_2,\\Gamma}$ (energy form with input and output on $\\Gamma$), then $g_2=\\Phi^*g_1$ for a smooth diffeomorphism with $\\Phi|_\\Gamma=\\mathrm{Id}$. Corollaries: full-boundary uniqueness, and uniqueness of smooth positive scalar conductivities on smooth Euclidean domains from one same-patch measurement. The October 5 companion adds a smooth unitary connection on a trivial Hermitian rank-two bundle, determined up to gauge fixed on the patch. Not shown: dimension two, nonsmooth metrics, stability estimates, or reconstruction.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This entry draws on two manuscripts: the smooth anisotropic uniqueness theorem (September 24, 2026, principal) and a stronger October 5 version that also recovers a unitary connection on a trivial rank-two bundle. The family's September 23 nonuniqueness paper for bounded measurable conductivities is a separate entry.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 and Corollaries 1.2-1.3 of 'Smooth anisotropic uniqueness in the Calderon problem from one boundary patch' were read against the problem as the manuscript and its references describe it. The theorem covers arbitrary smooth metrics, any nonempty proper open patch for both input and observation, nonorientable manifolds and several boundary components. The release's only Lean formalization in this family (Conductivity) concerns the separate bounded-measurable nonuniqueness entry, so this entry is unreviewed. The proof's central step is a uniform estimate for transfer functionals on shrinking spheres; it was not refereed here. Given its standing in inverse problems, this claim warrants expert review before anything else.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The anisotropic Calderon problem in dimension three and higher is the central open uniqueness problem of inverse boundary problems, open for smooth metrics since 1989 and the subject of a large literature; the same-patch partial-data scalar problem it also settles was a major open problem of its own. Placed level with the Blaschke conjecture (55), above Schiffer's conjecture (53), below Kadison's similarity problem (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Smooth anisotropic uniqueness in the Calderon problem from one boundary patch","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Determination of a metric and a unitary connection from one boundary patch","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026/paper.pdf","kind":"paper"},{"label":"Lee-Uhlmann (1989), Determining anisotropic real-analytic conductivities by boundary measurements","url":"https://doi.org/10.1002/cpa.3160420804","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ordinary-double-point-volume-gap","name":"The ordinary-double-point gap conjecture for normalized volumes of klt singularities","shortName":"ODP normalized-volume gap","problemNumber":null,"field":"Birational geometry, K-stability","fieldGroup":"Geometry & topology","statement":"For a klt germ $x\\in X$ of dimension $n$ with zero boundary, the normalized volume $\\widehat{\\mathrm{vol}}(x,X)=\\inf_vA_X(v)^n\\mathrm{vol}(v)$ over valuations centred at $x$ is at most $n^n$, with equality exactly at smooth points (Liu-Xu). Spotti and Sun, motivated by volume densities of tangent cones of Kahler-Einstein limits, conjectured that the ordinary double point comes next. It was known for surfaces (Li-Liu), threefolds (Liu-Xu) and local complete intersections (Liu), with a different sharp bound for toric germs (Moraga-Suss). Does every singular $n$-dimensional klt germ satisfy $\\widehat{\\mathrm{vol}}(x,X)\\le2(n-1)^n$, with equality exactly for an ordinary double point?","posedBy":"Cristiano Spotti and Song Sun","yearPosed":2017,"ageNote":"Spotti-Sun, Pure Appl. Math. Q. 13 (2017), Conjectures 1.2 and 5.5 (arXiv 1705.00377); listed in Xu-Zhuang, Open problems in K-stability of Fano varieties (2026).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $n\\ge2$ and a singular closed point of a normal complex algebraic $n$-fold with $K_X$ $\\mathbb Q$-Cartier, klt with zero boundary, $\\widehat{\\mathrm{vol}}(x,X)\\le2(n-1)^n$, with equality iff the germ is analytically an ordinary double point; the point need not be isolated. The companion proves dimension four (bound 162), the base case. Corollary 1.2 gives a strict anticanonical volume bound below $2n^n$ for singular K-semistable $\\mathbb Q$-Fano $n$-folds, so $(-K_V)^n=2n^n$ off projective space only for the smooth quadric or $\\mathbb P^1\\times\\mathbb P^{n-1}$. Not shown: nonzero boundary, the next gaps, or sharpness of the singular global bound.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts, both dated September 24, 2026; the every-dimension paper uses the dimension-four paper as its base case.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of both manuscripts were read against the conjecture as cited. The proofs (cone reduction through stable degeneration, an orbifold rational-curve argument for small stabilizers, jet cones and adjoint jets, induction on dimension from the fourfold case) were not refereed. They rely on the published theory of normalized-volume minimizers (Blum, Xu, Xu-Zhuang, Li-Wang-Xu) as inputs. No Lean formalization accompanies this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A known conjecture of normalized-volume and K-stability theory, posed in 2017 and settled before only up to dimension three; important to specialists. Placed below Popa's logarithmic additivity conjecture (35) and The projective-space base conjecture for Lagrangian fibrations (38), above The Generalized Busemann-Petty Problem in dimensions 2 and 3 (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The ordinary-double-point gap in every dimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The normalized-volume gap in dimension four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf","kind":"paper"},{"label":"Spotti-Sun, Explicit Gromov-Hausdorff compactifications (arXiv)","url":"https://arxiv.org/abs/1705.00377","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generalized-amplitude-damping-classical-capacity","name":"The unassisted classical capacity of the generalized amplitude-damping channel","shortName":"Classical capacity of generalized amplitude damping","problemNumber":null,"field":"Quantum Shannon theory; channel capacities; Holevo additivity","fieldGroup":"Quantum information & computing","statement":"The generalized amplitude-damping channel $\\mathcal A_{\\gamma,\\nu}$ is the qubit channel modelling energy relaxation with damping $\\gamma\\in[0,1]$ toward a thermal state with excited population $\\nu\\in[0,1]$. By the Holevo-Schumacher-Westmoreland theorem its unassisted classical capacity is the regularized Holevo information $\\lim_n\\chi(\\mathcal A^{\\otimes n})/n$, and since Holevo additivity fails in general (Hastings) the one-use value need not be the answer. Leditzky, Kaur, Datta and Wilde (2018) listed the classical capacity of amplitude damping as open, and Khatri, Sharma and Wilde (2020) gave only upper bounds for the generalized channel; exact values were known only where King's unital theorem or Shor's entanglement-breaking theorem applies. What is the classical capacity of $\\mathcal A_{\\gamma,\\nu}$ for all $\\gamma,\\nu$, and does it equal the one-shot Holevo capacity?","posedBy":"Leditzky, Kaur, Datta and Wilde, Phys. Rev. A 97 (2018) (amplitude damping open); Khatri, Sharma and Wilde, Phys. Rev. A 102 (2020) (only bounds for the generalized channel)","yearPosed":2018,"ageNote":"The manuscript cites Leditzky et al. (2018) as naming the classical capacity of ordinary amplitude damping as open; that case was settled by Tang, Zhu, Bai and Wang (2026) for channels with a pure output, which the generalized channel lacks. For the generalized channel the cited sources give bounds (Khatri-Sharma-Wilde, abstract read here) rather than a posing sentence.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that for every $\\gamma,\\nu\\in[0,1]$ the unassisted classical capacity of $\\mathcal A_{\\gamma,\\nu}$ equals its one-shot Holevo capacity, given by $\\max_{p\\in[0,1]}[h((1-\\gamma)p+\\gamma\\nu)-g(\\gamma\\nu(1-\\nu)+\\gamma(1-\\gamma)(p-\\nu)^2)]$ in bits, with $g$ the entropy of a qubit state of given determinant, attained by two equiprobable pure signals of opposite phase and their independent products. It also claims additivity of one-shot Holevo capacity, minimum output entropy and regularized classical capacity under tensoring with any finite-dimensional channel. It does NOT give the quantum or private capacity of the channel, entanglement-assisted settings, or a closed form for the maximizing $p$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. The capacity theorem has a Lean formalization listed in the release's formalization catalogue.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. The abstract and main theorem were read against the question. formalization.yaml lists ComparatorChallenges/AmplitudeDamping.json with declaration OAI.GAD.main in OAI/InformationTheory/AmplitudeDamping/Main.lean. The comparator statement, read here, says for all $\\gamma,\\nu\\in[0,1]$: the n-use Holevo quantity is n times the one-use value, equals n times an attained one-variable maximum, the operational capacity (defined by codes with collective POVM decoding, not by regularization) equals the one-use Holevo value, and equiprobable phase-pair product ensembles attain it. This is the headline. Additivity with an arbitrary different partner channel, a secondary claim, is not formalized. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"Determining the classical capacity of a standard qubit noise model is a natural open problem in quantum Shannon theory, but a specialist one, and the ordinary amplitude-damping case had just been settled by others. Below Bosonic quantum communication beyond the thermal threshold (25), level with the quantum pyramids conjecture (15) and online shadow tomography (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Classical capacity and entropy inequalities for generalized amplitude damping","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean formalization: Holevo additivity and capacity of generalized amplitude damping","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/InformationTheory/AmplitudeDamping/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: AmplitudeDamping","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/AmplitudeDamping.lean","kind":"lean-proof"},{"label":"Leditzky, Kaur, Datta and Wilde (2018), where the amplitude-damping capacity is listed as open","url":"https://doi.org/10.1103/PhysRevA.97.012332","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"shelah-eventual-categoricity-conjecture","name":"Shelah's eventual categoricity conjecture for abstract elementary classes","shortName":"Shelah's eventual categoricity conjecture","problemNumber":null,"field":"Model theory; abstract elementary classes","fieldGroup":"Logic & foundations","statement":"Morley proved that a countable first-order theory categorical in one uncountable cardinal is categorical in all of them. Shelah's abstract elementary classes (AECs) generalise elementary classes to nonelementary settings such as $L_{\\omega_1,\\omega}$ sentences, without compactness. Shelah's eventual categoricity conjecture asks for an analogue: for each infinite cardinal $\\lambda$ there should be a threshold $\\mu(\\lambda)$ such that every AEC $K$ with $LS(K)\\le\\lambda$ that is categorical in some cardinal $\\ge\\mu(\\lambda)$ is categorical in every cardinal $\\ge\\mu(\\lambda)$. Prior proofs needed amalgamation, tameness, a successor starting cardinal, large cardinals or weak GCH. Is eventual categoricity provable in ZFC for all AECs?","posedBy":"Saharon Shelah","yearPosed":2009,"ageNote":"The manuscript cites Shelah, Classification Theory for Abstract Elementary Classes (2009), Conjecture N.4.2, and Boney-Vasey (2015). The conjecture is older than that book: the companion notes Shelah introduced AECs in the mid-1970s with these categoricity questions, but the release gives no earlier dated statement.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.2: for every infinite cardinal $\\lambda$ there is $\\mu(\\lambda)\\ge\\lambda$ such that every AEC $K$ with $LS(K)\\le\\lambda$ categorical in some $\\kappa\\ge\\mu(\\lambda)$ is categorical in every $\\kappa'\\ge\\mu(\\lambda)$; equivalently (Theorem 1.3) an AEC categorical in unboundedly many cardinals is categorical on a tail. It applies to $L_{\\omega_1,\\omega}$ sentences in countable languages. It gives no explicit value for $\\mu(\\lambda)$; the prescribed $\\beth_{\\omega_1}$ threshold for $L_{\\omega_1,\\omega}$ and the Hanf threshold for AECs are left separate (the companion refutes the latter under CH).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 24, 2026: the ZFC eventual categoricity theorem (this entry) and a CH counterexample to the prescribed Hanf threshold (its own entry).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.2 was read against the conjecture. It gives, for every infinite $\\lambda$, a cardinal $\\mu(\\lambda)$ such that every AEC with $LS(K)\\le\\lambda$ categorical in one cardinal $\\ge\\mu(\\lambda)$ is categorical in all cardinals $\\ge\\mu(\\lambda)$, in ZFC, with no amalgamation, joint embedding, tameness or no-maximal-models hypothesis and any starting cardinal. This is the qualitative conjecture. It does not give a numerical threshold. The long proof (order presentations, working schemes, geometry built from unbounded categoricity) was not refereed and is not formalised; the family's Lean formalization covers only the CH companion.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":46,"significanceNote":"Eventual categoricity is the central test question of nonelementary model theory, open for decades and the target of the whole AEC program, though less known outside logic. Placed level with the Transitive Algebra Problem (46), below the Generator problem for von Neumann algebras (50), and above Wall's manifold-realization question for Poincare duality groups (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Eventual categoricity for abstract elementary classes","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A CH obstruction to a prescribed categoricity threshold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf","kind":"paper"},{"label":"Boney and Vasey, Categoricity and infinitary logics (2015)","url":"https://arxiv.org/abs/1508.03316v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"parity-not-in-qac0","name":"Parity is not in QAC0 (Moore's parity conjecture)","shortName":"Parity not in QAC0","problemNumber":null,"field":"Quantum circuit complexity; constant-depth lower bounds","fieldGroup":"Quantum information & computing","statement":"$\\mathrm{QAC}^0$ is the class of constant-depth quantum circuits on polynomially many qubits built from arbitrary one-qubit unitaries and Toffoli gates with unboundedly many controls, gates in a layer acting on disjoint qubits. Classically parity is not in $\\mathrm{AC}^0$ (Furst-Saxe-Sipser, Ajtai, Hastad). Moore introduced these quantum classes in 1999 and related coherent parity to unbounded fanout; he conjectured that parity cannot be computed in $\\mathrm{QAC}^0$. Prior lower bounds needed restrictions on the number of ancillas or on the entangling depth. Can a constant-depth quantum circuit of this kind, with polynomially many zero-initialized ancillas, compute the parity of $n$ input bits with bounded error?","posedBy":"Cristopher Moore (Quantum circuits: fanout, parity, and counting, arXiv quant-ph/9903046, 1999)","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims: for every fixed depth $d$, every $c\\ge1$ and every $0<\\varepsilon\\le1/2$, for all large $n$ no depth-$d$ circuit of one-qubit unitaries and unbounded-arity Toffoli gates on at most $n^c$ qubits, ancillas starting in $|0\\rangle$ and one output qubit measured, computes parity with probability at least $1/2+\\varepsilon$ on every input. Taking $\\varepsilon=1/6$ gives bounded error $2/3$. The companion paper gives a second proof via regular trajectories and pruning; via Xu-Li reductions the same obstruction holds for strict majority. It does not cover circuits with intermediate measurements, postselection or primitive fanout gates, and gives no explicit size-depth tradeoff beyond polynomial size.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. Two manuscripts dated the same day give two different proofs of the same bound.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstracts, introduction and Theorem 1.1 of the principal manuscript, read against Moore's conjecture as cited. The proofs were not refereed. Scope stated by the paper: unitary circuits with no intermediate measurement, postselection or primitive fanout gate; one measured output qubit, garbage unrestricted; Moore's original clean-ancilla model is covered as a special case. Lean-checked: the release's formalization catalogue lists comparator config ComparatorChallenges/QACParity.json with declaration OAI.QAC.parity_lower_bound (plus RegularParity.json, OAI.QAC.parity_lower_bound_polynomial_size). The statement was read here: for every depth d, exponent c >= 1 and 0 < eps <= 1/2, for all large n, no circuit of at most d disjoint layers of one-qubit unitaries and multi-control Toffolis on at most n^c qubits succeeds with probability >= 1/2 + eps on every input, success being the Born probability of the parity value on the measured qubit summed over all garbage. That is the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The best-known open lower-bound question in quantum circuit complexity, open since 1999 and attacked by Fourier, spectral and light-cone methods that all needed ancilla or depth restrictions. Placed below explicit depth-three circuit lower bounds beyond the square-root exponent (48) and the 3SUM hypothesis entry (50), above Feige's hypergraph Moore-bound conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Product-projection localization and the QAC0 parity lower bound","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Regular trajectories, pruning and quantum parity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf","kind":"paper"},{"label":"Moore, Quantum circuits: fanout, parity, and counting (1999)","url":"https://arxiv.org/abs/quant-ph/9903046","kind":"problem-record"},{"label":"Lean Comparator challenge QACParity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/QACParity.lean","kind":"lean-proof"},{"label":"Lean Comparator challenge RegularParity (polynomial-size 2/3 specialization)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/RegularParity.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/InformationTheory/QuantumCircuit/Parity.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/InformationTheory/QuantumCircuit/Parity.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lins-de-melo-pugh-quintic-lienard","name":"The Lins-de Melo-Pugh conjecture for quintic Lienard systems: at most two limit cycles","shortName":"Lins-de Melo-Pugh conjecture, degree five","problemNumber":null,"field":"Lienard systems; limit cycles","fieldGroup":"Differential equations","statement":"Lins, de Melo and Pugh conjectured that the classical Lienard system $\\dot x=y-F(x)$, $\\dot y=-x$, with $F$ a real polynomial of degree $n\\ge1$, has at most $\\lfloor (n-1)/2\\rfloor$ limit cycles. The conjecture fails for every $n\\ge6$ (Dumortier-Panazzolo-Roussarie in degree seven, De Maesschalck-Dumortier in degree six, De Maesschalck-Huzak in all degrees $n\\ge6$) and holds for $n\\le4$ (Li and Llibre for the quartic case); Rychkov proved it for odd quintic $F$. That left degree five. Does every classical Lienard system with $\\deg F\\le5$ have at most two limit cycles?","posedBy":"A. Lins, W. de Melo and C. C. Pugh, On Lienard's equation (Lecture Notes in Mathematics 597, Springer, 1977)","yearPosed":1977,"ageNote":"The manuscript cites an August 2026 preprint by Chen, Li, Zhang and Zhang whose introduction reports the unrestricted degree-five case as open.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $\\deg F\\le5$, with no parity or coefficient-sign assumption, the system $\\dot x=y-F(x)$, $\\dot y=-x$ has at most two limit cycles in $\\mathbb R^2$, counted as geometric images regardless of multiplicity, stability or hyperbolicity, and some member of the class has exactly two. This confirms the conjectured value $\\lfloor (5-1)/2\\rfloor=2$. It concerns only the classical form with restoring force exactly $x$; generalized Lienard systems and other degrees are not treated. The paper also shows that a proposed degree-five four-cycle example of Hernandez Rosales has only two cycles.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This manuscript, dated September 24, 2026, is the companion in a two-manuscript family and comes with a Lean formalization in the release's lean/ library. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. The release's formalization catalogue lists comparator config ComparatorChallenges/QuinticLienard.json with declaration OAI.QuinticLienard.main in OAI/Analysis/LienardCycles/Main.lean, permitted axioms propext, Quot.sound and Classical.choice. The comparator statement file was read: it defines the vector field (y - F(x), -x) for a real polynomial F, periodic orbits as ranges of nonconstant periodic solutions, and limit cycles as periodic orbits having an open neighbourhood that contains no other periodic orbit; it states that for deg F <= 5 the set of limit cycles has at most 2 elements and that some such F has exactly 2. That is the paper's headline claim, with no parity, sign or hyperbolicity assumption. Not rebuilt here: the Lean build and axiom check were not run. The paper's Theorem 1.1 was also read against the conjecture as quoted in its introduction.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A specific conjecture well known in the limit-cycle and Lienard community, and the last open degree of a conjecture known to fail from degree six on. Placed level with the Hilbert transform UMD dependence problem (25), below Stein's dimension-free Riesz transform problem (38), and far below the existential Hilbert sixteenth problem from the same family (85).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf","sourceName":"OpenAI math release: Two limit cycles for quintic Lienard systems","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI.QuinticLienard.main","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/LienardCycles/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: QuinticLienard","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/QuinticLienard.lean","kind":"lean-proof"},{"label":"Lean scope note for family 143","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/143.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"spherical-perceptron-free-energy","name":"The finite-temperature free energy of the spherical perceptron with bounded continuous potentials (Gyorgyi-Reimann continuous-RSB formula)","shortName":"Spherical perceptron free energy","problemNumber":null,"field":"Spin glasses; perceptron models","fieldGroup":"Mathematical physics","statement":"For $x$ uniform on the sphere of radius $\\sqrt N$ in $\\mathbb R^N$, $M=\\lfloor\\alpha N\\rfloor$ standard Gaussian patterns and a single-pattern potential $\\phi$, the pressure is $p_N=N^{-1}\\log\\int e^{\\beta\\sum_a\\phi(g^a\\cdot x/\\sqrt N)}d\\sigma_N(x)$. Gardner (1988) began the study of such storage volumes; Shcherbina-Tirozzi proved Gardner's replica-symmetric formula for nonnegative margins in a restricted regime. Gyorgyi and Reimann (2000) derived a continuous replica-symmetry-breaking formula for the finite-temperature model, which in general is not replica-symmetric and not a Gaussian spin glass. Does $p_N$ converge, for every density and temperature and nonconvex potentials, to that variational formula?","posedBy":"Geza Gyorgyi and Peter Reimann, Beyond storage capacity in a single model neuron: continuous replica symmetry breaking (J. Stat. Phys., 2000), after Elizabeth Gardner (1988)","yearPosed":2000,"ageNote":"Gardner's storage-volume problem dates to 1988; the precise finite-temperature formula the theorem matches is Gyorgyi-Reimann 2000, as the manuscript cites.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for all $\\alpha,\\beta>0$ and $\\phi\\in C_b(\\mathbb R)$, $\\mathbb Ep_N$ and $p_N$ (in probability) converge to $\\inf_m\\{\\alpha V_{\\beta\\phi}(m)+S(m)\\}$ over nondecreasing $m:[0,1]\\to[0,1]$, with $V$ a Brownian stochastic-control value and $S$ the spherical entropy; this matches the Gyorgyi-Reimann continuous-RSB functional. Companions: the analogous formula for the Gaussian Ising perceptron with any bounded Borel log-potential at all densities and positive temperatures, and for spherical models with bi-orthogonally invariant disorder without outliers. Not shown: hard constraints or zero temperature (storage capacity), discontinuous potentials in the spherical case, or properties of the optimizer.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This entry draws on three manuscripts dated September 24, 2026: the spherical free energy (principal), the Ising perceptron free energy for bounded Borel log-potentials, and the spherical model with bi-orthogonally invariant disorder.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of 'The free energy of the spherical random perceptron' were read against the Gyorgyi-Reimann formula as the manuscript cites it; Gyorgyi-Reimann was not read. The theorem needs a bounded continuous potential, so the discontinuous error-counting cost of the classical storage problem is not covered directly; hence partial. Lean: lean/ComparatorChallenges/PerceptronFreeEnergy.lean (solution module OAI.Probability.Perceptron.Main, present at the pinned commit) is not in the formalization catalogue; it was found through lean/docs/222.md. Its statement SphericalPerceptronFreeEnergy.main was read: for alpha, beta > 0 and bounded continuous phi, the infimum over monotone trials of alpha times a Brownian control value plus the spherical entropy is a finite real p, the expected pressure converges to p, and the pressure converges to p in probability. That is the headline of the principal paper. Not rebuilt here. The Ising companion's Lean challenge covers only finiteness of its variational value.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A rigorous Parisi-type formula for a classical non-Gaussian mean-field model, filling a recognized gap since Gardner's work, but for soft continuous potentials only. Below the Krauth-Mezard capacity of the Ising perceptron (35) and the Mezard-Parisi formula for diluted spin glasses (32), below the Gardner transition entry (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf","sourceName":"OpenAI math release: The free energy of the spherical random perceptron","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The free energy of the Ising random perceptron","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf","kind":"paper"},{"label":"Companion: The spherical perceptron with bi-orthogonally invariant disorder","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf","kind":"paper"},{"label":"Lean challenge statement: PerceptronFreeEnergy.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PerceptronFreeEnergy.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Probability/Perceptron/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/Perceptron/Main.lean","kind":"lean-proof"},{"label":"Gyorgyi-Reimann (2000), continuous replica symmetry breaking in a single model neuron","url":"https://arxiv.org/abs/cond-mat/0003360","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kuhn-lloyd-chromatic-fixed-point-loss","name":"The Kuhn-Lloyd question: does the chromatic fixed-point loss of a p-group equal the shortest cyclic subnormal chain length?","shortName":"Kuhn-Lloyd chromatic fixed-point loss equality","problemNumber":null,"field":"Equivariant stable homotopy theory, chromatic homotopy","fieldGroup":"Geometry & topology","statement":"For a finite $p$-group $G$, a subgroup $H$ and $n\\ge0$, let $r_n(G,H)$ be the least $r$ such that $K(n+r)_*(\\Phi^HX)=0$ implies $K(n)_*(\\Phi^GX)=0$ for every finite $p$-local genuine $G$-spectrum $X$, where $K(i)$ is Morava K-theory. This controls the inclusions among primes in the Balmer spectrum of finite $G$-spectra. Let $\\ell(G,H)$ be the shortest length of a subnormal chain from $H$ to $G$ with cyclic quotients; then $r_n\\le\\ell$. The abelian case was settled by Barthel et al., and Kuhn and Lloyd proved equality for central extensions of elementary abelian 2-groups by $C_2$. They expressed hope that $r_n(G,H)=\\ell(G,H)$ in general and asked whether the loss is independent of the height $n$. Is $r_n(G,H)=\\ell(G,H)$ for all finite $p$-groups?","posedBy":"Nicholas J. Kuhn and Christopher J. R. Lloyd","yearPosed":2024,"ageNote":"Kuhn-Lloyd, Amer. J. Math. 146 (2024), 769-812, Section 2.2, as cited by the manuscript; phrased as a 'cautious hope' plus an explicit question on height independence, not a formal conjecture.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every prime $p$, finite $p$-group $G$, subgroup $H$ and $n\\ge0$, $r_n(G,H)=\\ell(G,H)$, the shortest cyclic subnormal chain length; sharpness is witnessed by finite spectra, and the case $n=0$ includes rational detection. Combined with Balmer-Sanders reductions, a corollary determines the finite-height prime inclusions in the Balmer spectrum of finite genuine $G$-spectra for every finite group $G$. Not shown: anything about infinite height beyond what Balmer-Sanders already give, or non-finite spectra.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 24, 2026).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against the Kuhn-Lloyd question as the manuscript quotes it; the theorem states the hoped-for equality for every prime, every finite $p$-group and every height, hence height independence. The proof (profinite character sources, relative integration, telescopic estimates) was not refereed. No Lean formalization accompanies this manuscript.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"A recent specialist question whose answer completes the Balmer spectrum of finite equivariant spectra for all finite groups, a goal of tensor-triangular geometry since Balmer-Sanders. Placed well below Hopkins' chromatic splitting conjecture (40), level with the superexponential van der Waerden entry (22) and below Siegel-Yao ball packing (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026.pdf","sourceName":"OpenAI math release: Cyclic length and chromatic fixed-point loss","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kuhn-Lloyd, Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups (AJM 2024)","url":"https://preprint.press.jhu.edu/ajm/sites/default/files/AJM-kuhn-lloyd-FINAL.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"quillen-conjecture","name":"Quillen's conjecture on the poset of elementary abelian p-subgroups","shortName":"Quillen's conjecture","problemNumber":null,"field":"Finite group theory; subgroup complexes and homotopy","fieldGroup":"Algebra","statement":"Let $G$ be a finite group, $p$ a prime, and $\\mathcal A_p(G)$ the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen showed that a nontrivial normal $p$-subgroup ($O_p(G)\\ne1$) makes $\\mathcal A_p(G)$ contractible, and proved the converse for solvable groups, $p$-rank at most two and groups of Lie type in defining characteristic. Aschbacher and Smith reduced the problem for $p>5$ to unitary-group cases and later work (Piterman, Smith, Diaz Ramos) narrowed it further, settling the rational form for odd primes, but the prime $2$ remained open. Conjecture: if $\\mathcal A_p(G)$ is contractible, must $O_p(G)\\ne1$?","posedBy":"Daniel Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Advances in Mathematics 28 (1978), Conjecture 2.9","yearPosed":1978,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every finite group $G$ and prime $p$, if $O_p(G)=1$ then $\\widetilde H_*(\\mathcal A_p(G);\\mathbb Q)\\ne0$ (augmented, so the empty poset counts). This gives Quillen's conjecture and its stronger rational-homology form, and acyclicity over any field characterizes $O_p(G)\\ne1$. Priority: the manuscript credits Diaz Ramos (arXiv:2303.15613v6, July 2026) with the rational form at all odd primes, so the new content is chiefly the prime $2$ (via Piterman's component eliminations) plus a direct proof of the unitary dimension property. It does not prove the Quillen dimension property (homology in top degree) in general.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Quillen's Conjecture 2.9 as the manuscript cites it. Theorem 1.1 states that $O_p(G)=1$ implies nonzero augmented reduced rational homology of $\\mathcal A_p(G)$, for every finite group and prime, which implies non-contractibility (the Aschbacher-Smith rational form). The proof relies on published and preprint reductions (Aschbacher-Smith, Piterman-Smith, Piterman 2026 arXiv:2607.27500, Diaz Ramos 2026 arXiv:2303.15613v6) plus new frame-cycle constructions; it uses the classification of finite simple groups through those reductions. Not refereed here. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"Quillen's conjecture has been a central open problem linking finite group theory and topology since 1978, with decades of reductions through the classification. Famous across finite group theory, less known outside it, and the odd-prime rational case had just been settled by Diaz Ramos. Placed level with Kaplansky's direct finiteness conjecture (52), just above Serre's positivity conjecture (50) and below the Alperin weight conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rational-homology-and-Quillens-conjecture-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Rational homology and Quillen's conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Quillen 1978, Homotopy properties of the poset of nontrivial p-subgroups of a group","url":"https://doi.org/10.1016/0001-8708(78)90058-0","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"noncommutative-rational-formula-pit-polynomial-hitting-sets","name":"Polynomial-size black-box identity testing for noncommutative rational formulas","shortName":"Polynomial hitting sets for NC rational formulas","problemNumber":null,"field":"Algebraic complexity; derandomization of polynomial identity testing","fieldGroup":"Theoretical computer science","statement":"A noncommutative rational formula is a formula with $+$, $\\times$ and inverse gates over noncommuting variables, evaluated on tuples of square matrices where every inverse is defined. Hrubes and Wigderson reduced its identity testing to noncommutative singularity, solved deterministically in the white-box model by operator scaling. In the black-box model, Forbes-Shpilka gave quasipolynomial matrix hitting sets for division-free formulas, and Arvind, Chatterjee and Mukhopadhyay gave deterministic quasipolynomial-size rational matrix hitting sets for rational formulas of any inversion height, asking in their conclusion for polynomial matrix dimension. Is there a deterministic polynomial-time construction of a polynomial-size list of matrix tuples hitting every nonzero rational formula of size $s$ in $n$ variables?","posedBy":"V. Arvind, Abhranil Chatterjee and Partha Mukhopadhyay, Black-box identity testing of noncommutative rational formulas in deterministic quasipolynomial time, arXiv:2309.15647 (2023), conclusion","yearPosed":2023,"ageNote":"The manuscript cites the 2025 revision of the arXiv preprint (first posted September 2023) for the request for polynomial matrix dimension.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims deterministic polynomial-time construction of polynomial-size rational matrix hitting lists for noncommutative rational formulas over $\\mathbb Q$, with no bound on inverse nesting or constant heights, each nonzero formula receiving an invertible value. Companions: one rational tuple of dimension at most $2ns^2$ hits every nonzero division-free formula over every characteristic-zero field, and an analogous tuple over $\\mathbb F_p$ works in each positive characteristic. It does NOT cover general noncommutative circuits, and the rational result is over $\\mathbb Q$ only.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The family has three manuscripts: the rational-formula hitting lists (principal), a single rational hitting point for division-free formulas over every characteristic-zero field (same date), and a positive-characteristic version (4 October). The release formalizes the principal theorem and the single-point theorem in Lean.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against the question: one deterministic machine, given $1^n,1^s$, outputs in polynomial bit time a polynomial-size list of rational matrix tuples of a common dimension, one of which gives every admissible nonzero rational formula of size at most $s$ a defined invertible value. ComparatorChallenges/RationalHitting.lean (OAI.RationalHitting.main, OAI/Computability/RationalHitting/Main.lean, in formalization.yaml) states exactly this, with a TM0 Turing machine, explicit binary output encoding, polynomial bounds on dimension, output length and steps, and inverse gates requiring two-sided inverses. Permitted axioms propext, Quot.sound, Classical.choice. The companion's FormulaHitting challenge (OAI.NCHitting.universal_hitting, not in formalization.yaml) was also read; it states the single-tuple hitting property but not the construction-time bound. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"Polynomial-size black-box identity testing for noncommutative formulas is a recognized derandomization target improving the quasipolynomial bounds of Forbes-Shpilka and Arvind-Chatterjee-Mukhopadhyay, known mainly within algebraic complexity. Above the Polynomial-Time Low-Degree Conjecture entry (30), below the CVP hardness entry (35) and permanent formula lower bounds (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026.pdf","sourceName":"OpenAI math release: Polynomial Hitting Lists for Noncommutative Rational Formulas","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: One Rational Matrix Hitting Point for Noncommutative Formulas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026.pdf","kind":"paper"},{"label":"Companion: Uniform Matrix Hitting Points in Every Positive Characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026/uniform-matrix-hitting-points-positive-characteristic.pdf","kind":"paper"},{"label":"Lean: polynomial hitting lists for rational formulas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/RationalHitting/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: RationalHitting","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/RationalHitting.lean","kind":"lean-proof"},{"label":"Lean: single rational hitting tuple for division-free formulas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/FormulaHitting/Hitting.lean","kind":"lean-proof"},{"label":"Arvind, Chatterjee and Mukhopadhyay (2023), quasipolynomial black-box rational identity testing","url":"https://arxiv.org/abs/2309.15647","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"carleson-schrodinger-convergence-endpoint","name":"The endpoint case of Carleson's pointwise convergence problem for the Schrodinger equation: a.e. convergence for data in H^{n/(2(n+1))}","shortName":"Endpoint of Carleson's Schrodinger convergence problem","problemNumber":null,"field":"Harmonic analysis; dispersive equations and pointwise convergence","fieldGroup":"Analysis","statement":"Carleson asked for which Sobolev exponents $s$ the free Schrodinger evolution $e^{it\\Delta}f$ converges to $f$ almost everywhere as $t\\to0$ for every $f\\in H^s(\\mathbb R^n)$. In one dimension the answer is $s\\ge1/4$ (Carleson; Dahlberg-Kenig). Bourgain's 2016 counterexample showed that $s\\ge n/(2(n+1))$ is necessary for $n\\ge2$ but did not rule out equality; Du, Guth and Li (planar) and Du and Zhang ($n\\ge3$) proved convergence for every $s>n/(2(n+1))$, leaving only the endpoint. Does $e^{it\\Delta}f\\to f$ almost everywhere for every $f\\in H^{n/(2(n+1))}(\\mathbb R^n)$, $n\\ge2$ (in the plane, for $H^{1/3}$)?","posedBy":"Lennart Carleson (the convergence problem); the endpoint left open after Bourgain (2016), Du-Guth-Li (2017) and Du-Zhang (2019)","yearPosed":1980,"ageNote":"Carleson, Some analytic problems related to statistical mechanics (Lecture Notes in Math. 779, 1980), cited by the manuscripts. The endpoint became the remaining case only after Du-Guth-Li and Du-Zhang; the manuscripts cite no separate statement of the endpoint question.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Planar theorem: for every $f\\in H^{1/3}(\\mathbb R^2)$ there is a full-measure set $E_f$ on which the Gaussian-regularised evolution has a limit for every $0 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 24, 2026: the planar endpoint $H^{1/3}(\\mathbb R^2)$ and the endpoint in every dimension $n\\ge3$.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorems of both manuscripts were read against Carleson's problem and the endpoint left by Bourgain and Du-Zhang. The evolution is defined by taking the Gaussian-regularisation limit first, on one full-measure spatial set and for all $0 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'Topological triviality of mu-constant families of surface singularities' (September 24, 2026).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the problem as the manuscript frames it: for a holomorphic family $F$ on $\\mathbb C^3\\times\\Delta$ with isolated critical points and constant Milnor number there is, after shrinking, a homeomorphism germ $\\Phi(x,t)=(\\phi_t(x),t)$, jointly continuous with continuous inverse, with $\\phi_t(0)=0$ and $f_t\\circ\\phi_t=f_0$. Topological right-triviality implies constancy of the embedded topological type, so this answers the surface case in full. There is no Lean formalization. The proof uses semistable reduction, the threefold log minimal model program, the Fernández de Bobadilla-Pełka equimultiplicity theorem, Okuma's simultaneous resolution and Langer's orbifold Bogomolov-Miyaoka-Yau inequality; it was not refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The surface case is the one gap left by the Lê-Ramanujam theorem of 1976 and a standard open problem of equisingularity theory for fifty years. Famous within singularity theory. Placed below Zariski's multiplicity conjecture (52), level with Kuznetsov's rationality conjecture (45), above the Lipman-Zariski conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/main.pdf","sourceName":"OpenAI math release: Topological triviality of mu-constant families of surface singularities","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lê and Ramanujam, The invariance of Milnor's number implies the invariance of the topological type (1976)","url":"https://doi.org/10.2307/2373614","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"almost-linear-general-graph-matching","name":"Maximum-cardinality matching in general graphs in almost-linear time","shortName":"Almost-linear general matching","problemNumber":null,"field":"Graph algorithms; matching","fieldGroup":"Theoretical computer science","statement":"Given a simple undirected graph with $n$ vertices and $m$ edges, find a maximum-cardinality matching. Edmonds' blossom algorithm gave polynomial time; Micali and Vazirani (1980) achieved $O(m\\sqrt n)$, which remained the best combinatorial bound for sparse general graphs, while algebraic methods give $O(n^\\omega)$. After almost-linear-time exact max flow (Chen, Kyng, Liu, Peng, Probst Gutenberg and Sachdeva, 2022) gave almost-linear bipartite matching, the general-graph case, where odd cycles (blossoms) defeat the flow formulation, remained open. Can a maximum matching in a general graph be found in $m^{1+o(1)}$ time?","posedBy":"Long-standing open algorithmic question (no single poser cited in the manuscript); benchmark bound Micali and Vazirani (1980), bipartite case settled by Chen et al. (2022)","yearPosed":null,"ageNote":"The manuscript gives no reference that poses the question; it describes the history from Edmonds through Micali-Vazirani (1980) to almost-linear bipartite matching (2022). Beating O(m sqrt n) for sparse general graphs has been open since 1980, so yearPosed could defensibly be 1980; left null pending a citable statement.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a single uniform randomized algorithm finds a maximum-cardinality matching of any simple graph in $(n+m)^{1+o(1)}$ word operations on every path, with success probability at least $2/3$ and explicit output. Corollary: the same guarantees for deciding and constructing $f$-factors (prescribed-degree spanning subgraphs). It is randomized (no deterministic version), unweighted (maximum-weight general matching is not addressed), and the $o(1)$ is not made explicit.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the general-graph matching question. Theorem 1.1 states one uniform randomized word-RAM algorithm that halts within $Cp^{1+\\eta(p)}$ instructions on every computation path ($p=n+m$, $\\eta\\to0$) and outputs an explicit maximum matching with probability at least $2/3$. It is Monte Carlo: correctness of the output is not certified. The algorithm uses the deterministic almost-linear flow theorem (van den Brand et al.) as a black box. Not refereed here. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Maximum matching is among the most studied problems in algorithms, and the O(m sqrt n) barrier for sparse general graphs stood since 1980; almost-linear time would be a headline result in the field, though the question lacks a single named poser. Placed level with the Schonhage-Strassen conjecture (48), below the FPRAS for counting perfect matchings in general graphs (55), and above the DFT n log n lower bound (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/main.pdf","sourceName":"OpenAI math release: Almost-Linear-Time Maximum-Cardinality Matching in General Graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"birkhoff-conjecture-near-boundary","name":"The near-boundary Birkhoff conjecture: billiards integrable near the boundary are ellipses","shortName":"Near-boundary Birkhoff conjecture","problemNumber":null,"field":"Dynamical systems; convex billiards and integrability","fieldGroup":"Analysis","statement":"In a planar billiard in a strictly convex domain $\\Omega$, a caustic is a curve such that a trajectory tangent to it stays tangent after every reflection. Ellipses are integrable: confocal ellipses foliate a neighbourhood of the boundary by caustics. The Birkhoff conjecture, traditionally attributed to Birkhoff and first printed by Poritsky (1950), asserts that every integrable strictly convex billiard is an ellipse. Lazutkin found a Cantor family of caustics near the boundary for every smooth positively curved table; Bialy proved rigidity under integrability of the whole phase space; Avila-De Simoi-Kaloshin and Kaloshin-Sorrentino proved local versions near ellipses. Is a smooth, strictly convex, positively curved billiard table an ellipse whenever it is integrable near the boundary?","posedBy":"Traditionally attributed to G. D. Birkhoff (Dynamical Systems, 1927); first printed formulation by Hillel Poritsky, Ann. of Math. 51 (1950), as the manuscripts cite","yearPosed":1950,"ageNote":"The manuscripts attribute the conjecture to Birkhoff and cite his 1927 book for the setting, with Poritsky's 1950 paper as the early printed formulation; the near-boundary form is not attributed to a separate poser.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (phase foliations): if a $T$-invariant open annulus containing $(\\mathbb R/L\\mathbb Z)\\times(0,\\eta)$ in the billiard phase space is homeomorphically foliated by individually invariant essential curves, then $\\Omega$ is an ellipse. The companion proves the same conclusion from a continuous collar of smooth closed convex caustics, via analytic regularity of the boundary and complex continuation. Only continuity across leaves is assumed. Not covered: integrability only on a Cantor set, rational integrability, or integrability away from the boundary; the full Birkhoff conjecture (global integrability as typically defined) is reduced to but not identical with this case.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of each September 24 manuscript was read against the Birkhoff conjecture as the papers state it. One proves the ellipse conclusion when a full neighbourhood of the boundary is continuously foliated by smooth closed convex caustics; the other derives such a collar from a continuous foliation of a grazing phase annulus by invariant essential curves, and invokes the first. The proofs were not refereed. Scope stated by the papers: $C^\\infty$ boundary with positive curvature, a full collar (a Cantor family of caustics does not suffice), no assumption on dynamics away from the boundary. No Lean formalization exists.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A strong case of the Birkhoff conjecture, the best-known rigidity problem for billiards, open in this form despite the local results of Avila-De Simoi-Kaloshin and Kaloshin-Sorrentino. Placed below the closed geodesics on spheres entry (55) and Sinai's standard map conjecture (58), level with Brennan's conjecture (45) and Vinogradov's least quadratic nonresidue (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Continuous Phase Foliations Create Analytic Caustic Collars","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Rigidity of Smooth Billiards with a Continuous Caustic Collar","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"planar-fpp-no-bigeodesics","name":"No bigeodesics in planar first-passage percolation under a minimum-of-four second-moment condition","shortName":"No bigeodesics in planar FPP","problemNumber":null,"field":"First-passage percolation; infinite geodesics","fieldGroup":"Probability & statistics","statement":"Give the nearest-neighbor edges of $\\mathbb Z^2$ i.i.d. nonnegative weights with a continuous law and let $T(x,y)$ be the passage-time metric. A bigeodesic is a doubly infinite path $(v_i)_{i\\in\\mathbb Z}$ every finite segment of which is a geodesic. Furstenberg asked, as recorded by Kesten (1986, Remark 9.22), whether such paths can exist; the planar no-bigeodesics conjecture (Damron-Hanson, Conjecture 1) asserts that almost surely there are none for i.i.d. continuous weights. Earlier results excluded bigeodesics only in fixed directions, under curvature or differentiability assumptions on the limit shape, or in restricted geodesic graphs. Is it true that almost surely planar first-passage percolation with i.i.d. continuous weights has no bigeodesic?","posedBy":"Hillel Furstenberg, as recorded by Harry Kesten; stated as a conjecture by Damron and Hanson (2017)","yearPosed":1986,"ageNote":"Kesten's Saint-Flour notes (Lecture Notes in Math. 1180, 1986, Remark 9.22) attribute the question to Furstenberg; the date of Furstenberg's original question is not given. Damron and Hanson state the conjecture explicitly (Comm. Math. Phys. 349, 2017, Conjecture 1).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if the edge-weight law on $[0,\\infty)$ has no atoms and the minimum of four independent copies has finite second moment, then almost surely there is no bigeodesic, all paths excluded simultaneously on one probability-one event, with no assumption of differentiability, strict convexity or curvature of the limit shape. The law may be singular continuous, have bounded support or infinite mean. It does not cover laws failing the moment condition, higher dimensions, or directed models.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the manuscript states it. It assumes nonatomic weights with $E[\\min(X_1,\\dots,X_4)^2]<\\infty$ and the paper says explicitly that it does not assert the moment-free formulation; hence Partial. The structural inputs (Ahlberg-Hoffman's Busemann theory) are cited theorems from the literature, replaced at one step by an appendix argument. The proof was not refereed. This manuscript has no Lean formalization; the family's Lean work (lean/docs/212.md) covers only the companion limit-shape paper.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The no-bigeodesics question is a central open problem of first-passage percolation, tied to the ground-state question for disordered ferromagnets; the result covers the natural moment class but not the moment-free statement, so it sits below the parent. Placed below The Benjamini-Schramm criticality conjecture for bond percolation (79) and Infinitely many closed geodesics on every Riemannian sphere (55), and level with Local smooth isometric immersion of surfaces in three-space (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/main.pdf","sourceName":"OpenAI math release: No bigeodesics in planar first-passage percolation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Strict convexity and differentiability of the planar exponential FPP limit shape","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"binary-corrected-elliott-conjecture","name":"The binary corrected Elliott conjecture for ordinary averages","shortName":"Binary corrected Elliott conjecture","problemNumber":null,"field":"Analytic number theory; pretentious multiplicative functions","fieldGroup":"Number theory","statement":"Elliott (1992) conjectured that correlations $\\frac1N\\sum_{n\\le N}f_1(n+h_1)\\cdots f_k(n+h_k)$ of multiplicative functions bounded by one tend to zero when one factor does not pretend to be a twisted character. Matomaki, Radziwill and Tao showed the original hypothesis is insufficient for complex functions and corrected it: with $D(f,\\chi n^{it};N)^2=\\sum_{p\\le N}(1-\\mathrm{Re}\\,f(p)\\overline{\\chi(p)}p^{-it})/p$, require $\\inf_{|t|\\le N}D(f,\\chi n^{it};N)\\to\\infty$ for every fixed Dirichlet character $\\chi$. Tao proved the logarithmically averaged binary case, and Tao-Teravainen and Klurman-Mangerel-Teravainen proved it along most scales. For multiplicative $f_1,f_2:\\mathbb{N}\\to\\{|z|\\le1\\}$ with one factor uniformly nonpretentious and fixed distinct $h_1,h_2$, does $\\frac1N\\sum_{n\\le N}f_1(n+h_1)f_2(n+h_2)\\to0$?","posedBy":"P. D. T. A. Elliott (Notas Soc. Mat. Chile, 1992); corrected form by Matomaki, Radziwill and Tao (2015)","yearPosed":1992,"ageNote":"Elliott's 1992 conjecture; the corrected nonpretentiousness hypothesis used here is from Matomaki-Radziwill-Tao (Algebra Number Theory 2015), which the paper cites as equivalent to its condition.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For multiplicative $f_1,f_2:\\mathbb{N}\\to\\{|z|\\le1\\}$, at least one satisfying $\\inf_{|t|\\le N}D(f_j,\\chi n^{it};N)\\to\\infty$ for every fixed Dirichlet character, and fixed distinct $h_1,h_2\\ge0$: $\\frac1N\\sum_{n\\le N}f_1(n+h_1)f_2(n+h_2)\\to0$. Also for fixed nonproportional affine forms and in fixed residue classes. Functions may be complex and need not be completely multiplicative. Qualitative only, no rate; the $k$-point Elliott conjecture for $k\\ge3$ is not addressed.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single manuscript, dated September 24, 2026. The release also publishes an abridged summary of the model's reasoning for this family (reasoning_traces/ordinary-two-point-correlations.pdf).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.2 and Corollary 1.3 were read against the binary corrected Elliott conjecture with ordinary averages. The challenges are not in the formalization catalogue (lean/formalization.yaml); lean/ComparatorChallenges/OrdinaryTwoPointCorrelations.json and OrdinaryElliott.json exist with solution modules OAI.NumberTheory.TwoPointCorrelations.FinalMain and OAI.NumberTheory.OrdinaryCorrelations.Elliott.Main present at the pinned commit, permitted axioms propext, Quot.sound and Classical.choice. The statements OAI.OrdinaryCorrelations.binary_corrected_elliott and OAI.OrdinaryTwoPointCorrelations.binary_corrected_elliott and affine_corrected_elliott were read: multiplicative one-bounded f1, f2, one of them with inf over |t| <= N of the distance to chi(n) n^{it} tending to infinity for every Dirichlet character, imply the shifted (and affine) correlation averages tend to 0. That is the headline. Not rebuilt here. The paper makes no rate claim for general functions.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"The binary corrected Elliott conjecture is the natural generalisation of two-point Chowla to all bounded multiplicative functions, known to everyone working on pretentious methods but less famous than Chowla itself, whose ordinary case it contains qualitatively. Below two-point Chowla (58), level with the linear list Hadwiger conjecture (48), above Vinogradov's least quadratic nonresidue conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf","sourceName":"OpenAI math release: Ordinary two-point correlations of multiplicative functions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI/NumberTheory/TwoPointCorrelations/FinalMain.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/TwoPointCorrelations/FinalMain.lean","kind":"lean-proof"},{"label":"Lean proof: OAI/NumberTheory/OrdinaryCorrelations/Elliott/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/OrdinaryCorrelations/Elliott/Main.lean","kind":"lean-proof"},{"label":"Abridged summary of the model's reasoning (release reasoning_traces)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/reasoning_traces/ordinary-two-point-correlations.pdf","kind":"other"},{"label":"Matomaki, Radziwill, Tao, An averaged form of Chowla's conjecture (corrected Elliott condition, arXiv:1503.05121)","url":"https://arxiv.org/abs/1503.05121v3","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mutually-unbiased-bases-dimension-six","name":"Zauner's conjecture: at most three mutually unbiased bases in dimension six","shortName":"Mutually unbiased bases in dimension six","problemNumber":null,"field":"Mutually unbiased bases; complex Hadamard matrices","fieldGroup":"Quantum information & computing","statement":"Orthonormal bases $\\mathcal B,\\mathcal C$ of $\\mathbb C^d$ are mutually unbiased if $|\\langle b,c\\rangle|^2=1/d$ for all $b\\in\\mathcal B$, $c\\in\\mathcal C$. Let $N(d)$ be the largest number of pairwise unbiased bases. Always $N(d)\\le d+1$, with equality in prime-power dimensions (Ivanovic; Wootters-Fields), so $d=6$ is the first open case. Three bases exist in $\\mathbb C^6$ from a tensor-product construction; numerical searches (Butterley-Hall, Brierley-Weigert) and restricted exclusions (Grassl; Jaming et al. for the Fourier family) found no fourth. Zauner conjectured in his 1991 diploma thesis that three is the maximum; Klappenecker and Rotteler state $N(6)=3$ as Conjecture 1. Is $N(6)=3$?","posedBy":"Gerhard Zauner (diploma thesis, Vienna, 1991); stated as Conjecture 1 by Klappenecker and Rotteler (2004)","yearPosed":1991,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (computer-assisted): there are no four pairwise mutually unbiased orthonormal bases in $\\mathbb C^6$, hence $N(6)=3$, the lower bound coming from the explicit tensor-product triple. The proof covers every second basis (an arbitrary complex Hadamard matrix) by certified phase balls and excludes two further bases in each by a clique search with rounding margins. It is conditional on the stated binary64 arithmetic and on the complete execution it documents. The companion paper gives an exact rational certificate only for $N(6)\\le5$, and that weaker bound is what the release's Lean formalization proves.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The upper bound is a computer-assisted exhaustive cover of all complex Hadamard matrices of order six, certified under stated binary64 arithmetic and compiler conditions; the manuscript says it rests on a fresh complete execution and that a separate September 7 execution reported in its source manuscript was not authenticated.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Zauner's conjecture; it states that under the arithmetic and compiler conditions of its appendix, a documented complete execution excludes four pairwise unbiased bases, so $N(6)=3$. The conclusion is conditional on that computation. The manuscript points to an accompanying verification archive (verification/computation/README.md, verification/REPRODUCE.md, verification/code/proof_run.py); fetching those paths from the manuscript folder at the pinned commit returned 404, and the folder's README has no verification section, so the computation was not re-run here. Lean: lean/docs/266.md says the linked formalization (MUBSix) proves only that every family has at most five members, plus the companion's Fourier vanishing; it does not establish the bound of three, so this entry is Unreviewed, not Lean-checked.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Whether four mutually unbiased bases exist in dimension six is the best known open problem about mutually unbiased bases, open since 1991 and listed among standard open problems in quantum information. Placed level with the word problem for finite-rank Artin groups (45), above Singular Bernoulli convolutions at non-Pisot parameters (42), below Upper Bounds for High-Dimensional Sphere Packing (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf","sourceName":"OpenAI math release: The maximum number of mutually unbiased bases in dimension six","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Exact Fourier certificates for complex Hadamard matrices of order six (N(6) <= 5)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf","kind":"paper"},{"label":"Lean (five-basis bound only): OAI.MUB6.fourier_and_family_bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/MutuallyUnbiased/Main.lean","kind":"lean-proof"},{"label":"Lean scope note for this family (lean/docs/266.md)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/266.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"barnette-conjecture","name":"Barnette's conjecture","shortName":"Barnette's conjecture","problemNumber":null,"field":"Graph theory: Hamiltonian cycles in planar graphs","fieldGroup":"Combinatorics","statement":"A graph is cubic if every vertex has degree three, and polyhedral if it is planar and 3-vertex-connected, that is, the graph of a convex polyhedron. Tait conjectured that every cubic polyhedral graph is Hamiltonian; Tutte disproved this in 1946. Barnette's conjecture restricts to bipartite graphs. Known before this work: all faces of size 4 or 6 (Goodey), arbitrary faces in one face color class with only quadrilaterals and hexagons in the other two (Feder-Subi), and all faces of size at most 8 (Schnieders, 2025). Does every finite simple cubic bipartite planar 3-vertex-connected graph have a Hamiltonian cycle?","posedBy":"David Barnette; recorded by Branko Grunbaum as Unsolved Problem 5 in Recent Progress in Combinatorics (Proceedings of the Third Waterloo Conference, May 1968), Academic Press, 1969","yearPosed":1969,"ageNote":"The manuscript cites Grunbaum's record in the 1969 proceedings of the May 1968 Waterloo conference; it gives no separate date for Barnette's own formulation.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle. In the dual triangulation (Stein; Alt-Payne-Schmidt-Wood) the paper splits the vertices into two induced trees with a prescribed facial pattern, using Tutte states, a signed exponential sum over pairs of states, and gluing along separating triangles. Corollaries: the Hamiltonian cycle can avoid any prescribed edge (a known equivalent strengthening, via Kelmans, Hertel and Gorsky-Steiner-Wiederrecht), and every three-edge path in a cubic 3-connected Pfaffian bipartite graph lies in a Hamiltonian cycle. The result is confined to the bipartite polyhedral class.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source; the proof was not refereed. Lean: the challenge BarnetteHamiltonian (OAI.Barnette.main, solution module OAI/Combinatorics/Hamiltonian/Main.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Statement read here: every finite simple graph that is 3-regular, bipartite, planar (a crossing-free embedding in $\\mathbb R^2$ by injective arcs with disjoint interiors) and 3-vertex-connected (at least four vertices, connected after deleting any two) has a Hamiltonian cycle, a single closed walk through every vertex once. That is the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"One of the best-known open problems on Hamiltonicity, open since 1969 and the subject of a long line of face-size special cases. Scored below the Harary-Hill conjecture (55) and Ryser's conjecture (50), and above Gyarfas's tree-cover conjecture (32), as a famous conjecture within graph theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Paired states and Hamiltonian cycles in cubic bipartite planar graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/Combinatorics/Hamiltonian/Main.lean (Barnette.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Hamiltonian/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bass-trace-conjecture","name":"The Bass trace conjecture for complex group rings of all discrete groups","shortName":"Bass trace conjecture","problemNumber":null,"field":"K-theory of group rings; Hattori-Stallings traces","fieldGroup":"Algebra","statement":"For a discrete group $G$ and an idempotent matrix $e$ over the group ring $\\mathbb CG$, the Hattori-Stallings trace assigns to each conjugacy class $C$ the coefficient $\\sum_{g\\in C}\\sum_i e_{ii}(g)$, giving a map $K_0(\\mathbb CG)\\to\\bigoplus_{C}\\mathbb C\\,C$. Bass (1976) conjectured that for the integral group ring the trace is supported at the identity; the complex form asks that it vanish on every conjugacy class of infinite-order elements. Berrick, Chatterji and Mislin (2004) also stated an $\\ell^1$ form (finite support on finite-order classes for idempotents over $\\ell^1(G)$), proved when the Bost assembly map is rationally surjective. Known for many classes of groups but open in general. Does the Hattori-Stallings trace of every finitely generated projective $\\mathbb CG$-module vanish on infinite-order conjugacy classes, for every group $G$?","posedBy":"H. Bass, Euler characteristics and characters of discrete groups (Inventiones 1976); complex and l1 forms: Berrick, Chatterji and Mislin, Math. Annalen (2004), Conjectures 2.1 and 2.2","yearPosed":1976,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (24 September): for every discrete group $G$, every $x\\in K_0(\\mathbb CG)$ and every infinite-order $g$, $\\mathrm{HS}_G(x)([g])=0$; the integral (identity-support) form for $\\mathbb ZG$ is deduced in Section 9. The 5 October companion proves the $\\ell^1$-Bass conjecture (BCM 2004, Conjecture 2.2): the trace of every idempotent over $\\ell^1(G)$ is supported on finitely many finite-order classes, with no assembly hypothesis, which implies the complex form. Neither manuscript addresses the reduced $C^*$-algebra (the Kadison-Kaplansky question), where this release separately claims a counterexample.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization of the complex group-ring theorem accompanies it (Comparator challenge BassTrace, listed in lean/formalization.yaml). The l1 strengthening is a separate 5 October manuscript without Lean.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of both manuscripts (TeX source), read against the Bass conjecture as cited (BCM 2004 Conjecture 2.1; Chatterji-Mislin 2011 Conjecture 2). Lean: Comparator challenge BassTrace, declaration OAI.BassTrace.RightProjective.bassTraceModules_vanishing_and_support (listed in lean/formalization.yaml). Its statement was read here: for every group $G$ and every class $x$ in a module-level $K_0(\\mathbb C[G])$, the Hattori-Stallings trace vanishes at every infinite-order conjugacy class and its support lies in finite-order classes. That is the complex headline. Not rebuilt here. The $\\ell^1$-Bass theorem of the 5 October companion is stronger and is NOT formalized; the manuscript says its proof does not use the complex theorem as input.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"The Bass trace conjecture is a standard open problem in the K-theory of group rings since 1976, implied by Baum-Connes and known only for broad but restricted classes of groups; a proof for all groups, with the l1 strengthening, would be a major event for that community. Placed level with the Schonhage-Strassen conjecture (48), below Kaplansky's direct finiteness conjecture (52) and above the group-ring determinant conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf","sourceName":"OpenAI math release: The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"The l1-Bass Conjecture for Discrete Groups (companion, stronger l1 form)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf","kind":"paper"},{"label":"Lean Comparator challenge BassTrace (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BassTrace.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"arnold-corank-problem","name":"Arnold's corank problem: is the Hessian corank of a hypersurface singularity an ambient topological invariant?","shortName":"Arnold's corank problem","problemNumber":null,"field":"Singularity theory; topological invariants of hypersurface germs","fieldGroup":"Geometry & topology","statement":"For a holomorphic germ $f:(\\mathbb{C}^N,0)\\to(\\mathbb{C},0)$ with a critical point at the origin, let $\\mathrm{corank}_0(f)=N-\\mathrm{rank}_{\\mathbb{C}}\\mathrm{Hess}_0(f)$. Fernandes, Jelonek and Sampaio proved that the corank is preserved by ambient homeomorphisms in three complex variables, and Sampaio proved that its parity is an ambient topological invariant in every dimension. Arnold's question, as recorded by Sampaio (2026, Problem 1): if a homeomorphism germ of $(\\mathbb{C}^N,0)$ carries $V(f)$ onto $V(g)$ for reduced holomorphic germs $f,g$ with critical points at the origin, must $\\mathrm{corank}_0(f)=\\mathrm{corank}_0(g)$?","posedBy":"V. I. Arnold, as stated in J. E. Sampaio, 'On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems' (preprint, 2026), Problem 1","yearPosed":null,"ageNote":"The manuscript cites the question only through Sampaio's August 2026 preprint, which attributes it to Arnold. It gives no date for Arnold's original formulation, so the year is left blank.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The weighted homogeneous germs $f_1,f_2$ on $\\mathbb{C}^N$ ($N>3$, divisible by eight) of the Zariski counterexample are ambiently homeomorphic, have isolated critical points and real coefficients, and have Hessian ranks $16E>0$ and $0$, so their coranks differ. Corank parity is preserved, as Sampaio's theorem requires. Nothing is said about corank in three variables (where it is invariant) or about a minimal dimension.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two manuscripts in the family, September 24 and September 27, 2026, which the September 27 paper describes as independent arguments. The September 24 paper ships Python scripts (standard library only) that check its finite integer arithmetic; the README says they do not cover the lattice-theoretic or topological arguments.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the corollary 'Negative answer to Arnold's corank problem' in the September 24 manuscript was read against the question as the manuscript states it from Sampaio 2026. It is a corollary of the same pair of germs as the Zariski counterexample, whose Hessian ranks are 16E and 0 for an integer E fixed by the construction. Not refereed; no Lean formalization. The paper notes the result is consistent with Sampaio's corank-parity theorem, since both ranks are even, and with the three-variable invariance theorem, since N > 3.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A question attributed to Arnold that has come into focus with Sampaio's recent work on corank invariance; known to specialists in topological singularity theory rather than the wider field, and settled here as a corollary of the Zariski counterexample. Level with the four-partite Ryser case (20) and McKean's entropy-production conjecture (20), well below Zariski's question itself (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Sampaio, On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems (2026)","url":"https://www.researchgate.net/publication/413623762","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"square-difference-free-power-saving","name":"A power saving in the Furstenberg-Sarkozy theorem (square-difference-free sets)","shortName":"Power saving for square-difference-free sets","problemNumber":null,"field":"Additive combinatorics; Furstenberg-Sarkozy theorem","fieldGroup":"Combinatorics","statement":"Let $s(N)$ be the largest size of a set $A\\subseteq\\{1,\\ldots,N\\}$ with no two elements differing by a nonzero perfect square. Answering a question of Lovasz, Furstenberg and Sarkozy proved $s(N)=o(N)$. Quantitative bounds improved through Pintz-Steiger-Szemeredi and Bloom-Maynard to Green and Sawhney's $s(N)\\ll N\\exp(-c\\sqrt{\\log N})$, while Ruzsa-type constructions give $s(N)\\gg N^{0.7334}$. Green and Sawhney asked whether a fixed power saving holds. Is there an absolute $c>0$ with $s(N)\\ll N^{1-c}$?","posedBy":"B. Green and M. Sawhney, New bounds for the Furstenberg-Sarkozy theorem (arXiv:2411.17448), Section 1.1; the underlying problem originates with Lovasz","yearPosed":2024,"ageNote":"Green and Sawhney's arXiv preprint (2411.17448) dates from November 2024; the manuscript's bibliography lists it as 2025. The underlying density question goes back to Lovasz, answered qualitatively by Furstenberg (1977) and Sarkozy (1978).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: absolute $c>0$, $C$ with $|A|\\le CN^{1-c}$ for every square-difference-free $A\\subseteq[N]$; consequently the square-difference graph on $[N]$ has chromatic number at least $C^{-1}N^c$. Companions: for an intersective $h$ of degree $k\\ge2$, a power saving $O_h(N^{1-c_k})$ with exponent depending only on $k$; and for $h$ with a unit root modulo every modulus, a power saving $C_hN^{1-c_h}$ for differences $h(p)$ at primes. The exponent is tiny and not computed, far from Ruzsa's lower-bound exponent $0.733$; the true exponent is not addressed.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization accompanies it (Comparator challenge SquareDifference).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Green and Sawhney's question as cited. Lean: Comparator challenge SquareDifference, declaration OAI.SquareDifference.power_saving. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: there exist reals $c>0$ and $C$ such that for every $N\\ge1$ every finite set of integers $A\\subseteq[1,N]$ with $a-b\\ne m^2$ for all $a,b\\in A$ and $m\\ge1$ has $|A|\\le CN^{1-c}$. That is exactly the headline. Not rebuilt here. The paper says the exponent is extremely small and not computed. The two 5 October companions (intersective polynomials, prime arguments) are not formalized; the prime-argument one depends on a companion zero-free half-plane theorem for Dirichlet L-functions.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"Whether square-difference-free sets admit a power saving was one of the best-known quantitative questions in additive combinatorics, with bounds creeping up from log savings for over forty years; a power saving would be a widely celebrated breakthrough. Placed level with Hindman's finite sums and products (50), just below Sidorenko's conjecture (52) and above the circulant Hadamard conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A power saving for square-difference-free sets","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge SquareDifference (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SquareDifference.lean","kind":"lean-proof"},{"label":"Power saving for intersective polynomial differences, exponent depending only on the degree (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf","kind":"paper"},{"label":"A Power Saving for Polynomial Differences at Prime Arguments (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf","kind":"paper"},{"label":"Green and Sawhney, New bounds for the Furstenberg-Sarkozy theorem (arXiv:2411.17448)","url":"https://arxiv.org/abs/2411.17448","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"one-phase-bernoulli-critical-dimension","name":"The critical dimension of the one-phase Bernoulli free boundary problem: is it seven?","shortName":"One-phase Bernoulli critical dimension","problemNumber":null,"field":"Free boundary problems; Alt-Caffarelli functional","fieldGroup":"Differential equations","statement":"For nonnegative $v\\in H^1(B)$ the one-phase Bernoulli (Alt-Caffarelli) energy is $J(v;B)=\\int_B(|\\nabla v|^2+\\mathbf 1_{\\{v>0\\}})\\,dx$. By Weiss's monotonicity formula and dimension reduction, the regularity of minimizing free boundaries is governed by the least dimension $d_*$ admitting a nonzero, nonflat, one-homogeneous global minimizer: free boundaries are smooth below $d_*$ and singular sets have dimension at most $n-d_*$. Caffarelli, Jerison and Kenig settled dimension three, Jerison and Savin (2015) proved flatness through dimension four, so $d_*\\ge5$, and De Silva and Jerison (2009) constructed a nonflat minimizing cone in dimension seven, so $d_*\\le7$, by analogy with the Simons cone for minimal surfaces. Is $d_*=7$, that is, is every one-homogeneous global minimizer in dimensions five and six flat?","posedBy":"Open after De Silva and Jerison (2009) and Jerison and Savin (2015); the manuscript names no single poser","yearPosed":null,"ageNote":"The manuscript calls it 'the critical-dimension question whether d* = 7', citing De Silva-Jerison (J. Reine Angew. Math. 2009) for d* <= 7 and Jerison-Savin (GAFA 2015) for d* >= 5, but gives no posing reference; the question has been open in this form at least since 2009.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every nonzero one-homogeneous global minimizer of the one-phase Bernoulli energy in $\\mathbb R^d$, $1\\le d\\le6$, is flat, $u=(x\\cdot e)_+$, and a nonflat one-homogeneous global minimizer exists in $\\mathbb R^7$, so $d_*=7$. Corollary 1.2: the interior free boundary of a local minimizer in $\\mathbb R^n$ is smooth for $n\\le6$, its singular set is locally finite for $n=7$ and has Hausdorff dimension at most $n-7$ for $n\\ge7$, sharp by products of the De Silva-Jerison cone. Method: a Hessian-shape test function and vector field on the spherical link, with an exactly verified polynomial inequality. Not shown: classification of stable but non-minimizing cones, two-phase or vector-valued versions, or rigidity of non-homogeneous entire stable solutions.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'The critical dimension for one-phase Bernoulli minimizers' (September 24, 2026). Its key inequality is checked by an exact finite certificate, with Python reproduction scripts shipped in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question: every nonzero one-homogeneous global minimizer in dimensions $1\\le d\\le6$ is flat and a nonflat one exists in dimension seven, so $d_*=7$. The new content is flatness in dimensions five and six; existence in dimension seven is De Silva and Jerison's 2009 cone. Lean: lean/ComparatorChallenges/BernoulliNonflatInSeven.json exists with solution module OAI.Analysis.BernoulliCone.Main present, not in the formalization catalogue; the statement read here asserts only the existence of a nonzero, one-homogeneous, nonflat global minimizer in $\\mathbb R^7$, the already known half, and its scope note puts flatness in dimensions at most six outside it. So this entry is unreviewed. The flatness proof rests on a universal tensor inequality verified by an exact finite certificate; the release's reproduction scripts were not run here and, by their own README, do not certify the correspondence between formulas and code.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The critical dimension is the main structural question of the Alt-Caffarelli theory, the free-boundary analogue of Simons's theorem on minimal cones, open between five and seven since 2015. Famous within free-boundary regularity. Placed below the planar Mumford-Shah conjecture (52) and C1 regularity of infinity-harmonic functions (42), level with the Lane-Emden conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf","sourceName":"OpenAI math release: The critical dimension for one-phase Bernoulli minimizers","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (existence half only): OAI/Analysis/BernoulliCone/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/BernoulliCone/Main.lean","kind":"lean-proof"},{"label":"Lean statement (existence half only): BernoulliNonflatInSeven.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BernoulliNonflatInSeven.lean","kind":"lean-proof"},{"label":"Jerison and Savin, Some remarks on stability of cones for the one-phase free boundary problem (2015)","url":"https://doi.org/10.1007/s00039-015-0335-6","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gigli-alexandrov-curvature-characterization","name":"Gigli's conjecture: Alexandrov curvature bounds are characterized by his distributional sectional curvature on noncollapsed RCD spaces","shortName":"Gigli's Alexandrov characterization conjecture","problemNumber":null,"field":"Metric geometry; RCD spaces and Alexandrov spaces","fieldGroup":"Geometry & topology","statement":"On an $\\mathrm{RCD}(K,N)$ metric measure space, Gigli (2019) defined a distributional Riemann curvature tensor $R(X,Y,Z,W)$ acting on test vector fields and test functions. Let $n\\ge2$, $\\kappa\\in\\mathbb R$ and $(M,d)$ complete and separable. Gigli conjectured that $(M,d)$ is an $n$-dimensional Alexandrov space of curvature at least $\\kappa$ if and only if $(M,d,\\mathcal H^n)$ has full support, is $\\mathrm{RCD}((n-1)\\kappa,n)$, and satisfies $R(X,Y,Y,X)(f)\\ge\\kappa\\int f(|X|^2|Y|^2-\\langle X,Y\\rangle^2)\\,d\\mathcal H^n$ for all test fields $X,Y$ and nonnegative test functions $f$. Is this characterization true?","posedBy":"Nicola Gigli, Conjecture 1.1 of 'Riemann curvature tensor on RCD spaces and possible applications' (2019)","yearPosed":2019,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every integer $n\\ge2$ and $\\kappa\\in\\mathbb R$, a complete separable metric space is an $n$-dimensional Alexandrov space with curvature at least $\\kappa$ if and only if, with $m=\\mathcal H^n$, it is a full-support unreduced $\\mathrm{RCD}((n-1)\\kappa,n)$ space whose distributional sectional curvature is at least $\\kappa$ in Gigli's original global test classes. The reference measure must be exactly Hausdorff measure; collapsed or weighted spaces are not covered, and dimension one is excluded. The companion proves that weak Hessian upper bounds hold along every minimizing geodesic in any full-support $\\mathrm{RCD}(K,N)$ space, $1 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript uses a companion paper of the same date (weak Hessian bounds along every geodesic in RCD spaces) as an input to the converse direction.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Gigli's Conjecture 1.1. The only formal statement in the family is ComparatorChallenges/WeakHessian.json (OAI.WeakHessian.every_geodesic, listed in formalization.yaml), which formalizes the companion's lemma: a weak Hessian upper bound on a full-support RCD(K,N) space passes to a distributional bound along every constant-speed minimizing geodesic. That is one input to the converse direction, not the characterization, so under the tier rule this entry is unreviewed. The rest of the argument (forward direction, heat-regularized index inequality, frames along paths) is unformalized.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A recent conjecture by one author, natural and known within the RCD and Alexandrov-space community but specialist. Placed below lang-plaut-problem (38) and kalton-problem-2-bi-lipschitz-c0 (28), and just above stw-problem-xxi-uniform-property-gamma (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf","sourceName":"OpenAI math release: Gigli's distributional curvature characterization of Alexandrov spaces","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Weak Hessian bounds along every geodesic in RCD spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf","kind":"paper"},{"label":"Lean proof of the companion lemma (OAI.WeakHessian.every_geodesic)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/WeakHessian/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"brannen-simplex-projection-body-maximum","name":"Brannen's simplex conjecture for projection-body volume","shortName":"Brannen's simplex conjecture","problemNumber":null,"field":"Convex geometry; projection bodies, reverse affine isoperimetric inequalities","fieldGroup":"Geometry & topology","statement":"For a convex body $K\\subset\\mathbb R^d$ with projection body $\\Pi K$ (support function $h_{\\Pi K}(u)=\\mathrm{vol}_{d-1}(\\mathrm{proj}_{u^\\perp}K)$), the ratio $R_d(K)=|\\Pi K|/|K|^{d-1}$ is affine invariant. The simplex has $R_d=c_d=(d+1)d^d/d!$. Brannen (1996) conjectured that simplices maximize $R_d$ among all convex bodies, i.e. $|\\Pi K|\\le c_d|K|^{d-1}$. Is the simplex the maximizer of normalized projection-body volume in every dimension?","posedBy":"N. S. Brannen, Volumes of projection bodies, Mathematika 43 (1996)","yearPosed":1996,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1: for $K=T_{10}\\times T_{10}\\subset\\mathbb R^{20}$, $R_{20}(K)/c_{20}=121\\binom{20}{10}/(21\\cdot2^{20})>1$, so the simplex bound fails in dimension 20; iterating the product identity $R_{r+s}(A\\times B)=R_r(A)R_s(B)$ gives $R_n(K_n)\\ge\\lambda^nc_n$ for some $\\lambda>1$ and all large $n$. Priority: Feng, Hu, Liu and Xu (preprint, August 2026) already disproved the conjecture in every dimension $d\\ge9$, and Chen et al. proved it in dimension 3; the paper cites both and presents its result as a direct product construction with an exact value. It does not determine the optimal upper constant or the maximizers.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1 was read against Brannen's conjecture as the paper states it; it claims $R_{20}(T_{10}\\times T_{10})/c_{20}=22355476/22020096>1$. The proof (a product identity for projection bodies of polytopes plus an exact simplex computation) was not refereed. The Lean challenges ComparatorChallenges/ProjectionCounterexample.lean (declaration OAI.ProjectionCounterexample.universal_simplex_upper_bound_false, in lean/formalization.yaml) and ComparatorChallenges/ProjectionVolume.lean (OAI.Paper092.product_counterexample) were read here: the first asserts that not every convex body in $\\mathbb R^{20}$ has ratio at most that of the 20-simplex, the second gives the exact value for the product of two 10-simplices. This states the disproof. Not rebuilt here. The paper's exponential-excess corollary is not formalized. The conjecture had already been refuted for every $d\\ge9$ by Feng, Hu, Liu and Xu (August 2026), as the paper itself says.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A specialist conjecture from 1996 about the maximum of projection-body volume, and already refuted by human authors a month before this paper, so the question answered here was essentially closed. Placed with Optimal Partial Plank Coverings (15) and below Ehrhart's Volume Conjecture (25) and the Generalized Busemann-Petty Problem in Dimensions 2 and 3 (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A product counterexample to the simplex maximum for projection-body volume","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.ProjectionCounterexample.universal_simplex_upper_bound_false)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/ProjectionBody/Counterexample.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.Paper092.product_counterexample)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/ProjectionVolume/ProductCounterexample.lean","kind":"lean-proof"},{"label":"Comparator statement: ProjectionCounterexample.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ProjectionCounterexample.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"restricted-geometric-langlands-char-p","name":"The Gaitsgory-Raskin full-support conjecture for restricted geometric Langlands in positive characteristic","shortName":"Restricted geometric Langlands, char p","problemNumber":null,"field":"Geometric Langlands program; l-adic sheaves in positive characteristic","fieldGroup":"Number theory","statement":"Let $X$ be a smooth projective connected curve over an algebraically closed field $k$ of characteristic $p>0$, $G$ a connected reductive group, $\\ell\\ne p$. Gaitsgory and Raskin constructed, under the characteristic hypotheses of the restricted theory of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky, a $\\mathrm{QCoh}(Y)$-linear equivalence from $\\mathrm{Shv}_{\\mathrm{Nilp}}(\\mathrm{Bun}_G(X))$ onto $\\mathrm{IndCoh}_{\\mathrm{Nilp}}(Y')$ for an open-and-closed union $Y'$ of connected components of the restricted stack $Y=\\mathrm{LS}^{\\mathrm{restr}}_{\\check G}(X)$, and proved $Y'=Y$ for $GL_n$. Their Conjecture 1.3.10 asserts that no component is missing. Is $Y'=Y$, so that the restricted Langlands functor is an equivalence onto all of $\\mathrm{IndCoh}_{\\mathrm{Nilp}}(Y)$?","posedBy":"Dennis Gaitsgory and Sam Raskin","yearPosed":2025,"ageNote":"Gaitsgory-Raskin, Geometric Langlands in positive characteristic from characteristic zero, arXiv:2508.02237 (2025), Conjecture 1.3.10, as cited by the manuscript.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem (full support): under either regime, $Y'=Y$, so $\\mathbb L^{\\mathrm{restr}}_G:\\mathrm{Shv}_{\\mathrm{Nilp}}(\\mathrm{Bun}_G(X))\\to\\mathrm{IndCoh}_{\\mathrm{Nilp}}(Y)$ is a $\\mathrm{QCoh}(Y)$-linear equivalence; a rational-coefficient form is also proved over $\\overline{\\mathbb F}_q$. Consequences: nonzero localized categories at every spectral component, the full arithmetic trace formula in the finite-field regime, and nonzero Hecke eigenobjects for every parameter. It does not remove the characteristic hypotheses, and does not assert bounded constructibility or perversity of eigenobjects (the three October 5 eigensheaf companions construct perverse tame eigensheaves in specific settings).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has eight manuscripts (September 24 to October 5); this entry rests on the September 24 restricted-equivalence paper.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: The full-support theorem was read against Gaitsgory-Raskin's conjecture. It is proved in two regimes: over $\\overline{\\mathbb F}_q$ under the four characteristic conditions referenced by Gaitsgory-Raskin (with a finite-field descent of $X$ and $G$), and over any algebraically closed field of characteristic $p$ assuming additionally that $p$ is very good and does not divide $|W_G|$. The primed equivalence and the characteristic-zero input are cited from Gaitsgory-Raskin. Because the second regime carries extra hypotheses, Partial. The proof was not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A one-year-old conjecture by leading experts that completes the positive-characteristic restricted geometric Langlands equivalence; important inside the geometric Langlands community but technical and recent. Placed below Modularity of elliptic curves over imaginary quadratic fields (58) and the Ramanujan-Arthur decomposition entry of this family, and above The index conjecture for slc log Calabi-Yau pairs (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The Restricted Geometric Langlands Equivalence in Positive Characteristic","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Constructible tame Hecke eigensheaves in positive characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf","kind":"paper"},{"label":"Companion: Tame Hecke Eigensheaves with Several Marked Points","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026/Tame-Hecke-Eigensheaves-with-Several-Marked-Points.pdf","kind":"paper"},{"label":"Companion: Frobenius Structures on Tame Hecke Eigensheaves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf","kind":"paper"},{"label":"Gaitsgory-Raskin, arXiv:2508.02237 (Conjecture 1.3.10)","url":"https://arxiv.org/abs/2508.02237","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"higher-dimensional-duke-prime-degree-packets","name":"The higher-dimensional Duke theorem for torus packets of totally real fields of prime degree at least five (with primitive quartic and sextic cases)","shortName":"Higher-dimensional Duke theorem, prime degree","problemNumber":null,"field":"Homogeneous dynamics; equidistribution of periodic torus orbits","fieldGroup":"Number theory","statement":"Ideal classes of a totally real field $K$ of degree $n$ give compact orbits of the diagonal group $A_n$ on $X_n=\\mathrm{SL}_n(\\mathbb Z)\\backslash\\mathrm{SL}_n(\\mathbb R)$, grouped into packets by local homothety type. Duke proved equidistribution for $n=2$; Einsiedler, Lindenstrauss, Michel and Venkatesh proved the cubic case and described the extension to higher prime degree as conditional on subconvexity; Lemke Oliver, Thorner and Zaman handled prime degree outside a sparse exceptional set of fields. For a fixed degree $n$ and discriminants tending to infinity, do the volume-weighted packet measures on $X_n$ converge to Haar probability measure, with no escape of mass?","posedBy":"Einsiedler, Lindenstrauss, Michel and Venkatesh (Distribution of periodic torus orbits and Duke's theorem for cubic fields, Annals of Math. 2011, Section 1.6.3), following Linnik and Duke","yearPosed":2011,"ageNote":"The manuscripts locate the higher-degree question in ELMV 2011, Section 1.6.3, building on ELMV 2009 (Duke Math. J.) and Duke 1988; Linnik's ergodic method goes back to the 1960s.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Principal Theorem 1.1: for each fixed prime $n\\ge5$ and any totally real fields $K_i$ of degree $n$ with full lattices $M_i$ whose multiplier-order discriminant $D(M_i)\\to\\infty$, the packet measures converge weakly to Haar measure on $X_n$ with no escape of mass, for every local homothety type, nonmaximal orders of unbounded index and every field (no exceptional set). Companions: the same for Picard packets of arbitrary orders in primitive totally real quartic fields, and for ideal-class packets of maximal orders in primitive totally real sextic fields. Not shown: individual orbits, rates, composite degrees with proper intermediate fields, nonmaximal sextic orders.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of all three manuscripts, read against the higher-dimensional Duke problem as they state it; proofs not refereed. Lean-checked on the release's Comparator challenge DukePrimeDegree with solution module OAI.NumberTheory.DukePrimeDegree.MainUnconditional, both fetched at the pinned commit; the challenge is not listed in lean/formalization.yaml. The statement was read here and not rebuilt: it defines full lattices, multiplier discriminants, local homothety, packet orbits and volume-weighted packet measures from scratch, and asserts weak convergence to a Haar probability plus tightness for every prime degree $d+1\\ge5$, which is the principal theorem. The quartic and sextic companions are not formalized. Composite degrees with intermediate fields remain open, hence partial.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"The higher-dimensional Duke problem is a well-known goal of the Einsiedler-Lindenstrauss-Michel-Venkatesh program, and an unconditional prime-degree answer removes a subconvexity barrier; it is still a set of degree cases, not the full problem. Placed near the Mezard-Parisi diluted spin glass formula (32), above Bugeaud's Problem 10.61 (22) and below Patterson's cubic Gauss sum conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Equidistribution of primitive quartic torus packets for arbitrary orders","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf","kind":"paper"},{"label":"Companion: Equidistribution of primitive sextic torus packets","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf","kind":"paper"},{"label":"Lean Comparator challenge DukePrimeDegree (not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DukePrimeDegree.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/DukePrimeDegree/MainUnconditional.lean","kind":"lean-proof"},{"label":"Einsiedler, Lindenstrauss, Michel, Venkatesh, Duke's theorem for cubic fields, Annals 173 (2011)","url":"https://doi.org/10.4007/annals.2011.173.2.5","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"three-machine-precedence-scheduling","name":"Three-machine unit-job scheduling with precedence constraints (Garey-Johnson OPEN8): is P3|prec,p_j=1|Cmax in P?","shortName":"P3|prec,pj=1|Cmax","problemNumber":null,"field":"Scheduling theory; computational complexity","fieldGroup":"Theoretical computer science","statement":"Given a directed acyclic graph on $n$ unit-length jobs and a deadline $T$, can the jobs be scheduled on $m$ identical machines, at most $m$ per time slot, so that every precedence $u\\to v$ has $u$ strictly before $v$ and all jobs finish by $T$? For $m=2$ this is polynomial (Fujii-Kasami-Ninomiya 1969, Coffman-Graham 1972); with $m$ part of the input it is NP-complete (Ullman 1975). The fixed case $m=3$, $P3\\mid\\mathrm{prec},p_j=1\\mid C_{\\max}$, is listed as open problem OPEN8 in Garey and Johnson's book; the best exact algorithm known ran in time $2^{O(\\sqrt n\\log n)}$ (Nederlof-Swennenhuis-Wegrzycki). Is three-machine unit-job scheduling with precedence constraints solvable in polynomial time, or is it NP-complete?","posedBy":"Michael R. Garey and David S. Johnson, Computers and Intractability (1979), Appendix A13, OPEN8","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a uniform deterministic algorithm constructs a minimum-makespan schedule for unit-length jobs with arbitrary precedence constraints on three identical machines, and decides feasibility for a deadline $1\\le T\\le n$ exactly, in $O((L+2)^{150020})$ Turing machine steps. The method decomposes schedules into gaps whose job sets have bounded-size global Boolean descriptions and searches them by dynamic programming. The exponent is astronomically large and no practical claim is made. It does not settle $m\\ge4$ fixed machines, which remains open.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the problem as stated there (explicit DAG, unit jobs, three identical machines, no release dates); it claims a deterministic algorithm producing a minimum-makespan schedule and deciding feasibility for a given deadline in $O((L+2)^{150020})$ multitape Turing machine steps. The proof was not refereed. The challenge is not in the formalization catalogue (lean/formalization.yaml); it is found through lean/docs/124.md, and its solution module OAI.Computability.Scheduling.Main exists at the pinned commit. The statement ComparatorChallenges/ThreeMachine.lean was read here: it encodes instances and answers as bit strings and asserts one fixed multitape machine and constant $C$ such that, for every nonempty acyclic instance and every deadline option with $1\\le T\\le n$, the machine halts within $C(L+2)^{150020}$ steps with a correct output (an optimal schedule, or a feasible schedule or a correct 'infeasible'). This states the headline claim, polynomiality included. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"One of the dozen open problems in Garey and Johnson's 1979 list, famous across scheduling and complexity theory and open for 47 years. Placed level with the p-converse in Selmer corank one (45) and the word problem for Artin groups (45), below deterministic polynomial factorization over prime fields (55), and above the CVP polynomial-factor hardness entry (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.ThreeMachine.main_theorem)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/Scheduling/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: ThreeMachine.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ThreeMachine.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"supercritical-defocusing-nls-random-data-blowup","name":"Almost-sure global well-posedness for high-regularity random data in energy-supercritical defocusing NLS (Deng-Nahmod-Yue Open Problem 3), refuted on the twelve-torus","shortName":"Stable blowup for supercritical defocusing NLS","problemNumber":null,"field":"Dispersive PDE; nonlinear Schrodinger equation; random data; blowup","fieldGroup":"Differential equations","statement":"Consider the defocusing nonlinear Schrodinger equation $i\\partial_t u+\\Delta u=|u|^{p-1}u$ on the torus $\\mathbb T^d$ with odd $p$, in the energy-supercritical range $s_{cr}=d/2-2/(p-1)>1$, where the conserved energy does not control the regularity needed for global classical solutions. Deng, Nahmod and Yue (2020) observed that if classical solutions with random initial data of high regularity, such as Gaussian Fourier series $\\sum_n g_n\\langle n\\rangle^{-\\alpha}e^{in\\cdot x}$ with $\\alpha$ large, were almost surely global, then blowup for these equations would be non-generic and unstable, as the known whole-space blowup examples are. Their Open Problem 3 asks: in the energy-supercritical case, does almost-sure global well-posedness hold for random initial data of high regularity?","posedBy":"Yu Deng, Andrea R. Nahmod and Haitian Yue, Random tensors, propagation of randomness, and nonlinear dispersive equations (arXiv:2006.09285v1, 2020), Section 9.2.3, Open Problem 3","yearPosed":2020,"ageNote":"The manuscript cites Open Problem 3 of Deng-Nahmod-Yue and says its own eventual-alpha, $H^m$ quantified formulation is the one considered there, not a verbatim restatement. The posing text was read here in arXiv v1. The wider question of whether energy-supercritical defocusing NLS can blow up goes back to Bourgain's surveys.","solveType":"disproved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"Claims: there are an odd $p$ (which can be taken arbitrarily large), an integer $k>8$ and a nonempty open set $\\mathcal U\\subset H^k(\\mathbb T^{12})$ such that every solution of the defocusing NLS with data in $\\mathcal U$ blows up in finite time $T$ with $(T-t)^{1/(p-1)}|u(t,x_*)|\\to c_0>0$, a type-I self-similar singularity with logarithmic phase. Since Gaussian measures charge open sets, for every $\\alpha>k+6$ the Gaussian Fourier datum blows up with positive probability, so almost-sure global existence for high-regularity random data fails for this $(d,p)$. It does NOT give probability-one blowup, a bound uniform in $\\alpha$, an explicit $p$, or anything for low dimensions such as $d\\le 11$; the question for other pairs $(d,p)$ stays open.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. The proof is computer assisted: the release ships an exact rational-arithmetic verifier (Python standard library only) for the finite comparisons behind the self-similar profile and its spectral stability; the README says these checks do not verify the analytic arguments linking them to the blowup theorem. The main theorem and its Gaussian corollary also have a Lean comparator challenge in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 were read against Open Problem 3. The Lean challenge lean/ComparatorChallenges/DefocusingNLS.json (solution module OAI.MathematicalPhysics.DefocusingNLS.MainTheorems, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/371.md. Its statement was read here: stable_blowup gives, for every lower bound, an odd $p\\ge3$ above it, an integer $k>8$ and a nonempty open set of Fourier-side $H^k(\\mathbb T^{12})$ data whose classical solutions blow up at the self-similar rate $(T-t)^{-1/(p-1)}$ at a point and do not extend continuously past $T$; gaussian_blowup and gaussian_not_almost_sure_global give positive Gaussian probability of that set for every $\\alpha>k+6$. This is the headline. The Sobolev product is a challenge definition whose summability proof the solution must supply. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":33,"significanceNote":"Whether energy-supercritical defocusing NLS blows up, and whether such blowup is stable, has been a central question in dispersive PDE since Merle, Raphael, Rodnianski and Szeftel found the first (finite-codimension) blowup in 2022. An open set of blowup data refutes the almost-sure global existence hoped for in Deng-Nahmod-Yue Open Problem 3, but only for one high dimension and a large unspecified power. Placed just above Carleson's Schrodinger endpoint problem (32), below the Boussinesq blowup entry (45) and above IPM blowup with smooth forcing (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Stable self-similar blowup for a supercritical defocusing Schrodinger equation on the torus","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean formalization: stable blowup and Gaussian blowup on the twelve-torus","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/DefocusingNLS/MainTheorems.lean","kind":"lean-proof"},{"label":"Lean comparator statement: DefocusingNLS","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DefocusingNLS.lean","kind":"lean-proof"},{"label":"Deng-Nahmod-Yue, Random tensors... (arXiv:2006.09285v1), Open Problem 3","url":"https://arxiv.org/abs/2006.09285v1","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-821-large-totient-fibers","name":"Erdős's totient-fiber conjecture: $\\varphi(m)=n$ has more than $n^{1-\\varepsilon}$ solutions for infinitely many $n$","shortName":"Erdos Problem #821: large totient fibers","problemNumber":821,"field":"Multiplicative number theory; Euler's totient function","fieldGroup":"Number theory","statement":"Let $g(n)=\\#\\{m\\ge1:\\varphi(m)=n\\}$ count the preimages of $n$ under Euler's totient function. An elementary bound gives $g(n)\\ll_\\eta n^{1+\\eta}$; Pillai showed $\\limsup g(n)=\\infty$ and Erdős (1935) showed $g(n)>n^c$ infinitely often for some $c>0$. Pomerance (1980) formulated the question as whether the supremum $C$ of exponents with $g(n)>n^c$ infinitely often equals $1$, attributing it to Erdős, and showed it would follow from enough primes $p$ whose predecessor $p-1$ has no prime factor above $p^\\varepsilon$. The best exponent known was $0.7157$ (Lichtman 2022), after Baker-Harman's $0.7039$. Is it true that for every $\\varepsilon>0$ there are infinitely many $n$ with $g(n)>n^{1-\\varepsilon}$?","posedBy":"Paul Erdős (1956, as attributed by Pomerance 1980); Erdős Problem #821","yearPosed":1956,"ageNote":"The manuscript follows Pomerance (1980, p. 84), who attributes the conjecture to Erdős (1956); erdosproblems.com lists it as Problem #821 citing Erdős (1974). Erdős's work on totient multiplicities and smooth shifted primes goes back to 1935.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $\\varepsilon>0$ there are infinitely many $n$ with $g(n)>n^{1-\\varepsilon}$, which is optimal in the exponent because $g(n)\\ll_\\eta n^{1+\\eta}$. The input is Theorem 1.2: for fixed $0<\\delta<1/4$, at least $x^{1-o(1)}$ primes $p\\in(2x,5x]$ have $P^+(p-1)\\le x^\\delta$; in particular there are infinitely many primes with $P^+(p-1)\\le p^\\varepsilon$ for every $\\varepsilon>0$. It does not give a positive proportion of such primes (that stronger statement is in the Poisson-Dirichlet companion), and the $o(1)$ is not made explicit.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This manuscript proves the smooth shifted-prime count and the totient consequence; its graph and kernel estimates are reused by the Poisson-Dirichlet companion (separate entry for the Ford-Konyagin-Luca conjecture).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 and the introduction were read against the conjecture as Pomerance formulates it and as erdosproblems.com states Problem #821. No Lean formalization exists for this family. The deduction from smooth shifted primes to large fibers is the classical Erdős-Pomerance argument (Pomerance's Theorem B), written out in full; the new input is the $x^{1-o(1)}$ count of primes with $x^\\delta$-smooth predecessors, which goes far beyond the previous exponent 0.2843 and should be treated as unverified until specialists examine it. Not refereed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"A classical Erdős conjecture on Euler's function, open for about seventy years, with a long chain of partial exponents (Erdős, Pomerance, Friedlander, Baker-Harman, Lichtman). Far above a typical Erdős problem because it hinges on smooth shifted primes. Placed just above Convergence of the Flint Hills series (35) and Jacobsthal's quadratic-bound question: is h(k) = O(k^2)? (34), and below the Ford-Konyagin-Luca entry (40), whose positive-proportion statement is stronger.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Weighted dilation graphs, smooth shifted primes and totient fibers","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The Poisson-Dirichlet law for prime predecessors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: Prime predecessors with an even number of prime factors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf","kind":"paper"},{"label":"Erdős Problem #821 (erdosproblems.com)","url":"https://www.erdosproblems.com/821","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"off-diagonal-ramsey-log-exponent","name":"The logarithmic exponent of the off-diagonal Ramsey numbers r(s,t) for fixed s at least 5","shortName":"Log exponent of r(s,t), s >= 5","problemNumber":null,"field":"Extremal and probabilistic combinatorics: Ramsey numbers","fieldGroup":"Combinatorics","statement":"For fixed $s$, the Ramsey number $r(s,t)$ is the least $N$ such that every graph on $N$ vertices contains $K_s$ or an independent set of size $t$. Ajtai, Komlos and Szemeredi (1980) proved $r(s,t)=O_s(t^{s-1}/(\\log t)^{s-2})$, and Erdos asked for the true order of growth: for $s=3$ it is $t^2/\\log t$ (Kim 1995), Erdos Problem #166 asked for $r(4,t)\\gg t^3/(\\log t)^{O(1)}$ (Mattheus-Verstraete), and Erdos Problem #986 asked for $r(s,t)\\gg t^{s-1}/(\\log t)^{c(s)}$, which Bradac proved in 2026 with $c=2s-4$. That left the power of $\\log t$ between $s-2$ and $2s-4$. For fixed $s\\ge5$, what is the exponent $\\theta_s$ with $r(s,t)=t^{s-1}/(\\log t)^{\\theta_s+o(1)}$; in particular, is the Ajtai-Komlos-Szemeredi upper bound sharp up to $(\\log t)^{o(1)}$?","posedBy":"Classical problem of P. Erdos on the order of r(s,t) for fixed s (Erdos Problems #165, #166, #986; general case per Chung and Graham, 1998); upper bound by Ajtai, Komlos and Szemeredi (1980)","yearPosed":null,"ageNote":"erdosproblems.com, citing Chung and Graham, dates Erdos's general conjecture on the polynomial order of r(s,t) to 1947. The sharper question of the exact logarithmic exponent is a natural refinement left open by Bradac (2026); the manuscripts give no source posing it as stated, so yearPosed is left empty.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"For every fixed $s\\ge5$: there is $C_s$ such that for every $\\varepsilon>0$ and all large $t$, $t^{s-1}/(\\log t)^{s-2+\\varepsilon}\\le r(s,t)\\le C_s t^{s-1}/(\\log t)^{s-2}$, so $r(s,t)=t^{s-1}/(\\log t)^{s-2+o(1)}$ and the classical upper bound is sharp up to $(\\log t)^{o(1)}$. The lower bound comes from a random ordered point-hyperplane flag graph in $\\mathrm{PG}(s-1,q)$ (Bradac's construction) with a new entropy and compression analysis of its independent sets. It does NOT give $r(s,t)=\\Theta(t^{s-1}/(\\log t)^{s-2})$, and it does not cover $s=4$, where the gap between $t^3/(\\log t)^4$ and $t^3/(\\log t)^2$ remains.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The family has two manuscripts dated 24 September 2026: one for s = 5 and one for every fixed s >= 6, each self-contained.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of both TeX sources, read against the Erdos problems and the state of the art they cite (Bradac 2026, Ajtai-Komlos-Szemeredi 1980); the proofs were not refereed. Lean: the Comparator challenges RamseyFive (OAI.SharpRamseyFive.main, solution module OAI/Combinatorics/RamseyFive/Main.lean) and SharpLogRamsey (OAI.SharpLogRamsey.main, solution module OAI/Combinatorics/SharpRamsey/Main.lean) are not in the release's formalization catalogue, but their JSON and solution files exist at the pinned commit. Their statements were read here: with r(s,t) defined as the least N such that every graph on Fin N has an s-clique or a t-independent set, they assert an absolute C with $t^{s-1}/(\\log t)^{s-2+\\varepsilon}\\le r(s,t)\\le C t^{s-1}/(\\log t)^{s-2}$ eventually for every $\\varepsilon>0$, and convergence of the logarithmic exponent to $s-2$, for s = 5 and for every s >= 6 respectively. Together they state the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":44,"significanceNote":"Off-diagonal Ramsey asymptotics are a central theme of extremal combinatorics, and this pins the order of r(s,t) for every fixed s >= 5 to within a subpolynomial factor of the logarithm, beyond what Bradac's 2026 breakthrough reached. It is a sharpening of a problem whose polynomial order was already known, so it is scored below the full asymptotic question: above the Burr-Erdos hypercube Ramsey conjecture (40), below the linear list Hadwiger conjecture (48) and the FPRAS for perfect matchings (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"The sharp logarithmic exponent of r(5,t) (companion, s = 5)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean: OAI/Combinatorics/SharpRamsey/Main.lean (SharpLogRamsey.main, s >= 6)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/SharpRamsey/Main.lean","kind":"lean-proof"},{"label":"Lean: OAI/Combinatorics/RamseyFive/Main.lean (SharpRamseyFive.main, s = 5)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/RamseyFive/Main.lean","kind":"lean-proof"},{"label":"Erdos Problem #986","url":"https://www.erdosproblems.com/986","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"affine-bernstein-dimensions-three-to-nine","name":"The Euclidean-complete affine Bernstein problem in dimensions three through nine","shortName":"Affine Bernstein in dimensions 3-9","problemNumber":null,"field":"Geometric PDE; affine maximal hypersurfaces","fieldGroup":"Differential equations","statement":"Let $u$ be smooth with positive-definite Hessian on an open convex $\\Omega\\subset\\mathbb R^n$ and solve the affine maximal equation $U^{ij}w_{ij}=0$, $U^{ij}=\\det(D^2u)(D^2u)^{-1}_{ij}$, $w=(\\det D^2u)^{-(n+1)/(n+2)}$, the Euler equation of affine area. Chern asked whether an entire such surface ($n=2$) must be an elliptic paraboloid; Calabi proved it under Euclidean and affine completeness, and Trudinger and Wang proved the Euclidean-complete surface case in 2000, reduced higher dimensions to a uniform strict-convexity estimate, and exhibited a singular nonquadratic weak solution in dimension ten. For $n\\ge3$, if the graph of $u$ is complete for the induced Euclidean metric, must $\\Omega=\\mathbb R^n$ and $u$ be quadratic, and in which dimensions?","posedBy":"Shiing-Shen Chern (Affine minimal hypersurfaces, 1977 seminar, published 1979) for surfaces; the higher-dimensional Euclidean-complete form follows Trudinger and Wang (Invent. Math. 2000)","yearPosed":1979,"ageNote":"Chern's question concerns surfaces and was posed at a 1977 seminar published in 1979. The manuscripts call the higher-dimensional statement the Euclidean-complete affine Bernstein conjecture without naming who first stated it for n >= 3.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims: for $3\\le n\\le9$, every smooth affine maximal graph with positive-definite Hessian over an open convex domain whose graph is complete for the induced Euclidean metric is entire and quadratic, so an elliptic paraboloid; no growth or affine-completeness assumption. Via Trudinger-Wang it extends to affine-complete immersed hypersurfaces. The companion constructs a smooth nonquadratic entire solution with positive-definite Hessian in dimension ten, so the range cannot extend to $n=10$. Dimensions $n\\ge11$ are not addressed, and the dimension-ten example asserts no uniform Hessian bound or affine completeness.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The dimension-ten counterexample is a separate manuscript dated 5 October 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of both manuscripts, read against the problem as framed there. The proofs were not refereed. Lean-checked on the release's own Comparator challenge AffineBernstein together with its solution module OAI.Analysis.AffineBernstein.Main, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.AffineBernstein.affine_bernstein) takes 3 <= n <= 9, a nonempty open convex domain, a smooth u with positive-definite coordinate Hessian solving the classical affine maximal equation, and sequential completeness of the induced Euclidean path metric, and concludes that the domain is all of R^n and u is a positive-definite quadratic plus affine term, the graph an affine image of the standard paraboloid. That is the headline claim of this entry. Not formalized: the affine-complete immersed-hypersurface corollary and the dimension-ten counterexample.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"Chern's affine Bernstein problem is well known in geometric PDE and affine differential geometry, but its higher-dimensional form is a specialist question. Scored as a partial with a sharp range. Below De Giorgi's conjecture for Allen-Cahn (58) and C1 regularity of infinity-harmonic functions (42), level with the (B)-conjecture for even log-concave measures (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf","sourceName":"OpenAI math release: The affine Bernstein theorem in dimensions three through nine","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf","kind":"paper"},{"label":"Lean Comparator challenge AffineBernstein (not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/AffineBernstein.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Analysis/AffineBernstein/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/AffineBernstein/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"benjamini-schramm-pc-less-than-pu","name":"The Benjamini-Schramm nonuniqueness conjecture: p_c < p_u for bond percolation on nonamenable quasi-transitive graphs","shortName":"Benjamini-Schramm pc < pu conjecture","problemNumber":null,"field":"Percolation on nonamenable graphs","fieldGroup":"Probability & statistics","statement":"Let $G$ be an infinite connected locally finite quasi-transitive graph and consider Bernoulli bond percolation with critical parameter $p_c(G)$ (infinite clusters appear) and uniqueness threshold $p_u(G)$ (the infinite cluster becomes unique almost surely). On a regular tree of degree at least three, $p_c 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 24, 2026) is the whole family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against Conjecture 6 of Benjamini-Schramm; they state the conjecture for bond percolation in full generality (quasi-transitive, nonamenable via positive vertex isoperimetric constant) and Hutchcroft's stronger $p_c0$, that the critical two-point operator is bounded, $p_c 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 24, 2026: the principal existence theorem and a companion giving a uniform polynomial-time implementation on finite rational metrics, which uses the principal theorem.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question as the manuscript frames it after Bubeck-Coester-Rabani; it gives $O(\\log^2(k+1))$ with an absolute constant on every metric space with at least $k+1$ points. The proof was not refereed. The challenge lean/ComparatorChallenges/KServer.json (solution module OAI.Combinatorics.KServer.Main, present at the pinned commit) is not in the formalization catalogue; its statement KServer.lean was read here: there is $C>0$ such that for all $k\\ge2$, every metric space with an injection of $k+1$ points and every initial configuration, some history-dependent randomized policy satisfies expected cost $\\le C(\\log(k+1))^2\\cdot\\mathrm{OPT}+B$ for every finite request list, with $B=0$ when initial positions are distinct. This states the headline claim. UniformKServer.json covers the companion. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Randomized k-server is the flagship problem of online algorithms; after the O(log k) conjecture fell in 2023, a polylog(k) bound on general metrics was the main open question, with one withdrawn claim. Placed level with Smale's mean value conjecture (K=1) (45) and Brennan's conjecture (45), below the 3SUM hypothesis (50), and above Kahn's flow conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026.pdf","sourceName":"OpenAI math release: Squared-logarithmic randomized k-server on arbitrary metrics","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Uniform computation of the squared-logarithmic k-server bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026.pdf","kind":"paper"},{"label":"Lean proof (OAI.KServer.main_theorem)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/KServer/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: KServer.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KServer.lean","kind":"lean-proof"},{"label":"Lean proof, uniform algorithm (OAI.UniformKServer.uniform_squared_logarithmic_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/UniformKServer/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: UniformKServer.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/UniformKServer.lean","kind":"lean-proof"},{"label":"Bubeck, Coester, Rabani: The randomized k-server conjecture is false! (arXiv 2211.05753)","url":"https://arxiv.org/abs/2211.05753","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"scalar-curvature-extension-conjecture","name":"The scalar-curvature extension conjecture for Ricci flow in all dimensions","shortName":"Scalar-curvature extension conjecture","problemNumber":null,"field":"Geometric analysis; Ricci flow singularities","fieldGroup":"Geometry & topology","statement":"A smooth Ricci flow $\\partial_tg=-2\\mathrm{Ric}$ on a closed manifold extends past a finite time unless the full curvature becomes unbounded (Hamilton), and a Ricci bound suffices (Sesum). The scalar-curvature extension conjecture asks whether scalar curvature alone detects singularities: is it true that every finite-time singularity of a Ricci flow on a closed manifold of dimension at least four has unbounded scalar curvature, i.e. does $\\sup_{M\\times[0,T)}|R|<\\infty$ always allow the flow to be extended past $T$?","posedBy":"Stated as open in R. H. Bamler, Convergence of Ricci flows with bounded scalar curvature, Ann. of Math. 188 (2018), p. 756; known cases surveyed by C. Li and Y. Li (2026)","yearPosed":null,"ageNote":"The manuscripts cite the conjecture from Bamler (2018) and a 2026 survey without naming who first posed it or when; it was known for Type I flows (Enders-Muller-Topping 2011) and Kahler-Ricci flow (Zhang 2010).","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for some $q\\ge10$ there is a smooth Ricci flow on the closed manifold $S^2\\times S^{q+1}$ with $\\sup|R|<\\infty$ and $\\max_M|\\mathrm{Rm}|\\to\\infty$ at the finite maximal time; a corollary gives two-sided power-law curvature blowup with arbitrarily large exponents in one fixed dimension. This refutes the unrestricted all-dimensions conjecture. It relies on Stolarski's construction and gives no explicit dimension; it does not touch dimension four, which the companion proves true.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it; it claims a Ricci flow on $S^2\\times S^{q+1}$, $q\\ge10$, with bounded scalar curvature and $\\max|\\mathrm{Rm}|\\to\\infty$ at a finite maximal time. Proof not refereed; no Lean formalization. The construction imports Stolarski's (2019) doubly warped conical-singularity flows and their quantitative profile estimates as stated inputs; the new work is the scalar bound. The paper says no dimension-four conclusion is asserted.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A well-known conjecture in Ricci flow, studied by Bamler, Zhang, Simon and others, but within one community. Placed below Universal volume growth bounds from positive intermediate curvature (38), level with the Projected Lax Conjecture (35), above Interior Curvature Estimates for the Graphical Scalar Curvature Equation (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Bounded scalar curvature and smooth extension of four-dimensional Ricci flow","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"modified-integer-round-up-conjecture","name":"The Modified Integer Round-Up Conjecture for bin packing (Scheithauer-Terno)","shortName":"Modified integer round-up conjecture","problemNumber":null,"field":"Bin packing and cutting stock; configuration LP integrality gap","fieldGroup":"Algorithms & optimization","statement":"In one-dimensional bin packing (cutting stock), the Gilmore-Gomory configuration linear program assigns nonnegative weights to feasible bin patterns and minimizes their total subject to covering the item demands; its value $LP(I)$ is a lower bound for the optimum number of bins $OPT(I)$. The integer round-up property $OPT(I)=\\lceil LP(I)\\rceil$ fails (Marcotte 1985-86), but all known instances satisfied $OPT(I)\\le\\lceil LP(I)\\rceil+1$. Scheithauer and Terno (1995, 1997) investigated this modified bound, now known as the Modified Integer Round-Up Conjecture. Does every bin packing instance satisfy $OPT(I)\\le\\lceil LP(I)\\rceil+1$?","posedBy":"Guntram Scheithauer and Johannes Terno","yearPosed":1995,"ageNote":"Scheithauer and Terno, Eur. J. Oper. Res. 84 (1995) and Oper. Res. Lett. 20 (1997), as cited by the manuscript, which calls the bound the Modified Integer Round-Up Conjecture.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every integer $c\\ge0$ there are a finite rational instance $I$, every item larger than $1/6$ (so at most five items per bin), and an integer $B$ with $LP(I)=B$ and $OPT(I)>B+c$; the same LP value holds for the stronger formulation whose patterns are subsets of individual item copies. So the additive integrality gap of the configuration LP is unbounded and the Modified Integer Round-Up Conjecture is false. Not shown: a gap growing like $\\log OPT$ in the instance size, matching the Hoberg-Rothvoss upper bound; the paper says it obtains no such lower bound.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'Additive hardness and unbounded configuration gaps in bin packing' (September 24, 2026); the same reduction also answers the additive-approximation question, entered separately.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: for every $c\\ge0$ it gives a rational instance with all sizes above $1/6$, integral $LP(I)=B$ and $OPT(I)>B+c$; taking $c=1$ contradicts the conjecture with no ceiling ambiguity. Lean: formalization.yaml lists comparator ComparatorChallenges/BinPackingGap.json, declaration OAI.BinPackingGap.main_results in OAI/Computability/BinPacking/Main.lean. The statement was read here: its first conjunct says that for every $c$ there are an instance with $5B$ items, sizes in $(1/6,1)$, $B+c 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 24, 2026. The paper says it builds on Rohleder's variational principle for tangent vector fields.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Burdzy's conjecture; it covers bounded simply connected planar domains with $C^\\infty$ boundary only, which is why this is Partial. The challenge HotSpots is not in the formalization catalogue (lean/formalization.yaml); it is linked from lean/docs/369.md, its solution module exists at the pinned commit, and its statement was read here: for a nonempty open bounded simply connected planar domain with smooth boundary and any nonzero $u$ smooth up to the boundary in the first positive Neumann eigenspace (defined via the Rayleigh quotient), $\\nabla u\\neq0$ in $\\Omega$ and $\\inf_{\\partial\\Omega}u 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization accompanies it (Comparator challenge BassTorsionFree).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction, Corollary 1.2 and the torsion-free section of the TeX source, read against the idempotent conjecture as cited. Lean: Comparator challenge BassTorsionFree, declaration OAI.TorsionFreeBass.torsion_free_corollary. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: for torsion-free $G$, the trace of every idempotent matrix over $\\mathbb C[G]$ equals the rank of its augmentation, the $K_0$ trace equals the augmentation rank with range $\\mathbb Z$, and for every commutative domain $R$ of characteristic zero every idempotent of $R[G]$ is $0$ or $1$. The last clause is exactly the headline. Not rebuilt here. The paper notes its argument does not cover the reduced $C^*$-algebra.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":56,"significanceNote":"Kaplansky's idempotent conjecture is one of the classic group-ring conjectures, open for decades and a standard consequence of Baum-Connes; a proof over all characteristic-zero coefficients would be widely noticed in algebra and operator algebras. Placed between Kaplansky's direct finiteness conjecture (52) and the zero-divisor conjecture (62), close to the Alperin weight conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf","sourceName":"OpenAI math release: The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge BassTorsionFree (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BassTorsionFree.lean","kind":"lean-proof"},{"label":"The l1-Bass Conjecture for Discrete Groups (related companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"euclidean-prefix-discrepancy-sqrt-d","name":"Prescribed-order Euclidean prefix discrepancy O(sqrt d), independent of length (Bansal-Jiang-Meka-Singla-Sinha Conjecture 6.3)","shortName":"Euclidean prefix discrepancy sqrt d","problemNumber":null,"field":"Discrepancy theory; vector balancing","fieldGroup":"Combinatorics","statement":"Given vectors $v_1,\\dots,v_N$ in the Euclidean unit ball of $\\mathbb R^d$ in a fixed order, choose signs $\\varepsilon_i\\in\\{\\pm1\\}$ to keep every prefix sum $\\sum_{i\\le k}\\varepsilon_iv_i$ small. Banaszczyk proved a bound $O(\\sqrt d+\\sqrt{\\log N})$, which grows with the length when $N$ is large compared with $d$. Bansal, Jiang, Meka, Singla and Sinha (2021, Section 6, Conjecture 6.3), following Banaszczyk, conjectured the length-free square-root bound. Is there an absolute $C$ such that every such sequence has signs with all prefix sums of norm at most $C\\sqrt d$?","posedBy":"Nikhil Bansal, Haotian Jiang, Raghu Meka, Sahil Singla and Makrand Sinha, Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing, arXiv 2111.07049 (2021), Conjecture 6.3","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: an absolute constant $C$ such that every finite sequence in the Euclidean unit ball of $\\mathbb R^d$, in prescribed order, has one signing with every prefix of norm at most $C\\sqrt d$, independent of $N$. This removes the $\\sqrt{\\log N}$ term from Banaszczyk's bound and is optimal up to the constant. It is existential: no efficient or online algorithm is given, so the algorithmic and online versions of the question remain open.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the prescribed-order signing question as the paper cites it from Bansal et al.; the manuscript's citation was not checked against the 2021 preprint's own wording. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator SteinitzBergstrom, declaration OAI.EuclideanSteinitzBergstrom.main); its SignedPrefixBound part, read here, states exactly the length-free $C\\sqrt d$ prefix-signing bound for every finite sequence in the unit ball. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A recent, precisely stated conjecture in vector balancing, known within discrepancy theory and theoretical computer science. Placed below the Euclidean Steinitz-Bergstrom entry it implies (38) and the matrix Spencer entry (30), level with the stable coordinate conjecture in dimension five (22), and above the online Spencer vector-balancing question (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf","sourceName":"OpenAI math release: The Euclidean Steinitz-Bergstrom theorem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.EuclideanSteinitzBergstrom.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Steinitz/Main.lean","kind":"lean-proof"},{"label":"Bansal, Jiang, Meka, Singla, Sinha 2021 (arXiv 2111.07049v1)","url":"https://arxiv.org/abs/2111.07049v1","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"chai-invariant-ideal-conjecture-lubin-tate","name":"Chai's conjecture on stabilizer-invariant ideals of the Lubin-Tate deformation ring","shortName":"Chai's invariant-ideal conjecture","problemNumber":null,"field":"Arithmetic of formal groups; Lubin-Tate space","fieldGroup":"Number theory","statement":"Let $\\Gamma_n$ be the height-$n$ Honda formal group over $\\overline{\\mathbb F}_p$ and $E_0=W(\\mathbb F_{p^n})[[u_1,\\ldots,u_{n-1}]]$ its Lubin-Tate deformation ring, on which the Morava stabilizer group $A_n^\\times$ acts by changing the marking. The height ideals $I_h=(p,u_1,\\ldots,u_{h-1})$, $1\\le h\\le n$, cut out the loci where the deformed formal group has height at least $h$; with $0$ they are invariant primes. Chai (1996), studying the group action on the closed fiber of Lubin-Tate space, proposed that invariant irreducible formal loci should be height loci; Hovey and Strickland asked the same about invariant primes (Problem 16.8), and Barthel, Heard and Naumann made it precise as Chai's Hope for finite residue fields. For an open subgroup $U$ of the stabilizer, are $0,I_1,\\ldots,I_n$ the only proper $U$-stable primes of $E_0$, and $0,I_1,\\ldots,I_n,E_0$ the only $U$-invariant radical ideals?","posedBy":"Ching-Li Chai (Duke Math. J. 1996); formulated as Chai's Hope 4.1 by Barthel, Heard and Naumann (2022); compare Hovey-Strickland Problem 16.8","yearPosed":1996,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for any open subgroup $U\\subseteq A_n^\\times$, the proper $U$-stable primes of $E_0$ are exactly $0,I_1,\\ldots,I_n$ and the $U$-invariant radical ideals are exactly $0,I_1,\\ldots,I_n,E_0$; the radical ideals invariant under the full extended group $G_n$ are the same list. This is Chai's Hope in the finite-residue-field formulation, at every prime and height, extended to open subgroups. Through Barthel-Heard-Naumann it yields the Hovey-Strickland conjecture (separate entry). It is stated for the Honda formal group over finite-field coefficients and does not treat other residue fields or the full scope of Chai's closed-fiber questions.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The invariant-ideal classification is the main step of the single manuscript in this family, which then deduces the Hovey-Strickland conjecture (a separate catalog entry).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Chai's question and the Barthel-Heard-Naumann formulation (Hope 4.1 and Remark 4.2) as the manuscript cites them. No Lean formalization exists for this family. The manuscript says it proves the standard finite-residue-field formulation and extends it from the full stabilizer to every open subgroup; the core is a local orbit-density theorem at characteristic-p points of height between 1 and n-1, using perfectoid methods, Le Bras's theorem on Banach-Colmez sheaves and the Fargues-Fontaine curve. Chai's unpublished characteristic-zero result and the Gross-Hopkins period map handle primes not containing p. Not refereed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":26,"significanceNote":"A long-standing question about Lubin-Tate space known to arithmetic geometers and chromatic homotopy theorists, open since 1996 and settled before only in low height. Its main interest is that it drives the Hovey-Strickland conjecture (30 in its separate entry), so it is scored just below that. Placed level with the Robert-Tikuisis Conjecture (C1) (25) and below Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms) (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stabilizer-Orbits-and-Thick-Tensor-Ideals-of-Dualizable-Kn-Local-Spectra-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Barthel-Heard-Naumann, On conjectures of Hovey-Strickland and Chai (Selecta Math. 2022)","url":"https://doi.org/10.1007/s00029-022-00766-2","kind":"problem-record"},{"label":"Chai, The group action on the closed fiber of the Lubin-Tate moduli space (Duke 1996)","url":"https://doi.org/10.1215/S0012-7094-96-08230-7","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"finite-lattice-representation-problem","name":"The finite lattice representation problem: is every finite lattice the congruence lattice of a finite algebra?","shortName":"Finite lattice representation problem","problemNumber":null,"field":"Universal algebra; congruence lattices and subgroup intervals","fieldGroup":"Algebra","statement":"Gratzer and Schmidt (1963) proved that every algebraic lattice is the congruence lattice of some algebra, but the representing algebra may be infinite even when the lattice is finite. The finite lattice representation problem asks whether every finite lattice $L$ is isomorphic to $\\mathrm{Con}(A)$ for some finite algebra $A$ (any finite signature). Palfy and Pudlak (1980) showed this holds for all finite lattices if and only if every finite lattice is an interval $[H,G]$ in the subgroup lattice of a finite group. Positive results cover small lattices and lattices of width at most two, and the decision version (given a lattice table, decide representability) was recorded separately. Is every finite lattice the congruence lattice of a finite algebra?","posedBy":"Finite form of the Gratzer-Schmidt representation theorem; the manuscripts do not name who first posed it","yearPosed":null,"ageNote":"The manuscripts cite Gratzer-Schmidt (Acta Sci. Math. Szeged 24, 1963) for the infinite theorem, Palfy-Pudlak (Algebra Universalis 11, 1980) for the group equivalence, and McNulty for the formulations, but do not say who posed the finite problem or when.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Principal manuscript, Theorem 1.1: (i) a finite lattice is representable iff it admits a finite colored-graph witness; (ii) no Turing machine decides representability from the order table, so not every finite lattice is representable; (iii) there is a nonrepresentable lattice of minimum size, larger than 7, specified by a finite description involving a Busy Beaver constant (its size is not computed); (iv) recognizing full subgroup intervals of finite groups is undecidable. The companion constructs a finite lattice that is no interval $[D,G]$ in a finite group, hence (Palfy-Pudlak) a nonrepresentable one. No explicit small counterexample is exhibited.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated September 24, 2026: the principal one gives a colored-graph characterization, undecidability of representability and of full subgroup-interval recognition, and a minimal counterexample described via a Busy Beaver constant; the companion gives a direct construction of a nonrepresentable lattice through subgroup intervals.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript and Theorems 1.1-1.2 of the companion were read against the problem; both claim a finite lattice that is not the full congruence lattice of any finite algebra, the companion via a lattice that is no finite subgroup interval. Proofs (over 170 and 80 pages, using the classification of finite simple groups and Larsen-Pink) were not refereed. A Lean challenge exists, lean/ComparatorChallenges/FiniteCongruenceGraph.json (solution module OAI.Algebra.Universal.GraphCriterion present), not in the formalization catalogue; its statement, read here, is only the colored-graph criterion (a finite lattice is representable iff it has a finite colored-graph witness). That criterion does not imply the negative answer, so the entry is left unreviewed. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The finite lattice representation problem is widely called the most famous open problem of universal algebra, open for six decades and tied to finite group theory through Palfy-Pudlak. Famous within one community, placed level with Wall's manifold-realization question (42), below the transitive algebra problem (46), and above Kahn's flow conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Finite congruence lattices: characterization and undecidability","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A negative solution to the finite lattice representation problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean proof of the graph criterion only (OAI.FiniteCongruence.graph_criterion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/Universal/GraphCriterion.lean","kind":"lean-proof"},{"label":"Comparator statement: FiniteCongruenceGraph.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/FiniteCongruenceGraph.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"saxl-conjecture","name":"Saxl's conjecture: the tensor square of the staircase representation contains every irreducible representation of the symmetric group","shortName":"Saxl's conjecture","problemNumber":null,"field":"Representation theory of symmetric groups; Kronecker coefficients","fieldGroup":"Algebra","statement":"For partitions $\\lambda,\\mu,\\nu$ of $n$ the Kronecker coefficient $g(\\lambda,\\mu,\\nu)=\\dim\\mathrm{Hom}_{S_n}(S^\\nu,S^\\lambda\\otimes S^\\mu)$ is the multiplicity of the Specht module $S^\\nu$ in $S^\\lambda\\otimes S^\\mu$. Let $\\rho_m=(m,m-1,\\ldots,1)$ be the staircase partition of $N_m=m(m+1)/2$. At the UCLA Combinatorics Seminar on 20 March 2012 Saxl proposed that $S^{\\rho_m}\\otimes S^{\\rho_m}$ contains every irreducible representation of $S_{N_m}$ (recorded by Pak, Panova and Vallejo, Conjecture 1.2). Known cases covered hooks, two-row shapes, double and triple hooks, partitions comparable to $\\rho_m$ in dominance order, and random partitions with high probability. Is $g(\\rho_m,\\rho_m,\\lambda)>0$ for every $m\\ge1$ and every partition $\\lambda\\vdash N_m$?","posedBy":"Jan Saxl","yearPosed":2012,"ageNote":"Saxl proposed it at the UCLA Combinatorics Seminar on 20 March 2012, as recorded by Pak, Panova and Vallejo (Adv. Math. 288 (2016), Conjecture 1.2 and footnote 2; arXiv 1304.0738), cited by both manuscripts.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $m\\ge1$ and every $\\lambda\\vdash N_m=m(m+1)/2$, $g(\\rho_m,\\rho_m,\\lambda)>0$, so the tensor square of the staircase Specht module contains every irreducible representation of $S_{N_m}$. The proof gives the stronger statement that every constituent occurs inside one explicitly generated cyclic submodule of the tensor square, extending Ikenmeyer's triangular arrangement of alternating tensors. Not shown: values or bounds for these Kronecker coefficients, or a combinatorial interpretation of them.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript is 'A Cyclic Polytabloid Proof of Saxl's Conjecture' (September 24, 2026); the companion 'Universal Tensor Squares for Symmetric Groups' of the same date uses its cyclic statement and is entered separately.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the conjecture as Pak, Panova and Vallejo record it; it is the conjecture verbatim. Lean: formalization.yaml lists comparator ComparatorChallenges/Saxl.json, declaration OAI.Saxl.saxl_conjecture in OAI/RepresentationTheory/Saxl/Main.lean. The statement file was read here: Specht modules are orbit spans of polytabloids in a complex word space, the Kronecker multiplicity is the dimension of the space of intertwining maps from $S^\\mu$ into $S^{\\rho_m}\\otimes S^{\\rho_m}$, and SaxlConjecture asserts positivity for every $m\\ge1$ and every Young diagram of size $m(m+1)/2$. That states the headline. Not rebuilt here. The paper's stronger cyclic-submodule statement is not formalised.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"Saxl's conjecture is a well-known problem on Kronecker coefficients with a sizeable literature of partial cases (Pak-Panova-Vallejo, Ikenmeyer, Bessenrodt, Luo-Sellke) since 2012. Famous within algebraic combinatorics and representation theory. Placed below Rota's unimodality conjecture for matroid flats (36), level with Dean's conjecture for k=5 (32), above Unimodality of Kazhdan-Lusztig polynomials of matroids (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A Cyclic Polytabloid Proof of Saxl's Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion manuscript: Universal Tensor Squares for Symmetric Groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf","kind":"paper"},{"label":"Lean proof: OAI/RepresentationTheory/Saxl/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RepresentationTheory/Saxl/Main.lean","kind":"lean-proof"},{"label":"Lean statement: ComparatorChallenges/Saxl.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/Saxl.lean","kind":"lean-proof"},{"label":"Pak, Panova and Vallejo, Kronecker products, characters, partitions, and the tensor square conjectures","url":"https://arxiv.org/abs/1304.0738","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"minimal-model-existence-generalized-lc-pairs","name":"Existence of minimal models and Mori fibre spaces for projective log canonical pairs, in generalized-pair form","shortName":"Minimal model existence for lc pairs","problemNumber":null,"field":"Birational geometry; minimal model program","fieldGroup":"Geometry & topology","statement":"The minimal model conjecture in its existence form asserts that every projective log canonical pair $(X,B)$ over an algebraically closed field of characteristic zero has a minimal model (a birational model with nef $K_Y+B_Y$ and no worse discrepancies) when $K_X+B$ is pseudo-effective, and a Mori fibre space otherwise. Birkar-Cascini-Hacon-McKernan proved this for klt pairs with big boundary or big adjoint; the general case was open in dimension at least five, and for generalized pairs (Birkar-Zhang) carrying a fixed nef part. Does every projective (generalized) log canonical $\\mathbb Q$-pair admit a minimal model or a Mori fibre space?","posedBy":"Central conjecture of the minimal model program; the manuscript cites its statement as Conjecture 1.1 of Gongyo (2011); generalized pairs of Birkar and Zhang","yearPosed":null,"ageNote":"The conjecture is part of the minimal model program from the 1980s (Mori, Kawamata, Shokurov and others); the manuscript names no single poser and cites Gongyo's 2011 manuscript for the precise statement and Han-Li (2022) for the generalized-pair framework. No year is given because the sources do not fix one.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a projective generalized lc $\\mathbb Q$-pair $(X,B+M)$ over an algebraically closed field of characteristic zero there is a birational map $X\\dashrightarrow Y$ extracting no divisors, with $(Y,B_Y+M_Y)$ generalized lc and log discrepancies not decreasing (strictly on contracted divisors), such that $K_Y+B_Y+M_Y$ is nef if $K_X+B+M$ is pseudo-effective, and otherwise $Y$ has a Mori fibre space structure with relative Picard number one. With $M=0$ this gives the ordinary lc case. Companion: smooth projective varieties with $\\kappa_\\sigma(K_X)=1$ have $\\mathbb Q$-factorial terminal minimal models. Not shown: termination of arbitrary flip sequences, real coefficients, abundance (claimed in a separate release paper), or Q-factorial models in general.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript and the numerical-dimension-one companion are both dated September 24, 2026. The release's log abundance manuscript (already in the catalog) uses this paper's ordinary-pair corollary.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 of 'Minimal models and Mori fibre spaces for generalized log canonical Q-pairs' read against the existence conjecture as the paper cites it (Gongyo, Conjecture 1.1). No Lean formalization. The paper states its scope: rational boundary and rational nef data, no Q-factoriality, existence of some terminating program only (not termination of every MMP), and no semiampleness or nonvanishing. The terminating transfer is credited to Tsakanikas-Xie and earlier NQC MMP results; the new part is a local phase theorem for cyclic Gorenstein klt germs and a global contradiction for an infinite MMP.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":68,"significanceNote":"Existence of minimal models is, with abundance, one of the two central conjectures of the minimal model program, open from dimension five for lc pairs since BCHM. Placed below the log abundance entry (75), which also claims good minimal models, and above Iitaka subadditivity (62) and the Jacobian Conjecture band (65). Far above termination of flips for lc fourfolds (34).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Minimal models and Mori fibre spaces for generalized log canonical Q-pairs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion manuscript: Minimal models in numerical dimension one","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Minimal-models-in-numerical-dimension-one-September-24-2026/paper.pdf","kind":"paper"},{"label":"Related release manuscript: Log abundance in characteristic zero (uses this paper)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hovey-strickland-conjecture","name":"The Hovey-Strickland conjecture: the thick tensor ideals of dualizable K(n)-local spectra","shortName":"Hovey-Strickland conjecture","problemNumber":null,"field":"Chromatic homotopy theory; tensor-triangular geometry","fieldGroup":"Geometry & topology","statement":"Fix a prime $p$ and a height $n\\ge1$, and let $\\mathcal D_{(n,p)}$ be the homotopy category of dualizable $K(n)$-local spectra, with tensor product $L_{K(n)}(X\\wedge Y)$. A thick tensor ideal is a full subcategory closed under retracts, cofiber sequences, suspensions and tensoring with every object. For $0\\le k\\le n$ the localization of a finite type-$k$ spectrum generates a thick tensor ideal $\\mathcal D_k$, giving a chain $\\mathcal D_0\\supsetneq\\mathcal D_1\\supsetneq\\cdots\\supsetneq\\mathcal D_n\\supsetneq0$. By analogy with the Hopkins-Smith thick subcategory theorem for finite spectra, Hovey and Strickland (1999) conjectured that this chain is everything. Barthel, Heard and Naumann proved it at height two and showed it follows from Chai's invariant-ideal conjecture. For every prime and every height, are the thick tensor ideals of $\\mathcal D_{(n,p)}$ exactly the $n+2$ ideals $\\mathcal D_0,\\ldots,\\mathcal D_n,0$, so that the Balmer spectrum is a chain of $n+1$ points?","posedBy":"Mark Hovey and Neil Strickland (Morava K-theories and localisation, Mem. AMS 1999, Section 12); restated by Barthel, Heard and Naumann (2022)","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.2: for every prime $p$ and $n\\ge1$ the thick tensor ideals of the dualizable $K(n)$-local category are exactly $\\mathcal D_0\\supsetneq\\cdots\\supsetneq\\mathcal D_n\\supsetneq\\mathcal D_{n+1}=0$, so there are $n+2$ of them and each $\\mathcal D_k$ does not depend on the chosen type-$k$ spectrum; a corollary identifies the Balmer spectrum with the $n+1$ proper members of the chain, with its specialization order. It is deduced from Theorem 1.1 (Chai's conjecture, a separate entry) by the known Barthel-Heard-Naumann implication. It concerns dualizable objects only and does not classify localizing or other subcategories of the whole $K(n)$-local category. Height two was already known.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. One manuscript proves Chai's invariant-ideal conjecture as its main step and deduces the Hovey-Strickland conjecture from it through the implication already established by Barthel, Heard and Naumann; the same manuscript underlies the separate catalog entry for Chai's conjecture.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 (stabilizer-invariant ideals) and Theorem 1.2 (Hovey-Strickland conjecture) were read against the conjecture as Hovey-Strickland and Barthel-Heard-Naumann state it. The release has no Lean formalization for this family (no lean/docs page at the pinned commit). The spectral classification is not proved from scratch: it uses the Barthel-Heard-Naumann implication from invariant ideals to thick tensor ideals (their Theorems 4.9 and 4.15), resting on Mathew's descent and Balmer's nilpotence criterion. The new input is the classification of invariant primes in characteristic p via perfectoid and Fargues-Fontaine curve methods; characteristic-zero primes use the Gross-Hopkins period map. The proof was not refereed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A well-known structural conjecture of chromatic homotopy theory and tensor-triangular geometry, open since 1999 and settled before only at height two. Famous within that community, little known beyond it. Placed below Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms) (40), level with Smith-Toda complexes V(n) at every height when the prime may vary (30), and above the separate entry for Chai's invariant-ideal conjecture (26) that implies it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stabilizer-Orbits-and-Thick-Tensor-Ideals-of-Dualizable-Kn-Local-Spectra-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Barthel-Heard-Naumann, On conjectures of Hovey-Strickland and Chai (Selecta Math. 2022)","url":"https://doi.org/10.1007/s00029-022-00766-2","kind":"problem-record"},{"label":"Hovey-Strickland, Morava K-theories and localisation (Mem. AMS 1999)","url":"https://doi.org/10.1090/memo/0666","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"contingency-tables-arbitrary-margins","name":"Polynomial-time approximate counting and uniform sampling of contingency tables with arbitrary margins","shortName":"Counting and sampling contingency tables","problemNumber":null,"field":"Approximate counting and sampling (FPRAS, Markov chains)","fieldGroup":"Theoretical computer science","statement":"For margins $r\\in\\mathbb{Z}_{\\ge0}^m$, $c\\in\\mathbb{Z}_{\\ge0}^n$ with equal totals, let $\\Omega(r,c)$ be the nonnegative integer $m\\times n$ matrices with these row and column sums (optionally with individual cell bounds $b_{ij}$, zero bounds allowed). Exact counting is #P-complete even for two rows (Dyer-Kannan-Mount 1997). Polynomial approximate counting and sampling were known for two rows, for a fixed number of rows, for sufficiently large or dense margins, and for sparse margins, but not jointly polynomial in both dimensions and the binary length of the margins; Cryan, Dyer and Randall (2010) named unrestricted cell-bounded counting as open. Is there a fully polynomial randomized approximation scheme for $|\\Omega|$, and a polynomial-time almost-uniform sampler, for arbitrary binary-encoded margins (and cell bounds) with both dimensions part of the input?","posedBy":"Dyer, Kannan and Mount (1997) and the subsequent literature; the unrestricted cell-bounded case named open by M. Cryan, M. Dyer and D. Randall, SIAM J. Comput. (2010)","yearPosed":1997,"ageNote":"No single paper is cited as posing the problem in full generality. Dyer-Kannan-Mount (1997) proved #P-completeness and gave the first dense-case algorithms; Cryan, Dyer and Randall (2010) stated the unrestricted cell-bounded problem as open, and Arman, Gao and Wormald (2021) reported that no polynomial sampler for arbitrary margins was known.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"FPRAS manuscript: one randomized algorithm counts nonnegative integer matrices with prescribed row sums, column sums and entry bounds $b_{ij}$ (zeros allowed) within relative error $\\varepsilon$ with probability $1-\\delta$, using unbiased bits and a polynomial bound in the input length, $\\varepsilon^{-1}$ and $\\log\\delta^{-1}$ on every execution. Sampling manuscript: for ordinary tables, an almost-uniform sampler (TV error $2^{-k}$) in worst-case polynomial bit time and an exactly uniform sampler in expected polynomial bit time in $m,n,\\log(N+1)$. Exponents are deliberately large; the results are complexity classifications, not practical algorithms, and the exact sampler is polynomial only in expectation.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The family has two manuscripts, both dated 24 September 2026: an FPRAS for cell-bounded tables and exact and almost-uniform samplers for ordinary tables.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of both TeX sources, read against the open problem as described in their history sections (Dyer-Kannan-Mount, Cryan-Dyer-Randall, Arman-Gao-Wormald); the proofs were not refereed. Lean: the Comparator challenge ContingencyTables (OAI.ContingencyTables.counting, exactSampling, boundedSampling; solution module OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean) is not in the release's formalization catalogue, but its JSON and solution file exist at the pinned commit. Its statement was read here: it fixes a randomized Post-Turing machine model with binary encodings and asserts (i) a cell-bounded counter whose every execution halts within a fixed polynomial in input length, 1/eps and log(1/delta), outputs zero on infeasible inputs, and is within relative error eps with probability at least 1 - delta; (ii) a bounded-time sampler with total-variation error at most 2^-k; (iii) an exact sampler with uniform limiting law and polynomial expected running time. This states the headline claims. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"A long-standing open problem in approximate counting, with statistical motivation from exact conditional tests, that resisted Markov chain, dynamic programming and geometric methods for three decades. Scored below the FPRAS for perfect matchings in general graphs (55), whose problem is more famous, and well above approximating two-terminal network reliability (25); level with the Alon-Krivelevich-Sudakov coloring conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf","sourceName":"OpenAI math release: An FPRAS for cell-bounded contingency tables","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Exact uniform sampling of contingency tables with arbitrary margins (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf","kind":"paper"},{"label":"Lean: OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/ContingencyTables/UnconditionalMain.lean","kind":"lean-proof"},{"label":"Cryan, Dyer, Randall, Approximately counting integral flows and cell-bounded contingency tables (SICOMP 2010)","url":"https://doi.org/10.1137/060650544","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mumford-shah-conjecture-planar-interior","name":"The planar Mumford-Shah conjecture (interior regularity of the discontinuity set)","shortName":"Mumford-Shah conjecture (planar, interior)","problemNumber":null,"field":"Calculus of variations; free-discontinuity problems","fieldGroup":"Differential equations","statement":"For a bounded domain $\\Omega\\subset\\mathbb R^2$ and $g\\in L^\\infty(\\Omega)$, the Mumford-Shah functional is $\\int_{\\Omega\\setminus K}|\\nabla u|^2+\\mathcal H^1(K)+\\int_\\Omega|u-g|^2$, minimized over closed sets $K$ and functions $u$ smooth off $K$. Minimizers exist (De Giorgi, Carriero, Leaci; Ambrosio), $K$ is regular off a closed $\\mathcal H^1$-null set (David; Ambrosio-Fusco-Pallara), and Bonnet reduced the question to classifying global minimizers. Mumford and Shah (1989) conjectured that $K$ has only the geometry suggested by image segmentation. Is the discontinuity set of every minimizer, inside $\\Omega$, locally a finite union of $C^{1}$ arcs that are regular, end at crack tips, or meet in threes at $120^\\circ$?","posedBy":"David Mumford and Jayant Shah (Comm. Pure Appl. Math. 42, 1989)","yearPosed":1989,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a reduced absolute minimizer $(u,K)$ with $g\\in L^\\infty$, every $x\\in K$ has a neighborhood where $K$ is one $C^{1,\\alpha}$ arc through $x$, one arc ending at $x$, or three arcs meeting at $x$ at $120^\\circ$; only finitely many connected components of $K$ meet any $U\\Subset\\Omega$. Corollary: $\\nabla u\\in L^{4,\\infty}_{\\mathrm{loc}}$, sharp because of crack tips. The key step excludes bounded components in blow-up limits. It does not treat behaviour at $\\partial\\Omega$, higher dimensions, or minimizers in weaker senses than absolute ones.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. For the endpoint and compactness theory the manuscript invokes the De Lellis-Focardi monograph; its new step excludes bounded components of blow-up limits by a translation and harmonic-interaction argument.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the planar Mumford-Shah conjecture. It is stated for reduced absolute minimizers on bounded Lipschitz domains with bounded measurable data and gives, near every point of $K$, a $C^{1,\\alpha}$ arc, an arc ending at the point, or a $120^\\circ$ triple junction, plus local finiteness of the global components of $K$. The manuscript states it makes no assertion about finiteness up to $\\partial\\Omega$. The proof was not refereed. No Lean formalization of this manuscript is in the release (no lean/docs/366.md at the pinned commit). The manuscript itself cites a human preprint posted two days earlier, F. Deangelis, 'Solution of the Mumford-Shah conjecture', arXiv:2609.26732v1 (22 September 2026), which it credits with a proof of the global classification and the same interior structure by a different argument.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"The Mumford-Shah conjecture, open since 1989, is the best known regularity problem for free-discontinuity functionals, with a long line of partial results by Bonnet, David, Ambrosio-Fusco-Pallara and De Lellis-Focardi. Placed level with Sidorenko's conjecture (52), above the Kakeya maximal conjecture in R3 (48), below stationary varifold almost-everywhere regularity (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026.pdf","sourceName":"OpenAI math release: Interior regularity of planar Mumford-Shah minimizers","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Concurrent human proof: Deangelis, Solution of the Mumford-Shah conjecture (arXiv:2609.26732)","url":"https://arxiv.org/abs/2609.26732","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"shortest-superstring-factor-2-approximation","name":"A polynomial-time factor-2 approximation for shortest common superstring (the question behind the Tarhio-Ukkonen greedy conjecture)","shortName":"Factor-2 approximation for shortest common superstring","problemNumber":null,"field":"Approximation algorithms; string algorithms","fieldGroup":"Algorithms & optimization","statement":"Given a finite set $\\mathcal S$ of strings, the shortest common superstring problem asks for a shortest string containing every member of $\\mathcal S$ as a contiguous substring; it is NP-hard and MAX-SNP-hard. Tarhio and Ukkonen (1988) conjectured that the greedy maximum-overlap merging procedure is a 2-approximation. Proven guarantees improved from 3 (Blum, Jiang, Li, Tromp, Yannakakis) through 2.5 (Sweedyk), $2\\tfrac{11}{23}$ (Mucha) and below 2.466 (Englert, Matsakis, Vesely), with factor 2 the long-standing target. Is there a polynomial-time algorithm that always outputs a common superstring of length at most twice the optimum (in particular, is greedy one)?","posedBy":"Jorma Tarhio and Esko Ukkonen (greedy factor-2 conjecture); the algorithm-independent factor-2 question is cited in the manuscript without a named poser","yearPosed":1988,"ageNote":"Tarhio and Ukkonen, Theor. Comput. Sci. 57 (1988), stated the conjecture for Greedy. The manuscript says it 'answers affirmatively the question of whether a polynomial-time factor-2 approximation exists' but does not name who first posed that general form.","solveType":"proved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a deterministic algorithm, polynomial in the total encoded input size (including symbol labels), outputs for every finite family of explicitly represented strings a common superstring $T$ with $|T|\\le2\\,\\mathrm{OPT}$. The proof builds a balanced hierarchical graph on all substrings from forced occurrence counts (a lower bound $W$ on the optimum), decomposes it into periodic layers and connects them at extra cost at most $W$. It does not show that Greedy or the Collapsing procedure attains factor 2 (Greedy is reported false by Shibata 2026), and gives nothing below factor 2. An August 2026 preprint of Chukhin, Kulikov, Mihajlin and Smal reported $7/3$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 24, 2026), with a Lean formalization listed in the release's formalization catalogue.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the factor-2 question. The guarantee is for a new algorithm, not Greedy; the manuscript itself cites a September 2026 preprint (Shibata) reporting a counterexample to the Greedy conjecture with ratio at least 9/4. Lean: formalization.yaml lists OAI.Superstring.main (lean/OAI/Computability/Superstring/Main.lean, comparator Superstring.json). Its statement was read here: there is a function on instances with a polynomial-time finite-alphabet Turing machine implementation in the bit encoding whose output is a common superstring of every instance and has length at most 2 times the optimum in symbols. This is the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Factor 2 for shortest superstring has been the recognised target of string approximation since the greedy conjecture of 1988, with a long sequence of incremental constants; scored as the algorithm-independent version. Below the NP-hardness of uniform sparsest cut (45) and the Sakoda-Sipser problem (45), above the Goemans-Linial integrality gap entry (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A Polynomial-Time 2-Approximation for Shortest Common Superstring","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI/Computability/Superstring/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/Superstring/Main.lean","kind":"lean-proof"},{"label":"Lean statement: Superstring.lean (comparator challenge)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/Superstring.lean","kind":"lean-proof"},{"label":"Tarhio and Ukkonen, A greedy approximation algorithm for constructing shortest common superstrings (1988)","url":"https://doi.org/10.1016/0304-3975(88)90167-3","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"scalar-curvature-extension-dimension-four","name":"The scalar-curvature extension conjecture for Ricci flow in dimension four","shortName":"Scalar-curvature extension, dimension 4","problemNumber":null,"field":"Geometric analysis; four-dimensional Ricci flow","fieldGroup":"Geometry & topology","statement":"A smooth Ricci flow on a closed manifold can stop only if the full curvature blows up. Bamler and Zhang developed heat-kernel and compactness theory for flows with bounded scalar curvature and proved smooth extension in dimension four when $H_2(M;\\mathbb F)=0$ for all fields; Simon continued such flows through orbifold limits. In dimension four: if a smooth Ricci flow on a closed four-manifold has uniformly bounded scalar curvature on $[0,T)$ with $T<\\infty$, does it extend smoothly on the same manifold past $T$?","posedBy":"Stated as open in R. H. Bamler, Ann. of Math. 188 (2018), p. 756, with the four-dimensional theory of Bamler-Zhang (2017, 2019) and M. Simon (2020)","yearPosed":null,"ageNote":"The manuscripts do not say when or by whom the dimension-four case was first posed; Bamler-Zhang's partial result (Adv. Math. 2017) shows it was open by then.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a smooth Ricci flow on a closed four-manifold with $\\sup_{M\\times[0,T)}|R|<\\infty$, $T<\\infty$, extends smoothly past $T$; Corollary: at a finite maximal time $\\limsup\\max R=+\\infty$. This removes the $H_2$ hypothesis of Bamler-Zhang and gives smooth continuation rather than an orbifold flow. The companion proves $E=o(M)$ for the renormalized Einstein-Hilbert functional on degenerating trees of Ricci-flat spaces. It is specific to dimension four; in high dimension the family's other paper shows the statement is false.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the problem as the paper states it; it claims smooth extension on the same closed four-manifold whenever $|R|$ is bounded up to a finite $T$. The long proof (blowup trees of Ricci-flat pieces, a renormalized Einstein-Hilbert functional) was not refereed and is not formalized. It depends on the companion's elliptic inequality on degenerating Ricci-flat trees as a stated input, and on Bamler-Zhang's estimates.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The four-dimensional case is the main open instance of the scalar-curvature question and the focus of Bamler-Zhang's work; a proof would be a substantial result in Ricci flow. Placed level with the CAT(0) conjecture for Artin groups (40), below the hot spots conjecture for smooth simply connected planar domains (42), above Universal volume growth bounds from positive intermediate curvature (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Bounded scalar curvature and smooth extension of four-dimensional Ricci flow","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Path selection and an elliptic inequality on degenerating Ricci-flat trees","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf","kind":"paper"},{"label":"A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"glr-ramanujan-arthur-decomposition","name":"The Gaitsgory-Lafforgue-Raskin Ramanujan-Arthur decomposition of cuspidal automorphic functions over function fields","shortName":"GLR Ramanujan-Arthur decomposition","problemNumber":null,"field":"Automorphic forms over function fields; Arthur parameters","fieldGroup":"Number theory","statement":"Let $F$ be the function field of a smooth projective geometrically connected curve over $\\mathbb F_q$, $G$ a split connected semisimple group, and $C_{\\mathbb Q}$ the space of rational-valued cuspidal functions on $G(F)\\backslash G(\\mathbb A_F)/G(\\mathcal O_{\\mathbb A})$. For a nilpotent orbit $\\mathcal O$ of the dual group with Jacobson-Morozov map $\\phi_{\\mathcal O}$, let $C_{\\mathcal O}$ be the functions whose Hecke eigenvalues at every place $x$ have Satake class in $q_x^{1/2}$-twisted $\\phi_{\\mathcal O}$ times a compact element of the centralizer, the same orbit at every place. Gaitsgory, Lafforgue and Raskin conjectured (Conjectures 3.4.5 and 3.4.6, the rational and the $\\ell$-adic Ramanujan-Arthur conjecture) that the cuspidal space decomposes accordingly. Is $\\bigoplus_{\\mathcal O}C_{\\mathcal O}=C$?","posedBy":"Dennis Gaitsgory, Vincent Lafforgue and Sam Raskin","yearPosed":2026,"ageNote":"Gaitsgory-Lafforgue-Raskin, Tempered vs generic automorphic functions and the canonical filtration on automorphic functions, arXiv:2603.26925 (2026), Section 3.4, as cited. The underlying prediction is Arthur's (1989); this precise form is new.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $X,G,\\ell$ and every effective divisor $D$, the rational and $\\overline{\\mathbb Q}_\\ell$ cuspidal spaces at full level $D$ are the direct sums of their orbit-indexed summands; $D=0$ gives GLR Conjectures 3.4.5-3.4.6. Corollary 1.2: for split adjoint absolutely simple $G$, a cuspidal representation generic and unramified at one place is tempered at every unramified place. The October 5 companion constructs, from this, a commuting algebraic $SL_2$ enhancing each occurring cuspidal excursion parameter at full level; it does not assert ellipticity, Arthur packets or multiplicities, so V. Lafforgue's Arthur conjecture is not settled.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. This entry rests on the September 24 Ramanujan-Arthur decomposition paper; the October 5 global Arthur enhancement paper builds on it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against GLR Conjectures 3.4.5-3.4.6. The theorem is stated for every curve, every split connected semisimple $G$, every full level $D$ and every $\\ell$, with no characteristic hypothesis; at $D=0$ it is the conjecture. Inputs are V. Lafforgue's excursion parameters, L. Lafforgue's purity theorem for $GL_n$, and the Ciubotaru-Harris criterion for the corollary. The proof was not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A precise 2026 form of Arthur's prediction for the non-tempered cuspidal spectrum over function fields, proved for all split semisimple groups; recent as a stated conjecture but at the heart of the Langlands program. Placed below Modularity of elliptic curves over imaginary quadratic fields (58) and The p-converse to Gross-Zagier-Kolyvagin in Selmer corank at most one (45), and above The index conjecture for slc log Calabi-Yau pairs (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Global Arthur Enhancements of Cuspidal Excursion Parameters","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf","kind":"paper"},{"label":"Companion: Rationality of the Canonical Unramified Arthur Filtration","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf","kind":"paper"},{"label":"Gaitsgory-Lafforgue-Raskin, arXiv:2603.26925","url":"https://arxiv.org/abs/2603.26925","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-gallai-cycle-decomposition","name":"The Erdos-Gallai cycle decomposition conjecture (Erdos Problem #184)","shortName":"Erdos-Gallai cycle decomposition","problemNumber":184,"field":"Extremal graph theory; graph decompositions","fieldGroup":"Combinatorics","statement":"Erdos and Gallai conjectured that the edges of every graph on $n$ vertices can be partitioned into $O(n)$ edge-disjoint cycles and single edges; single edges are needed because forests have no cycles, and $K_{3,n-3}$ shows more than $n$ parts may be necessary. Erdos, Goodman and Posa (1966, Section 5) record the problem and an $O(n\\log n)$ bound. Conlon, Fox and Sudakov improved this to $O(n\\log\\log n)$ (2014) and Bucic and Montgomery to $O(n\\log^*n)$; linear bounds were known for random graphs and graphs of linear minimum degree. Is there an absolute constant $C$ such that every $n$-vertex graph decomposes into at most $Cn$ cycles and edges?","posedBy":"Paul Erdos and Tibor Gallai, recorded by Erdos, Goodman and Posa (Canad. J. Math. 18, 1966)","yearPosed":1966,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: an absolute constant $C$ exists such that the edges of every finite simple graph on $n$ vertices partition into at most $Cn$ simple cycles and single edges. Corollary 1.2: every Eulerian graph partitions into at most $Cn$ cycles, and under a quasirandom cut condition into $\\Delta/2+\\delta n$ cycles. The constant is not computed or optimized; the conjectured sharp values (Erdos suggested $n-1$; $K_{3,n-3}$-type lower bounds of about $3n/2$) are not addressed.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The argument builds on the expansion and path-closing methods of Bucic and Montgomery.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture and against erdosproblems.com/184; it gives an absolute constant $C$ with every finite simple graph on $n$ vertices partitioned into at most $Cn$ cycles and single edges. Lean: lean/formalization.yaml lists comparator CycleDecomposition with declaration OAI.ErdosGallai.erdos_gallai. The statement ComparatorChallenges/CycleDecomposition.lean was read here: there is a real $C>0$ such that for every $n$ and every SimpleGraph on Fin n there is a $k\\le Cn$ and $k$ pairwise disjoint edge sets, each the edge set of a cycle (Walk.IsCycle) or a single edge, whose union is the edge set. This is exactly the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"A well-known Erdos conjecture from the 1960s, with successive improvements by Conlon-Fox-Sudakov and Bucic-Montgomery; far above a typical Erdos problem but within one community. Placed above Gyarfas's tree cover conjecture (32), below Ryser's conjecture (50) and Sidorenko's conjecture (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf","sourceName":"OpenAI math release: A linear cycle-and-edge decomposition of every graph","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.ErdosGallai.erdos_gallai)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/CycleDecomposition/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: CycleDecomposition.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CycleDecomposition.lean","kind":"lean-proof"},{"label":"Erdos Problem #184","url":"https://www.erdosproblems.com/184","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"quantum-heisenberg-ferromagnet-spontaneous-magnetization","name":"Spontaneous magnetization of the three-dimensional quantum Heisenberg ferromagnet at low temperature (Lieb's Problem A)","shortName":"Heisenberg ferromagnet ordering","problemNumber":null,"field":"Quantum spin systems; phase transitions","fieldGroup":"Mathematical physics","statement":"The nearest-neighbour quantum Heisenberg ferromagnet on $\\mathbb Z^d$ has Hamiltonian $H=-\\sum_{|x-y|=1}\\mathbf S_x\\cdot\\mathbf S_y$ with spin-$S$ matrices at each site. Mermin and Wagner excluded order at positive temperature for $d\\le2$; Frohlich-Simon-Spencer proved ordering for the classical model in $d\\ge3$ and Dyson-Lieb-Simon for quantum antiferromagnets, but the reflection-positivity method gives no infrared bound for the quantum ferromagnet. Lieb listed the problem in 1999 (Problem A), and it was still described as open in 2025. Does the quantum Heisenberg ferromagnet in dimension $d\\ge3$, for every spin $S$, order at sufficiently low positive temperature, i.e. have an equilibrium state with nonzero spontaneous magnetization at zero field?","posedBy":"Elliott H. Lieb (1999, Problem A of his list of statistical-mechanics problems); obstruction noted by Dyson, Lieb and Simon (1978)","yearPosed":1999,"ageNote":"The manuscript cites Lieb (1999) Problem A as the explicit posing and Seiringer's 2025 account, which states it with a volume-averaged finite-volume two-point order parameter. Dyson-Lieb-Simon (1978, pp. 339-340) already explained why their method did not reach the ferromagnet.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $d\\ge3$ and $S\\in\\{1/2,1,3/2,\\dots\\}$ there is $\\beta_0(d,S)$ such that for every $\\beta\\ge\\beta_0$ the zero-field nearest-neighbour dynamics has a translation-invariant $\\beta$-KMS state with $\\omega(S_0^z)\\ge S/4$. The proof uses an exchange-cycle (random interchange) representation with boundary pins, stable-polynomial negative dependence and a determinant identity for conditioned walks. Not shown: long-range order in the finite-volume two-point formulation, the critical temperature, or anything in $d\\le2$. The companion manuscripts in the family prove Bloch's law and the spherical magnetization law in $d=3$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has four manuscripts; this entry's principal one is dated September 24, 2026, and the three Bloch-law manuscripts (October 5, 2026) build on its finite-set algebra and lattice geometry.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of 'Spontaneous magnetization in the quantum Heisenberg ferromagnet' read against Lieb's problem as the manuscript cites it. Lean: the challenge lean/ComparatorChallenges/Heisenberg.json (solution_module OAI.MathematicalPhysics.Heisenberg.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml. The statement OAI.Heisenberg.spontaneous_magnetization was read here: for $d\\ge3$ and $\\ell=2S\\ge1$, finite-volume dynamics converge, and for all large $\\beta$ there is a translation-invariant $\\beta$-KMS state with $\\omega(S^z_0)$ real and at least $\\ell/8=S/4$. That is the headline claim. Not rebuilt here. The paper itself says it resolves the problem in the KMS-state formulation and that the finite-volume two-point criterion in Seiringer's account is a different formulation it does not address.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"A central open problem of rigorous statistical mechanics for decades: ordering of the quantum Heisenberg ferromagnet in three dimensions, where the classical and antiferromagnetic cases fell in the 1970s. Level with Bose-Einstein condensation of the dilute hard-sphere gas (58), above Simon's Problem 2 on 2D Anderson localization (55), below the Haldane gap conjecture (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Spontaneous magnetization in the quantum Heisenberg ferromagnet","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion manuscript: Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf","kind":"paper"},{"label":"Companion manuscript: The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf","kind":"paper"},{"label":"Companion manuscript: The first lattice correction to Bloch's law","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf","kind":"paper"},{"label":"Lean formalization (Heisenberg spontaneous magnetization)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Heisenberg/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"catalan-constant-irrational","name":"Irrationality of Catalan's constant","shortName":"Catalan's constant is irrational","problemNumber":null,"field":"Diophantine approximation; irrationality of L-values","fieldGroup":"Number theory","statement":"Catalan's constant is $G=\\beta(2)=\\sum_{j\\ge0}(-1)^j/(2j+1)^2=L(2,\\chi_{-4})$, the simplest even value of the Dirichlet beta function. Odd beta values are rational multiples of powers of $\\pi$, but nothing was known about the arithmetic of $G$ itself. Rivoal and Zudilin (2003) proved that infinitely many even beta values are irrational and that at least one of $\\beta(2),\\beta(4),\\ldots,\\beta(14)$ is, a list later cut to $\\beta(2),\\ldots,\\beta(10)$; such family results never single out $G$. Apery-like recurrences and continued fractions for $G$ (Zudilin 2003) converge fast, but the integer linear forms they give do not tend to zero once denominators are cleared. Is Catalan's constant irrational?","posedBy":"Long-standing open problem with no single poser; the manuscript frames it through Rivoal and Zudilin (Math. Ann. 2003) and Zudilin (2003)","yearPosed":null,"ageNote":"The manuscript names no poser and no date for the question. Zhi-Wei Sun's preprint on the same question (arXiv:2609.04176) calls it a long-standing open problem about a constant introduced by Catalan in the nineteenth century. Year left blank rather than guessed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: $G$ is irrational. Under the hypothesis $G\\in\\mathbb Q$ the paper builds mixed determinants $\\Delta_N$ of size $48N$ from moments of two kernels, whose $\\zeta(2)$ terms cancel by Taylor contact of Chebyshev rows, so all $\\Delta_N$ are rational; a finite certificate gives $\\Delta_p\\ne0$ for large primes $p$, prime-by-prime denominator bounds give $\\liminf\\mathcal L_p>-2.29084$, and a real-place integral estimate gives $\\limsup\\mathcal L_N<-2.2909$, a contradiction. Corollaries: $4G$, the minimal volume of a two-cusped orientable hyperbolic 3-manifold, and every arithmetic hyperbolic 3-orbifold volume over $\\mathbb Q(i)$ are irrational. Not shown: an irrationality measure, transcendence, or anything about $\\beta(4),\\beta(6),\\ldots$. An earlier preprint by Zhi-Wei Sun (arXiv:2609.04176, 3 September 2026) also claims irrationality; the manuscript says it uses nothing from it, so the first claimed proof may not be this one.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, using on average about three hours of ChatGPT Pro thinking compute per result, with outputs grouped into families and manuscripts. The README's two exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region write-up, which was human-edited for readability) do not concern this family. The manuscript is credited to OpenAI alone, names no human author and has no acknowledgements. A Lean formalization of the irrationality statement is in the release's lean/ library (lean/docs/005.md). The release does not say how much human review happened before publication.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source (Catalan's constant is irrational) against the posed question, and the Lean statement. The result is not in the release's formalization catalogue (formalization.yaml), but lean/docs/005.md links ComparatorChallenges/Catalan.json, whose solution_module OAI.NumberTheory.Catalan.Main exists at the pinned commit; theorem OAI.InternalCatalan.catalan_irrational, permitted axioms propext, Quot.sound and Classical.choice. The comparator statement, read here, is Irrational (sum over j in N of (-1)^j / (2j+1)^2) over the reals, which is exactly the headline. Not rebuilt here; the comparator run and axiom check were not reproduced. The paper's argument rests on exact finite certificates (a nonvanishing certificate and rational trial coefficients for the real-place bound); the text says supplementary programs reproduce them, and I did not find such programs in the manuscript's folder. It proves qualitative irrationality only.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"One of the most famous individual irrationality questions, alongside zeta(5), and widely known outside number theory; decades of Apery-style attempts stalled on denominators. Placed with the Baum-Connes and Kothe conjecture entries at 60, below Kaplansky's zero-divisor conjecture at 62 and the Jacobian Conjecture at 65, because it is a single constant rather than a structural conjecture.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Catalan's constant is irrational","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI.InternalCatalan.catalan_irrational (solution module)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/Catalan/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Catalan.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/Catalan.lean","kind":"lean-proof"},{"label":"Lean scope note for family 005","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/005.md","kind":"lean-proof"},{"label":"Zhi-Wei Sun, Catalan's constant is irrational, arXiv:2609.04176 (earlier claimed proof, 3 Sep 2026)","url":"https://arxiv.org/abs/2609.04176","kind":"paper"},{"label":"Rivoal and Zudilin, Diophantine properties of numbers related to Catalan's constant (2003)","url":"https://doi.org/10.1007/s00208-003-0420-2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"simon-problem-7-continuum-phase-transition","name":"Simon's Problem 7 (1984): a continuum phase transition for a stable tempered pair potential, on an open interval of densities","shortName":"Simon's continuum phase transition problem","problemNumber":null,"field":"Classical statistical mechanics; continuum phase transitions","fieldGroup":"Mathematical physics","statement":"Fix a pair potential $v$ on $\\mathbb R^3$ that is stable, $\\sum_{i0$ and $f'_-(\\beta_c,\\rho)>f'_+(\\beta_c,\\rho)$. The companion gives a potential with divergent core and $o(r^{-3})$ tail, a common $\\beta_c\\in(1/2,3/2)$ on an open density interval, and says it does not meet a fixed power margin. Not shown: a transition for every sufficiently large $\\rho$ as Simon's wording asks, for Lennard-Jones or any finite-range potential, or identification of the phases. Earlier continuum transitions (Widom-Rowlinson, Lebowitz-Mazel-Presutti, recent Kac-type models) used several species, many-body terms or box-dependent interactions.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 24, 2026: one with a divergent repulsive core and an $o(r^{-3})$ tail, and one with a bounded continuous potential and an explicit power tail. Both have Lean formalizations of their main theorems.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the algebraic-decay manuscript was read against Simon's Problem 7 as printed in 1984. The potential is bounded, continuous, stable and satisfies $|\\phi(r)|\\le Cr^{-3-1/32}$ for $r\\ge1$, so it meets Simon's stability and temperedness conditions; Simon's kinetic term only adds a smooth function of $\\beta$. Lean: lean/ComparatorChallenges/RadialDensityInterval.json exists with solution module OAI.Probability.RadialTransition.DensityInterval present at the pinned commit; it is not listed in lean/formalization.yaml. Its statement OAI.RadialTransition.densityInterval was read here and states the headline (bounded continuous stable potential with that decay, a density interval around $5p/3$, one common $\\beta_c\\in[7/8,9/8]$ with a strict corner of the canonical free energy). Not rebuilt here. The companion's ContinuumTransition and RadialTransition statements are in the catalogue. The papers say the potential is engineered and is not Lennard-Jones, and do not identify the phases.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"Existence of a phase transition for a continuum gas with a pair potential is a long-standing problem of rigorous statistical mechanics, listed by Simon in 1984; scored for an engineered potential on a density interval rather than all large densities or a physical potential. Below Simon's Problem 2 on 2D localization (55) and the XY critical log correction (50), level with planar first-passage limit-shape convexity (38), above Solovej's generalized ionization conjecture (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf","sourceName":"OpenAI math release: A radial continuum phase transition with algebraic decay","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A continuum temperature singularity for a radial pair potential","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf","kind":"paper"},{"label":"Lean statement: RadialDensityInterval.lean (comparator challenge)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/RadialDensityInterval.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Probability/RadialTransition/DensityInterval.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/RadialTransition/DensityInterval.lean","kind":"lean-proof"},{"label":"Lean (companion): OAI/Probability/ContinuumTransition/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/ContinuumTransition/Main.lean","kind":"lean-proof"},{"label":"Simon, Fifteen problems in mathematical physics (1984), Problem 7","url":"https://math.caltech.edu/SimonPapers/R27.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"restriction-conjecture-r3-positive-curvature","name":"The Fourier restriction conjecture in three dimensions for positively curved surfaces: bounded-data and diagonal forms","shortName":"3D restriction, positively curved surfaces","problemNumber":null,"field":"Harmonic analysis; Fourier restriction","fieldGroup":"Analysis","statement":"For a compact smooth positively curved surface $\\Sigma\\subset\\mathbb R^3$, such as the sphere, let $E_\\Sigma f(x)=\\int_\\Sigma f(\\xi)e^{2\\pi i x\\cdot\\xi}\\,d\\sigma(\\xi)$. Stein's restriction conjecture in $\\mathbb R^3$ asks that $E_\\Sigma$ map $L^q(\\Sigma)$ to $L^p(\\mathbb R^3)$ whenever $p>3$ and $p\\ge 2q'$. Its bounded-data form asks for $L^\\infty(\\Sigma)\\to L^p(\\mathbb R^3)$ for all $p>3$ (Guth reached $p>13/4$), and its diagonal form for $L^p(\\Sigma)\\to L^p(\\mathbb R^3)$ for all $p>3$ (Wang and Wu reached $p>22/7$). The constant function on the sphere shows $p=3$ fails. Do the bounded-data and diagonal estimates hold for every $p>3$?","posedBy":"Elias M. Stein; the manuscripts date the problem to Stein's work and cite his Some problems in harmonic analysis (1979) and the Tomas-Stein theory","yearPosed":1979,"ageNote":"1979 is the date of the Stein problem survey the manuscripts cite. The restriction phenomenon is usually traced to Stein's unpublished work of the late 1960s; the manuscripts do not give that date.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Principal manuscript, Theorem 1.1: for every compact smooth positively curved $\\Sigma\\subset\\mathbb R^3$, possibly with smooth boundary, and every $33$, $s>2-6/p$. Companion: for the sphere, $L^\\infty(S^2)\\to L^p(\\mathbb R^3)$ for every $p>3$, and through Bourgain factorization the strict mixed-norm range $L^a\\to L^b$ with $b>3$, $b>2a'$. Not shown: the scaling-line endpoint $p=2q'$ that the full conjecture includes, surfaces with vanishing or mixed-sign curvature, and higher dimensions.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This family consists of two manuscripts dated September 24, 2026; the diagonal-extension manuscript is taken as principal, since its theorem contains the sphere bounded-data case. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: both main theorems were read against the restriction conjecture as the manuscripts describe it. The ranges are the full open ranges, and the sphere paper says explicitly that it makes no assertion on the scaling line. The proofs (a packet propagation theorem through an elliptic capacity, epsilon removal by sparse balls, chart decomposition for general elliptic surfaces) were not refereed. No Lean formalization. Dependence: the diagonal theorem takes as inputs the companion's packet theorem and the three-dimensional Kakeya maximal theorem from family 074, both unreviewed. The sphere paper says it also gives an independent route to that Kakeya estimate, so families 074 and 077 stand or fall partly together.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"The three-dimensional restriction conjecture is one of the central problems of harmonic analysis, with decades of exponent records (Tomas-Stein, Wolff, Tao, Bourgain-Guth, Guth, Wang-Wu). Reaching the full open range for positively curved surfaces is close to the whole three-dimensional problem, missing the scaling-line endpoint. Placed above the sum-product conjecture over the reals (55) and the Kakeya entries of family 074, below the Jacobian Conjecture (65) and the 67 percent critical-line result (68).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf","sourceName":"OpenAI math release: Diagonal Fourier extension for positively curved surfaces in three dimensions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Elliptic capacity propagation and Fourier restriction to the sphere","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf","kind":"paper"},{"label":"Input (family 074): The Kakeya maximal conjecture in three dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"david-semmes-riesz-higher-codimension","name":"The David-Semmes problem in higher codimension: bounded Riesz transforms force uniform rectifiability for 1 < n < d-1","shortName":"David-Semmes problem, higher codimension","problemNumber":null,"field":"Harmonic analysis; geometric measure theory, uniform rectifiability","fieldGroup":"Analysis","statement":"Let $\\mu$ be an $n$-Ahlfors-David regular measure on $\\mathbb R^d$ and $R_\\mu$ the $n$-dimensional Riesz transform with kernel $x/|x|^{n+1}$. David and Semmes asked whether boundedness of $R_\\mu$ on $L^2(\\mu)$ implies that $\\mu$ is uniformly $n$-rectifiable. The answer is yes for $n=1$ (Mattila-Melnikov-Verdera in the plane, curvature methods in general) and for $n=d-1$ (Nazarov-Tolsa-Volberg, 2014), whose maximum-principle argument has no known higher-codimension analogue; Mas and Tolsa characterized uniform rectifiability through $\\rho$-variation, a stronger hypothesis. Is every $n$-AD-regular measure on $\\mathbb R^d$ with $L^2$-bounded Riesz transform uniformly $n$-rectifiable when $1 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 24, 2026) is the whole family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.3 was read against the David-Semmes question in the range left open; for $d\\ge4$ and $2\\le n\\le d-2$ (all of $10$ and $M$ such that every regular $n$-AD-regular measure with truncations bounded by $C_R$ has, at every support point and admissible radius, an $M$-Lipschitz image of the $n$-ball of radius $r$ carrying mass at least $\\theta r^n$ in the ball. This states the headline (big pieces of Lipschitz images, the manuscript's definition of uniform rectifiability). Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The David-Semmes Riesz-transform problem is a central question of quantitative geometric measure theory; its codimension-one case was an Acta Mathematica paper. This entry is the remaining range, which completes it. Placed level with Brennan's conjecture (45) and Vinogradov's least quadratic nonresidue conjecture (45), above Stein's dimension-free weak-(1,1) Riesz transform problem (38), and below the Kakeya maximal conjecture in three dimensions (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Riesz transforms and uniform rectifiability in higher codimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.RieszRectifiability.quantitative_higher_codimension_riesz_rectifiability)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/RieszRectifiability/Uniformity/Rectifiability.lean","kind":"lean-proof"},{"label":"Comparator statement: RieszQuantitative.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/RieszQuantitative.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"p-converse-selmer-corank-one-every-prime","name":"The p-converse to Gross-Zagier-Kolyvagin in Selmer corank at most one, at every prime","shortName":"Selmer converse in corank 0 and 1 at every prime","problemNumber":null,"field":"Arithmetic of elliptic curves; Heegner points","fieldGroup":"Number theory","statement":"Let $E/\\mathbb{Q}$ be an elliptic curve, $p$ a prime, and $s_p(E)$ the $\\mathbb{Z}_p$-corank of the full $p^\\infty$-Selmer group. Gross-Zagier and Kolyvagin show that analytic rank $r\\le 1$ forces $\\mathrm{rank}\\,E(\\mathbb{Q})=r$ and finite Sha. The $p$-converse predicts $\\mathrm{ord}_{s=1}L(E,s)=s_p(E)$ (Keller-Yin, Conjecture A). Known cases (Skinner-Urban, Skinner, Zhang, Burungale-Castella-Skinner, Castella-Wan, Keller-Yin and others) need ordinary or supersingular reduction, residual irreducibility or Eisenstein conditions, CM, or $p>3$. If $s_p(E)=r\\in\\{0,1\\}$, does it follow for every $E$ and every $p$ that the analytic rank and Mordell-Weil rank equal $r$ and Sha is finite?","posedBy":"Converse problem developed by Skinner-Urban, Skinner and Wei Zhang; stated as the p-converse conjecture (Conjecture A) by Timo Keller and Mulun Yin","yearPosed":2014,"ageNote":"The manuscript cites Skinner-Urban (2014) and Zhang (2014) as the first converse theorems and Keller-Yin (arXiv 2410.23241, 2024) for the general conjecture. 2014 is the earliest dated reference it gives for the question; the question is older in folklore.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For every elliptic curve $E/\\mathbb{Q}$, every prime $p$ and $r\\in\\{0,1\\}$: $s_p(E)=r$ implies $\\mathrm{ord}_{s=1}L(E,s)=\\mathrm{rank}\\,E(\\mathbb{Q})=r$ and the whole of $Sha(E/\\mathbb{Q})$ is finite, with no restriction on reduction type, CM, residual representation, torsion or isogenies. It does not address coranks $\\geq 2$ and does not evaluate the BSD leading term (that is the companion entry). As an application it rederives that $X^3+Y^3=\\ell Z^3$ has rank one for primes $\\ell\\equiv 4,7,8 \\pmod 9$.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The prime p = 2 is taken from the release's Goldfeld-density manuscript (family 006).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem of the Selmer converse manuscript was read against the converse question as the manuscript and Keller-Yin state it, for coranks zero and one only. The proof was not refereed. No Lean formalization is listed for this manuscript. The case p = 2 and the auxiliary twist-density input come from another unreviewed release manuscript (family 006). The cube-sum corollary (Sylvester's primes 4, 7, 8 mod 9) was already proved by Yin and by Burungale-Tian, as the paper says, so it is not counted here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A central problem inside the BSD community with a long run of conditional results; not known outside it. Ties Connes' Rigidity Conjecture (45) and sits below the elliptic curve rank records (50) and well below the BSD formula entry it feeds.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf","sourceName":"OpenAI math release: The Selmer converse for elliptic curves at every prime","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Uses (family 006): Goldfeld's analytic density conjecture and the 2-converse","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf","kind":"paper"},{"label":"Same family: Exact Birch-Swinnerton-Dyer Formula from Low Selmer Corank","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/exact-bsd-low-selmer-corank.pdf","kind":"paper"},{"label":"Keller and Yin, p-converse theorems (Conjecture A), arXiv 2410.23241","url":"https://arxiv.org/abs/2410.23241","kind":"problem-record"},{"label":"Goldfeld's conjecture entry, which proves the p = 2 case","url":"https://vibemathed.com/problem/goldfeld-conjecture-quadratic-twists","kind":"other"},{"label":"Revised edition (October 7, 2026): Exact Birch-Swinnerton-Dyer Formula from Low Selmer Corank","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","kind":"paper"},{"label":"Revised edition (October 7, 2026): Goldfeld's analytic density conjecture and the 2-converse for elliptic curves","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","kind":"paper"},{"label":"Earlier edition (September 24, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"witnessed-symmetric-choice-strictly-stronger-cpt","name":"Lichter and Schweitzer's question: does witnessed symmetric choice strictly increase the expressive power of CPT with counting?","shortName":"Witnessed symmetric choice beats CPT","problemNumber":null,"field":"Finite model theory; descriptive complexity","fieldGroup":"Logic & foundations","statement":"Choiceless polynomial time with counting (CPT) computes invariantly with hereditarily finite sets over the input atoms and cannot choose an element of a set merely because it is nonempty. Lichter and Schweitzer introduced the extension CPT+WSC by witnessed symmetric choice: a program may pick an element of an orbit provided it constructs automorphisms certifying that the choice is symmetric, so the answer stays isomorphism invariant. In Section 7 of their paper they asked whether this extension is strictly more expressive than CPT itself. Is there a Boolean query on finite structures definable in CPT+WSC but not in CPT with counting?","posedBy":"Moritz Lichter and Pascal Schweitzer","yearPosed":2022,"ageNote":"Lichter and Schweitzer, Choiceless Polynomial Time with Witnessed Symmetric Choice (LICS 2022; arXiv 2205.14003, version 3, 2023), Section 7, cited by the manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: over a vocabulary of eight binary relations there is a CPT+WSC sentence with a single WSC operator that returns a Boolean value on every finite structure and whose true-model class is not defined by any CPT sentence, even with counting and unbounded set rank; so CPT+WSC is strictly more expressive than CPT in Lichter and Schweitzer's sense. Not shown: whether CPT+WSC captures polynomial time, or the effect of nested WSC operators, both of which the paper leaves open.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. This entry's manuscript (September 24, 2026) is the companion of the CPT noncapture paper in the same family and has a Lean formalization of its main theorem.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Lichter-Schweitzer's Section 7 question. Lean: lean/ComparatorChallenges/WitnessedChoice.json exists with solution module OAI.Computability.WitnessedChoice.Main present at the pinned commit; it is not listed in lean/formalization.yaml. Its statement WitnessedChoice.main asserts a sentence with exactly one WSC occurrence, Boolean on all inputs, whose true-model class differs from that of every CPT sentence. This is the headline claim, relative to the challenge file's own encoding of both formalisms, which was not audited here. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A young, specialist question from the paper that introduced witnessed symmetric choice; it becomes much less surprising once CPT is known not to capture polynomial time. Below Simon's extendable shellability conjecture (25) and the Belinskaya theorem for flows (20), level with the Simonovits product conjecture (18).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean statement: WitnessedChoice.lean (comparator challenge)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/WitnessedChoice.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Computability/WitnessedChoice/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/WitnessedChoice/Main.lean","kind":"lean-proof"},{"label":"Lichter and Schweitzer, Choiceless Polynomial Time with Witnessed Symmetric Choice (arXiv 2205.14003)","url":"https://arxiv.org/abs/2205.14003","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"zariski-multiplicity-conjecture","name":"Zariski's multiplicity conjecture: does embedded topology determine the multiplicity of a hypersurface singularity?","shortName":"Zariski's multiplicity conjecture","problemNumber":null,"field":"Singularity theory; topology of hypersurface singularities","fieldGroup":"Geometry & topology","statement":"For a reduced convergent power series $f\\in\\mathbb{C}\\{z_1,\\ldots,z_N\\}$ with $f(0)=0$, the multiplicity of the hypersurface germ $V(f)$ is $\\mathrm{ord}_0 f$, the least degree of a monomial in $f$. Two germs are ambiently homeomorphic if a homeomorphism of neighbourhoods of $0$ in $\\mathbb{C}^N$ fixing $0$ carries one zero set onto the other. For plane curves the embedded topology determines multiplicity, and by A'Campo and Le a germ ambiently homeomorphic to a smooth one is smooth. Zariski asked in 1971 whether this holds in general: if $(\\mathbb{C}^N,V(f),0)$ and $(\\mathbb{C}^N,V(g),0)$ are homeomorphic, must $V(f)$ and $V(g)$ have the same multiplicity?","posedBy":"Oscar Zariski, 'Some open questions in the theory of singularities', Bull. AMS 77 (1971)","yearPosed":1971,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Two independent counterexample constructions. September 24: for some finite $N>3$ divisible by eight, reduced weighted homogeneous real polynomials $f_1,f_2$ on $\\mathbb{C}^N$ with isolated critical points, $\\mathrm{ord}_0f_1=2$, $\\mathrm{ord}_0f_2=3$, and $(\\mathbb{C}^N,V(f_1),0)\\cong(\\mathbb{C}^N,V(f_2),0)$; the germs are also topologically right equivalent, so the right-equivalence version and its mod-2 version (Sampaio 2025, Conjectures 2 and Z mod 2) fail too. September 27: reduced convergent germs $f_4,f_5$ on $\\mathbb{C}^4$ with isolated critical points, orders $4$ and $5$, ambiently homeomorphic zero sets. Not addressed: hypersurfaces in $\\mathbb{C}^3$, and metric (bi-Lipschitz) or family versions, where positive results stand.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two manuscripts in the family, September 24 and September 27, 2026, which the September 27 paper describes as independent arguments. The September 24 paper ships Python scripts (standard library only) that check its finite integer arithmetic; the README says they do not cover the lattice-theoretic or topological arguments.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of each manuscript was read against Zariski's question as posed (reduced germs, ambient homeomorphism of pairs, multiplicity as order). The proofs were not refereed. No Lean formalization is listed in lean/formalization.yaml and lean/docs has no page for this family. The September 24 construction runs through Levine's classification of simple knots, strong approximation for special unitary groups and the Thom-Sebastiani and Sakamoto formulas; the ambient dimension N is not computed, only shown to exist and be divisible by eight. Its Python scripts check the finite integer arithmetic only, and were not run here. Both papers state that the surface case in C^3 is untreated.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"Zariski's 1971 question is the best-known open problem in the topology of hypersurface singularities, with a survey literature and many partial positive results (plane curves, families, quasihomogeneous surfaces, bi-Lipschitz multiplicity two). Famous well beyond one seminar but mainly inside algebraic and complex geometry. Placed just below Bloch's conjecture for surfaces with p_g = 0 (55) and above Kuznetsov's rationality conjecture (45) and the Campana-Peternell conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf","sourceName":"OpenAI math release: Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Ambiently homeomorphic isolated hypersurface germs in C^4 with multiplicities four and five","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026/paper.pdf","kind":"paper"},{"label":"Verification scripts for the finite arithmetic (September 24 paper)","url":"https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/verification","kind":"other"},{"label":"Zariski, Some open questions in the theory of singularities, Bull. AMS 77 (1971)","url":"https://doi.org/10.1090/S0002-9904-1971-12729-5","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ostmann-inverse-goldbach-conjecture","name":"Ostmann's inverse Goldbach conjecture","shortName":"Ostmann's inverse Goldbach conjecture","problemNumber":null,"field":"Additive number theory; sumsets and the primes","fieldGroup":"Number theory","statement":"Two subsets of $\\mathbb N_0$ are asymptotically equal if their symmetric difference is finite. Ostmann conjectured that the set $\\mathcal P$ of primes is asymptotically additively indecomposable. Is it true that there are no sets $A,B\\subseteq\\mathbb N_0$, each with at least two elements, such that $A+B$ differs from $\\mathcal P$ in only finitely many elements?","posedBy":"Hans-Heinrich Ostmann (Additive Zahlentheorie, 1956, p. 13); restated by Elsholtz and Harper (2015, Conjecture 1.2)","yearPosed":1956,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $A,B\\subseteq\\mathbb N_0$ have $|A|,|B|\\ge2$, then $(A+B)\\triangle\\mathcal P$ is infinite. After the Laffer-Mann reduction to two infinite summands, the proof combines residue restrictions at every prime, decorrelation from translated multiplicative characters, prime coverage and a positive statistic averaged over arrangements; it does not use Green and Harper's conditional inverse-sieve route. In particular, if $A+B$ contains every large prime it contains infinitely many composites. No quantitative version is claimed.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Ostmann's conjecture. lean/docs/013.md lists OstmannPrimes.json (OAI.Ostmann.main, module OAI.NumberTheory.Ostmann.Main) and OstmannComplete.json (OAI.Ostmann.inverseGoldbach and OAI.Ostmann.twoInfiniteSummandsImpossible, module OAI.NumberTheory.Ostmann.Complete). Both solution files exist at the pinned commit; neither challenge is in the formalization catalogue formalization.yaml. The statement was read here: for all sets A, B of natural numbers with at least two elements each, the symmetric difference of A+B and the primes is infinite. That is exactly the conjecture. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A named conjecture of additive number theory, open since 1956 and attacked with sieve methods by Hornfeck, Elsholtz, Green-Harper and Croot-Mao-Pohoata-Yip. Well known in that community, little known beyond it. Placed just below The Largest Sum-Free Subset of the Lattice Cube (30), above Goldbach's conjecture for the Liouville function (22), far below The Sum-Product Conjecture over the Reals (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf","sourceName":"OpenAI math release: The additive indecomposability of the primes","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Ostmann.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/Ostmann/Main.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.Ostmann.inverseGoldbach)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/Ostmann/Complete.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kuznetsov-rationality-conjecture","name":"Kuznetsov's rationality conjecture for cubic fourfolds","shortName":"Kuznetsov's conjecture on cubic fourfolds","problemNumber":null,"field":"Algebraic geometry: rationality, derived categories, cubic fourfolds","fieldGroup":"Geometry & topology","statement":"For a smooth complex cubic fourfold $X\\subset\\mathbf P^5$, the Kuznetsov component $\\mathrm{Ku}(X)$ is the right orthogonal of $\\mathcal O_X,\\mathcal O_X(1),\\mathcal O_X(2)$ in $D^b(X)$, a K3 category. All known rational cubics (Pfaffian, containing a plane with a section, divisors $\\mathcal C_{26},\\mathcal C_{38},\\mathcal C_{42}$) have $\\mathrm{Ku}(X)$ equivalent to the derived category of a K3 surface, and by Addington-Thomas this matches Hassett's lattice condition for associated K3 surfaces. Kuznetsov conjectured that $X$ is rational if and only if $\\mathrm{Ku}(X)\\simeq D^b(S)$ for a projective K3 surface $S$. Is that true?","posedBy":"Alexander Kuznetsov (Derived categories of cubic fourfolds, 2010, Conjecture 1.1)","yearPosed":2010,"ageNote":"Kuznetsov's paper appeared in the 2010 Progress in Mathematics volume (arXiv 2008). The Hodge-theoretic associated-K3 form goes back to Hassett (2000) and Harris-Hassett folklore; the manuscript cites Huybrechts' 2025 survey for its explicit form.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that for every sufficiently large admissible Hassett discriminant $d$ a very general cubic fourfold of discriminant $d$ is irrational although its Kuznetsov component is the derived category of a K3 surface and it has an associated polarized K3 in Hassett's Hodge-theoretic sense. This disproves Kuznetsov's conjecture and the sufficiency of the associated-K3 criterion (and of Fano variety birational to a K3 Hilbert square). The threshold is ineffective; no specific discriminant is shown irrational, the necessity direction is untouched, and stable rationality is not decided. These are not the first irrational cubic fourfolds: the paper cites Katzarkov-Kontsevich-Pantev-Yu (very general cubic) and Fay (some special divisors).","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2 and 1.3 of the TeX source read against the posed problem. For every admissible $d$ above an ineffective $d_0$, a very general cubic in $\\mathcal C_d$ is irrational yet has $\\mathrm{Ku}(X)\\simeq D^b(S)$; this refutes the categorical-implies-rational direction, which is enough to disprove the conjecture. The categorical equivalence comes from published work (Addington-Thomas; Bayer, Lahoz, Macri, Nuer, Perry, Stellari); the new part is irrationality, via an additive invariant of birational maps computed by weak factorization with Gromov-Witten operators and by Sarkisov links. Not refereed. No Lean formalization is listed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A central conjecture of the rationality programme for cubic fourfolds, famous within algebraic geometry and derived categories but younger and narrower than the classical problems. Placed with Connes' rigidity conjecture and Smale's mean value conjecture at 45, above the period-index conjecture at 30, below Griffiths' ample-bundle conjecture at 55.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026.pdf","sourceName":"OpenAI math release: Irrational cubic fourfolds with geometric K3 categories","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kuznetsov, Derived categories of cubic fourfolds (arXiv 0808.3351), Conjecture 1.1","url":"https://arxiv.org/abs/0808.3351","kind":"problem-record"},{"label":"Huybrechts, The K3 category of a cubic fourfold: an update (2025), Conjecture 2.1","url":"https://doi.org/10.1007/s13366-025-00788-9","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hilbert-16-uniform-bound-limit-cycles","name":"The existential Hilbert sixteenth problem: a uniform bound on the number of limit cycles of planar polynomial vector fields","shortName":"Existential Hilbert 16th problem","problemNumber":null,"field":"Planar polynomial vector fields; limit cycles","fieldGroup":"Differential equations","statement":"Hilbert's sixteenth problem asks, in its second part, for the maximum number and the relative position of the limit cycles of a real planar polynomial vector field $\\dot x=P(x,y)$, $\\dot y=Q(x,y)$ of degree $d$. Its existential form, which Ilyashenko's centennial survey separates from finiteness for each individual field, asks whether that maximum is finite at all. Even the quadratic case $d=2$ was open. Is there, for every $d\\ge1$, a finite number $H(d)$ such that every real planar polynomial vector field of degree at most $d$ has at most $H(d)$ limit cycles?","posedBy":"David Hilbert (1900), Problem 16, second part; the uniform-boundedness form as distinguished by Yu. Ilyashenko, Centennial history of Hilbert's 16th problem (Bull. AMS, 2002)","yearPosed":1900,"ageNote":"Posed at the 1900 Paris congress; the manuscript cites the 1902 Bulletin AMS translation. Finiteness for each single field (Dulac's problem) is a different, weaker statement, proved by Ecalle and Ilyashenko around 1991-1992. The uniform bound was open in every degree d >= 2.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for each $d\\ge1$ there is a finite integer $B(d)$ such that every real planar polynomial vector field of degree at most $d$ has at most $B(d)$ limit cycles in the whole plane, with no restriction on coefficients, location, stability or hyperbolicity. The proof works with matching systems of local passages and shows that each fixed auxiliary system has finitely many isolated solutions, uniformly as coefficients vary. It does NOT give an explicit or effective $B(d)$, so it says nothing about the value of $H(2)$, about Smale's question whether $H(d)\\le Kd^q$, or about the relative positions of cycles asked for in Hilbert's full second part. The exact quintic Lienard bound in the same family is a separate entry.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This family consists of two manuscripts dated September 24, 2026; the uniform-bound manuscript is the principal one. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the principal manuscript were read against the problem as the manuscript's own historical section describes it. The theorem states the uniform bound in full generality: every degree, the whole plane, no genericity, cycles counted as geometric images regardless of stability or hyperbolicity. The proof (eleven sections of asymptotic separation, implicit-function calculus, subanalytic preparation and a projection-counting argument) was not refereed. There is no Lean formalization of this manuscript; the family's Lean file covers only the quintic Lienard companion. The manuscript says the bound is not effective. It also notes that Yeung (2025) reported a gap in one leading-term argument of Ilyashenko's 1991 monograph on individual finiteness, and states that its own proof does not use individual finiteness as a premise. A claim of this size should wait for an expert reader before it is shown at any tier.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":85,"significanceNote":"Hilbert's sixteenth problem is among the most famous problems on Hilbert's list and is also Smale's thirteenth problem; a uniform bound was unknown even for quadratic fields after more than a century. Scored just below the Navier-Stokes Millennium entry (87), because only the non-effective existential part is answered, above the unforced 3D Euler blowup entry (80), and far above the Jacobian Conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf","sourceName":"OpenAI math release: Uniform bounds for planar polynomial limit cycles","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Two limit cycles for quintic Lienard systems","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"phangara-s-hadamard-order-six","name":"Phangara's conjecture: every order-six S-Hadamard matrix is equivalent to Tao's cubic matrix","shortName":"Order-six S-Hadamard classification","problemNumber":null,"field":"Complex Hadamard matrices; S-Hadamard matrices","fieldGroup":"Combinatorics","statement":"Lisonek (2019) called a complex Hadamard matrix $H$ an S-Hadamard matrix if its entrywise square $H^{\\circ2}$ is also Hadamard. Tao's cubic matrix of order six, with cube-root-of-unity entries, is an example. Phangara (MSc thesis, Simon Fraser University, 2026, Conjecture 3.1.6) conjectured that it is the only one in order six. Is every S-Hadamard matrix of order six equivalent, under row and column permutations and unit phase multiplications, to Tao's cubic matrix?","posedBy":"Jasleen Phangara (Conjecture 3.1.6, MSc thesis, Simon Fraser University, 2026)","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Proposition 3.1: if $H$ and its entrywise square are both complex Hadamard matrices of order six, then $H$ is equivalent to Tao's cubic matrix; in particular all its dephased entries are cube roots of unity. The manuscript notes that uniqueness of the cube-root class was already known from Butson classifications (Lampio-Ostergard-Szollosi; Liang-Chen-Long-Qiu); the new step derives cube-root entries from the two Hadamard conditions for arbitrary phases. It says nothing about other orders.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The result is Proposition 3.1 of the companion Fourier-certificate manuscript, used there as a lemma.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Proposition 3.1 of the manuscript was read against Phangara's conjecture as the manuscript quotes it; it states that if $H$ and $H^{\\circ2}$ are order-six Hadamard matrices then $H$ is equivalent to Tao's matrix. The proof is a short Newton-identity argument on row ratios. The Lean challenge HadamardCubeFiber formalizes only a supporting cancellation lemma (lean/docs/266.md says so), not this classification, so the entry is Unreviewed. The thesis itself was not opened here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A conjecture from a 2026 master's thesis, settled as a lemma by a short argument and close to known Butson classifications. Placed level with the lowest catalog tier such as Positivity on Deligne-Mumford stacks without the torsion-free hypothesis (5), below Thue-Morse first-occurrence formulas (6).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf","sourceName":"OpenAI math release: Exact Fourier certificates for complex Hadamard matrices of order six","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"entropy-rate-dimension-self-similar-measures","name":"The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture)","shortName":"Entropy-rate dimension / exact overlaps conjecture","problemNumber":null,"field":"Fractal geometry; dimension of self-similar measures","fieldGroup":"Analysis","statement":"Let $\\varphi_i(x)=r_ix+t_i$ ($0<|r_i|<1$) be finitely many similarities of $\\mathbb R$ and $p$ a positive probability vector, with self-similar measure $\\mu=\\sum_ip_i(\\varphi_i)_*\\mu$. Always $\\dim_H\\mu\\le\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$, where $h_{\\mathrm{RW}}=\\lim H(G_n)/n$ is the entropy rate of the random composed map $G_n$ and $\\chi=-\\sum p_i\\log|r_i|$. The exact overlaps conjecture, associated with Simon (1996) and central since Hochman's 2014 work, asks whether equality $\\dim_H\\mu=\\min\\{1,H(p)/\\chi\\}$ holds whenever no two distinct words of equal length give the same map. The entropy-rate dimension conjecture (Varju's survey, Conjecture 3) extends this to systems with exact overlaps. Does $\\dim_H\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$ hold for every finite self-similar system on the line?","posedBy":"Karoly Simon (exact overlaps question, 1996); entropy-rate form as Conjecture 3 in Peter Varju's survey (arXiv:2509.22042)","yearPosed":1996,"ageNote":"The manuscript associates the set version of the exact-overlap question with Simon (1996) and cites the entropy-rate formulation as Conjecture 3 of Varju's survey (arXiv:2509.22042, cited as v2, 2026). The year given is the earlier origin.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every finite system of contracting similarities of $\\mathbb R$ and every positive probability vector, $\\dim_H\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$, with no separation or arithmetic assumption; exact overlaps, repeated maps, negative and unequal ratios are allowed. Without exact overlaps this gives $\\min\\{1,H(p)/\\chi\\}$ (the exact overlaps conjecture in dimension one) and the attractor formula $\\dim_HK=\\min\\{1,s\\}$ with $\\sum|r_i|^s=1$. It says nothing about absolute continuity (for instance of Bernoulli convolutions) and does not treat higher-dimensional or non-conformal systems.","claimIssueNote":null,"solveDate":"2026-09-24","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof uses Feng-Hu exact dimensionality as its only external dimension theorem.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against both formulations. It states $\\dim_H\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$ for every finite indexed family with $0<|r_i|<1$ and every strictly positive weight vector, signed and unequal ratios and exact overlaps allowed; with no exact overlaps $H(G_n)=nH(p)$, giving the exact overlaps conjecture on the line. Lean: lean/formalization.yaml lists comparator SelfSimilar with declaration OAI.EntropyRateDimension.entropy_rate_dimension. The statement ComparatorChallenges/SelfSimilar.lean was read here: for a finite system with nonzero ratios of modulus below one and positive weights summing to one, and any probability measure satisfying the self-similarity equation, the lower Hausdorff dimension (infimum of dimH over positive-measure Borel sets) equals min(1, entropy rate / Lyapunov exponent), both in base 2, the entropy rate defined as the infimum of $H(G_{n})/n$ over complete affine maps. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The exact overlaps conjecture is the best known open problem in the dimension theory of self-similar measures, the subject of Hochman's Annals work and Varju's ICM-level program, and the entropy-rate form strengthens it. Placed level with Bloch's conjecture for surfaces with pg = 0 (55), above Singular Bernoulli convolutions at non-Pisot parameters (42), below Sinai's standard-map conjecture (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf","sourceName":"OpenAI math release: The entropy-rate dimension formula for self-similar measures on the line","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.EntropyRateDimension.entropy_rate_dimension)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MeasureTheory/SelfSimilar/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SelfSimilar.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SelfSimilar.lean","kind":"lean-proof"},{"label":"Comparator statement: SelfSimilarCorollaries.lean (no-overlap and attractor forms)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SelfSimilarCorollaries.lean","kind":"lean-proof"},{"label":"Varju, Entropy rates in the dimension theory of self-similar measures (Conjecture 3)","url":"https://arxiv.org/abs/2509.22042","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dolgachev-weisfeiler-affine-fibration","name":"The Dolgachev-Weisfeiler affine-fibration conjecture","shortName":"Dolgachev-Weisfeiler conjecture","problemNumber":null,"field":"Affine algebraic geometry, affine fibrations","fieldGroup":"Geometry & topology","statement":"An $\\mathbb A^r$-fibration is a morphism whose fiber over every point is affine $r$-space over the residue field. Dolgachev and Weisfeiler (1974, Section 3.8.5) predicted that an affine morphism to a normal, locally noetherian, integral base all of whose fibers are $\\mathbb A^r$ over their residue fields is Zariski-locally trivial, that is, locally a product with $\\mathbb A^r$. Is every such affine-space fibration over a normal integral base, for instance over $\\mathbb A^1$ or $\\mathbb A^2$ over $\\mathbb C$, a Zariski-locally trivial $\\mathbb A^r$-bundle?","posedBy":"B. Yu. Weisfeiler and I. V. Dolgachev, Unipotent group schemes over integral rings, Math. USSR-Izv. 8 (1974), Section 3.8.5","yearPosed":1974,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 7.1: for the fourfold $X=\\mathrm{Spec}\\,A$ of the cancellation counterexample, the maps $\\pi:X\\to\\mathbb A^1$ (given by $p$) and $\\rho=(p,H):\\mathbb A^5\\to\\mathbb A^2$ are smooth surjective morphisms of relative dimension three whose fiber over every point is $\\mathbb A^3$ over the residue field, yet neither is Zariski-locally trivial. Local triviality would make $A$ a locally polynomial algebra over $\\mathbb C[p]$ and hence, by Bass-Connell-Wright and Quillen-Suslin, a polynomial ring, contradicting Theorem 1.1. So the conjecture fails in relative dimension three over the smooth bases $\\mathbb A^1$ and $\\mathbb A^2$. Relative dimension two is not addressed.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, the introduction's paragraph on affine-space fibrations and Corollary 7.1 with its proof outline, read against the conjecture as the manuscript cites it (Dolgachev-Weisfeiler 1974, Section 3.8.5). The deduction itself is short given Theorem 1.1. Not Lean-checked as stated: the release formalizes the non-polynomiality and stabilization of A (Comparator challenge ComplexCancellation, which the deduction rests on), but its scope note says the further consequences are not included, and the fibration statement has no Lean counterpart. The formal check covers the key input only.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A well-known structural conjecture in affine algebraic geometry, famous within that community and tied to the cancellation and residual-coordinate problems, but little known outside it. Tied with the Campana-Peternell conjecture (35), above the Banks-Martin conjecture (25), below the Kuznetsov rationality conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An explicit failure of complex affine-space cancellation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Weisfeiler-Dolgachev 1974, Math. USSR-Izv. 8","url":"https://doi.org/10.1070/IM1974v008n04ABEH002127","kind":"problem-record"},{"label":"Lean Comparator challenge ComplexCancellation (covers the key input only)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ComplexCancellation.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"iyengar-ma-walker-dutta-char-p-domains","name":"The Iyengar-Ma-Walker Dutta multiplicity conjecture for complete local domains of positive characteristic","shortName":"IMW conjecture, char p domains","problemNumber":null,"field":"Commutative algebra; Dutta multiplicity of short complexes","fieldGroup":"Algebra","statement":"Over a $d$-dimensional Noetherian local ring $R$, a short complex is a finite free complex in degrees $0,\\dots,d$ with finite-length homology and nonzero zeroth homology. In characteristic $p>0$ its Dutta multiplicity is $\\chi_\\infty(F)=\\lim_n p^{-nd}\\sum_i(-1)^i\\ell(H_i(\\Phi^nF))$, where $\\Phi^n$ raises differential entries to $p^n$-th powers. Iyengar, Ma and Walker (2022, Conjecture 8.1) conjectured $\\chi_\\infty(F)\\ge e(R)$ for short complexes over complete local rings, which would imply Lech's conjecture (their Proposition 8.3). For every complete Noetherian local domain of characteristic $p>0$ and every short complex $F$, is $\\chi_\\infty(F)\\ge e(R)$?","posedBy":"Srikanth B. Iyengar, Linquan Ma and Mark E. Walker, Multiplicities and Betti numbers in local algebra via lim Ulrich points, Algebra & Number Theory 16 (2022), Conjecture 8.1","yearPosed":2022,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Corollary of the uniform complex estimate: for a complete Noetherian local domain $R$ of characteristic $p>0$ and every short complex $F$ over $R$, the Dutta multiplicity exists and $\\chi_\\infty(F)\\ge e(R)$. This is the complete-domain, positive-characteristic case of IMW Conjecture 8.1; the general conjecture for all complete local rings (and mixed characteristic) is not claimed. The paper notes it is a consequence of, not an input to, its proof of Lech's conjecture.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary (Dutta domain case) of the Lech manuscript was read against IMW Conjecture 8.1 as the paper cites it; it is the complete-domain, positive-characteristic case only. The challenge is not in the formalization catalogue (lean/formalization.yaml); it is found through lean/docs/194.md, and its solution module OAI.RingTheory.Multiplicity.Main (importing DuttaDomain) exists at the pinned commit. ComparatorChallenges/DuttaDomain.lean was read here: for every complete Noetherian local domain $D$ of characteristic $p$ and every short complex $F$ (free, finite, bounded in degrees $-d..0$, finite-length homology, nonzero $H^0$) it asserts finite-length homology of all Frobenius iterates, convergence of the Dutta sequence, and $e(D)\\le\\chi_\\infty(F)$. This states the corollary. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A recent specialist conjecture, case of domains in positive characteristic only, mainly of interest as a route to Lech's conjecture. Placed level with Carlson's associated-prime depth conjecture (15) and the BFLN initial form conjecture (15), below the Arnold corank problem (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Lech's multiplicity conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Lech.dutta_domain)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RingTheory/Multiplicity/DuttaDomain.lean","kind":"lean-proof"},{"label":"Comparator statement: DuttaDomain.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DuttaDomain.lean","kind":"lean-proof"},{"label":"Iyengar, Ma, Walker 2022 (DOI)","url":"https://doi.org/10.2140/ant.2022.16.1213","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"artin-k-pi-1-conjecture","name":"The K(pi,1) conjecture for Artin groups of finite rank","shortName":"Artin K(pi,1) conjecture","problemNumber":null,"field":"Geometric group theory, Artin groups","fieldGroup":"Geometry & topology","statement":"For a Coxeter matrix $M$ on a finite set $S$, the Artin group $A_M$ is the fundamental group of the quotient of the complexified hyperplane complement of the Coxeter group $W$ by $W$, and of the finite Salvetti complex $X(W,S)$, which has one cell per spherical subset of $S$. Deligne (1972) proved asphericity when $W$ is finite; Charney-Davis proved the FC and dimension-two cases, Paolini-Salvetti the affine case, Huang-Przytycki dimension three. The $K(\\pi,1)$ conjecture, attributed to Arnol'd, Brieskorn, Pham and Thom and listed as Problem 1 in Charney's Artin-group problem list, asserts asphericity in general. Is the Salvetti complex (equivalently the hyperplane-complement quotient) a $K(A_M,1)$ for every Coxeter matrix?","posedBy":"Attributed to Arnol'd, Brieskorn, Pham and Thom; stated in Brieskorn's Bourbaki seminar (LNM 317, 1973); Problem 1 in R. Charney, Problems related to Artin groups","yearPosed":1973,"ageNote":"The manuscript cites Brieskorn's Seminaire Bourbaki expose (1971/72, published 1973, Section 2) for the question and Paolini for the conventional attribution to Arnol'd, Brieskorn, Pham and Thom; it gives no earlier date.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every Coxeter matrix on a finite set, with arbitrary finite or infinite labels and connected or disconnected diagrams, the universal cover of the full Salvetti complex $X(W,S)$ is contractible, so $\\pi_n(X)=0$ for $n\\ge2$. The proof contracts the poset of spherical cosets by a well-ordered vertex filtration, using a framed zigzag-algebra braid action whose layer multiplicities define a harmonic height. Consequences recorded: torsion-freeness, the center $Z(A)\\cong\\mathbb Z^k$, and Dobrinskaya's monoid equivalence. Infinite rank is not covered.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the conjecture as cited (Brieskorn 1973, Charney-Davis 1995). The categorical and harmonic-height arguments were not refereed. Lean-checked on the release's Comparator challenge HarmonicArtin (declaration OAI.HarmonicArtin.salvetti_cover_contractible, listed in lean/formalization.yaml). Its statement was read here: for every Mathlib CoxeterMatrix on a finite type it asserts ContractibleSpace of the geometric realization of the nerve of a face preorder on pairs (Artin element, spherical subset). That is a combinatorial model of the Salvetti universal cover; the identification of this model with the topological universal cover is classical and is not part of the formal statement. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"The K(pi,1) conjecture is the central open problem about Artin groups, open for half a century with a long chain of partial results (Deligne, Charney-Davis, Paolini-Salvetti). Just below Cannon's conjecture (62), which has similar standing in geometric group theory, tied with Kothe's conjecture (60), below the Jacobian Conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Harmonic heights and the Artin K(pi,1) conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge HarmonicArtin (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/HarmonicArtin.lean","kind":"lean-proof"},{"label":"Parabolic intersections in Artin groups (companion, same harmonic-height method)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf","kind":"paper"},{"label":"Charney, Problems related to Artin groups (Problem 1)","url":"https://people.brandeis.edu/~charney/papers/Artin_probs.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"katok-entropy-rigidity-conjecture","name":"Katok's entropy rigidity conjecture in dimension at least three","shortName":"Katok entropy rigidity conjecture","problemNumber":null,"field":"Smooth dynamics; geodesic flows in negative curvature","fieldGroup":"Analysis","statement":"Let $(M,g)$ be a closed connected Riemannian manifold with strictly negative sectional curvature and geodesic flow $\\phi_t$ on the unit tangent bundle. The variational principle gives $h_{m_L}(\\phi_1)\\le h_{top}(\\phi_1)$, where $m_L$ is normalized Liouville measure. Katok (1982) proved for surfaces that equality forces constant curvature, and proved the higher-dimensional statement within the conformal class of a locally symmetric metric; later work gave only local rigidity near real or complex hyperbolic metrics. Katok conjectured that in every dimension equality of the Liouville entropy and the topological entropy characterizes locally symmetric metrics. Does $h_{m_L}=h_{top}$ hold for a negatively curved closed manifold only when $g$ is locally symmetric?","posedBy":"Anatole Katok","yearPosed":1982,"ageNote":"Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems 2 (1982), p. 347, cited by the manuscript as the formulation; the surface case is proved there.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a closed connected smooth Riemannian manifold of dimension $n\\ge3$ with strictly negative sectional curvature, normalized Liouville measure has maximal entropy for the geodesic flow if and only if the metric is locally symmetric (real, complex, quaternionic or Cayley hyperbolic, any scale). When the metric is not locally symmetric, the measure of maximal entropy and Liouville measure are mutually singular. It does not treat nonpositive curvature, Anosov geodesic flows without negative curvature, or the related volume-entropy (Besson-Courtois-Gallot) setting.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family is a single manuscript (September 23, 2026).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Katok's conjecture. For $n\\ge3$ and strictly negative sectional curvature it states $h_{m_L}(\\phi_1)=h_{top}(\\phi_1)$ if and only if $\\nabla R=0$, that is, the universal cover is a rank-one symmetric space of noncompact type at some scale. This is the conjecture in all dimensions where it was open (Katok proved $n=2$). The proof derives global smoothness of the stable and unstable distributions from entropy equality via normal forms, formal symmetry algebras and quantitative realization, then applies Benoist-Foulon-Labourie; it was not refereed. There is no Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"Katok's conjecture has been the guiding rigidity question for geodesic flows since 1982, open in every dimension above two except near symmetric metrics. Placed below Sinai's standard map entropy conjecture (58) and Rokhlin's multiple-mixing problem (62), level with the Lipschitz isomorphism problem for separable Banach spaces (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Entropy equality and local symmetry in negative curvature","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Katok, Entropy and closed geodesics (1982)","url":"https://doi.org/10.1017/S0143385700001656","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"separable-quotient-problem","name":"The separable quotient problem for Banach spaces","shortName":"Separable quotient problem","problemNumber":null,"field":"Banach space theory; set-theoretic functional analysis","fieldGroup":"Analysis","statement":"Every infinite-dimensional Banach space has a separable infinite-dimensional closed subspace. The separable quotient problem asks for the dual statement: does every infinite-dimensional Banach space $X$ (real or complex) admit a bounded linear surjection onto a separable infinite-dimensional Banach space, equivalently a closed subspace $M$ with $X/M$ separable and infinite dimensional? Positive answers were known for separable spaces (Johnson-Rosenthal), dual spaces (Argyros-Dodos-Kanellopoulos), spaces of density below $\\mathfrak b$ (Saxon-Sanchez Ruiz), and, consistently, densities at least $\\aleph_\\omega$ under large cardinals (Dodos-Lopez-Abad-Todorcevic). Does every infinite-dimensional Banach space have a separable infinite-dimensional quotient?","posedBy":"Traditionally attributed to Stefan Banach and Stanislaw Mazur; early written formulation by H. P. Rosenthal","yearPosed":1969,"ageNote":"The manuscript cites Brech (2025, p. 74): the problem is not stated explicitly in Banach's 1932 book, and Rosenthal's 1969 paper is an early written formulation. It is commonly dated to the 1930s by attribution.","solveType":"independent","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $\\mathbb F\\in\\{\\mathbb R,\\mathbb C\\}$, ZFC proves that if $2^{\\aleph_0}$ is real-valued measurable then every infinite-dimensional Banach space over $\\mathbb F$ has a separable infinite-dimensional quotient, and that under CH there is a counterexample of density $\\aleph_1$. So the problem is independent of ZFC if ZFC plus a measurable cardinal is consistent; restricted to density $\\aleph_1$ it is independent relative to ZFC alone (Corollary 1.2, with Martin's axiom). It does not give a ZFC answer, does not remove the measurable-cardinal strength for the unrestricted statement, and leaves open which other axioms decide it.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family is a single manuscript (September 23, 2026).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the problem. ZFC proves (i) if the continuum is real-valued measurable, every infinite-dimensional Banach space over $\\mathbb R$ or $\\mathbb C$ has a separable infinite-dimensional quotient, and (ii) under CH some space has none; hence independence relative to a measurable cardinal, and full independence for density $\\aleph_1$ relative to ZFC. The challenge lean/ComparatorChallenges/SeparableQuotientNegative.json and its solution module OAI.Analysis.SeparableQuotients.Main exist at the pinned commit (found via lean/docs/323.md) but are not in the formalization catalogue. The statement was read here: CH implies the separable-quotient assertion fails over both $\\mathbb R$ and $\\mathbb C$. That is narrower than the headline: the positive direction and the independence conclusion are not formalised. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The independence result is relative to the consistency of a measurable cardinal. Listed as Unreviewed rather than Lean-checked because its formal statement covers only one direction of the independence claim (CH implies a counterexample).","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The separable quotient problem is among the oldest and best-known open problems of Banach space theory. Placed just below the Lipschitz isomorphism problem for separable Banach spaces (52) and the hyperinvariant subspace problem on Hilbert space (52), and above the Transitive Algebra Problem (46).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Relative independence of the separable quotient problem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.SeparableQuotient.negative_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/SeparableQuotients/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SeparableQuotientNegative.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SeparableQuotientNegative.lean","kind":"lean-proof"},{"label":"Brech, The bounding number b and quotients in Banach spaces (survey, 2025)","url":"https://www.ime.usp.br/~brech/publ/BrechSurveyB.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"zarankiewicz-crossing-number-conjecture","name":"Zarankiewicz's crossing-number conjecture (Turan's brick factory problem)","shortName":"Zarankiewicz crossing-number conjecture","problemNumber":null,"field":"Topological graph theory; crossing numbers","fieldGroup":"Combinatorics","statement":"Turan's brick factory problem (1944) asks for the least number of crossings in a plane drawing of the complete bipartite graph $K_{m,n}$, crossings counted as points with edges drawn as simple arcs. Zarankiewicz (1954/55) gave a drawing with $\\lfloor\\frac m2\\rfloor\\lfloor\\frac{m-1}2\\rfloor\\lfloor\\frac n2\\rfloor\\lfloor\\frac{n-1}2\\rfloor$ crossings and a proof of optimality that turned out to contain a gap, leaving the formula as a conjecture. Kleitman proved it when $\\min(m,n)\\le6$, Woodall checked $K_{7,7}$ and $K_{7,9}$, and flag algebras gave about $0.91$ of the value asymptotically. Is $\\mathrm{cr}(K_{m,n})=\\lfloor\\frac m2\\rfloor\\lfloor\\frac{m-1}2\\rfloor\\lfloor\\frac n2\\rfloor\\lfloor\\frac{n-1}2\\rfloor$ for all positive $m,n$?","posedBy":"Paul Turan (brick factory problem, 1944, recounted in 1977); formula and flawed proof by Kazimierz Zarankiewicz (Fund. Math. 41, 1954/55)","yearPosed":1944,"ageNote":"Turan's 1977 note dates the problem to his 1944 labour-camp brickyard. Zarankiewicz's paper is cited by the manuscript as 1955 (Fund. Math. volume 41); the gap in his argument made the formula a conjecture.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims $\\mathrm{cr}(K_{m,n})=\\lfloor\\frac m2\\rfloor\\lfloor\\frac{m-1}2\\rfloor\\lfloor\\frac n2\\rfloor\\lfloor\\frac{n-1}2\\rfloor$ for all positive $m,n$. The lower bound comes from a linear-algebra inequality $\\sum_{i,j}\\dim(U_i\\cap V_j)\\le\\lfloor m^2/4\\rfloor N$ for subspaces with $U_i\\cap V_i=0$, proved by interpolation, applied through signed intersections of cycles; Zarankiewicz's drawing gives the upper bound. Crossings are counted as points. It does NOT address rectilinear, pair or odd crossing numbers of $K_{m,n}$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. The manuscript is credited to OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem: $\\mathrm{cr}(K_{m,n})=d_md_n$ with $d_r=\\lfloor r/2\\rfloor\\lfloor(r-1)/2\\rfloor$ for all positive $m,n$, ordinary point count, unrestricted drawings. formalization.yaml lists OAI.Zarankiewicz.mainTarget_proof (comparator BipartiteCrossing) as the main result. Its statement MainTarget says that for all $m,n>0$ some admissible drawing of $K_{m,n}$ has exactly $d_md_n$ crossing points and every admissible drawing has at least that many, with the same admissibility conditions as the complete-graph file (simple continuous arcs, finitely many proper crossings, no triple points). That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. Hebbar and Mangam (2018, Int. J. Eng. Technol.) published earlier claims of both formulas, which the manuscripts cite and treat as upper-bound constructions only. ","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Turan's brick factory problem is the founding problem of crossing-number theory, open for eighty years after a published proof was found to be wrong. Tied with the Harary-Hill conjecture, above the uniform Turan density of the tetrahedron at 48 and Erdos Problem 571 at 52, below Erdos Problem 3 at 65.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The crossing number of complete bipartite graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The crossing number of complete graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean: main declaration mainTarget_proof","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Crossing/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: crossing number of complete bipartite graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BipartiteCrossing.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/165.md","kind":"lean-proof"},{"label":"Zarankiewicz, On a problem of P. Turan concerning graphs","url":"https://doi.org/10.4064/fm-41-1-137-145","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kadison-kaplansky-conjecture","name":"The Kadison-Kaplansky Projection Conjecture","shortName":"Kadison-Kaplansky conjecture","problemNumber":null,"field":"Operator algebras; group C*-algebras","fieldGroup":"Analysis","statement":"For a torsion-free discrete group $H$, let $C_r^*(H)$ be its reduced group $C^*$-algebra. The Kadison-Kaplansky conjecture asserts that $C_r^*(H)$ contains no projections other than $0$ and $1$. Pimsner and Voiculescu proved it for free groups, and it follows from surjectivity of the coefficient-free Baum-Connes map, so it holds for amenable and hyperbolic groups. Does every torsion-free discrete group have a projectionless reduced $C^*$-algebra?","posedBy":"Richard Kadison, from Irving Kaplansky's question on idempotents in simple C*-algebras","yearPosed":null,"ageNote":"The manuscript cites Kadison's 2008 recollection (Notices AMS) for the origin and gives no posing year. Kadison's free-group question dates from the mid-twentieth century.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"There is a finitely generated torsion-free group $G=G_0*\\mathbb{F}_N$ and a projection $e\\in C_r^*(G)$ with $0<\\tau(e)<1/2$, so $e\\ne 0,1$. It is built from a matrix projection with small unnormalized trace over a graphical small-cancellation group, compressed to a scalar projection by free products. Not shown: a counterexample to the algebraic Kaplansky idempotent conjecture (the group ring has no nontrivial idempotents), or a finitely presented group.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript cites a further release manuscript (OpenAI, 2026, on Bass-type trace results) for the group ring having no nontrivial idempotents.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem was read against the projection form of the conjecture as the manuscript states it. The construction was not refereed. No Lean formalization is listed. The paper separates this from the algebraic idempotent conjecture: the group ring $\\mathbb{C}[G]$ of its group has no nontrivial idempotents, by a companion release result.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"A classic named conjecture of operator algebras, open since Kadison and Kaplansky; famous within the field but downstream of Baum-Connes. Above Connes' Rigidity Conjecture (45), below Baum-Connes from the same family (60) and non-sofic groups (60).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A torsion-free counterexample to the Kadison-Kaplansky projection conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Same family: An irrational-trace counterexample to reduced Baum-Connes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf","kind":"paper"},{"label":"Same family: A torsion-free counterexample to reduced Baum-Connes injectivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"naor-sublinear-dimension-reduction-lp","name":"Naor's sublinear-dimension question for finite subsets of Lp, 1 < p < infinity, p not 2","shortName":"Sublinear dimension reduction in Lp","problemNumber":null,"field":"Metric embeddings, Banach space geometry","fieldGroup":"Analysis","statement":"The Johnson-Lindenstrauss lemma embeds every $n$-point subset of Hilbert space into $O(\\log n)$ dimensions with distortion $1+\\varepsilon$, and Johnson and Lindenstrauss (1984, Problem 3) asked what analogues hold in other Banach spaces. In $L_1$ polynomial dimension is necessary (Brinkman-Charikar), and Ball's exact bound for $L_p$ is $\\binom n2$ coordinates. Naor (ICM 2018) remarked that for no fixed $p\\in[1,\\infty)\\setminus\\{2\\}$ was it even known whether every $n$-point subset of $\\ell_p$ embeds with bounded distortion into a normed space of dimension $o(n)$. For fixed $p\\ne2$ and distortion $D$, can every $n$-point subset of $L_p$ be embedded with distortion $D$ in dimension $o(n)$?","posedBy":"Assaf Naor (after William B. Johnson and Joram Lindenstrauss)","yearPosed":2018,"ageNote":"Naor, Metric dimension reduction: a snapshot of the Ribe program, Proc. ICM 2018, remark after Question 13; the underlying question is Johnson-Lindenstrauss (1984), Problem 3.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for fixed $11$, every $n$-point subset of any $L_p$ embeds with distortion $D$ into $\\ell_p^d$ (same exponent, coordinate target, nonlinear map) with $d\\le\\exp(C_{p,D}(\\log n)^{\\gamma(p)})=n^{o(1)}$, where $\\gamma(p)=2-p$ for $p<2$ and $1-2/p$ for $p>2$; exact embeddings need dimension of order $n^2$. This answers the $o(n)$ question for every $p\\in(1,\\infty)\\setminus\\{2\\}$. Not shown: $p=1$; Naor's Question 13 itself ($O(\\log n)$ dimension in some normed space), which Naor-Ren's 2026 lower bound already excludes for $\\ell_p^d$ targets when $p>2$; optimal exponents; an efficient algorithm.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The single manuscript (September 23, 2026) is the whole family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against Naor's remark (read in the ICM text) and Johnson-Lindenstrauss Problem 3. The proof (weighted-moment distributions by convex separation, electrical-flow localization, stable and signed Poisson increments, Maurey sampling) was not refereed. lean/formalization.yaml lists comparator SubpolynomialLp, declaration OAI.SubpolynomialLp.source_main (file OAI/Analysis/LpDimension/Main.lean). Its statement was read here: with $d_p(n,D)$ the least $d$ such that every injective $n$-point family in any $L_p(\\mu)$ embeds into $\\ell_p^d$ with distortion $D$, for $11$ there is $C$ with $\\log n/\\log(1+2D)\\le d_p(n,D)\\le\\exp(C(\\log n)^{\\gamma(p)})$; at $D=1$, $\\lfloor(n-1)/4\\rfloor^2\\le d_p(n,1)\\le\\binom n2$ for $n\\ge9$; and the log-ratio tends to 0 or 2. That states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A recognised question of the Ribe program and metric dimension reduction, answered strongly (subpolynomial, not just sublinear), but known mainly to Banach-space and metric-embedding specialists. Placed below The David-Semmes problem in higher codimension (45) and the planar GNRS entry (45), near Stein's dimension-free weak-(1,1) Riesz transform problem (38) and the HRT Conjecture (33).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Subpolynomial dimension reduction in Lp","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.SubpolynomialLp.source_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/LpDimension/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SubpolynomialLp.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SubpolynomialLp.lean","kind":"lean-proof"},{"label":"Naor, Metric dimension reduction: a snapshot of the Ribe program (arXiv)","url":"https://arxiv.org/abs/1809.02376","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"long-time-boltzmann-grad-limit-potentials","name":"Long-time Boltzmann-Grad limit for stable finite-range radial potentials","shortName":"Long-time Boltzmann limit, potentials","problemNumber":null,"field":"Kinetic theory; derivation of the Boltzmann equation","fieldGroup":"Mathematical physics","statement":"Lanford (1975) derived the Boltzmann equation from Newtonian hard spheres in the Boltzmann-Grad limit, but only for a short time; King and Pulvirenti-Saffirio-Simonella did the same for potentials. Deng, Hani and Ma (arXiv 2024) extended the hard-sphere derivation to every finite interval on which the Boltzmann solution is regular with Gaussian decay, and discussed the extension to smooth interaction potentials, where encounters last positive time and several particles interact at once, as a further problem. For a stable, finite-range radial potential, does the Boltzmann-Grad limit hold on every such regular kinetic interval, not just for short times?","posedBy":"Yu Deng, Zaher Hani and Xiao Ma, 'Long time derivation of the Boltzmann equation from hard sphere dynamics' (arXiv 2408.07818, Section 1.4.2)","yearPosed":2024,"ageNote":"The manuscript cites version 3 (2025) of the Deng-Hani-Ma preprint, first posted in 2024. The underlying question of deriving Boltzmann beyond Lanford's short time is decades older.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1: for a stable finite-range radial potential, a grand-canonical initial law with pair exclusion and $C^1$ density with spatially summable Gaussian bounds on it and its gradient, and any $T$ for which the Boltzmann equation with that potential's scattering has a classical solution with uniform Gaussian velocity bound on $[0,T]$, all fixed-order rescaled factorial marginals converge in $L^1$ to $f(t)^{\\otimes s}$ uniformly on $[0,T]$; empirical averages converge in probability. It assumes, rather than proves, existence of the regular kinetic solution, so it gives no global-in-time statement, and it does not treat long-range or non-radial potentials or give a rate.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. A companion of the same date proves long-time Gaussian fluctuations for hard spheres (separate entry); both build on the Deng-Hani-Ma layered expansion.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1 of the principal manuscript was read against the further problem as the paper reports it from Deng-Hani-Ma Section 1.4.2. It covers stable radial potentials vanishing beyond unit range, $C^2$ except a possible repulsive singularity at the origin, with attractive wells allowed, for $C^1$ initial densities with summable Gaussian bounds and initial pair exclusion, on every interval where a classical Boltzmann solution with uniform Gaussian decay is assumed to exist. The wording of the Deng-Hani-Ma problem itself was not read here. The proof was not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Extending the long-time derivation of the Boltzmann equation from hard spheres to genuine interaction potentials was a natural next problem after Deng-Hani-Ma, known within kinetic theory. Placed below Global classical solutions of the 3D relativistic Vlasov-Maxwell system (55) and The Bogoliubov quantum-depletion law for the dilute Bose gas (35), above McKean Entropy-Production Conjecture (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Boltzmann-Grad limit for stable radial potentials on regular kinetic intervals","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Hard-sphere fluctuations on the regular Boltzmann lifespan","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf","kind":"paper"},{"label":"Deng-Hani-Ma, Long time derivation of the Boltzmann equation (arXiv 2408.07818)","url":"https://arxiv.org/abs/2408.07818","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hindman-finite-sums-and-products","name":"Hindman's finite sums and products conjecture","shortName":"Hindman's finite sums and products conjecture","problemNumber":null,"field":"Arithmetic Ramsey theory","fieldGroup":"Combinatorics","statement":"For finite $A\\subset\\mathbb N$ let $\\mathrm{FS}(A)$ and $\\mathrm{FP}(A)$ be the sums and products of its nonempty subsets. The Folkman-Rado-Sanders theorem gives monochromatic $\\mathrm{FS}(A)$ for arbitrarily large $A$ in every finite coloring, and the multiplicative analogue follows, but the two together are different: Hindman (1980) found a finite coloring with no infinite set whose sums and products are monochromatic, and even the pattern $\\{x,y,x+y,xy\\}$ was open for colorings of $\\mathbb N$ in more than two colors (Moreira proved $\\{x,x+y,xy\\}$; Alweiss proved the full statement over $\\mathbb Q$). Hindman (1979) conjectured the finite version. For every finite coloring of $\\mathbb N$ and every $m$, is there an $m$-element set $A$ with $\\mathrm{FS}(A)\\cup\\mathrm{FP}(A)$ monochromatic?","posedBy":"Neil Hindman (Trans. Amer. Math. Soc. 247, 1979); also Hindman-Strauss, Question 17.18","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every finite coloring of $\\mathbb N$ and every $m\\ge1$ there are $a_1<\\dots 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof cites the Green-Tao-Ziegler inverse theorem, Tao-Ziegler concatenation, quantitative polynomial equidistribution and nilpotent recurrence as external inputs, adapting Alweiss's rational argument to the integers.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Hindman's conjecture; it gives, for every $r$-coloring of $\\mathbb N$ and every $m$, distinct $a_1<\\dots0$, $\\widehat f\\ge0$ and $f(x)\\le0$ for $|x|\\ge r$, giving density at most $\\mathrm{vol}(B(0,r/2))$. Numerically the bound appeared to match the best packings in dimensions 2, 8 and 24, and their Conjecture 7.3 asserts that in these dimensions an auxiliary function exists attaining the optimal density exactly. Viazovska (dimension 8) and Cohn-Kumar-Miller-Radchenko-Viazovska (dimension 24) constructed such functions; in the plane the optimal density $\\pi/(2\\sqrt3)$ is classical (Thue), but no sharp auxiliary function was known. Does an admissible function exist that makes the Cohn-Elkies bound equal to $\\pi/(2\\sqrt3)$ in dimension two?","posedBy":"Henry Cohn and Noam Elkies (Conjecture 7.3, Annals of Mathematics, 2003)","yearPosed":2003,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 constructs a real radial Schwartz function on $\\mathbb R^2$ with $\\widehat f(0)=1$, $f(0)=2/\\sqrt3$, $\\widehat f\\ge0$ and $f\\le0$ outside the unit disk, so the Cohn-Elkies bound equals the triangular packing density $\\pi/(2\\sqrt3)$. Its zeros beyond radius one lie at integral squared radii, which recovers uniqueness of the triangular packing among periodic equality cases. It does not satisfy the stronger zero-set condition of Cohn-Elkies Conjecture 8.1, and it gives no new packing bound (the planar density was already known).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The certificate is a radial Schwartz function whose global sign conditions are proved with rigorous interval arithmetic (python-flint) and analytic estimates; the manuscript's README gives the verifier command.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Cohn-Elkies Conjecture 7.3 in dimension two; it gives a real radial Schwartz $f$ with $\\widehat f(0)=1$, $f(0)=2/\\sqrt3$, $\\widehat f\\ge0$ and $f(x)\\le0$ for $|x|\\ge1$, which rescales to the requested witness. The manuscript notes that its witness does not meet the extra zero-set requirement of Cohn-Elkies Conjecture 8.1. Lean: lean/ComparatorChallenges/PlanarPacking.json exists with solution_module OAI.Analysis.PlanarPacking.Main, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement PlanarPacking.lean was read here: there exists a Schwartz map on the Euclidean plane, radial, with Fourier transform (kernel $e^{-2\\pi i\\langle x,\\xi\\rangle}$) equal to 1 at 0, value $2/\\sqrt3$ at 0, Fourier transform real and nonnegative everywhere, and $f(x)\\le0$ for $\\|x\\|\\ge1$. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A named conjecture in the sphere-packing community since 2003, the planar member of the trio whose dimension 8 and 24 cases were Viazovska's breakthroughs; in the plane the density was already known, so only the certificate was open. Placed above Tightness of the Cohn-Elkies Bound in Dimension 36 (25), level with the Colin de Verdiere chromatic conjecture (30), below Upper Bounds for High-Dimensional Sphere Packing (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A sharp Fourier certificate for planar circle packing","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.sharp_fourier_certificate)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/PlanarPacking/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: PlanarPacking.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PlanarPacking.lean","kind":"lean-proof"},{"label":"Related in family: Universal optimality of the triangular lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"small-cohen-macaulay-module-conjecture","name":"Hochster's small Cohen-Macaulay module conjecture for complete local domains","shortName":"Small Cohen-Macaulay module conjecture","problemNumber":null,"field":"Commutative algebra; homological conjectures","fieldGroup":"Algebra","statement":"A small (maximal) Cohen-Macaulay module over a Noetherian local ring $(R,\\mathfrak m)$ is a nonzero finitely generated module $M$ with $\\mathrm{depth}\\,M=\\dim R$. Hochster conjectured in the early 1970s that every complete Noetherian local domain has one. It holds in dimension at most two, and in some graded positive-characteristic cases in dimension three; big Cohen-Macaulay modules and algebras exist in all characteristics, but they are not finitely generated, and the existence question was open even for local rings of affine three-dimensional domains over algebraically closed fields. Does every complete Noetherian local domain admit a small Cohen-Macaulay module?","posedBy":"Melvin Hochster","yearPosed":1973,"ageNote":"Hochster, Cohen-Macaulay modules (Conference on Commutative Algebra, Lawrence 1972; LNM 311, 1973) and his 1975 CBMS lectures, as cited; Hochster (2017) records the conjecture and his later expectation of counterexamples.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a three-dimensional complete Noetherian normal local domain $R\\supset\\mathbb C$ with residue field $\\mathbb C$ having no small Cohen-Macaulay module; the uncompleted vertex localization of the same section ring, essentially of finite type over $\\mathbb C$, also has none. The mechanism is Theorem 1.2: if the completed cone $R(X,H)$ over a smooth projective surface has a small Cohen-Macaulay module, then $K_X^2-2c_2(X)\\le6H^2$, applied to a Hirzebruch-Kummer surface with $K_X\\equiv5H$. It does not treat positive or mixed characteristic, the non-domain forms, or dimension three with other residue fields. Priority: Liang Chen posted a claimed counterexample of the same kind on 21 September 2026, two days before this manuscript (arXiv:2609.24142): the completed section ring of a degree-six Hirzebruch-Kummer surface with $K_Y\\sim4H$, resting on his stated Main claim.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 23, 2026) has no companion in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture; it gives a three-dimensional complete Noetherian normal local domain containing $\\mathbb C$ with residue field $\\mathbb C$ and no nonzero finitely generated module of depth three, which contradicts the domain form as posed. The obstruction (Theorem 1.2) is $K_X^2-2c_2(X)\\le6H^2$ for completed section rings of surfaces, and the example uses a Hirzebruch-Kummer surface from Anghel's 2026 preprint, with the numerical checks redone in the paper. A reader should know that the manuscript itself reports overlapping recent work: Anghel (graded case, September 14, 2026) and Chen (arXiv 2609.24142, September 21, 2026). Not refereed. The 7 October 2026 update added the supporting Comparator challenge SurfaceConeCandidateSupport (read here): some complete Noetherian integrally closed local $\\mathbb C$-domain of Krull dimension 3 has residue field $\\mathbb C$. It does not state the absence of a small Cohen-Macaulay module, so the entry stays unreviewed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The small Cohen-Macaulay module conjecture is one of Hochster's central homological conjectures and well known across commutative algebra, though Hochster himself had come to expect counterexamples. Placed level with Brennan's conjecture (45) and the word problem for Artin groups (45), below Kaplansky's zero-divisor conjecture (62), and above the Campana-Peternell conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A Complete Local Domain without a Small Cohen-Macaulay Module","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Hochster, Homological conjectures and lim Cohen-Macaulay sequences (2017), Conjectures 2.1-2.2","url":"https://sites.lsa.umich.edu/hochster/wp-content/uploads/sites/1337/2024/08/DSlim2.pdf","kind":"problem-record"},{"label":"Chen, arXiv:2609.24142 (21 Sept 2026): earlier claimed counterexample, ChatGPT-assisted","url":"https://arxiv.org/abs/2609.24142","kind":"independent"},{"label":"Lean statement (supporting result only): SurfaceConeCandidateSupport","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/SurfaceConeCandidateSupport.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"brauchart-hardin-saff-log-energy-sphere","name":"The Brauchart-Hardin-Saff conjecture on the linear term of optimal logarithmic energy on the 2-sphere","shortName":"Brauchart-Hardin-Saff log-energy conjecture","problemNumber":null,"field":"Discrete energy on the sphere; asymptotics of optimal logarithmic energy","fieldGroup":"Analysis","statement":"Let $E_{\\log}(n)$ be the minimum over $n$ points on the unit sphere $S^2$ of $-\\sum_{i\\ne j}\\log\\|y_i-y_j\\|$ (Smale's seventh problem concerns near-minimizers). The leading terms $(\\tfrac12-\\log2)n^2-\\tfrac n2\\log n$ are classical. Brauchart, Hardin and Saff (2012, Conjecture 4) conjectured that the next term is $C_{\\mathrm{BHS}}n+o(n)$ with the explicit constant $C_{\\mathrm{BHS}}=2\\log2+\\tfrac12\\log\\tfrac23+3\\log(\\sqrt\\pi/\\Gamma(1/3))$ coming from the triangular lattice. Betermin and Sandier (2018) proved that the linear coefficient exists and equals this value exactly when the triangular lattice minimizes the planar renormalized energy. Does $E_{\\log}(n)=(\\tfrac12-\\log2)n^2-\\tfrac n2\\log n+C_{\\mathrm{BHS}}n+o(n)$ hold?","posedBy":"Johann S. Brauchart, Douglas P. Hardin and Edward B. Saff (Conjecture 4, 2012)","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 1.3: $E_{\\log}(n)=(\\tfrac12-\\log2)n^2-\\tfrac n2\\log n+C_{\\mathrm{BHS}}n+o(n)$ with the Brauchart-Hardin-Saff constant, obtained from the triangular minimality of the planar Coulomb renormalized energy and the Betermin-Sandier equivalence. It is the logarithmic, $d=2$ case only; the Riesz-energy cases of the Brauchart-Hardin-Saff conjectures and higher spheres are not addressed, and no explicit point configurations are produced.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The result is Corollary 1.3 of the Coulomb renormalized-energy manuscript: it applies the Betermin-Sandier equivalence to that manuscript's main theorem after a normalization check.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary 1.3 of the principal manuscript was read against Conjecture 4 of Brauchart-Hardin-Saff in dimension two; it states the full asymptotic with the conjectured constant, derived from Theorem 1.1 (the Sandier-Serfaty conjecture) through Betermin-Sandier's Theorem 1.5. Its correctness therefore rests on Theorem 1.1, which is unreviewed. No Lean formalization covers it. The manuscript says this is an asymptotic value only, not a construction of near-optimal point sets or an algorithm for Smale's seventh problem.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A named conjecture known in the discrete-energy and potential-theory community since 2012, but already known to be equivalent to the Sandier-Serfaty conjecture, so it adds little standing of its own. Placed level with the Colin de Verdiere chromatic conjecture (30), below Solovej's generalized ionization conjecture (32) and the Sandier-Serfaty conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Triangular minimality for planar Coulomb renormalized energy","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Universal optimality of the triangular lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"covering-density-linear-bound-question","name":"Is the worst-case translative covering density of convex bodies linear in the dimension?","shortName":"Linear covering density question","problemNumber":null,"field":"Discrete and convex geometry; translative and lattice coverings","fieldGroup":"Geometry & topology","statement":"For a convex body $K\\subset\\mathbb R^n$ let $\\theta_T(K)$ be the least density of a covering of $\\mathbb R^n$ by translates of $K$. Rogers (1957) proved $\\theta_T(K)\\le n\\log n+n\\log\\log n+5n$ for every convex body, and lower bounds of order $n$ are known for balls. Naszodi recorded the question whether a universal linear bound holds: is there an absolute constant $C$ with $\\theta_T(K)\\le Cn$ for every convex body in every dimension? Equivalently, what is the order of the worst-case covering density in dimension $n$?","posedBy":"M. Naszodi, Flavors of Translative Coverings, New Trends in Intuitive Geometry, Bolyai Soc. Math. Stud. 27 (2018), Section 3 (arXiv 1603.04481)","yearPosed":2016,"ageNote":"Recorded in Naszodi's survey, arXiv 1603.04481 (2016), published 2018; the manuscript does not say who first asked it. Rogers' n log n upper bound dates from 1957.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Translative paper, Theorem 1.1: for absolute $c>0$ and all large $n$ there is a centrally symmetric convex body (even a rational polytope) with $\\theta_T(K_n)>cn\\log n$, so no universal linear bound exists; with Rogers' bound the worst-case order is $n\\log n$. It extends the lattice lower bound of Li and Liu (2026) to arbitrary translative coverings. The companion proves every convex body has a single-lattice covering of density at most $Cn\\log n$, removing the $(\\log\\log n)^{O(1)}$ loss of Li-Liu, so the lattice supremum also has order $n\\log n$. Constants are not optimized; Euclidean balls are not treated.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the translative paper (centrally symmetric $K_n$ with $\\theta_T(K_n)>cn\\log n$) was read against the question as the paper states it with the Naszodi reference. Proofs not refereed. The challenge ComparatorChallenges/CoveringDensity.lean is not in lean/formalization.yaml; it is found through lean/docs/092.md and its solution module OAI.Geometry.CoveringDensity.Main exists at the pinned commit. Its statement was read here: absolute $0 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: number fields (September 23, 2026, this entry's principal) and global function fields including characteristic two (October 5, 2026).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the number-field manuscript was read against the conjecture; it states the full conjecture over number fields with abstract normal subgroups and no finite-generation assumption, using the established reduction (cases already known) and supplying the remaining anisotropic outer A, trialitarian and $E_6$ cases. The function-field companion's abstract states the same over every global function field. Neither proof was refereed; there is no Lean formalization. The number-field paper relies on cited prior work (Segev-Seitz, Rapinchuk-Potapchik, Prasad-Rapinchuk's metaplectic kernel) as inputs.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The Margulis-Platonov conjecture is a standard open problem of the arithmetic theory of algebraic groups since 1979, with the hardest anisotropic cases open. Placed below the blockwise Alperin weight conjecture (55) and the word problem for finite-rank Artin groups (45), level with Wall's manifold-realization question for Poincare duality groups (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Margulis-Platonov conjecture over number fields","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The Margulis-Platonov conjecture over global function fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/margulis-platonov-global-function-fields.pdf","kind":"paper"},{"label":"Margulis, Finiteness of quotient groups of discrete subgroups (1979)","url":"https://doi.org/10.1007/BF01077485","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"nagata-biran-szemberg-maximal-seshadri","name":"The qualitative Nagata-Biran-Szemberg conjecture: maximal multipoint Seshadri constants for many very general points","shortName":"Maximal multipoint Seshadri constants","problemNumber":null,"field":"Algebraic geometry; Seshadri constants and positivity","fieldGroup":"Geometry & topology","statement":"Let $X$ be a smooth projective variety of dimension $n$ with an ample line bundle $L$, and let $\\varepsilon(X,L;p)$ be the multipoint Seshadri constant at $r$ points, the infimum of $L\\cdot C/\\sum_i\\mathrm{mult}_{p_i}C$ over curves $C$. It is always at most $(L^n/r)^{1/n}$. Generalizing Nagata's plane conjecture, and motivated by Biran's symplectic packing stability, Szemberg and coauthors conjectured that on surfaces equality holds at $r$ very general points for every sufficiently large $r$ (Strycharz-Szemberg and Szemberg 2004, in a stronger form with an explicit threshold; Syzdek-Szemberg 2010, Conjecture 4.3), and Roé and Ross stated the same eventual maximality in every dimension over uncountable algebraically closed fields (2009, Conjecture 1.3). Harbourne had it when $rL^2$ is a square. Is there, for every polarized $(X,L)$, a threshold $r_0$ such that $\\varepsilon(X,L;p)=(L^n/r)^{1/n}$ at very general $p$ for every $r\\ge r_0$?","posedBy":"Beata Strycharz-Szemberg and Tomasz Szemberg (surfaces, stronger form); Wioletta Syzdek and Tomasz Szemberg (qualitative form); Joaquim Roé and Julius Ross (all dimensions)","yearPosed":2004,"ageNote":"2004 is the stronger Strycharz-Szemberg-Szemberg conjecture (Serdica Math. J. 30), which contains the qualitative one; the manuscripts cite Syzdek-Szemberg (2010) for the qualitative surface statement and Roé-Ross (2009) for arbitrary dimension. Biran's packing stability (1999) is the symplectic precedent.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Surfaces (Theorem 1.1, principal): for every smooth integral complex projective surface $S$ and ample $L$ there is $r_0(S,L)$ such that for every $r\\ge r_0$, at $r$ very general points, $L-\\sqrt{L^2/r}\\sum E_i$ is nef on the blowup, so $\\varepsilon(S,L;p)=\\sqrt{L^2/r}$; Corollary 5.5 extends this to uncountable algebraically closed fields of characteristic zero. Companion, $n\\ge3$ over $\\mathbb C$: $\\varepsilon(X,L;p)=(L^n/r)^{1/n}$ at very general points for every $r\\ge r_0(X,L)$. Companion, positive characteristic, $n\\ge2$: the same equality at the geometric generic tuple. Not shown: an explicit or uniform threshold, the higher-cycle Seshadri analogues Roé-Ross distinguish, or very general points (rather than the geometric generic tuple) in positive characteristic. The sharp plane case $r\\ge10$ is the separate Nagata entry.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has three manuscripts on this question: surfaces (September 23, 2026), complex dimension at least three (October 5, 2026) and positive characteristic (October 5, 2026). The higher-dimensional paper says it adapts the planar mechanism and does not use the surface theorem as an input.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of each of the three manuscripts was read against the conjecture as stated by Syzdek-Szemberg and Roé-Ross. The proofs were not refereed. lean/formalization.yaml lists a main result for the surface manuscript only (comparator MaximalSeshadriConstants, declaration OAI.MaximalSeshadri.Geometry.maximalSeshadriConstants). Its statement, read here: for a smooth integral surface given as a closed subscheme of complex projective space and an ample line bundle $L$, there is $r_0$ such that for every $r\\ge r_0$ a countable family of proper closed sets of distinct $r$-tuples, with a tuple outside all of them, has the property that outside them a point blowup exists on which the boundary class is nef and the Seshadri constant equals $\\sqrt{L^2/r}$. That is the headline claim for surfaces over $\\mathbb C$. Not rebuilt here. The dimension $n\\ge3$ and positive-characteristic manuscripts are not formalized and are unreviewed; in positive characteristic the result is at the geometric generic tuple, not at very general points. The thresholds are not explicit, so the stronger 2004 form with a specified threshold is not addressed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A well known conjecture in the Seshadri-constant community, the qualitative extension of Nagata's conjecture to all polarized varieties, with Biran's packing stability as its symplectic shadow. Less famous than Nagata itself. Placed below the Kuga-Satake Hodge conjecture (48) and Nagata's conjecture (62 in this batch), and level with Voiculescu's free entropy dimension question (38), above the Campana-Peternell conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf","sourceName":"OpenAI math release: Maximal Seshadri constants on arbitrary polarized surfaces","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.MaximalSeshadri.Geometry.maximalSeshadriConstants)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/AlgebraicGeometry/Seshadri/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: MaximalSeshadriConstants.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/MaximalSeshadriConstants.lean","kind":"lean-proof"},{"label":"Companion: Maximal Multipoint Seshadri Constants in Higher Dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026/main.pdf","kind":"paper"},{"label":"Companion: Maximal multipoint Seshadri constants in positive characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026/seshadri-positive-characteristic.pdf","kind":"paper"},{"label":"Companion: Nagata's conjecture for plane curves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf","kind":"paper"},{"label":"Roé and Ross, An inequality between multipoint Seshadri constants (2009)","url":"https://doi.org/10.1007/s10711-008-9315-4","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hoering-split-tangent-integrability","name":"Hoering's conjecture: tangent-bundle summands on rationally connected manifolds are integrable","shortName":"Hoering's integrability conjecture","problemNumber":null,"field":"Algebraic geometry; rationally connected varieties, foliations","fieldGroup":"Geometry & topology","statement":"Let $X$ be a smooth rationally connected projective complex manifold with a holomorphic splitting $T_X=E_1\\oplus E_2$ into positive-rank subbundles. Hoering (2007) proved that if one summand is integrable then $X\\cong X_1\\times X_2$ compatibly, and conjectured (2008, Conjecture 1.2) that at least one summand is always integrable; Campana-Peternell had settled small ranks on Fano manifolds, and Hoering (2026, Conjecture 1.5) states that both summands are integrable in the smooth rationally connected case, after proving it for Fano type. Is every summand of such a splitting integrable, so that $X$ is a compatible product?","posedBy":"A. Hoering, The structure of uniruled manifolds with split tangent bundle, Osaka J. Math. 45 (2008), Conjecture 1.2; both-summand form: Hoering, arXiv:2602.15427 (2026), Conjecture 1.5","yearPosed":2008,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: on a smooth connected rationally connected projective manifold of dimension at least two, both summands of every holomorphic splitting $T_X=E_1\\oplus E_2$ into positive-rank subbundles are integrable. Corollary 1.2 (with Hoering's 2007 product theorem): $X\\cong X_1\\times X_2$ with $E_i=\\mathrm{pr}_i^*T_{X_i}$. Smoothness is needed (Hoering 2026 gives a singular counterexample); splittings with more than two summands are not treated.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization accompanies it (Comparator challenge SplitTangentIntegrability).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and Corollary 1.2 of the TeX source, read against Hoering's conjectures as cited. Lean: Comparator challenge SplitTangentIntegrability, declaration OAI.SplitTangent.integrability_main. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: for a compact connected complex manifold of dimension at least 2 with a projective embedding for which a dense Zariski-open set of pairs of points lie on rational curves, both summands of every holomorphic tangent splitting with positive ranks are integrable. That is the headline integrability claim; the product corollary (via Hoering 2007, Theorem 1.4) is not formalized. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A specialist conjecture within the split-tangent-bundle programme, posed in 2008 and restated in 2026 after the Fano-type case; it is known mainly to a small group of algebraic geometers. Placed below the Campana-Peternell conjecture in dimension six (28) and above Schafhauser-Tikuisis-White Problem XXI (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026/main.pdf","sourceName":"OpenAI math release: Integrability of split tangent bundles on rationally connected manifolds","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge SplitTangentIntegrability (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SplitTangentIntegrability.lean","kind":"lean-proof"},{"label":"Universal-cover splitting for compact Kahler manifolds (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"baum-connes-conjecture","name":"The Baum-Connes Conjecture (coefficient-free, reduced)","shortName":"Baum-Connes conjecture","problemNumber":null,"field":"Operator algebras; noncommutative geometry; K-theory","fieldGroup":"Analysis","statement":"For a countable discrete group $G$, the Baum-Connes assembly map $\\mu^G:K_*^G(\\underline{E}G)\\to K_*(C_r^*(G))$ sends equivariant $K$-homology of the classifying space for proper actions to the $K$-theory of the reduced group $C^*$-algebra. Baum and Connes conjectured that it is an isomorphism. It was proved for a-T-menable groups (Higson-Kasparov) and hyperbolic groups (Mineyev-Yu, Lafforgue); Higson-Lafforgue-Skandalis disproved the version with coefficients, but the coefficient-free group conjecture stayed open. Is $\\mu^G$ an isomorphism for every countable discrete group $G$?","posedBy":"Paul Baum and Alain Connes (preprint circulated 1982, published 2000); reformulated with Nigel Higson (1994)","yearPosed":1982,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Three counterexamples. (1) A finitely generated group with a projection in $C_r^*(G)$ of irrational canonical trace, so $\\mu_0^G$ is not surjective. (2) A finitely generated torsion-free group whose degree-zero reduced assembly map kills a class of infinite order, so rational injectivity fails (the reduced form of the strong Novikov conjecture). (3) The Kadison-Kaplansky counterexample also gives a torsion-free group with a class outside the assembly image. Not shown: failure of maximal assembly, a finitely presented counterexample, or failure of the classical Novikov conjecture.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorems of the three family manuscripts were read against the coefficient-free reduced conjecture as stated by Baum-Connes-Higson. The constructions were not refereed. No Lean formalization is listed. The surjectivity failure rests on Luck's theorem that traces of assembly classes are rational. The groups are finitely generated; finite presentation is not claimed. Maximal assembly and the classical Novikov conjecture are not disproved.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"The organizing conjecture of noncommutative geometry for forty years, with consequences for Novikov and Kadison-Kaplansky; the version with coefficients was already known false, which keeps it below the Jacobian Conjecture (65). Ties non-sofic groups (60) and Kothe (60), above Connes' Rigidity (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An irrational-trace counterexample to reduced Baum-Connes","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A torsion-free counterexample to reduced Baum-Connes injectivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf","kind":"paper"},{"label":"Companion: A torsion-free counterexample to the Kadison-Kaplansky projection conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"unique-games-conjecture","name":"The Unique Games Conjecture","shortName":"Unique Games Conjecture","problemNumber":null,"field":"Hardness of approximation, PCPs","fieldGroup":"Theoretical computer science","statement":"A Unique Games instance is a graph with a finite label set $K$ and, on each edge, a permutation of $K$; a labeling satisfies an edge when the label at one endpoint is the image of the other endpoint's label under that permutation. Khot (2002) conjectured that for every $\\varepsilon,\\delta>0$ there is a finite alphabet size $q=q(\\varepsilon,\\delta)$ such that it is NP-hard to distinguish instances over $q$ labels in which a $1-\\varepsilon$ fraction of the constraints can be satisfied from instances in which at most a $\\delta$ fraction can. Under this hypothesis, optimal inapproximability was known for Max-Cut (the Goemans-Williamson ratio), Vertex Cover (factor $2$) and many other problems, but only conditionally. Is the Unique Games Conjecture true?","posedBy":"Subhash Khot, On the Power of Unique 2-Prover 1-Round Games, STOC 2002","yearPosed":2002,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every fixed $\\varepsilon,\\delta\\in(0,1/2)$ there are $s\\ge1$ and a deterministic polynomial-time reduction from 3SAT to explicit unweighted, simple bipartite Unique Games over $K=\\mathbb F_2^s$ whose constraints are translations $a(v)=a(u)+c_e$, with value at least $1-\\varepsilon$ on satisfiable formulas and at most $\\delta$ on unsatisfiable ones. This is the conjecture in its usual edge-value form. The route keeps the Barak-Kothari-Steurer degree-two matrix shortcode and its Khot-Minzer-Safra inverse theorem and changes the noise through a nonlinear map to push completeness to $1-\\varepsilon$. Section 7 transfers known UG-hardness reductions (cut, covering, CSP, ordering, deletion, clustering) to NP-hardness; those reductions are credited to their authors. Companion manuscripts give direct NP-hardness proofs for Max-Cut beyond $\\alpha_{GW}$, Vertex Cover below factor $2$, and every constant factor for Min-UnCut and directed feedback vertex set.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The release also includes an abridged summary of the model's reasoning for this family (reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf), and Lean formalizations of the main theorem and of the Max-Cut and Vertex Cover companions.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the principal manuscript, read against Khot's conjecture as cited (Khot 2010 survey, Conjecture 2.5; translation form via KKMO Corollary 13). The proof was not refereed. Lean: formalization.yaml lists OAI.UniqueGamesTheorem.theorem11 (lean/OAI/Computability/UniqueGames/Theorem.lean, comparator ComparatorChallenges/UniqueGamesTheorem.lean). Its statement was read: for all real 0 < eps, delta < 1/2 there is a reduction from binary-encoded 3SAT to nonempty simple bipartite translation Unique Games over a fixed alphabet identified with F_2^s, computable in polynomial time by a Mathlib TM2 machine, with completeness at least 1-eps and soundness at most delta. That is the headline claim, with no extra hypotheses; permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The Max-Cut and Vertex Cover companions also have Lean main results (OAI.OptimalMaxCut.main, OAI.VertexCover.every_fixed_factor_below_two) whose statements match their headlines; also not rebuilt. The paper names Hastad's parity hardness and the Khot-Minzer-Safra Grassmann expansion theorem as external inputs; since theorem11 carries no hypotheses, a passing comparator check would mean those inputs are formalized too, which was not audited here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":74,"significanceNote":"Khot's Unique Games Conjecture (2002) is the central open conjecture of hardness of approximation: it is the hypothesis behind optimal inapproximability for Max-Cut, Vertex Cover and, via Raghavendra, every constraint satisfaction problem, and is known across all of theoretical computer science. Placed above the Jacobian Conjecture (65) and the APSP hypothesis (55), below the Navier-Stokes Millennium problem entry (87). Sharp Hardness for MAX-3-CUT (25) is one of its conditional consequences.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Unique Games Theorem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"A Direct Proof of Optimal Max-Cut Hardness","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf","kind":"paper"},{"label":"The Factor-Two Hardness Threshold for Vertex Cover","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/paper.pdf","kind":"paper"},{"label":"Constant-factor hardness of Min-UnCut","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-Min-UnCut-September-23-2026/paper.pdf","kind":"paper"},{"label":"Constant-factor hardness of directed feedback vertex set","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-directed-feedback-vertex-set-September-23-2026/paper.pdf","kind":"paper"},{"label":"Reasoning summary: ordinary NP-hardness at the basic semidefinite threshold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf","kind":"paper"},{"label":"Lean: Unique Games theorem (theorem11)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/UniqueGames/Theorem.lean","kind":"lean-proof"},{"label":"Lean: optimal Max-Cut hardness","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/MaxCut/Main.lean","kind":"lean-proof"},{"label":"Lean: Vertex Cover factor-two hardness","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/VertexCover/Main.lean","kind":"lean-proof"},{"label":"Khot 2002, On the Power of Unique 2-Prover 1-Round Games","url":"https://doi.org/10.1145/509907.510017","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"solomon-yau-least-volume-conjecture","name":"The Solomon-Yau least-volume conjecture for minimal hypersurfaces of round spheres","shortName":"Solomon-Yau least-volume conjecture","problemNumber":null,"field":"Minimal hypersurfaces in spheres; volume gaps","fieldGroup":"Geometry & topology","statement":"The equator is the least-volume closed minimal hypersurface of the unit sphere $S^{m+1}$. The minimal Clifford products $S^k(\\sqrt{k/m})\\times S^{m-k}(\\sqrt{(m-k)/m})$, $1\\le k 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Yau's Problem 31 and Ge-Li Conjecture 1.12(ii). It covers every $m\\ge2$ and smooth closed connected minimal immersions, with volume counted on the domain; the reduction from immersed to embedded uses known multiplicity bounds (Nguyen 2023, Ge-Li 2022). The release has no Lean formalization for this family (lean/docs/349.md does not exist at the pinned commit). The proof is a min-max argument with a dimension induction on cone regularity; it was not refereed here. The same paper proves the lower bound in Perdomo's average-curvature conjecture as a step, which Nifa (2026 preprint) also proves.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A question from Yau's problem list on the volume spectrum of minimal hypersurfaces, whose two-dimensional case needed the Marques-Neves Willmore machinery; known across geometric analysis but not a headline problem. Below Yau's nodal-set and uniformization conjecture entries at 58, level with the He-Schramm rigidity conjecture at 32 and the Long-time Boltzmann-Grad limit entry at 30.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Solomon-Yau least-volume theorem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Ge and Li, A lower bound for L2 length of second fundamental form on minimal hypersurfaces (Conjecture 1.12)","url":"https://arxiv.org/abs/2103.07747","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"schonhage-strassen-n-log-n-conjecture","name":"The Schonhage-Strassen conjecture that n log n is optimal for integer multiplication on multitape Turing machines","shortName":"Schonhage-Strassen n log n conjecture","problemNumber":null,"field":"Algorithms; bit complexity of integer multiplication","fieldGroup":"Theoretical computer science","statement":"Schonhage and Strassen (1971) showed that two $n$-bit integers can be multiplied in $O(n\\log n\\log\\log n)$ time on a multitape Turing machine and proposed $n\\log n$ as the optimal order of growth. Successive improvements (Furer; De-Kurur-Saha-Saptharishi; Harvey, van der Hoeven and Lecerf) culminated in the unconditional $O(n\\log n)$ algorithm of Harvey and van der Hoeven, matching the conjectured optimum. Is $n\\log n$ optimal in this model, that is, does every fixed multitape Turing machine computing the product of two $n$-bit integers need time $\\Omega(n\\log n)$?","posedBy":"A. Schonhage and V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing 7 (1971), Section 1","yearPosed":1971,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: one fixed deterministic multitape Turing machine multiplies two $n$-bit integers exactly in worst-case time $O(n(\\lg n)^{1-\\kappa})$ with $\\kappa=2^{-182}$, for every $n$, so $T(n)/(n\\lg n)\\to0$ and the $n\\log n$ lower bound fails in this model. Corollaries: exact division and integer square root in the same time, and (with Harvey-van der Hoeven's reduction) transposition of $k\\times k$ binary matrices in $O(k^2(\\lg k)^{1-\\kappa})$. Not shown: any practical algorithm (constants are astronomically large), or anything about Boolean circuit size or other machine models; the true complexity of multiplication remains open.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it, and the README cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction (Theorem 1.1) and the history section's lower-bound scope subsection of the TeX source, read against the conjecture as the manuscript cites it. The manuscript makes the model explicit: one deterministic Turing machine with a fixed finite alphabet and a fixed number of one-dimensional tapes, exact for every input length, worst-case time. It notes that Schonhage's linear-time storage-modification-machine result and Groff's unit-cost RAM result are in other models. The algorithm and its 11 sections of tape and precision analysis were not refereed. No Lean formalization. The README cautions that unformalized results could have issues.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"The n log n conjecture for integer multiplication has organised the complexity of arithmetic since 1971; Harvey and van der Hoeven's matching upper bound was widely publicised, and beating it would overturn the standing expectation of optimality. Famous across algorithms. Below the 3SUM hypothesis (50) and the matrix multiplication exponent (55), above Kalai's conjecture for tight trees (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Integer multiplication below n log n","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Schonhage-Strassen 1971, Schnelle Multiplikation grosser Zahlen","url":"https://doi.org/10.1007/BF02242355","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fgs-sharp-connected-matching-conjecture","name":"The Furedi-Gyarfas-Simonyi Sharp Connected-Matching Conjecture","shortName":"Furedi-Gyarfas-Simonyi connected matching","problemNumber":null,"field":"Extremal graph theory, matchings","fieldGroup":"Combinatorics","statement":"A connected matching in a graph is a matching whose edges are pairwise joined by at least one edge. Furedi, Gyarfas and Simonyi (2005) conjectured that every graph with independence number two on $4t-1$ vertices has a connected matching of size $t$. It was verified for $t\\le17$ by them and for $t\\le22$ by Chen and Deng, and Cambie showed it follows from Hadwiger's conjecture. Does every graph $G$ with $\\alpha(G)=2$ on $4t-1$ vertices have a connected matching of size $t$?","posedBy":"Zoltan Furedi, Andras Gyarfas and Gabor Simonyi, Connected matchings and Hadwiger's conjecture, Combin. Probab. Comput. 14 (2005)","yearPosed":2005,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"From Theorem 1.1 of the Hadwiger paper: deleting at most three vertices from its graphs gives graphs with $\\alpha=2$ on $4t-1$ vertices whose connected-matching number stays below $t$ for all large $t$, so the sharp conjecture fails. The weaker conjecture of the same authors, that $\\mathrm{cm}(G)\\ge c_0|V(G)|$ for some absolute $c_0>0$, is not settled: the bound $\\mathrm{cm}0$ there is a fixed alphabet size for which it is NP-hard to distinguish satisfiable 2-to-1 games from games of value at most $\\delta$. Khot, Minzer and Safra (2018) proved the version with completeness $1-\\varepsilon$; perfect completeness, which is what several coloring and satisfiable-CSP reductions need, was known only with soundness about $23/24$ (Austrin-O'Donnell-Tan-Wright), and Fei-Minzer-Wang had it for 4-to-1 games. Is it NP-hard, for every fixed $\\delta>0$, to distinguish satisfiable 2-to-1 games from those of value at most $\\delta$?","posedBy":"Subhash Khot, On the power of unique 2-prover 1-round games (STOC 2002)","yearPosed":2002,"ageNote":"Khot's original statement allows fibers of size at most two; the manuscript proves the exact-two version, which implies it.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every fixed rational $\\delta\\in(0,1)$ there are $q=q(\\delta)$ and a deterministic polynomial-time reduction from 3-SAT to unweighted 2-to-1 games over $[2q],[q]$ with exact two-element fibers, value 1 on YES and at most $\\delta$ on NO instances. Corollaries make the Guruswami-Sinop maximum $k$-colorable subgraph hardness unconditional and recover Hastad's satisfiable Not-Two result. It does NOT prove the Rich 2-to-1 variant of Braverman-Khot-Minzer, the Unique Games Conjecture, or anything about alphabet size as a function of $\\delta$ beyond existence.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. Its external inputs are a perfect-completeness PCP, projection-game parallel repetition and the Khot-Minzer-Safra Grassmann expansion theorem, credited to their authors.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against Khot's conjecture: for every fixed rational $\\delta\\in(0,1)$, a deterministic polynomial-time reduction from 3-SAT to exact 2-to-1 games over alphabets $[2q],[q]$ with value 1 on satisfiable formulas and at most $\\delta$ otherwise. formalization.yaml lists ComparatorChallenges/PerfectCompleteness.json, declaration OAI.PerfectCompleteness.Theorem11.exists_reduction; the challenge JSON names solution module OAI.Computability.PerfectCompleteness.Main (present at the pinned commit). The statement was read here: for every rational $0<\\delta<1$ there is a reduction with a fixed alphabet $q\\ge2$, projection tables with exactly two preimages per right label, a nonempty unweighted edge list, a TM2 machine computing it in polynomial time from a binary 3SAT encoding, value exactly 1 on satisfiable inputs and at most $\\delta$ otherwise. That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The perfect-completeness 2-to-1 conjecture has been a well-known PCP question since 2002 and is the hypothesis behind several coloring and satisfiable-CSP hardness results; the imperfect version (Khot-Minzer-Safra 2018) was a major event. Below The Unique Games Conjecture (74) and the 3SUM hypothesis (50), above Polynomial-Factor Hardness for the Closest Vector Problem (35) and Sharp Hardness for MAX-3-CUT (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Perfect completeness for 2-to-1 games","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: Theorem 1.1 reduction (formalization.yaml file)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/PerfectCompleteness/Theorem11.lean","kind":"lean-proof"},{"label":"Lean comparator statement: perfect-completeness 2-to-1 reduction","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PerfectCompleteness.lean","kind":"lean-proof"},{"label":"Khot (2002), On the power of unique 2-prover 1-round games","url":"https://doi.org/10.1145/509907.510017","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"connes-rigidity-property-t-lattices","name":"Connes' rigidity conjecture for lattices in higher-rank and other property (T) local-field groups (W*-superrigidity of lattice group factors)","shortName":"Connes rigidity for property (T) lattices","problemNumber":null,"field":"Von Neumann algebras; rigidity of lattices","fieldGroup":"Analysis","statement":"Connes asked whether nonisomorphic ICC groups with Kazhdan's property (T) have nonisomorphic group von Neumann algebras, and what the fundamental groups of these factors are (Problems 1-2). Popa (ICM 2006) displayed the stronger stable form: an isomorphism $L(\\Gamma)^t\\cong L(\\Lambda)$, with $\\Lambda$ any ICC group, should force $t=1$ and come from a group isomorphism up to a character and inner conjugacy. The motivating examples are lattices such as $\\mathrm{SL}_3(\\mathbb Z)$; for general property (T) groups the conjecture was refuted in 2026 by wreath-like counterexamples. For ICC groups commensurable with lattices in products of higher-rank or rank-one property (T) groups over local fields, is $L(\\Gamma)$ W*-superrigid in Popa's stable sense?","posedBy":"Alain Connes, Noncommutative Geometry (1994), Ch. V, Appendix B.epsilon, Problems 1-2 (rigidity question from about 1980); stable form by Sorin Popa (ICM 2006)","yearPosed":1980,"ageNote":"The manuscripts cite Connes 1980 (countable fundamental group of property (T) factors) and the problems as recorded in Connes' 1994 book, plus Popa's 2006 ICM formulation. The lattice class is the setting the conjecture was formulated around; the general property (T) form was refuted in 2026 (catalog entries connes-rigidity-conjecture and connes-rigidity-icc-property-t).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Principal (Theorem 1.2): for $\\Gamma$ in the class $\\mathscr K$ (ICC groups commensurable with lattices in finite products of noncompact property (T) groups $\\mathbf H_i(k_i)^+$ over $\\mathbb R$, $\\mathbb C$ or $p$-adic fields), any ICC $\\Lambda$ and arbitrary scalar 2-cocycles, every bifinite $L_\\mu(\\Gamma)$-$L_\\omega(\\Lambda)$ correspondence is a summand of finite sums of models built from finite-index subgroup isomorphisms and finite-dimensional projective representations. Companion: $L_\\omega(\\Lambda)^t\\cong L_\\nu(\\Pi)^s$ for $\\Lambda\\in\\mathscr K$ and ICC $\\Pi$ iff $t=s$ and the groups are isomorphic with cohomologous cocycles, every such isomorphism being implemented canonically; these factors have trivial fundamental group (e.g. $L(\\mathrm{SL}_3(\\mathbb Z))$). Not shown: lattices in rank-one groups without property (T), or general property (T) groups.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 23, 2026: the geometric arithmetic exhaustion theorem for finite correspondences (principal) and an algebraic companion that derives stable canonical recovery, trivial fundamental groups and all finite-index factor neighbors from it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts were read against Connes' problems and Popa's formulation as the manuscripts cite them; the cited texts themselves were not read. The headline theorem rests on a long geometric argument (boundary measurements, local-field branches, locality, gluing) that was not refereed. There is no Lean formalization for this family (lean/docs/286.md does not exist at the pinned commit). Resolution is partial because Connes' question concerns all ICC property (T) groups, where the answer is negative, and this proves it for the lattice class only. Given the claim's weight in operator algebras, it warrants expert review before anything else.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":44,"significanceNote":"W*-superrigidity of higher-rank lattices such as SL3(Z), and triviality of their fundamental groups, are among the best-known open problems of von Neumann algebra theory since Connes around 1980. Scored as the lattice case of Connes' rigidity conjecture, just below that entry (45), above Connes' rigidity for ICC property (T) groups (35), below the free group factor isomorphism problem (68) and Kadison's similarity problem (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Arithmeticity of twisted finite correspondences for lattices over local fields","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The arithmetic category and stable recovery of lattice factors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026/paper.pdf","kind":"paper"},{"label":"Connes, Noncommutative Geometry (1994), Ch. V App. B (problems)","url":"https://alainconnes.org/wp-content/uploads/book94bigpdf.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"goldfeld-conjecture-quadratic-twists","name":"Goldfeld's conjecture on analytic ranks of quadratic twists","shortName":"Goldfeld's conjecture for quadratic twists","problemNumber":null,"field":"Arithmetic of elliptic curves; arithmetic statistics","fieldGroup":"Number theory","statement":"For an elliptic curve $E/\\mathbb{Q}$ and squarefree $d$ let $E^{(d)}$ be the quadratic twist and $a(E^{(d)})=\\mathrm{ord}_{s=1}L(E^{(d)},s)$ its analytic rank. Goldfeld (1979, Conjecture B) conjectured that the average analytic rank over the twist family is $1/2$; the density form predicts that analytic ranks 0 and 1 each occur for half of all twists. Known before: positive proportions for special curves (James, Vatsal, Kriz-Li for curves with a rational 3-isogeny), conditional means under GRH (Heath-Brown, Fiorilli), and Smith's theorem that 2-power Selmer coranks 0 and 1 each have density 1/2, giving Goldfeld's conjecture under BSD. For every $E/\\mathbb{Q}$, do analytic ranks 0 and 1 each have density $1/2$ among the twists, and does the mean analytic rank tend to $1/2$?","posedBy":"Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields, Number Theory Carbondale 1979, LNM 751, p. 113, Conjecture (B)","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Density paper: for every $E/\\mathbb{Q}$ and $j\\in\\{0,1\\}$, the proportion of squarefree $0<|d|\\le X$ with $a(E^{(d)})=j$ tends to $1/2$; on a density-one set analytic and Mordell-Weil ranks agree and Sha is finite. The main new input is the 2-converse: if the $2^\\infty$-Selmer corank is 0 or 1 it equals the analytic and Mordell-Weil ranks and Sha is finite. Mean paper: $\\frac{1}{\\#\\mathcal D(Y)}\\sum_{d}a(E^{(d)})\\to1/2$, via a tail bound on high-rank twists, plus algebraic-rank moments. Both use signed squarefree twists ordered by $|d|$, not Goldfeld's discriminant ordering; no higher analytic-rank moments are claimed.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated 23 September 2026: the density paper (principal) and the mean analytic rank paper, which adds a tail estimate. The density paper's key input is the pointwise 2-converse, combined with Smith's human Selmer-distribution theorem. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of both manuscripts were read against Goldfeld's Conjecture (B) as the manuscripts cite it. The proofs were not refereed. No Lean formalization: neither manuscript is in lean/formalization.yaml and lean/docs/006.md does not exist at the pinned commit. Conditional inputs: the density theorem uses Smith's distribution of 2-power Selmer coranks (arXiv 2503.17619, cited as version 1); the mean uses the density theorem plus a new tail estimate. Scope the papers state: twists are counted over signed squarefree $d$ ordered by $|d|$, whereas Goldfeld's Conjecture (B) orders quadratic-field discriminants; the mean paper says it resolves the conjecture 'in this counting convention'. The density manuscript cites companions dated 24 September and the mean paper cites a 'V2' of the density paper, so texts were revised after their dates.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Goldfeld's conjecture is a central prediction of arithmetic statistics, known throughout the elliptic-curves community, and was unconditionally known only for special families. Above the elliptic curve rank records (50) and the p-converse in Selmer corank at most one (45), below Artin's primitive root conjecture (63) and the BSD formula in Selmer corank at most one (78), which builds on this result.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","sourceName":"OpenAI math release: Goldfeld's analytic density conjecture and the 2-converse for elliptic curves","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Same family: The mean analytic rank of quadratic twists of elliptic curves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026/paper.pdf","kind":"paper"},{"label":"Goldfeld 1979, Conjectures on elliptic curves over quadratic fields","url":"https://doi.org/10.1007/BFb0062705","kind":"problem-record"},{"label":"The BSD formula entry that depends on this result","url":"https://vibemathed.com/problem/bsd-formula-selmer-corank-one","kind":"other"},{"label":"The Selmer p-converse entry, whose p = 2 case is proved here","url":"https://vibemathed.com/problem/p-converse-selmer-corank-one-every-prime","kind":"other"},{"label":"Earlier edition (September 23, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): The mean analytic rank of quadratic twists of elliptic curves","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"borsuk-conjecture-dimension-nine","name":"The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine","shortName":"Borsuk's conjecture fails in dimension 9","problemNumber":null,"field":"Discrete geometry; Borsuk partition problem","fieldGroup":"Combinatorics","statement":"Borsuk (1933) asked whether every bounded set of positive diameter in $\\mathbb R^d$ can be covered by $d+1$ subsets of strictly smaller diameter. Kahn and Kalai disproved this in high dimensions in 1993, and the smallest known failing dimension then fell through 946, 561, 323, 321, 298, 65 (Bondarenko) and 64 (Jenrich-Brouwer, 2014) to 63 (Grinsztajn, 2026). What is the smallest dimension $d$ in which Borsuk's assertion fails, and can the record of 63 be lowered substantially?","posedBy":"K. Borsuk, Drei Satze uber die n-dimensionale euklidische Sphare, Fundamenta Mathematicae 20 (1933)","yearPosed":1933,"ageNote":"Borsuk posed the partition question in 1933; the smallest-failing-dimension question became the live form of the problem after the Kahn-Kalai disproof of 1993.","solveType":"disproved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: the compact set of rank-one orthogonal projectors $uu^T$, $u\\in\\mathbb R^4$ a unit vector, with the Frobenius metric, lies in the nine-dimensional affine space of trace-one symmetric $4\\times4$ matrices, has diameter $\\sqrt2$ and cannot be covered by ten sets of smaller diameter, so Borsuk's assertion fails in dimension 9 (the record was 63). A corollary extends this to compact counterexamples in every dimension $d\\ge9$. Not shown: the smallest failing dimension (dimensions 4 to 8 remain open) or the exact Borsuk number of the set. The witness is a continuum, not a finite point set.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The main theorem has a Lean formalization in the release (Comparator challenge BorsukNine).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Borsuk's question. Lean-checked on the release's Comparator challenge BorsukNine (declaration OAI.BorsukNine.main_theorem, listed in lean/formalization.yaml). Its statement was read here: the image of the unit sphere of R^4 under $u\\mapsto uu^T$, inside the Euclidean space of 4x4 matrices, is compact, lies in the trace-one symmetric matrices, has diameter $\\sqrt2$, and admits no cover by ten subsets each of diameter less than $\\sqrt2$. That is the headline; that trace-one symmetric matrices form a nine-dimensional affine space is elementary and not itself stated. Not rebuilt here. The paper says it does not determine the smallest failing dimension.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Scored on the question answered, the smallest failing dimension of Borsuk's conjecture, as for the existing dimension-63 entry: the conjecture itself fell in 1993. The drop from 63 to 9 is far larger than any earlier step, but the score is on the question, not the step. Tied with the Borsuk dimension-63 record (30) and the Hadamard matrix of order 668 (30), below Erdos's planar unit distance conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A nine-dimensional counterexample to Borsuk's covering assertion","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge BorsukNine (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BorsukNine.lean","kind":"lean-proof"},{"label":"Borsuk 1933, Drei Satze uber die n-dimensionale euklidische Sphare","url":"https://doi.org/10.4064/fm-20-1-177-190","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gyarfas-tree-cover-conjecture","name":"Gyarfas's conjecture on covering edge-colored complete graphs by r-1 monochromatic trees","shortName":"Gyarfas tree-cover conjecture","problemNumber":null,"field":"Ramsey-type covering problems; monochromatic trees","fieldGroup":"Combinatorics","statement":"Color each edge of a finite complete graph with one of $r$ colors. A monochromatic tree cover is a family of monochromatic trees (overlaps and single vertices allowed) whose vertex sets cover the graph. The $r$ monochromatic stars at any vertex always give a cover by $r$ trees. Gyarfas conjectured that $r-1$ trees always suffice; Erdos, Gyarfas and Pyber recorded that this covering formulation is equivalent to the intersecting case of Ryser's conjecture. Can the vertices of every $r$-edge-colored complete graph be covered by at most $r-1$ monochromatic trees?","posedBy":"Andras Gyarfas; recorded by Erdos, Gyarfas and Pyber (J. Combin. Theory Ser. B 51, 1991)","yearPosed":1991,"ageNote":"The manuscripts cite Erdos-Gyarfas-Pyber (1991, Section 3) and Milicevic (2017, Conjecture 1); they give no earlier source, so 1991 is the earliest date the literature here supports.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For every sufficiently large prime $q$ (September 27 paper), and for $r=s^n+1$ with $s\\equiv2\\pmod3$ a large prime and $n$ large odd (September 23 paper), there is a finite complete graph with an $r$-edge-coloring whose vertices need exactly $r$ monochromatic trees to cover them; the same holds for connected monochromatic subgraphs. Built from the Ryser counterexamples by coloring each pair of hyperedges by a part where they meet. Not addressed: small $r$, or the distinct vertex-disjoint tree-partition conjecture of Erdos-Gyarfas-Pyber.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two manuscripts: the extension-field construction (September 23, 2026) and the balanced prime-order construction (September 27, 2026); the later paper says its proof is independent, and the earlier imports only three elementary lemmas from it.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the tree-cover corollary in both manuscripts was read against the conjecture as stated by Milicevic (Conjecture 1) and Erdos-Gyarfas-Pyber. The Lean main results for the September 23 paper (RyserCovering, RyserOddExtensions; and BalancedRyser for the companion, not in the formalization catalogue) were read; they state the hypergraph counterexamples only. The transfer to edge-colored complete graphs is a short standard argument proved in the papers and not formalised. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A well-known conjecture in Ramsey-type covering problems, but recorded since 1991 as equivalent to the intersecting case of Ryser's conjecture, so its fall is carried by Ryser's. Scored as a conjecture famous within one community: level with Dean's conjecture for k = 5 (32) and the Seymour second-neighbourhood outdegree-7 case (32), below Ryser's conjecture (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to Ryser's covering conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Balanced counterexamples to Ryser's conjecture at prime orders (September 27, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf","kind":"paper"},{"label":"Lean proof: OAI/Combinatorics/Ryser/Construction/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Ryser/Construction/Main.lean","kind":"lean-proof"},{"label":"Lean proof: OAI/Combinatorics/Ryser/OddExtensions.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Ryser/OddExtensions.lean","kind":"lean-proof"},{"label":"Erdos, Gyarfas, Pyber, Vertex coverings by monochromatic cycles and trees (JCTB 1991)","url":"https://doi.org/10.1016/0095-8956(91)90007-7","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"zariski-cancellation-complex-dimension-four","name":"The Zariski cancellation problem for complex affine space in dimension four","shortName":"Zariski cancellation over C, dimension 4","problemNumber":null,"field":"Affine algebraic geometry","fieldGroup":"Geometry & topology","statement":"Zariski's cancellation problem asks whether affine space is determined by its cylinder: if $X$ is an affine variety with $X\\times\\mathbb A^1\\cong\\mathbb A^{n+1}$, must $X\\cong\\mathbb A^n$? Algebraically, over $\\mathbb C$: if $A$ is a finitely generated complex algebra with $A[w]\\cong\\mathbb C^{[n+1]}$, is $A\\cong\\mathbb C^{[n]}$? It holds for $n=1$ (Abhyankar-Eakin-Heinzer) and $n=2$ (Fujita, Miyanishi-Sugie). Gupta (2014) showed it fails for every $n\\ge3$ in positive characteristic, and the characteristic-zero case in dimensions $n\\ge3$ is listed as open in Gaifullin-Petrov (July 2026) and Gupta's ICM 2022 survey. Does cancellation hold for complex affine space in dimension $n\\ge3$, in particular for $n=4$?","posedBy":"Oscar Zariski; the manuscript cites Abhyankar-Eakin-Heinzer (1972), Fujita (1979), Gupta's ICM 2022 survey and Gaifullin-Petrov (2026) for the problem and its status","yearPosed":null,"ageNote":"The manuscript gives no date for the posing. The survey literature it cites (Gupta) traces the problem to Zariski at the 1949 Paris colloquium, in a birational form; the affine form is treated in Abhyankar-Eakin-Heinzer (1972), the earliest cited source. Left blank rather than choose between 1949 and 1972.","solveType":"disproved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: with $P=\\mathbb C[p,s,u,F,J]$, $x=s^2+u^3+p^2F$ and $H=x^2F-(1+2sx)J-p^2J^2-pu$, the algebra $A=P/(H)$ is an integral complex domain of dimension four with $A[w]\\cong\\mathbb C^{[5]}$ but $A\\not\\cong\\mathbb C^{[4]}$. The stabilization uses that $H$ is a residual coordinate (El Kahoui-Ouali, Dutta-Lahiri); non-polynomiality is proved by degenerating to a graded algebra, lifting a putative additive action through a torus bundle over a quadric, and excluding it by a Mason-Stothers abc argument. It answers only $n=4$: dimension three is not treated, and the paper notes that adjoining variables to a counterexample does not by itself give counterexamples in higher dimensions.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the cancellation problem as the manuscript states it. The degeneration and rigidity arguments were not refereed. Lean-checked on the release's Comparator challenge ComplexCancellation (declaration OAI.ComplexCancellation.main, file OAI/Algebra/AffineCancellation/Main.lean, listed in lean/formalization.yaml). Its statement was read here: it defines the explicit H in MvPolynomial (Fin 5) over C and asserts that A = P/(H) is of finite type, a domain, of Krull dimension 4, that Polynomial A is C-algebra isomorphic to MvPolynomial (Fin 5) C, and that A is not isomorphic to MvPolynomial (Fin 4) C. That is the headline claim exactly. Not rebuilt here. The release's Lean scope note says the stable-coordinate and line-bundle consequences are not formalized.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"Zariski's cancellation problem is one of the central questions of affine algebraic geometry; Gupta's positive-characteristic disproof was celebrated, and characteristic zero in dimension at least three was the remaining famous case. Only dimension four is answered. Tied with Kothe's conjecture (60) and Baum-Connes (60), below the Jacobian Conjecture (65), its sibling problem.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An explicit failure of complex affine-space cancellation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge ComplexCancellation (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ComplexCancellation.lean","kind":"lean-proof"},{"label":"Gupta, The Zariski cancellation problem and related problems (ICM 2022)","url":"https://doi.org/10.4171/ICM2022/151","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"eilenberg-ganea-conjecture","name":"The Eilenberg-Ganea conjecture","shortName":"Eilenberg-Ganea conjecture","problemNumber":null,"field":"Geometric group theory; cohomological and geometric dimension of groups","fieldGroup":"Geometry & topology","statement":"For a group $G$, the cohomological dimension $\\mathrm{cd}\\,G$ is the projective dimension of the trivial $\\mathbb Z[G]$-module $\\mathbb Z$, and the geometric dimension $\\mathrm{gd}\\,G$ is the least dimension of a $K(G,1)$ CW complex; always $\\mathrm{cd}\\,G\\le\\mathrm{gd}\\,G$. Eilenberg and Ganea (1957) proved $\\mathrm{cd}\\,G=\\mathrm{gd}\\,G$ whenever $\\mathrm{cd}\\,G\\ge3$, and Stallings and Swan settled dimension one. Bestvina and Brady (1997) showed that at least one of the Eilenberg-Ganea conjecture and Whitehead's asphericity conjecture is false, without deciding which. Does every group of cohomological dimension two admit a two-dimensional classifying space?","posedBy":"Samuel Eilenberg and Tudor Ganea, On the Lusternik-Schnirelmann category of abstract groups, Ann. of Math. 65 (1957)","yearPosed":1957,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a finitely generated residually finite group $G$ with $\\mathrm{cd}\\,G=2$ and $\\mathrm{gd}\\,G=3$; no two-dimensional $K(G,1)$ exists regardless of the number of cells. $G$ is a Bestvina-Brady group (the kernel of the height map of a right-angled Artin group); it is of type FL and FP$_\\infty$ but not finitely presented. The obstruction compares an acyclic two-dimensional level complex with any hypothetical aspherical presentation, using small SU(2) labelings and a degree argument. It does not give a finitely presented counterexample, and by itself it does not decide Whitehead's asphericity conjecture.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture as the paper states it (with the 1957 reference); it claims a finitely generated, residually finite group $G$, the height kernel of a right-angled Artin group over an acyclic flag triangulation of the presentation complex of $\\langle x,y\\mid x^2=y^5,x^2=(xy^{-1})^3\\rangle$, with $\\mathrm{cd}\\,G=2$ and $\\mathrm{gd}\\,G=3$. The proof was not refereed. The challenge is not in the formalization catalogue (lean/formalization.yaml); it is found through lean/docs/249.md, and its solution module OAI.Topology.EilenbergGanea.Main exists at the pinned commit. The statement ComparatorChallenges/EilenbergGanea.lean was read here: it defines the group as the kernel of the height map on the Artin group of an explicit seed graph and asserts it is finitely generated, residually finite, has cohomological dimension 2 (projective dimension of the trivial module), has a 3-dimensional classifying CW space, and has no 2-dimensional one in any universe, with no bound on cells. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"A classical problem of group cohomology and topology open since 1957, known to every geometric group theorist and tied to Whitehead's asphericity conjecture through Bestvina-Brady. Placed level with Kaplansky's quasitrace problem (55), below Cannon's Conjecture (62) and Kaplansky's zero-divisor conjecture (62), and above Wall's manifold-realization question for Poincare duality groups (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A finitely generated counterexample to the Eilenberg-Ganea conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.EilenbergGanea.source_main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Topology/EilenbergGanea/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: EilenbergGanea.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/EilenbergGanea.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bogomolov-pop-reconstruction-conjecture","name":"The Bogomolov-Pop conjecture: function fields over algebraically closed fields are determined by their pro-l abelian-by-central Galois groups","shortName":"Bogomolov-Pop reconstruction conjecture","problemNumber":null,"field":"Birational anabelian geometry","fieldGroup":"Number theory","statement":"Let $K/k$ be a function field of transcendence degree at least two over an algebraically closed field $k$ of characteristic different from a prime $\\ell$, and let $\\Pi_K^c$ be the maximal pro-$\\ell$ quotient of its absolute Galois group in which commutators are central, with abelianization $\\Pi_K^a$. Bogomolov (1991) proposed that this small quotient determines the field; the precise Isom form, with its scalar and Frobenius ambiguities, is Conjecture 1 in Topaz (2016). Known cases had constants an algebraic closure of a finite field (Bogomolov-Tschinkel, Pop) or extra inertia data (Pop). Is the natural map from field isomorphisms of perfect closures, modulo Frobenius, to bracket-compatible isomorphisms $\\Pi_L^a\\to\\Pi_K^a$ modulo $\\mathbb Z_\\ell^\\times$ a bijection?","posedBy":"Fedor Bogomolov; Isom form recorded by Adam Topaz","yearPosed":1991,"ageNote":"Bogomolov, On two conjectures in birational algebraic geometry (1991), as cited by the manuscript; the precise Isom formulation is Topaz, Reconstructing function fields from rational quotients of mod-l Galois groups, Math. Ann. 366 (2016), Introduction, Conjecture 1.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (principal paper): for function fields $K/k$, $L/l$ of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from $\\ell$, the map from isomorphisms of perfect closures modulo Frobenius to bracket-compatible isomorphisms $\\Pi_L^a\\to\\Pi_K^a$ modulo $\\mathbb Z_\\ell^\\times$ is bijective; equal characteristic and transcendence degree are forced. Companions: the same reconstruction from $K_1^M/\\ell$, $K_2^M/\\ell$ and their product (removing Topaz's rational-subgroup data and transcendence degree at least five), and from $K_1^M/p$ with Steinberg relations in characteristic $p$. Not shown: transcendence degree one, or fields with non-algebraically-closed constants.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript (September 23, 2026) proves the pro-l Galois form; two companions (October 5, 2026) prove analogues from mod-l Milnor K-theory (away from the characteristic) and from mod-p Milnor K-theory in characteristic p.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the conjecture as cited; Theorem 1.1 states the bijection for arbitrary algebraically closed constants of characteristic different from $\\ell$, transcendence degree at least two, including surfaces and $\\ell=2$. Lean: the challenge BogomolovPopInjectivity (theorem OAI.BogomolovPop.main_graph_injective) exists at the pinned commit but is not in the formalization catalogue; its scope page says it proves only injectivity of the reconstruction map, and existence of a field isomorphism for every admissible Galois isomorphism is outside it. Because the formal statement covers one direction, not the headline, the tier is unreviewed. Not rebuilt.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"The central conjecture of Bogomolov's birational anabelian program, open for 35 years beyond finite-field constants, with major partial work by Bogomolov-Tschinkel and Pop; famous in anabelian geometry, little known outside. Placed below the section conjecture for modular curves (42) and the local p-adic section conjecture (50), above Siegel-Yao ball packing (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Bogomolov-Pop reconstruction theorem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Reconstruction of Function Fields from Mod-l Milnor K-Theory","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf","kind":"paper"},{"label":"Companion: Reconstruction from Milnor K-theory modulo the characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf","kind":"paper"},{"label":"Lean formalization (injectivity only): BogomolovPop/Injectivity.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/BogomolovPop/Injectivity.lean","kind":"lean-proof"},{"label":"Topaz, Reconstructing function fields from rational quotients of mod-l Galois groups (Conjecture 1)","url":"https://doi.org/10.1007/s00208-015-1327-4","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"morse-number-arnold-conjecture","name":"The Morse-number Arnold conjecture for nondegenerate Hamiltonian diffeomorphisms","shortName":"Morse-number Arnold conjecture","problemNumber":null,"field":"Symplectic topology; Hamiltonian fixed points, Morse theory","fieldGroup":"Geometry & topology","statement":"For a closed connected symplectic manifold $(M,\\omega)$ let $\\mathrm{Morse}(M)$ be the least number of critical points of a Morse function on $M$. Floer theory and its virtual extensions show that a nondegenerate Hamiltonian diffeomorphism has at least the sum of the Betti numbers many fixed points, but the Morse number can be larger. The Morse-number form of the Arnold conjecture asks: if $H$ is a one-periodic Hamiltonian whose time-one map is nondegenerate, is the number of fixed points with contractible orbits at least $\\mathrm{Morse}(M)$?","posedBy":"V. I. Arnold, First steps in symplectic topology (1986); recorded as Conjecture 1.1 in R. Golovko, On variants of Arnold conjecture, Arch. Math. (Brno) 56 (2020)","yearPosed":1986,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are a closed symplectic 12-manifold and a one-periodic Hamiltonian with all fixed points nondegenerate and contractible and $\\#\\mathrm{Fix}=\\chi(M)+128<\\mathrm{Morse}(M)$; Corollary 1.2 makes $\\mathrm{Morse}(M)-\\#\\mathrm{Fix}$ arbitrarily large (the manifold depends on the deficit). The mechanism: the augmentation ideal of $A_5^k$ needs two generators while $d(A_5^k)$ is unbounded, realized by relative handle cancellation on $X\\times\\mathbb R^8$. The fixed-point count still exceeds the homological (Floer) bounds. The examples are not simply connected; the October 5 companions give simply connected Kahler examples below the stable Morse number.","claimIssueNote":"The release says: 'Ma [Theorem 1.5] asserted that the minimum number of fixed points over all Hamiltonian diffeomorphisms always equals Crit(M). Theorem 1.1 contradicts that assertion' (degenerate paper) and 'Ma's preprint asserts that the minimum number of fixed points of a nondegenerate Hamiltonian diffeomorphism always equals the ordinary Morse number ... Theorem 1.1 contradicts that unrestricted assertion' (Morse-number paper). The other side: R. Ma, Proofs on Arnold conjectures, arXiv 0808.0613v7 (2013), Theorem 1.5, claims these lower bounds hold in general. Ma's preprint is unpublished; no referee has ruled between them.","solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against Golovko's Conjecture 1.1 as the paper states it. The paper claims twelve-dimensional examples $M=X\\times(S^2)^4$, $\\pi_1(X)=A_5^k$, with all fixed points nondegenerate and contractible and $\\#\\mathrm{Fix}=\\chi(M)+128<\\mathrm{Morse}(M)$, with arbitrarily large deficit. The proof was not refereed and is not formalized. Graded contested because Ma's 2013 preprint claims the opposite; see the claim note.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The Morse-number form is a natural strengthening of the homological Arnold conjecture, discussed by Arnold and by Eliashberg-Gromov, but known mainly to symplectic specialists. Placed below Vicente's Gromov width question (45), level with the Projected Lax Conjecture (35), above Arnold's corank problem (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A nondegenerate counterexample to the Morse-number Arnold bound","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kadison-similarity-problem","name":"Kadison's Similarity Problem","shortName":"Kadison similarity problem","problemNumber":null,"field":"Operator algebras","fieldGroup":"Analysis","statement":"Kadison (1955) asked whether every bounded unital homomorphism $\\pi$ from a unital $C^*$-algebra $A$ into the bounded operators $B(H)$ on a Hilbert space is similar to a $*$-homomorphism, that is, whether there is a bounded invertible $S$ such that $S\\pi(\\cdot)S^{-1}$ is a $*$-homomorphism. Known cases included nuclear algebras (Bunce, Christensen), cyclic representations and algebras without tracial states (Haagerup) and II$_1$ factors with property $\\Gamma$ (Christensen). Haagerup showed similarity is equivalent to complete boundedness, and Kirchberg showed the problem is equivalent to the derivation problem. Is every bounded representation of a $C^*$-algebra on Hilbert space similar to a $*$-representation?","posedBy":"Richard V. Kadison, On the orthogonalization of operator representations, Amer. J. Math. 77 (1955)","yearPosed":1955,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every unital $C^*$-algebra $A$, every complex Hilbert space $H$ and every bounded complex-linear unital homomorphism $\\pi:A\\to B(H)$ there is a bounded invertible $S$ with $S\\pi(\\cdot)S^{-1}$ a $*$-homomorphism. It goes through Theorem 1.2, an absolute-constant commutator bound $\\|[Y^{(h)},X]\\|\\le C\\,g_P(Y)\\|X\\|$ over all von Neumann algebras $P$, all $Y$ and all matrix amplifications, combined with Kirchberg's derivation criterion. Corollary 1.3: every von Neumann algebra is hyperreflexive with constant at most $2C$, a statement known to be equivalent to a positive answer (Eleftherakis-Paulsen). No bound on the condition number of $S$ and no value of $C$ is given.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1, Theorem 1.2 and Corollary 1.3, read against Kadison's 1955 problem as cited. The proof was not refereed. The theorem is unrestricted: no separability, nuclearity or normality assumption, arbitrary Hilbert space. Lean-checked on the release's own Comparator challenge KadisonSimilarity together with its solution module, both present at the pinned commit; the challenge is not listed in the release's formalization catalogue, the statement was read here but not independently audited, and the development was not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Kadison's similarity problem (1955) is one of the best-known open problems in operator algebras, worked on by Haagerup, Christensen, Pisier and Kirchberg for seventy years. Placed just below Kothe's conjecture (60) and the sofic groups question (60), above the Cycle Double Cover Conjecture (55) and well above Connes' Rigidity Conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Kadison's similarity theorem through uniform derivation estimates","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kadison 1955, Amer. J. Math. 77","url":"https://doi.org/10.2307/2372645","kind":"problem-record"},{"label":"Lean Comparator challenge KadisonSimilarity (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KadisonSimilarity.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tingley-problem","name":"Tingley's problem: does every surjective isometry between unit spheres of Banach spaces extend to a linear isometry?","shortName":"Tingley's problem","problemNumber":null,"field":"Functional analysis; geometry of Banach spaces","fieldGroup":"Analysis","statement":"For a real normed space $X$ let $S_X$ be its unit sphere with the inherited metric. Mazur and Ulam (1932) proved that surjective isometries between real normed spaces are affine, but a unit sphere has empty interior, so this does not apply. Tingley (1987) asked whether every surjective isometry $f:S_X\\to S_Y$ between unit spheres of real Banach spaces extends to a real-linear isometry of $X$ onto $Y$. Positive answers were known for many classical spaces ($C_0$, $L^p$, von Neumann algebras and their preduals, unital $C^*$-algebras) and for every two-dimensional space (Banakh 2022), but not in general, even in finite dimension. Does every surjective isometry between the unit spheres of two real Banach spaces extend to a surjective real-linear isometry?","posedBy":"Daryl Tingley, Isometries of the unit sphere, Geometriae Dedicata 22 (1987)","yearPosed":1987,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $X,Y$ are nonzero real Banach spaces and $f:S_X\\to S_Y$ is a surjective isometry, then $T(0)=0$, $T(x)=\\|x\\|f(x/\\|x\\|)$ is a surjective real-linear isometry, the unique linear extension of $f$. No separability, reflexivity, smoothness, strict convexity or finite-dimensionality is assumed. For complex spaces only real linearity is concluded. The proof uses a maximal-defect argument, common supports and Darbo's fixed-point theorem. Incomplete normed spaces are not treated.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Tingley's question as the paper states it from Tingley (1987); it claims that for nonzero real Banach spaces the radial map $T(x)=\\|x\\|f(x/\\|x\\|)$ is a surjective real-linear isometry and the unique linear extension. The proof was not refereed. lean/formalization.yaml lists a main result (comparator TingleySphereIsometry, declaration OAI.Tingley.tingley_sphere_isometry_full, file OAI/Analysis/SphereIsometry/Extension.lean). ComparatorChallenges/TingleySphereIsometry.lean was read here: for complete, nontrivial real normed spaces of any size and a surjective isometry between unit spheres, it asserts a linear isometric equivalence agreeing with $f$ on the sphere, given by the radial formula, and unique among real-linear maps extending $f$. This states the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A well-known problem of Banach space geometry since 1987 with a large literature of special cases, open even in finite dimension three. Placed level with Stein's dimension-free weak (1,1) Riesz transform problem (38) and just below Banach's isometric conjecture (40), above Kalton's Problem 2 on bi-Lipschitz copies of c0 (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A positive solution to Tingley's problem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Tingley.tingley_sphere_isometry_full)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/SphereIsometry/Extension.lean","kind":"lean-proof"},{"label":"Comparator statement: TingleySphereIsometry.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/TingleySphereIsometry.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"donaldson-tamed-to-compatible-conjecture","name":"Donaldson's tamed-to-compatible conjecture for almost complex four-manifolds","shortName":"Donaldson's tamed-to-compatible conjecture","problemNumber":null,"field":"Symplectic geometry of four-manifolds; almost complex structures","fieldGroup":"Geometry & topology","statement":"Let $X$ be a closed smooth four-manifold with a smooth almost complex structure $J$. A symplectic form $\\omega$ tames $J$ if $\\omega(v,Jv)>0$ for all nonzero tangent vectors $v$, and is compatible with $J$ if moreover $\\omega(Ju,Jv)=\\omega(u,v)$. Donaldson asked whether a $J$ tamed by some symplectic form must also admit a compatible symplectic form. Known cases: integrable $J$ (complex surfaces, via Buchdahl, Lamari and Li-Zhang), $\\mathbb{CP}^2$ (Gromov), generic tamed $J$ when $b_2^+=1$ (Taubes), $S^2\\times S^2$ and some rational surfaces (Li-Zhang), and conditional results under $h_J^-=b_2^+-1$. If an almost complex structure on a closed four-manifold is tamed by a symplectic form, is it compatible with some symplectic form?","posedBy":"S. K. Donaldson, Two-forms on four-manifolds and elliptic equations, in Inspired by S. S. Chern (2006), Section 5.2, Question 2","yearPosed":2006,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if a smooth almost complex structure $J$ on a closed connected smooth four-manifold is tamed by a symplectic form, there is a smooth symplectic form compatible with the same $J$. The October 6 edition corrects the cone corollary of the first edition, which had claimed $\\mathcal K_J^t=\\mathcal K_J^c+H_J^-$ in general via Li-Zhang: it now characterizes the compatible cone by strict intersection positivity on the closure of the invariant part of the taming cone, proves that equality only when $h_J^-=b_2^+-1$ (so when $b_2^+=1$ every taming class still contains a compatible form), and gives a four-torus example where the inclusion is strict. Not shown: that an arbitrary taming class contains a compatible form, anything in dimensions above four, or Donaldson's separate a priori estimate conjecture for the prescribed-volume equation.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The main theorem has a Lean formalization in the release (Comparator challenge TamingCompatibility).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Donaldson's question as cited; the October 6 edition leaves Theorem 1.1 unchanged and revises only the cone corollary (see the result note), so the headline and the Candidate rating stand. Lean-checked on the release's Comparator challenge TamingCompatibility (declaration OAI.TamingCompatibility.taming_implies_compatibility, listed in lean/formalization.yaml). Its statement was read here: for a compact connected Hausdorff second-countable smooth manifold modelled on R^4 and a smooth almost complex structure $J$ with $J^2=-1$, if some symplectic two-form tames $J$ then some symplectic two-form tames $J$ and is $J$-invariant. Two-forms, smoothness and closedness are encoded by hand through pullbacks along smooth chart maps rather than Mathlib's differential forms; that encoding was read but not audited in depth. It states the headline claim, with the same $J$ and no cohomology class prescribed. Not rebuilt here. The Lean file is unchanged at the new commit; lean/formalization.yaml still lists the September 23 manuscript.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A well-known conjecture of four-dimensional symplectic geometry, posed by Donaldson in 2006 and attacked through Taubes's curve-counting methods and the Li-Zhang cone comparison; famous within symplectic topology. Tied with Vicente's question on the Gromov width of polar products (45), below the strong hyperkahler SYZ conjecture (48) and well below Arnold's nearby Lagrangian conjecture (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf","sourceName":"OpenAI math release: Taming implies compatibility on four-manifolds","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Lean Comparator challenge TamingCompatibility (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/TamingCompatibility.lean","kind":"lean-proof"},{"label":"Earlier edition (September 23, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"anderson-extended-states-simon-problem-1","name":"Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher","shortName":"Anderson extended states, d >= 3 (Simon's Problem 1)","problemNumber":null,"field":"Random Schrodinger operators; delocalization","fieldGroup":"Mathematical physics","statement":"Let $H_\\lambda=-\\Delta+\\lambda V$ on $\\ell^2(\\mathbb{Z}^d)$ with iid site potentials $V_x$ uniform on $[-1,1]$. Anderson (1958) predicted localization at strong disorder; physics predicts that for $d\\ge 3$ and weak disorder extended states survive. Absolutely continuous spectrum was proved on the Bethe lattice and on trees (Klein, Aizenman-Sims-Warzel, Aizenman-Warzel), but never on $\\mathbb{Z}^d$. Simon's Problem 1 asks: for $d\\ge 3$ and suitable small disorder, does $H_\\lambda$ almost surely have an energy range of purely absolutely continuous spectrum?","posedBy":"Barry Simon (Problem 1 in 'Schrodinger operators in the twenty-first century'), formalizing the extended-states prediction going back to Anderson","yearPosed":2000,"ageNote":"The physical prediction goes back to Anderson (1958); 2000 is Simon's formulation, which the manuscript cites as the problem.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For each $d\\ge 3$ there is $\\lambda_d>0$ such that for every $0<\\lambda<\\lambda_d$, almost surely a fixed open interval $I_d$ near the bottom band edge (for $d\\ge 4$, $-2d+(1/200,1/100)$) lies in the spectrum and the spectral restriction to $I_d$ is purely absolutely continuous and nonzero. $I_d$ does not depend on $\\lambda$. Not shown: a.c. spectrum in the bulk of the band, the location of a mobility edge, quantum diffusion, uniformity in $d$, or other laws.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem was read against Simon's Problem 1 as the manuscript paraphrases it; Simon's text itself was not opened. The proof was not refereed. No Lean formalization exists for this manuscript. The paper itself says it does not locate the mobility edge and does not resolve Simon's diffusion question (Problem 3), and that for d = 3 the disorder threshold is only existential.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":65,"significanceNote":"The extended-states conjecture is widely called the biggest open problem in the spectral theory of random operators, open since Anderson. Ties the Jacobian Conjecture (65), below critical percolation in every dimension (78), above the 2D localization entry from the same family (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Same family: Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kirchberg-embedding-problem","name":"Kirchberg's embedding problem: does every separable C*-algebra embed in a norm ultrapower of the Cuntz algebra O2?","shortName":"Kirchberg's embedding problem","problemNumber":null,"field":"Operator algebras; model theory of C*-algebras","fieldGroup":"Analysis","statement":"Kirchberg and Phillips proved that every separable unital exact C*-algebra embeds unitally into the Cuntz algebra $\\mathcal O_2$. Kirchberg asked how far this universality extends beyond exact algebras once $\\mathcal O_2$ is replaced by its norm ultrapower $\\mathcal O_2^\\omega$. Goldbring and Sinclair (2015) named this Kirchberg's embedding problem and showed its equivalence with existential closedness statements; it is distinct from Connes' embedding problem and from the tensor-norm form of Kirchberg's QWEP conjecture. Does every separable unital C*-algebra admit a unital embedding into $\\mathcal O_2^\\omega$ for a free ultrafilter $\\omega$ on $\\mathbb N$?","posedBy":"Eberhard Kirchberg, as recorded and named by Isaac Goldbring and Thomas Sinclair, On Kirchberg's embedding problem (J. Funct. Anal., 2015)","yearPosed":2015,"ageNote":"Year of the first citable statement, Goldbring-Sinclair 2015, which recounts that Kirchberg considered the question after the Kirchberg-Phillips embedding theorem (2000); when Kirchberg first raised it is not dated there.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $G=\\mathbb Z[1/2]^3\\rtimes(\\mathrm{SL}_3(\\mathbb Z)\\times\\mathbb Z)$ with $(M,k)\\cdot v=2^kMv$, the full group C*-algebra $C^*(G)$ is separable and unital and has no unital embedding into $B^\\omega$ for any nonzero unital nuclear $B$ and any free ultrafilter $\\omega$; in particular none into $\\mathcal O_2^\\omega$. The obstruction combines property (T) of $\\mathbb Z^3\\rtimes\\mathrm{SL}_3(\\mathbb Z)$ with a finiteness lemma for approximate representation types in nuclear algebras. Via Goldbring-Sinclair it follows that no exact or nuclear unital C*-algebra is existentially closed. It says nothing about Connes' embedding problem or the QWEP conjecture.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript dated September 23, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 and the consequences section were read against the problem as Goldbring-Sinclair state it. The Lean challenge lean/ComparatorChallenges/NuclearUltrapower.lean (solution module OAI.Analysis.NuclearUltrapower.Main, which exists at the pinned commit) is not in the formalization catalogue; it was found through lean/docs/292.md. Its statement main_no_embedding was read: there is a separable full group C*-algebra of the explicit group (dyadic 3-vectors by SL3(Z) times Z) with no injective unital *-homomorphism into the norm ultrapower of any nonzero unital nuclear C*-algebra (nuclearity as min = max tensor norm) for any free ultrafilter, and in particular none into that of the universal Cuntz algebra on two isometries. That is the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A named question at the meeting point of C*-algebra classification and continuous model theory, studied by Goldbring, Sinclair, Fox and Hart. Known within that community, much less widely than Connes' embedding problem. Placed below Kadison's similarity problem (58) and the Toms-Winter conjecture (48), above the spatial Kadison-Kastler problem for separable C*-algebras (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An explicit obstruction to nuclear norm-ultrapower embeddings","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean challenge statement: NuclearUltrapower.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/NuclearUltrapower.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Analysis/NuclearUltrapower/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/NuclearUltrapower/Main.lean","kind":"lean-proof"},{"label":"Goldbring-Sinclair, On Kirchberg's embedding problem (2015)","url":"https://doi.org/10.1016/j.jfa.2015.02.016","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"naor-markov-type-superreflexivity","name":"Naor's question: does nontrivial Markov type force an equivalent uniformly smooth norm (superreflexivity)?","shortName":"Markov type and superreflexivity","problemNumber":null,"field":"Banach space geometry; Ribe program; metric invariants","fieldGroup":"Analysis","statement":"A normed space $X$ has Markov type $p$ with constant $K$ if for every finite stationary reversible Markov chain $(Z_t)$ on a set $S$, every $f:S\\to X$ and every $t\\ge1$, $\\mathbb E\\|f(Z_t)-f(Z_0)\\|^p\\le K^p\\,t\\,\\mathbb E\\|f(Z_1)-f(Z_0)\\|^p$. Ball introduced Markov type for Lipschitz extension. Naor, Peres, Schramm and Sheffield showed that uniform smoothness of power type $p$ implies Markov type $p$, so by Pisier's renorming every superreflexive space has Markov type $p$ for some $p>1$. Markov type is a metric property, so a converse would give a metric characterization of superreflexivity in the Ribe program. Naor (2012, Question 4) asked: can a Banach space have nontrivial Markov type (Markov type $p$ for some $p>1$) without admitting an equivalent uniformly smooth norm?","posedBy":"Assaf Naor, An introduction to the Ribe program, Japanese Journal of Mathematics 7 (2012), Question 4","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that every real Banach space with Markov type $p$ for at least one $p>1$ admits an equivalent uniformly convex norm, hence is superreflexive and (by Pisier) admits an equivalent uniformly smooth norm. With the known converse (Naor-Peres-Schramm-Sheffield plus Pisier) this characterizes superreflexivity by nontrivial Markov type, answering Naor's question negatively. The proof builds, from a nonreflexive space finitely representable in a non-superreflexive one, a spreading norm and explicit reversible walks whose image has linear displacement. It does NOT address complex scalars separately, the exact Markov type exponent of a given space, or Ball's Markov type 2 questions for specific spaces.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. The main theorem has a Lean formalization listed in the release's formalization catalogue.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. The main theorem was read against Naor's Question 4. formalization.yaml lists ComparatorChallenges/MarkovType.json with declaration OAI.MarkovSuperreflexivity.hasNontrivialMarkovType_iff_hasEquivalentUCNorm in OAI/Analysis/MarkovType/Main.lean. The comparator statement, read here, says that for every real Banach space (CompleteSpace), having Markov type $p$ for some $p>1$, with the inequality required over all finite reversible chains and all times $n\\ge1$, is equivalent to admitting an equivalent uniformly convex norm. Given Pisier's duality between uniformly convex and uniformly smooth renormings, which the formal statement does not itself include, this is the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A named question in the Ribe program, which seeks metric characterizations of linear properties of Banach spaces; superreflexivity already had metric characterizations (Bourgain's trees, Mendel-Naor Markov convexity), so this adds a natural one rather than the first. Placed with Kalton's Problem 2 (28), below Naor's sublinear dimension question (35) and Banach's isometric conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Nontrivial Markov Type Forces Superreflexivity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean formalization: nontrivial Markov type iff equivalent uniformly convex norm","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/MarkovType/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: MarkovType","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/MarkovType.lean","kind":"lean-proof"},{"label":"Naor, An introduction to the Ribe program (2012), Question 4","url":"https://doi.org/10.1007/s11537-012-1222-7","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"euclidean-ramsey-sets-classification","name":"The classification problem for finite Euclidean Ramsey sets","shortName":"Euclidean Ramsey classification","problemNumber":null,"field":"Euclidean Ramsey theory","fieldGroup":"Combinatorics","statement":"A finite set $A$ in Euclidean space is Ramsey if for every number of colors $r$ there is a dimension $D$ such that every $r$-coloring of $\\mathbb R^D$, with no regularity assumed, contains a monochromatic congruent copy of $A$ at its original scale. Erdos, Graham, Montgomery, Rothschild, Spencer and Straus introduced the notion in 1973 and proved that every Ramsey set is spherical. Known Ramsey sets include nondegenerate simplices (Frankl-Rodl), sets with a soluble transitive isometry group and cyclic trapezoids (Kriz), and vertex sets of regular polytopes (Cantwell). Graham conjectured that every spherical set is Ramsey (1994); Leader, Russell and Walters conjectured that a set is Ramsey exactly when it embeds in a finite transitive set (2012). Which finite sets are Ramsey: is there a necessary and sufficient criterion?","posedBy":"Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey theorems I, J. Combin. Theory Ser. A 14 (1973)","yearPosed":1973,"ageNote":"The manuscript frames the characterization as the problem of the area since the 1973 paper, with two conjectured answers: Graham's (1994, sphericity) and Leader-Russell-Walters' (2012, subtransitivity). It does not quote an explicit question from the 1973 paper. Graham's conjecture was disproved by Palvolgyi in September 2026, before this manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a finite $A\\subset\\mathbb R^d$ of at least two points with full affine span is Ramsey iff some $(d+1)\\times(d+1)$ matrix $P$ over $F\\otimes_{\\mathbb Q}F$, with $F$ the field generated by the coordinates, satisfies $(p_i\\otimes1)^TP(1\\otimes p_i)=0$ for every $p_i=(1,a_i)$ and $m_F(P_{\\alpha\\beta})=\\delta_{\\alpha\\beta}$ on the spatial block. Consequences: every subtransitive set and every set of at most five points on a circle is Ramsey; nine circle points with algebraically independent parameters, and twelve points forming three rotated squares, are not. The criterion is exact algebra over the coordinate field; the paper says it is not a procedure for deciding the property from numerical coordinates. The Leader-Russell-Walters disproof is a separate entry. Graham's spherical conjecture had already fallen to Palvolgyi's ChatGPT-found seven-point circle set (arXiv:2609.23327), which this manuscript cites; see graham-spherical-ramsey-conjecture.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 and the consequences section of the TeX source; the proof was not refereed. Lean-checked on the Comparator challenge EuclideanRamsey, listed in the release's formalization catalogue (OAI.EuclideanRamsey.classification, OAI/Combinatorics/EuclideanRamsey/Main.lean). Its statement: for an injective family of $s\\ge2$ points in $\\mathbb R^d$, $d\\ge1$, with full affine span, being Ramsey (for every $r\\ge2$ some $D\\ge1$ such that every $r$-coloring of $\\mathbb R^D$ has a monochromatic family with the same pairwise distances) is equivalent to the existence of the tensor matrix over the coordinate field. That is Theorem 1.1. Further challenges for this family state cosphericity of Ramsey sets, the subtransitive and five-circle-point sufficiency results, the quadratic-independence criterion, the nine-point example (these five are not in the catalogue; their solution files exist at the pinned commit) and the twelve-point example (GrahamSpherical, catalogued). Statements read here; nothing rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Classifying Euclidean Ramsey sets has organized Euclidean Ramsey theory since 1973, with well-known conjectured answers by Graham and by Leader, Russell and Walters. Scored with Brennan's conjecture (45) and Vinogradov's least quadratic nonresidue conjecture (45), below the Hadwiger-Nelson five-color result (55), a question known well outside the area.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A classification of finite Euclidean Ramsey configurations","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/Combinatorics/EuclideanRamsey/Main.lean (classification)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/EuclideanRamsey/Main.lean","kind":"lean-proof"},{"label":"Lean: OAI/Combinatorics/SphericalRamsey/Main.lean (twelve-point non-Ramsey example)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/SphericalRamsey/Main.lean","kind":"lean-proof"},{"label":"Lean: OAI/Combinatorics/EuclideanRamsey/Spherical.lean (Ramsey sets are cospherical)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/EuclideanRamsey/Spherical.lean","kind":"lean-proof"},{"label":"Palvolgyi, A cyclic non-Ramsey heptagon (arXiv:2609.23327), the disproof of Graham's conjecture cited here","url":"https://arxiv.org/abs/2609.23327","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lech-multiplicity-conjecture","name":"Lech's multiplicity conjecture","shortName":"Lech's conjecture","problemNumber":null,"field":"Commutative algebra; Hilbert-Samuel multiplicity","fieldGroup":"Algebra","statement":"For a nonzero Noetherian local ring $(A,\\mathfrak a)$ of dimension $d$, the Hilbert-Samuel multiplicity is $e(A)=\\lim_N d!\\,\\ell_A(A/\\mathfrak a^N)/N^d$. Lech (1960) proved the inequality below when the source has dimension at most two and in complete-intersection cases; Ma proved it in equal characteristic for dimension three, for standard graded sources over perfect fields, and a bound $e(R)\\le\\max\\{1,d!/2^d\\}e(S)$. Lech asked whether multiplicity can decrease under flat maps. If $(R,\\mathfrak m)\\to(S,\\mathfrak n)$ is a flat local homomorphism of Noetherian local rings, is $e(R)\\le e(S)$?","posedBy":"Christer Lech, Note on multiplicities of ideals, Ark. Mat. 4 (1960); Inequalities related to certain couples of local rings, Acta Math. 112 (1964)","yearPosed":1960,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every flat local homomorphism $(R,\\mathfrak m)\\to(S,\\mathfrak n)$ of nonzero Noetherian local rings, $e(R)\\le e(S)$, in all dimensions and characteristics. The key input is a uniform lower bound on the zeroth homology of finite free complexes with entries in high powers of a parameter ideal, valid for generalized length functions, applied with Frobenius perfection in characteristic $p$ and perfectoid lengths in mixed characteristic. The paper does not claim the stronger Iyengar-Ma-Walker conjecture for all complete local rings, only its complete-domain characteristic-$p$ case.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Lech's question as the paper states it from Lech (1960, 1964); it claims $e(R)\\le e(S)$ for every flat local homomorphism of nonzero Noetherian local rings, with no restriction on dimension, residue fields or characteristic. The proof was not refereed. It uses the perfectoid normalized lengths of Cai-Lee-Ma-Schwede-Tucker in mixed characteristic and Eisenbud-Schreyer vector bundles. The family's Lean challenge (DuttaDomain, found via lean/docs/194.md, not in formalization.yaml) states only a supporting characteristic-$p$ comparison over complete local domains, not Lech's inequality, so this entry is left Unreviewed; that statement is entered separately.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"One of the best-known open problems of commutative algebra, open since 1960 with steady partial progress by Lech, Hanes, Ma and others. Placed below Zariski's multiplicity conjecture (52), level with Vinogradov's least quadratic nonresidue (45) and Brennan's conjecture (45), and well above Carlson's associated-prime depth conjecture (15).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Lech's multiplicity conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean (supporting lemma only): OAI.Lech.dutta_domain","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RingTheory/Multiplicity/DuttaDomain.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kakeya-maximal-conjecture-r3","name":"The Kakeya maximal conjecture in three dimensions","shortName":"Kakeya maximal conjecture in R^3","problemNumber":null,"field":"Harmonic analysis; Kakeya problems","fieldGroup":"Analysis","statement":"For $\\omega\\in S^2$ and $0<\\delta<1$ let $K_\\delta f(\\omega)$ be the supremum, over unit tubes of radius $\\delta$ in direction $\\omega$, of the average of $|f|$ on the tube. The Kakeya maximal conjecture in $\\mathbb R^n$ asks that $\\|K_\\delta f\\|_{L^n(S^{n-1})}\\le C_\\varepsilon\\delta^{-\\varepsilon}\\|f\\|_{L^n(\\mathbb R^n)}$ for every $\\varepsilon>0$; it implies the Kakeya set conjecture. In $\\mathbb R^3$ Wang and Zahl proved the set conjecture in 2025, but their union estimate carries a density power $\\lambda^{K(\\varepsilon)}$ where the maximal bound needs $\\lambda^3$, a gap stated after their theorem. Does the Kakeya maximal estimate hold in $\\mathbb R^3$ for every $\\varepsilon>0$?","posedBy":"Maximal form of the Kakeya conjecture (Cordoba, Bourgain); the manuscripts cite the formulations of Katz and Tao (2002) and Zahl's 2025 survey, Conjecture 1.3''","yearPosed":null,"ageNote":"The manuscripts do not attribute the maximal conjecture to one poser or year. The earliest sources they cite for the maximal formulation are Cordoba's planar maximal estimates (1977) and Bourgain's higher-dimensional work (1991).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $\\varepsilon>0$ there is $C_\\varepsilon$ such that $\\|K_\\delta f\\|_{L^3(S^2)}\\le C_\\varepsilon\\delta^{-\\varepsilon}\\|f\\|_{L^3(\\mathbb R^3)}$ for all $0<\\delta<1$ and $f\\in L^3(\\mathbb R^3)$. This is the $n=3$ case of the maximal conjecture; it strengthens the Wang-Zahl set theorem by getting the cubic dependence on shading density. The paper also derives Nikodym maximal and local curved Kakeya estimates by cited transfers. It does not treat dimension four or higher for the maximal operator; the four-dimensional Hausdorff case is a separate entry. The same release's local smoothing result in three dimensions (family 079, entry local-smoothing-conjecture-r3) also yields this Kakeya maximal estimate, by the established Bochner-Riesz to restriction to Kakeya chain in its Section 11.1.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This family consists of two manuscripts dated September 23 and 24, 2026; this entry draws on the September 23 one. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the maximal conjecture as stated in the paper's introduction and, independently, in the companion four-dimensional paper (which quotes the Katz-Tao formulation). It is the full three-dimensional case, with exactly the epsilon loss the conjecture allows. The proof (critical exponents for a density-sensitive multiplicity inequality, equality configurations, plate structure) was not refereed. No Lean formalization exists for this family. Inputs are published theorems: the Guth-Wang-Zahl set estimate, multilinear Kakeya with Guth's endpoint, and the Ren-Wang planar Furstenberg theorem. The four-dimensional paper and the family 077 restriction papers use this theorem, so an error would propagate; the sphere-restriction companion in family 077 also claims an independent route to it.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"With the three-dimensional Kakeya set conjecture settled by Wang and Zahl in 2025, the maximal version was the natural remaining three-dimensional target in harmonic analysis. Placed level with the uniform Turan density of the tetrahedron (48), above Stein's dimension-free Riesz transform problem (38), below the sum-product conjecture over the reals (55) and below the four-dimensional Kakeya case from this family.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Kakeya maximal conjecture in three dimensions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Zahl, A survey of the Kakeya conjecture, 2000-2025 (arXiv 2512.09397)","url":"https://arxiv.org/abs/2512.09397","kind":"problem-record"},{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/local-smoothing-conjecture-r3","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"log-brunn-minkowski-conjecture","name":"The logarithmic Brunn-Minkowski conjecture for origin-symmetric convex bodies","shortName":"Log-Brunn-Minkowski conjecture","problemNumber":null,"field":"Convex geometry; Lp Brunn-Minkowski theory","fieldGroup":"Geometry & topology","statement":"For origin-symmetric convex bodies $K,L\\subset\\mathbb R^n$ with support functions $h_K,h_L$ and $0\\le\\lambda\\le1$, the logarithmic combination is the Wulff body $(1-\\lambda)K+_0\\lambda L=\\{x:\\langle x,u\\rangle\\le h_K(u)^{1-\\lambda}h_L(u)^\\lambda\\ \\forall u\\in S^{n-1}\\}$. Boroczky, Lutwak, Yang and Zhang (2012, Problem 1.1) conjectured the log-Brunn-Minkowski inequality $|(1-\\lambda)K+_0\\lambda L|\\ge|K|^{1-\\lambda}|L|^\\lambda$, which strengthens the multiplicative Brunn-Minkowski inequality and implies the symmetric $L_p$ Brunn-Minkowski inequality for $0 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 23, 2026) is the whole family; it also derives the (B)-conjecture, listed as a separate entry.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Problem 1.1 of Boroczky-Lutwak-Yang-Zhang; it is the conjecture verbatim, for arbitrary origin-symmetric convex bodies in every dimension, with no smoothness or unconditionality assumption. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator LogBrunnMinkowski, declaration OAI.LogBrunnMinkowski.main). The comparator statement ComparatorChallenges/LogBrunnMinkowski.lean was read here: for $n\\ge1$, compact convex origin-symmetric $K,L\\subset\\mathbb R^n$ with nonempty interior and $0\\le t\\le1$, it asserts $\\mathrm{vol}(K)^{1-t}\\mathrm{vol}(L)^t\\le\\mathrm{vol}(W)$ where $W$ is the Wulff body of $h_K^{1-t}h_L^t$. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript itself notes that Stancu announced the all-dimensional inequality in an Oberwolfach report (2018) with details deferred.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The log-Brunn-Minkowski conjecture has been a central open problem of the Brunn-Minkowski theory since 2012, with a large literature of partial cases. Placed level with Ryser's conjecture (50) and the Kadison-Kaplansky conjecture (50), below the symmetric Mahler conjecture (60), which is older and better known, and well above Courtade's conjecture on volumes of Minkowski sums (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The logarithmic Brunn-Minkowski conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.LogBrunnMinkowski.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/LogVolume/BrunnMinkowski.lean","kind":"lean-proof"},{"label":"Comparator statement: LogBrunnMinkowski.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/LogBrunnMinkowski.lean","kind":"lean-proof"},{"label":"Boroczky-Lutwak-Yang-Zhang, The log-Brunn-Minkowski inequality (2012)","url":"https://doi.org/10.1016/j.aim.2012.07.015","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"circulant-hadamard-conjecture","name":"The circulant Hadamard conjecture: real circulant Hadamard matrices exist only in orders 1 and 4","shortName":"Circulant Hadamard conjecture","problemNumber":null,"field":"Design theory; Hadamard matrices and difference sets","fieldGroup":"Combinatorics","statement":"A real Hadamard matrix of order $n$ is an $n\\times n$ matrix $H$ with entries $\\pm1$ and $HH^T=nI_n$; it is circulant if every row is the cyclic shift of the first. Order 4 has the example with first row $(-1,1,1,1)$, and order 1 is trivial. The circulant Hadamard conjecture, traditionally attributed to Ryser (1963), asserts that there are no others. Turyn showed any further order is $4u^2$ with $u$ odd and not a prime power; field descent and later methods of Schmidt and Leung-Schmidt, and the 2017 computation of Logan and Mossinghoff, excluded all but 4489 orders up to $4\\cdot10^{30}$. Several claimed proofs have appeared. Is every real circulant Hadamard matrix of order 1 or 4?","posedBy":"Herbert J. Ryser (traditional attribution)","yearPosed":1963,"ageNote":"The manuscript says the conjecture is traditionally attributed to Ryser and cites his Carus monograph Combinatorial Mathematics (1963) and Leung-Schmidt (2012).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a real circulant Hadamard matrix of order $n\\ge1$ exists if and only if $n\\in\\{1,4\\}$. Section 2 reproves Turyn's restriction $n=4u^2$ with $u$ odd, and the main argument excludes every odd $u>1$ through an alternating product of character values over the primes dividing $u$. Corollary 1.2 deduces the Barker-sequence classification. It says nothing about the (non-circulant) Hadamard conjecture on existence in every order divisible by 4, nor about complex or group-developed variants.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 23, 2026) is the whole family; the Barker-sequence consequence is listed as a separate entry.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as stated in the manuscript and in Ryser's tradition; it is the full conjecture with no restriction on the order. The proof (group-ring and cyclotomic arguments) was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator CirculantHadamard, declaration OAI.CirculantHadamard.exists_iff_order_one_or_four). The comparator statement CirculantHadamard.lean was read here: for every $n>0$, a real $n\\times n$ matrix with $\\pm1$ entries, $H_{ij}=h(j-i)$ with index arithmetic modulo $n$, and $HH^T=nI$ exists if and only if $n=1$ or $n=4$. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript lists five earlier claimed complete proofs (Oh-Hashi 2016, Orozco Lopez 2019, Morris 2023, Gallardo 2024, Manjhi-Kumar 2025) without assessing them.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"A sixty-year-old conjecture, well known in design theory and coding, with a long record of partial exclusions and failed proofs. Placed just above Kahn's flow conjecture (40) and Sendov's Conjecture (40), level with Wall's manifold-realization question (42), and above the Hadamard matrix of order 668 (30), a single case of the better-known general Hadamard conjecture.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The circulant Hadamard conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.CirculantHadamard.exists_iff_order_one_or_four)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/LinearAlgebra/CirculantHadamard/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: CirculantHadamard.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CirculantHadamard.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"he-schramm-removable-implies-rigid","name":"He-Schramm rigidity conjecture: circle domains with conformally removable boundary are rigid","shortName":"He-Schramm rigidity (removable implies rigid)","problemNumber":null,"field":"Geometric function theory; conformal rigidity and removability","fieldGroup":"Analysis","statement":"A circle domain $\\Omega\\subseteq\\hat{\\mathbb C}$ is conformally rigid if every conformal equivalence from $\\Omega$ onto another circle domain is the restriction of a Mobius transformation. A compact set $E$ is conformally removable if every orientation-preserving homeomorphism of the sphere that is conformal off $E$ is Mobius. He and Schramm proved rigidity for countably connected circle domains and for boundaries of $\\sigma$-finite length, and conjectured that a circle domain is rigid if and only if its boundary is conformally removable. Rajala (2025) disproved the rigid-implies-removable direction. Is every circle domain whose boundary is conformally removable conformally rigid?","posedBy":"Zheng-Xu He and Oded Schramm, in their work on rigidity of circle domains (Invent. Math. 1994)","yearPosed":1994,"ageNote":"The manuscript attributes the conjecture to He and Schramm right after citing their 1994 rigidity paper but does not give a precise location; 1994 is that paper's year. Younsi (2016) studied its formulations.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims that every circle domain whose boundary is conformally removable is conformally rigid, with no bound on the number of complementary components and no boundary-extension hypothesis on the map. An intermediate theorem shows that a function continuous across a compact totally disconnected removable set with finite Dirichlet energy off it is globally Sobolev. Together with Rajala's counterexample this settles both directions of the He-Schramm equivalence: removable implies rigid, the converse fails. It does NOT characterize rigid circle domains.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. The manuscript is credited to OpenAI with no human author named. It uses the quantitative transfer and selection theorem of the companion Koebe manuscript.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture as the manuscript states it; it proves exactly the removable-implies-rigid direction, the other direction having been disproved by Rajala. The Lean challenge ComparatorChallenges/KoebeCircleDomains.json (solution module OAI.Analysis.CircleDomains.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/071.md. It compares two theorems, OAI.Problem047.koebe_circle_domain and OAI.Problem047.removability_implies_rigidity, with permitted axioms propext, Quot.sound and Classical.choice. The statement was read here; not rebuilt here. removability_implies_rigidity says that for circle domains U and V with conformally removable frontier of U (compact, and every orientation-preserving sphere homeomorphism conformal off it is Mobius), every conformal equivalence f from U onto V agrees on U with a Mobius map. That is the headline claim. The proof depends on the companion Koebe manuscript, also unreviewed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A well-known conjecture within conformal geometry, already half disproved by Rajala in 2025, so the open question was one direction. Below its sibling Koebe's circle-domain conjecture at 60; level with famous-in-one-community entries such as the Colin de Verdiere chromatic conjecture at 30, and below Free entropy dimension invariance at 38.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Removable Boundaries and Rigidity of Circle Domains","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Koebe's Circle-Domain Conjecture (supplies the transfer theorem used here)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean: solution module (koebe_circle_domain, removability_implies_rigidity)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/CircleDomains/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: circle-domain theorem and removable-boundary rigidity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KoebeCircleDomains.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/071.md","kind":"lean-proof"},{"label":"He and Schramm (1994), rigidity of circle domains with sigma-finite boundary","url":"https://doi.org/10.1007/BF01231761","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"forcing-conjecture-bipartite","name":"The forcing conjecture for bipartite graphs","shortName":"Forcing conjecture","problemNumber":null,"field":"Quasirandom graphs, graph limits","fieldGroup":"Combinatorics","statement":"A graph $F$ with $m$ edges is $p$-forcing if every sequence of graphs $G_n$ with edge density tending to $p$ and $t(F,G_n)\\to p^m$ is quasirandom of density $p$, and forcing if this holds for every $p\\in(0,1)$. Chung, Graham and Wilson showed the four-cycle is forcing; Skokan and Thoma asked which bipartite graphs can replace it. Conlon, Fox and Sudakov formulated the forcing conjecture: a graph is forcing if and only if it is bipartite and contains a cycle. The conjecture implies Sidorenko's conjecture. Is every bipartite graph containing a cycle forcing?","posedBy":"Question of J. Skokan and L. Thoma (Graphs Combin., 2004); conjecture formulated by D. Conlon, J. Fox and B. Sudakov (GAFA, 2010, Conjecture 2)","yearPosed":2004,"ageNote":"Dated to Skokan and Thoma's 2004 question as the manuscript describes it; the precise conjecture is stated in Conlon, Fox and Sudakov (2010).","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 1.3: for the same connected bipartite $H$ (which contains cycles) there are $p\\in(0,1)$ and a nonconstant graphon $W$ with $t(K_2,W)=p$ and $t(H,W)=p^{66}$; sampling gives finite graphs with edge and $H$-densities tending to $p$ and $p^{66}$ but four-cycle density tending to a limit above $p^4$. So $H$ is not $p$-forcing at that $p$, and the conjecture fails. Since the forcing conjecture implies Sidorenko's, this follows from the Sidorenko counterexample; no separate idea is claimed.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The disproof is Corollary 1.3 of the Sidorenko counterexample manuscript, obtained by interpolating between kernels of the same mean and sampling; it is not covered by the Lean formalization.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary 1.3 and its proof outline in 'A counterexample to Sidorenko's conjecture', read against the forcing conjecture as quoted from Conlon-Fox-Sudakov. The argument is a short consequence of Theorem 1.1, which is Lean-checked; the corollary itself is not formalized and was not refereed. The manuscript notes it exhibits one density p only and does not claim failure of p-forcing at every density, which is enough to refute the conjecture as stated.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A well-known conjecture in the quasirandomness community, but it implies Sidorenko's conjecture, so it falls with that entry rather than standing as a separate challenge. Placed below the Sidorenko entry (52), just under the Colin de Verdiere chromatic conjecture (30), and level with the Kara-Por-Wood big-line-big-clique conjecture (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to Sidorenko's conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: Sidorenko counterexample (challenge not in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Sidorenko/RandomHost.lean","kind":"lean-proof"},{"label":"Conlon, Fox and Sudakov 2010, An approximate version of Sidorenko's conjecture","url":"https://doi.org/10.1007/s00039-010-0097-0","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"free-group-factor-isomorphism-problem","name":"The free group factor isomorphism problem","shortName":"Free group factor isomorphism problem","problemNumber":null,"field":"Operator algebras: II_1 factors and free probability","fieldGroup":"Analysis","statement":"For a discrete group $G$ let $L(G)=\\lambda(G)''$ be its group von Neumann algebra on $\\ell^2(G)$, and let $\\mathbb F_n$ be the free group on $n$ generators. Murray and von Neumann showed the free group factors are not hyperfinite but could not distinguish their ranks. Voiculescu, Radulescu and Dykema built interpolated factors $L(\\mathbb F_r)$, $10}$. It also shows free entropy dimension depends on the generating tuple. It does not identify $L(\\mathbb F_2)$ with other non-free group factors.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The release also publishes an abridged summary of the model's reasoning for this family (reasoning_traces/free-group-factor-isomorphism.pdf).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the TeX source read against the posed problem; a trace-preserving normal *-isomorphism $L(\\mathbb F_n)\\cong L(\\mathbb F_{n+1})$ for $n\\ge3$, hence $L(\\mathbb F_2)\\cong L(\\mathbb F_3)$, answering it affirmatively with the Radulescu-Dykema dichotomy. Lean-checked on the release's own Comparator challenge InterpolatedFactors (OAI.FreeGroupFactorMain.Interpolation.allInterpolatedIsomorphic) together with its solution module, both present at the pinned commit; the challenge is not listed in the release's formalization catalogue, the statement was read here but not independently audited, and the development was not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":68,"significanceNote":"The most famous open problem in von Neumann algebras for over half a century, with an answer expected by many to go the other way. Scored just above the Jacobian conjecture at 65 and well above Connes' rigidity conjecture at 45 and the sofic groups question at 60, level with the proportion of zeta zeros on the critical line at 68.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf","sourceName":"OpenAI math release: An isomorphism of the free group factors","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean comparator statement InterpolatedFactors.lean (not in formalization.yaml main results at this commit)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/InterpolatedFactors.lean","kind":"lean-proof"},{"label":"Lean scope note for family 287","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/287.md","kind":"other"},{"label":"Abridged reasoning summary released by OpenAI for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/reasoning_traces/free-group-factor-isomorphism.pdf","kind":"other"},{"label":"Murray and von Neumann, On rings of operators IV (1943)","url":"https://doi.org/10.2307/1969107","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"rokhlin-multiple-mixing-problem","name":"Rokhlin's multiple-mixing problem for a single transformation","shortName":"Rokhlin's multiple-mixing problem","problemNumber":null,"field":"Ergodic theory; mixing and joinings","fieldGroup":"Analysis","statement":"Let $T$ be an invertible measure-preserving transformation of a probability space $(\\Omega,\\mathcal F,\\mu)$. $T$ is mixing if $\\mu(A\\cap T^{-n}B)\\to\\mu(A)\\mu(B)$ as $|n|\\to\\infty$, and mixing of order $k$ if $\\mu(\\bigcap_{i=1}^k T^{-t_i}A_i)\\to\\prod_i\\mu(A_i)$ whenever $t_1<\\cdots 11/12 zero-free region write-up, which was human-edited for readability) do not concern this family. The manuscript is credited to OpenAI alone, names no human author and has no acknowledgements. A Lean formalization of the main theorem is in the release's lean/ library (lean/docs/145.md). The release does not say how much human review happened before publication.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source against Rokhlin's question, and the Lean statement. The result is not in the release's formalization catalogue (formalization.yaml), but lean/docs/145.md links ComparatorChallenges/Rokhlin.json, whose solution_module OAI.Dynamics.MultipleMixing.Main exists at the pinned commit; theorem OAI.Rokhlin.mixing_all_finite_orders, permitted axioms propext, Quot.sound and Classical.choice. The comparator statement, read here, takes an arbitrary measurable space with a probability measure, a measurable equivalence T preserving it, and mixing defined over integer times with |n| to infinity, and concludes for every k >= 3 that the measure of the intersection of the k translated sets tends to the product of measures as the vector of positive successive gaps tends to infinity (atTop on the product order, so every gap diverges). That is the headline claim, on arbitrary rather than standard probability spaces. Not rebuilt here; the comparator run and axiom check were not reproduced.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"The best known open problem in abstract ergodic theory, open since 1949 with a 2024 survey titled for its 75 years, and only structured special cases settled. Tied with Kaplansky's zero-divisor conjecture and Cannon's conjecture at 62, above Thompson's group F at 58, below the Hilbert-Smith conjecture at 65, since its reach outside dynamics is narrower.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Rokhlin's multiple-mixing problem for one transformation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI.Rokhlin.mixing_all_finite_orders","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Dynamics/MultipleMixing/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Rokhlin.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/Rokhlin.lean","kind":"lean-proof"},{"label":"Lean scope note for family 145","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/145.md","kind":"lean-proof"},{"label":"Ryzhikov, Multiple mixing, 75 years of Rokhlin's problem (arXiv:2411.07234)","url":"https://arxiv.org/abs/2411.07234","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"turyn-merit-factor-conjecture","name":"Turyn's conjecture that binary merit factors are bounded","shortName":"Turyn's merit factor conjecture","problemNumber":null,"field":"Binary sequences; aperiodic autocorrelation","fieldGroup":"Combinatorics","statement":"For a binary word $A=(a_0,\\dots,a_{N-1})\\in\\{-1,1\\}^N$ with aperiodic autocorrelations $C_u(A)=\\sum_{j=0}^{N-1-u}a_ja_{j+u}$, the merit factor is $F(A)=N^2/(2\\sum_{u=1}^{N-1}C_u(A)^2)$. Equivalently $1/F(A)=\\|P_A\\|_4^4/N^2-1$ for the polynomial $P_A(z)=\\sum a_jz^j$ on the unit circle. The best proven constructions, from modified Legendre sequences, reach limiting merit factor about $6.34$ (Jedwab, Katz and Schmidt). Turyn's conjecture, equivalently Erdos's $L^4$-norm conjecture, asserts that $F(A)$ is bounded over all binary words of all lengths. Is the merit factor of binary sequences bounded?","posedBy":"Richard J. Turyn (name as used by Downarowicz and Lacroix 1998); equivalently Erdos's L4-norm conjecture","yearPosed":null,"ageNote":"The manuscript cites Downarowicz and Lacroix (Theoret. Comput. Sci. 1998), who call the boundedness conjecture Turyn's conjecture, equivalently Erdos's L4-norm conjecture. Neither the manuscript nor that citation as quoted gives the year Turyn stated it, so the year is left blank.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Section 8 deduces from Theorem 1.1 (real-sign polynomials of every large length with maximum $(1+o(1))\\sqrt N$) that $\\max_{A\\in\\{-1,1\\}^N}F(A)\\to\\infty$ as $N\\to\\infty$ through all integers, disproving Turyn's conjecture in this all-length form. Via Downarowicz and Lacroix it also gives a uniquely ergodic binary Morse shift with simple spectrum and absolutely continuous zero-coordinate spectral measure with $L^2$ density. The construction is existential: no explicit sequences, no rate and no algorithm are given.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript is dated September 23, 2026. Two companions dated October 5, 2026 strengthen it: one adds a lower bound of sqrt(N)/16, the other makes the polynomials two-sided ultraflat; the latter says it reuses lemmas of a 'version-2' refinement of the principal paper.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Section 1.2 and Section 8 of the manuscript were read against Turyn's conjecture; the paper claims the largest merit factor at length $N$ tends to infinity through all integer lengths, deduced from flatness in every finite $L^p$ sense. The challenge LittlewoodFiniteFlatness is not in the formalization catalogue (lean/formalization.yaml); it is linked from lean/docs/076.md and its solution module exists at the pinned commit. Its statement was read here: one family of real-sign polynomials, one for each length, with $\\int |\\,|P_N|/\\sqrt N-1|^p\\to0$ for every finite $p>0$. This is narrower than the headline: merit factors are not defined in the Lean, and the passage from $p=4$ flatness to unbounded merit factor is a standard identity left informal. Not rebuilt here. The paper says its theorem conflicts with nonflatness claims in preprints of el Abdalaoui (arXiv:1609.03435 and two others) and examines specific issues in those arguments in its Appendix A; no public response to this release was found or searched for here. Listed as Unreviewed rather than Lean-checked because its formal statement covers only the L^p flatness step, not the unbounded merit factor.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The merit factor problem is the central open question of low-autocorrelation binary sequence design, with the best proven constructions stuck near 6.34 (Jedwab, Katz and Schmidt) and consequences for spectral questions on Morse shifts (Downarowicz and Lacroix). Field-famous rather than broadly famous: placed level with the Bilu-Linial signing conjecture (35) and Crouzeix's conjecture (35), below its companion Erdos question on Littlewood polynomial maxima (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Asymptotically minimal maxima of real Littlewood polynomials","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.AsymptoticallyMinimalLittlewood.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Littlewood/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: AsymptoticallyMinimalLittlewood.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/AsymptoticallyMinimalLittlewood.lean","kind":"lean-proof"},{"label":"Lean proof of finite-exponent flatness (OAI.AsymptoticallyMinimalLittlewoodFiniteFlatness.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Littlewood/FiniteFlatness.lean","kind":"lean-proof"},{"label":"Comparator statement: LittlewoodFiniteFlatness.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/LittlewoodFiniteFlatness.lean","kind":"lean-proof"},{"label":"Companion: Ultraflat real Littlewood polynomials (two-sided, Oct 5)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf","kind":"paper"},{"label":"Companion: Nearly minimal maxima and positive minima of Littlewood polynomials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"toms-winter-conjecture","name":"The Toms-Winter conjecture: strict comparison implies Jiang-Su stability for simple nuclear C*-algebras","shortName":"Toms-Winter conjecture","problemNumber":null,"field":"Operator algebras; regularity in the Elliott classification program","fieldGroup":"Analysis","statement":"For simple separable unital infinite-dimensional nuclear C*-algebras $A$, the Toms-Winter conjecture asserts the equivalence of three regularity properties: finite nuclear dimension, Jiang-Su stability $A\\cong A\\otimes\\mathcal Z$, and strict comparison of positive elements by traces. Winter and Castillejos-Evington-Tikuisis-White-Winter proved finite nuclear dimension equivalent to $\\mathcal Z$-stability, and Rordam proved that $\\mathcal Z$-stability gives strict comparison. The remaining implication, strict comparison implies $\\mathcal Z$-stability, was known only under conditions on the tracial boundary (finitely many extreme traces, compact finite-dimensional boundary, Lin's condition) or with uniform property $\\Gamma$. Does strict comparison imply $\\mathcal Z$-stability for every simple separable nuclear non-elementary C*-algebra?","posedBy":"Andrew Toms and Wilhelm Winter","yearPosed":2010,"ageNote":"The release cites Winter-Zacharias, The nuclear dimension of C*-algebras (Adv. Math. 2010), Conjecture 9.3, and Schafhauser-Tikuisis-White's 99 problems (Conjecture 5, Problem XVIII). The conjecture circulated in Toms and Winter's work a few years earlier.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: strict comparison implies $A\\cong A\\otimes\\mathcal Z$ for every separable simple nuclear non-elementary $A$, unital or not. More generally (Theorem 1.2) every separable nuclear algebra whose Cuntz semigroup is almost unperforated and fully almost divisible is $\\mathcal Z$-stable, which answers the implication (iii) to (ii) of the nonsimple Toms-Winter problem (STW Problem LXXV) and the abstract Cuntz-semigroup question (STW Problem XXVI). Separability and nuclearity remain hypotheses; nonnuclear and non-separable algebras are not treated.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has four manuscripts. This entry's principal is 'Cuntz comparison and Jiang-Su absorption' (September 23, 2026); 'Tracial projection methods and uniform property Gamma' (same date) gives an independent proof in the unital case.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the conjecture. It states that every separable simple nuclear non-elementary C*-algebra with strict comparison (in the extended-functional convention, which it shows equals almost unperforation of the Cuntz semigroup) is $\\mathcal Z$-stable, including nonunital algebras with unbounded traces. With the known equivalence of finite nuclear dimension and $\\mathcal Z$-stability this completes the conjecture. The tracial-projection companion proves the unital case independently. Neither proof was refereed. The family's Lean comparator (UniformGamma) states only the companion's Problem XXI result, not this implication, so the entry is unreviewed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"The Toms-Winter conjecture is the central regularity conjecture of the Elliott classification program, with the comparison implication the last open piece after 2021. Placed below Kaplansky's quasitrace problem (55) and the Kadison similarity problem (58), and level with the Linear List Hadwiger Conjecture (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Cuntz comparison and Jiang-Su absorption","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Tracial projection methods and uniform property Gamma (unital route)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026/paper.pdf","kind":"paper"},{"label":"Companion: Nuclear dimension and Jiang-Su stability without elementary subquotients","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026/paper.pdf","kind":"paper"},{"label":"Schafhauser, Tikuisis and White, Nuclear C*-algebras: 99 problems","url":"https://arxiv.org/abs/2506.10902v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"group-ring-determinant-conjecture","name":"The group-ring Determinant Conjecture (Luck)","shortName":"Determinant Conjecture","problemNumber":null,"field":"L2-invariants; Fuglede-Kadison determinants over group von Neumann algebras","fieldGroup":"Algebra","statement":"For a discrete group $G$ and a matrix $A$ over $\\mathbb Z[G]$, let $T_A$ act on $\\ell^2(G)^n$ by the regular representation and let $\\mu_A$ be the spectral measure of $T_A^*T_A$ with respect to the group trace. A nonsingular integer matrix has $|\\det|\\ge1$; the Determinant Conjecture (Luck 2002, Conjecture 13.2) asks for the analogue $\\int_{(0,\\infty)}\\log t\\,d\\mu_A(t)\\ge0$, which for invertible $A$ says the Fuglede-Kadison determinant is at least $1$. It was known for amenable groups (Schick) and sofic groups (Elek-Szabo), and underlies approximation and torsion results for $L^2$-invariants. Does the Determinant Conjecture hold for every group?","posedBy":"Wolfgang Luck, L2-Invariants: Theory and Applications to Geometry and K-Theory (2002), Conjecture 13.2","yearPosed":2002,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a finitely generated group $G_D$, $n\\ge1$ and $A\\in M_n(\\mathbb Z[G_D])$, invertible over $\\mathbb Q[G_D]$ with bounded inverse, whose Fuglede-Kadison determinant lies strictly between $0$ and $1$; the log integral is finite and negative. The group is nonsofic by Elek-Szabo. It disproves the unrestricted conjecture only; it says nothing about torsion-free groups, groups satisfying the Atiyah conjecture, or the determinant approximation conjecture, and it relies on the characteristic-two direct-finiteness counterexample.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof takes the characteristic-two direct-finiteness counterexample of the companion manuscript as its only nonstandard input.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Conjecture 13.2 of Luck as the paper quotes it; it claims a finitely generated group $G_D$ and $A\\in M_n(\\mathbb Z[G_D])$ invertible over $\\mathbb Q[G_D]$ with $0<\\det_{\\mathcal N(G_D)}(T_A)<1$. The proof was not refereed; it depends on the characteristic-two companion (also Lean-checked). lean/formalization.yaml lists a main result for this manuscript (comparator GroupRingDeterminant, declaration OAI.GroupRingDeterminant.main). ComparatorChallenges/GroupRingDeterminant.lean was read here: it asserts a finitely generated group, $n\\ge1$, an integral matrix invertible over $\\mathbb Q[G]$, an invertible bounded operator $T$ matching the left-regular action coordinatewise, and $0<\\exp(\\tfrac12\\mathrm{Re}\\,\\tau(\\log T^*T))<1$. This states the headline for invertible matrices, the case that disproves the conjecture; the spectral-integral reformulation is not included. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A structural conjecture of L2-invariant theory, known to specialists as the gateway to approximation and torsion results; long believed true since it held for every sofic group. Placed level with the Projected Lax Conjecture (35) and the Popa log-additivity conjecture (35), below the Kadison-Kaplansky projection conjecture (50) and above the Artin parabolic intersection conjecture (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A Counterexample to the Group-Ring Determinant Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.GroupRingDeterminant.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/GroupDeterminants/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: GroupRingDeterminant.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/GroupRingDeterminant.lean","kind":"lean-proof"},{"label":"Input: Kaplansky counterexample in characteristic two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"artin-parabolic-intersection-conjecture","name":"The Parabolic Intersection Conjecture for Artin groups","shortName":"Parabolic Intersection Conjecture","problemNumber":null,"field":"Group theory, Artin groups","fieldGroup":"Algebra","statement":"A parabolic subgroup of an Artin group $A_M$ is a conjugate $gA_Xg^{-1}$ of a standard subgroup generated by a subset $X$ of the generators. The Parabolic Intersection Conjecture, formulated precisely in Godelle (2023, Conjecture 1) and Cumplido's survey (Conjecture 4), asserts that the intersection of two parabolic subgroups is parabolic. It was known for right-angled, spherical (Cumplido-Gebhardt-Gonzalez-Meneses-Wiest), large-type, some two-dimensional and some affine and FC types. Is the intersection of any two parabolic subgroups of an arbitrary finite-rank Artin group again parabolic?","posedBy":"E. Godelle, On parabolic subgroups of Artin-Tits groups, J. Algebra 632 (2023), Conjecture 1; also M. Cumplido, survey arXiv:2509.08382, Section 3, Conjecture 4","yearPosed":2023,"ageNote":"Year of the formulation the manuscript cites (Godelle, J. Algebra 2023; arXiv 2022). Pairwise closure was studied in special classes earlier, from at least Cumplido et al. (2019).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every Coxeter matrix on a finite set, all $X,Y$ and $g,h$, there are $Z,k$ with $gA_Xg^{-1}\\cap hA_Yh^{-1}=kA_Zk^{-1}$. Corollary 1.2 (via Moller-Paris-Varghese): arbitrary intersections of parabolics are parabolic, already attained by at most $|S|$ members, and every subset has a unique parabolic closure. Corollary 1.4: irreducible Artin groups with an infinite label are acylindrically hyperbolic (already known from Kato-Oguni) with weakly malnormal proper parabolics. The proof uses a relative harmonic height for a framed zigzag braid action and does not use asphericity of the Salvetti complex.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 with its corollaries, read against the conjecture as cited (Godelle 2023, Cumplido survey). The stabilizer and height arguments were not refereed. Lean-checked on the release's Comparator challenge ArtinParabolicIntersections, found through lean/docs/254.md; this challenge is not in the release's formalization catalogue, but its solution module OAI.Topology.ArtinGroups.ParabolicIntersections.Main exists at the pinned commit. Its statement was read here: theorem unconditional_parabolic_intersections asserts that for every finite Coxeter matrix the intersection of two conjugates of standard subgroups is a conjugate of a standard subgroup; further theorems state the arbitrary-intersection, closure and non-clique dynamics corollaries. That covers the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A named conjecture famous within the Artin-group community, the subject of a steady line of special-case results and a recent survey, but not widely known outside it. Tied with the Matrix Spencer conjecture partial (30) and the Colin de Verdiere chromatic conjecture (30), above the Banks-Martin conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Parabolic Intersections in Artin Groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge ArtinParabolicIntersections (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ArtinParabolicIntersections.lean","kind":"lean-proof"},{"label":"Godelle, On parabolic subgroups of Artin-Tits groups (arXiv)","url":"https://arxiv.org/abs/2207.06528v1","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"double-dimer-cle4-half-plane","name":"The double-dimer CLE4 scaling-limit conjecture in the half-plane Temperleyan setting","shortName":"Double-dimer loops to CLE4 (half-plane)","problemNumber":null,"field":"Dimer model; conformal loop ensembles","fieldGroup":"Probability & statistics","statement":"Superposing two independent uniform dimer coverings of a planar lattice graph produces doubled edges and a collection of loops. Kenyon proved that dimer height fluctuations converge to the Gaussian free field, Kenyon and Wilson matched boundary pairing probabilities with GFF contour lines, and Kenyon, Dubédat, Basok-Chelkak and Bai-Wan identified topological loop observables with those of nested $CLE_4$. The double-dimer conjecture asks that the loops themselves, as curves, converge to nested $CLE_4$; topological observables alone do not exclude thin excursions or retraced arcs. For the standard Temperleyan dimer law on the square lattice in the upper half-plane, does the full double-dimer loop ensemble converge to nested $CLE_4$ in the topology of loop collections, matching every macroscopic loop as a curve?","posedBy":"Kenyon and collaborators (conjecture motivated by Kenyon's GFF limit and Kenyon-Wilson); the manuscript cites no single posing source","yearPosed":null,"ageNote":"The manuscript calls it 'the double-dimer CLE4 scaling-limit conjecture' and cites Kenyon-Wilson (Trans. AMS 2011) and Kenyon (Comm. Math. Phys. 2014) as evidence, without naming who first posed it or when.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for two independent dimer coverings of the Temperleyan square lattice in the upper half-plane with the standard infinite-volume law, the loop collection $F(\\mathcal L_\\delta)$, mapped to the disk, is tight and converges in law as $\\delta\\to0$ to $F(\\mathcal L)$ with $\\mathcal L$ nested $CLE_4$, matching every macroscopic loop as an unparametrized curve with all nesting generations. New inputs: a bound on disjoint traversals of a slab and control of signed crossing counts, from a column transfer matrix. Not shown: bounded or general simply connected domains, non-Temperleyan boundary conditions, or other lattices.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript, 'The curve scaling limit of half-plane double dimers' (September 23, 2026).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture: for the half-plane Temperleyan law the compactified loop ensembles are tight and converge to standard nested $CLE_4$ in a loop-collection topology defined up to reparametrization, along the full mesh limit. This is one domain and one boundary condition, which the paper itself calls the half-plane Temperleyan form, so the entry is partial. There is no Lean formalization. The proof imports topological convergence from Basok and Izyurov's 2025 preprint (Theorem 4.2(2)) and adds transfer-matrix estimates for slab traversals and signed crossings; none of this was refereed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"Convergence of double-dimer loops to CLE4 is a well-known conjecture of planar statistical mechanics; this is its half-plane Temperleyan case, upgrading known observable convergence to curve convergence, so it is scored below the full conjecture. Just below the Ellipsoid Fitting Conjecture (30), well below Cardy's formula for square-lattice bond percolation (48) and the Rohde-Schramm FK-SLE conjecture (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The curve scaling limit of half-plane double dimers","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kenyon, Conformal invariance of loops in the double-dimer model (2014)","url":"https://doi.org/10.1007/s00220-013-1881-0","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mezard-parisi-formula-diluted-spin-glasses","name":"The Mezard-Parisi formula for diluted spin glasses: the Panchenko-Talagrand equality conjecture","shortName":"Mezard-Parisi diluted formula","problemNumber":null,"field":"Spin glass theory; diluted mean-field models","fieldGroup":"Mathematical physics","statement":"In diluted spin glasses such as the Viana-Bray model each spin has a finite mean number of interactions. Mezard and Parisi proposed a hierarchical cavity description of the limiting free energy by nested laws of local fields, with replica symmetry breaking. Franz-Leone obtained replica-symmetric and one-step bounds for even arity by Guerra's interpolation, and Panchenko and Talagrand proved the arbitrary finite-level hierarchical upper bounds for Poisson even-arity Ising models with factorized interactions under a positivity condition, conjecturing equality. In that class, is the limiting pressure equal to the infimum of the Mezard-Parisi trial functional over all finite hierarchy depths and trial laws?","posedBy":"Dmitry Panchenko and Michel Talagrand, 'Bounds for diluted mean-fields spin glass models' (PTRF 2004), conjecture after Theorem 4","yearPosed":2004,"ageNote":"The physical prediction is Mezard-Parisi (2001); the precise variational equality conjecture proved here is the one stated by Panchenko-Talagrand in 2004.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Main theorem: for Poisson-diluted even-arity Ising models in the Panchenko-Talagrand class (factorized interactions $e^{\\theta}=a(1+b\\prod f_l)$ with the stated independence and positivity of moments of $-b$), with only first moments of the interaction and field, the limiting pressure exists and equals $\\inf_r$ of the infimum of the Mezard-Parisi hierarchical functional of depth $r$. Covers Viana-Bray, symmetric diluted even-spin models and soft random even-$K$ SAT at positive temperature. Not covered: odd arity (for example K-SAT with odd K), models outside the positivity class, fixed-degree random regular graphs, identification of the optimal hierarchy depth, or zero temperature beyond the stated ground-state limits.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript; its main theorem is formalized in Lean (see verification).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem was read against the Panchenko-Talagrand conjecture as the paper quotes it; the paper claims it for their Poisson even-arity Ising class with factorized interactions and their positivity condition, needing only first moments. lean/docs/221.md points to ComparatorChallenges/DilutedSpin.json (theorem OAI.DilutedSpinGlass.mezard_parisi, solution module OAI.Probability.DilutedSpin.Main); the solution file exists at the pinned commit and the challenge is not in the formalization catalogue formalization.yaml. The statement was read here: for every even arity $p\\ge2$, positive density, integrable interactions and field, and the factorization, independence, finite-moment and positivity hypotheses, the finite-$N$ pressure converges to the infimum over depths $r$ of the infimum of the explicit finite-depth cavity functional. That is the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"The diluted analogue of the Parisi formula is a well-known open problem in mathematical spin glass theory, open since Panchenko-Talagrand's 2004 bounds; famous within that community. Placed near Full-RSB in the Sherrington-Kirkpatrick spin glass (35), above Talagrand's critical Sherrington-Kirkpatrick overlap conjecture (30) and The Gardner Transition in the Ising pure p-spin glass (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Mezard-Parisi formula for diluted spin glasses","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.DilutedSpinGlass.mezard_parisi)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/DilutedSpin/Main.lean","kind":"lean-proof"},{"label":"Panchenko-Talagrand, Bounds for diluted mean-fields spin glass models (2004)","url":"https://doi.org/10.1007/s00440-004-0342-2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"triangular-lattice-universal-optimality","name":"Universal optimality of the triangular lattice in the plane (Cohn-Kumar)","shortName":"Universal optimality of the triangular lattice","problemNumber":null,"field":"Discrete geometry; energy minimization and universal optimality","fieldGroup":"Geometry & topology","statement":"For a locally finite configuration $\\mathcal C\\subset\\mathbb R^2$ of density one and a potential $g$, the energy per point is the lower limit of $N_R^{-1}\\sum_{x\\ne y\\in\\mathcal C\\cap B_R}g(|x-y|^2)$. Among lattices of covolume one the triangular lattice minimizes inverse-power sums (Rankin, Cassels, Ennola, Diananda) and Gaussian theta sums (Montgomery 1988), but these say nothing about non-lattice competitors. Cohn and Kumar (2007, Conjecture 9.4) conjectured that the triangular lattice, like $E_8$ and the Leech lattice, is universally optimal, with sharp Fourier auxiliary functions certifying it; Cohn, Kumar, Miller, Radchenko and Viazovska proved the analogue in dimensions 8 and 24. Does the triangular lattice of covolume one minimize energy per point among all density-one planar configurations, for every completely monotone function $g$ of squared distance?","posedBy":"Henry Cohn and Abhinav Kumar (Conjecture 9.4, J. Amer. Math. Soc. 2007)","yearPosed":2007,"ageNote":"The lattice-only form goes back to Rankin (1953) and Cassels (1959); the conjecture against all configurations is Cohn-Kumar 2007, as cited by the manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: the density-one triangular lattice $A$ minimizes the lower energy per point $E_g$ among all locally finite planar configurations of centered-disk density one, for every nonnegative smooth completely monotone $g$ of squared distance, including Gaussians and all inverse powers, with infinite energies allowed and no separation assumption. The proof builds sharp radial Schwartz minorants $f_\\alpha\\le e^{-\\pi\\alpha|x|^2}$ with $\\widehat f_\\alpha\\ge0$ touching at all nonzero lattice and dual-lattice points, and integrates over Gaussians. Section 8 transfers this, following Petrache-Serfaty, to triangular minimality of the logarithmic and Riesz $0 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a second manuscript, 'An atomic certificate for triangular-lattice universal optimality' (September 26, 2026), giving a separate construction of the sharp Gaussian minorants; the Lean formalization in the release is attached to that companion. Both manuscripts ship finite checkers (Python, python-flint) for their numerical inputs.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the Cohn-Kumar question. It proves $E_g(\\mathcal C)\\ge\\sum_{a\\in A\\setminus 0}g(|a|^2)=E_g(A)$ for every smooth completely monotone $g\\ge0$ and every locally finite configuration of centered-disk density one, infinite energies included. The manuscript states plainly that it does not construct a sharp auxiliary function for each mixed potential, the stronger per-potential quantifier in Cohn-Kumar's Conjecture 9.4; it identifies the minimum value, not all minimizers. Lean: lean/ComparatorChallenges/TriangularEnergy.json exists with solution_module OAI.Analysis.Triangular.Energy.Universal, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement TriangularEnergy.lean was read here: for every smooth nonnegative completely monotone $g$ on $(0,\\infty)$ and every locally finite $\\mathcal C$ with centered disk count over $\\pi R^2$ tending to 1, the lattice energy (an extended-real series) is at most the liminf of disk energies of $\\mathcal C$ and equals that of the lattice. This states the headline energy comparison. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Universal optimality of the triangular lattice is the planar case of the Cohn-Kumar conjecture, open since 2007 with the lattice-only form going back to Rankin in 1953, and the remaining member of the trio whose dimension 8 and 24 cases were celebrated. Placed level with the Hadwiger-Nelson problem (55) and Bloch's conjecture for surfaces with pg = 0 (55), above Upper Bounds for High-Dimensional Sphere Packing (50), below the symmetric Mahler conjecture (60).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Universal optimality of the triangular lattice","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An atomic certificate for triangular-lattice universal optimality","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.AtomicTriangular.universal_energy_minimum)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Triangular/Energy/Universal.lean","kind":"lean-proof"},{"label":"Comparator statement: TriangularEnergy.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/TriangularEnergy.lean","kind":"lean-proof"},{"label":"Lean scope note for this family (lean/docs/090.md)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/090.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kaplansky-zero-divisor-conjecture","name":"Kaplansky's zero-divisor conjecture","shortName":"Kaplansky zero-divisor conjecture","problemNumber":null,"field":"Group rings and geometric group theory","fieldGroup":"Algebra","statement":"For a group $G$ and a field $K$, the group algebra $K[G]$ consists of finite formal sums of elements of $G$. If $g\\in G$ has finite order $m>1$ then $(1-g)(1+g+\\cdots+g^{m-1})=0$, so torsion forces zero divisors. The domain conclusion is known for unique-product groups, torsion-free elementary amenable groups and several geometric classes. Kaplansky's zero-divisor conjecture asserts the converse. If $G$ is torsion-free and $K$ is a field, is $K[G]$ free of nonzero zero divisors?","posedBy":"Graham Higman (1940 thesis) and Irving Kaplansky (1956 conference, published 1957, Problem 6)","yearPosed":1940,"ageNote":"The manuscript traces the question to Higman's 1940 thesis and to Kaplansky's 1956 conference problem list, published 1957.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims a counterexample: a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space and nonzero $\\alpha,\\beta\\in\\mathbb F_2[G]$ with $\\alpha\\beta=0$. The group is the fundamental group of a cone complex over two random labelled graphs, with cancellation forced by odd label intersections; torsion-freeness comes from asphericity. It works in characteristic two only and does not disprove the zero-divisor conjecture over $\\mathbb Q$ or $\\mathbb C$, nor the Atiyah conjecture.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The release includes a Lean 4 formalization of the main theorem.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source against the posed problem, and the Lean statement. formalization.yaml lists OAI.TorsionFreeZeroDivisors.main (file lean/OAI/Algebra/GroupRing/Main.lean, comparator ComparatorChallenges/TorsionFreeZeroDivisors.json, axioms propext, Quot.sound, Classical.choice). The comparator statement asserts a group $G$ that is finitely presented, torsion-free in the ordinary sense, has a finite connected 2-dimensional CW complex with contractible covering as $K(G,1)$, and has nonzero $\\alpha,\\beta$ in MonoidAlgebra (ZMod 2) G with $\\alpha\\beta=0$. That is the full headline claim. Not rebuilt here; the comparator run was not reproduced.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"One of Kaplansky's three famous group-ring conjectures, open for over eighty years; the sister unit conjecture fell to Gardam in 2021 to wide attention. Scored just above the Kothe conjecture and the sofic groups question at 60, below the Jacobian conjecture at 65 because it lives mainly in group rings and geometric group theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A Torsion-Free Group Algebra with Zero Divisors","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: lean/OAI/Algebra/GroupRing/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/GroupRing/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: TorsionFreeZeroDivisors.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/TorsionFreeZeroDivisors.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"nakayama-conjecture-dominant-dimension","name":"The Nakayama conjecture on dominant dimension","shortName":"Nakayama conjecture","problemNumber":null,"field":"Representation theory of algebras; homological conjectures","fieldGroup":"Algebra","statement":"Let $\\Lambda$ be a finite-dimensional algebra over a field with minimal injective resolution $0\\to\\Lambda\\to I^0\\to I^1\\to\\cdots$. Its dominant dimension is infinite when every $I^j$ is projective. The classical Nakayama conjecture asserts that infinite dominant dimension forces $\\Lambda$ to be self-injective. Its relatives are the generalized Nakayama conjecture of Auslander and Reiten (every indecomposable injective occurs as a summand of some $I^j$), the strong Nakayama conjecture (a module $V$ with $\\mathrm{Ext}^i(V,\\Lambda)=0$ for all $i\\ge0$ is zero), the Auslander-Gorenstein conjecture and the Wakamatsu tilting conjecture; the finitistic dimension conjecture implies the classical one. Must a finite-dimensional algebra of infinite dominant dimension be self-injective?","posedBy":"Tadasi Nakayama (classical Nakayama conjecture); formulations recalled from Y. Zhang (2018) in the manuscript","yearPosed":null,"ageNote":"The manuscript does not cite Nakayama's original paper. It recalls the formulations from Zhang (Algebra Colloquium 2018) and cites Mueller (1968) for the dominant-dimension connection; the conjecture is usually dated to Nakayama in the late 1950s, which the curator should confirm from the original.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Corollary 1.2: for the symmetric pair $(A,M)$ of the Tachikawa counterexample and every field extension $K$ of $\\mathbb F_2(q,H_1,H_2)$, the finite-dimensional algebra $\\Gamma_K=\\mathrm{End}(A_K\\oplus M_K)^{op}$ has all injective terms projective but infinite self-injective dimension, has a nonzero simple $S$ with $\\mathrm{Ext}^i(S,\\Gamma_K)=0$ for all $i\\ge0$, and has a projective-injective Wakamatsu tilting module that is not tilting. Hence the classical, generalized and strong Nakayama conjectures, the Auslander-Gorenstein conjecture and the Wakamatsu tilting conjecture all fail. Corollary 1.3: $\\Gamma_K$ has infinite little and big finitistic dimension. It does NOT give characteristic-zero examples.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. The Nakayama-type counterexamples are Corollary 1.2 of the Tachikawa manuscript, obtained from its symmetric counterexample through the classical Mueller correspondence between pairs (A, M) and algebras of large dominant dimension.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Corollary 1.2 of the Tachikawa manuscript was read against the classical Nakayama conjecture: for $\\Gamma_K=\\mathrm{End}_{A_K}(A_K\\oplus M_K)^{op}$, every term of the minimal injective resolution is projective while the injective dimension of $\\Gamma_K$ is infinite, so $\\Gamma_K$ has infinite dominant dimension and is not self-injective. This consequence has no Lean formalisation: lean/docs/199.md says the endomorphism-algebra and related homological consequences are not included; the formalised Tachikawa statement covers only the symmetric counterexample. The release README warns that some unformalised results could have issues. Readers should know that a counterexample here also gives infinite finitistic dimension (Corollary 1.3), contradicting the finitistic dimension conjecture; the release treats that in a separate family (An algebra of infinite little finitistic dimension).","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The Nakayama conjecture is one of the oldest and best known homological conjectures for finite-dimensional algebras, sitting just below the finitistic dimension conjecture in that hierarchy. Above the Auslander-Reiten conjecture entry (45) and 'Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture?' (30), below the Koethe Conjecture (60), which is known across ring theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to Tachikawa's second conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An explicit counterexample to the Auslander-Reiten conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean (Tachikawa counterexample only, not the Nakayama corollary)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RingTheory/Tachikawa/Counterexample.lean","kind":"lean-proof"},{"label":"Mueller (1968), The classification of algebras by dominant dimension","url":"https://doi.org/10.4153/CJM-1968-037-9","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"serre-positivity-conjecture","name":"Serre's positivity conjecture for intersection multiplicities","shortName":"Serre's positivity conjecture","problemNumber":null,"field":"Commutative algebra; intersection multiplicities over regular local rings","fieldGroup":"Algebra","statement":"Let $(R,\\mathfrak m)$ be a regular local ring of dimension $d$ and $M,N$ finitely generated $R$-modules with $\\ell(M\\otimes_R N)<\\infty$. Serre defined the intersection multiplicity $\\chi^R(M,N)=\\sum_{i}(-1)^i\\,\\ell(\\mathrm{Tor}_i^R(M,N))$ and conjectured that it vanishes when $\\dim M+\\dim N0$ for all nonzero $M,N$ over every regular local ring with $\\ell(M\\otimes_R N)<\\infty$ and $\\dim M+\\dim N=\\dim R$?","posedBy":"J.-P. Serre, Local Algebra, Chapter V (cited by the manuscript in the 2000 English translation, Springer Monographs in Mathematics)","yearPosed":null,"ageNote":"The manuscript cites Serre's Local Algebra only through its 2000 English translation, so it gives no date for the conjecture itself. The French original (Algebre locale, multiplicites) is a 1960s lecture-notes volume, usually dated 1965; the curator should confirm and set yearPosed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $(R,\\mathfrak m)$ is a commutative Noetherian regular local ring of dimension $d$ and $M,N$ are nonzero finitely generated modules with $\\ell(M\\otimes_R N)<\\infty$ and $\\dim M+\\dim N=d$, then $\\chi^R(M,N)>0$. This covers ramified mixed characteristic and needs no lift to an unramified ring. Consequences recorded: the Kurano-Roberts symbolic-power containment $P\\cap Q^{(r)}\\subseteq\\mathfrak m^{r+1}$ in ramified mixed characteristic, and a strict-transform inequality $\\chi^A(A/P,A/Q)\\ge e(A/P)e(A/Q)$ with an equality criterion. It does not address Serre's other questions beyond positivity (vanishing and nonnegativity were already known), and gives no explicit lower bound beyond $\\chi\\ge1$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it, and the README cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Serre's positivity conjecture as the manuscript states it (Chapter V of Local Algebra). The theorem states the full conjecture for every commutative Noetherian regular local ring, with no restriction on characteristic, ramification or liftability. The proof (normalized lengths from perfectoid methods, rational K-theory descent, and an asymptotic rank quotient) was not refereed. No Lean formalization. The manuscript itself records, in a footnote, a public post by Editan dated 9 December 2025 and credited to Editan, Copilot and Gemini that asserts the prime-quotient form of the same conjecture; the manuscript treats it as a same-scope claim, not an established theorem.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"Serre's positivity conjecture is the last open part of his multiplicity conjectures and one of the central homological conjectures of commutative algebra, open since the 1960s with Gabber's nonnegativity as the main prior step. Famous across commutative algebra but rarely known outside it. Tied with Ryser's conjecture (50), just below Zariski's multiplicity conjecture (52) and the blockwise Alperin weight conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Positivity of Serre's Intersection Multiplicity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Serre, Local Algebra (Springer 2000 translation), Chapter V","url":"https://doi.org/10.1007/978-3-662-04203-8","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kaplansky-quasitrace-problem","name":"Kaplansky's quasitrace problem: is every 2-quasitrace on a unital C*-algebra a trace?","shortName":"Kaplansky's quasitrace problem (QT = T)","problemNumber":null,"field":"Operator algebras; traces and quasitraces on C*-algebras","fieldGroup":"Analysis","statement":"A 1-quasitrace on a unital $C^*$-algebra $A$ is a function $\\tau:A\\to\\mathbb C$ with $\\tau(x^*x)=\\tau(xx^*)\\ge0$, $\\tau(h+ik)=\\tau(h)+i\\tau(k)$ for self-adjoint $h,k$, and linear on every abelian $C^*$-subalgebra; it is a 2-quasitrace if it extends to such a function on $M_2(A)$. Blackadar and Handelman showed 2-quasitraces correspond to dimension functions, and the question whether every one is linear is equivalent to Kaplansky's 1951 question whether every $AW^*$-factor of type II$_1$ is a von Neumann algebra. Haagerup proved it for exact algebras, and Milhoj and Rordam showed it is equivalent to asking whether the minimal tensor product of two unital simple stably finite $C^*$-algebras is stably finite. Is every normalized 2-quasitrace on every unital $C^*$-algebra a tracial state?","posedBy":"Irving Kaplansky (1951, the type II1 AW*-factor question, per Gow's account); quasitrace form via Blackadar and Handelman (J. Funct. Anal. 1982); tensor form Milhoj and Rordam (2023, Question 2.5)","yearPosed":1951,"ageNote":"The manuscript dates Kaplansky's introduction of AW*-algebras to 1951 through Gow's 2026 account and treats the factor question and the quasitrace problem as equivalent (Milhoj-Rordam Section 2, Gow Theorem 4.11). The quasitrace formulation dates from Blackadar and Handelman (1982); the manuscript does not cite Kaplansky's own paper.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are a separable unital $C^*$-algebra $A=M_{48}(B)$ with a normalized 2-quasitrace and positive contractions $a,b$ with $\\tau(a+b)-\\tau(a)-\\tau(b)\\ge1/144$ for every normalized 2-quasitrace $\\tau$, so none is a trace. $B$ is generated by 24 shifts between vector bundles with $\\sum x_i^*x_i=1$ and $\\sum x_ix_i^*\\le\\frac23$, Chern-class obstructions ruling out proper infiniteness in every matrix size. Hence a type II$_1$ $AW^*$-factor that is not a von Neumann algebra exists. Corollary: two unital simple stably finite algebras, one $C_r^*(\\mathbb F_2)$, with properly infinite minimal tensor product (answering Milhoj-Rordam Question 2.5), and a separable stably finite algebra with no tracial state. Not shown: a simple or exact example (exact is impossible by Haagerup), or an explicit II$_1$ factor.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, using on average about three hours of ChatGPT Pro thinking compute per result, with outputs grouped into families and manuscripts. The README's two exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region write-up, which was human-edited for readability) do not concern this family. The manuscript is credited to OpenAI alone, names no human author and has no acknowledgements. Lean formalizations of the main theorem and of the stable-finiteness corollary are in the release's lean/ library (lean/docs/294.md). The release does not say how much human review happened before publication.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the TeX source against the posed problem, and both Lean statements. formalization.yaml lists ComparatorChallenges/KaplanskyQuasitrace.json, declaration OAI.Kaplansky.kaplansky_quasitrace_counterexample in OAI/Analysis/Kaplansky/Main.lean, permitted axioms propext, Quot.sound and Classical.choice. Its statement, read here, defines 1-quasitraces with the three axioms above (linearity on closed commutative nonunital star-subalgebras), 2-quasitraces by extension to 2x2 matrices, and asserts a separable C*-algebra with a normalized 2-quasitrace and positive contractions a, b such that every normalized 2-quasitrace has Re(tau(a+b) - tau(a) - tau(b)) >= 1/144. That is the headline: no normalized 2-quasitrace there is additive. A second challenge, ComparatorChallenges/KaplanskyStableFiniteness.json (solution module OAI.Analysis.Kaplansky.StableFiniteness exists at the pinned commit), is not in the formalization catalogue; read here, it states Corollary 1.2 (i) to (iii) with the tensor product modelled concretely on l2(F2, H). Neither was rebuilt here. On paper, the AW*-factor reading and Corollary 1.2 (i) rest on the equivalences in Milhoj-Rordam and Gow's 2026 preprint; the constructed algebra is necessarily nonexact.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"A central open problem of C*-algebra theory since Blackadar and Handelman, equivalent to Kaplansky's 1951 AW*-factor question, with Haagerup's exact case a landmark of its own. Placed between the Kadison-Kaplansky projection conjecture at 50 and Kadison's similarity problem at 58, and level with the Griffiths conjecture at 55: famous across operator algebras but little known outside them.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI.Kaplansky.kaplansky_quasitrace_counterexample","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Kaplansky/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: KaplanskyQuasitrace.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KaplanskyQuasitrace.lean","kind":"lean-proof"},{"label":"Lean proof: OAI.KaplanskyConsequences.stable_finiteness_bundle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Kaplansky/StableFiniteness.lean","kind":"lean-proof"},{"label":"Lean comparator statement: KaplanskyStableFiniteness.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KaplanskyStableFiniteness.lean","kind":"lean-proof"},{"label":"Lean scope note for family 294","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/294.md","kind":"lean-proof"},{"label":"Milhoj and Rordam, Around traces and quasitraces (arXiv:2309.17412), Question 2.5","url":"https://arxiv.org/abs/2309.17412","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gottschalk-surjunctivity-conjecture","name":"Gottschalk's surjunctivity conjecture","shortName":"Gottschalk surjunctivity","problemNumber":null,"field":"Symbolic dynamics on groups; cellular automata","fieldGroup":"Algebra","statement":"For a group $G$ and a finite alphabet $A$, a cellular automaton on $A^G$ is a map $\\tau(x)(g)=\\phi((x(gu))_{u\\in M})$ given by a finite memory set $M\\subset G$ and a local rule $\\phi:A^M\\to A$. $G$ is surjunctive if every injective cellular automaton on $A^G$, for every finite $A$, is surjective. Gromov and Weiss proved that sofic groups are surjunctive, and surjunctivity implies Kaplansky's direct finiteness over finite fields. Gottschalk (1973) conjectured that every group is surjunctive. Is every group surjunctive?","posedBy":"Walter Gottschalk, Some general dynamical notions, Recent Advances in Topological Dynamics, LNM 318 (1973)","yearPosed":1973,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The group $G$ and elements $a,b\\in K[G]$ of the characteristic-two direct-finiteness counterexample ($K$ finite, $G$ finitely presented with odd torsion) define a cellular automaton on $K^G$ that is injective but not surjective, so $G$ is not surjunctive. The odd-characteristic companion (September 26) gives the same conclusion over a field of order $p^4$ for a finitely generated group with torsion. The automata are linear; no torsion-free non-surjunctive group is claimed in these papers (the torsion-free companion does not discuss automata).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The cellular automaton is obtained from the direct-finiteness counterexample by the standard algebra-to-automaton transfer, proved in full in Section 6 of the principal manuscript.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the September 23 manuscript's Section 6 was read against Gottschalk's conjecture as stated there; the coefficients of $b$ give a linear cellular automaton on $K^G$ ($K$ finite) with left inverse from $a$, injective and not surjective. The proof was not refereed. The Lean statements attached to this manuscript do not include the automaton; the formal statement that does is the odd-characteristic companion's ComparatorChallenges/OddKaplansky.lean (listed in formalization.yaml, declaration OAI.OddKaplansky.main_theorem), read here: for the least prime factor $p$ of $(\\binom{1200}{600}!)^2+1$, a field of order $p^4$, a finitely generated group with a nontrivial torsion element, and $a,b$ with $ab=1$, $ba\\ne1$, such that the convolution map $x\\mapsto(g\\mapsto\\sum_u b_u x(gu))$ on all configurations $G\\to K$ is injective and not surjective. That states a counterexample to the conjecture (alphabet $K$, memory set the support of $b$). Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"A well-known problem of symbolic dynamics since 1973, central to the sofic groups programme of Gromov and Weiss. Placed below the sofic groups question (60) and Kaplansky's direct finiteness (52), level with Kakeya maximal in three dimensions (48) and the binary corrected Elliott conjecture (48), above Wall's manifold-realization question (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Counterexample in Odd Characteristic (Lean-checked automaton)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.OddKaplansky.main_theorem)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/OddKaplansky/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: OddKaplansky.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/OddKaplansky.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dixmier-unitarizability-problem","name":"Dixmier's unitarizability problem for discrete groups","shortName":"Dixmier's unitarizability problem","problemNumber":null,"field":"Group theory; amenability and uniformly bounded representations","fieldGroup":"Algebra","statement":"A representation $\\pi:G\\to GL(H)$ of a discrete group on a Hilbert space is uniformly bounded if $\\sup_g\\|\\pi(g)\\|<\\infty$, and unitarizable if some bounded invertible $S$ makes every $S\\pi(g)S^{-1}$ unitary; $G$ is unitarizable if every uniformly bounded representation is. Day and Dixmier (1950) showed amenable groups are unitarizable, and groups containing a nonabelian free subgroup are not (Ehrenpreis-Mautner, Pytlik-Szwarc). Partial converses were known via Pisier's quantitative criterion, Epstein-Monod, Osin, Monod-Ozawa and Vergara. Dixmier asked whether unitarizability characterizes amenability: is every unitarizable discrete group amenable?","posedBy":"Jacques Dixmier, Les moyennes invariantes dans les semi-groupes et leurs applications, Acta Sci. Math. (Szeged) 12 (1950), section 5, p. 221","yearPosed":1950,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a discrete group is amenable if and only if every uniformly bounded representation on every complex Hilbert space is similar to a unitary one. For every nonamenable $G$ and $\\varepsilon>0$ there is a representation with $\\sup_g\\|\\pi(g)\\|\\le1+\\varepsilon$ not similar to a unitary one, on a separable space if $G$ is countable. This settles Dixmier's question for all discrete groups. Not shown: anything for nondiscrete locally compact groups, or a bound on similarity degree beyond what the theorem gives.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript (September 23, 2026) proves the main equivalence without companion inputs, per its INPUTS.md; only a side consequence for group C*-algebras uses the release's Kadison similarity manuscript.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Unitarizability implies amenability for discrete groups' was read against Dixmier's question as cited (1950, section 5). The proof (a bounded non-inner operator cocycle built from sparse assignments and random-sign matrix weights) was not refereed. Lean: lean/formalization.yaml lists comparator config ComparatorChallenges/DixmierAllDiscrete.json, declaration OAI.Dixmier.current_main_theorem in OAI/Analysis/Unitarizability/AllDiscrete.lean, permitted axioms propext, Classical.choice and Quot.sound. The statement file was read: for every discrete group, amenability (a positive normalized left-invariant mean on bounded functions) is equivalent to every uniformly bounded representation on every complete complex inner product space being similar to an isometric one, and every nonamenable group has a nonunitarizable representation with bound at most 1 + epsilon, on a separable space when the group is countable. That is the headline claim. A second challenge, ComparatorChallenges/Dixmier.json (countable groups, separable witness with bound 101, solution module OAI/Analysis/Unitarizability/UniverseTransport.lean, present at the commit), is not in the formalization catalogue; its statement was read here. Neither was rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Dixmier's 1950 question is one of the classic open problems linking amenability and operator theory, studied by Pisier, Monod, Ozawa, Epstein and others for seventy years, and closely tied to Kadison's similarity problem. Placed just below Kadison's Similarity Problem (58) and the Thompson's group F entry (58), tied with the four-dimensional Kakeya set entry (55), above the Kadison-Kaplansky Projection Conjecture (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Unitarizability implies amenability for discrete groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI.Dixmier.current_main_theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Unitarizability/AllDiscrete.lean","kind":"lean-proof"},{"label":"Lean comparator statement: DixmierAllDiscrete","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DixmierAllDiscrete.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Dixmier (countable groups, not in catalogue)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/Dixmier.lean","kind":"lean-proof"},{"label":"Lean scope note for family 251","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/251.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"aov-perfect-matching-entropy-conjecture","name":"The Anari-Oveis Gharan-Vinzant perfect-matching entropy conjecture","shortName":"AOV perfect-matching entropy conjecture","problemNumber":null,"field":"Approximate counting; matching polytope; entropy","fieldGroup":"Theoretical computer science","statement":"Let $G$ be a graph on $n=2m$ vertices with $N(G)\\ge1$ perfect matchings and perfect-matching polytope $P(G)$. For $x\\in P(G)$ put $F(x)=-\\sum_e x_e\\log x_e$ and $B(x)=-\\sum_e(1-x_e)\\log(1-x_e)$. Anari, Oveis Gharan and Vinzant proposed a convex-programming approach to counting, in which $\\max_{x\\in P(G)}(F(x)+B(x))$ estimates $\\log N(G)$; for bipartite graphs the Schrijver and Gurvits inequalities make it accurate. Is it true for every graph that $\\max_{x\\in P(G)}(F(x)+B(x))\\le\\log N(G)+O(n)$, that is, that the binary-coordinate entropy of a perfect-matching mean exceeds $\\log N(G)$ by at most a linear function of the number of vertices?","posedBy":"Nima Anari, Shayan Oveis Gharan and Cynthia Vinzant, Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids, FOCS 2018, Conjecture 6","yearPosed":2018,"ageNote":"Conjecture 6 of the FOCS 2018 version, as the manuscript cites it; the arXiv version (1807.00929) appeared in July 2018.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a loopless multigraph on $2m\\ge2$ vertices with a perfect matching and every $x\\in P(G)$, boundary included, $F(x)-(2-2/m)B(x)\\le H(x)\\le F(x)$, where $H(x)$ is the maximum entropy of a law on perfect matchings with marginals $x$. Hence $\\log N(G)\\le\\max_{x\\in P(G)}(F+B)\\le\\log N(G)+3m-2$, the conjectured linear error with an explicit constant. Also proved: the sharp bound $|\\mathrm{supp}\\,x|-\\dim F_x\\le3m-2$ for the minimal face, and lower counts such as $N(G)\\ge e^{-2(m-1)}k^m$ for $k$-regular graphs with all odd cuts of size at least $k$. The paper notes the bipartite coefficient-one inequality $H\\ge F-B$ fails for a nonbipartite eight-vertex example.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the counting corollary of the manuscript were read against Conjecture 6 as the manuscript states it; the conjecture text in the FOCS paper was not opened here. The proof was not refereed. Lean: lean/formalization.yaml lists OAI.MatchingEntropy.entropy_main (Comparator MatchingEntropy). Its statement, read here, is an earlier coefficient-eight bound, $F(x)-H(x)\\le 8m(1-e^{-F(x)/m})$ for every $x$ in the polytope; with $H\\le\\log N$ and $B\\le m$ this already gives the conjecture's linear form with constant $9m$, though not the paper's sharp $3m-2$. The sharp pointwise bound $F-(2-2/m)B\\le H\\le F$ is the separate challenge MatchingEntropyBounds (OAI.MatchingEntropyBounds.Refined.pointwise_entropy, solution module OAI.Combinatorics.MatchingEntropy.Main, present at the pinned commit), which is not in the formalization catalogue; its statement was read here. Neither development was rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A named conjecture from a FOCS paper, known to the approximate-counting and polynomial-method community and not much beyond; it mattered as a route to deterministic counting of nonbipartite matchings. Level with Approximate Counting for Spin Systems on Planar Graphs (18), below Anari's Bethe permanent conjecture (22) and The Optimal Approximation Ratio for Permanents of PSD Matrices (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf","sourceName":"OpenAI math release: Entropy and Face Dimension of the Perfect-Matching Polytope","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Same family: A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf","kind":"paper"},{"label":"Lean (catalogued): OAI.MatchingEntropy.entropy_main","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/PerfectMatching/Main.lean","kind":"lean-proof"},{"label":"Lean Comparator challenge MatchingEntropyBounds (sharp pointwise bound)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/MatchingEntropyBounds.lean","kind":"lean-proof"},{"label":"Anari, Oveis Gharan and Vinzant, arXiv 1807.00929","url":"https://arxiv.org/abs/1807.00929","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-littlewood-polynomials-fixed-gap","name":"Erdos's question on the maximum modulus of real Littlewood polynomials: is it at least (1 + c) sqrt(N)?","shortName":"Erdos's fixed-gap question, real signs","problemNumber":null,"field":"Littlewood polynomials; extremal problems on the unit circle","fieldGroup":"Analysis","statement":"For signs $\\epsilon_k\\in\\{-1,1\\}$ let $P(z)=\\sum_{k=0}^{N-1}\\epsilon_k z^k$. Parseval gives $\\|P\\|_{L^2(|z|=1)}=\\sqrt N$, so $\\max_{|z|=1}|P(z)|\\ge\\sqrt N$. Erdos asked whether this is far from sharp: is there an absolute constant $c>0$ with $\\max_{|z|=1}|P(z)|\\ge(1+c)\\sqrt N$ for every such polynomial of large length? The analogue for complex unimodular coefficients was refuted by Kahane's ultraflat polynomials (1980); the Rudin-Shapiro polynomials give $O(\\sqrt N)$ and Balister et al. gave two-sided flatness up to constants, but the constant $1+c$ for real signs stayed open. Is $\\liminf_N \\min_{\\epsilon}\\max_{|z|=1}|P(z)|/\\sqrt N>1$?","posedBy":"Paul Erdos","yearPosed":1957,"ageNote":"Erdos, Some unsolved problems, Michigan Math. J. 4 (1957), Problem 22, as cited by the manuscript; also Hayman-Lingham, Research problems in function theory, Problems 4.13 and 4.31.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: $\\min_{\\epsilon}\\max_{|z|=1}|\\sum_{k0$ as in Erdos's question exists. The construction relaxes to real coefficients in $[-1,1]$ with small defect, using sampled quadratic-phase modes and a hypergraph interval packing, then rounds to signs by a Lovett-Meka partial-coloring argument. It is existential and gives no rate. The principal paper gives no uniform lower bound; the October 5 companions add $|P|\\ge\\sqrt N/16$ and then full two-sided ultraflatness $(1\\pm\\epsilon)\\sqrt N$ at every large length.","claimIssueNote":"This result conflicts with published claims. Preprints by el Abdalaoui (arXiv:1609.03435 and two others) claim the opposite, that such flat polynomials do not exist; the release's manuscript argues in its Appendix A that those arguments have specific gaps. Until the conflict is settled in public, the entry is Contested.","solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript is dated September 23, 2026. Two companions dated October 5, 2026 strengthen it: one adds a lower bound of sqrt(N)/16, the other makes the polynomials two-sided ultraflat; the latter says it reuses lemmas of a 'version-2' refinement of the principal paper.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 ($\\lim_N m_N=1$ through all integers $N$) was read against Erdos's real-sign question; it answers it in the negative. lean/formalization.yaml lists a main result for this manuscript (comparator AsymptoticallyMinimalLittlewood, declaration OAI.AsymptoticallyMinimalLittlewood.main). The comparator statement was read here: for every $\\eta>0$ there is $N_0$ such that for every $N\\ge N_0$ there are real signs with $|\\sum_{k0$, does every sufficiently large $n$-point planar set have a point determining at least $n^{1-\\varepsilon}$ distinct distances?","posedBy":"Paul Erdős","yearPosed":1957,"ageNote":"Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), Problem 16, as cited by the manuscript; the name 'weak pinned conjecture' follows arXiv:2505.06590 (Conjecture 4.7). erdosproblems.com lists the question as Problem 604.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every fixed $s>0$, $F_n(s)\\to0$, where $F_n(s)$ is the largest fraction, over $n$-point planar sets, of ordered distinct pairs $(x,y)$ such that at least $n^s$ points of the set lie at distance $\\|y-x\\|$ from $x$. Corollary 1.2: for every fixed $\\varepsilon>0$ the fraction of pins with fewer than $n^{1-\\varepsilon}$ distinct distances tends to $0$ uniformly, so every large set has such a pin, and in fact all but $o(n)$ points are. No explicit rate is given. Not shown: the stronger form $\\gg n/\\sqrt{\\log n}$ in Erdős's question, the averaged conjecture $\\sum_x|D_x|\\gg n^2/\\sqrt{\\log n}$, or any quantitative form of the $n^{o(1)}$ loss.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript is 'The weak pinned planar distance theorem' (September 23, 2026); its family also contains a separate unit-distance manuscript of the same date, entered separately.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against Erdős's question as recorded on erdosproblems.com (Problem 604) and in the manuscript. Corollary 1.2 gives, for every fixed $\\varepsilon>0$, that all but $o(n)$ points of any $n$-point planar set determine at least $n^{1-\\varepsilon}$ distances, uniformly over configurations, with no rate. Lean: formalization.yaml lists comparator ComparatorChallenges/PinnedDistances.json, declaration OAI.WeakPinned.main in OAI/Geometry/PinnedDistances/Main.lean. The statement file was read here: main says that for each $s>0$ the supremum over $n$-point planar sets of the fraction of ordered pairs $(x,y)$ whose distance fiber at $x$ has at least $n^s$ points tends to $0$. That is the paper's Theorem 1.1, which is stronger than the headline; the headline follows by the five-line pigeonhole count printed as Corollary 1.2, which is not itself formalised. Not rebuilt here. The proof uses real-closed-field transfer to number fields and the product formula.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The weak form of Erdős's 500 dollar pinned-distance problem; the exponent had stood at Katz-Tardos's 0.864 for twenty years, and the pinned question is a central problem of incidence geometry after Guth-Katz. Placed below the Hadwiger-Nelson five-color step (55) and the Falconer distance conjecture (60), just above Erdős's planar unit distance conjecture as the catalog scores it (40), far above Erdős Problem 654 (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The weak pinned planar distance theorem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion manuscript: A power saving for planar unit distances","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean proof: OAI/Geometry/PinnedDistances/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/PinnedDistances/Main.lean","kind":"lean-proof"},{"label":"Lean statement: ComparatorChallenges/PinnedDistances.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PinnedDistances.lean","kind":"lean-proof"},{"label":"Erdős Problem 604 (erdosproblems.com)","url":"https://www.erdosproblems.com/604","kind":"problem-record"},{"label":"Erdős, Some unsolved problems (1957), Problem 16","url":"https://doi.org/10.1307/mmj/1028997963","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kurosh-problem-division-rings","name":"Kurosh's problem for division rings: are algebraic division rings locally finite?","shortName":"Kurosh problem for division rings","problemNumber":null,"field":"Noncommutative ring theory; division algebras","fieldGroup":"Algebra","statement":"A division ring $D$ with center $F$ is algebraic over $F$ if every element satisfies a nonzero polynomial over $F$, and locally finite if every finite subset generates a division subring of finite dimension over $F$. In 1941, in connection with Burnside's problem, Kurosh asked whether a division ring that is algebraic and finitely generated as an algebra over a central subfield must be finite-dimensional over it. Golod's nil algebras answer the analogous question for general algebras but contain nilpotents, so they do not touch division rings; Jacobson and Kaplansky settled the bounded-degree and polynomial-identity cases. Is every division ring that is algebraic over its center locally finite?","posedBy":"A. G. Kurosh, Ringtheoretische Probleme, die mit dem Burnsideschen Problem uber periodische Gruppen in Zusammenhang stehen (1941), Problem (K)","yearPosed":1941,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: there is a countable division ring $D$ of characteristic zero with center $F$ and $x,y\\in D$ such that $D=F[x,y]=F(x,y)$, $[D:F]=\\infty$, and every element of $D$ is algebraic over $F$ (with no uniform degree bound). So algebraicity over the center does not imply local finiteness. It does NOT give an example in positive characteristic or over a prescribed center, and polynomial degrees grow without bound, as they must by Jacobson's theorem.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. The construction is a countable inductive scheme using symbol and cyclic algebras, formal series in two noncommuting variables and a lifting theorem that makes each scheduled element algebraic.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against Kurosh's Problem (K): a countable division ring of characteristic zero, algebraic over its center $F$, with $D=F[x,y]$ for two elements and $[D:F]=\\infty$. Since $F$ is central, this is a negative answer to the problem as Kurosh posed it (over a central subfield). No Lean formalisation exists for this family. The release README warns that some unformalised results could have issues.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Kurosh's division-ring problem has been a classical open problem of noncommutative algebra since 1941, untouched by Golod's methods and listed among the main questions about division algebras. Below the Koethe Conjecture (60), above the word problem for finite-rank Artin groups (45) and the finitely presented nil-algebra entry (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A Counterexample to Kurosh's Division-Ring Problem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kurosh (1941), Ringtheoretische Probleme (Izv. Akad. Nauk SSSR)","url":"https://www.mathnet.ru/rus/im3832","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fontaine-mazur-two-dimensional-p-2","name":"The odd regular two-dimensional Fontaine-Mazur conjecture over Q at p = 2","shortName":"Fontaine-Mazur at p = 2","problemNumber":null,"field":"Galois representations; modularity lifting at p = 2","fieldGroup":"Number theory","statement":"The Fontaine-Mazur conjecture (1995) predicts that irreducible $p$-adic Galois representations unramified outside finitely many primes and de Rham at $p$ come from algebraic geometry. For odd two-dimensional representations of $G_{\\mathbb Q}$ with distinct Hodge-Tate weights it takes the form of modularity up to Tate twist. Building on Kisin and Emerton, Pan (2022) and Zhang (2025) established this for every odd prime, and at $p=2$ Tung removed local restrictions only under a nonsolvable residual image. Let $r:G_{\\mathbb Q}\\to\\mathrm{GL}_2(\\overline{\\mathbb Q}_2)$ be continuous, irreducible, odd, unramified outside finitely many primes, and de Rham at $2$ with distinct Hodge-Tate weights. Is $r$ a Tate twist of the Galois representation of a classical cuspidal eigenform, with no condition on the residual representation?","posedBy":"Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations (1995); the p = 2 two-dimensional case is the part left open after Pan and Zhang, as the manuscript explains","yearPosed":1995,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every continuous, irreducible, odd $r:G_{\\mathbb Q}\\to\\mathrm{GL}_2(\\overline{\\mathbb Q}_2)$ unramified outside finitely many primes and de Rham at $2$ with distinct Hodge-Tate weights is a Tate twist of the representation of a classical cuspidal eigenform; no residual hypothesis. With the odd-prime results this gives the regular odd two-dimensional case over $\\mathbb Q$ at every prime. It does not address equal Hodge-Tate weights, even representations, higher dimension or other number fields, so the general Fontaine-Mazur conjecture stays open. Companions prove 2-adic pro-modularity without a de Rham hypothesis (October 4) and Emerton's dimension conjecture at $p=2$ (October 5), entered separately.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript says this theorem supplies the weight-two modularity input to the release's Hilbert's tenth problem over Q manuscript. It cites several 2025-2026 works (Thorne, Zhang, Emerton-Gee-Pan-Zhu, Pan) as inputs.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the September 23 manuscript was read against the odd regular two-dimensional case of the Fontaine-Mazur conjecture as the paper formulates it; it claims modularity up to Tate twist for every such 2-adic representation, including scalar and reducible residual representations. The proof was not refereed. It depends on recent external results whose status was not checked here, notably Thorne's modularity of ordinary weight-{0,1} representations (cited as 2026), Zhang (2025), Pan (2022), Paskunas-Tung (2021) and the release's own companion results. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The last open prime in the odd regular two-dimensional Fontaine-Mazur conjecture over the rationals, a long-standing target of modularity lifting since Kisin and Emerton. Placed level with the p-converse in Selmer corank one at every prime (45) and Vinogradov's least quadratic nonresidue (45), below modularity of elliptic curves over imaginary quadratic fields (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf","sourceName":"OpenAI math release: Fontaine-Mazur modularity at the prime 2","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Companion: Unrestricted pro-modularity at the prime two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-4-2026/two-adic-promodularity.pdf","kind":"paper"},{"label":"Companion: The Dimension of the Two-Adic Hecke Algebra at Odd Level","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf","kind":"paper"},{"label":"Hilbert's tenth problem over Q, whose proof uses this result","url":"https://vibemathed.com/problem/hilberts-tenth-problem-over-q","kind":"other"},{"label":"Earlier edition (September 23, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf","kind":"paper"},{"label":"Revised edition (October 6, 2026): Unrestricted pro-modularity at the prime two","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gromov-codimension-two-width-conjecture","name":"Gromov's codimension-two Urysohn width conjecture for positive scalar curvature, with the macroscopic dimension conjecture for universal covers","shortName":"Gromov's codimension-two width conjecture","problemNumber":null,"field":"Scalar curvature; Urysohn width and macroscopic dimension","fieldGroup":"Geometry & topology","statement":"For a metric space $X$ the Urysohn $d$-width $\\mathrm{UW}_d(X)$ is the infimum, over continuous maps $f:X\\to K$ to simplicial complexes of dimension at most $d$, of $\\sup_y\\operatorname{diam}f^{-1}(y)$. Gromov proposed that positive scalar curvature collapses two dimensions at the curvature scale: a complete Riemannian $n$-manifold with $\\mathrm{Scal}\\ge1$ should map to an $(n-2)$-dimensional polyhedron with all point inverses of diameter bounded by a constant depending only on $n$, and the universal cover of a closed manifold with positive scalar curvature should have macroscopic dimension at most $n-2$. Known: closed and complete three-manifolds (Gromov-Lawson, Liokumovich-Maximo, Liokumovich-Wang), spin cases under the Strong Novikov Conjecture (Bolotov, Bolotov-Dranishnikov), and the aspherical consequence in dimensions four and five (Chodosh-Li, Gromov). Is $\\mathrm{UW}_{n-2}(M)\\le C_n$ for every complete $n$-manifold with $\\mathrm{Scal}\\ge1$?","posedBy":"Mikhael Gromov (Large Riemannian manifolds, 1986, Section 2.A(c); Positive curvature, macroscopic dimension, spectral gaps and higher signatures, 1996, Section 2 1/2; A dozen problems, 2017)","yearPosed":1986,"ageNote":"The spectral companion dates the quantitative formulation to Gromov 1986, Section 2.A(c); the principal manuscript cites the 1996 macroscopic-dimension paper and Gromov's 2017 problem list (filling-radius Conjecture A-minus).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Principal manuscript, Theorem 1.1: for every $n\\ge4$ there is $C_n$ such that every connected complete boundaryless Riemannian $n$-manifold with $\\mathrm{Scal}\\ge1$ admits a continuous map to a simplicial complex of dimension at most $n-2$ whose entire fibers have diameter at most $C_n$; no orientability, spin, compactness or bounded-geometry assumption. Corollaries: Gromov's filling-radius bound $\\mathrm{FillRad}\\le C_n/(2\\sigma)$ for closed manifolds with $\\mathrm{Scal}\\ge\\sigma^2$, $\\dim_{mc}\\le n-2$ for universal covers of closed positive-scalar-curvature manifolds in every $n\\ge2$, and an independent proof that closed aspherical manifolds of dimension $\\ge4$ carry no positive scalar curvature. The October 5 companions extend the width bound to the spectral condition $-4\\Delta+\\mathrm{Scal}\\ge1$ ($n\\ge4$, and $n=3$ with a graph target and bound $500/\\sqrt\\lambda$). Not shown: the Lipschitz form $\\dim_{MC}$, or any explicit constant.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introductions and main theorems of all three manuscripts, read against Gromov's conjecture as they quote it; the proofs were not refereed. No Lean formalization exists for this family (lean/docs/336.md is absent at the pinned commit). Scope limits the papers state: the uniform width theorem is for $n\\ge4$ (dimension three rests on Liokumovich-Wang and the spectral three-manifold companion); the macroscopic-dimension corollary is the continuous invariant $\\dim_{mc}$, which the paper distinguishes from Dranishnikov's Lipschitz invariant $\\dim_{MC}$; the constant $C_n=4D+2$ is not made explicit. The proof is long (a countable-label minimal-hypersurface cutting scheme with singular-set shielding in appendices of over 200 kB of TeX), so independent checking will take time.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Gromov's codimension-two width and macroscopic dimension conjectures are central organising problems of scalar-curvature geometry, open since 1986 and 1996 and known in full only in dimension three; the width form implies the aspherical conjecture. Placed just below the Gromov-Lawson aspherical conjecture (60), which is better known outside the field, and well above Gromov's integral scalar-curvature bound for simplicial volume (38) and the universal volume growth question (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Positive scalar curvature and uniform codimension-two width","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Spectral scalar curvature and uniform Urysohn width (October 5, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf","kind":"paper"},{"label":"Companion: Spectral scalar curvature and Urysohn width in dimension three (October 5, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf","kind":"paper"},{"label":"Gromov, A dozen problems, questions and conjectures about positive scalar curvature (2017)","url":"https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/dozen_problems_Sept_2017.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"alperin-weight-conjecture","name":"The blockwise Alperin weight conjecture","shortName":"Alperin weight conjecture","problemNumber":null,"field":"Modular representation theory of finite groups","fieldGroup":"Algebra","statement":"Let $p$ be a prime and $G$ a finite group. A $p$-weight of $G$ is a pair $(Q,\\phi)$ with $Q$ a $p$-subgroup and $\\phi$ an irreducible character of $N_G(Q)/Q$ of $p$-defect zero; each weight is assigned to a $p$-block of $G$ by Brauer induction. For a block $B$, let $l(B)$ be the number of irreducible Brauer characters (simple modules) in $B$ and let $W_p(B)$ be the set of $G$-conjugacy classes of weights assigned to $B$. Alperin conjectured that simple modules are counted locally, block by block. Does $l(B)=|W_p(B)|$ hold for every prime $p$, every finite group $G$ and every $p$-block $B$?","posedBy":"J. L. Alperin, Weights for finite groups (Arcata conference 1986, proceedings 1987)","yearPosed":1987,"ageNote":"Formulated at the 1986 Arcata conference; the original account appeared in the 1987 proceedings, per the manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims $l(B)=|W_p(B)|$ for every prime $p$, finite group $G$ and $p$-block $B$, including the $Q=1$ case. Consequences derived: $l(B)\\ge l(b)$ for the Brauer correspondent $b$, equality for abelian defect groups, the Sylow normalizer inequality for Brauer character counts, and principal-block bounds of Hung, Sambale and Tiep. The proof uses chain inversion and powers modulo commutators to reduce to counting tuples in groups with trivial $O_p$, then proves the needed $p$-power divisibility via moduli of marked genus-one covers. It is an equality of cardinalities only: no natural or equivariant bijection, and no inductive condition for simple groups, is established.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, which was posed some 4,000 open research problems during an evaluation, with the vast majority of results obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each. This manuscript is not among the README's named exceptions. It is credited to OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1 was read against the posed conjecture and states the numerical blockwise form exactly, with no restriction on prime, group or defect group, using Brauer's central-character block induction. There is no Lean formalization for this manuscript in the release. Nothing was re-run here. A reader should know the proof route is unusual: it avoids the classification of finite simple groups and the inductive blockwise weight condition, reducing instead to a divisibility statement for tuple counts proved through high-index genus-one curve covers and inseparable genus change in characteristic p. The equivariant or canonical refinements of the conjecture are not claimed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"One of the handful of central local-global conjectures of modular representation theory, open for almost forty years and the target of a long reduction program through finite simple groups; famous within algebra rather than across mathematics. Scored with the Cycle Double Cover Conjecture at 55, just below the Köthe Conjecture and Gromov and Weiss's Question on Sofic Groups at 60.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Blockwise Alperin Weight Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"purely-cosmetic-surgery-conjecture","name":"The purely cosmetic surgery conjecture for knots in the three-sphere","shortName":"Purely cosmetic surgery conjecture","problemNumber":null,"field":"Low-dimensional topology: Dehn surgery on knots","fieldGroup":"Geometry & topology","statement":"For a knot $K\\subset S^3$ and a slope $r\\in\\mathbb{Q}\\cup\\{\\infty\\}$, let $S^3_r(K)$ be the oriented manifold obtained by Dehn surgery. Two distinct slopes $r\\ne s$ are purely cosmetic if $S^3_r(K)$ and $S^3_s(K)$ are orientation-preservingly homeomorphic. Gordon conjectured (as part of his cosmetic filling conjecture) that this never happens for a nontrivial knot; the question is Bleiler's Problem 1.81(A) in Kirby's list. Known before this work: the meridional case (Gordon-Luecke), same-sign pairs (Wu), the reduction to opposite slopes $\\pm 2$ or $\\pm 1/q$ (Ni-Wu, Hanselman), exclusion of $\\pm 1/q$ (Daemi-Lidman-Miller Eismeier), and verification for many families and all knots up to about 19 crossings. Does every nontrivial knot $K\\subset S^3$ satisfy: $S^3_r(K)\\cong S^3_s(K)$ orientation-preservingly implies $r=s$?","posedBy":"C. McA. Gordon, Dehn surgery on knots, Proc. ICM Kyoto 1990 (published 1991), Conjecture 6.1; recorded as Bleiler's Problem 1.81(A) in Kirby's 1997 problem list","yearPosed":1990,"ageNote":"Gordon stated the conjecture in his 1990 ICM address, published in the proceedings in 1991 (the manuscript cites the 1991 volume). Kirby's 1997 list records it as Problem 1.81(A), attributed to Bleiler.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every nontrivial smooth knot $K\\subset S^3$ and all $r,s\\in\\mathbb{Q}\\cup\\{\\infty\\}$, an orientation-preserving homeomorphism $S^3_r(K)\\cong S^3_s(K)$ forces $r=s$; the homeomorphism need not preserve the surgery core or the knot exterior. After the cited reduction to $g(K)=2$ and slopes $\\pm 2$, the paper derives a contradiction by comparing a nonzero integer instanton count $\\Omega$ on a parameterized cobordism with a one-dimensional count of solutions with spinors whose boundary would force $2^\\eta\\Omega=0$. It does not address chirally cosmetic (orientation-reversing) surgeries or knots in other three-manifolds.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as stated by Gordon and in Kirby's list; the statement matches the full conjecture, including the meridional and zero slopes. The proof was not refereed. The argument takes as an input the reduction (Proposition 1.2) that any purely cosmetic pair must be the slopes $\\pm 2$ on a genus-two knot, which the manuscript draws from Daemi, Lidman and Miller Eismeier (arXiv:2410.21248) combined with Hanselman and Ni-Wu; the new work rules out this last case with an instanton count compared against a nonabelian (SO(3)) monopole cobordism in the Pidstrigach-Tyurin, Okonek-Teleman and Feehan-Leness tradition, an area with a history of delicate analytic gaps. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A named conjecture central to Dehn surgery theory for 35 years, on Kirby's list and the target of Heegaard Floer, instanton, quantum and computer-assisted work. Famous within low-dimensional topology but not beyond it. Scored above the Kinoshita conjecture and Kirby Problem 4.37 (35) and level with Brennan's conjecture (45), below the Kervaire conjecture (55) and Cannon's conjecture (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Purely cosmetic surgery on knots in the three-sphere","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Daemi, Lidman, Miller Eismeier, Filtered instanton homology and cosmetic surgery (arXiv:2410.21248)","url":"https://arxiv.org/abs/2410.21248","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"long-time-hard-sphere-fluctuations","name":"Long-time nonequilibrium fluctuations of the hard-sphere gas (finite-dimensional laws)","shortName":"Long-time hard-sphere fluctuations","problemNumber":null,"field":"Kinetic theory; fluctuation theory","fieldGroup":"Mathematical physics","statement":"Beyond the Boltzmann equation, the fluctuations of the empirical measure of a dilute hard-sphere gas are expected to be Gaussian and governed by the linear fluctuating Boltzmann equation. Bodineau, Gallagher, Saint-Raymond and Simonella proved this for short times away from equilibrium and for long times at equilibrium. After deriving the Boltzmann equation on every regular kinetic interval, Deng, Hani and Ma identified long-time nonequilibrium fluctuations as a further application of their method. Do the fluctuations of a hard-sphere gas converge to the Gaussian solution of the linear fluctuating Boltzmann equation throughout every interval on which the Boltzmann solution is regular, without an equilibrium or small-data assumption?","posedBy":"Yu Deng, Zaher Hani and Xiao Ma (arXiv 2408.07818, Section 1.4.3)","yearPosed":2024,"ageNote":"The manuscript cites the 2025 version of the Deng-Hani-Ma preprint, first posted in 2024.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"For a deterministic grand-canonical hard-sphere gas in $\\mathbb R^3$ with smooth initial density having summable Gaussian bounds on it and its gradient, on every $[0,T]$ where the Boltzmann solution has uniform Gaussian velocity decay, the empirical measure centered at its exact microscopic expectation and scaled by $\\sqrt{\\mu_\\varepsilon}$ has finite-dimensional distributions converging to those of the Gaussian solution of the linear fluctuating Boltzmann equation. It does not prove process-level convergence or tightness, does not treat periodic domains (the BGSS setting) or potentials, and assumes the regular kinetic solution exists.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. Companion of the stable-potential Boltzmann-Grad paper of the same date; it retains the Deng-Hani-Ma layered collision histories.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem of the manuscript was read against the further application as the paper reports it from Deng-Hani-Ma Section 1.4.3; the Deng-Hani-Ma text itself was not read. The theorem is a finite-dimensional central limit theorem in whole space, not convergence of the fluctuation process (no path-space tightness), which is why the entry is partial. The proof was not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":24,"significanceNote":"Long-time nonequilibrium fluctuations are a recognized next step in the rigorous kinetic theory program, but the question as posed is recent and the result covers finite-dimensional laws only. Placed below the potential-case entry of this family (30) and The Bogoliubov quantum-depletion law for the dilute Bose gas (35), and just above McKean Entropy-Production Conjecture (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Hard-sphere fluctuations on the regular Boltzmann lifespan","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The Boltzmann-Grad limit for stable radial potentials on regular kinetic intervals","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf","kind":"paper"},{"label":"Deng-Hani-Ma, Long time derivation of the Boltzmann equation (arXiv 2408.07818)","url":"https://arxiv.org/abs/2408.07818","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"free-entropy-dimension-invariance","name":"Voiculescu's question: is free entropy dimension an invariant of the generated von Neumann algebra?","shortName":"Invariance of free entropy dimension","problemNumber":null,"field":"Operator algebras: free probability, free entropy","fieldGroup":"Analysis","statement":"Voiculescu's free entropy dimension $\\delta(X_1,\\ldots,X_n)$ of a self-adjoint tuple in a tracial von Neumann algebra, together with its variants $\\delta_0$, $\\delta^*$ and $\\delta^\\star$, equals $n$ for a free semicircular $n$-tuple generating $L(\\mathbb F_n)$. If $\\delta$ depended only on the von Neumann algebra generated, it would distinguish the free group factors. Does the free entropy dimension of a finite self-adjoint generating tuple depend only on the tracial von Neumann algebra it generates?","posedBy":"Dan Voiculescu (Free entropy, Bull. London Math. Soc. 2002, Section 2.6)","yearPosed":2002,"ageNote":"Year from the survey the manuscript cites. The question is older, implicit in Voiculescu's free entropy papers from the 1990s, which the manuscript does not cite for it.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims a negative answer for $\\delta$, $\\delta_0$, $\\delta^*$ and $\\delta^\\star$: each takes every integer value $n\\ge2$ on finite self-adjoint generating tuples of the single factor $L(\\mathbb F_2)$, obtained by transporting semicircular and group-algebra tuples through the new isomorphisms. It does not compute free entropy dimension for other algebras or address the microstates-free equality questions.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. This entry is a corollary in the free group factor manuscript.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary (dependence on von Neumann generators) of the TeX source read against Voiculescu's question; for every $n\\ge2$ it gives generating tuples of $L(\\mathbb F_2)$ with $\\delta=\\delta_0=n$ and others with $\\delta^*=\\delta^\\star=n$, a negative answer for all four dimensions. It is a short transport argument that depends entirely on the unrefereed isomorphism theorem. The Lean comparator for the family does not state this corollary.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A central question in free probability, but tied to and answered as a consequence of the free group factor problem, so scored well below that entry (68). Placed just below Connes' rigidity conjecture at 45 and Kahn's flow conjecture at 40, above the Hyperkahler period-index conjecture at 22.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf","sourceName":"OpenAI math release: An isomorphism of the free group factors","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Voiculescu, Free entropy (Bull. London Math. Soc. 2002)","url":"https://doi.org/10.1112/S0024609301008992","kind":"problem-record"},{"label":"A second route via the generator problem (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ising-bulk-universality-weak-interactions","name":"Conformal universality of bulk spin and energy correlations for weakly interacting square-lattice Ising models","shortName":"Ising universality under weak interactions","problemNumber":null,"field":"Two-dimensional statistical mechanics; Ising model; conformal invariance","fieldGroup":"Probability & statistics","statement":"The critical nearest-neighbor Ising model has rigorous conformally covariant scaling limits of its spin and energy correlations (Chelkak-Hongler-Izyurov, Hongler-Smirnov). Universality predicts the same limits, up to two field normalizations, for models with weak finite-range even multispin perturbations at their own critical temperature. Rigorous results covered energy correlations in the plane and on cylinders (Giuliani-Greenblatt-Mastropietro, Antinucci-Giuliani-Greenblatt) and boundary spins in the half-plane, but not bulk spins. Giuliani's ICM 2022 survey (Section 2, Open Problems 1-2) asks for mixed spin and energy limits. For weak even finite-range perturbations, do the critical mixed bulk spin and energy correlations converge, after two normalizations, to the nearest-neighbor conformal limits?","posedBy":"Alessandro Giuliani, Scaling limits and universality of Ising and dimer models (ICM 2022), Section 2 and Open Problems 1-2","yearPosed":2022,"ageNote":"The universality expectation is decades older (conformal field theory of Belavin-Polyakov-Zamolodchikov, 1984); the citable statement as an open problem used by the manuscript is Giuliani's 2022 survey.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"For sufficiently small square-symmetric finite-range even multispin perturbations of either sign, with one critical-temperature branch and two field normalizations, mixed bulk spin and energy correlations converge to the nearest-neighbor conformal limits in every bounded simply connected $C^2$ Jordan domain with free, plus or minus boundary conditions, and in the periodic plane state; energies are centered in their actual states. Companion: weakly perturbed contour energies have a domain-independent temperature at which the Dobrushin interface converges to chordal $\\mathrm{SLE}_3$. Not shown: perturbations without square symmetry, multiply connected domains, boundary spin observables, or strong perturbations.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This entry draws on three manuscripts of September 23, 2026: the bulk correlation paper, a companion proving chordal SLE3 limits for weakly perturbed contour energies, and a technical companion on buffered comparison and stopping bands. The family's three October 5 random-bond manuscripts are a separate entry.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and main theorem statement of 'Conformal universality of bulk Ising correlations under weak interactions' were read against Giuliani's program as the manuscript summarizes it; the paper itself says it covers the square-symmetric model in smooth simply connected domains and the plane, while the broader program includes other topologies and boundary observables. Giuliani's survey was not read here. Lean challenge BufferedIsing (lean/docs/218.md) covers only a finite-graph likelihood comparison lemma from the companion 'Buffered comparison and stopping-band resolution in critical Ising'; its scope note says the stopping-band theorem is outside it. It does not state this entry's headline, so the entry is unreviewed, not Lean-checked.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A step in a recognized universality program of mathematical statistical mechanics, scored for the special case proved (square symmetry, simply connected domains, small perturbations). Well below the Rohde-Schramm conjecture (52) and Cardy's formula for square-lattice bond percolation (48), below the Gardner transition in the Ising pure p-spin glass (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Conformal universality of bulk Ising correlations under weak interactions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: SLE3 universality for weak finite-range Ising interactions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026/paper.pdf","kind":"paper"},{"label":"Companion: Buffered comparison and stopping-band resolution in critical Ising","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026/paper.pdf","kind":"paper"},{"label":"Giuliani, Scaling limits and universality of Ising and dimer models (ICM 2022)","url":"https://arxiv.org/abs/2110.09372","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"choiceless-polynomial-time-noncapture","name":"The Blass-Gurevich-Shelah conjecture: choiceless polynomial time with counting does not capture polynomial time","shortName":"CPT with counting does not capture PTIME","problemNumber":null,"field":"Finite model theory; descriptive complexity","fieldGroup":"Logic & foundations","statement":"Immerman and Vardi showed that fixed-point logic captures polynomial time on ordered finite structures; Chandra-Harel and Gurevich asked whether some logic captures polynomial time on unordered structures, and Gurevich conjectured not. Blass, Gurevich and Shelah introduced choiceless polynomial time (CPT), which computes with hereditarily finite sets over the input atoms under polynomial resource bounds without arbitrary choices, and its extension by a cardinality operation. CPT with counting was long the most prominent candidate logic for polynomial time: it defines the Cai-Furer-Immerman queries that defeat fixed-point logic with counting. Blass, Gurevich and Shelah conjectured in their 1997 preprint that the counting extension is still a proper fragment of polynomial time. Is there a polynomial-time, isomorphism-invariant query on finite structures that CPT with counting cannot define?","posedBy":"Andreas Blass, Yuri Gurevich and Saharon Shelah","yearPosed":1997,"ageNote":"The manuscript cites the 1997 preprint version of Blass, Gurevich and Shelah, Choiceless Polynomial Time (Ann. Pure Appl. Logic, 1999), Introduction, for the explicit conjecture; later accounts through Dawar-Gradel-Kullmann-Pago (2025) list it as open.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Main theorem: over a fixed vocabulary of eight binary relations, consistency over $\\mathbb F_3$ of an explicitly defined linear system (normalisation equations $\\sum_{a\\in X_t}\\lambda_a=1$ and consistency equations $\\mu_y=\\sum_{a\\in X_t}\\lambda_aC(y,a)$) is a polynomial-time query on all finite structures, with no promise, that no CPT program with counting defines, allowing hereditarily finite sets of unbounded rank. Priority: Shelah (2000) claimed noncapture for counting extensions of choiceless computation; later accounts kept the problem open, and the manuscript gives a separate proof not relying on that claim. Not shown: Gurevich's conjecture that no logic captures polynomial time, or anything about rank-based or other extensions.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. This entry's principal manuscript is dated September 23, 2026; a companion (September 24) proves a separate separation for witnessed symmetric choice (its own entry). Both have Lean formalizations of their main theorems.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem was read against the Blass-Gurevich-Shelah conjecture. Lean: formalization.yaml lists OAI.CPTSeparation.main (lean/OAI/ModelTheory/Choiceless/Separation.lean, comparator ChoicelessPolynomialTime.json). Its statement was read here: the linear-consistency query is isomorphism invariant, is computed by a polynomial-time finite-alphabet Turing machine on every ordered encoding, and is not definable in the formalised hereditarily-finite-set language with cardinality and polynomial bounds on stages and objects. This is the headline claim, but it rests on the challenge file's own encoding of CPT with counting (about 550 lines), whose fidelity to the Blass-Gurevich-Shelah definition was not audited here. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Whether CPT with counting captures polynomial time has been a central question of descriptive complexity since 1997, the leading candidate logic for PTIME; famous within finite model theory, less so outside it. Below the Sakoda-Sipser problem (45) and the question whether the Partition Principle implies Choice (45), above the Lane-Emden conjecture (40) and pointwise multiple ergodic averages for mixing maps (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Choiceless polynomial time with counting does not capture polynomial time","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI/ModelTheory/Choiceless/Separation.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/ModelTheory/Choiceless/Separation.lean","kind":"lean-proof"},{"label":"Lean statement: ChoicelessPolynomialTime.lean (comparator challenge)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ChoicelessPolynomialTime.lean","kind":"lean-proof"},{"label":"Blass, Gurevich and Shelah, Choiceless Polynomial Time (1999)","url":"https://doi.org/10.1016/S0168-0072(99)00005-6","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"thompson-group-f-amenability","name":"The amenability problem for Thompson's group F (Geoghegan's conjecture)","shortName":"Amenability of Thompson's group F","problemNumber":null,"field":"Geometric group theory; amenability","fieldGroup":"Algebra","statement":"Thompson's group $F$ is the group, under composition, of increasing piecewise linear homeomorphisms of $[0,1]$ with finitely many pieces, dyadic rational breakpoints and slopes in $2^{\\mathbb Z}$. A discrete group is amenable if its bounded real functions carry a positive normalized left-invariant mean. $F$ has no nonabelian free subgroup (Brin-Squier), so the free-subgroup obstruction does not apply, yet it is not elementary amenable. Geoghegan conjectured in 1979 that $F$ is not amenable. Is Thompson's group $F$ nonamenable?","posedBy":"Ross Geoghegan (1979), as recorded by J. W. Cannon, W. J. Floyd and W. R. Parry, Introductory notes on Richard Thompson's groups (Enseign. Math., 1996), p. 227","yearPosed":1979,"ageNote":"Complete claims in both directions have appeared and been withdrawn or disputed (Shavgulidze 2009 for amenability, with errors identified by Moore; Moore's own withdrawn claim; Akhmedov 2021 for nonamenability), as the manuscript recounts.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: Thompson's group $F$ is not amenable. The proof finds a fixed finite set $S\\subset F$ and $b>0$ with $|hA\\triangle A|\\ge b|A|$ for some $h\\in S$, for every nonempty finite $A\\subset F$, so no Folner sets exist. It converts approximate invariance of finite averages into an approximate fixed point of a Lipschitz self-map of a Hilbert ball, contradicting a Benyamini-Sternfeld displacement map that the paper constructs explicitly. It confirms Geoghegan's conjecture. The formal version gives no explicit expansion constant for a standard generating set, and the paper does not address other questions about $F$ such as its exact isoperimetric profile.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. This single-manuscript family is dated September 23, 2026 and comes with a Lean formalization in the release's lean/ library. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. The formalization catalogue lists comparator config ComparatorChallenges/ThompsonNonamenability.json with declaration OAI.ThompsonNonamenability.thompson_F_nonamenable_composition in OAI/GroupTheory/Thompson/Main.lean, permitted axioms propext, Quot.sound and Classical.choice. The comparator statement file was read: it defines F concretely as strictly increasing homeomorphisms of [0,1] with finitely many dyadic knots, each piece affine with slope an integer power of 2; it defines an invariant mean as a positive, normalized, left-invariant real-linear functional on bounded real functions on the group; and it states that there is a group structure on F whose product is composition and that no invariant mean exists. That is the headline claim. Not rebuilt here: the Lean build and axiom check were not run. Given the problem's history of withdrawn claims, an expert reading is still wanted even with a formal proof; the Lean statement was read for fidelity here, not audited independently.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"One of the best known open problems in geometric group theory, open since 1979, with several published and withdrawn claims in both directions. Placed just below Gromov and Weiss's question on sofic groups and the Kothe conjecture (both 60), above Connes' rigidity conjecture (45), and below the Jacobian Conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Thompson's group F is nonamenable","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: thompson_F_nonamenable_composition","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/Thompson/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: ThompsonNonamenability","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ThompsonNonamenability.lean","kind":"lean-proof"},{"label":"Lean scope note for family 248","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/248.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"boone-higman-conjecture","name":"The Boone-Higman conjecture: groups with solvable word problem embed in finitely presented simple groups","shortName":"Boone-Higman conjecture","problemNumber":null,"field":"Combinatorial group theory; word problem and embedding theorems","fieldGroup":"Algebra","statement":"Higman's embedding theorem (1961) characterizes finitely generated subgroups of finitely presented groups as the recursively presented groups. Boone and Higman (1974) proved that a finitely generated group has solvable word problem if and only if it embeds in a simple group that in turn embeds in a finitely presented group, and asked whether the simple overgroup itself can be chosen finitely presented. Thompson later obtained a finitely generated simple overgroup, and the conjecture was proved for hyperbolic groups, Baumslag-Solitar and free-by-cyclic groups, $\\mathrm{Out}(F_n)$ and mapping class groups, among others. Does every finitely generated group with solvable word problem embed in a finitely presented simple group?","posedBy":"William W. Boone and Graham Higman (J. Austral. Math. Soc. 18, 1974, p. 43)","yearPosed":1974,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group. The route is Theorem 1.2: every such group embeds in a finitely presented group with a faithful two-transitive action on a countable set with finitely generated point stabilizers, built as a mapping torus of an affine Steinberg group over finitely presented $\\mathbb F_2$-algebras, which Zaremsky's criterion for twisted Brin-Thompson groups turns into a finitely presented simple overgroup. The higher-finiteness strengthening is the companion's separate entry; no explicit bound on the size of the presentation is given.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has three manuscripts dated September 23, 2026: this one, a companion proving the stronger type $F_\\infty$ version, and a companion constructing a universal group of type $F_\\infty$.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the Boone-Higman question; it states the equivalence of decidable word problem and embedding in a finitely presented simple group for every finitely generated group, with no finite-presentation assumption on the input. Lean: lean/ComparatorChallenges/BooneHigman.json exists with solution_module OAI.GroupTheory.BooneHigman.Main, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement BooneHigman.lean was read here: for a finitely generated group, having some finite generating family whose word-equals-identity predicate is ComputablePred is equivalent to an injective homomorphism into a finitely presented group satisfying IsSimpleGroup, in the same universe. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The Boone-Higman conjecture has been a central open problem of combinatorial group theory since 1974, with a recent survey devoted to partial progress. Placed level with the Kadison-Kaplansky conjecture (50), above the word problem for finite-rank Artin groups (45), below the amenability problem for Thompson's group F (58) and Cannon's Conjecture (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Finite algebraic envelopes and the Boone-Higman conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.FiniteAlgebraicEnvelopes.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/BooneHigman/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: BooneHigman.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BooneHigman.lean","kind":"lean-proof"},{"label":"Companion: Simple F-infinity overgroups of groups with decidable word problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf","kind":"paper"},{"label":"Companion: A universal group of type F-infinity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"nagata-conjecture-plane-curves","name":"Nagata's conjecture on plane curves through very general points","shortName":"Nagata's conjecture","problemNumber":null,"field":"Algebraic geometry; linear systems of plane curves","fieldGroup":"Geometry & topology","statement":"Let $p_1,\\dots,p_r$ be very general points of the complex projective plane. Nagata, in his work on Hilbert's fourteenth problem, conjectured that for $r\\ge 10$ every nonzero effective plane curve $C$ of degree $d$ with $\\mathrm{mult}_{p_i}C\\ge m_i$ satisfies $\\sum_{i=1}^r m_i < d\\sqrt r$. He proved it when $r$ is a perfect square at least $16$. The bound fails at $r=9$ (a cubic through nine points), and partial results (Roé, Harbourne, Ciliberto-Miranda degenerations, the $117/37$ bound at ten points) left every nonsquare $r\\ge10$ open. Equivalently, the multipoint Seshadri constant of a line at $r\\ge10$ very general points is $1/\\sqrt r$. Does the strict inequality $\\sum_i m_i 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (September 23, 2026) is self-contained; the Seshadri-constant companions in the same family cite it.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Nagata's conjecture; it gives the strict, nonhomogeneous form for every $r\\ge10$, with one countable union of proper closed exceptional sets chosen simultaneously for all degrees and multiplicity vectors. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator Nagata, declaration OAI.Nagata.nagata_conjecture). The comparator statement ComparatorChallenges/Nagata.lean was read here: for every $r\\ge10$ it asserts a countable family of proper Zariski-closed subsets of ordered distinct configurations in $\\mathbb P^2(\\mathbb C)$, with a configuration outside all of them, such that outside them every nonzero effective plane curve (a cycle of homogeneous prime ideals, repeated components allowed) satisfies $\\sum m_i 11/12 zero-free region whose write-up was human-edited). All three manuscripts are authored 'OpenAI' and name no human author. The matrix-potential paper and the equality-case paper (both dated 2026-10-05) build on the scalar paper's action and continuation framework.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the scalar paper were read against the conjecture. formalization.yaml lists ComparatorChallenges/LiebThirring.json with declaration OAI.SharpLiebThirring.sharp_lieb_thirring (file OAI/Analysis/LiebThirring/Main.lean, present at the pinned commit). The statement was read here: for every 1/2 < gamma < 3/2 and every a.e. nonnegative W in L^(gamma+1/2), the supremum over finite orthonormal families of weak negative-energy eigenfunctions of the sum of k_i^(2 gamma) is at most the explicit constant times the potential integral; the optimal constant and the one-bound-state constant both equal that value; and the sech^2 potential attains it. This is the headline claim. Not rebuilt here. The matrix-potential and equality-case companions are not formalized.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"The one-dimensional Lieb-Thirring conjecture has been open since 1976 and is a standard named problem in mathematical physics and spectral theory, though less famous than the three-dimensional exponent-one case tied to stability of matter, and its exponent-one case had just been settled by Read and Schulz. Placed above Solovej's generalized ionization conjecture (32), below The Sandier-Serfaty conjecture (42) and well below The ionization conjecture, Simon's Problem 9 (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Sharp one-dimensional Lieb-Thirring constants","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Sharp one-dimensional Lieb-Thirring inequalities for matrix potentials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf","kind":"paper"},{"label":"Companion: Equality cases in the sharp one-dimensional matrix Lieb-Thirring inequality","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf","kind":"paper"},{"label":"Lean proof (OAI.SharpLiebThirring.sharp_lieb_thirring)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/LiebThirring/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"nearby-lagrangian-conjecture","name":"Arnold's nearby Lagrangian conjecture","shortName":"Nearby Lagrangian conjecture","problemNumber":null,"field":"Symplectic topology; exact Lagrangians in cotangent bundles","fieldGroup":"Geometry & topology","statement":"Let $Q$ be a closed connected smooth manifold and $T^*Q$ its cotangent bundle with the canonical symplectic form. A closed embedded Lagrangian $L\\subset T^*Q$ is exact if the canonical one-form restricts to an exact form on $L$. Known constraints are strong: Abouzaid (2012) and Kragh (2013) proved the projection $L\\to Q$ is a homotopy equivalence, Abouzaid and Kragh (2018) a simple homotopy equivalence, and the conjecture holds for $T^*S^2$ (Hind) and $T^*T^2$ (Dimitroglou Rizell, Goodman, Ivrii). Is every closed exact embedded Lagrangian in $T^*Q$ Hamiltonian isotopic to the zero section?","posedBy":"V. I. Arnold (historical setting in Arnold, First steps in symplectic topology, 1986); modern formulation as in Abouzaid (2012) and Kragh (2013)","yearPosed":1986,"ageNote":"The manuscript attributes the conjecture to Arnold and cites his 1986 survey for the broader historical setting, while taking the precise formulation from Abouzaid (2012) and Kragh (2013). It does not date the first statement more exactly; 1986 is the earliest date the manuscript's own references support.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for some sufficiently large even $N$, $Q=S^9\\times S^{N-1}$ admits a closed exact smoothly embedded Lagrangian $L\\subset T^*Q$, diffeomorphic to $Q$ and with $Q$ and $L$ simply connected, that no compactly supported Hamiltonian isotopy carries to the zero section. $L$ is cut out by a generating family whose data at infinity form a nontrivial class $a\\in\\pi_9$ of the stable smooth tube space with trivial associated spherical fibration; a Morse and h-cobordism argument shows such data cannot come from a family with one nondegenerate critical point per fibre, as an isotopy to the zero section would force. Not shown: an explicit $N$, counterexamples in low dimension, or any failure of the homotopy-theoretic consequences already proved.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, using on average about three hours of ChatGPT Pro thinking compute per result, with outputs grouped into families and manuscripts. The README's two exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region write-up, which was human-edited for readability) do not concern this family. The manuscript is credited to OpenAI alone, names no human author and has no acknowledgements. This family has no Lean formalization (there is no lean/docs/340.md at the pinned commit). The release does not say how much human review happened before publication.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract and Theorem 1.1 of the TeX source against the conjecture as the manuscript states it (every closed exact embedded Lagrangian in T*Q is Hamiltonian isotopic to the zero section). The theorem claims the negation for one base, so it addresses the conjecture as posed for all closed Q. No formalization exists for this family and the argument was not checked. Scope the paper itself states: N is some sufficiently large even integer, not given explicitly, so the counterexample lives in very high dimension; it says nothing about low-dimensional bases or specific bases such as spheres or tori. The proof leans on Waldhausen's tube fibration and Rognes's computation of low-degree smooth Whitehead groups to produce a nonzero class in the ninth homotopy group of the stable tube space with trivial image in BG. Given that the field broadly expected the conjecture to hold, an expert reading is especially wanted.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"The central open conjecture of symplectic topology about cotangent bundles, open for four decades and the target of the Abouzaid-Kragh programme; a disproof was not expected. Tied with Cannon's conjecture and Kaplansky's zero-divisor conjecture at 62, below the Hilbert-Smith conjecture at 65, above the Griffiths conjecture entry at 55, which is famous within a smaller community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to the nearby Lagrangian conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Abouzaid, Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (Invent. Math. 2012)","url":"https://doi.org/10.1007/s00222-011-0365-0","kind":"problem-record"},{"label":"Kragh, Parametrized ring-spectra and the nearby Lagrangian conjecture (Geom. Topol. 2013)","url":"https://doi.org/10.2140/gt.2013.17.639","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stw-problem-xxi-uniform-property-gamma","name":"Schafhauser-Tikuisis-White Problem XXI: tracial real rank zero implies uniform property Gamma","shortName":"STW Problem XXI (uniform property Gamma)","problemNumber":null,"field":"Operator algebras; uniform tracial completions","fieldGroup":"Analysis","statement":"Let $A$ be a unital separable simple infinite-dimensional nuclear stably finite C*-algebra with traces, $B$ its uniform tracial completion, and $B^\\omega$ the uniform tracial ultrapower along a free ultrafilter. Uniform property $\\Gamma$ asks for central projections that halve every limit trace on every coefficient. Schafhauser, Tikuisis and White asked (Problem XXI, in their completion-and-halving formulation) whether real rank zero of $B^\\omega$ forces uniform property $\\Gamma$. Does real rank zero of the uniform tracial ultrapower imply uniform property $\\Gamma$?","posedBy":"Christopher Schafhauser, Aaron Tikuisis and Stuart White","yearPosed":2025,"ageNote":"Schafhauser, Tikuisis and White, Nuclear C*-algebras: 99 problems (arXiv 2506.10902, 2025), Section 6, Problem XXI, Definition 16.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $A$ unital, separable, simple, infinite-dimensional, nuclear and stably finite with $T(A)\\neq\\emptyset$, if $B^\\omega$ has real rank zero then there is a projection $p\\in B^\\omega\\cap B'$ with $\\lambda(px)=\\lambda(x)/2$ for all $x\\in B$ and all limit traces $\\lambda$, so $A$ has uniform property $\\Gamma$; it also identifies all traces on $B$. It does not show that real rank zero of $B^\\omega$ holds in general, and Toms (2026) has shown nuclearity alone does not give uniform property $\\Gamma$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry's principal is 'Tracial projection methods and uniform property Gamma' (September 23, 2026), one of four manuscripts in the family; the same manuscript also gives the unital Toms-Winter route recorded in the Toms-Winter entry.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Problem XXI and answers it positively in the STW formulation. lean/formalization.yaml lists comparator UniformGamma, declaration OAI.ComparatorModel.CurrentMain.main. The statement was read here: for every separable, topologically simple, infinite-dimensional, nuclear (unique C*-norm on tensor products), stably finite unital C*-algebra with a tracial state, and every free ultrafilter at which the family ultrapower of the uniform tracial completion has real rank zero, there is a central projection halving every limit trace on every element of the completion. The claim is wrapped in existential quantifiers over five auxiliary construction classes (norms on quotients, null ideals, Cauchy algebras, limit traces), which the proof must supply. It states the headline of Problem XXI; the strict-comparison results of the same paper are not in this statement. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. Listed as Unreviewed rather than Lean-checked because its formal statement takes the key constructions as supplied hypotheses.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A numbered problem from a 2025 problem list, technical and recent. Placed level with Unit-Area Triangles in Planar Sets of Large Measure (15) and Improved Approximation Ratios for Multiway Cut (15), above a typical Erdos problem (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Tracial projection methods and uniform property Gamma","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.ComparatorModel.CurrentMain.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/TracialSplitting/Challenge.lean","kind":"lean-proof"},{"label":"Comparator statement: UniformGamma.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/UniformGamma.lean","kind":"lean-proof"},{"label":"Schafhauser, Tikuisis and White, Nuclear C*-algebras: 99 problems","url":"https://arxiv.org/abs/2506.10902v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kobayashi-canonical-ampleness-conjecture","name":"Kobayashi's conjecture: a compact Kahler manifold with no entire curves has ample canonical bundle","shortName":"Kobayashi canonical-ampleness conjecture","problemNumber":null,"field":"Complex geometry; Kobayashi hyperbolicity","fieldGroup":"Geometry & topology","statement":"A compact complex manifold $X$ is Kobayashi hyperbolic exactly when every holomorphic map $\\mathbb C\\to X$ is constant (Brody). Kobayashi conjectured that hyperbolicity forces positivity of the canonical bundle: a compact Kahler hyperbolic manifold should have ample $K_X$, and hence be projective. Known results needed metric hypotheses: negative or quasi-negative holomorphic sectional curvature (Wu-Yau, Tosatti-Yang, Diverio-Trapani), or Kahler hyperbolicity in Gromov's sense (Chen-Yang); without rational curves $K_X$ is known to be nef (Ou, Cao-Horing), but bigness was open. Is $K_X$ ample for every compact Kahler manifold of positive dimension that contains no nonconstant entire curve?","posedBy":"Shoshichi Kobayashi","yearPosed":1970,"ageNote":"Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (Marcel Dekker, 1970), cited by the manuscript as the source of the theory; Diverio's 2020 survey discusses the conjecture.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $X$ is a compact connected Kahler manifold of positive dimension and every holomorphic map $\\mathbb C\\to X$ is constant, then $K_X$ is ample and $X$ is projective. Corollaries: $K_X^{\\otimes m}$ is globally generated for $m\\ge n+2$ (using the release's Fujita freeness result), and a finite cover splits as a ball-quotient-type factor times a hyperbolic factor when the universal cover is semialgebraic. It does not cover singular spaces, non-Kahler compact complex manifolds, or the converse direction (Green-Griffiths-Lang type statements).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family is a single manuscript (September 23, 2026). Its two corollaries cite other results from the same release (Fujita freeness and a semialgebraic-cover classification); the main theorem does not use them.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Kobayashi's conjecture. It states that every compact connected Kahler manifold of positive dimension in which every holomorphic map from $\\mathbb C$ is constant has ample $K_X$, so a power of $K_X$ embeds $X$ projectively. That is the conjecture in the smooth compact Kahler category. The proof (extremal holomorphic discs with an area penalty, a boundary-normalised frame, a Hessian identity giving a volume lower bound, then Aubin-Yau and Morse inequalities) was not refereed. There is no Lean formalization. The singular and non-Kahler settings are outside its scope, as the paper says.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Kobayashi's conjecture is one of the standard open problems of complex hyperbolicity, open since 1970 with progress only under curvature hypotheses. Placed below Fujita's freeness conjecture (60) and the Iitaka subadditivity conjecture (62), and above the Campana-Peternell conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026/canonical-ampleness.pdf","sourceName":"OpenAI math release: Canonical ampleness of compact hyperbolic Kahler manifolds","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"robert-tikuisis-conjecture-c1","name":"Robert-Tikuisis Conjecture (C1): finite nuclear dimension implies Jiang-Su stability without elementary subquotients","shortName":"Robert-Tikuisis Conjecture (C1)","problemNumber":null,"field":"Operator algebras; nuclear dimension of nonsimple C*-algebras","fieldGroup":"Analysis","statement":"For simple algebras, finite nuclear dimension and Jiang-Su stability are equivalent away from the elementary case. Robert and Tikuisis studied the nonsimple setting, where the natural exclusion is that no ideal subquotient $I/J$ is a nonzero elementary algebra $\\mathcal K(H)$ (nowhere scattered algebras). Their Conjecture (C1) predicts that a separable nuclear C*-algebra of finite nuclear dimension with no nonzero elementary ideal subquotients absorbs the Jiang-Su algebra $\\mathcal Z$. Does finite nuclear dimension imply $A\\cong A\\otimes\\mathcal Z$ for every separable nuclear C*-algebra with no elementary subquotients?","posedBy":"Leonel Robert and Aaron Tikuisis","yearPosed":2017,"ageNote":"Robert and Tikuisis, Nuclear dimension and Z-stability of non-simple C*-algebras, Trans. Amer. Math. Soc. 369 (2017) (arXiv 2013), cited by the manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for separable nuclear $A$ with no nonzero elementary ideal subquotients, $\\dim_{nuc}A<\\infty$, $A\\cong A\\otimes\\mathcal Z$ and $\\dim_{nuc}A\\le1$ are equivalent. Theorem 1.3: $\\dim_{nuc}(A_0\\otimes\\mathcal Z)\\le1$ for every separable nuclear $A_0$. Theorem 1.4 answers Kirchberg-Rordam's Question 6.2 (full orthogonal positive elements in a finite maximal tensor power of a unital algebra without characters). It does not address the strict-comparison part of the nonsimple problem, which the Cuntz-comparison companion treats.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry's principal is 'Nuclear dimension and Jiang-Su stability without elementary subquotients' (September 23, 2026), one of four manuscripts in the family.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Conjecture (C1). For separable nuclear algebras with no nonzero elementary ideal subquotients it proves finite nuclear dimension, $\\mathcal Z$-stability and nuclear dimension at most 1 equivalent; the forward implication is (C1). The proof was not refereed and is not formalised.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A named conjecture within the operator-algebra regularity community, extending the simple theory to nonsimple algebras. Placed level with Non-MF groups and non-finite full group C*-algebras (25) and the Burger-Ozawa-Thom question on strong Ulam stability (25), below the Matrix Spencer Conjecture for C*-Algebra Contractions (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Nuclear dimension and Jiang-Su stability without elementary subquotients","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Cuntz comparison and Jiang-Su absorption","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026/paper.pdf","kind":"paper"},{"label":"Robert and Tikuisis, Nuclear dimension and Z-stability of non-simple C*-algebras","url":"https://arxiv.org/abs/1308.2941v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"rwre-directional-zero-one-law","name":"The directional zero-one law for random walks in iid random environments in dimension at least three","shortName":"Directional zero-one law for RWRE","problemNumber":null,"field":"Random walks in random environment","fieldGroup":"Probability & statistics","statement":"Consider nearest-neighbour random walk on $\\mathbb Z^d$ in an environment of iid transition rows, and for a direction $\\ell\\ne0$ let $A_\\ell=\\{X_n\\cdot\\ell\\to+\\infty\\}$. Kalikow (1981) proved $P_0(A_\\ell\\cup A_{-\\ell})\\in\\{0,1\\}$ for the annealed law, and the question whether $P_0(A_\\ell)$ itself must be 0 or 1 goes back to him. Zerner and Merkl (2001) proved it in dimension two for iid strictly elliptic environments; Bramson, Zeitouni and Zerner (2006) showed it fails for some stationary mixing environments in $d\\ge3$ and posed the iid case as Open Problem 3. Is the annealed escape probability $P_0(A_\\ell)$ always 0 or 1 for iid elliptic environments in every dimension $d\\ge3$?","posedBy":"Steven A. Kalikow; the iid higher-dimensional case stated as Open Problem 3 by Maury Bramson, Ofer Zeitouni and Martin Zerner","yearPosed":1981,"ageNote":"Kalikow, Generalized random walk in a random environment, Ann. Probab. 9 (1981); Bramson-Zeitouni-Zerner (Ann. Probab. 34, 2006), Open Problem 3, for iid uniformly elliptic environments in d >= 3.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $d\\ge3$, iid nearest-neighbour rows that are almost surely strictly positive (no uniform lower bound, no moment condition) and any fixed real $\\ell\\ne0$, $P_0(A_\\ell)\\in\\{0,1\\}$. The October companion extends the law to stationary ergodic finite-range-dependent environments under uniform ellipticity, and the family's ballisticity paper gives, for uniformly elliptic iid environments in $d\\ge3$, the hemisphere description of the transient directions. Not shown: ballisticity or a velocity under strict ellipticity alone, jumps beyond nearest neighbours, or general mixing environments (where the law fails by Bramson-Zeitouni-Zerner).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has three manuscripts (September 23 and October 5, 2026).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the principal manuscript were read against the question as cited. The proof (regeneration in a real direction, finite spatial radii under coexistence, and an entropy bound on contacts between regeneration arrangements) was not refereed. The challenge lean/ComparatorChallenges/DirectionalWalk.json (theorem OAI.DirectionalZeroOne.directional_zero_one, solution module OAI.Probability.DirectionalWalk.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml; it is found through lean/docs/220.md. Its statement was read here: for $d\\ge3$, any probability law on nearest-neighbour transition rows whose entries are almost surely all positive, the iid product environment, and any real $\\ell\\ne0$, the annealed probability from the origin that $X_n\\cdot\\ell\\to\\infty$ is 0 or 1. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"A classical question of random walks in random environment, open for about forty-five years in dimensions three and up, well known in that community but not outside it. Placed below the Ballisticity entry of the same family (42) and The Ibragimov-Iosifescu conjecture (40), near Feige's Conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A directional zero-one law under strict ellipticity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A directional zero-one law for finite-range-dependent random environments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf","kind":"paper"},{"label":"Lean proof module (OAI.DirectionalZeroOne.directional_zero_one)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/DirectionalWalk/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: DirectionalWalk.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DirectionalWalk.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strong-cosmic-censorship-near-kerr-l2","name":"Strong cosmic censorship near Kerr in Christodoulou's square-integrable-connection formulation","shortName":"Strong cosmic censorship near Kerr (L2 connection)","problemNumber":null,"field":"General relativity; Einstein vacuum equations; Cauchy horizons","fieldGroup":"Mathematical physics","statement":"Penrose's strong cosmic censorship conjecture asserts that for generic asymptotically flat vacuum initial data the maximal globally hyperbolic development is inextendible, so that general relativity is deterministic. Kerr's smooth inner Cauchy horizon makes the regularity of extensions essential: Dafermos and Luk proved that near Kerr the development does extend continuously ($C^0$) across part of the Cauchy horizon, so the $C^0$ formulation fails. Christodoulou proposed the formulation at the threshold of locally square-integrable connection coefficients. For generic smooth vacuum data near the two-ended data of a rotating subextremal Kerr black hole ($0<|a|0$, $0<|\\mathfrak a|0$ such that, among smooth complete two-ended vacuum data $d$ with tenth weighted seminorm $p_{10}(d-d_*)<\\varepsilon$ from the Kerr bridge data, those whose full maximal globally hyperbolic development has a future extension with continuous nondegenerate metric in $W^{1,2}_{\\mathrm{loc}}$ form a meagre set in the weighted smooth topology. No symmetry is imposed and extensions need not be vacuum. Companions prove the weaker $C^2$ and $C^1$ versions. This is strong cosmic censorship in Christodoulou's formulation near Kerr only; it is not the global conjecture, and it does not treat one-ended or nonrotating data.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has three manuscripts dated the same day; the principal one relies on intermediate results of the C1 and C2 companions but not on their headline theorems. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Definition 1.1 and Theorem 1.2 of the principal manuscript were read against strong cosmic censorship as Penrose and Christodoulou formulate it, as the manuscript cites them; the companions' abstracts and introductions were skimmed. The proofs were not refereed. No Lean formalization: none of the three manuscripts is in lean/formalization.yaml and lean/docs/264.md does not exist at the pinned commit. Scope the paper itself states: genericity is local, inside a neighbourhood of each fixed Kerr bridge whose size depends on M and a; no extremal or nonrotating case, no uniformity as a tends to 0 or M, and no genericity outside these neighbourhoods; inextendibility is not asserted to be open. Only two-ended data are treated, not one-ended collapse data.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Strong cosmic censorship is one of the central conjectures of mathematical general relativity; the near-Kerr case in the square-integrable formulation is the target the Dafermos-Luk programme left open after showing the continuous version fails. Scored as a local partial below the full conjecture. Above relativistic Vlasov-Maxwell global regularity (55) and Simon's Problem 2 on 2D Anderson localization (55), below Simon's Problem 1 on extended states (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf","kind":"paper"},{"label":"Companion: Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf","kind":"paper"},{"label":"Christodoulou 2009, The Formation of Black Holes in General Relativity","url":"https://doi.org/10.4171/068","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"b-conjecture-even-log-concave-measures","name":"The (B)-conjecture for even log-concave measures: log-concavity of t -> mu(e^t K)","shortName":"(B)-conjecture for log-concave measures","problemNumber":null,"field":"Convex geometry; log-concave measures","fieldGroup":"Geometry & topology","statement":"Let $\\mu$ be an even log-concave measure on $\\mathbb R^n$ and $K$ an origin-symmetric convex body. The (B)-conjecture asks whether $t\\mapsto\\mu(e^tK)$ is log-concave on $\\mathbb R$. Cordero-Erausquin, Fradelizi and Maurey (2004) proved it for the Gaussian measure, and the question for general even log-concave measures remained open; Saroglou (2016) showed that the logarithmic Brunn-Minkowski inequality in a given dimension implies it for every even log-concave density in that dimension. Is $t\\mapsto\\mu(e^tK)$ log-concave for every even log-concave measure $\\mu$ and every origin-symmetric convex body $K$?","posedBy":"Not named in the manuscript; it cites Cordero-Erausquin, Fradelizi and Maurey (2004) for the conjecture's Gaussian case","yearPosed":null,"ageNote":"The manuscript cites Cordero-Erausquin, Fradelizi and Maurey, J. Funct. Anal. 214 (2004), whose title already names the (B)-conjecture, and Saroglou (Mathematika 2016) for the transfer; it does not say who first posed it or when. The wider literature attributes the Gaussian question to Banaszczyk.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 8.1: for every $n\\ge1$, every even log-concave Radon measure $\\mu$ on $\\mathbb R^n$ (finite or infinite total mass, possibly supported on a subspace) and origin-symmetric convex bodies $K,L$, the log-Brunn-Minkowski inequality holds for $\\mu$; taking $L$ a dilate of $K$ gives log-concavity of $t\\mapsto\\mu(e^tK)$. This is the scalar-dilation form; the stronger diagonal (B)-property for general linear dilations is not claimed except where Saroglou's equivalence for cubes applies.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry is Corollary 8.1 of the log-Brunn-Minkowski manuscript (September 23, 2026), obtained by combining its main theorem with Saroglou's published transfer theorem and a short support-subspace reduction written by the model.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary 8.1 was read; it gives $\\mu(W[h_K^{1-\\lambda}h_L^\\lambda])\\ge\\mu(K)^{1-\\lambda}\\mu(L)^\\lambda$ for every even log-concave Radon measure, hence the scalar-dilation (B)-property. The deduction cites Saroglou's transfer theorem (Mathematika 2016, Theorem 3.1) and adds a reduction for measures supported on proper subspaces. The Lean formalization (comparator LogBrunnMinkowski, OAI.LogBrunnMinkowski.main, listed in lean/formalization.yaml) states only the Lebesgue volume inequality, read here; the transfer to measures and the (B)-conjecture are not formalized. Not rebuilt here. The proof was not refereed. Listed as Unreviewed rather than Lean-checked because its formal statement covers only the Lebesgue inequality, and the step to the (B)-conjecture uses Saroglou's published transfer.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The (B)-conjecture for general even log-concave measures is a well-known question within convex geometry and asymptotic geometric analysis, but since 2016 it was known to follow from the log-Brunn-Minkowski conjecture. Placed below its parent, the log-Brunn-Minkowski conjecture (50), level with the Hadamard matrix of order 668 (30) and above Courtade's conjecture (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The logarithmic Brunn-Minkowski conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof of the volume inequality (OAI.LogBrunnMinkowski.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/LogVolume/BrunnMinkowski.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kaplansky-direct-finiteness-conjecture","name":"Kaplansky's direct finiteness conjecture","shortName":"Kaplansky direct finiteness","problemNumber":null,"field":"Group rings; direct and stable finiteness","fieldGroup":"Algebra","statement":"A unital ring is directly finite if $ab=1$ implies $ba=1$. For a field $K$ and a group $G$, the group algebra $K[G]$ consists of finite formal sums $\\sum c_g g$. In characteristic zero $K[G]$ is directly finite (and stably finite) for every group, by Kaplansky's trace argument. In positive characteristic it was known for free-by-amenable groups (Ara-O'Meara-Perera) and for all sofic groups (Elek-Szabo). Kaplansky asked whether the characteristic-zero theorem persists in positive characteristic. For every field $K$, every group $G$ and all $a,b\\in K[G]$, does $ab=1$ imply $ba=1$?","posedBy":"Irving Kaplansky, Fields and Rings (Chicago Lectures in Mathematics), Part II, Section 3, Problem 2, as the manuscripts cite","yearPosed":1969,"ageNote":"The manuscripts cite the second edition (1972) of Fields and Rings, pp. 122-123; the problem list appears in the first edition (1969). Curator may prefer 1972, the edition the manuscripts actually cite.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 (September 23): a finite field $K$ of characteristic two, a finitely presented group $G$ containing odd-order torsion, and $a,b\\in K[G]$ with $ab=1$, $ba\\ne1$. By Elek-Szabo, $G$ is not sofic. Companions: for a specified odd prime $p$ (the least prime factor of $(\\binom{1200}{600}!)^2+1$), a field of order $p^4$ and a finitely generated group with torsion (September 26); and a finitely presented torsion-free group with a finite two-dimensional classifying complex and $a,b,c\\in\\mathbb F_2[G]$ with $ab=1$, $ac=0$, $c\\ne0$ (October 4). No counterexample in characteristic zero is possible, and none is claimed.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. A reasoning summary for this result (family 197, characteristic two) is published with the release. Companions give an odd-characteristic counterexample (September 26) and a torsion-free counterexample over $\\mathbb F_2$ (October 4).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the September 23 manuscript was read against Kaplansky's question as the companions cite it; it claims a finite field $K$ of characteristic two, a finitely presented group $G$ with an element of odd prime order, and $a,b\\in K[G]$ with $ab=1\\ne ba$, specified by a terminating prescription that has not been executed. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator KaplanskyFinitelyPresented, declaration OAI.KaplanskyCounterexample.finitelyPresented_counterexample); ComparatorChallenges/KaplanskyFinitelyPresented.lean was read here and asserts exactly that: a finite field of characteristic 2, a finitely presented group with an element of odd prime order, and $a,b$ in the monoid algebra with $ab=1$, $ba\\ne1$. This states the headline claim. A second challenge, KaplanskyDirectFiniteness (finitely generated group version), is not in the catalogue but its solution module exists. Not rebuilt here. The nonsoficity of $G$ is outside the Lean statements. The torsion-free companion (October 4) is not formalized.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"One of Kaplansky's group-ring problems, open for over fifty years and closely tied to soficity and surjunctivity; less famous than the zero-divisor and unit conjectures. Placed below Kaplansky's zero-divisor conjecture (62) and Gromov and Weiss's sofic groups question (60), level with Sidorenko's conjecture (52), and above the Kadison-Kaplansky projection conjecture (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.KaplanskyCounterexample.finitelyPresented_counterexample)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RingTheory/DirectFiniteness/FinitelyPresented.lean","kind":"lean-proof"},{"label":"Comparator statement: KaplanskyFinitelyPresented.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KaplanskyFinitelyPresented.lean","kind":"lean-proof"},{"label":"Comparator statement: KaplanskyDirectFiniteness.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KaplanskyDirectFiniteness.lean","kind":"lean-proof"},{"label":"Companion: A Torsion-Free Group Algebra That Is Not Directly Finite","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf","kind":"paper"},{"label":"Companion: Counterexample in Odd Characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf","kind":"paper"},{"label":"Lean proof, odd characteristic (OAI.OddKaplansky.main_theorem)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/OddKaplansky/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cotype-cotype-conjecture","name":"The cotype-cotype conjecture: finite cotype of a Banach space and its dual implies K-convexity (approximation property case)","shortName":"Cotype-cotype conjecture","problemNumber":null,"field":"Banach space geometry; Rademacher type and cotype, K-convexity","fieldGroup":"Analysis","statement":"A Banach space $X$ is $K$-convex if the Rademacher projections on $L_2(\\{-1,1\\}^s;X)$ are uniformly bounded; by Maurey-Pisier and Pisier this is equivalent to nontrivial Rademacher type. If $X$ is $K$-convex then $X$ and $X^*$ have finite cotype. Pisier (1980-81) proved the converse when $X$ has the bounded approximation property and the cotype exponents $q,r$ of $X,X^*$ satisfy a restriction, and asked whether the restriction can be removed (Remark 2(iii)). Without any approximation hypothesis the converse fails (Pisier 1983). Szarek and Tomczak-Jaegermann (2009) call this the cotype-cotype conjecture, and Gupta, Misra and Ray (2025, Conjecture 1.5) state the BAP form. If $X$ has the (bounded) approximation property and both $X$ and $X^*$ have finite cotype, with arbitrary exponents, must $X$ be $K$-convex?","posedBy":"G. Pisier, Seminaire d'analyse fonctionnelle 1980-81, expose VII, Remark 2(iii); named the cotype-cotype conjecture by Szarek and Tomczak-Jaegermann (2009)","yearPosed":1981,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a nonzero real Banach space $X$ with the approximation property is $K$-convex if and only if $C_q(X)<\\infty$ and $C_r(X^*)<\\infty$ for some $q,r\\in[2,\\infty)$, possibly different; equivalently such $X$ has nontrivial Rademacher type. Corollary 1.2 makes the bound on $K(X)$ uniform in the cotype data and gives dimension-free metric-entropy duality for the corresponding convex bodies; a further consequence settles a Gupta-Misra-Ray tensor-norm conjecture (their Conjecture 1.4) under AP. Real scalars only; constants are existential. It does not address spaces failing AP, where the statement is false.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Pisier's question and the BAP formulation of Gupta-Misra-Ray. The result assumes only the ordinary approximation property, which is weaker than BAP, so the BAP conjecture is covered; the conjecture without an approximation hypothesis is false (Pisier 1983), as the paper says. The family is in lean/formalization.yaml (ComparatorChallenges/Cotype.json, declaration OAI.Cotype.mainTarget_proved, file OAI/Analysis/Cotype/Main.lean). The comparator statement was read here: for every nontrivial real Banach space with the approximation property, K-convexity (uniform bound on finite-cube Rademacher projections) is equivalent to finite cotype of $X$ and of its continuous dual. This is the headline claim. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":26,"significanceNote":"A forty-five-year-old question of Pisier in the local theory of Banach spaces, known to specialists in type, cotype and K-convexity but not far outside them. Below Talagrand's operator cotype problem at 32 and Kalton's Problem 2 at 28, and below the Pietsch metric-entropy duality conjecture at 36, of which the paper gives a special case.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The cotype-cotype conjecture under the approximation property","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/Analysis/Cotype/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Cotype/Main.lean","kind":"lean-proof"},{"label":"Pisier, Semi-groupes holomorphes et K-convexite (suite), 1980-81","url":"https://www.numdam.org/item/SAF_1980-1981____A7_0/","kind":"problem-record"},{"label":"Gupta, Misra and Ray, Conjecture 1.5 (arXiv 2305.13270)","url":"https://arxiv.org/abs/2305.13270","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"donaldson-hypersymplectic-deformation-conjecture","name":"Donaldson's hypersymplectic deformation conjecture: deforming hypersymplectic four-manifolds to hyperkahler triples in fixed cohomology","shortName":"Donaldson hypersymplectic conjecture","problemNumber":null,"field":"Differential geometry: hypersymplectic and hyperkahler four-manifolds","fieldGroup":"Geometry & topology","statement":"A hypersymplectic structure on a closed oriented four-manifold $X$ is a triple $\\omega=(\\omega_1,\\omega_2,\\omega_3)$ of closed two-forms whose pointwise span is three-dimensional and positive definite for the wedge product. Donaldson (2006, Section 5.3, Question 3) asked whether a compact oriented four-manifold carrying such a triple admits a hyperkahler structure, and proposed a continuity method keeping the cohomology classes fixed. Fine and Yao formulated the normalized deformation problem as a conjecture: after normalizing $\\int_X\\omega_i\\wedge\\omega_j=\\delta_{ij}$, can every hypersymplectic triple be deformed through hypersymplectic triples, keeping each class $[\\omega_i]$ fixed, to a hyperkahler triple, i.e. one with $\\omega_i\\wedge\\omega_j=2\\delta_{ij}\\mu$ for a volume form $\\mu$? Known before this work: convergence of the hypersymplectic flow for simple-type structures on $T^4$ (Huang-Wang-Yao), $T^3$-invariant triples on $T^4$ (Fine-He-Yao) and triples preserved by a circle action. Does every normalized hypersymplectic triple on a closed four-manifold deform, with its cohomology classes fixed, to a hyperkahler triple?","posedBy":"S. K. Donaldson, Two-forms on four-manifolds and elliptic equations (2006), Section 5.3, Question 3; deformation form stated as Conjecture 1.1 by J. Fine and C. Yao (Duke Math. J. 2018)","yearPosed":2006,"ageNote":"Donaldson's question appeared in the 2006 Chern memorial volume (arXiv math/0607083). The cohomology-preserving deformation statement is Fine and Yao's Conjecture 1.1 (arXiv 1704.07620, 2017; Duke Math. J. 167, 2018).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for a closed connected oriented smooth four-manifold $X$ and a smooth hypersymplectic triple with $\\int_X\\omega_i\\wedge\\omega_j=\\delta_{ij}$, there is a smooth path $\\omega(t)$ of hypersymplectic triples with $\\omega(0)=\\omega$, $[\\omega_i(t)]=[\\omega_i]$, ending at forms with $\\omega_i(1)\\wedge\\omega_j(1)=2\\delta_{ij}\\mu_1$, parallel and self-dual for a hyperkahler metric. An arbitrary positive triple is reduced to this case by a constant linear change. Corollaries: every nonzero combination $\\sum c_i\\omega_i$ is Kahler for some hyperkahler metric, and any closed four-manifold with a positive closed triple is diffeomorphic to the K3 manifold or $T^4$. Not shown: long-time existence or convergence of the hypersymplectic flow, or any statement in higher dimensions.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The manuscript reuses the current-estimate and density-splitting machinery of the companion release paper on Donaldson's tamed-to-compatible conjecture, reproving the parts it needs.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source, read against Donaldson's Question 3 and Fine-Yao Conjecture 1.1; the theorem states the full normalized deformation with all three classes fixed and a hyperkahler endpoint, and the paper says it resolves that conjecture. The proof was not refereed. It does not go through the hypersymplectic flow; convergence or uniqueness of that flow is left open. Key inputs are Bauer's bound for symplectic four-manifolds with torsion first Chern class, Preiss's rectifiability theorem, Riviere-Tian regularity of integral cycles, and the companion release paper Taming implies compatibility on four-manifolds (itself unreviewed), whose needed proofs are reproduced in Sections 3 and 4. The October 7 edition leaves Theorem 1.1 unchanged; it drops the citation of Li-Zhang's comparison of taming and compatible cones and obtains the prescribed taming class from the paper's own class criterion (under its definite-pair and $b^+=3$ hypotheses), so it does not rely on the general cone equality that the revised Taming paper narrowed. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A question Donaldson raised in his 2006 program on two-forms and elliptic equations, made precise by Fine and Yao and attacked through the hypersymplectic and G2-Laplacian flows; famous within four-dimensional and G2 geometry, and it would show hypersymplectic four-manifolds are smoothly K3 or the torus. Below Donaldson's tamed-to-compatible conjecture (45), level with the projective-space base conjecture for Lagrangian fibrations (38), above the hyperkahler period-index conjecture (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf","sourceName":"OpenAI math release: Deforming hypersymplectic four-manifolds to hyperkahler triples","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Donaldson, Two-forms on four-manifolds and elliptic equations (arXiv:math/0607083)","url":"https://arxiv.org/abs/math/0607083","kind":"problem-record"},{"label":"Fine and Yao, Hypersymplectic 4-manifolds and the G2-Laplacian flow (arXiv:1704.07620)","url":"https://arxiv.org/abs/1704.07620","kind":"problem-record"},{"label":"Earlier edition (September 23, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"little-finitistic-dimension-conjecture","name":"The finitistic dimension conjecture for finite-dimensional algebras","shortName":"Little finitistic dimension conjecture","problemNumber":null,"field":"Representation theory of finite-dimensional algebras; homological algebra","fieldGroup":"Algebra","statement":"For a finite-dimensional algebra $A$ over a field, the little finitistic dimension is $\\mathrm{findim}\\,A=\\sup\\{\\mathrm{pd}_AN: N\\text{ finitely generated},\\ \\mathrm{pd}_AN<\\infty\\}$. The finitistic dimension conjecture, publicized by Bass, asserts that $\\mathrm{findim}\\,A<\\infty$ for every finite-dimensional (more generally, Artin) algebra. It was proved for monomial algebras, radical-cube-zero algebras and representation dimension at most three (Igusa-Todorov), and it implies several other homological conjectures; Huisgen-Zimmermann showed the little and big dimensions can differ but both stayed finite in all examples. Is the little finitistic dimension of every finite-dimensional algebra finite?","posedBy":"Hyman Bass (publicized the finitistic dimension questions)","yearPosed":1960,"ageNote":"The manuscript says the finitistic-dimension questions 'were publicized by Bass in 1960' (Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. AMS 1960).","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a finite-dimensional complex algebra $A$ with, for each $m\\ge1$, a finite-dimensional left module $N_m$ with $2m-2\\le\\mathrm{pd}\\,N_m<\\infty$; so $\\mathrm{findim}\\,A=\\infty$ and the big finitistic dimension is infinite too, refuting the conjecture for Artin algebras. By Rickard's theorem injective left $A$-modules do not generate the derived category. Corollary (left-right asymmetry): an algebra $\\Lambda$ with both finitistic dimensions infinite on the left and zero on the right. The construction passes through a finitely presented group algebra with Abels-type central involutions. A companion OpenAI manuscript (Tachikawa family) independently gives characteristic-two examples.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture. The proof was not refereed. lean/formalization.yaml lists OAI.LittleFinitistic.Main.exists_counterexample (comparator LittleFinitistic). Its statement ComparatorChallenges/LittleFinitistic.lean was read: there is a finite-dimensional $\\mathbb C$-algebra $A$ whose little finitistic dimension (supremum of projective dimensions of finitely generated left modules of finite projective dimension, Mathlib's projectiveDimension) is infinite, with, for every $m\\ge1$, a finitely generated module $N$ with $2m-2\\le\\mathrm{pd}\\,N<\\infty$. This states the headline claim. A second catalogued result, OAI.LittleFinitistic.extreme_left_right_asymmetry (FinitisticAsymmetry), states the corollary on left-right asymmetry and injective generation; read here too. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The finitistic dimension conjecture is the central homological conjecture for finite-dimensional algebras, open since Bass in 1960 and implying several other conjectures in the area. Placed below Kaplansky's zero-divisor conjecture (62) and The blockwise Alperin weight conjecture (55), and well above Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture? (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An algebra of infinite little finitistic dimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.LittleFinitistic.Main.exists_counterexample)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/Finitistic/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: LittleFinitistic.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/LittleFinitistic.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.LittleFinitistic.extreme_left_right_asymmetry)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/FinitisticAsymmetry/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: FinitisticAsymmetry.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/FinitisticAsymmetry.lean","kind":"lean-proof"},{"label":"Related (family 199): A counterexample to Tachikawa's second conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stationary-varifold-almost-everywhere-regularity","name":"Almost-everywhere regularity of stationary integral varifolds","shortName":"Almost-everywhere regularity of varifolds","problemNumber":null,"field":"Geometric measure theory: regularity of stationary varifolds","fieldGroup":"Geometry & topology","statement":"Let $V$ be a stationary integral $m$-varifold in an open set $U\\subset\\mathbb R^{m+n}$: an integer-multiplicity rectifiable $m$-dimensional measure whose first variation of area vanishes. A support point is regular if near it $V$ is a positive integer multiple of a single smooth embedded minimal submanifold. Allard (1972) proved that the regular set is relatively open and dense and that density-one points are regular, but at points with higher-multiplicity planar tangents his theorem does not apply. Simon's notes record the question, and Brena, Decio and De Lellis (2025, Conjecture 1.2) conjecture a positive answer. Is $\\mathcal H^m(\\mathrm{Sing}\\,V)=0$ for every stationary integral $m$-varifold, in every dimension and codimension?","posedBy":"Leon Simon, Introduction to Geometric Measure Theory (notes, 2018), Ch. 5 Remark 6.4; stated as Conjecture 1.2 by Brena, Decio and De Lellis, Rend. Lincei Mat. Appl. 36 (2025)","yearPosed":2018,"ageNote":"2018 is the earliest record the manuscript cites (Simon's revised notes); it gives Allard's 1972 theory as background but does not say the question was posed there, so no earlier year is given.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every stationary integral $m$-varifold in an open subset of $\\mathbb R^{m+n}$, $m,n\\ge1$, has $\\mathcal H^m(\\mathrm{Sing}\\,V)=0$, with no stability, minimizing, orientability, density-bound or codimension hypothesis. Corollary 1.2 gives the same on round spheres. The main new estimate is a signed excess bound in flat cylinders. The companion A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold (5 Oct 2026) claims the stronger bound $\\dim_{\\mathcal H}\\mathrm{Sing}\\,V\\le m-1$, which implies this one and is a separate entry; it uses this paper's signed-excess theorem as an input. Neither paper treats non-Euclidean ambient metrics, where De Lellis-Hirsch-Lihn-Spolaor construct exotic examples.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and Corollary 1.2 of the TeX source, read against the question as Simon and Brena-Decio-De Lellis state it according to the manuscript's citations; the proof was not refereed. The result is for Euclidean open sets and, by a separate cone argument, round spheres; it says nothing about general Riemannian metrics. The release has no Lean formalization for this family, and the README cautions that unformalized results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Whether stationary integral varifolds are regular almost everywhere is the basic question left open by Allard's 1972 regularity theory, with recent partial progress by Brena, De Lellis, Franceschini, Hirsch and Spolaor. Placed level with Kadison's similarity problem (58) and Thompson group F amenability (58), above the C-infinity Caratheodory conjecture (55), below the Hilbert-Smith conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Almost-everywhere regularity of stationary integral varifolds","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf","kind":"paper"},{"label":"Brena, Decio, De Lellis, Remarks and conjectures on stationary varifolds (2025)","url":"https://doi.org/10.4171/RLM/1077","kind":"problem-record"},{"label":"Simon, Introduction to Geometric Measure Theory (2018 notes)","url":"https://math.stanford.edu/~lms/ntu-gmt-text.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"campana-abelianity-conjecture","name":"Campana's abelianity conjecture for special compact Kahler manifolds","shortName":"Campana's abelianity conjecture","problemNumber":null,"field":"Complex algebraic geometry; fundamental groups of Kahler manifolds","fieldGroup":"Geometry & topology","statement":"Campana's classification program splits compact Kahler manifolds into special ones and fibrations over orbifold bases of general type. A manifold $X$ is special if it has no Bogomolov sheaf: no line bundle $L\\subset\\Omega^p_X$ with $\\kappa(X,L)=p\\ge1$. Rationally connected manifolds, compact tori and manifolds of Kodaira dimension zero are special. Campana (2004, Conjecture 7.1) conjectured that the fundamental group of a special manifold is virtually abelian; Campana and Claudon proved it for threefolds. It would imply, for instance, that every compact Kahler manifold with $\\kappa=0$ has virtually abelian $\\pi_1$. Is the fundamental group of every special compact Kahler manifold virtually abelian?","posedBy":"Frederic Campana (Conjecture 7.1, Ann. Inst. Fourier 54, 2004)","yearPosed":2004,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every special compact Kahler manifold has virtually abelian fundamental group, in every dimension. Corollaries: compact Kahler manifolds with $\\kappa=0$ have virtually abelian $\\pi_1$; Campana's Conjecture S and the bimeromorphic Iitaka uniformization (finite etale covers bimeromorphic to tori when $\\gamma d=n$ or the universal cover is $\\mathbb C^n$); holomorphic convexity of universal covers of special manifolds. It does not prove the orbifold or quasi-projective versions in general: the companions give a two-step nilpotent bound for linear images in the open case (credited first to Cao-Deng-Hacon-Paun) and one conditional fourfold orbifold case.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two companions in the family build on it: 'Two-step monodromy of special quasi-projective varieties' (September 24, 2026), an independent proof of a result the manuscript credits to Cao-Deng-Hacon-Paun, and 'A conditional abelianity theorem for special fourfold pairs with a half-weight divisor' (October 5, 2026), which uses this theorem.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Campana's Conjecture 7.1 (in the Claudon-Horing appendix form the manuscript cites); it states that every connected smooth compact Kahler manifold that is special has a finite-index abelian subgroup in its full topological $\\pi_1$, with no linearity, projectivity or residual finiteness assumption. The proof was not refereed. No Lean formalization of this family is in the release (no lean/docs/057.md at the pinned commit). The manuscript cites another release manuscript (Orbifold and logarithmic Iitaka subadditivity) only to record a known specialness implication, not as a proof input.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Campana's abelianity conjecture is a central prediction of his classification program, open since 2004 and known only in dimension three; it contains the long-standing question for Kodaira dimension zero. Placed above the Campana-Peternell conjecture (35) and the projective-base conjecture for Lagrangian fibrations (38), level with the Kuznetsov rationality conjecture (45), below Iitaka's subadditivity conjecture (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The abelianity conjecture for special compact Kahler manifolds","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Two-step monodromy of special quasi-projective varieties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026/paper.pdf","kind":"paper"},{"label":"Companion: A conditional abelianity theorem for special fourfold pairs with a half-weight divisor","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor-October-5-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"one-sample-matroid-prophet-inequality","name":"Constant-competitive single-sample prophet inequalities on general matroids","shortName":"One-sample matroid prophet inequality","problemNumber":null,"field":"Online algorithms; prophet inequalities and matroid secretary problems","fieldGroup":"Theoretical computer science","statement":"In a matroid prophet inequality, independent nonnegative values $V_e$ on the elements of a known matroid $M$ arrive one at a time and an online rule must irrevocably accept or reject each, keeping the accepted set independent; it is compared with $\\mathbb E[\\mathrm{OPT}_M(V)]$. With known distributions, Kleinberg and Weinberg (2012) achieve 1/2. Azar, Kleinberg and Weinberg (2014) introduced the limited-information model in which the rule only gets one independent sample from each unknown distribution, and related it to order-oblivious matroid secretary algorithms. For general matroids only a doubly logarithmic loss in the rank was known with one sample (Feldman-Svensson-Zenklusen), and constant guarantees needed polylogarithmically many samples (Fu et al. 2024). Is there a single-sample rule achieving a constant fraction of the expected offline optimum on every matroid?","posedBy":"Azar, Kleinberg and Weinberg (SODA 2014) introduced single-sample prophet inequalities; Fu, Lu, Tang, Wu, Wu and Zhang (EC 2024) describe the general-matroid single-sample question as open","yearPosed":2014,"ageNote":"The manuscript does not quote an explicit statement of the question. 2014 is the Azar-Kleinberg-Weinberg framework it cites; the abstract of Fu et al. (2024, also cited) records that no constant-factor single-sample algorithm was known for general matroids and ties the question to the matroid secretary conjecture.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a measurable online rule, independent of the value laws, which for every finite matroid and every independent family of samples $S_e$ and values $V_e$ with common laws and $\\mathbb E[\\mathrm{OPT}_M(V)]<\\infty$ accepts an independent set with expected value at least $2^{-310}\\mathbb E[\\mathrm{OPT}_M(V)]$, against an adversary that chooses the order after seeing samples, values and the full random seed (the proof gives $2^{-293}$). A corollary gives a constant-competitive matroid secretary rule against adversarial reordering after a random observation prefix. The constant is tiny and no efficient implementation is asserted. Recent preprints (Singla 2026, Huang 2026) settle the random-order matroid secretary problem; the paper notes those keep uniformly random arrivals and do not give this result.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question. It gives ratio $2^{-310}$ with one sample per element, for every finite known matroid and arbitrary laws with finite expected optimum, even when the arrival order may depend on all samples, values and the rule's seed. The Lean challenge lean/ComparatorChallenges/MatroidProphet.json (declaration OAI.MatroidProphet.one_sample, solution module OAI/Probability/MatroidProphet/Main.lean present at the pinned commit) is not in the formalization catalogue; its statement was read here and states exactly this: a feasible rule with finite seed law, for any independent sample/value pairs with equal laws and any measurable order depending on everything, gets at least $2^{-310}$ of the expected optimum. That is the headline claim. Not rebuilt here. No running-time or polynomial-oracle bound is claimed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A recognised open question in online algorithms and mechanism design, closely tied to the matroid secretary conjecture but narrower in audience. Level with the Cerny conjecture for one-cluster automata entry at 25, above Tight Bound for Online Vertex Cover under Edge Arrivals at 15, and below Polynomial worst-case trace reconstruction at 42.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf","sourceName":"OpenAI math release: One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/Probability/MatroidProphet/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/MatroidProphet/Main.lean","kind":"lean-proof"},{"label":"Lean (supporting hidden-vector and secretary forms): OAI/Probability/MatroidSecretary/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/MatroidSecretary/Main.lean","kind":"lean-proof"},{"label":"Fu, Lu, Tang, Wu, Wu and Zhang, Sample-Based Matroid Prophet Inequalities (2024)","url":"https://arxiv.org/abs/2406.12799","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hadwiger-conjecture","name":"Hadwiger's Conjecture","shortName":"Hadwiger's conjecture","problemNumber":null,"field":"Graph coloring and graph minors","fieldGroup":"Combinatorics","statement":"For a graph $G$ let $h(G)$, the Hadwiger number, be the largest $t$ such that $K_t$ is a minor of $G$. Hadwiger (1943) conjectured that every graph with no $K_{t+1}$ minor is $t$-colourable, that is, $\\chi(G)\\le h(G)$. It was known for $t\\le5$ (Hadwiger and Dirac for $t=3$, Wagner plus the Four Colour Theorem for $t=4$, Robertson-Seymour-Thomas for $t=5$). The special case of graphs with independence number at most two (equivalently, every such $m$-vertex graph has a $K_{\\lceil m/2\\rceil}$ minor, Plummer-Stiebitz-Toft) and the fractional weakening $\\chi_f(G)\\le h(G)$ discussed by Reed and Seymour (1998) were also open. Does every finite graph satisfy $\\chi(G)\\le h(G)$?","posedBy":"Hugo Hadwiger, Uber eine Klassifikation der Streckenkomplexe, Vierteljahrsschr. Naturforsch. Ges. Zurich 88 (1943); fractional weakening discussed by Reed and Seymour (1998)","yearPosed":1943,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are arbitrarily large $m$ and $m$-vertex graphs with $\\alpha(G)\\le2$ and connected-matching number below $m/100$. With the counting bound $h(G)\\le(|V(G)|+4\\,\\mathrm{cm}(G)+2)/3$, Corollary 1.2 gives $h(G)<26m/75+2/3 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The single manuscript (September 23, 2026) is the whole family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against the question as the manuscript cites it from Gromov. The construction (positive rank-one metric decomposition, oscillatory corrections with controlled normal geometry, smoothing) was not refereed. The challenge lean/ComparatorChallenges/SurfaceImmersion.json (theorem OAI.ClosedSurfaceR4.FiniteOrderSmoothing.smooth_isometric_immersion, solution module OAI.Geometry.SurfaceImmersion.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml; it is found through lean/docs/333.md. Its statement was read here: for every compact Hausdorff second-countable smooth manifold modelled on $\\mathbb R^2$ without boundary and every smooth Riemannian metric, there is a smooth map to $\\mathbb R^4$ whose differential preserves the inner product on every tangent space. This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"A classical question of the isometric embedding problem going back to Chern and recorded by Gromov, well known to geometric analysts but narrower than Nash-type or curvature questions. Placed just below Local smooth isometric immersion of surfaces in three-space (45), level with The Lipman-Zariski conjecture (42).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Smooth isometric immersions of closed surfaces into Euclidean four-space","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof module (OAI.Geometry.SurfaceImmersion.Main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/SurfaceImmersion/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SurfaceImmersion.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SurfaceImmersion.lean","kind":"lean-proof"},{"label":"Gromov, Spaces and questions (author manuscript)","url":"https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/SpacesandQuestions.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ryser-conjecture","name":"Ryser's conjecture on covers of r-partite hypergraphs","shortName":"Ryser's conjecture","problemNumber":null,"field":"Extremal hypergraph theory; covers and matchings","fieldGroup":"Combinatorics","statement":"An $r$-partite $r$-uniform hypergraph $H$ has disjoint vertex parts $V_1,\\ldots,V_r$ and every edge has exactly one vertex in each part. Let $\\tau(H)$ be its covering number (least size of a vertex set meeting every edge) and $\\nu(H)$ its matching number. Ryser's conjecture asserts $\\tau(H)\\le(r-1)\\nu(H)$. For $r=2$ this is Konig's theorem; Aharoni proved $r=3$; in the intersecting case ($\\nu=1$) it was known through $r=5$ (Gyarfas, Tuza), and truncated projective planes show it would be tight whenever $r-1$ is a prime power. Does every $r$-partite $r$-uniform hypergraph satisfy $\\tau(H)\\le(r-1)\\nu(H)$?","posedBy":"Herbert J. Ryser (attributed); an equivalent formulation appears in J. R. Henderson's 1971 Caltech thesis","yearPosed":1971,"ageNote":"Year of Henderson's thesis, the earliest written source the manuscripts cite; they point to Best and Wanless, 'What did Ryser conjecture?' (2018), for the history of the attribution.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Ryser's conjecture is false, already for intersecting hypergraphs. September 23: for every sufficiently large prime $s\\equiv2\\pmod3$ and every sufficiently large odd $n$ (threshold depending on $s$), there is an intersecting $(s^n+1)$-partite $(s^n+1)$-uniform hypergraph with $\\tau=s^n+1$, against the conjectured $s^n$. September 27: for every sufficiently large prime $q$, an intersecting $(q+1)$-partite $(q+1)$-uniform hypergraph with $\\tau=q+1$ and exactly $q+1$ nonisolated vertices in each part. Not given: an explicit rank or example, or anything for small $r$ (cases $r=4,5$ remain open in general).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two manuscripts: the extension-field construction (September 23, 2026) and the balanced prime-order construction (September 27, 2026); the later paper says its proof is independent, and the earlier imports only three elementary lemmas from it.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of both manuscripts was read against Ryser's conjecture as posed. lean/formalization.yaml lists two main results for the September 23 paper: OAI.RyserCoveringCounterexample.exists_prime_eventually_counterexampleRank (one prime p > 5, p = 2 mod 3, with counterexamples at rank p^n + 1 for all large prime n) and OAI.RyserOdd.eventualOddFailures_and_infinite (every large such prime, every large odd n, infinitely many ranks). Both challenge statements were read: finite r-partite hypergraphs, intersecting, matching number 1, cover number r. The balanced companion's challenge (BalancedRyser, OAI.Balanced.main_result) is not in the formalization catalogue; its statement was read and matches that paper's theorem. None was rebuilt here. All thresholds are existential and not numerical, so no explicit counterexample is given.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"Ryser's conjecture is one of the central open problems of hypergraph matching theory, with a fifty-year literature of special cases and near-extremal constructions; most experts expected it to be true. A disproof of the whole conjecture, scored on the conjecture itself. Below the Erdos-Sos conjecture (58) and the cycle double cover conjecture (55), above Kahn's flow conjecture (40) and far above the four-partite matching-number-two case (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to Ryser's covering conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Balanced counterexamples to Ryser's conjecture at prime orders (September 27, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf","kind":"paper"},{"label":"Lean proof: OAI/Combinatorics/Ryser/Construction/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Ryser/Construction/Main.lean","kind":"lean-proof"},{"label":"Lean proof: OAI/Combinatorics/Ryser/OddExtensions.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Ryser/OddExtensions.lean","kind":"lean-proof"},{"label":"Lean proof (balanced companion): OAI/Combinatorics/BalancedRyser/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/BalancedRyser/Main.lean","kind":"lean-proof"},{"label":"Best and Wanless, What did Ryser conjecture? (arXiv:1801.02893)","url":"https://arxiv.org/abs/1801.02893","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"finitely-presented-burnside-problem","name":"The finitely presented Burnside problem: is every finitely presented periodic group finite?","shortName":"Finitely presented Burnside problem","problemNumber":null,"field":"Group theory; Burnside-type finiteness problems","fieldGroup":"Algebra","statement":"A group is periodic if every element has finite order. Burnside (1902) asked how such conditions constrain finitely generated groups. Golod (1964) built infinite finitely generated periodic groups, Novikov and Adian infinite groups of bounded exponent, and Grigorchuk an infinite finitely generated 2-group, but none of these is finitely presented, and Ol'shanskii-Sapir's finitely presented torsion-by-cyclic groups have an infinite cyclic quotient. The finite-presentation version of Burnside's question asks whether a group given by finitely many generators and finitely many relators (in the ordinary sense, not within a variety) can be infinite while every element has finite order. Is every finitely presented periodic group finite?","posedBy":"Recorded by Ol'shanskii and Sapir (Publ. IHES 2003, p. 45) and by Haettel and Osajda, in the line of Burnside's 1902 finiteness questions","yearPosed":null,"ageNote":"The manuscripts cite Burnside (1902) for the family of questions and Ol'shanskii-Sapir (2003) and Haettel-Osajda for explicit statements of the finitely presented version; neither gives an original posing date, and the question is older than 2003.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: there is a finitely presented graded $\\mathbb F_2$-algebra $R$ such that $\\mathrm{St}_{12}(R)$ is infinite, ordinarily finitely presented and periodic; Corollary 1.2 adds Kazhdan's property (T), so it is nonamenable. The companion claims $\\mathrm{St}_{12}(R)$ is residually finite and that the kernel $G$ of the map to $\\mathrm{St}_{12}(\\mathbb F_2)$ is an infinite finitely presented residually finite 2-group, necessarily of unbounded exponent. It does NOT give a finitely presented infinite group of bounded exponent (the bounded Burnside question for finite presentations), and element orders depend on the element.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The second (5 October 2026) builds on the first's algebra and local-rule hierarchy to add residual finiteness.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against the question: an explicit unital $\\mathbb F_2$-algebra $R$ for which the Steinberg group $\\mathrm{St}_{12}(R)$ is infinite, ordinarily finitely presented and periodic. The Lean challenge ComparatorChallenges/PeriodicGroup.json (theorem OAI.SourceBurnside.thm_main, solution module OAI.GroupTheory.PeriodicGroups.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/247.md. Its statement was read here: a ring $R$ that is an $\\mathbb F_2$-algebra with the Steinberg group on 12 indices (presented by the addition and commutator relations) infinite, finitely presented and periodic, together with the bare statement that some infinite finitely presented periodic group exists. That is the headline claim; no uniform exponent is asserted. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The residual-finiteness companion and the nil-algebra conclusions are not formalised.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"Whether finitely presented periodic groups are finite is one of the best known open questions of combinatorial group theory, the finite-presentation residue of the Burnside problems that resisted Golod, Novikov-Adian, Ol'shanskii and Grigorchuk style constructions. Level with the Koethe Conjecture (60), just below Kaplansky's zero-divisor conjecture (62) and Cannon's Conjecture (62), above the word problem for finite-rank Artin groups (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An infinite finitely presented periodic group","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An infinite finitely presented residually finite 2-group","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf","kind":"paper"},{"label":"Lean: solution module for the periodic Steinberg group","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/PeriodicGroups/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: infinite finitely presented periodic group","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PeriodicGroup.lean","kind":"lean-proof"},{"label":"Ol'shanskii and Sapir (2003), Non-amenable finitely presented torsion-by-cyclic groups","url":"https://doi.org/10.1007/s10240-002-0006-7","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bizeul-log-sobolev-subgaussian-log-concave","name":"Bizeul's conjecture: a dimension-free log-Sobolev inequality for log-concave measures with subgaussian linear marginals","shortName":"Bizeul's log-Sobolev conjecture","problemNumber":null,"field":"Asymptotic convex geometry; functional inequalities for log-concave measures","fieldGroup":"Geometry & topology","statement":"Call a centered random vector $X$ linearly $a$-subgaussian if $\\sup_{|\\theta|=1}\\mathbb E\\exp(\\langle X,\\theta\\rangle^2/a^2)\\le2$. A log-Sobolev inequality forces such tails, and Bakry-Emery gives one under uniform convexity, but log-concavity alone does not. Bobkov's criterion gives a constant of order $na^2$ and Bizeul improved this to order $\\sqrt n\\,a^2$, with the dimension-free bound for rotationally invariant laws. Bizeul conjectured that subgaussian linear marginals suffice; Klartag and Lehec list it as Conjecture 76. Is there a universal $C$ such that every centered log-concave probability measure $\\mu$ on $\\mathbb R^n$ that is linearly $a$-subgaussian satisfies $\\mathrm{Ent}_\\mu(f^2)\\le Ca^2\\int|\\nabla f|^2d\\mu$?","posedBy":"Pierre Bizeul (On the log-Sobolev constant of log-concave vectors, formulated 2023, J. Funct. Anal. 290 (2026), Conjecture 2); Klartag and Lehec, Bull. AMS (2025), Conjecture 76","yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is an absolute $C$ such that for every $n$, every centered log-concave probability $\\mu$ on $\\mathbb R^n$ with a density and linear subgaussian parameter $a$, $\\mathrm{Ent}_\\mu(f^2)\\le Ca^2\\int|Df|^2d\\mu$ for all $f\\in C_c^\\infty$; this includes uniform measures on convex bodies and needs no smoothness or curvature. By Otto-Villani it gives the transport-entropy inequality $W_2(\\nu,\\mu)^2\\le Ca^2H(\\nu|\\mu)$ with the same constant. Not given: a value of $C$, or anything for non-log-concave measures.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against Bizeul's Conjecture 2 as the paper states it; proof not refereed. No Lean formalization exists for this family. The paper states its scope: centered log-concave laws with a Lebesgue density (degenerate laws on lower-dimensional subspaces are reduced to this), the specific $\\psi_2$ marginal normalization above, and an unspecified absolute constant $C$.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A recent conjecture in the KLS circle of problems, given prominence by its listing in the Klartag-Lehec Bulletin survey; it is a natural question for specialists in high-dimensional convexity rather than a long-standing one. Placed below the (B)-conjecture for even log-concave measures (30) and well below the KLS conjecture for quadratic forms (45), above strong log-concavity of Chernoff's density (10).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hadwiger-nelson-not-five-colorable","name":"The Hadwiger-Nelson problem: the Euclidean plane is not five-colorable","shortName":"Plane is not five-colorable","problemNumber":null,"field":"Euclidean Ramsey theory; chromatic number of the plane","fieldGroup":"Combinatorics","statement":"Color every point of the plane $\\mathbb{R}^2$ so that any two points at distance exactly $1$ receive different colors, with no regularity imposed on the color classes. The Hadwiger-Nelson problem asks for the least number of colors $\\chi(\\mathbb{R}^2)$ for which this is possible. Isbell's hexagonal coloring gives $\\chi(\\mathbb{R}^2)\\le 7$, the Moser spindle gives $\\ge 4$, and de Grey (2018) gave a finite unit-distance graph that is not four-colorable, so $5\\le\\chi(\\mathbb{R}^2)\\le 7$. Falconer (1981) showed that five colors are needed if the classes must be measurable. Is the plane five-colorable, that is, can $\\chi(\\mathbb{R}^2)=5$?","posedBy":"Edward Nelson (1950, unpublished, with John Isbell's upper bound seven); published by Hugo Hadwiger (Elemente der Mathematik, 1961)","yearPosed":1950,"ageNote":"The manuscript dates the problem to Nelson in 1950 via Soifer's historical account; Hadwiger's 1961 'Ungeloste Probleme Nr. 40' is the published formulation it cites.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Every coloring of $\\mathbb{R}^2$ with five colors, with arbitrary (possibly non-measurable) color classes, has two points at distance $1$ of the same color. With the classical seven-coloring this gives $6\\le\\chi(\\mathbb{R}^2)\\le 7$. The route is a transfer theorem showing that a proper $k$-coloring exists if and only if a 'weak measurable' $k$-coloring (monochromatic unit pairs of measure zero) exists, for every finite $k$, followed by a geometric obstruction to weak measurable five-colorings. It does not decide between six and seven, and it does not give a finite non-five-colorable unit-distance graph (one exists by de Bruijn-Erdos compactness, but none is exhibited).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single manuscript, dated September 23, 2026; the release's overview counts it as one family.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the Hadwiger-Nelson problem as posed (arbitrary color classes, unit distance in the Euclidean plane). lean/formalization.yaml lists a main result for this paper: OAI.EuclideanFiveColor.no_proper_five_coloring in OAI/Geometry/PlaneColoring/Five.lean, with comparator ComparatorChallenges/EuclideanFiveColor. Its challenge statement was read: there is no function from the complex plane to Fin 5 giving distinct colors to every pair at norm distance exactly 1, with no measurability hypothesis. That is the headline claim, not a narrower one. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The proof passes through a transfer theorem (a proper k-coloring exists iff a weak measurable k-coloring exists, in ZFC), which is the step a reader should look at first. The paper says six versus seven remains open, and notes that a May 2026 public manuscript announcing chi = 7 rests on a false density bound.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The chromatic number of the plane is one of the best-known open problems in combinatorial geometry, and de Grey's 2018 step from four to five was widely covered. This is the next step, five to six, for arbitrary colorings, and leaves six versus seven open, so it is scored as a partial below the full problem. Above Erdos's planar unit distance conjecture (40) and the unit-distance independence ratio question (20), level with the sum-product conjecture over the reals (55), below Hadwiger's conjecture (70).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Euclidean plane is not five-colorable","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof: OAI/Geometry/PlaneColoring/Five.lean (no_proper_five_coloring)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/PlaneColoring/Five.lean","kind":"lean-proof"},{"label":"Comparator statement: EuclideanFiveColor.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/EuclideanFiveColor.lean","kind":"lean-proof"},{"label":"Hadwiger, Ungeloste Probleme Nr. 40, Elemente der Mathematik 16 (1961)","url":"https://www.e-periodica.ch/cntmng?pid=edm-001%3A1961%3A16%3A%3A215","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"yau-uniformization-conjecture","name":"Yau's uniformization conjecture","shortName":"Yau's uniformization conjecture","problemNumber":null,"field":"Kahler geometry, several complex variables","fieldGroup":"Geometry & topology","statement":"In his 1982 problem survey Yau asked whether a complete noncompact Kahler manifold with positive holomorphic bisectional curvature must be biholomorphic to $\\mathbb C^n$, a noncompact counterpart of the Frankel conjecture (settled by Mori and Siu-Yau). It was known under extra hypotheses: maximal volume growth (Chau-Tam, Liu, Lee-Tam), curvature decay or pinching, and recently for surfaces with positive sectional curvature (Datar-Pingali-Seshadri). Is every complete connected noncompact Kahler manifold of complex dimension $n$ with strictly positive holomorphic bisectional curvature biholomorphic to $\\mathbb C^n$?","posedBy":"S.-T. Yau, Survey on partial differential equations in differential geometry, Seminar on Differential Geometry, Ann. of Math. Stud. 102 (1982), Section 9(b), p. 45","yearPosed":1982,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if a connected noncompact complex manifold of dimension $n\\ge1$ carries a smooth complete Kahler metric with pointwise strictly positive holomorphic bisectional curvature, then it is biholomorphic to $\\mathbb C^n$; no curvature bounds, volume growth, noncollapsing or topological hypotheses are assumed. The proof runs a Kahler-Ricci flow built from exhaustions, proves Harnack inequalities by a disc-minimization argument, obtains shrinking holomorphic charts and assembles them by polynomial-automorphism dynamics into a biholomorphism onto all of $\\mathbb C^n$. It does not say the metric is flat, and it does not treat nonnegative curvature.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against Yau's 1982 question as cited. The flow, Harnack and dynamics arguments, which are long and analytic, were not refereed. No Lean main result. The theorem uses pointwise strict positivity, which is the conjecture's hypothesis; versions with nonnegative curvature are not claimed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Yau's uniformization conjecture is the best-known open problem on noncompact Kahler manifolds of positive curvature, open since 1982 and the target of decades of work by Mok, Chau-Tam, Liu and others. Tied with Thompson's group F amenability (58) and the Kadison similarity problem (58), just below Cannon's conjecture (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Uniformization of complete Kahler manifolds with positive bisectional curvature","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Yau 1982, Survey on PDE in differential geometry (Ann. of Math. Stud. 102)","url":"https://doi.org/10.1515/9781400881918-002","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"milne-rationality-conjecture","name":"Milne's rationality conjecture for abelian varieties","shortName":"Milne's rationality conjecture","problemNumber":null,"field":"Arithmetic geometry: Hodge classes, reduction and rational Tate classes","fieldGroup":"Number theory","statement":"Let $A$ be an abelian variety of dimension $d$ over $\\overline{\\mathbb Q}$ with good reduction $A_0$ at a $p$-adic place, and $\\gamma$ a rational Hodge class in $H^{2r}(A(\\mathbb C),\\mathbb Q(r))$. Deligne's absolute-Hodge theorem gives compatible specializations of $\\gamma$ to $\\ell$-adic cohomology of $A_0$ for $\\ell\\ne p$ and to crystalline cohomology, but the reduction may acquire divisors that do not lift. Milne conjectured that the specialized class nevertheless behaves rationally, which he showed equivalent (for CM abelian varieties) to the existence of a good theory of rational Tate classes. For every list of divisors $D_1,\\dots,D_{d-r}$ on $A_0$, is the intersection number of the specialized $\\gamma$ with $D_1\\cdots D_{d-r}$ one and the same rational number in every $\\ell$-adic and in the crystalline realization?","posedBy":"J. S. Milne (weak form in his 2000 Langlands-Rapoport notes; pairing form in 2007 AIM notes and Conjecture 4.1 of Rational Tate classes, Mosc. Math. J. 2009)","yearPosed":2000,"ageNote":"The manuscript cites a weak-rationality formulation in Milne's April 2000 notes (Conjecture 6.1(A)) and the pairing formulation in his July 2007 AIM notes and his 2009 paper (Conjecture 4.1). Year is the earliest cited form.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every abelian variety over $\\overline{\\mathbb Q}$ with good reduction at a $p$-adic place, every residue characteristic $p$ including $2$, every rational Hodge class $\\gamma$ of codimension $r$ and every divisors $D_1,\\dots,D_{d-r}$ on the reduction, the $\\ell$-adic traces ($\\ell\\ne p$) and the crystalline trace of $\\gamma_0\\cdot D_1\\cdots D_{d-r}$ equal one rational number. Also claims each specialized Hodge class lies in one fixed good theory of rational Tate classes, and (using the CM Hodge theorem) is represented by a rational algebraic cycle in all realizations. It does not prove the Hodge or Tate conjecture for general abelian varieties.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The main theorem is proved independently of the release's Hodge theorem for CM abelian varieties; only the algebraic-specialization corollary uses that theorem, which the README lists as produced outside the fixed procedure.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against Milne's conjecture as cited (pairing form). The proof was not refereed. No Lean formalization exists for this family at the pinned commit (no lean/docs/001.md, no comparator challenge). Inputs the paper relies on include Kisin-Zhou and Pappas-Rapoport integral models and a 2026 uniformization result of Gleason, Lim and Xu. The corollary that every specialized Hodge class is represented by one rational algebraic cycle rests on the release's unrefereed Hodge theorem for CM abelian varieties (a README exception); Theorem 1.1 itself does not.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A specialist conjecture in the theory of motives over finite fields, important to Milne's program on rational Tate classes and Langlands-Rapoport but little known outside it. Placed below the Tate conjecture for abelian varieties over finite fields (60) and the Hodge standard conjecture for abelian varieties (50), just below the period-index conjecture (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf","sourceName":"OpenAI math release: Milne's rationality conjecture for abelian varieties","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Milne, Rational Tate classes, Mosc. Math. J. 9 (2009), author's version","url":"https://www.jmilne.org/math/articles/2009b.pdf","kind":"problem-record"},{"label":"Milne, Towards a proof of the conjecture of Langlands and Rapoport (2000 notes)","url":"https://www.jmilne.org/math/articles/2000.pdf","kind":"problem-record"},{"label":"Earlier edition (September 23, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"artin-groups-word-problem","name":"The word problem for finite-rank Artin groups","shortName":"Word problem for Artin groups","problemNumber":null,"field":"Combinatorial group theory, Artin groups","fieldGroup":"Algebra","statement":"An Artin group is given by generators $\\sigma_s$, $s\\in S$, and braid relations equating the alternating words of length $m_{st}$ in $\\sigma_s,\\sigma_t$. Garside, Brieskorn-Saito and Deligne solved the word problem for spherical (finite-type) Artin groups, and it is known for further classes such as large type (Holt-Rees) and diagrams without $A_3$ or $B_3$ subdiagrams (Blasco-Garcia et al. 2026), but no algorithm was known in general. Charney's problem list asks (Problem 10) for solvable word and conjugacy problems in infinite type, and Blasco-Garcia, Cumplido, Holt, Morris-Wright and Rees pose the general decidability question explicitly. Is the word problem decidable for every Artin group with finitely many generators?","posedBy":"R. Charney, Problems related to Artin groups, Problem 10 (word and conjugacy problems); posed explicitly in Blasco-Garcia, Cumplido, Holt, Morris-Wright and Rees, J. Algebra 703 (2026)","yearPosed":null,"ageNote":"Charney's problem list is undated in the family's references, and the 2026 paper restates a long-standing question; the manuscripts give no original date. Charney's Problem 10 also asks for the conjugacy problem, which is not addressed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 1.3 of the parabolic-intersections manuscript: there is an algorithm that, given a finite Coxeter matrix $M$, a subset $X$ and words $u,v$, decides whether $v\\in uA_Xu^{-1}$; with $X=\\varnothing$ this decides the word problem for every finite-rank Artin group. The algorithm uses the framed zigzag-algebra action, shown faithful in Section 8, computed exactly over a cyclotomic field, and tests one transported projective by Gaussian elimination. No complexity bound is asserted, and the conjugacy problem is not treated.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary 1.3, its discussion in the introduction and the start of its proof in Section 8, read against the question as the manuscript cites it. The faithfulness of the action, on which everything rests, was not refereed. Not Lean-checked: the release's Comparator challenge for this manuscript (ArtinParabolicIntersections) states the intersection, closure and dynamics results but not the membership algorithm or faithfulness, so this corollary has no formal counterpart. It is a corollary inside a manuscript whose headline is a different conjecture.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Decidability of the word problem for all Artin groups is a classic open question of combinatorial group theory, as basic as the K(pi,1) conjecture and better known outside the specialty than the parabolic or CAT(0) questions. Below the K(pi,1) conjecture (60), tied with Vinogradov's least quadratic nonresidue conjecture (45) and the Kuznetsov rationality conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Parabolic Intersections in Artin Groups","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Charney, Problems related to Artin groups (Problem 10)","url":"https://people.brandeis.edu/~charney/papers/Artin_probs.pdf","kind":"problem-record"},{"label":"Blasco-Garcia et al. 2026, Rewriting in Artin groups without A3 or B3 subdiagrams","url":"https://doi.org/10.1016/j.jalgebra.2026.02.017","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"banach-simple-lebesgue-spectrum","name":"Banach's problem: a probability-preserving transformation with simple Lebesgue spectrum","shortName":"Banach's simple Lebesgue spectrum problem","problemNumber":null,"field":"Ergodic theory; spectral theory of measure-preserving maps","fieldGroup":"Analysis","statement":"For an invertible measure-preserving transformation $T$ of a probability space, the Koopman operator $U_Tg=g\\circ T$ is unitary on $L^2$ and leaves the constants invariant. Banach's problem, in its probability-preserving form, asks whether some $T$ has simple Lebesgue spectrum on the orthogonal complement $L^2_0$ of the constants: equivalently, whether there is $f\\in L^2_0$ whose iterates $\\{f\\circ T^n\\}_{n\\in\\mathbb Z}$ form an orthonormal basis of $L^2_0$. Rokhlin (1949) asked for ergodic automorphisms with simple, or at least finite-multiplicity, Lebesgue spectrum. Known constructions gave Lebesgue components, finite-measure flows, simple non-Lebesgue spectrum, or infinite-measure examples, but not this. Does such a probability-preserving transformation exist?","posedBy":"Stefan Banach (question recorded by Ulam); the probability-space form was asked by V. A. Rokhlin","yearPosed":1949,"ageNote":"The manuscript cites Rokhlin (1949, Section 4, no. 7) for the probability-space question and Ulam's 1960 collection (Section 6, p. 76) for the question attributed to Banach, which it describes as a distinct real-line formulation. The date of Banach's own question is not given.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are a $C^\\infty$ volume-preserving diffeomorphism $T$ of $\\mathbb T^3$ and a real $f\\in L^2_0(\\mathbb T^3)$ such that $\\{f\\circ T^n\\}_{n\\in\\mathbb Z}$ is an orthonormal basis of $L^2_0$; hence $T$ is ergodic and $U_T$ on $L^2_0$ is unitarily equivalent to multiplication by $w$ on $L^2(S^1)$. Mixing, zero entropy and zero Lyapunov exponents follow; mixing of all orders uses a separate multiple-mixing manuscript. The manuscript states this is distinct from the historical real-line question attributed to Banach and does not address that formulation; $f$ is only $L^2$, not smooth.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the probability-preserving form of the problem. The proof was not refereed. The Lean challenge ComparatorChallenges/ThreeTorus.json (theorem OAI.ThreeTorus.simple_lebesgue_spectrum, solution module OAI.Dynamics.ThreeTorus.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/144.md. Its statement was read here: a bijection $T$ of $(\\mathbb R/\\mathbb Z)^3$, smooth with smooth inverse via local lifts, preserving Haar measure, and a real $f$ in the complex mean-zero $L^2$ whose bilateral iterates are orthonormal with dense span, with $T$ ergodic and a linear isometry onto $L^2$ of the circle conjugating the Koopman operator to multiplication by the first character. This states the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"Banach's simple Lebesgue spectrum problem is one of the classical open problems of spectral ergodic theory, usually listed beside Rokhlin's multiple-mixing problem. Placed below Rokhlin's multiple-mixing problem (62) and Sinai's conjecture on positive metric entropy of the standard map (58), and well above Kalton's Problem 2 (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A smooth three-torus diffeomorphism with simple Lebesgue spectrum","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.ThreeTorus.simple_lebesgue_spectrum)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Dynamics/ThreeTorus/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: ThreeTorus.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ThreeTorus.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"shafarevich-holomorphic-convexity","name":"The Shafarevich conjecture on holomorphic convexity of universal covers","shortName":"Shafarevich holomorphic convexity","problemNumber":null,"field":"Complex algebraic geometry; universal covers and fundamental groups","fieldGroup":"Geometry & topology","statement":"Shafarevich asked whether the universal cover of a complete algebraic variety is holomorphically convex. In the usual smooth projective formulation: for every smooth connected projective complex variety $X$, is the universal cover $\\widetilde X$, with the lifted complex structure, holomorphically convex, i.e. is the holomorphic hull $\\{v:|g(v)|\\le\\sup_K|g|\\ \\forall g\\in\\mathcal O(\\widetilde X)\\}$ of every compact $K$ compact? A direct form for varieties with large fundamental group (Kollar) asks whether $\\widetilde X$ is Stein when it contains no positive-dimensional compact analytic subvariety. The conjecture was proved for elliptic surfaces, for virtually nilpotent fundamental groups, and for fundamental groups with a faithful linear representation (Eyssidieux-Katzarkov-Pantev-Ramachandran). Does it hold in general?","posedBy":"Igor Shafarevich","yearPosed":1974,"ageNote":"The surface manuscript cites Shafarevich's Basic Algebraic Geometry (1974, p. 407) via Lasell-Ramachandran; the fourfold manuscript cites Kollar (1995) for the large-fundamental-group formulation.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Surface manuscript, Theorem 1.1: a smooth connected projective complex surface $X$ containing a connected nodal curve $Z_0$ with smooth rational components such that $\\pi_1(Z_0)\\to\\pi_1(X)$ has infinite image; a component of its preimage in $\\widetilde X$ is a closed noncompact union of compact rational curves, so hulls are noncompact and $\\widetilde X$ is not holomorphically convex. By the linear Shafarevich theorem $\\pi_1(X)$ has no faithful finite-dimensional complex representation. Fourfold manuscript, Theorem 1.1: a smooth projective fourfold with large fundamental group (no positive-dimensional compact subvariety in $\\widetilde X$) whose universal cover is not Stein, hence not holomorphically convex; this disproves the large-fundamental-group form as well. Neither paper addresses quasi-projective or Kahler variants beyond these.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts: the surface counterexample (September 23) and the fourfold with large fundamental group (October 5).","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the surface manuscript and Theorem 1.1 of the fourfold manuscript were read against the conjecture as posed. The surface counterexample is an infinite Nori string: a connected nodal curve of rational components with infinite fundamental-group image, whose lift is a noncompact connected union of compact curves. The fourfold argument is topological (a lifted surface with nonzero third homology cannot be Stein). Both rely on cited inputs (Stover-Toledo ramified covers, Zarhin, Hamm-Le Lefschetz). The proofs were not refereed. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"The Shafarevich conjecture has been a guiding problem in complex geometry for fifty years, with major partial results (linear Shafarevich) and earlier proposed counterexamples that stayed conditional. Placed below Yau's uniformization conjecture (58) and Are all word-hyperbolic groups residually finite? (66), and above Kuznetsov's rationality conjecture for cubic fourfolds (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A surface counterexample to Shafarevich holomorphic convexity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A projective fourfold with large fundamental group and non-Stein universal cover","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lipman-zariski-conjecture","name":"The Lipman-Zariski conjecture in characteristic zero","shortName":"Lipman-Zariski conjecture","problemNumber":null,"field":"Algebraic geometry; singularities and derivation modules","fieldGroup":"Geometry & topology","statement":"On a smooth complex variety of dimension $d$ the tangent sheaf is locally free of rank $d$. Lipman (1965), following a question of Zariski, proved that local freeness of the derivation module forces normality and asked for the converse of smoothness: if $A$ is a finitely generated integral $\\mathbb C$-algebra and $\\mathrm{Der}_{\\mathbb C}(A)$ is locally free, is $A$ regular? It was known for hypersurfaces, local complete intersections, graded rings, isolated singularities in dimension at least three, klt and log canonical spaces, and several classes of normal surface singularities, but the surface case was open. Is every complex algebraic variety with locally free tangent sheaf nonsingular?","posedBy":"Joseph Lipman, after Oscar Zariski","yearPosed":1965,"ageNote":"Lipman, Free derivation modules on algebraic varieties, Amer. J. Math. 87 (1965), as cited by the manuscript; the attribution to Zariski is in the conjecture's standard name, not stated in the manuscript.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are a finitely generated normal integral $\\mathbb C$-algebra $A$ of dimension two and a maximal ideal $\\mathfrak m$ with $\\mathrm{Der}_{\\mathbb C}(A)\\cong A^{\\oplus2}$ and $A_{\\mathfrak m}$ not regular; the surface can be smooth away from that point. By Jorder's theorem no local tangent frame there commutes. The paper also answers negatively, in characteristic zero, the derivation-blowup question of Barajas, Chavez-Martinez and Romano-Velazquez (2026, Question 1). It does not address higher-dimensional non-isolated cases left open by earlier results or give a counterexample with a commuting frame (impossible by Jorder).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 23, 2026) has no companion in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture; it gives a finitely generated normal two-dimensional $\\mathbb C$-algebra $A$ with $\\mathrm{Der}_{\\mathbb C}(A)\\simeq A^{\\oplus2}$ globally free and a maximal ideal at which $A$ is not regular, which contradicts the conjecture as posed. The construction starts from an analytic Gorenstein surface singularity (exceptional curve of genus 21, self-intersection $-10$) and algebraizes it; the paper checks that the example lies outside the Graf, Bergner-Graf, log canonical and Biswas-Gurjar-Kolte criteria. Not refereed. No Lean formalization in the release.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"The Lipman-Zariski conjecture has been a standard open problem of singularity theory since 1965, with a long line of partial results from Scheja-Storch to Greb-Kebekus-Kovacs-Peternell and Graf; it is famous within algebraic geometry but less widely than the homological conjectures. Placed below the small Cohen-Macaulay module conjecture (45) and Brennan's conjecture (45), level with the Wall manifold-realization question (42), and above the CAT(0) conjecture for Artin groups (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A singular normal affine surface with free tangent sheaf","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lipman (1965), Free derivation modules on algebraic varieties","url":"https://doi.org/10.2307/2373252","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"k3-surfaces-oka","name":"Are all complex K3 surfaces Oka manifolds?","shortName":"K3 surfaces are Oka","problemNumber":null,"field":"Complex geometry: Oka theory and holomorphic flexibility","fieldGroup":"Geometry & topology","statement":"A complex manifold $X$ is Oka if maps from Stein manifolds to $X$ satisfy the Oka principle; by Forstneric's theorem this is equivalent to the convex approximation property: for every $m\\ge1$, every holomorphic map to $X$ from a neighbourhood of a compact convex $K\\subset\\mathbb C^m$ can be approximated uniformly on $K$ by holomorphic maps $\\mathbb C^m\\to X$. Forstneric and Larusson's 2011 survey asked (Section 8, Problem C) whether K3 surfaces are Oka. Known before this work: elliptic and Kummer K3 surfaces are dominable by $\\mathbb C^2$ (Buzzard-Lu), Kummer surfaces are strongly dominable, Kummer and elliptic K3 surfaces are Oka-1 (Alarcon-Forstneric), the Kobayashi pseudodistance vanishes on every K3 surface (Kamenova-Lu-Verbitsky), and Xie-Zhao proved that smooth $(2,2,2)$ hypersurfaces in $(\\mathbb P^1)^3$ are Oka and that Oka K3 periods are dense, conjecturing that all K3 surfaces are Oka. Is every complex K3 surface, projective or not, an Oka manifold?","posedBy":"F. Forstneric and F. Larusson, Survey of Oka theory, New York J. Math. 17a (2011), Section 8, Problem C; conjectured explicitly by S.-Y. Xie and S. Zhao (arXiv:2608.24392, 2026)","yearPosed":2011,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every complex K3 surface $X$, every $m\\ge1$, every compact convex $K\\subset\\mathbb C^m$ and every holomorphic $f$ near $K$ with values in $X$, $f$ is a uniform limit on $K$ of entire maps $\\mathbb C^m\\to X$; hence $X$ is Oka. Corollaries: dense immersed entire curves with prescribed first jet (Zariski dense if projective, the K3 case of Campana's prediction); Enriques surfaces and hence all minimal compact complex surfaces of Kodaira dimension zero are Oka; and, using the companion global-spherical-shell paper, a minimal class VII surface is Oka exactly when it is Hopf or Enoki. Not shown: the Oka property for other surfaces of non-negative Kodaira dimension, or for higher-dimensional hyperkahler manifolds.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source against Problem C of the Forstneric-Larusson survey and the Xie-Zhao conjecture; the theorem states the convex approximation property in every source dimension for every complex K3 surface, including nonprojective ones, which is equivalent to Oka by Forstneric's theorem. The proof was not refereed. It uses Chen-Gounelas genus-one curve families, the Xie-Zhao period framework and Verbitsky's corrected orbit alternatives. The further classification of class VII surfaces in Section 9 depends on the companion release paper on global spherical shells (itself unreviewed), but the K3 theorem does not. No Lean formalization is supplied for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A well-known question of Oka theory since Forstneric and Larusson's 2011 survey, with steady partial progress (Kummer and elliptic cases, dense Oka periods) and a target of recent work; famous within holomorphic flexibility and complex surfaces but not beyond. Level with the Campana-Peternell conjecture in dimension six (28), above the hyperkahler period-index conjecture (22), well below the Global Spherical Shell conjecture (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Every complex K3 surface is Oka","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Forstneric and Larusson, Survey of Oka theory (New York J. Math. 17a, 2011)","url":"https://nyjm.albany.edu/j/2011/17a-2.html","kind":"problem-record"},{"label":"Xie and Zhao, Periods of Oka K3 surfaces are dense (arXiv:2608.24392)","url":"https://arxiv.org/abs/2608.24392","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hyperbolic-groups-residually-finite","name":"Are all word-hyperbolic groups residually finite?","shortName":"Residual finiteness of hyperbolic groups","problemNumber":null,"field":"Geometric group theory; hyperbolic groups","fieldGroup":"Algebra","statement":"A group is residually finite if every nonidentity element survives in some finite quotient. Is every word-hyperbolic group residually finite? The question goes back to Gromov's theory of hyperbolic groups (1987) and is formulated, with its consequences, by Agol, Groves and Manning (2009, Section 5): a positive answer would make every quasiconvex subgroup of every hyperbolic group separable. Kapovich and Wise showed it is equivalent to every nontrivial hyperbolic group having a nontrivial finite quotient, and to every hyperbolic group being virtually torsion-free. Agol's theorem gives residual finiteness for cubulated hyperbolic groups, and non-residually-finite nonpositively curved examples (Wise, Burger-Mozes) were known only outside the hyperbolic class. Is every word-hyperbolic group residually finite?","posedBy":"Mikhail Gromov's program on hyperbolic groups; formulated by Ian Agol, Daniel Groves and Jason Fox Manning","yearPosed":1987,"ageNote":"The manuscript places the question in the program initiated by Gromov, Hyperbolic groups (1987), and cites Agol-Groves-Manning (Geom. Topol. 2009, Section 5) for its formulation; it does not quote a first statement. 1987 is the date of Gromov's paper.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a torsion-free word-hyperbolic group $G$, the fundamental group of a finite Euclidean triangle complex, that is not residually finite; one member of a fixed finite family of nontrivial elements lies in the finite residual, without identifying which. Corollary 1.2: every finite-dimensional representation of $G$ over every commutative field kills the finite residual, so $G$ is linear over no field. With Kapovich-Wise this gives a nontrivial hyperbolic group with no nontrivial finite quotient and a hyperbolic group that is not virtually torsion-free (curator's reading of the paper's stated consequences). It says nothing about skew fields or infinite-dimensional representations, and does not settle whether all hyperbolic groups are sofic.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against the question as posed. The construction was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator TorsionFreeHyperbolic, declaration OAI.Release075.main). Its statement, read here: there exist a group $G$ that is torsion-free, word-hyperbolic (a finite set whose Cayley graph is connected and has uniformly $\\delta$-thin geodesic triangles) and not Group.ResiduallyFinite. That is the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The release's own Lean scope note (lean/docs/252.md) says nonlinearity over every field is not part of the formal statement; Corollary 1.2 is a short deduction via Malcev's theorem in the paper only.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":66,"significanceNote":"Whether hyperbolic groups are residually finite is widely regarded as the central open question of geometric group theory since Gromov, equivalent to virtual torsion-freeness and behind the separability program used in Agol's proof of the virtual Haken conjecture. Placed just above the Hilbert-Smith Conjecture (65) and the Jacobian Conjecture (65), above Cannon's Conjecture (62) and Gromov and Weiss's Question on Sofic Groups (60), below the free group factor isomorphism problem (68).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A torsion-free hyperbolic group that is not residually finite","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.Release075.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/Hyperbolic/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: TorsionFreeHyperbolic.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/TorsionFreeHyperbolic.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"deligne-drinfeld-conjecture","name":"The Deligne-Drinfeld conjecture: the Grothendieck-Teichmuller Lie algebra is free on odd generators","shortName":"Deligne-Drinfeld conjecture","problemNumber":null,"field":"Grothendieck-Teichmuller theory; multiple zeta values","fieldGroup":"Algebra","statement":"Let $\\mathfrak{grt}_1$ be the graded space of Lie polynomials $\\psi(x,y)$ in two letters satisfying antisymmetry, the hexagon (three-term) relation and the pentagon relation in the infinitesimal pure braid Lie algebra $\\mathfrak t_4$, with the Ihara bracket. Ihara and Drinfeld produced nonzero elements $\\sigma_{2k+1}$ in each odd weight $2k+1\\ge3$; Brown's mixed Tate motives theorem (2012) shows they can be chosen to generate a free Lie subalgebra, and Naef-Willwacher checked equality through weight 29. The Deligne-Drinfeld conjecture asserts that $\\mathfrak{grt}_1$ is exactly a free Lie algebra with one generator in each odd weight $3,5,7,\\dots$. Is $\\mathfrak{grt}_1\\otimes\\mathbb Q$ freely generated by one element in each odd weight at least three?","posedBy":"Pierre Deligne and Vladimir Drinfeld","yearPosed":1990,"ageNote":"Deligne, Le groupe fondamental de la droite projective moins trois points (MSRI Publ. 16, 1989), and Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Qbar/Q) (Algebra i Analiz 1990; English 1991), both cited by the manuscript. 1990 is the date of Drinfeld's paper; the manuscript does not single out a year.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are homogeneous $\\sigma_{2k+1}\\in W_{2k+1}$ such that $e_{2k+1}\\mapsto\\sigma_{2k+1}$ is a graded Lie isomorphism $\\mathrm{Lie}_{\\mathbb Q}\\langle e_3,e_5,\\dots\\rangle\\cong(W,\\{\\,,\\})$, with $W$ closed under the Ihara bracket, and the induced map on weight completions is a continuous isomorphism. Via Willwacher's theorem this gives $H^0(\\mathrm{GC}_2)\\cong\\mathrm{Lie}\\langle e_3,e_5,\\dots\\rangle$. With Brown's theorem it identifies $\\mathfrak{grt}_1$ with the motivic Lie algebra. Generators are not canonical, and no other graph-complex degrees are computed.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 23, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture; it asserts a graded Lie isomorphism from the free Lie algebra on $e_3,e_5,\\dots$ onto the rational solution space $W$ of (1.2)-(1.4) with the Ihara bracket, and the completed isomorphism. The key new input is the all-weight dimension upper bound (Corollary 5.4), proved by a characteristic-two degeneration of the pentagon. lean/formalization.yaml lists a main result (comparator DeligneDrinfeld, declaration OAI.DeligneDrinfeld.main). The comparator statement was read here: $W$ is defined as the rational solutions of the three equations, and the statement gives a weight-graded linear equivalence from the free Lie algebra on odd weights to $W$ intertwining the bracket with the Ihara bracket, plus continuous inverse maps on completions. This states the headline. lean/docs/008.md says the graph-complex consequence is not formalized. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"A central conjecture linking Drinfeld associators, the absolute Galois group, mixed Tate motives and Kontsevich's graph complex, open for over thirty-five years and resistant even after Brown's 2012 breakthrough; famous across several communities but specialist in statement. Placed level with the Kuga-Satake Hodge conjecture entry (48), below the Alperin weight conjecture (55), and above the Artin groups CAT(0) conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Deligne-Drinfeld conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.DeligneDrinfeld.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/Drinfeld/Completion.lean","kind":"lean-proof"},{"label":"Comparator statement: DeligneDrinfeld.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DeligneDrinfeld.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stable-coordinate-conjecture-dimension-five","name":"The Stable Coordinate Conjecture in ambient dimension five","shortName":"Stable Coordinate Conjecture, dimension 5","problemNumber":null,"field":"Polynomial automorphisms, coordinates","fieldGroup":"Algebra","statement":"A polynomial $f\\in k^{[n]}$ is a coordinate if it is part of a polynomial coordinate system, that is, the image of a variable under an automorphism. The Stable Coordinate Conjecture, recorded by Makar-Limanov, van Rossum, Shpilrain and Yu (2004, p. 342), asks whether stabilization can create coordinates: if $f\\in\\mathbb C^{[n]}$ becomes a coordinate of $\\mathbb C^{[n+1]}$ after adjoining one variable, must $f$ already be a coordinate of $\\mathbb C^{[n]}$?","posedBy":"L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T. Yu, The stable equivalence and cancellation problems, Comment. Math. Helv. 79 (2004), p. 342","yearPosed":2004,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Corollary 1.2: the explicit $H\\in P=\\mathbb C[p,s,u,F,J]$ is a coordinate of $P[w]$ (by the Dutta-Lahiri theorem, since $H$ is a residual coordinate over $\\mathbb C[p]$) but not a coordinate of $P$, because otherwise $P/(H)\\cong\\mathbb C^{[4]}$, contradicting Theorem 1.1. So the conjecture fails over $\\mathbb C$ in ambient dimension five after one stabilization. Lower ambient dimensions are not addressed. A four-variable counterexample, the first possible dimension, was given in a later OpenAI math release manuscript (5 October 2026), see stable-coordinate-conjecture-four-variables.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, the introduction and Corollary 1.2 with its proof, read against the conjecture as the manuscript cites it. The deduction is two lines given Theorem 1.1 and the published Dutta-Lahiri theorem. Not Lean-checked as stated: the release's Lean scope note says the stable-coordinate consequence is not formalized; only the non-polynomiality and stabilization of P/(H) are (Comparator challenge ComplexCancellation), and the step that H itself is a coordinate of P[w] is not in the formal statement.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A recorded conjecture in the polynomial-automorphism community, a close relative of the cancellation problem but much less widely known. Tied with the Inverse Generator Problem (22) and Anari's Bethe permanent conjecture (22), below the Banks-Martin conjecture (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An explicit failure of complex affine-space cancellation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Makar-Limanov, van Rossum, Shpilrain, Yu 2004, Comment. Math. Helv. 79","url":"https://doi.org/10.1007/s00014-003-0796-3","kind":"problem-record"},{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/stable-coordinate-conjecture-four-variables","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-138-superexponential-van-der-waerden","name":"Erdos Problem #138: do the van der Waerden numbers grow superexponentially, W(k)^(1/k) -> infinity?","shortName":"Superexponential van der Waerden numbers","problemNumber":null,"field":"Ramsey theory, arithmetic progressions","fieldGroup":"Combinatorics","statement":"Let $W(k)$ be the least $N$ such that every two-colouring of $\\{1,\\dots,N\\}$ contains a monochromatic $k$-term arithmetic progression. Classical lower bounds are exponential: Berlekamp gave $W(p+1)>p2^p$ for primes $p$, and the local lemma gives roughly $2^k/k$. Erdos asked whether the growth is in fact superexponential: is it true that $W(k)^{1/k}\\to\\infty$?","posedBy":"Paul Erdos","yearPosed":1980,"ageNote":"Erdos, A survey of problems in combinatorial number theory, Ann. Discrete Math. 6 (1980), p. 90, item (2), as cited by the manuscript. Recorded as Erdos Problem 138, whose companion question $W(k+1)-W(k)\\to\\infty$ is already in the catalog.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is an absolute $K_0$ with $W_r(k)>k^{ck\\lfloor\\log_2 r\\rfloor}$, $c=10^{-5}$, for all $k\\ge K_0$ and $r\\ge2$; hence $W_r(k)^{1/k}\\to\\infty$ for each fixed $r$. Prior work in 2026: Fox-Hunter proved superexponential growth for three colours, and Campos-Fox-Schildkraut proved $W_2(k)\\ge(1-o(1))k2^{k-1}$, settling $W_2(k)/2^k\\to\\infty$ but not the limit asked here. Not shown: the companion question $W(k+1)-W(k)\\to\\infty$, or any upper bound improvement.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 23, 2026) with a Lean formalization of its headline bound listed in the release's formalization catalogue.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Erdos's question; the theorem implies $W_r(k)^{1/k}\\to\\infty$ for each fixed $r\\ge2$, including $r=2$. Lean: formalization.yaml lists OAI.QuantitativeVanDerWaerden.uniform_lower_bound (OAI/Combinatorics/ProgressionColoring/Main.lean). The comparator statement was read: it asserts an absolute $K$ with $k^{k\\lfloor\\log_2 r\\rfloor/100000} 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (September 23, 2026) gives a structural proof via an optimal similarity and a common positive boundary density; a second manuscript (September 26, 2026) gives a separate direct proof of the finite-matrix inequality. The manuscripts cite the scalar proofs of Jin, of Lorist and Schwenninger, and of Luo.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the complete inequality in Corollaries 6.1 and 6.3 of the principal manuscript and Theorem 1.1 of the direct companion were read against the complete conjecture. lean/formalization.yaml lists main results for both manuscripts (CompleteCrouzeix, DirectCrouzeix, StructuralCrouzeix). The CompleteCrouzeix comparator was read here: for all $n,m\\ge1$, $d$, $A\\in M_n$ and $B_j\\in M_m$, $\\|\\sum_k A^k\\otimes B_k\\|\\le2\\sup_{W(A)}\\|\\sum_k z^kB_k\\|$, and 2 is the least such universal constant. This states the headline claim for matrices. The operator version, challenge CrouzeixHilbert, is not in the formalization catalogue; it is linked from lean/docs/325.md, its solution module exists at the pinned commit, and its statement (arbitrary complex Hilbert space, plus holomorphic and rational extensions and sharpness) was read here. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The complete form is the natural strengthening of Crouzeix's conjecture and was open even after the scalar proofs of summer 2026, with a published case only for base dimension three. But it stood in the shadow of the scalar conjecture, which carries the fame. Placed below Crouzeix's conjecture (35), level with the strong specialist band at 30 such as log-concavity of matroid flats (30), and above the Lorist-Schwenninger remark question (4).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The complete Crouzeix theorem: optimal similarity and a common positive boundary representation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A direct proof of the complete Crouzeix inequality (Sept 26)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.CompleteCrouzeix.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/NumericalRange/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: CompleteCrouzeix.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CompleteCrouzeix.lean","kind":"lean-proof"},{"label":"Comparator statement: CrouzeixHilbert.lean (operators on any Hilbert space)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CrouzeixHilbert.lean","kind":"lean-proof"},{"label":"Lean proof of the operator version (OAI.CrouzeixHilbert.hilbert)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/Crouzeix/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strong-kadison-kastler-conjecture","name":"The strong Kadison-Kastler conjecture","shortName":"Strong Kadison-Kastler conjecture","problemNumber":null,"field":"Operator algebras; perturbation theory of von Neumann algebras","fieldGroup":"Analysis","statement":"For von Neumann algebras $M,N\\subseteq B(H)$ the Kadison-Kastler distance $d(M,N)$ is the Hausdorff distance between their unit balls in operator norm. Kadison and Kastler (1972) proposed that sufficiently close von Neumann algebras should be spatially isomorphic via a unitary close to the identity. This was proved for injective algebras (Christensen, Johnson, Raeburn-Taylor) and for certain crossed-product $\\mathrm{II}_1$ factors (Cameron et al.). Is there, for every $\\varepsilon>0$, a $\\delta>0$ such that any two von Neumann algebras on the same Hilbert space with $d(M,N)<\\delta$ satisfy $uMu^*=N$ for a unitary $u$ with $\\|u-1\\|<\\varepsilon$?","posedBy":"R. V. Kadison and D. Kastler, Perturbations of von Neumann algebras. I. Stability of type, Amer. J. Math. 94 (1972)","yearPosed":1972,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $\\varepsilon>0$ there is $\\delta(\\varepsilon)>0$, independent of the algebras, their type, representations and the Hilbert space, such that unital von Neumann algebras $M,N\\subseteq B(H)$ with $d(M,N)<\\delta$ satisfy $uMu^*=N$ with $\\|u-1\\|<\\varepsilon$. The proof combines amplification estimates, a modular and core analysis for type III, and reductions across types. It concerns von Neumann algebras only; the companions show that one-sided near inclusions and close separable C*-algebras behave differently. No explicit $\\delta(\\varepsilon)$ is given.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it with the 1972 reference; it claims a tolerance $\\delta(\\varepsilon)$ uniform over all unital von Neumann algebras, representations and Hilbert spaces. The proof was not refereed. The challenge ComparatorChallenges/StrongKadisonKastler.lean is not in lean/formalization.yaml; it is found through lean/docs/289.md and its solution module OAI.Analysis.KadisonKastler.StrongStability exists at the pinned commit. Its statement was read here: for every $\\varepsilon>0$ there is $\\delta>0$ such that for every complex Hilbert space $H$ and von Neumann algebras $M,N$ on $H$ with Hausdorff distance of unit balls below $\\delta$ there is a unitary $v$ with $vMv^*=N$ and $\\|v-1\\|<\\varepsilon$. This states the headline. Not rebuilt here. The paper cites the release's similarity companion for a related commutator estimate.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The Kadison-Kastler perturbation problem has organized decades of work in operator algebras since 1972 and is among the field's classical questions, a step below the similarity problem. Placed below Kadison's Similarity Problem (58) and the Kadison-Kaplansky Projection Conjecture (50), level with Connes' Rigidity Conjecture (45), above Voiculescu's free entropy dimension question (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Universal strong Kadison-Kastler stability","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf","kind":"paper"},{"label":"Close Separable C*-Algebras Without Spatial Conjugacy","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.KadisonKastler.universal_strong_stability)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/KadisonKastler/StrongStability.lean","kind":"lean-proof"},{"label":"Comparator statement: StrongKadisonKastler.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/StrongKadisonKastler.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"smith-toda-complexes-every-height","name":"Smith-Toda complexes V(n) at every height when the prime may vary","shortName":"Smith-Toda complexes at every height","problemNumber":null,"field":"Stable homotopy theory; chromatic homotopy","fieldGroup":"Geometry & topology","statement":"At a prime $p$, a Smith-Toda complex $V(n)$ is a finite $p$-local spectrum whose Brown-Peterson homology is $BP_*/(p,v_1,\\ldots,v_n)$ as a $BP_*BP$-comodule, each generator killed to its first power. Adams, Smith and Toda constructed $V(1)$, $V(2)$, $V(3)$ for $p>2,4,6$; Nave proved that $V((p+1)/2)$ does not exist for $p\\ge7$; and Culver-Zhang (2023 version) recorded the existence of $V(4)$ as open at every prime. Hopkins-Smith periodicity realizes suitable higher powers, not first powers. The Smith-Toda realization problem asks for which $n$ and $p$ the complex $V(n)$ exists. For every height $n$, does $V(n)$ exist at some prime $p$?","posedBy":"The Smith-Toda realization problem (Smith 1970, Toda 1971); V(4) recorded open by Culver and Zhang (2023 arXiv version)","yearPosed":1971,"ageNote":"The manuscript frames this as the Smith-Toda realization problem, citing Smith (1970) and Toda (1971), and does not name a poser for the all-height, varying-prime form; 1971 is the date of Toda's paper. The V(4) record is from the September 2023 version of Culver-Zhang.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $n\\ge0$ there are a prime $p$ and a finite $p$-local spectrum $X$ with $BP_*X\\cong BP_*/(p,v_1,\\ldots,v_n)$ as comodules with the canonical coaction and generator in degree zero. The companion constructs $V(4)$ at $p=1009$, answering the existence question Culver-Zhang recorded as open at every prime. Partial for the realization problem: no explicit or effective prime for $n\\ge5$, no statement that $V(n)$ exists for all large $p$, no multiplicative structure, and nothing on the gap between Nave's nonexistence range and the constructed cases.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. A companion of the same date constructs V(4) explicitly at p = 1009 by a direct obstruction computation.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript and the companion's main theorem were read against the realization problem. The principal theorem gives, for each $n$, a prime depending on $n$ (no bound on it), via an ultraproduct of $p$-local categories and descent to one prime; the companion gives $V(4)$ at $p=1009$. Neither proof was refereed here. The paper states that it does not claim one prime for all heights, a bound on the prime, or a ring structure. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Existence of Smith-Toda complexes is a classical question in stable homotopy theory, stuck at V(3) since the early 1970s; famous within homotopy theory. Scored as a partial answer to the realization problem. Placed above The Campana-Peternell conjecture on Fano manifolds with nef tangent bundle in dimension six (28), below Is every word-hyperbolic group CAT(0)? (45) and The Eilenberg-Ganea conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Finite Smith-Toda Complexes at Varying Primes","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A finite Smith-Toda complex V(4) at the prime 1009","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf","kind":"paper"},{"label":"Culver-Zhang, Exotic Picard groups and chromatic vanishing (V(4) recorded open)","url":"https://doi.org/10.1016/j.topol.2023.108742","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"finitely-presented-nil-algebra-nilpotent","name":"Must a finitely presented nil algebra be nilpotent? (Ufnarovskij's question)","shortName":"Finitely presented nil algebras","problemNumber":null,"field":"Noncommutative ring theory; nil and radical algebras","fieldGroup":"Algebra","statement":"An associative algebra $A$ without identity is nil if every element has some power equal to zero, and nilpotent if $A^N=0$ for one $N$. Golod's construction gives finitely generated nil algebras that are not nilpotent, and Lenagan, Smoktunowicz and Young built such algebras with restricted growth, but these are not finitely presented. Smoktunowicz records as Ufnarovskij's question whether a finitely presented nil algebra must be nilpotent; Amitsur asked the analogous question for finitely presented Jacobson radical algebras, and the finite-presentation version of Kurosh's algebraic finiteness problem asks the same for finitely presented algebraic algebras. Is every finitely presented nil algebra nilpotent?","posedBy":"Ufnarovskij, as recorded by Smoktunowicz (Bull. LMS 2008, Question 8.1); also Lenagan, Smoktunowicz and Young (2012, Section 9); radical form due to Amitsur","yearPosed":null,"ageNote":"The manuscript attributes the question to Ufnarovskij through Smoktunowicz (2008) and gives no original date.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Corollary 3.8: there is an infinite-dimensional finitely presented nonunital associative $\\mathbb F_2$-algebra that is nil and Jacobson radical but not nilpotent; its unitization is finitely presented, algebraic and infinite-dimensional. This answers Ufnarovskij's nil question, Amitsur's radical question and the finite-presentation version of Kurosh's algebraic finiteness question negatively, over $\\mathbb F_2$. Every matrix over its positive-degree ideal is nilpotent. It does NOT give examples over other fields or in characteristic zero, and nilpotence indices are unbounded.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. The nil algebra is the positive-degree ideal of the same finitely presented graded algebra used to build the periodic Steinberg group.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Corollary 3.8 and the introduction of the principal manuscript were read against the question: an infinite-dimensional finitely presented nonunital nil associative $\\mathbb F_2$-algebra that is Jacobson radical but not nilpotent, whose unitization is finitely presented, algebraic and infinite-dimensional. This part has no Lean formalisation: lean/docs/247.md says the nil-algebra and radical-algebra conclusions are outside the formalised statements. The release README warns that some unformalised results could have issues.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A natural and recorded question in the Kurosh-Golod line of noncommutative ring theory, known to specialists rather than widely. Below the Koethe Conjecture (60) and the finitely presented Burnside problem entry (60), level with An Explicit Counterexample to the Rank-Two Poisson Conjecture (35), above the periodicity conjecture for finite-dimensional algebras (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An infinite finitely presented periodic group","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An infinite finitely presented residually finite 2-group","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf","kind":"paper"},{"label":"Smoktunowicz (2008), The Jacobson radical of rings with nilpotent homogeneous elements","url":"https://doi.org/10.1112/blms/bdn086","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"perfect-matchings-fpras-general-graphs","name":"A fully polynomial randomized approximation scheme for counting perfect matchings in general graphs","shortName":"FPRAS for perfect matchings in general graphs","problemNumber":null,"field":"Approximate counting; Markov chain Monte Carlo","fieldGroup":"Theoretical computer science","statement":"Counting perfect matchings exactly is #P-complete even for bipartite graphs (Valiant 1979). Jerrum and Sinclair (1989) gave an FPRAS for all matchings and for perfect matchings only when the ratio of near-perfect to perfect matchings is polynomially bounded; Jerrum, Sinclair and Vigoda (2004) removed that restriction in the bipartite case (the permanent of a nonnegative matrix). Stefankovic, Vigoda and Wilmes showed that chains of the Jerrum-Sinclair-Vigoda type cannot work on some nonbipartite graphs. Is there a fully polynomial randomized approximation scheme for the number $Z(G)$ of perfect matchings of an arbitrary graph $G$, that is, a randomized algorithm returning $\\widehat Z$ with $(1-\\varepsilon)Z(G)\\le \\widehat Z\\le(1+\\varepsilon)Z(G)$ with probability at least $1-\\delta$, in time polynomial in the input size, $\\varepsilon^{-1}$ and $\\log\\delta^{-1}$?","posedBy":"Mark Jerrum and Alistair Sinclair, Approximating the permanent, SIAM J. Comput. 18 (1989), Section 7(i)","yearPosed":1989,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a uniform classical randomized algorithm that, for any finite simple undirected graph $G$ and rationals $0<\\varepsilon<1$, $0<\\delta<1/2$, returns $\\widehat Z\\ge0$ with $\\Pr[(1-\\varepsilon)Z(G)\\le\\widehat Z\\le(1+\\varepsilon)Z(G)]\\ge1-\\delta$, returns 0 with certainty when $Z(G)=0$, and runs in worst-case bit time polynomial in the input length, $\\varepsilon^{-1}$ and $\\log\\delta^{-1}$. Corollaries give counting schemes for edge sets with prescribed degrees and for matchings of a given size. It is for unweighted simple graphs: it does not claim an FPRAS for hafnians with arbitrary nonnegative weights, and the polynomial's degree is not optimised.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The algorithm and its analysis are the model's; the paper says it uses a different state space from the Jerrum-Sinclair-Vigoda chain (products of perfect-matching spaces on an enlarged colored graph) to get around the Stefankovic-Vigoda-Wilmes obstruction. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the question as Jerrum and Sinclair raised it (Section 7(i) of their 1989 paper, as the manuscript cites it). The proof was not refereed. Lean: the release's Comparator challenge MatchingFPRAS (theorem OAI.MatchingFPRAS.thm_main, solution module OAI.Combinatorics.MatchingCount.Main, both present at the pinned commit) is not in the release's formalization catalogue (lean/formalization.yaml); it is reached through lean/docs/113.md. Its statement was read here: a single finite randomized Post-Turing machine, on an explicit binary encoding of any finite simple graph and rationals $0<\\varepsilon<1$, $0<\\delta<1/2$, halts on every random tape within $C(\\text{input length}+\\lceil\\varepsilon^{-1}\\rceil+\\lceil\\log_2\\delta^{-1}\\rceil+1)^d$ steps with a nonnegative rational, outputs 0 on every tape when there is no perfect matching, and is within relative error $\\varepsilon$ on at least a $1-\\delta$ fraction of tapes. That is the headline claim. The development was not rebuilt here and the statement has not been independently audited.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"Whether perfect matchings in general (nonbipartite) graphs admit an FPRAS has been the standing open problem of approximate counting since Jerrum and Sinclair raised it in 1989, and the bipartite case won Jerrum, Sinclair and Vigoda the Fulkerson Prize. Level with The Matrix Multiplication Exponent (55), above The 3SUM hypothesis (50), below the L = RL = BPL derandomization question (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf","sourceName":"OpenAI math release: A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Same family: Entropy and Face Dimension of the Perfect-Matching Polytope","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf","kind":"paper"},{"label":"Lean Comparator challenge MatchingFPRAS (statement)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/MatchingFPRAS.lean","kind":"lean-proof"},{"label":"Lean solution module OAI.Combinatorics.MatchingCount.Main","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/MatchingCount/Main.lean","kind":"lean-proof"},{"label":"Jerrum and Sinclair 1989, Approximating the permanent","url":"https://doi.org/10.1137/0218077","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"blaschke-conjecture","name":"The Blaschke conjecture: Blaschke manifolds are compact rank-one symmetric spaces","shortName":"Blaschke conjecture","problemNumber":null,"field":"Riemannian geometry; manifolds all of whose geodesics are closed","fieldGroup":"Geometry & topology","statement":"A connected closed Riemannian manifold $(M,g)$ of positive dimension is a Blaschke manifold if its injectivity radius equals its diameter, so every unit-speed geodesic minimizes up to the diameter. Round spheres, the real, complex and quaternionic projective spaces and the Cayley plane with their standard metrics are Blaschke. Blaschke posed the problem for surfaces (settled by Green, 1963); the sphere and real projective cases were settled by Berger, Kazdan, Weinstein and Yang, and Bott-Samelson theory fixes the cohomology type, but the metric question was open for the complex, quaternionic and Cayley types. Is every Blaschke manifold isometric, up to scaling, to a compact rank-one symmetric space?","posedBy":"Wilhelm Blaschke (surfaces); general metric form in Arthur Besse's monograph (1978)","yearPosed":1921,"ageNote":"Blaschke's Vorlesungen uber Differentialgeometrie I (1921) for surfaces; the higher-dimensional metric conjecture is as described in Besse, Manifolds all of whose geodesics are closed (1978), with McKay's 2016 progress summary, all cited by the manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $(M,g)$ is connected, closed, smooth, of positive dimension, with $\\mathrm{inj}(M,g)=\\mathrm{diam}(M,g)$, then $(M,g)$ is isometric up to scale to a round sphere, a standard real, complex or quaternionic projective space, or the Cayley plane. Theorem 1.2 gives $\\mathrm{Vol}(M,g)\\ge\\mathrm{Vol}(M_0,g_0)$ against the model at equal diameter, with equality only for isometry; the upper bound in quaternionic dimension $4r$, $r\\ge2$, is proved from characteristic classes and the geodesic space. It does not treat the weaker Besse-type questions on manifolds all of whose geodesics are closed without the Blaschke condition (SC or P-manifolds in general).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 23, 2026) has no companion in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture; it covers every connected closed smooth Blaschke manifold of positive dimension, including all quaternionic dimensions and the Cayley plane, with conclusion isometry up to scale to a compact rank-one symmetric space. The argument is a sharp volume lower bound with rigidity (via Szabo's classification of harmonic manifolds) plus a matching upper bound from topology; it cites results of Yang, Kramer-Stolz and a theorem attributed to Reznikov via Wilking, whose hypotheses it says it verifies. Not refereed. No Lean formalization in the release.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The Blaschke conjecture is a century-old landmark problem of global Riemannian geometry, with major partial results by Berger, Kazdan, Weinstein and Yang. Placed level with the infinitude of closed geodesics on spheres (55) and the Mahler conjecture for general convex bodies (55), below the symmetric Mahler conjecture (60), and above Sidorenko's conjecture (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The metric Blaschke theorem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"McKay, Summary of progress on the Blaschke conjecture (arXiv 1309.1326)","url":"https://arxiv.org/abs/1309.1326v5","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sandier-serfaty-triangular-renormalized-energy","name":"The Sandier-Serfaty conjecture: the triangular lattice minimizes planar Coulomb renormalized energy","shortName":"Sandier-Serfaty conjecture","problemNumber":null,"field":"Coulomb gases; renormalized energy of Ginzburg-Landau vortex patterns","fieldGroup":"Mathematical physics","statement":"Sandier and Serfaty introduced a renormalized energy $W$ for infinite configurations of planar point charges against a uniform neutralizing background, defined through curl-free fields $E$ with $\\operatorname{div}E=2\\pi(\\nu-m)$, the logarithmic self-energy removed and energy taken per unit area. It governs microscopic vortex patterns in the Ginzburg-Landau model and limits of two-dimensional Coulomb gases. They proved that a minimum exists, that it is approximated by periodic fields, and that among Bravais lattices the triangular lattice is optimal (Theorems 1 and 2 of their 2012 paper). Their Conjecture 1 asks: is the triangular lattice the minimizer of $W$ over the full admissible class of fields, not only among lattices?","posedBy":"Etienne Sandier and Sylvia Serfaty (Conjecture 1, Comm. Math. Phys. 313, 2012)","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every admissible curl-free field with unit background, $W(E)\\ge W(E_\\triangle)$, and $\\inf W=W(E_\\triangle)$ is finite, with no separation, periodicity or Voronoi assumption on the competitor. Theorem 1.2 gives the finite version: on every square torus with $n\\ge2$ points the renormalized energy is at least $nW(E_\\triangle)$. The proof builds one trial potential from Voronoi sectors on a minimizing torus configuration, with an exact identity leaving a nonnegative square, plus interval-arithmetic estimates. It does not classify minimizers (compact changes keep the energy per area) and does not treat the Riesz or higher-dimensional cases.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof combines a Voronoi-sector trial-potential argument with a finite interval-arithmetic certificate (C++ with Boost.Multiprecision) that the manuscript's README says to run; it uses no universal-optimality result as input. The companion 'Universal optimality of the triangular lattice' (September 23, 2026) reaches the logarithmic jellium minimum by a different route through the Petrache-Serfaty implication.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against Sandier-Serfaty Conjecture 1. It states $W(E)\\ge W(E_\\triangle)$ for every cutoff family and every field in the admissible class $\\mathcal A_1$ (density-one normalization, which the manuscript matches to the conjecture's background $m=2\\pi$), with the triangular value finite and equal to the infimum. The proof was not refereed and the interval certificate was not re-run here. No Lean formalization in the release covers this manuscript (lean/docs/090.md lists only the universal-optimality and circle-packing statements). The manuscript itself notes the minimum statement does not classify all minimizing configurations.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"A well-known conjecture in the Coulomb-gas and Ginzburg-Landau community, open since 2012 and tied to crystallization questions in mathematical physics. Placed level with Singular Bernoulli convolutions at non-Pisot parameters (42) and Wall's manifold-realization question (42), below the ionization conjecture (55) and the universal optimality of the triangular lattice (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Triangular minimality for planar Coulomb renormalized energy","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Universal optimality of the triangular lattice (Section 8, jellium minima)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lex-plus-powers-conjecture-char-zero","name":"The lex-plus-powers conjecture over fields of characteristic zero","shortName":"Lex-plus-powers conjecture (char. 0)","problemNumber":null,"field":"Commutative algebra; graded Betti numbers and syzygies","fieldGroup":"Algebra","statement":"With $S=k[x_1,\\ldots,x_n]$, $2\\le a_1\\le\\cdots\\le a_c$ and $P=(x_1^{a_1},\\ldots,x_c^{a_c})$, let $I$ be a homogeneous ideal containing a regular sequence of degrees $a_1,\\ldots,a_c$, and let $L$ be the lex-plus-powers ideal (lex segment plus $P$) with the same Hilbert function, which exists if the Eisenbud-Green-Harris conjecture holds. The lex-plus-powers conjecture, attributed to Charalambous and Evans, strengthens Bigatti-Hulett-Pardue extremality of lex ideals: $L$ should have the largest graded Betti numbers. It was known for ideals containing $P$ itself (Mermin-Murai) and for fast-growing degrees in characteristic zero (Caviglia-Sammartano). Is $\\beta_{p,j}(S/I)\\le\\beta_{p,j}(S/L)$ for all $p,j$?","posedBy":"Attributed to Hara Charalambous and E. Graham Evans; first printed in work of Evans and Richert per Francisco-Richert (2007); public formulations in Richert (2004) and Caviglia-Sammartano (2018)","yearPosed":null,"ageNote":"The manuscript gives no year for the original Charalambous-Evans formulation; it cites Francisco-Richert (2007) for its origins and Richert (2004) as an early public statement. The conjecture dates to the 1990s by common account, which could not be confirmed from the sources read.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims the lex-plus-powers conjecture over every characteristic-zero field, for regular sequences of any length with degrees at least two and no growth condition, with no restriction on the other generators of $I$; this contains the Eisenbud-Green-Harris statement. The key new step is a Hilbert-matching monomial ideal with no smaller Betti numbers (Mermin-Murai Conjecture 10.2). It does NOT treat positive characteristic.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. This manuscript takes the division-coefficient construction from its same-day companion (the Eisenbud-Green-Harris entry) and adds a resolution transfer to a monomial ideal; the final monomial comparison follows Mermin-Murai and is included for completeness.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 (Artinian, over $\\mathbb C$) and Corollary 1.2 (every characteristic-zero field, regular sequences of any length) were read against the conjecture: the lex-plus-powers ideal exists, has the same Hilbert function and has graded Betti numbers at least those of $S/I$ in every homological and internal degree. The non-Artinian case uses the published Caviglia-Kummini reduction. Positive characteristic is not addressed. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"The lex-plus-powers conjecture is well known among specialists in syzygies and Hilbert functions but less famous than the Eisenbud-Green-Harris conjecture it strengthens; characteristic zero only. Scored below the Eisenbud-Green-Harris entry (40), level with the quantum geometric Langlands irrational-level entry (36), above the Period-Index entry (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Artinian Lex-Plus-Powers Betti Theorem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Commuting Division-Coefficient Forms and the Artinian Eisenbud-Green-Harris Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lebrun-salamon-conjecture","name":"The LeBrun-Salamon conjecture: positive quaternionic-Kahler manifolds and contact Fano manifolds are homogeneous","shortName":"LeBrun-Salamon conjecture","problemNumber":null,"field":"Differential and algebraic geometry; quaternionic-Kahler and contact Fano","fieldGroup":"Geometry & topology","statement":"A Riemannian manifold of dimension $4m$, $m\\ge2$, is quaternionic-Kahler if its holonomy lies in $Sp(m)Sp(1)$; it is positive if its scalar curvature is positive. Wolf showed that the compact symmetric examples (the Wolf spaces) correspond to the adjoint varieties of simple complex Lie algebras, and Salamon's twistor construction attaches to every closed positive quaternionic-Kahler manifold a complex contact Fano manifold. LeBrun and Salamon proved finiteness and strong rigidity results and conjectured that the symmetric examples are the only ones. The algebraic form, the contact-Fano homogeneity conjecture, asks the same of every smooth complex projective contact Fano manifold, which should be an adjoint variety $\\mathbb P(\\mathcal O_{\\min}(\\mathfrak g))$. Known cases covered real dimension 8, 12 and 16 and low complex dimensions, or required extra hypotheses (reductive automorphism group, nonnegative sectional curvature). Is every closed connected positive quaternionic-Kahler manifold of dimension at least 8 a Wolf space, and is every contact Fano manifold an adjoint variety?","posedBy":"Claude LeBrun and Simon Salamon","yearPosed":1994,"ageNote":"LeBrun and Salamon, Strong rigidity of positive quaternion-Kahler manifolds, Invent. Math. 118 (1994), cited by the manuscript as the setting of the homogeneity conjecture; the general algebraic formulation is cited from Buczynski (Doc. Math. 2010, Conjecture 1.1).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every smooth connected complex projective contact Fano manifold $(X,F)$ of dimension $2n+1$, $n\\ge1$, is contact-isomorphic to the adjoint variety $Y_{\\mathfrak g}$ of a simple complex Lie algebra, with $L\\simeq\\mathcal O(1)|_{Y_{\\mathfrak g}}$. Corollaries: every such manifold is Kahler-Einstein; every smooth projective contact manifold with $b_2=1$ is an adjoint variety and with $b_2\\ge2$ is $\\mathbb P(T^*Z)$ (via Kebekus-Peternell-Sommese-Wisniewski and Demailly); every closed connected positive quaternionic-Kahler manifold of real dimension $4m\\ge8$ satisfies $\\nabla Rm=0$ and is homothetic to a Wolf space. It says nothing about negative or zero scalar curvature, about noncompact or incomplete metrics, or about compact Kahler contact manifolds that are not projective beyond the jet criterion it states.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family is a single manuscript (September 23, 2026). It cites earlier claimed proofs by Yasukura and by Kobayashi (2008) and records that Yasukura later identified errors in his.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2-1.4 of the manuscript were read against the conjecture as posed. Theorem 1.1 states that every smooth connected complex projective contact Fano manifold of dimension $2n+1\\ge3$ is contact-isomorphic to an adjoint variety; Corollary 1.4 deduces that every closed connected positive quaternionic-Kahler manifold of real dimension $4m\\ge8$ is homothetic to a Wolf space. This is the conjecture in full. The proof (contact-line smoothing, a descended symmetric section on $X\\times X$, a jet criterion for contact transitivity) was not refereed, and there is no Lean formalization. Readers should know that at least two earlier claimed proofs of the unrestricted classification (Yasukura's preprint versions, which the author later withdrew in part) did not survive, which the manuscript itself records.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":56,"significanceNote":"The LeBrun-Salamon conjecture has been a central open problem on special holonomy since 1994, with the algebraic contact-Fano form studied across algebraic geometry; known cases stopped at real dimension 16. Placed below Cannon's Conjecture (62) and the Yau Uniformization Conjecture (58), and above the Generator problem for von Neumann algebras (50).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Contact Fano manifolds and the LeBrun-Salamon conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"LeBrun and Salamon, Strong rigidity of positive quaternion-Kahler manifolds (1994)","url":"https://doi.org/10.1007/BF01231528","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"falconer-distance-conjecture","name":"The Falconer distance conjecture","shortName":"Falconer distance conjecture","problemNumber":null,"field":"Geometric measure theory; harmonic analysis","fieldGroup":"Analysis","statement":"For $E\\subset\\mathbb R^d$ let $\\Delta(E)=\\{|x-y|:x,y\\in E\\}$ be its distance set. Falconer (1985) showed that $\\dim_H E>(d+1)/2$ forces $\\Delta(E)$ to have positive Lebesgue measure, and lattice-type examples show the threshold cannot go below $d/2$. Later thresholds were $4/3$ in the plane (Wolff), $d/2+1/3$ (Erdogan), $9/5$ in $\\mathbb R^3$ and further improvements via decoupling (Du, Guth, Ou, Wang, Wilson, Zhang; Du, Zhang), and $5/4$ in the plane for pinned distances (Guth, Iosevich, Ou, Wang). Does every compact $E\\subset\\mathbb R^d$, $d\\ge2$, with $\\dim_H E>d/2$ have $\\mathcal L^1(\\Delta(E))>0$?","posedBy":"Kenneth Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985)","yearPosed":1985,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every integer $d\\ge2$ and every compact $E\\subset\\mathbb R^d$ with $\\dim_H E>d/2$, $\\mathcal L^1(\\Delta(E))>0$. No regularity beyond the strict dimension bound is assumed. The paper does NOT treat the endpoint $\\dim_H E=d/2$, does NOT prove a pinned version (some $y$ with $\\mathcal L^1(\\Delta_y(E))>0$) at the threshold $d/2$. It states that it uses no decoupling theorem and no earlier distance theorem at an improved threshold; it does use the Orponen-Shmerkin incidence theorem.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 23, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against Falconer's question as the introduction states it; it is the full conjecture, every $d\\ge2$, compact $E$, strict threshold $d/2$, with no packing-dimension, Ahlfors-regularity or pinned hypothesis. The proof was not refereed. Lean: the formalization catalogue lists only the planar case (PlanarFalconer, OAI.PlanarFalconer.mainTarget_proved). The all-dimensions statement is the release's Comparator challenge FalconerAllDimensions (OAI.Falconer.falconer_distance_conjecture), whose solution module OAI.MeasureTheory.Falconer.Campaign123PlanarFurstenbergProof exists at the pinned commit; that challenge is not in the formalization catalogue. Its statement was read here and states Theorem 1.1 exactly: for all $d\\ge2$ and compact $E$ in EuclideanSpace with $d/2<\\dim_H E$, the distance set has positive volume. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The paper makes no endpoint claim and no single-pin claim at $d/2$.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"One of the central problems of geometric measure theory and harmonic analysis since 1985, worked on by Wolff, Erdogan, Guth and many others; here claimed in every dimension including the plane. Placed level with the Baum-Connes conjecture (60), above the four-dimensional Kakeya set conjecture (55), and just below the three-dimensional restriction result for positively curved surfaces (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Falconer distance conjecture in all dimensions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator statement FalconerAllDimensions (not in formalization.yaml main results)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/FalconerAllDimensions.lean","kind":"lean-proof"},{"label":"Lean Comparator statement PlanarFalconer (the catalogued planar case)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PlanarFalconer.lean","kind":"lean-proof"},{"label":"Lean scope note for family 073","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/073.md","kind":"other"},{"label":"Falconer 1985, Mathematika 32","url":"https://doi.org/10.1112/S0025579300010998","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"universal-f-infinity-group","name":"A universal group of type F-infinity containing every finitely presented group (F-infinity Higman embedding)","shortName":"Universal group of type F-infinity","problemNumber":null,"field":"Geometric group theory; finiteness properties and embedding theorems","fieldGroup":"Algebra","statement":"Higman's embedding theorem puts every finitely generated recursively presented group into a finitely presented group, and there is a universal finitely presented group containing all finitely presented ones. Fournier-Facio and Zaremsky asked for higher-dimensional versions (Question 1.3 of 'Finiteness properties and Higman's rope trick', 2026): can finitely generated recursively presented groups be embedded into groups of type $F_n$, in particular type $F_\\infty$, and they showed that their rope-trick groups fail for this. Is there a single group of type $F_\\infty$ into which every finitely presented group embeds?","posedBy":"Francesco Fournier-Facio and Matthew C. B. Zaremsky (Question 1.3, arXiv:2607.21727)","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 constructs one group $H$ of type $F_\\infty$ containing every finitely presented group; by Higman's theorem its finitely generated subgroups are exactly the finitely generated recursively presented groups (Corollary 1.2). No word-problem hypothesis is imposed. The classifying space need not be finite-dimensional, so type $F$ is not claimed, and $H$ is not claimed simple.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. It is the third of three Boone-Higman family manuscripts, all dated September 23, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against Fournier-Facio-Zaremsky Question 1.3 as the manuscript quotes it; the $F_\\infty$ form is answered positively. Lean: lean/formalization.yaml lists comparator UniversalFInfinity with declaration OAI.UniversalFInfinity.universal_group_of_type_FInfinity. The statement UniversalFInfinity.lean was read here: there is a group H of type $F_\\infty$ (a Hausdorff connected CW complex with finitely many cells in each dimension, fundamental group H and a contractible covering space) such that every finitely presented group in the same universe embeds injectively in H. This states Theorem 1.1; the converse about recursively presented subgroups is not included. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A question stated in a July 2026 preprint, two months before it was answered; natural and clean but with little history. Placed level with the Arnold corank problem (20) and Effective log Iitaka fibration with finite coefficients (20), below Simple F-infinity overgroups (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A universal group of type F-infinity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.UniversalFInfinity.universal_group_of_type_FInfinity)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/UniversalGroup/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: UniversalFInfinity.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/UniversalFInfinity.lean","kind":"lean-proof"},{"label":"Fournier-Facio and Zaremsky, Finiteness properties and Higman's rope trick","url":"https://arxiv.org/abs/2607.21727","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-distinct-distances-higher-dimensions","name":"The higher-dimensional Erdos distinct-distances conjecture","shortName":"Higher-dimensional distinct distances","problemNumber":null,"field":"Combinatorial geometry; incidence geometry","fieldGroup":"Combinatorics","statement":"For a finite set $P\\subset\\mathbb R^d$ let $\\Delta(P)=\\{|p-q|:p\\ne q\\in P\\}$. Erdos introduced the distinct-distances problem in 1946; the integer grid $\\{1,\\ldots,t\\}^d$ shows that $n$ points can determine only $O(n^{2/d})$ distances when $d\\ge3$. Known lower bounds fell short of this exponent: $n^{0.5643}$ and $n^{2/d-2/(d(d+2))}$ (Solymosi-Vu), $n^{3/5}$ up to logarithms in $\\mathbb R^3$ via Guth-Katz, and $n^{2/3-o(1)}$ in $\\mathbb R^3$ (Tidor-Yu-Zakharov). For every fixed $d\\ge3$, is there $c_d>0$ such that every set of $n\\ge2$ points in $\\mathbb R^d$ determines at least $c_dn^{2/d}$ distinct distances?","posedBy":"Paul Erdos (1946, 'On sets of distances of n points'); the higher-dimensional form as recorded by Bardwell-Evans-Sheffer and Tidor-Yu-Zakharov","yearPosed":1946,"ageNote":"The manuscript says Erdos introduced the problem in 1946 and cites Bardwell-Evans-Sheffer (2019) and Tidor-Yu-Zakharov (2026) for the higher-dimensional conjecture; it does not quote the higher-dimensional form from the 1946 paper itself.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every integer $d\\ge3$ there is $c_d>0$ such that every set of $n\\ge2$ distinct points in $\\mathbb R^d$ determines at least $c_dn^{2/d}$ distinct distances, matching the integer grid up to the constant. No dependence of $c_d$ on $d$ is made explicit, nothing is claimed for pinned distances (many distances from one point), and the planar problem, whose conjectured order is $n/\\sqrt{\\log n}$, is untouched beyond reusing the Guth-Katz bound. Priority: Aksoy Yazici earlier claimed the same constant-factor bound in all dimensions $d\\ge3$, but withdrew the manuscript because a step was wrong; it is not an opposite claim.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript; it proves its concentration theorem and the needed incidence bounds, including the planar Guth-Katz input, in appendices.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture; it is the full constant-factor statement $|\\Delta(P)|\\ge c_dn^{2/d}$ for every $d\\ge3$ with no position assumption, not an $n^{2/d-o(1)}$ bound. The proof runs a minimal-counterexample induction on dimension through rigid-motion flats, Hilbert function estimates and approximate complete intersections; it was not refereed here. The paper itself notes an earlier claim of the same bound (Aksoy Yazici) that was withdrawn as incorrect. No Lean formalization exists for this family.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The distinct-distances problem is one of Erdos's most famous; after Guth-Katz settled the plane up to a logarithm, the higher-dimensional conjecture was a prominent open problem of combinatorial geometry with partial exponents only. Placed above Erdos's Planar Unit Distance Conjecture (40) and far above Erdos Problem #654 (10), below The Falconer distance conjecture (60).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The higher-dimensional Erdos distinct-distances conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Erdos, On sets of distances of n points (1946)","url":"https://doi.org/10.1080/00029890.1946.11991674","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"l-equals-rl-equals-bpl","name":"Derandomization of logarithmic space: is L = RL = BPL?","shortName":"L = RL = BPL","problemNumber":null,"field":"Space-bounded derandomization","fieldGroup":"Theoretical computer science","statement":"$\\mathsf L$ is the class of languages decided in deterministic logarithmic space. $\\mathsf{RL}$ allows a polynomial-time logarithmic-space machine fresh random bits with one-sided error (acceptance probability $0$ on no-inputs, at least $1/2$ on yes-inputs), and $\\mathsf{BPL}$ allows two-sided bounded error ($\\le1/3$ versus $\\ge2/3$). Aleliunas, Karp, Lipton, Lovasz and Rackoff, after giving a randomized logspace algorithm for undirected connectivity, asked whether polynomial-time randomized logarithmic space can be simulated in deterministic logarithmic space. The best unconditional results before this work were Saks-Zhou space $O(\\log^{3/2}n)$, improved by Hoza to $O(\\log^{3/2}n/\\sqrt{\\log\\log n})$, and Nisan's simultaneous polynomial time with $O(\\log^2n)$ space. Is $\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}$?","posedBy":"R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lovasz and C. Rackoff (1979)","yearPosed":1979,"ageNote":"The manuscript credits Aleliunas, Karp, Lipton, Lovasz and Rackoff (FOCS 1979) with asking the question explicitly; Gill (1977) had earlier set up the theory of probabilistic machines.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims $\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}$: every language decided with bounded one-sided or two-sided error by a randomized machine with polynomial worst-case running time and $O(\\log n)$ space is decided by a deterministic $O(\\log n)$-space machine. Quantitatively, acceptance probabilities can be approximated to within $2^{-q}$ in space $O(\\log(n+2)+q)$ and time $(n+2)^{O(1)}2^{O(q)}$. The method builds a hierarchy of copies of the configuration graph with a shared finite random environment and evaluates a family of $O(\\log n)$-bit estimators, more than three quarters of them accurate, by exhaustive enumeration. It does not give a pseudorandom generator with logarithmic seed, and it does not address randomized logspace without a polynomial time bound.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, which was posed some 4,000 open research problems during an evaluation, with the vast majority of results obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each. This manuscript is not among the README's named exceptions. It is credited to OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the posed question and states the full equality L = RL = BPL for the standard model (polynomial worst-case running time on every random tape, logarithmic space counting all live counters). Lean-checked on the release's Comparator challenge LogspaceEquality (declaration OAI.ExactDerandomization.exact_logarithmic_space_derandomization, solution module OAI.Computability.Logspace.Equality), added in the 7 October 2026 update and listed in lean/formalization.yaml. Its statement was read here: for finite transition-table machines over binary words, with read-only input heads, binary work tapes, one coin bit per step and space at most $c\\lceil\\log_2(n+2)\\rceil$ on every coin tape, the classes L, RL (polynomial clock, $\\ge1/2$ versus $0$) and BPL ($\\ge2/3$ versus $\\le1/3$) satisfy L = RL and RL = BPL. That is the headline. Not rebuilt here. The paper's quantitative approximation ($2^{-q}$ accuracy in space $O(\\log n+q)$) and its explicit compiler are not part of the Comparator statement. The paper concerns polynomial-time randomized logspace; it does not claim that randomized logspace machines without a time bound are derandomized.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":65,"significanceNote":"Whether randomness helps small-space computation is one of the central questions of complexity theory, open for nearly fifty years and resistant to the pseudorandom-generator program; widely believed true, which keeps it below the great separations. Scored with the Jacobian Conjecture at 65, above The APSP hypothesis and The Matrix Multiplication Exponent at 55, and above Bipartite Exact Matching in P at 25.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Exact Derandomization of Logarithmic Space: L = RL = BPL","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Lean Comparator challenge LogspaceEquality (OpenAI math release)","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/LogspaceEquality.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gromov-lawson-aspherical-conjecture","name":"The Gromov-Lawson conjecture: closed aspherical manifolds admit no metric of positive scalar curvature","shortName":"Gromov-Lawson aspherical conjecture","problemNumber":null,"field":"Scalar curvature; topology of manifolds","fieldGroup":"Geometry & topology","statement":"A closed manifold is aspherical if its universal cover is contractible. Gromov and Lawson (1983), with Dirac-operator methods, and Schoen and Yau, with minimal hypersurfaces, obstructed positive scalar curvature on tori and enlargeable manifolds; the conjecture that no closed aspherical manifold carries a metric of positive scalar curvature was open beyond dimension five (Chodosh-Li, Gromov). Hanke (2011, Conjecture 2.9) formulated the stronger homological form: if a closed oriented $n$-manifold $M$ admits positive scalar curvature, the classifying map sends its rational fundamental class to zero in $H_n(B\\pi_1(M);\\mathbb Q)$. Does any closed aspherical manifold admit a metric of positive scalar curvature, and are positive-scalar-curvature manifolds always rationally inessential?","posedBy":"M. Gromov and H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. IHES 58 (1983); homological form: B. Hanke (2011), Conjecture 2.9","yearPosed":1983,"ageNote":"The manuscript cites Gromov-Lawson 1983 for the obstruction programme and names the corollary the Gromov-Lawson aspherical conjecture; the homological form is Hanke 2011, Conjecture 2.9, also stated by Yau (2000) and Gromov (2023).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if a closed connected oriented smooth $M^n$, $n\\ge2$, admits positive scalar curvature then $(c_M)_*[M]=0$ in $H_n(B\\pi_1(M);\\mathbb Q)$. Corollaries: no closed aspherical manifold (orientable or not) admits positive scalar curvature; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat; a closed oriented manifold mapping with nonzero degree to an aspherical one has no positive scalar curvature; and nonnegative scalar curvature forces zero real simplicial volume. Prior cases: dimensions 4 and 5 (Chodosh-Li, Gromov). It does not give the quantitative scalar-curvature/simplicial-volume inequality (that is the companion entry).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and the aspherical corollaries of the TeX source, read against Hanke's Conjecture 2.9 and the aspherical conjecture as cited. Theorem 1.1 states rational inessentiality for every closed connected oriented smooth manifold of dimension at least 2 with positive scalar curvature, with no spin, fundamental-group or dimension hypothesis; the aspherical corollary covers nonorientable manifolds via the double cover. The proof uses the local torical theorem of Cecchini-Schick as its terminal input and a new coordinate-shortening scheme with circle stabilization. Not refereed here. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"The aspherical conjecture is one of the central problems of scalar-curvature geometry, open for four decades and only recently settled in dimensions four and five by Chodosh-Li and Gromov. Proving it in all dimensions, with Hanke's stronger homological form, would be widely called a landmark. Placed level with the Borel conjecture in dimension four (60), below Cannon's conjecture (62) and above the Eilenberg-Ganea conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Positive scalar curvature forces rational inessentiality","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"An integral scalar curvature bound for real simplicial volume (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf","kind":"paper"},{"label":"Gromov-Lawson 1983, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds","url":"https://doi.org/10.1007/BF02953774","kind":"problem-record"},{"label":"Independent route via Gromov's codimension-two width conjecture (OpenAI math release)","url":"https://vibemathed.com/problem/gromov-codimension-two-width-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"seymour-second-neighborhood-conjecture","name":"Seymour's second neighborhood conjecture","shortName":"Seymour's second neighborhood conjecture","problemNumber":null,"field":"Directed graphs; oriented graphs and tournaments","fieldGroup":"Combinatorics","statement":"An oriented graph is a finite digraph with no loops, multiple arcs or pairs of opposite arcs. For a vertex $v$ let $N_1^+(v)$ be its out-neighbours and $N_2^+(v)$ the vertices at directed distance exactly two. Seymour conjectured that every nonempty oriented graph has a vertex with $|N_1^+(v)|\\le|N_2^+(v)|$. The tournament case (Dean's conjecture) was proved by Fisher (1996); Kaneko and Locke proved it for minimum outdegree at most six, and later work treated tournaments missing a matching, star or clique, dense cases, and random orientations, with a best universal ratio of about 0.7155 (Huang-Peng). Does every nonempty finite oriented graph have such a vertex?","posedBy":"P. Seymour; recorded by N. Dean and B. J. Latka, Squaring the tournament - an open problem, Congressus Numerantium 109 (1995)","yearPosed":1995,"ageNote":"The manuscript cites Dean and Latka (1995) as the record of the conjecture and does not date Seymour's own statement; it is usually placed around 1990, so the problem is somewhat older than the year given.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every nonempty finite oriented graph has a vertex $v$ with $|N_1^+(v)|\\le|N_2^+(v)|$, with no connectivity, density or degree assumption. Via Seacrest's equivalences it also gives the vertex- and arc-weighted forms, and it yields short-cycle consequences, for example a directed triangle whenever minimum in- and outdegree are at least one third of the order. It does not address infinite digraphs or digraphs with 2-cycles, which the conjecture excludes.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The main theorem has a Lean formalization in the release (Comparator challenge SeymourSecondNeighborhood).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as recorded by Dean and Latka. Lean-checked on the release's Comparator challenge SeymourSecondNeighborhood (declaration OAI.SeymourSecondNeighborhood.exists_goodVertex, listed in lean/formalization.yaml). Its statement was read here: for every nonempty finite type with a loopless asymmetric relation, some vertex has at most as many first out-neighbours as second neighbours, where second neighbours exclude the vertex itself and its out-neighbours and are reached by a two-step path. That is exactly the headline claim, with no connectivity or degree hypothesis. Not rebuilt here. The minimal-counterexample and pruning argument was not refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A well-known conjecture in digraph theory, open for about three decades with a steady literature (Fisher's tournament theorem, Kaneko-Locke, dense-case and ratio results). Famous within graph theory, less so outside. Below the Petersen Coloring Conjecture (40), above Gyarfas's tree cover conjecture (32) and the two existing partial entries on this conjecture (32 and 30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A proof of Seymour's second-neighborhood conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge SeymourSecondNeighborhood (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SeymourSecondNeighborhood.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"yau-nodal-set-conjecture-smooth","name":"Yau's nodal-set conjecture for smooth metrics: is nodal measure comparable to sqrt(lambda)?","shortName":"Yau's nodal-set conjecture (smooth)","problemNumber":null,"field":"Spectral geometry; nodal sets of Laplace eigenfunctions","fieldGroup":"Geometry & topology","statement":"Let $(M,g)$ be a smooth closed connected Riemannian manifold of dimension $n$ and $u$ a real eigenfunction, $-\\Delta_gu=\\lambda u$ with $\\lambda>0$. Yau (1982) conjectured that the nodal set $Z_u$ satisfies $c_g\\sqrt\\lambda\\le\\mathcal H^{n-1}(Z_u)\\le C_g\\sqrt\\lambda$ with constants depending only on $(M,g)$. Donnelly and Fefferman proved both bounds for real-analytic metrics. For smooth metrics Logunov proved the lower bound in every dimension and a polynomial upper bound; on smooth surfaces the best upper bound was $C\\lambda^{3/4-\\beta}$ (Logunov-Malinnikova). Does the two-sided bound $\\mathcal H^{n-1}(Z_u)\\asymp\\sqrt\\lambda$ hold for every smooth metric, in particular the upper bound?","posedBy":"Shing-Tung Yau","yearPosed":1982,"ageNote":"Yau, Problem section, in Seminar on Differential Geometry, Annals of Mathematics Studies 102 (Princeton, 1982), pp. 669-706, cited by all three manuscripts.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Dimensions three and four (Theorems 1.1, 1.2 of the principal manuscript): a smooth metric arbitrarily close to the round metric on $S^3$, and one on $S^2\\times T^2$, with exact eigenfunctions whose nodal measure divided by $\\sqrt\\lambda$ is unbounded. Dimension five and up (companion): a metric on $S^4\\times S^1$ with nodal measure $\\gg\\lambda^{1/2+\\epsilon_0}$ for a fixed $\\epsilon_0>0$, and products give every dimension above five. Surfaces (companion): $\\mathcal H^1(Z_u)\\le C(M,g)\\sqrt\\lambda$, so with Bruning's lower bound Yau's conjecture holds in dimension two. So the upper bound is true for surfaces and false for smooth metrics in every dimension $\\ge3$; the lower bound (Logunov) is untouched, and real-analytic metrics are unaffected. No power excess is claimed in dimensions 3 and 4.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has three manuscripts, all dated September 23, 2026: the principal one constructs counterexamples in dimensions three and four, a companion gives power-law violations in dimension five and above, and a second companion proves the upper bound on every smooth surface.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorems of all three manuscripts were read against Yau's conjecture as they state it (fixed smooth metric, all eigenfunctions). Proofs not refereed. lean/formalization.yaml lists SmoothYau (OAI.YauCounterexamples.sphere_three and sphere_two_torus_two) and YauCounterexample (OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample). Their comparator statements were read: in every smooth neighborhood of the round metric on $S^3$ there is a metric with an eigenfunction sequence, $\\lambda_j\\to\\infty$, whose finite nodal measure over $\\sqrt{\\lambda_j}$ tends to infinity; likewise for one metric on $S^2\\times T^2$; and for one metric on $S^4\\times S^1$ the ratio of 4-dimensional nodal measure to $\\sqrt\\lambda$ tends to infinity. The last does not state the paper's power-law excess $\\lambda^{1/2+\\epsilon_0}$. The surface bound has a challenge, NodalLength.json (solution OAI.Analysis.NodalLength.Main present), not in the catalogue; NodalLength.lean states $\\mathcal H^1(Z_u)\\le C\\sqrt\\lambda$ on every compact connected smooth Riemannian surface. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Yau's nodal conjecture is one of the best known problems in spectral geometry; work on it (Donnelly-Fefferman, Logunov, Logunov-Malinnikova) drew major prizes. Placed level with Yau's uniformization conjecture (58) and the a.e. regularity of stationary varifolds (58), below the Falconer distance conjecture (60), and above Schiffer's Conjecture (53).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Smooth counterexamples to Yau's nodal upper bound in dimensions three and four","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Power-law violations of Yau's nodal upper bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf","kind":"paper"},{"label":"Companion: Sharp nodal length on smooth surfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean proof, S^3 (OAI.YauCounterexamples.sphere_three)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/SmoothYau/SphereMetric/ThreeSphere.lean","kind":"lean-proof"},{"label":"Lean proof, S^2 x T^2 (OAI.YauCounterexamples.sphere_two_torus_two)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/SmoothYau/ProductMetric/SphereProduct.lean","kind":"lean-proof"},{"label":"Lean proof, S^4 x S^1 (OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/NodalSets/Main.lean","kind":"lean-proof"},{"label":"Lean proof, surfaces (OAI.SharpNodal.Main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/NodalLength/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SmoothYau.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SmoothYau.lean","kind":"lean-proof"},{"label":"Comparator statement: YauCounterexample.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/YauCounterexample.lean","kind":"lean-proof"},{"label":"Comparator statement: NodalLength.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/NodalLength.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"harary-hill-conjecture","name":"The Harary-Hill conjecture on the crossing number of complete graphs","shortName":"Harary-Hill conjecture","problemNumber":null,"field":"Topological graph theory; crossing numbers","fieldGroup":"Combinatorics","statement":"The crossing number $\\mathrm{cr}(G)$ is the least number of crossing points in a drawing of $G$ in the plane, with vertices at distinct points and edges as simple arcs meeting in finitely many proper crossings. Hill's construction (with earlier upper bounds by Guy, 1960) draws $K_n$ with $Z(n)=\\frac14\\lfloor\\frac n2\\rfloor\\lfloor\\frac{n-1}2\\rfloor\\lfloor\\frac{n-2}2\\rfloor\\lfloor\\frac{n-3}2\\rfloor$ crossings, and Harary and Hill (1963) conjectured that this is optimal. It was verified only for $n\\le12$ (and reported for $n\\le14$), for restricted drawing classes, and asymptotically up to a factor of about $0.986$. Is $\\mathrm{cr}(K_n)=Z(n)$ for every $n$?","posedBy":"Frank Harary and Anthony Hill, On the number of crossings in a complete graph (Proc. Edinburgh Math. Soc., 1963); upper bound earlier in Guy (1960)","yearPosed":1963,"ageNote":"The manuscript cites Harary-Hill (1963) for the conjecture and Guy's 1960 note for the first published upper bound; the conjecture is also often called Guy's conjecture.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims $\\mathrm{cr}(K_n)=\\frac14\\lfloor\\frac n2\\rfloor\\lfloor\\frac{n-1}2\\rfloor\\lfloor\\frac{n-2}2\\rfloor\\lfloor\\frac{n-3}2\\rfloor$ for every $n$, by a lower bound from signed intersections of cycles and an even-node order detector, plus a matching two-page drawing. Crossings are counted as points, every repeated crossing of an edge pair included. It does NOT address the rectilinear crossing number, the pair crossing number or the odd crossing number, which are different parameters.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. The manuscript is credited to OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem: $\\mathrm{cr}(K_n)=Z(n)$ for all $n\\ge3$ under the ordinary point-count convention, with unrestricted vertex positions and edge routes. formalization.yaml lists OAI.Paper170.complete_graph_crossing_number (comparator CompleteCrossing) as the main result. Its statement defines admissible drawings in $\\mathbb R^2$ (injective vertices, injective continuous edge paths whose interiors avoid vertices, finitely many crossing points, each locally homeomorphic to two axes, no triple points), counts crossing points, takes the infimum over drawings, and asserts it equals the floor-product formula for $n\\ge3$. That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. Hebbar and Mangam (2018, Int. J. Eng. Technol.) published earlier claims of both formulas, which the manuscripts cite and treat as upper-bound constructions only. ","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The central open problem on crossing numbers for six decades, settled only up to n = 12 or 14 by computer and asymptotically to about 98.6 percent by flag algebras. Tied with its sibling Zarankiewicz conjecture, above the uniform Turan density of the tetrahedron at 48 and Erdos Problem 571 at 52, below Erdos Problem 3 at 65 and the Hadwiger Conjecture at 70.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The crossing number of complete graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The crossing number of complete bipartite graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean: main declaration complete_graph_crossing_number","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/CompleteCrossing/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: crossing number of complete graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CompleteCrossing.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/165.md","kind":"lean-proof"},{"label":"Harary and Hill (1963)","url":"https://doi.org/10.1017/S0013091500025645","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sinai-standard-map-positive-entropy","name":"Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters","shortName":"Sinai's standard map entropy conjecture","problemNumber":null,"field":"Dynamical systems and ergodic theory","fieldGroup":"Analysis","statement":"The standard map $f_k(x,y)=(x+y+k\\sin(2\\pi x),\\,y+k\\sin(2\\pi x))$ on $\\mathbb T^2$ preserves area $m$ and was introduced by Chirikov as a model of conservative chaos. For large $k$ it has hyperbolic sets (Duarte), full-dimension transitive sets for residual parameters (Gorodetski) and positive topological entropy (Oliveira), but also elliptic islands, and none of this gives positive area. Sinai's conjecture, in the formulation of Berger and Turaev (2019, Conjecture 0.4), asks: is the metric entropy $h_m(f_k)$, equivalently a positive Lyapunov exponent on positive area, positive for a set of parameters $k$ of positive Lebesgue measure?","posedBy":"Yakov Sinai; the manuscript cites the formulation of P. Berger and D. Turaev, On Herman's positive entropy conjecture, Adv. Math. 349 (2019), Conjecture 0.4","yearPosed":null,"ageNote":"The manuscript attributes the conjecture to Sinai but cites no work of Sinai or year for it; it cites Berger and Turaev (2019) for the formulation and Obata (2021) for the large-parameter form. The standard map itself is cited from Chirikov (1979).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is $k_0>0$ with $h_m(f_k)>0$ for every $k\\ge k_0$; equivalently, by Pesin's formula, the top Lyapunov exponent is positive on a set of positive area. Corollary 1.2: a positive-area ergodic component with nonzero exponents that splits into finitely many cyclic Bernoulli pieces. It does NOT claim ergodicity of area, positive exponents almost everywhere, a uniform lower bound on the entropy, or any explicit $k_0$, and says nothing about small or moderate $k$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 23, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the TeX source read against Sinai's conjecture as cited; positivity for every $k\\ge k_0$ is stronger than the positive-measure parameter set the conjecture asks for. The proof was not refereed. Lean: three Comparator challenges, StandardMapEntropy (OAI.StandardMapEntropy.main_entropy), StandardMapLyapunov and StandardMapComponents, with solution modules present at the pinned commit; none is in the formalization catalogue. The StandardMapEntropy statement was read here: it defines the same map and normalized area on the torus and asserts some $k_0>0$ with positive metric entropy for all $k\\ge k_0$, which is the headline. Metric entropy is defined in the challenge itself (finite measurable partitions, block entropies), since Mathlib has none, so that definition is the prover's own. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"Whether the standard map is chaotic on positive area for many parameters is one of the best-known open problems of smooth ergodic theory, with decades of partial structure but no positive-area result. Placed level with the amenability of Thompson's group F (58) and Kadison's similarity problem (58), above the four-dimensional Kakeya set conjecture (55), and below the Hilbert-Smith conjecture (65).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Positive Metric Entropy for the Standard Map at Large Parameters","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator statement StandardMapEntropy (not in formalization.yaml main results)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/StandardMapEntropy.lean","kind":"lean-proof"},{"label":"Lean Comparator statement StandardMapLyapunov","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/StandardMapLyapunov.lean","kind":"lean-proof"},{"label":"Lean Comparator statement StandardMapComponents","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/StandardMapComponents.lean","kind":"lean-proof"},{"label":"Lean scope note for family 146","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/146.md","kind":"other"},{"label":"Berger and Turaev 2019, Adv. Math. 349 (Conjecture 0.4)","url":"https://doi.org/10.1016/j.aim.2019.04.002","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"explicit-nonbipartite-ramanujan-graphs-every-degree","name":"Explicit nonbipartite Ramanujan graphs of every degree","shortName":"Explicit Ramanujan graphs, every degree","problemNumber":null,"field":"Spectral graph theory; expanders","fieldGroup":"Combinatorics","statement":"A $d$-regular graph is Ramanujan if every nonconstant adjacency eigenvalue $\\lambda$ satisfies $|\\lambda|\\le2\\sqrt{d-1}$, the Alon-Boppana optimum. Lubotzky, Phillips and Sarnak and Margulis built explicit Ramanujan families only for $d=p+1$, Morgenstern extended this to $d=q+1$ for prime powers $q$, and Marcus, Spielman and Srivastava proved existence of bipartite Ramanujan graphs of every degree. Existence of nonbipartite ones in every degree now follows from random-graph universality (Huang, McKenzie, Yau), but non-constructively. For every fixed $d\\ge3$, can nonbipartite $d$-regular Ramanujan graphs be constructed deterministically, in polynomial time, at every large size?","posedBy":"Alexander Lubotzky, Ralph Phillips and Peter Sarnak","yearPosed":1988,"ageNote":"The manuscript cites Lubotzky-Phillips-Sarnak (1988) and Morgenstern (1994); Morgenstern's abstract describes his construction as a partial solution to a problem raised by Lubotzky, Phillips and Sarnak. The manuscript itself does not quote a posing sentence.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $d\\ge3$ there are $n_0(d)$, $k_d$ and a deterministic algorithm that, on every even $n\\ge n_0(d)$, outputs the adjacency list of a simple connected nonbipartite $d$-regular graph on $n$ vertices whose nonconstant eigenvalues lie strictly between $\\pm2\\sqrt{d-1}$, in $O_d(n^{k_d})$ bit operations. The construction uses partial pairings, deterministic resolvent estimates, cleanup and an exact spectral repair step. It is not strongly explicit (no local adjacency queries), and the exponent may grow with $d$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 23, 2026.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Section 1.1 were read against the posed problem. The theorem gives, for each fixed $d\\ge3$, a deterministic algorithm that on even $n\\ge n_0(d)$ outputs a simple $d$-regular graph on $n$ vertices in $O_d(n^{k_d})$ bit operations with every nonconstant eigenvalue strictly inside $(-2\\sqrt{d-1},2\\sqrt{d-1})$. Scope limits stated by the paper: the exponent and threshold depend on $d$, running time is not polynomial jointly in $n$ and $d$, there is no local neighbour-query algorithm, and the Bilu-Linial signing conjecture is not resolved. Existence alone was already known from Huang-McKenzie-Yau. No Lean formalization exists for this family. The proof was not refereed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"Explicit Ramanujan graphs beyond prime-power-plus-one degrees have been sought since 1988, but the existence half fell to random-graph universality in 2024, leaving the deterministic construction as the open part. Placed level with the Bilu-Linial signing conjecture (35), which is the classical route to the same goal, and below the 3SUM hypothesis (50) and Subset Sum below 2^(n/2).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Deterministic nonbipartite Ramanujan graphs in every fixed degree","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"boone-higman-simple-f-infinity-overgroups","name":"Simple overgroups of type F-infinity for groups with solvable word problem (higher-finiteness Boone-Higman)","shortName":"Simple F-infinity overgroups","problemNumber":null,"field":"Geometric group theory; finiteness properties and simple groups","fieldGroup":"Algebra","statement":"A group has type $F_\\infty$ if it has a classifying space $K(H,1)$ with finitely many cells in each dimension; type $F_2$ is finite presentation. The Boone-Higman conjecture asks for a finitely presented simple overgroup of every finitely generated group with solvable word problem. In the discussion following Question 5.6 of their survey 'Progress around the Boone-Higman conjecture', Belk, Bleak, Matucci and Zaremsky raise the higher-finiteness strengthening. Does every finitely generated group with solvable word problem embed in a simple group of type $F_\\infty$?","posedBy":"James Belk, Collin Bleak, Francesco Matucci and Matthew Zaremsky (survey, arXiv:2306.16356, discussion after Question 5.6)","yearPosed":2025,"ageNote":"The manuscript cites version 3 of the survey (22 May 2025); the survey was first posted in 2023, and whether the question appears in that first version was not checked here.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: every finitely generated group with decidable word problem embeds in a nontrivial simple group of type $F_\\infty$; with the classical converse, these are exactly the finitely generated subgroups of simple groups of type $F_\\infty$. This implies the Boone-Higman conjecture. The classifying space may be infinite-dimensional; no torsion-freeness, finite cohomological dimension or type $F$ is claimed.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. It is the second of three Boone-Higman family manuscripts, all dated September 23, 2026.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the survey's question; it gives an injective homomorphism of any finitely generated group with decidable word problem into a nontrivial simple group of type $F_\\infty$. Lean: lean/ComparatorChallenges/SimpleOvergroups.json exists with solution_module OAI.GroupTheory.SimpleOvergroups.Main, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement SimpleOvergroups.lean was read here: type $F_\\infty$ is a Hausdorff connected CW complex of finite type with the group as fundamental group and a contractible covering space; the theorem gives a nontrivial simple group of that type with an injective homomorphism from G. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A recent strengthening raised in a survey rather than a long-standing named problem, though it contains the Boone-Higman conjecture. Scored as a question within one community: level with the Kraetzer integral means conjecture (25), above the Stable Coordinate Conjecture in dimension five (22), well below the Boone-Higman conjecture (50) whose standing comes from the base case.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Simple F-infinity overgroups of groups with decidable word problem","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.SimpleFInftyOvergroups.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/SimpleOvergroups/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SimpleOvergroups.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SimpleOvergroups.lean","kind":"lean-proof"},{"label":"Related in family: Finite algebraic envelopes and the Boone-Higman conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Belk-Bleak-Matucci-Zaremsky, Progress around the Boone-Higman conjecture","url":"https://arxiv.org/abs/2306.16356","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"polyakov-conjecture-2d-on-exponential-decay","name":"Polyakov's mass-generation conjecture: exponential decay at every temperature in the two-dimensional O(n) model, n at least 3","shortName":"Polyakov's conjecture for 2D O(n) models","problemNumber":null,"field":"Statistical mechanics; classical spin systems, nonlinear sigma models","fieldGroup":"Mathematical physics","statement":"In the classical nearest-neighbour $O(n)$ model on $\\mathbb Z^2$, spins $\\sigma_x\\in S^{n-1}$ carry the Gibbs weight $\\exp(\\beta\\sum_{x\\sim y}\\sigma_x\\cdot\\sigma_y)$. Mermin and Wagner showed there is no spontaneous magnetization at any $\\beta<\\infty$, and McBryan-Spencer gave algebraic upper bounds on correlations, but these allow a zero exponential rate. For $n=2$ the Berezinskii-Kosterlitz-Thouless phase (Frohlich-Spencer) has only power-law decay at low temperature. For $n\\ge3$ Polyakov's renormalization argument (1975) predicts mass generation at every positive temperature; rigorous proofs covered only high temperature (Aizenman-Simon) and large $n$ (Kupiainen). Do two-point correlations $\\langle\\sigma_x\\cdot\\sigma_y\\rangle$ decay exponentially in $|x-y|$ for every $n\\ge3$ and every finite $\\beta>0$?","posedBy":"Alexander M. Polyakov (Phys. Lett. B 59, 1975), predicting mass generation for n >= 3; recorded as an open conjecture by Aru, Garban and Sepulveda (Comm. Math. Phys., 2025)","yearPosed":1975,"ageNote":"Year of Polyakov's renormalization paper, which the manuscripts cite as the source of the prediction. The rigorous all-temperature formulation is the one Aru-Garban-Sepulveda (2025) record as open.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every integer $n\\ge3$ and finite $\\beta>0$ there are $A<\\infty$, $m>0$ with $0\\le\\langle\\sigma_x\\cdot\\sigma_y\\rangle_{G,b}\\le Ae^{-m\\|x-y\\|_2}$ for every finite subgraph $G$ of $\\mathbb Z^2$ with free boundary and every coupling array $b\\in[0,\\beta]^E$; Corollary 1.2 passes this to free-boundary box limits, giving finite susceptibility. The companion 'Sharp mass bounds' proves, for $O(4)$, a positive gap of the full transfer operator (all local observables) in the unique periodic state at every $\\beta>0$, and $m\\asymp\\sqrt\\beta e^{-\\pi\\beta}$ for large $\\beta$. Not shown: uniqueness of infinite-volume Gibbs states for general $n$, a full transfer gap for $n\\ne4$ in this paper, or the low-temperature correlation length for general $n$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript (September 23, 2026) is self-contained; a companion of the same date proves the stronger full transfer gap for O(4), and later companions in the family build the O(3) continuum limit and the exact O(4) mass asymptotic on these estimates.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of 'Exponential decay in two-dimensional classical O(n) models' were read against Polyakov's prediction as the paper and Aru-Garban-Sepulveda state it. The proof (a fourth-variation rotation inequality on annuli, a sign-cluster crossing bound and a reduction from n to 3 components) was not refereed. Lean: lean/formalization.yaml lists comparator config ComparatorChallenges/ClassicalON.json, declaration OAI.ClassicalON.main in OAI/Probability/ClassicalON/Main.lean, permitted axioms propext, Quot.sound and Classical.choice. The statement file ComparatorChallenges/ClassicalON.lean was read: for every n >= 3 and beta > 0 it gives A and m > 0 such that, on every finite nearest-neighbour subgraph of Z^2 with edge strengths in [0, beta], the normalized product-sphere Gibbs correlation of two spins lies between 0 and A exp(-m |x-y|). That is exactly Theorem 1.1, the headline claim; it does not cover infinite-volume limits or any transfer gap. Not rebuilt here. Scope the paper itself states: spin two-point function only, free boundary conditions and their subsequential limits, constants that may degrade as beta grows, no uniqueness of Gibbs states and no correlation-length asymptotic.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"Polyakov's prediction of a mass gap at every temperature for two-dimensional O(n) models with n at least 3 has been a flagship open problem of rigorous statistical mechanics since 1975, the lattice cousin of asymptotic freedom. Tied with Cannon's Conjecture (62), below Simon's Problem 1 on Anderson delocalization (65), above Kadison's Similarity Problem (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Exponential decay in two-dimensional classical O(n) models","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Sharp mass bounds for the two-dimensional O(4) model (full transfer gap at every temperature)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean proof: OAI.ClassicalON.main","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/ClassicalON/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: ClassicalON","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ClassicalON.lean","kind":"lean-proof"},{"label":"Lean scope note for family 215","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/215.md","kind":"lean-proof"},{"label":"Polyakov 1975, Phys. Lett. B 59","url":"https://doi.org/10.1016/0370-2693(75)90161-6","kind":"problem-record"},{"label":"Aru, Garban, Sepulveda 2025 (records the conjecture as open), arXiv 2212.06767","url":"https://arxiv.org/abs/2212.06767","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fujita-freeness-conjecture","name":"Fujita's freeness conjecture","shortName":"Fujita's freeness conjecture","problemNumber":null,"field":"Algebraic geometry; adjoint linear systems","fieldGroup":"Algebra","statement":"Let $X$ be a smooth complex projective variety of dimension $n$, $K_X$ its canonical bundle and $L$ an ample line bundle. Fujita conjectured that $K_X+mL$ is globally generated for every $m\\ge n+1$ (and very ample for $m\\ge n+2$); the bound is sharp for $(\\mathbb P^n,\\mathcal O(1))$. Known: curves by Riemann-Roch, surfaces by Reider, dimensions 3, 4, 5 by Ein-Lazarsfeld, Kawamata and Ye-Zhu; in general Angehrn-Siu's quadratic bound $m\\ge(n^2+n+2)/2$, later improved to $n(\\log\\log n+2.34)$ by Ghidelli-Lacini and to about $1.776n$ by Han. Is $K_X+(n+1)L$ globally generated for every smooth projective $X$ of every dimension $n$ and every ample $L$?","posedBy":"Takao Fujita, On polarized manifolds whose adjoint bundles are not semipositive, Algebraic Geometry, Sendai 1985, Adv. Stud. Pure Math. 10 (1987)","yearPosed":1987,"ageNote":"The cited source is the proceedings of the 1985 Sendai conference, published in 1987; the conjecture is often dated to the 1985 meeting.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every smooth connected complex projective variety $X$ of dimension $n\\ge1$ and every ample line bundle $L$, $K_X+(n+1)L$ is generated by global sections; hence $K_X+mL$ is globally generated for all $m\\ge n+1$, even when $L$ itself is not. The bound is sharp. The method minimises an exponential average of normalised section orders over bases and weighted monomial valuations, then lifts sections with Fujita's discrepancy-and-vanishing mechanism. It does not prove the very-ampleness conjecture ($m\\ge n+2$) or point separation.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against Fujita's conjecture as the manuscript cites it. The proof was not refereed. No Lean formalization: the manuscript is not in lean/formalization.yaml and lean/docs/038.md does not exist at the pinned commit. Scope the paper itself states: only the freeness (global generation) part is proved, not separation of points or tangents and not the very-ampleness half of Fujita's conjecture. Readers should know a previous claimed proof (Chan, arXiv 2411.07129) was withdrawn in November 2024; the manuscript cites that withdrawal. A brief web search on 7 October 2026 found no public dispute of this manuscript.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"Fujita's freeness conjecture is one of the best-known open problems of higher-dimensional algebraic geometry, open since the mid 1980s with only low-dimensional cases and nonlinear or non-sharp bounds known. Above Bloch's conjecture for surfaces with pg = 0 (55) and the Kuznetsov rationality conjecture (45), below the log abundance conjecture (75) and the Hodge conjecture for CM abelian varieties (72).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fujitas-freeness-conjecture-September-23-2026/Fujitas-freeness-conjecture-September-23-2026.pdf","sourceName":"OpenAI math release: Fujita's freeness conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Fujita 1987, Adv. Stud. Pure Math. 10","url":"https://doi.org/10.2969/aspm/01010167","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"anderson-localization-2d-simon-problem-2","name":"Simon's Problem 2: pure-point spectrum for the two-dimensional Anderson model at every disorder","shortName":"2D Anderson localization (Simon's Problem 2)","problemNumber":null,"field":"Random Schrodinger operators; Anderson localization","fieldGroup":"Mathematical physics","statement":"Let $H=\\Delta+v$ on $\\ell^2(\\mathbb{Z}^2)$, with nearest-neighbor hopping and independent site potentials $v_x$ uniform on $[-h,h]$. The scaling theory of Abrahams, Anderson, Licciardello and Ramakrishnan (1979) predicts no metallic behavior in two dimensions, at any disorder. Rigorous localization was known only at large disorder or at extreme energies (Frohlich-Spencer, Frohlich-Martinelli-Scoppola-Spencer, Aizenman-Molchanov) and near the band edge (Ding-Smart). Simon's Problem 2 asks: for every $h>0$, does $H$ almost surely have pure-point spectrum throughout its spectrum?","posedBy":"Barry Simon (Problem 2 in 'Schrodinger operators in the twenty-first century'), formalizing the two-dimensional localization conjecture of Abrahams, Anderson, Licciardello and Ramakrishnan","yearPosed":2000,"ageNote":"The physical prediction dates from 1979 (Abrahams et al.); the 2000 date is Simon's mathematical formulation, which the manuscript cites as the problem.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For each fixed $h>0$, the operator $H_v=\\Delta+v$ on $\\ell^2(\\mathbb{Z}^2)$ with iid $v_x$ uniform on $[-h,h]$ almost surely has pure-point spectral type with a complete orthonormal eigenbasis, spectrum $[-4-h,4+h]$ and dense eigenvalues. The null set may depend on $h$. Not shown: one event for all $h$, bounds uniform in energy, exponential decay of eigenfunctions, dynamical localization, or other single-site laws such as Bernoulli.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem was read against Simon's Problem 2 as the manuscript quotes it; Simon's text itself was not opened. The proof was not refereed. Lean: lean/docs/261.md describes a comparator challenge (PlanarAndersonSpectrum.lean) that states only that the spectrum is [-4-h, 4+h], explicitly not the pure-point claim; it is not in formalization.yaml and its solution module is absent at the pinned commit, so it counts for nothing. Scope stated by the paper: uniform single-site law only; no dynamical localization or eigenfunction decay is claimed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"A flagship conjecture of mathematical physics, open since the 1979 scaling theory, though less famous outside the field than extended states. Below the d >= 3 extended-states entry from the same family (65) and critical percolation (78), level with Griffiths' positivity conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Same family: Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean scope note for family 261 (supporting spectrum statement only)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/261.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gnrs-conjecture-planar-bounded-treewidth","name":"The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture","shortName":"GNRS conjecture: planar and bounded treewidth","problemNumber":null,"field":"Metric embeddings into L1; multicommodity flow-cut gaps","fieldGroup":"Theoretical computer science","statement":"Gupta, Newman, Rabinovich and Sinclair conjectured that for every proper minor-closed family of graphs there is a constant $C$ such that the shortest-path metric of every graph in the family, with arbitrary positive edge lengths, embeds into $L_1$ with distortion at most $C$; equivalently, the multicommodity flow-cut gap is uniformly bounded on the family. Constant distortion was known for series-parallel graphs (GNRS; sharp value 2 by Chakrabarti-Jaffe-Lee-Vincent) and for bounded outerplanarity, while the best bound for $n$-vertex planar graphs was $O(\\sqrt{\\log n})$ (Rao 1999). Do planar graph metrics, and for each fixed $k$ the metrics of graphs of treewidth at most $k$, embed into $L_1$ with distortion bounded independently of the graph and the edge lengths?","posedBy":"A. Gupta, I. Newman, Y. Rabinovich and A. Sinclair, Cuts, trees and l1-embeddings of graphs, Combinatorica 24 (2004)","yearPosed":2004,"ageNote":"The manuscripts cite only the 2004 Combinatorica paper. An earlier conference version of that paper exists but is not cited, so the conjecture may be somewhat older than the year given.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Planar paper, Theorem 1.1: a universal constant $C\\ge1$ such that every finite connected planar graph with positive edge lengths embeds into $L_1$ with distortion at most $C$; hence the undirected multicommodity flow-cut gap is uniformly bounded on planar graphs. Treewidth companion, Theorem 1.1: for each $k$, a constant $C_k$ (no explicit estimate) for graphs with tree decompositions of bag size at most $k$, hence also for graphs excluding a fixed planar minor. Together they give constant distortion for fixed-parameter almost-embeddable families. Not shown: the full GNRS conjecture for every minor-closed family (closure under clique sums is open), and no explicit value of $C$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. Both main theorems have Lean formalizations in the release (Comparator challenges PlanarL1 and BoundedTreewidthL1).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of both manuscripts (TeX source), read against the GNRS conjecture as cited. Lean-checked on the release's Comparator challenge PlanarL1 (declaration OAI.PlanarL1.planar_graph_metrics_embed_L1, listed in lean/formalization.yaml). Its statement was read here: it asserts one real $C\\ge1$ such that every connected graph on Fin n with a planar drawing (injective vertex points, simple arcs with pairwise disjoint interiors) and symmetric positive edge lengths has a map into $L^1$ of some measure space with $d\\le\\|f(x)-f(y)\\|\\le C d$ for the walk-length distance $d$. That is the planar headline. The companion challenge BoundedTreewidthL1 (OAI.BoundedTreewidthL1.main_theorem, also listed) states, for every bag-size bound $k\\ge2$, a constant $C(k)$ and an embedding into finite-dimensional $\\ell_1$ for every finite connected graph with a tree decomposition of bag size at most $k$; that is the treewidth headline. Neither was rebuilt here. The papers state that the full GNRS conjecture (general clique-sum closure) is not established.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The planar embedding conjecture is one of the best-known open problems in metric embeddings and approximation algorithms, and with bounded treewidth it is the most studied case of the GNRS conjecture; the full conjecture stays open, so this scores the special cases. Tied with Kalai's conjecture for tight trees (45), below the 3SUM hypothesis (50), above the CAT(0) conjecture for Artin groups (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Planar Graph Metrics Embed into L1 with Constant Distortion","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"L1 Embeddings of Graphs of Bounded Treewidth (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean Comparator challenge PlanarL1 (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PlanarL1.lean","kind":"lean-proof"},{"label":"Lean Comparator challenge BoundedTreewidthL1 (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/BoundedTreewidthL1.lean","kind":"lean-proof"},{"label":"Gupta-Newman-Rabinovich-Sinclair 2004, Cuts, trees and l1-embeddings of graphs","url":"https://doi.org/10.1007/s00493-004-0015-x","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"finitely-presented-simple-amenable-group","name":"Is there an infinite finitely presented simple amenable group?","shortName":"Finitely presented simple amenable group","problemNumber":null,"field":"Geometric group theory: amenable and simple groups","fieldGroup":"Algebra","statement":"A discrete group is amenable if it has Folner sets: for every finite $K\\subseteq G$ and $\\varepsilon>0$ there is a nonempty finite $D$ with $|gD\\,\\triangle\\,D|<\\varepsilon|D|$ for all $g\\in K$. Juschenko and Monod (2013) proved that topological full groups of minimal Cantor homeomorphisms are amenable, which with Matui's work gives infinite finitely generated simple amenable groups. Those derived groups are not finitely presented (Matui), and the local embeddability of full groups into finite groups (Grigorchuk-Medynets) explains why such examples cannot be. Does there exist an infinite group that is finitely presented, simple and amenable?","posedBy":"Recorded by J. O. Button, Largeness of LERF and 1-relator groups, Groups Geom. Dyn. 4 (2010), p. 729; raised again by Juschenko and Monod, Ann. of Math. 178 (2013), p. 776","yearPosed":2010,"ageNote":"The manuscript says the question predates Juschenko-Monod and cites Button (2010) as a record of it, without naming an original poser. It also appears as Problem 10 in Zaremsky's July 2026 problem list.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there exists an infinite finitely presented simple amenable group. It is an alternating subgroup of a polygon exchange group (piecewise translations of finitely many squares over a real quadratic ring, with two inclined edge directions besides the coordinate ones). Finite presentation comes from propagating relations over a fixed finite generating set with a homological argument in the style of Szymik-Wahl; amenability from correlated random polygonal partitions giving almost-invariant measures. No explicit bound on the sizes of the generating set or relators is given.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of the TeX source; the construction and proofs were not refereed. Lean-checked on the Comparator challenge SimpleAmenable, listed in the release's formalization catalogue (OAI.SimpleAmenable.main, OAI/GroupTheory/SimpleAmenable/Main.lean). Its statement: there exists a group that is infinite, finitely presented (Mathlib's Group.IsFinitelyPresented), simple (Mathlib's IsSimpleGroup) and satisfies the Folner condition as displayed in the problem statement. That is the headline claim, with amenability in its Folner form. The release's Lean notes say the paper's later general claims (central kernels, enlargements across families) are not formalized. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"A recognized existence question of geometric group theory, sharpened by the Juschenko-Monod theorem of 2013 and still on 2026 problem lists. Scored with the CAT(0) conjecture for Artin groups (40) and Wall's manifold-realization question (42), below the amenability problem for Thompson's group F (58).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An infinite finitely presented simple amenable group","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: OAI/GroupTheory/SimpleAmenable/Main.lean (main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/GroupTheory/SimpleAmenable/Main.lean","kind":"lean-proof"},{"label":"Button, Largeness of LERF and 1-relator groups (GGD 2010), where the question is recorded","url":"https://doi.org/10.4171/GGD/102","kind":"problem-record"},{"label":"Zaremsky, Some open problems (July 2026), Problem 10","url":"https://zaremsky.github.io/open_problems.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"barker-sequence-conjecture","name":"The Barker-sequence conjecture: no Barker sequence of length greater than 13","shortName":"Barker-sequence conjecture","problemNumber":null,"field":"Sequence design; aperiodic autocorrelation","fieldGroup":"Combinatorics","statement":"A Barker sequence of length $n>1$ is a sign sequence $a\\in\\{-1,1\\}^n$ whose aperiodic autocorrelations $C_a(t)=\\sum_{j=0}^{n-t-1}a_ja_{j+t}$ satisfy $|C_a(t)|\\le1$ for $1\\le t1$, a Barker sequence of length $n$ exists if and only if $n\\in\\{2,3,4,5,7,11,13\\}$. The new input is the even case, via the circulant Hadamard theorem; the odd case is the classical Turyn-Storer theorem. Section 5 records examples at every listed length. Nothing is claimed about nearly-Barker sequences or the asymptotic merit-factor problem.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. This entry is Corollary 1.2 of the circulant Hadamard manuscript (September 23, 2026): the new circulant Hadamard theorem settles the even case, and the odd case is classical (Turyn-Storer; Schmidt-Willms).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary 1.2 and Section 5 were read; the even case reduces to Theorem 1.1 by the classical periodic-autocorrelation argument, and the odd case is quoted from Turyn-Storer and Schmidt-Willms. The challenge lean/ComparatorChallenges/EvenBarker.json (solution module OAI.LinearAlgebra.Barker.Main, present at the pinned commit) is not in the formalization catalogue; its statement EvenBarker.lean was read here: a $\\pm1$ integer sequence of positive even length $n$ with all aperiodic autocorrelations at shifts $00$, after positive rescaling, isometric to round $S^4$, round $\\mathbb{RP}^4$ or Fubini-Study $\\mathbb{CP}^2$?","posedBy":"DaGang Yang, Rigidity of Einstein 4-manifolds with positive curvature (Invent. Math., 2000), Conjecture 1","yearPosed":2000,"ageNote":"The year is that of Yang's explicit statement, which the manuscript cites. The question is older in spirit: Berger's 1961 quarter-pinching rigidity is the first partial answer the manuscript lists.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: a connected smooth closed Einstein four-manifold with strictly positive sectional curvature, with no orientability assumption, is after positive rescaling isometric to round $S^4$, Fubini-Study $\\mathbb{CP}^2$ or round $\\mathbb{RP}^4$. The core step is half-conformal flatness ($W^+\\equiv0$ or $W^-\\equiv0$) from a weighted Hessian estimate and moment balance of the two Weyl blocks. Companions: a closed Einstein four-manifold with $\\mathrm{Ric}=3g$, $K\\ge0$ and one zero-curvature plane has universal cover $S^2(1/\\sqrt3)\\times S^2(1/\\sqrt3)$, giving the full nonnegative classification; and a simply connected closed four-manifold with $K\\ge0$ and small scale-invariant $L^2$ trace-free Ricci energy is diffeomorphic to $S^4$, $\\mathbb{CP}^2$ or $S^2\\times S^2$. Not shown: anything in higher dimensions or for complete noncompact metrics.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has three manuscripts: the strictly positive classification (September 23, 2026), a zero-plane rigidity theorem extending it to nonnegative curvature (October 4) and an L2 topological gap that uses that classification as a stated premise (October 5). The principal manuscript ships an exact-arithmetic verifier for its finite polynomial sign certificates; its coverage note says it checks only those finite claims, not the geometric or analytic reductions.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of 'Positively curved Einstein four-manifolds' were read against Yang's Conjecture 1 as the manuscript cites it; they match. The Lean challenge lean/ComparatorChallenges/EinsteinFour.lean (solution module OAI.Geometry.EinsteinFour.Classification, which exists at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/348.md. Its statement OAI.PositiveEinsteinFour.classification was read here: for every connected compact Hausdorff smooth four-manifold with a smooth metric that is Einstein (Ricci computed from chart Christoffel symbols) and has positive sectional curvature, some positive rescaling is distance-isometric to the round 4-sphere, Fubini-Study CP2 or round RP4. That is the headline claim. The formalization was not rebuilt here. The proof uses finite polynomial certificates checked by a shipped script, which was not run here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A long-standing rigidity question of four-dimensional Riemannian geometry, attacked since Berger's 1961 pinching theorem and only settled before under extra pinching, topological or symmetry hypotheses. Well known among geometric analysts but not beyond. Placed below the Blaschke conjecture (55) and the LeBrun-Salamon conjecture (56), above Beauville's splitting conjecture for two summands (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Positively curved Einstein four-manifolds","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Zero-Plane Rigidity for Einstein Four-Manifolds (nonnegative curvature)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf","kind":"paper"},{"label":"Companion: An L2 Einstein Gap for Nonnegatively Curved Four-Manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf","kind":"paper"},{"label":"Lean challenge statement: EinsteinFour.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/EinsteinFour.lean","kind":"lean-proof"},{"label":"Lean solution module: OAI/Geometry/EinsteinFour/Classification.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/EinsteinFour/Classification.lean","kind":"lean-proof"},{"label":"Yang (2000), Rigidity of Einstein 4-manifolds with positive curvature","url":"https://doi.org/10.1007/PL00005792","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cardy-formula-poisson-voronoi-percolation","name":"Cardy's formula for critical Poisson-Voronoi percolation (annealed crossing part of Schramm's Problem 2.12)","shortName":"Cardy's formula for Voronoi percolation","problemNumber":null,"field":"Percolation, conformal invariance","fieldGroup":"Probability & statistics","statement":"In critical planar Voronoi percolation, the cells of a Poisson point process of intensity $\\varepsilon^{-2}$ are coloured black or white independently with probability $1/2$. Cardy predicted, and Smirnov proved for site percolation on the triangular lattice, that crossing probabilities of a conformal quadrilateral converge to Cardy's hypergeometric function of its conformal modulus. Schramm (ICM 2006, Problem 2.12) asked for the corresponding conformal-invariance theorem for Voronoi percolation. Do the crossing probabilities of critical Poisson-Voronoi percolation in a Jordan quadrilateral converge to Cardy's formula, and is the scaling limit conformally invariant?","posedBy":"Oded Schramm","yearPosed":2006,"ageNote":"Schramm, Proceedings of the ICM 2006 (Madrid), Problem 2.12, as cited by the manuscript; Benjamini and Schramm had earlier studied conformal invariance of Voronoi percolation (1998).","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every bounded Jordan domain with four marked boundary points, the annealed probability of a black crossing in critical Poisson-Voronoi percolation converges to Cardy's function $F(x)$ of the conformal cross-ratio $x$. A corollary gives a quenched $L^2$ limit for fixed rectangles. The companions prove that the expected number of pivotal cells for a unit-square crossing is asymptotic to $c\\,\\varepsilon^{-3/4}$ and that conditional near-critical crossing-threshold laws for rational polygonal quads converge in probability over tessellations to the triangular-lattice law. Not shown: convergence of interfaces to SLE(6) or the full conformal invariance of Schramm's problem.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript (September 23, 2026) proves the annealed Cardy formula; two companions (October 5, 2026) take it as an input to prove a pivotal-count asymptotic and quenched near-critical universality of crossing thresholds.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Schramm's Problem 2.12 as the manuscript cites it. The paper itself limits the scope: the limit is of annealed crossing probabilities (averaged over points and colours); it does not claim convergence of interfaces or a quenched crossing limit in arbitrary domains. Both companions are conditional on the principal paper's theorem. No Lean formalization accompanies this family. Uses established box-crossing and quenched-annealed comparison inputs (Tassion, Ahlberg-Griffiths-Morris-Tassion, Vanneuville).","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Conformal invariance of Voronoi percolation is a well-known universality problem from Schramm's ICM list; the annealed Cardy formula is its main crossing-probability part, not the full scaling limit. Placed below the release's square-lattice bond Cardy entry (48), whose model is the more famous open case, and level with the Talagrand SK overlap conjecture (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Cardy's formula for critical Poisson-Voronoi percolation","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: From critical crossings to quenched near-critical universality in Voronoi percolation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf","kind":"paper"},{"label":"Companion: A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"linear-list-hadwiger-conjecture","name":"The Linear List Hadwiger Conjecture","shortName":"Linear List Hadwiger conjecture","problemNumber":null,"field":"Graph coloring and graph minors","fieldGroup":"Combinatorics","statement":"Kawarabayashi and Mohar (2007, Conjecture 1.3) proposed a linear relaxation of Hadwiger's conjecture for list colouring: there is an absolute constant $C$ such that every graph with no $K_t$ minor is $Ct$-choosable. Degeneracy gives $O(t\\sqrt{\\log t})$; the best list bounds before this work were superlinear (for example Postle's $O(t(\\log\\log t)^6)$), and $K_t$-minor-free graphs can have list chromatic number close to $2t$, so $C=1$ is impossible. Even the ordinary-colouring version, $\\chi(G)=O(h(G))$, was open. Is there an absolute constant $C$ with $\\chi_\\ell(G)\\le C\\,h(G)$ for every graph $G$?","posedBy":"Ken-ichi Kawarabayashi and Bojan Mohar, A relaxed Hadwiger's conjecture for list colorings, J. Combin. Theory Ser. B 97 (2007), Conjecture 1.3","yearPosed":2007,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is an absolute integer $C\\ge1$ such that every finite nonempty simple graph satisfies $\\chi_\\ell(G)\\le C\\,h(G)$; Corollary 1.2 gives the ordinary linear Hadwiger bound $\\chi(G)\\le C\\,h(G)$. The constant is not computed or optimised and must be at least $2$ by Steiner's examples. The proof shows $K_t$-minor-free graphs on at most $t^{11/10}$ vertices are $Lt$-choosable and then lifts this to all orders through Postle-style separability and a three-district recursion. It does not give $C$ close to $1$ or $2$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and the INPUTS.md file, read against Kawarabayashi-Mohar Conjecture 1.3 as cited. The proof was not refereed. INPUTS.md lists three results used without proof: the Reed-Seymour fractional bound, a connectivity-extraction theorem (Delcourt-Postle, credited to Girao-Narayanan) and a terminal-linkage theorem (Delcourt-Postle, after Bollobas-Thomason), plus LP duality. The manuscript is dated 23 September 2026 but cites work posted 1 October 2026, so the text was revised after its date. Lean-checked on the release's own Comparator challenge ListHadwiger (OAI.LinearListHadwiger.main_theorem) together with its solution module, both present at the pinned commit; the challenge is not listed in the release's formalization catalogue, the statement was read here but not independently audited, and the development was not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Whether chromatic number is linear in the Hadwiger number has been a main target of the minor-colouring program (Norin-Postle-Song, Delcourt-Postle, Postle for lists); this entry answers the list version and with it the ordinary one. Placed above Connes' Rigidity Conjecture (45) and the Petersen Coloring Conjecture (40), below the Cycle Double Cover Conjecture (55) and well below Hadwiger's conjecture itself.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A linear list-coloring bound in terms of the Hadwiger number","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"A counterexample to Hadwiger's conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Kawarabayashi-Mohar 2007, JCTB 97","url":"https://doi.org/10.1016/j.jctb.2006.11.002","kind":"problem-record"},{"label":"Lean Comparator challenge ListHadwiger (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ListHadwiger.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"eisenbud-green-harris-conjecture-char-zero","name":"The Eisenbud-Green-Harris conjecture over fields of characteristic zero","shortName":"Eisenbud-Green-Harris conjecture (char. 0)","problemNumber":null,"field":"Commutative algebra; Hilbert functions","fieldGroup":"Algebra","statement":"Let $S=k[x_1,\\ldots,x_n]$ be a standard graded polynomial ring over a field, let $2\\le a_1\\le\\cdots\\le a_c$, and let $I$ be a homogeneous ideal containing a regular sequence $f_1,\\ldots,f_c$ with $\\deg f_i=a_i$. Eisenbud, Green and Harris, from their work on higher Castelnuovo theory, proposed that the pure powers $x_1^{a_1},\\ldots,x_c^{a_c}$ can replace the regular sequence without changing the Hilbert function: there should be a monomial ideal containing $(x_1^{a_1},\\ldots,x_c^{a_c})$, indeed a lex-plus-powers ideal, with the same Hilbert function as $I$. Known cases required fast-growing degrees or regular sequences of special form. Does every such $I$ have the Hilbert function of a lex-plus-powers ideal?","posedBy":"David Eisenbud, Mark Green and Joe Harris, Higher Castelnuovo theory, Asterisque 218 (1993), Section 4","yearPosed":1993,"ageNote":"The conjecture is usually stated over an arbitrary field; this entry covers characteristic zero, which is all the manuscripts claim.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims the Eisenbud-Green-Harris conjecture over every field of characteristic zero, for homogeneous regular sequences of any positive length with all degrees at least two and with no growth condition on the degrees, via commuting division-coefficient linear forms whose ordered monomials form a basis of the whole polynomial algebra. It does NOT treat positive characteristic. The Betti-number strengthening (lex-plus-powers) is the companion's result, listed separately.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The principal manuscript constructs commuting linear forms with division-algebra coefficients and deduces the Hilbert-function statement; its companion of the same date uses that construction to prove the stronger lex-plus-powers Betti inequalities, listed as a separate entry.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Corollary 1.2 (Artinian case over $\\mathbb C$) and the characteristic-zero corollary of the principal manuscript were read against the conjecture: every homogeneous ideal containing a regular sequence of degrees at least two has the Hilbert function of a monomial ideal containing the corresponding pure powers, for regular sequences of any length. The passage from Artinian to general uses the published Caviglia-Maclagan reduction and a coefficient-field descent. Positive characteristic is not addressed. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The Eisenbud-Green-Harris conjecture is one of the best known open problems on Hilbert functions in commutative algebra, open for over thirty years with only special degree patterns known. Characteristic zero is the main case but not all of it, so scored below a full solution. Below Lech's multiplicity conjecture (45) and the finitistic dimension conjecture (50), level with the CAT(0) conjecture for Artin groups (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Commuting Division-Coefficient Forms and the Artinian Eisenbud-Green-Harris Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: The Artinian Lex-Plus-Powers Betti Theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf","kind":"paper"},{"label":"Eisenbud, Green and Harris, Higher Castelnuovo theory (1993)","url":"https://www.numdam.org/item/AST_1993__218__187_0/","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kadison-ringrose-conjecture","name":"The Kadison-Ringrose cohomology conjecture","shortName":"Kadison-Ringrose conjecture","problemNumber":null,"field":"Operator algebras: bounded Hochschild cohomology","fieldGroup":"Analysis","statement":"For a complex von Neumann algebra $M$, let $C_b^k(M,M)$ be the bounded complex $k$-linear maps $M^k\\to M$ (operator-norm boundedness, no complete boundedness), with the Hochschild differential $d$, and let $H_b^k(M,M)=\\ker d/\\operatorname{im} d$, the image taken without closure. In degree one vanishing is the Kadison-Sakai theorem that every derivation of $M$ is inner. Kadison and Ringrose proved vanishing for type I and hyperfinite algebras, Christensen, Effros and Sinclair whenever the type $\\mathrm{II}_1$ summand absorbs the hyperfinite $\\mathrm{II}_1$ factor, and later work handled Cartan subalgebras and property $\\Gamma$. Is $H_b^k(M,M)=0$ for every complex von Neumann algebra $M$ and every $k\\ge1$?","posedBy":"Richard V. Kadison and John R. Ringrose (Cohomology of operator algebras I, II, 1971); Sinclair and Smith trace the self-coefficient formulation to 1967","yearPosed":1967,"ageNote":"The manuscript cites Sinclair and Smith (PNAS 1998, p. 3376), who trace the self-coefficient formulation to 1967; the first published papers on the problem are Kadison and Ringrose, Acta Math. 1971.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: for every complex von Neumann algebra $M$ and every $k\\ge2$, every bounded $k$-cocycle $f\\in C_b^k(M,M)$ with $df=0$ equals $dg$ for some bounded $g\\in C_b^{k-1}(M,M)$, so $H_b^k(M,M)=0$. Combined with the Kadison-Sakai theorem in degree one this is the full conjecture. The key new input is a Liouville theorem for harmonic maps under a single unitary random walk built from Popa's free-independent sequences in type $\\mathrm{II}_1$ algebras. It concerns ordinary bounded cohomology with coefficients in $M$ itself; it says nothing new about coefficients in $B(H)$ or about completely bounded cohomology beyond what follows.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the conjecture as the manuscript states it (vanishing of $H_b^k(M,M)$ for all complex von Neumann algebras and all $k\\ge1$). The proof was not refereed. Theorem 1.1 covers every degree $k\\ge2$ with no separability or type restriction; degree one is the classical Kadison-Sakai inner-derivation theorem, which the paper cites rather than reproves. Lean-checked on the release's own Comparator challenge KadisonRingrose together with its solution module OAI.Analysis.BoundedHochschild.MainResult, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.BoundedHochschild.KadisonRingrose.main_result) takes any C*-algebra with a W*-algebra (predual) structure and says every bounded multilinear cocycle of degree n+2 with values in the algebra is the differential of a bounded cochain of degree n+1, which is the headline claim for all degrees at least two.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"A central problem of operator-algebra cohomology for over fifty years, the subject of a Sinclair-Smith monograph, open exactly for finite algebras without property Gamma or Cartan structure. Famous within operator algebras but less widely known than Kadison's similarity problem (58) or Kaplansky's quasitrace problem (55); placed just below Connes' rigidity conjecture (45) and above Voiculescu's free entropy dimension question (38).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Vanishing of higher bounded Hochschild cohomology","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Kadison and Ringrose, Cohomology of operator algebras I, Acta Math. 126 (1971)","url":"https://doi.org/10.1007/BF02392032","kind":"problem-record"},{"label":"Sinclair and Smith, The Hochschild cohomology problem for von Neumann algebras, PNAS 1998","url":"https://doi.org/10.1073/pnas.95.7.3376","kind":"problem-record"},{"label":"Lean Comparator challenge KadisonRingrose (not in formalization.yaml at this commit)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KadisonRingrose.lean","kind":"lean-proof"},{"label":"Lean solution module OAI/Analysis/BoundedHochschild/MainResult.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/BoundedHochschild/MainResult.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fpras-common-bases-two-matroids","name":"An FPRAS for counting common bases of two matroids","shortName":"Counting common bases of two matroids","problemNumber":null,"field":"Approximate counting; matroids; Markov chains","fieldGroup":"Theoretical computer science","statement":"Given independence oracles for two matroids $M_1,M_2$ of the same rank $r$ on $[n]$, estimate $|\\mathcal B(M_1)\\cap\\mathcal B(M_2)|$, the number of common bases. The problem contains counting bases of one matroid (FPRAS by Anari-Liu-Oveis Gharan-Vinzant) and counting perfect matchings of a bipartite graph (the Jerrum-Sinclair-Vigoda permanent FPRAS). For intersections only coarse or exponential-factor approximations were known, and Anari-Oveis Gharan-Vinzant's $2^{O(r)}$-factor method is fully polynomial only for $r=O(\\log n)$. Cryan, Guo and Mousa asked for a rapidly mixing chain sampling common bases. Is there a fully polynomial randomized approximation scheme for the number of common bases of two arbitrary matroids given by independence oracles?","posedBy":"Cryan, Guo and Mousa (Ann. Probab. 2021) asked for a fast sampler; the FPRAS question is Problem 1 of Section 13.2 of Kuikui Liu's dissertation (2023)","yearPosed":2021,"ageNote":"Year of the Cryan-Guo-Mousa journal version the manuscript cites; the question is likely older folklore in approximate counting, but the manuscript gives no earlier source.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: one randomized oracle algorithm, given independence oracles for rank-$r$ matroids on $[n]$ and rational $\\varepsilon,\\delta\\in(0,1)$, outputs $\\widehat Z$ within relative error $\\varepsilon$ of the number of common bases with probability at least $1-\\delta$, outputs zero when there are none, and uses polynomially many oracle calls and bit operations on every execution. Corollaries give FPRASes and almost-uniform samplers for common independent sets of prescribed, unrestricted or maximum size, even for different ranks. The method is a Jerrum-Sinclair-Vigoda style annealed chain with defect states, using the homogeneous Tutte theorem of Branden-Huh (Lorentzian polynomials). It gives no exact counting and no deterministic algorithm, and does not treat three or more matroids. The October companion gives an FPRAS for common integer bases of two polymatroids with binary capacities.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. A later manuscript in the family (5 October 2026) extends the method to common integer bases of two polymatroids with binary capacities and builds on this paper's transport and trace estimates.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against Liu's formulation. formalization.yaml lists ComparatorChallenges/CommonBasesFPRAS.json with declaration OAI.common_bases_fpras in OAI/Combinatorics/MatroidCounting/CommonBases.lean (file exists at the pinned commit). The challenge statement was read here: it fixes an explicit oracle register machine and asserts one program and polynomial $B$ in $n$, the binary input length, $\\lceil\\varepsilon^{-1}\\rceil$ and $\\log\\delta^{-1}$ such that, for all equal-rank matroids on $\\mathrm{Fin}\\,n$ with exact independence oracles, every run halts within $B$ oracle calls and bit operations, outputs zero when the count is zero, and at least a $1-\\delta$ fraction of random tapes give relative error at most $\\varepsilon$. That is the headline claim. Not rebuilt here. The polymatroid extension is not formalized.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Approximately counting common bases of two matroids generalizes the permanent FPRAS and single-matroid basis counting, and was a well-known open problem in approximate counting. Famous within that community. Placed level with White's Conjecture on Matroids (40), above Rota's Unimodality Conjecture for Matroid Flats (36) and well above Approximate Counting for Spin Systems on Planar Graphs (18).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/main.pdf","sourceName":"OpenAI math release: Approximate counting of common bases of two matroids","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An FPRAS for Common Integer Polymatroid Bases with Binary Capacities","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/polymatroid-fpras.pdf","kind":"paper"},{"label":"Lean proof (OAI.common_bases_fpras)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/MatroidCounting/CommonBases.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"coarse-novikov-conjecture-bounded-geometry","name":"The coarse Novikov conjecture for bounded-geometry spaces: is coarse assembly rationally injective?","shortName":"Coarse Novikov conjecture","problemNumber":null,"field":"Higher index theory; coarse geometry; operator K-theory","fieldGroup":"Geometry & topology","statement":"For a uniformly discrete metric space $X$ of bounded geometry, coarse K-homology $KX_*(X)=\\varinjlim_r K_*^{lf}(P_r(X))$ is the limit over Rips complexes, and the coarse assembly map $\\mu_X:KX_*(X)\\to K_*(C^*(X))$ sends a class to its index in the K-theory of the Roe algebra. Roe formulated the coarse index theory and its Novikov-type injectivity question, and Higson and Roe (1995) stated the coarse Baum-Connes conjecture that $\\mu_X$ is an isomorphism. Surjectivity was known to fail for some expanders (Higson-Lafforgue-Skandalis), while injectivity was proved under geometric hypotheses such as coarse embeddability in Hilbert space (Yu). The coarse Novikov conjecture asks: is $\\mu_X\\otimes\\mathbb Q$ injective for every bounded-geometry space $X$?","posedBy":"John Roe, Coarse cohomology and index theory on complete Riemannian manifolds (Memoirs AMS 497, 1993); Nigel Higson and John Roe, On the coarse Baum-Connes conjecture (1995), Conjecture 6.6","yearPosed":1993,"ageNote":"The manuscripts credit Roe (1993) with the Novikov-type formulation and Higson-Roe (1995, Conjecture 6.6) with the coarse Baum-Connes conjecture, of which rational injectivity is the Novikov half; no single posing sentence for the bounded-geometry injectivity statement is quoted.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims a uniformly discrete bounded-geometry space $X$, a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order class $\\alpha\\in KX_1(X)$ whose image under ordinary coarse assembly into $K_1$ of the reduced locally compact Roe algebra vanishes, so $\\mu_X$ fails to be injective integrally and rationally. The companion (5 October 2026) claims the same kind of class is killed already by maximal coarse assembly, which implies the reduced statement. Earlier kernel examples (Yu; Dranishnikov-Ferry-Weinberger) lacked bounded geometry or died under coarsening. It does NOT give a counterexample to the Novikov conjecture or to Baum-Connes for groups, nor a space with property A or coarsely embeddable into Hilbert space.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. Both manuscripts are credited to OpenAI with no human author named. The family has two manuscripts: the reduced counterexample (23 September 2026), taken as principal here, and a 5 October 2026 strengthening to maximal Roe algebras. The reduced result has a Lean comparator challenge in the release.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 of both manuscripts was read against the coarse Novikov conjecture. A Lean challenge lean/ComparatorChallenges/CoarseAssembly.json (solution module OAI.Topology.CoarseAssembly, present at the pinned commit) exists but is not in the formalization catalogue; found through lean/docs/307.md. Its statement was read here and does not settle the tier: the 6600-line challenge defines its own graphs, Roe algebra, K-theory and an assembly map from a bespoke analytic index family, and that family (separatedIndexFamily) is defined to be the zero map whenever its scale-wise indices are not compatible. In that case the asserted vanishing $\\mu(\\alpha)=0$ and non-injectivity hold trivially, so the formal statement need not express the headline. Under the strict tier rule the entry stays Unreviewed. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Rational injectivity of coarse assembly for bounded-geometry spaces was a standing conjecture of higher index theory, the coarse analogue of the Novikov conjecture, and counterexamples had only been found without bounded geometry or before coarsening. A bounded-geometry counterexample would be a major result for that community, though it says nothing about the group Novikov conjecture. Below the reduced Baum-Connes entry from the same release (60), level with Naor's sublinear dimension question (35), above the Smith-Toda complexes entry (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to the coarse Novikov conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Failure of rational injectivity for maximal coarse assembly (5 October 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/paper.pdf","kind":"paper"},{"label":"Lean formalization: reduced coarse assembly counterexample (statement scope unclear, see note)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Topology/CoarseAssembly.lean","kind":"lean-proof"},{"label":"Lean comparator statement: CoarseAssembly","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/CoarseAssembly.lean","kind":"lean-proof"},{"label":"Higson and Roe, On the coarse Baum-Connes conjecture (1995)","url":"https://doi.org/10.1017/CBO9780511629365.008","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"degenerate-arnold-conjecture-critical-number","name":"The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points","shortName":"Degenerate Arnold conjecture","problemNumber":null,"field":"Symplectic topology; Hamiltonian fixed points, Arnold conjecture","fieldGroup":"Geometry & topology","statement":"Arnold asked whether a Hamiltonian diffeomorphism $\\phi$ of a closed symplectic manifold $M$ must have at least as many geometrically distinct fixed points as a smooth function on $M$ has critical points. Without nondegeneracy assumptions this is the critical-number bound $\\#\\mathrm{Fix}(\\phi)\\ge\\mathrm{Crit}(M)$, with the weaker rational cup-length form $\\#\\mathrm{Fix}(\\phi)\\ge\\mathrm{cuplength}(M;\\mathbb Q)$ (unit included). These are known on tori and $\\mathbb{CP}^n$, and when $[\\omega]$ and $c_1$ vanish on $\\pi_2(M)$ (Rudyak-Oprea). Do the critical-number and cup-length bounds hold for every Hamiltonian diffeomorphism of every closed symplectic manifold?","posedBy":"V. I. Arnold, First steps in symplectic topology, Russian Math. Surveys 41 (1986); recorded as Conjectures 1.4 and 1.6 in R. Golovko, On variants of Arnold conjecture, Arch. Math. (Brno) 56 (2020)","yearPosed":1986,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: on the complex quadric threefold there is a smooth Hamiltonian diffeomorphism with exactly three fixed points, at least one degenerate, while every smooth function has at least four critical points and the rational cup length is four; this disproves both unrestricted degenerate bounds, and contractibility conditions cannot rescue them. The construction composes a Hamiltonian involution fixing $Q^2\\cong S^2\\times S^2$ with a short invariant flow, using a three-critical-point function on $S^2\\times S^2$. The count three is sharp by Gong (2025). It says nothing against the nondegenerate homological (Floer) bounds. Jiao (arXiv, 27 Sep 2026, after this paper) independently reports a two-fixed-point example on $S^2\\times S^2$.","claimIssueNote":"The release says: 'Ma [Theorem 1.5] asserted that the minimum number of fixed points over all Hamiltonian diffeomorphisms always equals Crit(M). Theorem 1.1 contradicts that assertion' (degenerate paper) and 'Ma's preprint asserts that the minimum number of fixed points of a nondegenerate Hamiltonian diffeomorphism always equals the ordinary Morse number ... Theorem 1.1 contradicts that unrestricted assertion' (Morse-number paper). The other side: R. Ma, Proofs on Arnold conjectures, arXiv 0808.0613v7 (2013), Theorem 1.5, claims these lower bounds hold in general. Ma's preprint is unpublished; no referee has ruled between them.","solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjectures as the paper states them (Arnold 1986, Golovko 2020). It claims a smooth Hamiltonian diffeomorphism of the quadric $Q^3$ with exactly 3 fixed points while $\\mathrm{Crit}(Q^3)=\\mathrm{cuplength}(Q^3;\\mathbb Q)=4$. The proof was not refereed. Lean: ComparatorChallenges/ArnoldCounterexample.lean (OAI.ArnoldCounterexample.main, in lean/formalization.yaml) was read here; it states a Hamiltonian $\\phi$ of the literal quadric with fixed set of size 3, $4\\le$ the minimum critical number over all smooth functions, and a degenerate fixed point. It covers the critical-number form; the cup-length value is not formalized. Not rebuilt here. Graded contested because an earlier preprint (Ma) claims the opposite; see the claim note.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The degenerate (cup-length and critical-number) Arnold conjecture is one of the best-known open problems of symplectic topology, proved in many settings since Conley-Zehnder and Hofer; a counterexample on a monotone manifold changes the picture. Placed with Donaldson's tamed-to-compatible conjecture (45), below Arnold's nearby Lagrangian conjecture (62) and above Arnold's corank problem (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Three fixed points on the symplectic quadric threefold","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.ArnoldCounterexample.main)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/Arnold/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: ArnoldCounterexample.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ArnoldCounterexample.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"colin-de-verdiere-chromatic-conjecture","name":"The Colin de Verdiere Chromatic Conjecture","shortName":"Colin de Verdiere chromatic conjecture","problemNumber":null,"field":"Spectral graph theory, graph coloring","fieldGroup":"Combinatorics","statement":"Colin de Verdiere (1990) introduced the graph invariant $\\mu(G)$: the maximum nullity of a real symmetric matrix that is negative on edges, zero on non-edges, has exactly one negative eigenvalue and satisfies the Strong Arnold Property. It characterizes outerplanar ($\\mu\\le2$), planar ($\\mu\\le3$) and linklessly embeddable ($\\mu\\le4$) graphs. He conjectured that $\\chi(G)\\le\\mu(G)+1$ for every graph; since $h(G)\\le\\mu(G)+1$, this is implied by Hadwiger's conjecture, and it holds when $\\mu(G)\\le4$. Does every finite graph satisfy $\\chi(G)\\le\\mu(G)+1$?","posedBy":"Yves Colin de Verdiere, Sur un nouvel invariant des graphes et un critere de planarite, J. Combin. Theory Ser. B 50 (1990)","yearPosed":1990,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are graphs of arbitrarily large order with $\\alpha(G)\\le2$ and $\\mu(G)+1<|V(G)|/2\\le\\chi(G)$; the fractional chromatic number also exceeds $\\mu(G)+1$. The proof shows every well-signed matrix on the graph with one negative eigenvalue has rank above $|V(G)|/2+1$, without using the Strong Arnold Property, so the Lovasz-Schrijver parameter $\\kappa$ also satisfies $\\kappa(G)+1<|V(G)|/2$. Since $h\\le\\mu+1$, these graphs also violate Hadwiger's inequality, by a route independent of the connected-matching companion. Nothing is claimed for small $\\mu$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the conjecture as cited (Colin de Verdiere 1990, 1998; van der Holst-Lovasz-Schrijver 1999). The construction and rank argument were not refereed. formalization.yaml lists no main result for this manuscript. Examples exist only at large order; the paper notes the conjecture remains true where mu is at most 4.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A conjecture well known within spectral and structural graph theory since 1990, but weaker than Hadwiger's and less studied on its own. Placed level with Babai's Minimal Cayley Graph Problem (30), below the Petersen Coloring Conjecture (40), and far below Hadwiger's conjecture itself.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to the Colin de Verdiere chromatic conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"A counterexample to Hadwiger's conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Colin de Verdiere 1990, JCTB 50","url":"https://doi.org/10.1016/0095-8956(90)90093-F","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hypercube-ramsey-conjecture","name":"The Burr-Erdős hypercube Ramsey conjecture","shortName":"Hypercube Ramsey conjecture","problemNumber":181,"field":"Graph Ramsey theory","fieldGroup":"Combinatorics","statement":"Let $Q_n$ be the $n$-dimensional hypercube graph, with $2^n$ vertices, and $R(H)$ the two-colour Ramsey number. Burr and Erdős (1975) called a family of graphs an $L$-set if its Ramsey numbers are bounded by a constant times the order, conjectured that bounded-degeneracy families are $L$-sets (proved by Lee, 2017), and asked separately whether the cubes form an $L$-set. Known bounds: $R(Q_n)\\ge3\\cdot2^{n-1}-1$, and upper bounds improved from $n2^{2n+5}$ (Fox-Sudakov) and $2^{2n+6}$ (Conlon-Fox-Sudakov) to $2^{(2-c)n}$ (Tikhomirov). Is $R(Q_n)\\le C2^n$ for an absolute constant $C$?","posedBy":"S. A. Burr and P. Erdős, On the magnitude of generalized Ramsey numbers for graphs, Infinite and Finite Sets (Keszthely, 1973), Colloq. Math. Soc. Janos Bolyai 10, Section 7","yearPosed":1975,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is an absolute constant $C$ with $R(Q_n)\\le C2^n$ for every $n\\ge0$; with the two-block lower bound $3\\cdot2^{n-1}-1$ this determines the order of $R(Q_n)$. Consequences: linear Ramsey bounds for subgraphs occupying a fixed fraction of a cube, such as Cartesian powers of paths and other subgraphs of $Q_d$. Not shown: any numerical value of $C$, whether $R(Q_n)/2^n$ converges, or the general Burr-Erdős question for other dense families.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it, and the README cautions that unformalized results could have issues.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Burr and Erdős's cube question as the manuscript cites it and against erdosproblems.com Problem 181, which states it as $R(Q_n)\\ll2^n$ and listed it as open when fetched on 7 October 2026. The statement matches. The proof is long (eighteen sections of regime analysis, patch tilings and resampling dynamics) and was not refereed. No Lean formalization; the README cautions that unformalized results could have issues.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"One of the best-known problems in graph Ramsey theory, open for fifty years and the subject of a line of celebrated bounds (Conlon-Fox-Sudakov, Lee, Tikhomirov); well above a typical Erdős problem. Tied with the four-uniform Erdős matching conjecture (40), above Erdős Problem #74 (38), below Sidorenko's conjecture (52).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The hypercube Ramsey number has linear order","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Erdős Problem #181 (erdosproblems.com)","url":"https://www.erdosproblems.com/181","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strong-hyperkahler-syz-conjecture","name":"The strong hyperkahler SYZ conjecture","shortName":"Strong hyperkahler SYZ conjecture","problemNumber":null,"field":"Complex and algebraic geometry; hyperkahler manifolds, Lagrangian fibrations","fieldGroup":"Geometry & topology","statement":"Let $X$ be a compact irreducible holomorphic symplectic (hyperkahler) Kahler manifold of dimension $2n$ with Beauville-Bogomolov-Fujiki form $q$. A nonzero class $c_1(L)$ that is nef and isotropic, $q(c_1(L))=0$, has numerical dimension $n$, and should come from a Lagrangian fibration $X\\to B$ with $\\dim B=n$. This was known for all known deformation types (Bayer-Macri, Markman, Yoshioka, Mongardi-Rapagnetta, Mongardi-Onorati, with Soldatenkov-Verbitsky's deformation theorem), and Verbitsky proved nonvanishing under a semipositive metric. Is every holomorphic line bundle $L$ on such an $X$ with $c_1(L)\\ne0$ nef and isotropic semiample?","posedBy":"Misha Verbitsky (GAFA 2010, Conjecture 1.7); earlier nef-isotropic forms in Hassett-Tschinkel (2001, Remark 3.12) and Sawon (2003, Conjecture 4.1)","yearPosed":2001,"ageNote":"Year of Hassett-Tschinkel's Remark 3.12, the earliest explicit nef-isotropic semiampleness formulation the manuscript cites; the 'strong' form with this name is Verbitsky's Conjecture 1.7 (2010).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $c_1(L)\\ne0$, $c_1(L)$ is in the closure of the Kahler cone and $q_X(c_1(L))=0$, then $L$ is semiample: some $L^m\\simeq f^*A$ for a surjective connected-fibre map $f:X\\to B$ onto a normal projective $n$-dimensional $B$ with $A$ ample, every fibre component Lagrangian of dimension $n$. Corollary 1.2: in each dimension there are finitely many deformation types with $b_2\\ge5$. Corollary 1.3 (projective $X$, via the companion): $B\\simeq\\mathbb P^n$. Not shown: existence of a nef isotropic class on a given $X$, or anything for $b_2\\le4$ in the finiteness corollary.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The manuscript (September 23, 2026) proves the main theorem without companion inputs; its projective-case corollary uses the same-day companion 'Projective-space bases of Lagrangian fibrations', which in turn uses this paper's main theorem.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2-1.3 of 'The strong hyperkahler SYZ conjecture' were read against Verbitsky's Conjecture 1.7 as cited. The proof (a degeneration to a flat cylinder in logarithmic coordinates, a special Lagrangian torus by the Zhang-McLean method, hyperkahler rotation to a nonprojective manifold, Greb-Lehn-Rollenske, and Soldatenkov-Verbitsky deformation back to the given pair) was not refereed. No Lean formalization. The theorem has no restriction on dimension, deformation type or second Betti number; the finiteness corollary for deformation types with b2 >= 5 also uses Engel-Filipazzi-Greer-Mauri-Svaldi. The paper does not use the release's contested log abundance results.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"A central conjecture of hyperkahler geometry, known for every known deformation type but open in general; together with the finiteness corollary it would reshape the classification question. Tied with the Kuga-Satake Hodge conjecture entry (48), just below the Hodge conjecture for products of K3 surfaces (50), above Connes' Rigidity Conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026/main.pdf","sourceName":"OpenAI math release: The strong hyperkahler SYZ conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Same family: Projective-space bases of Lagrangian fibrations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf","kind":"paper"},{"label":"Verbitsky 2010, Hyperkahler SYZ conjecture and semipositive line bundles (GAFA)","url":"https://doi.org/10.1007/s00039-009-0037-z","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"siegel-yao-ball-packing-conjecture","name":"Siegel-Yao Conjecture A: symplectic packings of balls into a ball in dimension at least six","shortName":"Siegel-Yao ball packing conjecture","problemNumber":null,"field":"Symplectic geometry, embedding problems","fieldGroup":"Geometry & topology","statement":"Let $B^{2n}(R)$ be the closed ball of capacity $R$ (capacity $\\pi$ times squared radius) in $(\\mathbb C^n,\\omega_0)$. Two obstructions to embedding $\\bigsqcup_iB^{2n}(R_i)$ symplectically into the interior of $B^{2n}(R)$ are known: volume, $\\sum_iR_i^n 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 23, 2026).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Conjecture A as the manuscript cites it; the theorem states the conjectured criterion as an equivalence for all $n\\ge3$, $k\\ge1$ and positive capacities. Lean: the challenge BallPacking (theorem OAI.HigherDimensionalBallPacking.main_theorem, solution module OAI.Geometry.BallPacking.Main) exists at the pinned commit but is not in the formalization catalogue; its statement was read here. It defines smooth symplectic embeddings on open neighbourhoods of the closed balls with disjoint images in the open target ball and states the iff with the two inequalities, which is the headline. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A recent named conjecture giving the complete answer to the classical ball-packing problem in dimensions six and up, a central theme of symplectic rigidity since Gromov and McDuff-Polterovich, but posed only recently. Placed below Vicente's Gromov-width question (45) and the Talagrand SK overlap conjecture (30), above Kuperberg's six-cylinder conjecture (20).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Symplectic Ball Packings in Higher Dimensions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean formalization: BallPacking/Main.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/BallPacking/Main.lean","kind":"lean-proof"},{"label":"Siegel-Yao, On symplectic packing problems in higher dimensions (Math. Ann. 2025)","url":"https://doi.org/10.1007/s00208-025-03221-7","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-3","name":"Erdős Problem #3: sets with divergent reciprocal sum contain arbitrarily long arithmetic progressions","shortName":"Erdős Problem #3","problemNumber":3,"field":"Additive combinatorics","fieldGroup":"Combinatorics","statement":"If $A\\subseteq\\mathbb N$ satisfies $\\sum_{n\\in A}1/n=\\infty$, must $A$ contain arbitrarily long arithmetic progressions? This extends Szemeredi's theorem to sparse sets, including the primes. The case of three-term progressions was settled by Bloom and Sisask (2020); for general $k$ the best bound before this work, $r_k(N)\\ll_kN\\exp(-(\\log\\log N)^{c_k})$ of Leng, Sah and Sawhney, falls short of the summability over dyadic intervals that the conjecture needs, where $r_k(N)$ is the largest size of a subset of $\\{1,\\ldots,N\\}$ with no $k$-term progression. Does every set of positive integers with divergent reciprocal sum contain $k$-term arithmetic progressions for every $k$?","posedBy":"Paul Erdős","yearPosed":1974,"ageNote":"The manuscript cites Erdos, Problems, Math. Balkanica 4 (1974), Problem 4.33.6. The density version goes back to Erdos and Turan (1936), and the problem is often called the Erdos-Turan conjecture on arithmetic progressions; erdosproblems.com lists it as Problem #3 with a 5000 dollar prize.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1 claims, for each fixed $k\\ge3$, $r_k(N)\\le C_kN\\exp(-c_k(\\log N)^{\\varepsilon_k})$ with unspecified positive constants depending on $k$; equivalently a density-$\\alpha$ subset of $[N]$ has a $k$-term progression once $\\log N\\ge A_k(2+\\log(1/\\alpha))^{A_k}$. Summing over dyadic blocks gives Erdos's conjecture, and even the stronger criterion with weights $(\\log(2+a))^B/a$, and recovers Green-Tao for dense subsets of the primes. The method is a density increment on triangular polynomial cells with controlled precision. No improvement of the three-term exponent is claimed, the exponents $\\varepsilon_k$ are not optimized, and it does not give an asymptotic formula for $r_k(N)$ (Erdos Problem #142).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, which was posed some 4,000 open research problems during an evaluation, with the vast majority of results obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each. This manuscript is not among the README's named exceptions. It is credited to OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Corollary 1.2 was read against the posed problem and states it exactly; it follows in a few lines from Theorem 1.1 by summing over dyadic intervals. Lean-checked on the release's own Comparator challenge ErdosReciprocal (OAI.Erdos3.manuscriptReciprocalProgressionTheorem) together with its solution module, both present at the pinned commit; the challenge is not listed in the release's formalization catalogue, the statement was read here but not independently audited, and the development was not rebuilt here. The quantitative density bound is outside the formal statement.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":65,"significanceNote":"Erdős's most famous conjecture on arithmetic progressions and one of his highest-prize problems, open for half a century and a long-standing benchmark for quantitative Szemerédi bounds. Scored with the Jacobian Conjecture at 65, above Erdős Problem #548: the Erdős–Sós conjecture at 58 and The Sum-Product Conjecture over the Reals at 55, and above Erdős Problem #1: sum-distinct sets at 50.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Quasipolynomial Bounds for Arithmetic Progressions","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Erdős Problem #3 on erdosproblems.com","url":"https://www.erdosproblems.com/3","kind":"problem-record"},{"label":"Lean comparator statement: Erdos reciprocal-sum progression theorem (not listed in formalization.yaml)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/ErdosReciprocal.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/159.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"calderon-bounded-measurable-conductivities","name":"Calderon's uniqueness question for bounded measurable scalar conductivities in dimension three","shortName":"Calderon problem for L-infinity conductivities","problemNumber":null,"field":"Inverse problems; elliptic PDE","fieldGroup":"Differential equations","statement":"Calderon (1980) posed the inverse conductivity problem for real scalar conductivities $\\gamma\\in L^\\infty(\\Omega)$ with $0 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This entry rests on one manuscript dated September 23, 2026; the family's two smooth-uniqueness manuscripts are a separate entry.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of 'Nonuniqueness for bounded measurable scalar conductivities in three dimensions' were read against Calderon's problem. Lean: formalization.yaml lists ComparatorChallenges/Conductivity.json, declaration OAI.ScalarConductivity.main_nonuniqueness in OAI/Analysis/Conductivity/Main.lean. The challenge statement was read: two measurable essentially bounded functions on the ball of radius 3 in R3, between constants 00$ almost surely. In one dimension transient walks can have zero speed (Solomon). Sznitman's conditions $(T)$ and $(T')$ and his effective criterion give ballisticity from quantitative slab-exit bounds, and Guerra and Ramirez proved that the polynomial condition implies $(T')$, but all such results start from quantitative assumptions. Fribergh and Kious (2016, Conjecture 1.1) formulated the conjecture with the bare one-direction transience hypothesis. In dimension $d\\ge2$, does almost-sure transience in a fixed direction $\\ell$ imply a deterministic limiting velocity $v$ with $v\\cdot\\ell>0$?","posedBy":"Folklore of Sznitman's ballisticity programme; stated as Conjecture 1.1 by Alexander Fribergh and Daniel Kious","yearPosed":2016,"ageNote":"The manuscript cites Fribergh-Kious, Probab. Theory Related Fields 165 (2016), Conjecture 1.1, for the formulation. The question is older, implicit in Sznitman's work on conditions (T) and (T') around 2001-2002, but the manuscript gives no earlier statement.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $d\\ge2$, iid uniformly elliptic nearest-neighbour rows (no independence within a row needed) and a fixed unit $\\ell$ with $P_0(A_\\ell)=1$, there is a deterministic $v$ with $v\\cdot\\ell>0$ and $X_n/n\\to v$ almost surely, so the walk is ballistic in direction $\\ell$. For $d\\ge3$ positive probability of $A_\\ell$ suffices, the velocity is unique, and the directions with positive transience probability are exactly the open hemisphere $\\{u:u\\cdot v>0\\}$, with probability one there and zero elsewhere. Not shown: the strictly (not uniformly) elliptic case, non-iid environments, or central limit theorems and slowdown estimates.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The family has three manuscripts (September 23 and October 5, 2026).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against Fribergh-Kious Conjecture 1.1 as the manuscript states it. The proof (record regeneration, a spatial-radius estimate, two-walk comparisons along Gaussian scales) was not refereed. lean/formalization.yaml lists comparator DirectionalBallisticity, declaration OAI.DirectionalTransience.directional_transience_implies_ballisticity (file OAI/Probability/Ballisticity/Main.lean). Its statement was read here: for all $d\\ge2$, every probability law on transition rows with a uniform lower bound $\\kappa>0$ on all entries, and every unit $\\ell$, if annealed-almost surely $X_n\\cdot\\ell\\to\\infty$ then there is $v$ with $v\\cdot\\ell>0$ and $X_n/n\\to v$ annealed-almost surely. That states the headline claim. A second challenge, VelocityHemisphere.json (solution module OAI.Probability.Ballisticity.VelocityHemisphere, present, not in formalization.yaml), states the $d\\ge3$ positive-probability and hemisphere refinement. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":42,"significanceNote":"Transience-implies-ballisticity is the best-known open problem of multidimensional random walks in random environment, the target of Sznitman's condition (T) programme for about twenty-five years, but known mostly within that area. Placed level with The asymptotic Gotsman-Linial conjecture (42), below No bigeodesics in planar first-passage percolation (45), above The Courtade-Kumar conjecture (40).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Directional-transience-implies-ballisticity-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Directional transience implies ballisticity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.DirectionalTransience.directional_transience_implies_ballisticity)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/Ballisticity/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: DirectionalBallisticity.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DirectionalBallisticity.lean","kind":"lean-proof"},{"label":"Comparator statement: VelocityHemisphere.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/VelocityHemisphere.lean","kind":"lean-proof"},{"label":"Fribergh-Kious, Local trapping for elliptic random walks in random environments (arXiv)","url":"https://arxiv.org/abs/1404.2060","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"relativistic-vlasov-maxwell-global-regularity","name":"Global classical solutions of the 3D relativistic Vlasov-Maxwell system for large data","shortName":"Relativistic Vlasov-Maxwell global regularity","problemNumber":null,"field":"Kinetic theory, nonlinear PDE","fieldGroup":"Differential equations","statement":"The relativistic Vlasov-Maxwell system couples a particle density $f(t,x,v)\\ge0$ on $\\mathbb R^3_x\\times\\mathbb R^3_v$, transported by $\\partial_tf+\\hat v\\cdot\\nabla_xf+(E+\\hat v\\times B)\\cdot\\nabla_vf=0$ with $\\hat v=v/\\sqrt{1+|v|^2}$, to Maxwell's equations with charge and current densities computed from $f$. Wollman proved local existence of classical solutions, DiPerna and Lions global weak solutions, and Glassey and Strauss showed that a classical solution continues as long as its momentum support stays bounded and proved global existence for small (dilute) data. Global classical existence was known only under smallness, symmetry or reduced dimension. For arbitrary smooth, compactly supported particle data and smooth finite-energy fields satisfying the constraints, does the classical solution exist for all time?","posedBy":"Classical open problem framed by R. Glassey and W. Strauss (1986 continuation criterion); no single poser named in the manuscript","yearPosed":null,"ageNote":"The manuscript calls this the large-data global classical regularity problem and traces it through Wollman (1984), Glassey and Strauss (1986, 1987) and DiPerna and Lions (1989), but cites no paper that states it as an open problem. It has been open at least since Glassey and Strauss's 1986 criterion; yearPosed is left blank.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that every smooth admissible datum ($f_0\\in C_c^\\infty$, $f_0\\ge0$, fields in $C_b^\\infty\\cap L^2$ satisfying $\\nabla\\cdot E_0=\\rho_{f_0}$, $\\nabla\\cdot B_0=0$) generates a unique global classical solution of the one-species relativistic Vlasov-Maxwell system in three dimensions, smooth on every $[0,T]$ with compact phase-space support there. The key estimate controls the signed momentum impulse along characteristics, integrating the retarded force along source trajectories before taking absolute values, which rules out momentum blow-up and lets the Glassey-Strauss criterion apply. It treats one species without background charge; it does not give uniform-in-time momentum bounds (the support may grow with $T$) or asymptotic behaviour, and it does not cover the non-relativistic or multi-species systems explicitly.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, which was posed some 4,000 open research problems during an evaluation, with the vast majority of results obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each. This manuscript is not among the README's named exceptions. It is credited to OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem and claims global existence, uniqueness and smoothness on every finite interval with no size or symmetry restriction. formalization.yaml lists OAI.RVM.global_classical_solution as the main result. Its 112-line comparator statement defines the system itself from Mathlib primitives (fderiv, Bochner integrals, Lp): admissible data are smooth, compactly supported, nonnegative f0 and smooth bounded-derivative L2 fields satisfying both Gauss constraints; the conclusion is a C1 classical solution with L2-continuous fields, compact phase support on finite horizons, smoothness on every slab, and uniqueness among classical solutions. That states the headline claim; the hand-written definitions were not audited against the paper here. Not rebuilt here. Scope the paper states: one species, no background charge; the proof uses the Glassey-Strauss criterion as stated by Luk and Strain.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The central open problem of collisionless plasma kinetics for four decades, resisted by the Glassey-Strauss program and known only for small, symmetric or lower-dimensional data, but famous within one PDE community rather than across mathematics. Scored with The Sum-Product Conjecture over the Reals at 55, above Finite-time blowup for the inviscid Boussinesq system with smooth forcing at 45, and well below Finite-time blowup for the 3D incompressible Euler equations from smooth data at 80.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Global classical solutions of the three-dimensional relativistic Vlasov-Maxwell system","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean: main declaration OAI.RVM.global_classical_solution","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/VlasovMaxwell/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: global classical Vlasov-Maxwell solutions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/VlasovMaxwell.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/362.md","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"periodic-tiling-conjecture-dimension-three","name":"The periodic tiling conjecture in dimension three","shortName":"Periodic tiling conjecture, dimension 3","problemNumber":null,"field":"Translational tilings of lattices and Euclidean space","fieldGroup":"Combinatorics","statement":"A finite nonempty $F\\subset\\mathbb Z^d$ is a translational tile if $A\\oplus F=\\mathbb Z^d$ for some $A$; the tiling is periodic if $A$ is invariant under a finite-index subgroup. The periodic tiling conjecture (Lagarias and Wang, 1996, also in a Euclidean form for bounded measurable tiles of $\\mathbb R^d$) asserts that every translational tile admits a periodic tiling. It holds in $\\mathbb Z$ (Newman) and $\\mathbb Z^2$ (Bhattacharya), and Greenfeld and Tao disproved it in sufficiently high dimension. Does every finite translational tile of $\\mathbb Z^3$, and every bounded tile of $\\mathbb R^3$, admit a periodic tiling?","posedBy":"Jeffrey C. Lagarias and Yang Wang","yearPosed":1996,"ageNote":"Lagarias and Wang, Tiling the line with translates of one tile, Invent. Math. 124 (1996), p. 346, as cited by the manuscript. The dimension-three case became the open frontier after Greenfeld and Tao's high-dimensional counterexample (Annals 2024).","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a finite nonempty $T\\subset\\mathbb Z^3$ that tiles $\\mathbb Z^3$ by translations, but no translation set of a tiling is invariant under a finite-index subgroup; and $T+[0,1]^3$ tiles $\\mathbb R^3$ up to null sets, with no tiling whose translation set is invariant under a full-rank lattice, even allowing real translations. With Bhattacharya's planar theorem, three is the least lattice dimension where periodic tiling fails. The tile is not claimed to be connected, and the result concerns full periodicity only, not the absence of every individual period.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 23, 2026. The paper says it follows the Sudoku strategy of Greenfeld and Tao and adds a new encoding keeping the finite factor cyclic.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture; it gives a finite tile of $\\mathbb Z^3$ that tiles but admits no fully periodic tiling, and its unit-cube thickening has the same property in $\\mathbb R^3$ with arbitrary real translations. lean/formalization.yaml lists a main result for this manuscript (comparator PeriodicTilingThree, declaration OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality). The comparator statement was read here: it asserts a nonempty finite $T\\subset\\mathbb Z^3$ with a tiling and no finite-index-invariant tiling, an almost-everywhere tiling of $\\mathbb R^3$ by its thickening with no tiling invariant under a full-rank lattice, and that 3 is the least $d$ admitting such a lattice tile. This states the headline claim, plus minimality. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"The periodic tiling conjecture is a well-known tiling problem whose high-dimensional failure was an Annals paper; the remaining low dimensions, with dimension three now the least possible, were the natural open frontier. Placed above the three-dimensional strongly aperiodic monotile question (35), just below the Artin groups CAT(0) conjecture (40), and below whole-conjecture entries such as Brennan's conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A translational tile with no fully periodic tiling in dimension three","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.PeriodicTilingThree.periodic_tiling_counterexample_and_minimality)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/PeriodicTiling/Counterexample.lean","kind":"lean-proof"},{"label":"Comparator statement: PeriodicTilingThree.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/PeriodicTilingThree.lean","kind":"lean-proof"},{"label":"Greenfeld and Tao, A counterexample to the periodic tiling conjecture (high dimension)","url":"https://arxiv.org/abs/2211.15847","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lagrangian-fibration-base-projective-space","name":"The projective-space base conjecture for Lagrangian fibrations of compact hyperkahler manifolds","shortName":"Base of a Lagrangian fibration is P^n","problemNumber":null,"field":"Complex and algebraic geometry; hyperkahler manifolds, Lagrangian fibrations","fieldGroup":"Geometry & topology","statement":"Let $f:X\\to B$ be a fibration with connected fibres from a compact irreducible holomorphic symplectic Kahler manifold of dimension $2n$ onto a normal base with $0<\\dim B<2n$. Matsushita showed $\\dim B=n$ and the fibres are Lagrangian; Hwang proved $B\\cong\\mathbb P^n$ when $X$ is projective and $B$ is smooth, Greb-Lehn extended this to Kahler $X$, and the conclusion was known for the $K3^{[n]}$, Kummer, OG6 and OG10 types and in dimension four (Ou, Huybrechts-Xu). The projective-space base conjecture asks: is the normal base $B$ of such a fibration always isomorphic to $\\mathbb P^n$, without assuming $B$ smooth?","posedBy":"Not named in the manuscript; it states the projective-space base conjecture in the setting of Matsushita (1999), Hwang (2008) and Greb-Lehn (2014)","yearPosed":null,"ageNote":"The manuscript calls this the projective-space base conjecture but cites no poser or year; Hwang's 2008 smooth-base theorem is the earliest cited partial result. Left null rather than guessed.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if $f:X\\to B$ is a projective surjective connected-fibre morphism from a compact IHS Kahler manifold of dimension $2n$ onto a normal projective $n$-dimensional $B$, with every fibre component $n$-dimensional and isotropic for $\\sigma$, then $B\\cong\\mathbb P^n$, with no smoothness assumption on $B$, no bound on $b_2$ and no restriction on deformation type. With Kim-Oguiso-Shinder it excludes codimension-one multiple fibres. Not shown: the case of a nonprojective $X$ (Kahler $X$ with a non-projective morphism).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored 'OpenAI' and name no human author. The manuscript (September 23, 2026) uses the same-day companion 'The strong hyperkahler SYZ conjecture' (its Theorem 1.1 and Lemma 4.2) as an input.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Projective-space bases of Lagrangian fibrations' and its introduction were read against the conjecture as the paper states it. The proof (local cotangent tensors and a Deligne-Mumford stratum, an Euler-characteristic obstruction via reduction to positive characteristic, a second Lagrangian fibration from deformations, then Hwang's theorem) was not refereed. No Lean formalization. It depends on the unreviewed SYZ companion. Scope the paper itself states: projective fibrations (relatively ample bundle) onto a normal projective base with equidimensional Lagrangian fibres, so X is projective; the nonprojective Kahler case of the conjecture is not claimed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A well-known conjecture of hyperkahler geometry since Hwang's smooth-base theorem, settled before only for known deformation types and in dimension four. Tied with Voiculescu's free entropy dimension question (38) and Stein's dimension-free Riesz transform problem (38), below the strong SYZ conjecture from this family (48), above the Hyperkahler Period-Index Conjecture (22).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf","sourceName":"OpenAI math release: Projective-space bases of Lagrangian fibrations","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Input: The strong hyperkahler SYZ conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cardy-formula-square-lattice-bond-percolation","name":"Cardy's formula and conformal invariance for critical bond percolation on the square lattice","shortName":"Cardy's formula on the square lattice","problemNumber":null,"field":"Percolation; conformal invariance","fieldGroup":"Probability & statistics","statement":"Cardy (1992) predicted that for critical two-dimensional percolation the probability of an open crossing between two boundary arcs of a conformal quadrilateral converges to an explicit conformally invariant function, $\\int_0^x[t(1-t)]^{-2/3}dt\\,/\\int_0^1[t(1-t)]^{-2/3}dt$ in the half-plane coordinate $x$. Smirnov (2001) proved it for site percolation on the triangular lattice, and Camia-Newman built the full loop limit there, but the proof uses the lattice's special symmetry. For the most standard model, bond percolation on $\\mathbb Z^2$ at $p=1/2$, only partial or disputed claims existed. Does Cardy's formula, and conformal invariance of the exploration interface (convergence to $\\mathrm{SLE}_6$), hold for critical bond percolation on the square lattice?","posedBy":"John Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992), as a physical prediction; the square-lattice case is the q = 1 case of Rohde-Schramm Conjecture 9.7 (2005)","yearPosed":1992,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.2: for critical bond percolation on $\\delta\\mathbb Z^2$ in a Jordan domain with four uniformly approximated marks, all boundary edges sampled and no external connections, the probability of an open crossing between the arcs $[a,b]$ and $[c,d]$ converges to Cardy's hypergeometric function of the cross-ratio. The $q=1$ case of Theorem 1.1 gives chordal $\\mathrm{SLE}_6$ convergence of the exploration interface and CLE$_6$ convergence of all nested loops. The crossing result is proved directly, not from the curve limit. It does not treat other lattices or general boundary conditions beyond those stated.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.2 (Cardy's formula) and the $q=1$ case of Theorem 1.1 of the September 23 manuscript were read against Cardy's prediction as the paper states it; they claim the crossing limit in every Jordan quadrilateral with uniformly approximated marks under the stated free-boundary convention, and $\\mathrm{SLE}_6$ convergence of the interface. The proof was not refereed. The manuscript itself notes overlapping prior public claims for square-lattice bond percolation (Tsai-Yam-Zhou, arXiv 1112.2017; Zhou, arXiv 2409.03235) that were not checked here. No Lean formalization exists.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"Conformal invariance of square-lattice bond percolation is a famous open problem of probability, repeatedly singled out since Smirnov's triangular-lattice proof as the natural next case. Placed just below the Rohde-Schramm entry from the same paper (52), level with Kakeya maximal in three dimensions (48) and the linear list Hadwiger conjecture (48), and below the Polyakov conjecture (62).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Square-lattice FK interfaces and nested loops for 1 <= q < 4","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf","kind":"paper"},{"label":"Companion: Natural Occupation Measures for Critical Square-Lattice FK Interfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"planar-unit-distances-power-saving","name":"A power saving over the Spencer-Szemerédi-Trotter n^(4/3) bound for planar unit distances (Erdős Problem 90)","shortName":"Unit distances: beating n^(4/3)","problemNumber":90,"field":"Combinatorial geometry; unit distances and incidences","fieldGroup":"Combinatorics","statement":"Let $u(n)$ be the largest number of pairs at distance exactly $1$ among $n$ points of the Euclidean plane. Erdős introduced the problem in 1946, showed $u(n)\\ge n^{1+c/\\log\\log n}$ and conjectured $u(n)=n^{1+o(1)}$ (erdosproblems.com Problem 90). Spencer, Szemerédi and Trotter proved $u(n)=O(n^{4/3})$ in 1984, and no better exponent was known; Valtr's metric with $n^{4/3}$ unit pairs shows that an improvement must use a special feature of the Euclidean metric, as the problem record notes. In 2026 an OpenAI construction disproved Erdős's conjecture with $u(n)\\ge n^{1+\\varepsilon}$ for infinitely many $n$ (Sawin: exponent $1.0141$). Can the upper exponent $4/3$ be lowered by a fixed amount: is $u(n)=O(n^{4/3-\\delta})$ for some absolute $\\delta>0$?","posedBy":"Paul Erdős (problem, 1946); the 4/3 upper bound is due to Spencer, Szemerédi and Trotter (1984)","yearPosed":1946,"ageNote":"The manuscript cites Erdős (Amer. Math. Monthly 1946) for the problem and Spencer-Szemerédi-Trotter (1984) for the exponent 4/3, and phrases its result as answering whether that exponent can be lowered, without a separate posing reference. The erdosproblems.com record of Problem 90 names O(n^(4/3)) as the best known upper bound.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there are absolute constants $0 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript is 'A power saving for planar unit distances' (September 23, 2026). It cites the earlier OpenAI construction disproving Erdős's n^(1+o(1)) conjecture and the human exposition by Alon et al.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the unit-distance problem as posed: it gives absolute $C$ and $1\\le\\beta<4/3$ with $u(n)\\le Cn^\\beta$ for all $n$, a power saving with no explicit $\\beta$. Lean: lean/ComparatorChallenges/PlanarUnitDistances.json exists and its solution module OAI.Geometry.UnitDistances.Main is present at the pinned commit; the challenge is not in the formalization catalogue (formalization.yaml). The statement file was read here: theorem main asserts that there are $C,\\beta$ with $0 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 and Section 5 of the TeX source read against the problem as stated. Theorem 1.2 proves the type $\\mathrm{II}_1$ factor case; the full answer for all von Neumann algebras with separable predual follows only through Willig's published direct-integral reduction and the earlier type I and properly infinite cases, which the paper cites and does not reprove. The proof was not refereed. Lean: FactorGeneration (OAI.Generator.single_generation_of_II1_separable_predual) is a main result in the formalization catalogue. Its statement was read here: a WOT-closed unital star-subalgebra of B(H) that is a factor, finite and without minimal projections, with separable predual, equals the von Neumann algebra generated by one element. That is Theorem 1.2, not the reduction to general algebras. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"A long-standing, well-known problem in von Neumann algebras, open for the free group factors for decades. Placed level with the Kadison-Kaplansky conjecture (50), below Kadison's similarity problem (58) and the free group factor isomorphism problem (68), and above Connes' rigidity conjecture (45) and the free entropy dimension question (38), which it implies.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Relative generation and the generator problem for finite factors","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator statement FactorGeneration","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/FactorGeneration.lean","kind":"lean-proof"},{"label":"Lean Comparator statement RelativeGeneration (not in formalization.yaml main results)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/RelativeGeneration.lean","kind":"lean-proof"},{"label":"Lean scope note for family 296","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/296.md","kind":"other"},{"label":"Sherman 2012, Canad. J. Math. 64 (history of the problem)","url":"https://doi.org/10.4153/CJM-2011-048-2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"talagrand-discrete-convexity-conjecture","name":"Talagrand's discrete convexity conjecture","shortName":"Talagrand's discrete convexity conjecture","problemNumber":null,"field":"Probabilistic combinatorics; Bernoulli product measures and covers","fieldGroup":"Combinatorics","statement":"Let $\\mu_p$ be the product Bernoulli measure on subsets of $[N]$. A family $\\mathcal A$ is $p$-small if there are sets $I$ with $\\sum p^{|I|}\\le1/2$ such that every member of $\\mathcal A$ contains one of them. For a family $\\mathcal D$ and an integer $k$, let $E_k(\\mathcal D)$ be the sets contained in no union of $k$ members of $\\mathcal D$. By analogy with his Gaussian convexity question, Talagrand (2010, Conjecture 7.1; also Research Problem 13.3.2 in his 2021 book) conjectured that a fixed number of members of a sufficiently likely family covers everything outside a $p$-small family. Is there a universal integer $k$ such that $\\mu_p(\\mathcal D)\\ge1-1/k$ implies that $E_k(\\mathcal D)$ is $p$-small, for every $N$, every $p\\in(0,1)$ and every family $\\mathcal D$?","posedBy":"Michel Talagrand (Conjecture 7.1, STOC 2010; Research Problem 13.3.2, Upper and Lower Bounds for Stochastic Processes, 2nd ed., 2021)","yearPosed":2010,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: with $k=2^{75}$, for every $N\\ge1$, $p\\in(0,1)$ and $\\mathcal D\\subseteq2^{[N]}$, $\\mu_p(\\mathcal D)\\ge1-1/k$ implies $E_k(\\mathcal D)$ is $p$-small. Consequences: a containment cover for large values of positive selector processes (recovering Park-Pham's theorem), and with Li's fractional result and the companion rounding theorem, $E_2(\\mathcal D)$ is $(p/C)$-small when $\\mu_p(\\mathcal D)>1/2$. Earlier, Hua-Song-Tudose derived a discrete statement with three unions at density $p^C$ and Park one with two unions at a $\\log\\log(1/p)$ loss. The constant $k$ is far from optimal and the Gaussian convexity question is a different problem.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Its main theorem is the essential input of the family's graph-decomposition manuscript (October 5, 2026).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Conjecture 7.1 as the manuscript states it, at the original density $p$ (no loss in $p$), with $k=2^{75}$ and no monotonicity on $\\mathcal D$. lean/formalization.yaml lists TalagrandDiscreteConvexity.json (declaration OAI.TalagrandDiscreteConvexity.talagrand_discrete_convexity, file OAI/Combinatorics/DiscreteConvexity/Main.lean). The statement TalagrandDiscreteConvexity.lean was read here; not rebuilt here. For every $N\\ge1$, $0 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: the main theorem and its p-adic reduction were read against the conjecture as stated by the manuscript's references. The proof, built on Witt groups of sheaf categories and a fixed signature lattice, was not refereed. No Lean formalization is listed. Scope stated by the paper: second-countable Hausdorff manifolds without boundary, and second-countable groups.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":65,"significanceNote":"The central open problem of transformation groups since the mid-twentieth century, famous across geometric topology, with only dimension 3 settled in the last forty years. Ties the Jacobian Conjecture (65), above Kothe (60) and Griffiths' positivity conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The Hilbert-Smith conjecture in every finite dimension","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"six-vertex-gff-limit-balanced","name":"Gaussian free field limit of the six-vertex height function throughout the regime a = b = 1, 0 < c <= 2","shortName":"Six-vertex GFF limit, balanced regime","problemNumber":null,"field":"Statistical mechanics; six-vertex model and random surfaces","fieldGroup":"Probability & statistics","statement":"In the square-lattice six-vertex model with weights $a=b=1$ and $c>0$, the ice rule makes the arrows a height function $h$ on faces. The Coulomb-gas picture predicts that in the critical range $|\\Delta|\\le1$, $\\Delta=1-c^2/2$ (that is $02$.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the prediction. For every $c\\in(0,2]$ it constructs the plane law $\\mathbb P_c$ as an iterated limit of balanced tori and proves convergence of $h_\\delta$ to $\\sigma(c)\\Gamma$ in finite-dimensional laws, in negative Holder, Besov and Sobolev restriction spaces, and in moments of increments. Scope limits the paper states: one zero-slope plane state only, no uniqueness among Gibbs states or slopes, no uniformity as $c\\to0$, and isotropic weights $a=b$ only. At $c=2$ and for the amplitude on $1\\le c<2$ it uses Duminil-Copin-Kozlowski-Lammers-Manolescu (2026), whose rotational-invariance input is cited as in preparation; the manuscript supplies its own endpoint rotation argument. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Gaussian free field fluctuations of the six-vertex model across its critical range are a flagship conjecture of two-dimensional statistical mechanics, long out of reach beyond the free-fermion point. Below the Rohde-Schramm FK interface entry at 52 and the XY critical decay entry at 50, which are older and more famous, and level with the Courtade-Kumar entry at 40.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf","sourceName":"OpenAI math release: The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Di Francesco, Saleur, Zuber, Coulomb gas picture and conformal invariance (1987)","url":"https://doi.org/10.1007/BF01009954","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sidorenko-conjecture","name":"Sidorenko's conjecture","shortName":"Sidorenko's conjecture","problemNumber":null,"field":"Extremal graph theory, homomorphism densities","fieldGroup":"Combinatorics","statement":"For finite simple graphs $H$ and $G$, let $t(H,G)$ be the probability that a uniformly random map $V(H)\\to V(G)$ is a homomorphism, and $p(G)=2|E(G)|/|V(G)|^2$ the edge density. Sidorenko conjectured that every bipartite graph $H$ with at least one edge satisfies $t(H,G)\\ge p(G)^{|E(H)|}$ for every $G$: a random graph of the same density minimises the number of copies of $H$. It was known for trees, even cycles, complete bipartite graphs, hypercubes, graphs with a vertex complete to the other side (Conlon-Fox-Sudakov), weakly norming graphs, tree-arrangeable graphs and many other classes. Does every bipartite graph satisfy Sidorenko's inequality?","posedBy":"Alexander Sidorenko, Inequalities for functionals generated by bipartite graphs (Diskret. Mat., 1991) and A correlation inequality for bipartite graphs (Graphs Combin., 1993)","yearPosed":1991,"ageNote":"The manuscript cites Sidorenko's 1991 and 1993 papers for the conjecture and places its roots in the supersaturation questions of Erdos, Lovasz, Spencer and Simonovits (1979 to 1984); the form with injective copies is often credited to Erdos and Simonovits.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for the explicit bipartite incidence graph $H$ of 22 triples on 13 points ($35$ vertices, $66$ edges, connected, degrees 4 to 6 on one side and 3 on the other) there is a finite simple graph $G$ with an edge and $t(H,G) 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two September 23 manuscripts cover $1\\le q<4$ and $01$, and Kolmogorov (1923) gave an integrable function whose Fourier series diverges almost everywhere. Between these, convergence was known in $L\\log L\\log\\log L$ (Sjolin 1969), in $L\\log L\\log\\log\\log L$ (Antonov 1996) and in the quasi-Banach space $\\mathrm{QA}$ of Arias de Reyna. Does $S_Nf(x)\\to f(x)$ for almost every $x$, along the full sequence $N=0,1,2,\\ldots$, for every $f$ in the Orlicz class $L\\log L(\\mathbb T)=\\{f:\\int|f|\\log(2+|f|)<\\infty\\}$?","posedBy":"Classical question after Carleson, Hunt, Sjolin and Antonov; stated as Conjecture 1 (with a weak-L1 maximal formulation) by Victor Lie (2017)","yearPosed":null,"ageNote":"The manuscript cites no first posing. The earliest explicit statement it cites is V. Lie, Pointwise convergence of Fourier series (I), JEMS 19 (2017), Definition 1.1 and Conjecture 1. The question is implicit in the work it lists from Sjolin (1969) and Antonov (1996) onward; the original posing date is not given in the literature read here.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every complex-valued $f\\in L\\log L(\\mathbb T)$ there is a null set $N_f$ with $\\lim_{N\\to\\infty}S_Nf(x)=f(x)$ for $x\\notin N_f$, along the full sequence. The key estimate is a good-set bound $\\int_{\\{M\\rho\\le e^{4h}\\}}\\sup_N|S_N\\rho|\\le Cd$ for bounded densities of height $e^d$, summed over height layers. The paper also deduces a weak-$L^1$ maximal inequality in Luxemburg norm. It does not address spaces strictly between $L\\log L$ and $L^1$ (where Konyagin's divergence results apply below $L\\sqrt{\\log L}$), nor lacunary or higher-dimensional analogues.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 23, 2026) has no companion in the release.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript (TeX source) was read against the question; it states almost-everywhere convergence of the ordinary symmetric partial sums along the full sequence for every complex-valued $f\\in L\\log L(\\mathbb T)$, which is the question as posed. Lean-checked on the release's Comparator challenge FourierLLogL (declaration OAI.FourierLLogL.paperMain, solution module OAI.Analysis.FourierLLogL.PaperMain), added in the 7 October 2026 update and listed in lean/formalization.yaml. Its statement was read here: for every a.e.-strongly measurable $f$ on $\\mathbb R/2\\pi\\mathbb Z$ with $\\int|f|\\log(2+|f|)<\\infty$ (Haar measure) there is a measurable null set outside which $\\sum_{|k|\\le N}\\widehat f(k)e^{ikx}\\to f(x)$ as $N\\to\\infty$. That is the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The corollary's Luxemburg-norm weak-$L^1$ maximal bound is not part of the Comparator statement. The proof (an entropy-compression argument for a dyadic model, transferred to Fourier sums) was not refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"Whether L log L suffices for almost everywhere convergence of Fourier series has been the natural endpoint question in the area since Carleson and Hunt, with decades of incremental spaces (Sjolin, Antonov, Arias de Reyna). Placed below the Bochner-Riesz conjecture in three dimensions (62) and the Falconer distance conjecture (60), level with Sidorenko's conjecture (52), and above Brennan's conjecture (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Almost-everywhere Fourier convergence in L log L","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Lie (2017), Pointwise convergence of Fourier series (I), Conjecture 1","url":"https://doi.org/10.4171/JEMS/703","kind":"problem-record"},{"label":"Lean Comparator challenge FourierLLogL (OpenAI math release)","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/ComparatorChallenges/FourierLLogL.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cartan-hadamard-isoperimetric-conjecture","name":"The generalized Cartan-Hadamard isoperimetric conjecture","shortName":"Cartan-Hadamard conjecture","problemNumber":null,"field":"Riemannian geometry; isoperimetric inequalities","fieldGroup":"Geometry & topology","statement":"Let $(M^n,g)$ be a Cartan-Hadamard manifold: complete, simply connected, with sectional curvature $\\le\\kappa\\le0$. The Euclidean Cartan-Hadamard conjecture (usually attributed to Aubin, also to Gromov and to Burago-Zalgaller) asserts that every bounded smooth domain $\\Omega\\subset M$ satisfies the Euclidean isoperimetric inequality $|\\partial\\Omega|\\ge n\\omega_n^{1/n}|\\Omega|^{(n-1)/n}$, with equality only for regions isometric to Euclidean balls. Its generalized form replaces the right side by the boundary area of the equal-volume geodesic ball in the simply connected space form of curvature $\\kappa$. It was known in dimension two (Weil, Beckenbach-Rado, Bol), three (Kleiner) and, for $\\kappa=0$, four (Croke). Does the comparison with the model ball of curvature $\\kappa$ hold in every dimension?","posedBy":"Thierry Aubin (Euclidean form, 1976); the curvature-kappa generalization is usually credited to Gromov and to Burago-Zalgaller","yearPosed":1976,"ageNote":"The manuscript does not attribute the conjecture; it cites Aubin's 1976 paper 'Problemes isoperimetriques et espaces de Sobolev' (the paper usually credited with the conjecture) only for the sharp Sobolev constant. The attribution and year come from the standard literature, not from this manuscript.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for $n\\ge2$, $\\kappa\\le0$ and $(M^n,g)$ complete simply connected with $\\mathrm{Sec}\\le\\kappa$, every measurable $E$ of finite volume and finite perimeter has $P(E)\\ge\\mathcal I_{\\sqrt{-\\kappa}}(\\mathrm{vol}(E))$, the model-ball profile. Theorem 1.2: for $\\mathrm{Sec}\\le0$, a bounded positive-volume set attains the Euclidean bound iff it agrees a.e. with an open region whose induced metric is a round Euclidean ball (the ambient manifold need not be flat). Corollaries give Faber-Krahn and Saint-Venant comparisons with the model ball. Equality rigidity for $\\kappa<0$ is not claimed. The companion proves the sharp Euclidean filling bound for integral $n$-cycles, $n\\ge2$, in any proper CAT(0) space and deduces the Euclidean inequality for domains in dimension $d\\ge3$. Recent preprints cited by the paper (Chen-Ghomi-Wang, dimensions 3 to 9) overlap in low dimensions.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The CAT(0) filling companion (same date) gives a second, independent route to the Euclidean case in dimension at least three.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 of the principal manuscript were read against the conjecture; Theorem 1.1 is the generalized comparison for every $n\\ge2$ and $\\kappa\\le0$, for all finite-volume finite-perimeter sets. The proof was not refereed. The Lean work covers the companion, not the principal manuscript. lean/docs/337.md points to two challenges, ComparatorChallenges/SharpCAT0Filling.json (theorem OAI.CAT0Fillings.sharp_integral_filling, solution module OAI.Geometry.CAT0Fillings.Main) and FillingCoefficient.json (OAI.SharpIntegralFillings.coefficient_optimal, module OAI.Geometry.IntegralFillings.Main); both solution files exist at the pinned commit, and neither challenge is in the formalization catalogue formalization.yaml. The statements were read here: every compactly supported integral $n$-cycle, $n\\ge2$, in a proper CAT(0) space has a compactly supported integral filling of mass at most $c_n M(T)^{(n+1)/n}$ with the Euclidean coefficient, and that coefficient is optimal. This is the Euclidean case in current form; the passage to domains in a manifold, the curvature-$\\kappa$ comparison and the equality rigidity are not formalized. Not rebuilt here. Listed as Unreviewed rather than Lean-checked because its formal statements cover only the companion paper's Euclidean CAT(0) filling bound.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"The Cartan-Hadamard conjecture is a well-known open problem of Riemannian geometry, open in high dimensions for about fifty years with only dimensions two to four settled before recent preprints. Placed below the symmetric Mahler conjecture (60) and Infinitely many closed geodesics on every Riemannian sphere (55), and above Local smooth isometric immersion of surfaces in three-space (45).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Generalized Cartan-Hadamard isoperimetry and Euclidean equality rigidity","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Sharp integral fillings in CAT(0) spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.CAT0Fillings.sharp_integral_filling)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/CAT0Fillings/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: SharpCAT0Filling.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/SharpCAT0Filling.lean","kind":"lean-proof"},{"label":"Lean proof (OAI.SharpIntegralFillings.coefficient_optimal)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/IntegralFillings/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: FillingCoefficient.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/FillingCoefficient.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"beauville-splitting-two-summands","name":"Beauville's splitting conjecture for two integrable summands","shortName":"Beauville splitting, two summands","problemNumber":null,"field":"Complex and Kahler geometry; split tangent bundles and foliations","fieldGroup":"Geometry & topology","statement":"Let $X$ be a compact Kahler manifold with a holomorphic splitting $T_X=\\bigoplus_i E_i$ such that partial sums of the summands are integrable. Beauville (2000, Section 2.3) conjectured that the universal cover of $X$ then splits as a product $\\prod Y_i$ compatibly with the decomposition, and proved it for Kahler-Einstein manifolds and surfaces. Later work covered line-bundle summands with integrable complement (Brunella-Pereira-Touzet), Hermitian-flat summands (Pereira-Touzet, Druel-Pereira-Pym-Touzet) and foliations with a compact leaf of finite holonomy. For two summands the hypothesis is that both are integrable. If $T_X=E_1\\oplus E_2$ with both summands integrable of positive rank, is the universal cover a compatible product $Y_1\\times Y_2$?","posedBy":"A. Beauville, Complex manifolds with split tangent bundle, in Complex Analysis and Algebraic Geometry (de Gruyter 2000), Section 2.3","yearPosed":2000,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: if a compact connected Kahler manifold has $T_X=E_1\\oplus E_2$ with both summands integrable holomorphic subbundles of positive rank, its universal cover is biholomorphic to $Y_1\\times Y_2$ (connected, simply connected) with $d\\Phi(\\pi^*E_i)=\\mathrm{pr}_i^*T_{Y_i}$. Corollary: non-uniruled projective manifolds with a two-summand splitting (integrability by Hoering 2007). It does not treat three or more summands, gives no finite-cover product, and assumes integrability (automatic integrability is the companion entry, for rationally connected manifolds).","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization accompanies it (Comparator challenge KahlerSplitting, listed in lean/formalization.yaml).","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Beauville's Section 2.3 as cited. Lean: Comparator challenge KahlerSplitting, declaration OAI.UniversalCoverSplitting.main (listed in lean/formalization.yaml). Its statement was read here: for a compact connected complex manifold of dimension $n\\ge2$ with a Kahler metric and a holomorphic splitting into ranks $r_1,r_2>0$ whose two projections are integrable, and any ordinary universal cover, there are connected simply connected complex manifolds $Y_1,Y_2$ of dimensions $r_1,r_2$ forming a compatible product. That is the headline; manifolds, metrics and covers are encoded in the challenge file itself and were read but not audited. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Beauville's conjecture is well known among complex geometers working on foliations and split tangent bundles; the two-summand case is its main instance but not the full statement. Placed just above the Campana-Peternell conjecture in dimension six (28), below the full Campana-Peternell Kodaira-dimension conjecture (35) and the group-ring determinant conjecture (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Universal-cover splitting for compact Kahler manifolds","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean Comparator challenge KahlerSplitting (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KahlerSplitting.lean","kind":"lean-proof"},{"label":"Integrability of split tangent bundles on rationally connected manifolds (companion)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026/main.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"goe-bulk-universality-random-regular-graphs","name":"GOE bulk universality for random regular graphs of fixed degree","shortName":"GOE bulk universality, random regular graphs","problemNumber":null,"field":"Random matrix theory; spectra of sparse random graphs","fieldGroup":"Probability & statistics","statement":"Let $A$ be the adjacency matrix of a uniform random $d$-regular graph on $n$ vertices with $d\\ge3$ fixed. Its eigenvalue density tends to the Kesten-McKay law, and Jakobson, Miller, Rivin and Rudnick predicted, with numerical evidence, that the local eigenvalue statistics in the bulk are those of the Gaussian Orthogonal Ensemble. Bulk universality was proved for growing degrees (Bauerschmidt-Huang-Knowles-Yau; Bourgade-Huang for $(\\log n)^{24}\\ll d\\le n^{1/2}$), and edge universality at fixed degree (Huang-McKenzie-Yau). Bourgade and Huang state the fixed-degree case as Conjecture 2.11. For fixed $d\\ge3$ and fixed bulk energy, does the unfolded eigenvalue point process converge to the GOE bulk process?","posedBy":"Dmitry Jakobson, Stephen D. Miller, Igor Rivin and Zeev Rudnick, Eigenvalue spacings for regular graphs, IMA Vol. 109 (1999); stated as Conjecture 2.11 by Bourgade and Huang (arXiv:2607.07617, 2026)","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims fixed-energy GOE bulk universality of the full microscopic eigenvalue point process for uniform random $d$-regular graphs, every fixed $d\\ge3$ including cubic graphs, at every energy in the open Kesten-McKay bulk. The companion extends this to sufficiently weak fixed iid uniform diagonal disorder on compact subintervals of the clean band. It does NOT address the spectral edges (already known), energy-averaged spacing statistics in the original JMRR form beyond what the point-process limit implies, or strong disorder.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscripts are credited to OpenAI with no human author named. The principal manuscript (23 September) proves the clean fixed-degree case; a companion (5 October) adapts its entropy and Gaussian insertion arguments to weak fixed Anderson disorder. No Lean formalization.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture: for every fixed $d\\ge3$ and fixed $E$ in the open Kesten-McKay bulk, Laplace functionals of the unfolded point process of a uniform simple labelled $d$-regular graph converge to those of the GOE bulk process, along all admissible $n$ with no conditioning. The paper says this proves Bourgade-Huang Conjecture 2.11 in a fixed-energy Laplace-functional form; the 1999 prediction was phrased via empirical nearest-neighbor spacings. Uses Huang-Yau rigidity and delocalization as input. No Lean formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Bulk GOE statistics for sparse fixed-degree random regular graphs is a well-known prediction in random matrix theory, open since 1999 and the last missing case of a long universality program. Level with the Courtade-Kumar conjecture (40), below the planar first-passage bigeodesics entry (45) and Cardy's formula for bond percolation (48).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026/paper.pdf","sourceName":"OpenAI math release: GOE bulk universality for fixed-degree random regular graphs","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Fixed-energy universality for weak Anderson disorder on random regular graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026/fixed-energy-universality-weak-anderson-disorder-random-regular-graphs.pdf","kind":"paper"},{"label":"Bourgade and Huang, Loop equations characterize random matrix statistics (Conjecture 2.11)","url":"https://arxiv.org/abs/2607.07617v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"depth-three-circuits-beyond-sqrt-n","name":"Explicit depth-three circuit lower bounds beyond the square-root exponent","shortName":"Depth-three lower bounds beyond 2^O(sqrt n)","problemNumber":null,"field":"Circuit complexity; depth-three Boolean circuits","fieldGroup":"Theoretical computer science","statement":"For a Boolean function $f$ on $\\{0,1\\}^n$, let $S_3(f)$ be the least number of gates in an unbounded fan-in OR-AND-OR (depth-three) circuit computing $f$. Hastad's switching lemma gives $S_3(\\text{parity})=2^{\\Theta(\\sqrt n)}$, and Paturi-Pudlak-Zane's bound $\\Omega(n^{1/4}2^{\\sqrt n})$ is tight for parity; all lower bounds for explicit functions remained of the form $2^{c\\sqrt n}$ for a constant $c$. Hastad, Jukna and Pudlak asked for an explicit function needing $2^{n^{1/2+\\varepsilon}}$ gates, and Gurumukhani-Paturi-Pudlak-Saks-Talebanfard discuss as open the weaker goal of a superconstant multiple of $\\sqrt n$ in the exponent. Is there an explicit function, for instance a language in polynomial time, with $S_3(f_n)\\ge 2^{\\omega(\\sqrt n)}$?","posedBy":"Discussed as open by Gurumukhani, Paturi, Pudlak, Saks and Talebanfard (CCC 2024); Hastad, Jukna and Pudlak (1995) posed the stronger 2^(n^(1/2+eps)) problem","yearPosed":2024,"ageNote":"The manuscript cites GPPST (CCC 2024, Section 1.1) for this open problem and notes it is weaker than Hastad-Jukna-Pudlak's 1995 Problem 2. The square-root barrier itself dates from Hastad's 1986 parity bound, so the question is older than its 2024 citation; the year here is the earliest posing the manuscript gives for exactly this question.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is a language $L$ decidable by a deterministic Turing machine in time $C(n+1)^a$ whose $n$-bit membership functions satisfy $\\log_2 S_3(f_n)/\\sqrt n\\to\\infty$, counting all gates of OR-AND-OR circuits with no fan-in restriction. The language and machine are fixed before the constant $A$. It does not give a fixed exponent $n^{1/2+\\varepsilon}$ (Hastad-Jukna-Pudlak's problem stays open), says nothing about AND-OR-AND circuits beyond what duality gives for the complement, and nothing about depth above three or Valiant's $2^{\\omega(n/\\log\\log n)}$ threshold.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the vast majority of results, this one included, were produced with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question as the manuscript and its cited sources pose it. The proof was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator DepthThree, declaration OAI.DepthThreeLowerBound.exists_polynomial_time_language_depth_three_lower_bound). The statement ComparatorChallenges/DepthThree.lean was read: there are a language $L$, a finite multitape Turing machine deciding it within $C(n+1)^a$ steps on every input, and for every real $A>0$ an $N$ such that every OR-AND-OR circuit (literal and constant inputs, arbitrary fan-in and sharing, all gates counted) computing $L$ on $n\\ge N$ bits has more than $2^{A\\sqrt n}$ gates. This states the headline claim. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"The square-root barrier for explicit depth-three circuits has stood since Hastad's 1986 parity bound and is one of the best known barriers in circuit lower bounds. Placed below The matrix multiplication exponent: is omega at most 9/4? (55) and L = RL = BPL (65), above Polynomial-Factor Hardness for the Closest Vector Problem (35).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf","sourceName":"OpenAI math release: Beyond the Square-Root Exponent for Depth-Three Boolean Circuits","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Lean proof (OAI.DepthThreeLowerBound...)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/DepthThree/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: DepthThree.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/DepthThree.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"relative-bicentralizer-conjecture","name":"Marrakchi's relative bicentralizer conjecture","shortName":"Relative bicentralizer conjecture","problemNumber":null,"field":"Operator algebras; type III von Neumann algebras, bicentralizers","fieldGroup":"Analysis","statement":"Connes asked whether the bicentralizer of a faithful normal state on a type $\\mathrm{III}_1$ factor with separable predual is always trivial; Haagerup settled the amenable case. Ando, Haagerup, Houdayer and Marrakchi built a relative bicentralizer and its flow for expected inclusions. Marrakchi formulated the relative bicentralizer conjecture and proved that, for separable preduals, it is equivalent to: for every inclusion $N\\subset M$ of von Neumann algebras with separable preduals and a faithful normal conditional expectation, there is an expected amenable subalgebra $P\\subset N$ with $P'\\cap c(M)=N'\\cap c(M)$, where $c(M)$ is the continuous core. Is the relative bicentralizer conjecture true?","posedBy":"A. Marrakchi, Kadison's problem for type III subfactors and the bicentralizer conjecture, Invent. Math. 239 (2025), Conjecture C (arXiv 2308.15163)","yearPosed":2023,"ageNote":"Posed in the arXiv preprint 2308.15163 (2023), published in Invent. Math. in 2025. The absolute problem it generalizes goes back to Connes (1980s).","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every unital inclusion $N\\subset M$ with separable preduals and a faithful normal conditional expectation there are an amenable $P\\subset N$ and an expectation $N\\to P$ with $P'\\cap c(M)=N'\\cap c(M)$; with Marrakchi's Theorem D this is the relative bicentralizer conjecture, including arbitrary centers and types, and gives an expected MASA corollary relevant to Kadison's problem for subfactors. Along the way it reproves Connes' bicentralizer conjecture for separable type $\\mathrm{III}_1$ factors; priority for that absolute case belongs to Houdayer and Marrakchi (arXiv 2609.11462, 10 Sep 2026), whose statement has no separability assumption. The companion gives a second absolute proof via bounded recovery. Non-separable preduals are not treated.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Marrakchi's Conjecture C via his Theorem D equivalence, as the paper states it. The proof was not refereed. Not Lean-checked: the family's only formal statement, ComparatorChallenges/BoundedRecovery.lean (OAI.BoundedRecovery.bounded_recovery), was read here and covers the companion's bounded-recovery lemma for a state with scalar centralizer, not the relative or the absolute bicentralizer statement, so under the tier rule this entry stays unreviewed. The argument uses as inputs Marrakchi's Theorem D equivalence and his 2026 binormal-state lemmas.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A recent conjecture of Marrakchi that generalizes Connes' bicentralizer problem to inclusions, important within type III von Neumann algebra theory; its absolute special case had just been announced solved by Houdayer and Marrakchi, which lowers the standing of what remained. Placed below Voiculescu's free entropy dimension question (38) and Connes' Rigidity Conjecture (45), near Connes' Rigidity Conjecture for ICC Property (T) Groups (35) and above the Burger-Ozawa-Thom question (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026.pdf","sourceName":"OpenAI math release: Expected amenable subalgebras preserving core commutants","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Bounded recovery for modular spectral averages","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026/Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf","kind":"paper"},{"label":"Lean proof of the companion lemma (OAI.BoundedRecovery.bounded_recovery)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/ModularRecovery/Recovery.lean","kind":"lean-proof"},{"label":"Marrakchi, Kadison's problem for type III subfactors and the bicentralizer conjecture (arXiv)","url":"https://arxiv.org/abs/2308.15163v3","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tachikawa-second-conjecture","name":"Tachikawa's second conjecture for self-injective algebras","shortName":"Tachikawa's second conjecture","problemNumber":null,"field":"Representation theory of algebras; homological conjectures","fieldGroup":"Algebra","statement":"Let $A$ be a finite-dimensional self-injective algebra over a field and $M$ a finite-dimensional $A$-module that is self-orthogonal, meaning $\\mathrm{Ext}^i_A(M,M)=0$ for every $i>0$. Tachikawa's second conjecture, from his 1973 monograph on quasi-Frobenius rings, asserts that such an $M$ must be projective. It is the self-injective case of the Auslander-Reiten conjecture and is tied to the Nakayama conjecture through dominant dimension and endomorphism algebras. Partial results covered radical cube zero, quantum complete intersections and conditional stratification criteria (Chen-Fang-Xi). Is every self-orthogonal finite-dimensional module over a finite-dimensional self-injective algebra projective?","posedBy":"Hiroyuki Tachikawa, Quasi-Frobenius Rings and Generalizations: QF-3 and QF-1 Rings (Lecture Notes in Mathematics 351, 1973)","yearPosed":1973,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: over $k=\\mathbb F_2(q,H_1,H_2)$ there are a finite-dimensional symmetric algebra $A$ and a nonprojective finite-dimensional module $M$ with $\\mathrm{Ext}^i_A(M,M)=0$ for all $i>0$, so Tachikawa's second conjecture fails, and $M$ is also a symmetric counterexample to the Auslander-Reiten conjecture. The conclusions persist under every field extension. It does NOT address Tachikawa's first conjecture separately, nor give characteristic-zero examples.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The Tachikawa manuscript reuses the companion's starting algebra and adds a transfer principle to a symmetric algebra.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture: a finite-dimensional symmetric (hence self-injective) algebra over $\\mathbb F_2(q,H_1,H_2)$ with a nonprojective finite-dimensional self-orthogonal module. formalization.yaml lists ComparatorChallenges/Tachikawa.json, declaration OAI.Tachikawa.main_theorem, file OAI/RingTheory/Tachikawa/Counterexample.lean. The statement was read here: over the fraction field of $\\mathbb F_2[X_0,X_1,X_2]$ there are a finite algebra with a linear isomorphism to its dual satisfying the symmetric-form identities, and a finite module that is not projective with all positive self-Ext groups trivial. That is the headline claim. lean/docs/199.md says the field-extension, endomorphism-algebra and homological consequences are not formalised. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A named conjecture of the 1970s, well known within the representation theory of algebras but narrower than the Auslander-Reiten conjecture it specialises. Below the Auslander-Reiten conjecture entry (45), above 'Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture?' (30) and the periodicity conjecture for finite-dimensional algebras (25).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: A counterexample to Tachikawa's second conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: An explicit counterexample to the Auslander-Reiten conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean: Tachikawa counterexample (formalization.yaml file)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/RingTheory/Tachikawa/Counterexample.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Tachikawa counterexample","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/Tachikawa.lean","kind":"lean-proof"},{"label":"Tachikawa (1973), Quasi-Frobenius Rings and Generalizations","url":"https://doi.org/10.1007/BFb0059997","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"koebe-circle-domain-conjecture","name":"Koebe's circle-domain conjecture (Kreisnormierungsproblem)","shortName":"Koebe's circle-domain conjecture","problemNumber":null,"field":"Geometric function theory; conformal uniformization","fieldGroup":"Analysis","statement":"A circle domain is a domain in the Riemann sphere $\\hat{\\mathbb C}$ each of whose complementary components is a closed round disk or a single point. Koebe proved that every finitely connected domain is conformally equivalent to a circle domain, and He and Schramm (1993) extended this to countably connected domains. Koebe's Kreisnormierungsproblem, posed in 1908, asks for the general case, with no restriction on the number or shape of the complementary components, which may be uncountably many. Is every domain $G\\subseteq\\hat{\\mathbb C}$ conformally equivalent to a circle domain?","posedBy":"Paul Koebe, Uber die Uniformisierung beliebiger analytischer Kurven, Dritte Mitteilung (Nachr. Ges. Wiss. Gottingen, 1908)","yearPosed":1908,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims that every domain in $\\hat{\\mathbb C}$ is conformally equivalent to a circle domain, with no countability or geometric condition on the complementary components. The method marks countably many components, collapses unmarked ones to points with small energy barriers, and shows marked ones become exactly round disks, using transboundary extremal length and a modulus-probability duality. Combined with Ntalampekos-Rajala it also gives a convergent exhaustion by finitely connected domains. It does NOT give uniqueness of the circle-domain model; the rigidity question is the companion's subject and is answered there only under conformal removability of the boundary.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. The manuscript is credited to OpenAI with no human author named.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against Koebe's problem as the manuscript and its references state it: every nonempty connected open subset of the sphere is conformally equivalent to a circle domain, with no condition on the complement. The Lean challenge ComparatorChallenges/KoebeCircleDomains.json (solution module OAI.Analysis.CircleDomains.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/071.md. It compares two theorems, OAI.Problem047.koebe_circle_domain and OAI.Problem047.removability_implies_rigidity, with permitted axioms propext, Quot.sound and Classical.choice. The statement was read here; not rebuilt here. koebe_circle_domain says every open connected U in the one-point compactification of C admits a set V and a map f with V a circle domain (open, connected, each complementary component a point or a Mobius image of the closed unit disk) and f a conformal equivalence of U onto V (nonzero complex derivative in charts at every point, with a conformal inverse). That is the headline claim. Uniqueness up to Mobius maps is not claimed here and is false in general.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"Open since 1908 and the central uniformization problem for infinitely connected planar domains; He and Schramm's countable case was a major 1993 result. Famous across complex analysis but narrower than the Hilbert-Smith Conjecture at 65; placed with the Kadison Similarity Problem and Thompson's group F amenability at 58 and just above them for its age.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Koebe's Circle-Domain Conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: Removable Boundaries and Rigidity of Circle Domains","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean: solution module (koebe_circle_domain, removability_implies_rigidity)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/CircleDomains/Main.lean","kind":"lean-proof"},{"label":"Lean comparator statement: circle-domain theorem and removable-boundary rigidity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/KoebeCircleDomains.lean","kind":"lean-proof"},{"label":"Lean: release scope note for this family","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/071.md","kind":"lean-proof"},{"label":"Koebe (1908), digitized article record","url":"https://eudml.org/doc/58725","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"talagrand-fractional-expectation-threshold","name":"Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant","shortName":"Fractional vs integral expectation thresholds","problemNumber":null,"field":"Probabilistic combinatorics; thresholds","fieldGroup":"Combinatorics","statement":"For an increasing family $\\mathcal F$ of subsets of a finite set $X$, call $\\mathcal F$ $p$-small if some family $\\mathcal G$ with every member of $\\mathcal F$ containing a member of $\\mathcal G$ has $\\sum_{S\\in\\mathcal G}p^{|S|}\\le1/2$; the expectation threshold $q(\\mathcal F)$ is the largest such $p$. The fractional expectation threshold $q_f(\\mathcal F)$ allows instead weights $g(S)\\in[0,1]$ with $\\sum_{S\\subseteq H}g(S)\\ge1$ for $H\\in\\mathcal F$ and $\\sum_S g(S)p^{|S|}\\le1/2$, so $q\\le q_f$. Frankston-Kahn-Narayanan-Park and Park-Pham proved the Kahn-Kalai comparisons with the threshold $p_c$; special cases of Talagrand's comparison were known for weights on singletons, pairs, cliques and bounded-size sets. Talagrand (2010, Conjecture 6.3) asked: is there a universal constant $L$ with $q_f(\\mathcal F)\\le L\\,q(\\mathcal F)$ for every increasing family?","posedBy":"Michel Talagrand (Conjecture 6.3, 'Are many small sets explicitly small?', STOC 2010)","yearPosed":2010,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: $q_f(\\mathcal F)\\le25\\cdot512^4\\,q(\\mathcal F)$ for every finite nonempty $X$ and nonempty proper increasing $\\mathcal F\\subseteq2^X$, independently of $|X|$, the sizes of minimal members and the support of the fractional cover; equivalently every fractional cover of cost at most $1/2$ at $r$ rounds to an integral cover of cost at most $1/2$ at $r/(25\\cdot512^4)$. Previously the best general loss was of order $\\log\\log|X|$ (Fischer-Person with Pham). The constant is explicit but not optimized.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. It belongs to a family with 'Talagrand's discrete-convexity conjecture' (same date) and 'Graph Decompositions at the Integral Expectation Threshold' (October 5, 2026), each answering a different posed problem.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Talagrand's Conjecture 6.3 as the manuscript states it. It gives $q_f(\\mathcal F)\\le25\\cdot512^4\\,q(\\mathcal F)$ for every nonempty proper increasing family on a finite nonempty ground set, both with budget $1/2$. lean/formalization.yaml lists TalagrandExpectationThreshold.json (declaration OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence, file OAI/Combinatorics/ExpectationThreshold/Main.lean). The statement TalagrandExpectationThreshold.lean was read here; not rebuilt here. It defines integral and fractional smallness with budget 1/2 (fractional weights in [0,1], including the empty set), $q$ and $q_f$ as suprema of feasible $p\\in[0,1]$, and states $q_f\\le25\\cdot512^4\\,q$ for nonempty, proper, increasing families. This states the headline. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A named Talagrand conjecture at the centre of the threshold theory that produced the Park-Pham proof of the Kahn-Kalai conjecture, with a line of partial results by DeMarco, Kahn, Frankston, Park, Pham, Fischer and Person. Level with Talagrand's convolution conjecture and Talagrand's Convexity Problem at 37, a little above for being the comparison the field isolated after Park-Pham; below Kahn's flow conjecture at 40.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf","sourceName":"OpenAI math release: Integral and fractional expectation thresholds are equivalent","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Family: Talagrand's discrete-convexity conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Family: Graph Decompositions at the Integral Expectation Threshold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf","kind":"paper"},{"label":"Lean proof (OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/ExpectationThreshold/Main.lean","kind":"lean-proof"},{"label":"Talagrand, Are many small sets explicitly small? (STOC 2010)","url":"https://doi.org/10.1145/1806689.1806693","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"auslander-reiten-conjecture","name":"The Auslander-Reiten conjecture for Artin algebras","shortName":"Auslander-Reiten conjecture","problemNumber":null,"field":"Representation theory of algebras; homological conjectures","fieldGroup":"Algebra","statement":"Let $R$ be an Artin algebra (for example a finite-dimensional algebra over a field) and $M$ a finitely generated $R$-module. Auslander and Reiten (1975), in their work on the generalized Nakayama conjecture, conjectured that $M$ must be projective if $\\mathrm{Ext}^i_R(M,M\\oplus R)=0$ for every $i>0$. It was proved for many classes (complete intersections, radical cube zero, Igusa-Todorov algebras, rings satisfying Auslander's condition), and Schulz's non-projective module without self-extensions lives over a skew field, outside this setting. The Gorenstein-projective conjecture (Luo-Huang) is its restriction to Gorenstein-projective modules. Is every finitely generated module $M$ over an Artin algebra with $\\mathrm{Ext}^i_R(M,M\\oplus R)=0$ for all $i>0$ projective?","posedBy":"Maurice Auslander and Idun Reiten, On a generalized version of the Nakayama conjecture (Proc. AMS, 1975)","yearPosed":1975,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims Theorem 1.1: over $k=\\mathbb F_2(q,H_1,H_2)$ there are a finite-dimensional triangular algebra $\\Lambda$ with eight simple modules and $\\mathrm{rad}(\\Lambda)^4\\ne0$, and a finite-dimensional nonprojective Gorenstein-projective module $Z$ with $\\mathrm{Ext}^i(Z,Z)=\\mathrm{Ext}^i(Z,\\Lambda)=0$ for all $i\\ge1$, persisting under every field extension (so also over an algebraically closed field of characteristic two). This disproves the Auslander-Reiten and Gorenstein-projective conjectures; $Z\\oplus\\Lambda$ also refutes the generator form. It does NOT touch the commutative Noetherian form of the conjecture, and gives no example in characteristic zero.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The two counterexamples share a ten-dimensional starting algebra; each manuscript states it contains its own complete proof.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture as posed: an explicit finite-dimensional algebra over $k=\\mathbb F_2(q,H_1,H_2)$ with a nonprojective module $Z$ satisfying the vanishing for every $i\\ge1$. formalization.yaml lists ComparatorChallenges/AuslanderReiten.json, declaration OAI.ArExplicit.Statement.main, file OAI/Algebra/AuslanderReiten/All.lean. The statement was read here: $q,H_1,H_2$ algebraically independent over $\\mathbb F_2$, and a finite-dimensional algebra with a finite-dimensional module that is not projective, is a cokernel in a totally acyclic complex of finite projectives, has vanishing positive Ext into itself and into the algebra, has semisimple quotient $k^8$ and nonzero fourth radical power, and keeps all of this after base change to every field extension. That is the headline claim, including the Gorenstein-projective form. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A central homological conjecture of the representation theory of Artin algebras since 1975, with a large literature of special cases. Above the catalog's 'Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture?' (30), which settles relations among these conjectures rather than any one, and the periodicity conjecture for finite-dimensional algebras (25); below the Nakayama conjecture entry (50) and the Koethe Conjecture (60).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: An explicit counterexample to the Auslander-Reiten conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A counterexample to Tachikawa's second conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean: Auslander-Reiten counterexample (formalization.yaml file)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/AuslanderReiten/All.lean","kind":"lean-proof"},{"label":"Lean comparator statement: Auslander-Reiten counterexample","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/AuslanderReiten.lean","kind":"lean-proof"},{"label":"Auslander and Reiten (1975), On a generalized version of the Nakayama conjecture","url":"https://doi.org/10.1090/S0002-9939-1975-0389977-6","kind":"problem-record"},{"label":"The equivalence of Tachikawa's second conjecture and Auslander-Reiten","url":"https://vibemathed.com/problem/tachikawa-second-conjecture-implies-auslander-reiten","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strong-thin-tree-conjecture","name":"The strong thin tree conjecture","shortName":"Strong thin tree conjecture","problemNumber":null,"field":"Graph theory; spanning trees, cuts and connectivity","fieldGroup":"Combinatorics","statement":"A spanning tree $T$ of a multigraph $G$ is $\\alpha$-thin if $|\\delta_T(S)|\\le\\alpha|\\delta_G(S)|$ for every nonempty proper vertex set $S$. Goddyn's thin tree conjecture asks whether for every $\\varepsilon>0$ sufficiently high edge connectivity guarantees an $\\varepsilon$-thin spanning tree; the strong form asks for the rate: is there a universal constant $C$ such that every $k$-edge-connected multigraph has a $C/k$-thin spanning tree? Known results gave $O(1/k)$ for planar and bounded-genus graphs (Oveis Gharan-Saberi) and $\\mathrm{poly}(\\log\\log n)/k$ in general (Anari-Oveis Gharan). Via Asadpour et al., thin trees bound the integrality gap of asymmetric TSP. Does every $k$-edge-connected finite multigraph have a spanning tree that is $C/k$-thin, with $C$ absolute?","posedBy":"Luis Goddyn (thin tree conjecture); the C/k strong form is from the thin-tree literature around asymmetric TSP","yearPosed":2004,"ageNote":"The manuscript cites Goddyn's online problem list 'Some Open Problems I Like', Problem 4, page dated 17 March 2004 (archived 2021). It does not cite a separate first statement of the strong C/k form; the conjecture may be older than the page.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: there is an absolute $C>0$ such that every finite loopless $k$-edge-connected multigraph with at least two vertices has a spanning tree $T$ with $|\\delta_T(S)|\\le(C/k)|\\delta_G(S)|$ for all cuts. The proof extracts tree packings on low-resistance edges and iterates a Marcus-Spielman-Srivastava sparsification. The companion gives a deterministic polynomial-time construction, polynomial in the binary input length. Neither paper gives an explicit constant or a new ATSP approximation ratio (constant-factor ATSP was already known by other methods), and spectrally thin trees are not claimed.","claimIssueNote":null,"solveDate":"2026-09-23","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript and its algorithmic companion are both dated September 23, 2026; the companion extends the hierarchy argument to a deterministic polynomial-time construction.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the conjecture; it gives the strong $C/k$ form for every $k\\ge1$ and every finite loopless multigraph on at least two vertices, which also implies Goddyn's original form. No value of $C$ is given. lean/formalization.yaml lists a main result for this manuscript (comparator StrongThinTree, declaration OAI.StrongThinTree.strongThinTree, file OAI/Combinatorics/ThinTrees/Main.lean). ComparatorChallenges/StrongThinTree.lean was read here: it states exactly the headline over finite multigraphs given by endpoint maps with parallel edges distinct. The companion's challenge AlgorithmicThinTrees.json is not in the formalization catalogue; its solution module OAI/Combinatorics/ThinTrees/AlgorithmicMain.lean exists at the pinned commit, and its statement (a stack machine that halts in polynomially many steps in the input length, including binary-encoded multiplicities, and outputs a $C/k$-thin tree) was read here. Neither was rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The thin tree conjecture is a well known open problem at the meeting point of graph theory and approximation algorithms, studied since the mid-2000s for its link to asymmetric TSP; its importance for ATSP was reduced once constant-factor ATSP was found by other means. Placed level with Brennan's conjecture (45), below Ryser's conjecture (50) and Sidorenko's conjecture (52), and above Gyarfas's tree cover conjecture (32).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf","sourceName":"OpenAI math release: The strong thin tree conjecture","collection":"openai-math-2026-10","collectionVersion":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","links":[{"label":"Companion: A polynomial-time construction of strong thin trees","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean proof (OAI.StrongThinTree.strongThinTree)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/ThinTrees/Main.lean","kind":"lean-proof"},{"label":"Comparator statement: StrongThinTree.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/StrongThinTree.lean","kind":"lean-proof"},{"label":"Lean proof of the algorithmic companion (not in formalization catalogue)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/ThinTrees/AlgorithmicMain.lean","kind":"lean-proof"},{"label":"Comparator statement: AlgorithmicThinTrees.lean","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/ComparatorChallenges/AlgorithmicThinTrees.lean","kind":"lean-proof"},{"label":"Goddyn, Some Open Problems I Like, Problem 4 (archived)","url":"https://web.archive.org/web/20211025204852/https://www.sfu.ca/~goddyn/Problems/problems.html","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gromov-width-symmetric-polar-products","name":"Vicente's question: is the Gromov width of every symmetric Lagrangian polar product equal to 4?","shortName":"Gromov width of symmetric polar products","problemNumber":null,"field":"Symplectic geometry: capacities of Lagrangian products","fieldGroup":"Geometry & topology","statement":"For an origin-symmetric convex body $K\\subset\\mathbb R^n$ put $K$ in position space and its polar $K^\\circ$ in momentum space of $(\\mathbb R^{2n},\\sum_j dq_j\\wedge dp_j)$. Artstein-Avidan, Karasev and Ostrover (2014) proved that the Hofer-Zehnder capacity of $K\\times K^\\circ$ is 4 and showed that Viterbo's volume-capacity conjecture for these products implies the symmetric Mahler conjecture. The Gromov width $c_G$ is the supremum of $\\pi r^2$ over symplectic embeddings of the radius-$r$ ball, so $c_G\\le4$; equality was known for Euclidean balls, the cube (Ramos-Sepe) and $\\ell_p$ balls (Karasev). Vicente noted that a positive answer immediately implies the Mahler conjecture. Is $c_G(K\\times K^\\circ)=4$ for every origin-symmetric convex body $K$?","posedBy":"Alejandro Vicente, Questions related to the Mahler and the Viterbo Conjecture, blog post, 13 January 2026 (question 1)","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Theorem 1.1: for every $n\\ge2$ and every origin-symmetric convex body $K\\subset\\mathbb R^n$, $c_G(\\mathrm{int}\\,K\\times\\mathrm{int}\\,K^\\circ)=4$, with explicit smooth symplectic embeddings of every ball of capacity $c<4$; no smoothness or strict convexity is assumed. The lower bound comes from a holomorphic ball principle (high-order vanishing gives large balls) and a uniform slice estimate for a conformal lens; the upper bound from a supporting-cylinder argument and nonsqueezing. Since symplectic embeddings preserve volume this reproves the symmetric Mahler inequality, but the paper makes no claim about equality cases. It says nothing about non-symmetric products or other capacities, and does not conflict with the Haim-Kislev-Ostrover counterexample to Viterbo's general conjecture.","claimIssueNote":null,"solveDate":"2026-09-22","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here: the abstract, introduction, background section and Theorem 1.1 of the TeX source, and Vicente's blog post itself, whose first listed question is exactly this one; the proof was not refereed. Lean-checked on the Comparator challenge SymmetricPolar (OAI.SymmetricPolar.symmetric_polar_main, OAI/Geometry/PolarProducts/Main.lean), listed in the release's formalization catalogue. Its statement, read here, says that for $n\\ge2$ and every compact convex symmetric $K$ with nonempty interior, the Gromov width of $\\mathrm{int}\\,K\\times\\mathrm{int}\\,K^\\circ$ (supremum over smooth embeddings preserving the standard form $\\omega_0$ of open balls $\\pi(|q|^2+|p|^2)2$ (for example $r=\\pi$), that is not Ramsey: a finite colouring of $\\mathbb R^n$ with no monochromatic congruent copy, in every dimension. Since the set lies on a circle, this disproves the conjecture that every spherical set is Ramsey. The colouring is non-constructive, built from a non-measurable derivation of $\\mathbb R$. The unverified appendix adds that almost every seven-point circular set is non-Ramsey and that the method cannot work below seven points. The OpenAI release's classification manuscript (23 September 2026) cites this paper as the disproof, adds a twelve-point spherical example by the same derivation-moment method, and proves every set of at most five circle points Ramsey; six circle points remain open between the two.","claimIssueNote":null,"solveDate":"2026-09-20","model":"ChatGPT (version not stated)","modelMaker":"OpenAI","humanCollaborators":["Dömötör Pálvölgyi"],"aiRole":"From the paper's AI disclosure: \"This manuscript was written entirely by ChatGPT. I (Dömötör Pálvölgyi) contributed nothing to the proofs or ideas, I only gave suggestions about the presentation.\" The author's closing note says the construction was found by ChatGPT, that he had tried similar field-automorphism constructions himself without success, and that the appendix contains further results by ChatGPT he has not verified. The model version is not stated.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Read here on 7 October 2026 from the arXiv source: Theorem 1 states that for any transcendental $r>2$ the seven points $(0,0)$ and $(j,\\pm\\sqrt{j(2r-j)})$, $j\\in\\{1,3,4\\}$, lie on the circle $(x-r)^2+y^2=r^2$ and are not Ramsey, which contradicts the conjecture as posed. The proof is one page: a field derivation $D$ with $D(r)=1$, weights $2,-2,2,-1$ whose zeroth, first, second and mixed moments vanish, and a weighted energy identity $\\sum\\lambda_p\\|D(p')\\|^2=-24/R$ invariant under congruence, closed by Rado's theorem. The moment conditions and the energy identity were recomputed by hand here and hold. The appeal to Rado and the existence of $D$ (which needs a transcendence basis) were read, not re-derived. The author states he contributed nothing to the proof; the appendix is marked unverified by him. No formal proof.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"The leading conjecture of Euclidean Ramsey theory for five decades, posed by Erdos, Graham and coauthors in the paper that founded the area, and well known across combinatorics though not far outside it. Level with the Burr-Erdos hypercube Ramsey conjecture (40) and Hopkins' chromatic splitting conjecture (40), below the classification problem for Euclidean Ramsey sets (45), which it is one half of, and above the Leader-Russell-Walters conjecture (28).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.23327","sourceName":"arXiv:2609.23327, Pálvölgyi (20 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, Euclidean Ramsey theorems I (1973)","url":"https://doi.org/10.1016/0097-3165(73)90011-3","kind":"problem-record"},{"label":"OpenAI release: classification of finite Euclidean Ramsey configurations (23 Sept 2026), cites this paper","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf","kind":"paper"},{"label":"Lean: OAI/Combinatorics/SphericalRamsey/Main.lean (twelve-point non-Ramsey example, OpenAI release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/SphericalRamsey/Main.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"katsumasa-roff-yoshinaga-conjecture-on-hausdorff-continuity-of-magnitude","name":"Katsumasa–Roff–Yoshinaga conjecture on Hausdorff continuity of magnitude","shortName":"Magnitude continuity at finite sets","problemNumber":null,"field":"Metric geometry; magnitude of metric spaces","fieldGroup":"Geometry & topology","statement":"Fix a finite-dimensional real normed space $U$ whose finite similarity matrices $Z_A=(e^{-\\|x-y\\|})_{x,y\\in A}$ are positive definite. For a nonempty finite subset $A\\subset U$, define $\\operatorname{Mag}A=\\mathbf{1}^{\\mathsf T}Z_A^{-1}\\mathbf{1}$.\n\nMust $\\operatorname{Mag}F_k\\to\\operatorname{Mag}F$ whenever $F_k,F\\subset U$ are nonempty finite sets and $d_H(F_k,F)\\to0$, without any uniform bound on $\\#F_k$? Equivalently, is magnitude continuous on the nonempty finite subsets of $U$ equipped with Hausdorff distance?","posedBy":"Hirokazu Katsumasa, Emily Roff and Masahiko Yoshinaga, Conjecture 1.3 of \"Is magnitude generically continuous for finite metric spaces?\", arXiv:2501.08745v2 (2025)","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"Yes, with a quantitative bound: magnitude is Hausdorff-continuous at every nonempty finite subset of a finite-dimensional positive definite normed space, with a local Lipschitz-type estimate in the Hausdorff distance whose constants depend on the number of points, the dimension and the least eigenvalue of the similarity matrix, and with compact approximants allowed. The paper also obtains convergence of aggregated weights and a continuity criterion in L_1 by local order oscillation, and, in the other direction, a compact convex subset of L_1[0,1] whose finite subsets approach a singleton with magnitudes tending to 3/2, which refutes an implication stated in Kalisnik and Lesnik's preprint from the one-point property to singleton continuity.","claimIssueNote":null,"solveDate":"2026-09-20","model":"GPT-6 Astra Pro; Claude Fable 5.1","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Mingchang Liu"],"aiRole":"The author and the AI models collaborated in developing and refining the proofs. The manuscript’s disclosure identifies the cut-deletion argument, order-oscillation estimates and Euclidean compact-limit calculation as parts of that collaboration. The models also assisted with checking arguments, comparing sources, reviewing the manuscript, and drafting and refining the exposition.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026. Conjecture 1.3 was read in arXiv:2501.08745v2: for a finite-dimensional positive definite normed space U, X maps to |X| is continuous in the Hausdorff topology on finite subsets of U. Liu's Theorem 1.1 (Zenodo record 22866753, 20 September, 18 pages, read here) states exactly that class and more: for a nonempty compact X within Hausdorff distance r of a finite F, with r below an explicit threshold, -2m^2 r/lambda^2 <= Mag X - Mag F <= 2nm^3 r/lambda^2, where lambda is the least eigenvalue of F's similarity matrix; the title's \"finite-dimensional subspaces of L_1\" is the conjecture's class by Meckes's Corollary 3.5, which the paper cites. The argument (cut deletion, pinching, signed cluster aggregation) was not checked step by step here and no specialist has read it: a two-day-old manuscript on Zenodo, not on arXiv. Prior work is partial and is cited correctly: Kalisnik and Lesnik at skew finite subsets of l_1^N, So on cluster types, Yoshinaga on fixed multiplicity strata.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"A named conjecture, stated in print in 2025 by three of the people who work on magnitude, on the stability question that matters for the invariant's use in topological data analysis; within a year it drew partial results from three other groups, which is what an active question looks like. Level with the Umans-Wang divisor conjecture at 18 less a little for a smaller field; above Koizumi-Liu at 12, a conjecture answered by one of its own authors in the negative.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://zenodo.org/records/22866753","sourceName":"Zenodo preprint, Mingchang Liu, Magnitude continuity at finite sets in finite-dimensional L_1 subspaces (v1, 20 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Katsumasa, Roff and Yoshinaga (2025), Conjecture 1.3","url":"https://arxiv.org/abs/2501.08745v2","kind":"problem-record"},{"label":"Kalisnik and Lesnik (2025), the skew-subset partial result","url":"https://arxiv.org/abs/2506.21128","kind":"paper"}],"relations":[],"submittedBy":"SilentIbis759","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"matroid-intersection-integrality-gap-is-at-most-k-1-frac-1-k","name":"The integrality gap of weighted $k$-matroid intersection","shortName":"Matroid intersection integrality gap","problemNumber":null,"field":"Approximation algorithms; matroid optimisation","fieldGroup":"Algorithms & optimization","statement":"Weighted $k$-matroid intersection asks for a maximum-weight set independent in each of $k$ matroids on a common ground set. The natural linear-programming relaxation optimises over the intersection of the $k$ matroid independence polytopes, and its integrality gap is conjectured to be at most $k-1$. That is known for $k \\le 3$; for $k \\ge 4$ the best general upper bound was $k$. Lee, Sviridenko and Vondrák conjectured further that for weighted $p$-matchoids the gap is exactly $p - 1 + 1/p$. How large is the gap?","posedBy":"The $k-1$ bound is the standard conjecture for $k$-matroid intersection; the $p-1+1/p$ form for weighted $p$-matchoids is Conjecture 1 of Lee, Sviridenko and Vondrák","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Partial on the headline conjecture, complete on a named part of another. For weighted $k$-matroid intersection the integrality gap is at most $k - 1 + 1/k$, improving the previous general bound of $k$ but not reaching the conjectured $k-1$. For weighted $p$-matchoids the gap is at most $p - 1 + 1/p$, with a deterministic LP-relative algorithm achieving the same factor; the paper states that this resolves the $p$-matchoid part of Lee, Sviridenko and Vondrák's Conjecture 1, and projective planes of order $p-1$ give tight instances whenever one exists.","claimIssueNote":null,"solveDate":"2026-09-18","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Yu Cong","Yajie Zhao"],"aiRole":"From the paper's AI Disclosure: the authors used GPT-5.6 Sol to assist with developing the integrality-gap proof and deriving the local-ratio algorithm.","verification":"unreviewed","verificationNote":"Checked here on 27 September 2026 against arXiv:2609.21477, posted 18 September. The abstract and introduction were read: the $k$-matroid bound improves from $k$ to $k-1+1/k$, short of the conjectured $k-1$, and the $p$-matchoid result attains $p-1+1/p$ with a deterministic LP-relative algorithm, which the paper says resolves the $p$-matchoid part of Lee, Sviridenko and Vondrák's Conjecture 1, projective planes of order $p-1$ giving tight instances where one exists. The AI Disclosure is one sentence and is quoted in the AI role field. The mathematics was not checked here; nine days old, no referee, no formalisation.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The integrality gap of the matroid intersection relaxation is a standard question in approximation algorithms, stuck at the trivial $k$ for every $k \\ge 4$ since Lee, Sviridenko and Vondrák's work; the matchoid half of their Conjecture 1 falls here outright. Level with the catalog's Ryser and Petrykowski entries at 20: a real named target in a working subfield, short of the headline conjecture.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.21477","sourceName":"Integrality-Gap Bounds for Weighted Matchoids and Matroid Intersection, arXiv:2609.21477 (18 September 2026)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"AmberWombat684","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"talagrand-operator-cotype-problem","name":"Talagrand's operator cotype problem","shortName":"Talagrand's operator cotype problem","problemNumber":null,"field":"Banach space theory; stochastic processes","fieldGroup":"Analysis","statement":"For a bounded operator $U$ between Banach spaces, write $C_q^r(U)$ for its Rademacher cotype-$q$ constant, $C_q^g(U)$ for its Gaussian cotype-$q$ constant and $\\|U\\|_{q,1}$ for its $(q,1)$-summing norm. Talagrand asked, as a research problem in his book on upper and lower bounds for stochastic processes, whether there is a universal constant $L$ with $C_q^r(U) \\le L\\max\\{C_q^g(U), \\|U\\|_{q,1}\\}$ for every such $U$. Is the Rademacher cotype of an operator controlled in this way?","posedBy":"Michel Talagrand, Research Problem 19 in Upper and Lower Bounds for Stochastic Processes (2nd edition, 2021)","yearPosed":2021,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"No: the answer is negative already for $q=2$, so no such universal constant exists.","claimIssueNote":null,"solveDate":"2026-09-17","model":"ChatGPT (GPT-5.6)","modelMaker":"OpenAI","humanCollaborators":["Xinglong Wu"],"aiRole":"Stated in the abstract itself, not only in a disclosure paragraph: \"The counterexample was discovered by ChatGPT (GPT-5.6).\"","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.19731: the abstract and introduction both state the problem as Talagrand's and give the negative answer at $q=2$, and the reference list confirms the source as Talagrand's Upper and Lower Bounds for Stochastic Processes, second edition, Springer 2021, where it is a numbered research problem. The mathematics was not checked here; five days old, no referee.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"A numbered research problem in Talagrand's own book, in the area his Fields-level work defined; specialists in Banach space geometry would recognise it, few outside would. Just below the catalog's Talagrand convolution conjecture at 37, which is older and more widely cited, and above the abelian-envelope entry at 22.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.19731","sourceName":"A Counterexample to Talagrand's Operator Cotype Problem, arXiv:2609.19731 (17 September 2026)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"weak-pach-tardos-conjecture-acyclic-matrix-patterns","name":"The weak Pach-Tardos conjecture for acyclic matrix patterns","shortName":"Weak Pach-Tardos conjecture","problemNumber":null,"field":"Extremal combinatorics; forbidden 0-1 matrices","fieldGroup":"Combinatorics","statement":"For a 0/1 pattern $P$, let $\\mathrm{Ex}(n,P)$ be the largest number of 1-entries in an $n \\times n$ 0/1 matrix containing no copy of $P$; $P$ is acyclic when its bipartite incidence graph is a forest. Pach and Tardos conjectured in 2005 that $\\mathrm{Ex}(n,P) \\le n\\,\\mathrm{polylog}(n)$ for every acyclic $P$. Pettie and Tardos refuted that in 2024, exhibiting acyclic patterns with $\\mathrm{Ex}(n,P) \\ge n\\,2^{\\Omega(\\sqrt{\\log n})}$. The weaker form survives and is, in the authors' words, arguably one of the main open problems in the area: is $\\mathrm{Ex}(n,P) \\le n^{1+o(1)}$ for every acyclic $P$?","posedBy":"János Pach and Gábor Tardos (2005); the weak form is what survives Pettie and Tardos's 2024 refutation of the original","yearPosed":2005,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Proved, in the weak form: for every acyclic pattern $P$, $\\mathrm{Ex}(n,P) \\le n^{1+O_P(1/\\log\\log n)}$, hence $n^{1+o(1)}$. Stated as Theorem 1 in the language of ordered bipartite graphs: for all $s,t \\ge 1$ and $\\varepsilon>0$ there is $n_0$ such that every $n \\times n$ ordered bipartite graph with $n \\ge n_0$ and at least $n^{1+\\varepsilon}$ edges contains a copy of every $s \\times t$ ordered bipartite tree. The paper does NOT restore the original polylog conjecture, which stays refuted.","claimIssueNote":"The title says \"Proof of the Pach-Tardos conjecture\" without qualification, but the conjecture in its published 2005 form was disproved by Pettie and Tardos in 2024. What is proved is the weaker $n^{1+o(1)}$ statement. The paper's own introduction is explicit about this; the title alone is not.","solveDate":"2026-09-17","model":"ChatGPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":["Lior Gishboliner","Xiangyu Li"],"aiRole":"The paper's declaration of AI use, in full: \"The proof was found by ChatGPT-6 Astra, with substantial input and guidance from the authors. The authors then rewrote the proof and take full responsibility for its correctness.\"","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.20726. The introduction was read and states the distinction this entry turns on: Pach and Tardos conjectured the polylog bound, Pettie and Tardos refuted it, and what is proved here is the weaker $n^{1+o(1)}$ form, restated as Theorem 1 for ordered bipartite trees and Corollary 2 as $\\mathrm{Ex}(n,P) \\le n^{1+O_P(1/\\log\\log n)}$. The AI declaration is quoted verbatim. The mathematics was not checked here; five days old, no referee.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"A 2005 conjecture at the centre of forbidden-matrix theory, which feeds Davenport-Schinzel theory, the Stanley-Wilf line of work and the twin-width parameter; the authors call the weak form arguably the main open problem of the area. The original form's refutation in 2024 raised rather than lowered the stakes on what remained. Level with the Bilu-Linial entry at 35 and below the uniform Turán density entry at 48, which is older and more widely followed.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.20726","sourceName":"Proof of the Pach-Tardos conjecture, arXiv:2609.20726 (17 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Pettie and Tardos, the 2024 refutation of the original form","url":"https://arxiv.org/abs/2407.02638","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"simonovits-sos-3ap-intersecting-families","name":"The Simonovits-Sós conjecture on 3AP-intersecting families","shortName":"3AP-intersecting families","problemNumber":null,"field":"Extremal set theory","fieldGroup":"Combinatorics","statement":"A family $F$ of subsets of $[n]$ is 3AP-intersecting if any two members meet in a set containing a non-trivial three-term arithmetic progression. Simonovits and Sós conjectured that the largest such family has size $2^{n-3}$, attained by fixing a progression. Before this work nothing better than the trivial bound $\\tfrac12 2^n$ was known. What is the maximum size?","posedBy":"Miklós Simonovits and Vera T. Sós, by personal communication to Chung, Graham, Frankl and Shearer, who recorded it in their 1986 paper","yearPosed":1986,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Partial: any 3AP-intersecting family has size at most $(\\tfrac12 - c)2^n$ for an absolute constant $c>0$, the first bound below the trivial one. The conjectured maximum $2^{n-3}$ is not established. The same bound is proved for $H$-intersecting families whenever $H$ is a 3-graph on $[n]$ with bounded codegrees, and a clique shows the codegree assumption cannot be removed.","claimIssueNote":null,"solveDate":"2026-09-16","model":"GPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":["Peter Keevash"],"aiRole":"From the paper's statement on AI use: the proof was found by GPT-6 Astra following a hint by the author, who then simplified and rewrote it.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.18870: the abstract calls this the first non-trivial progress towards the Simonovits-Sós conjecture and states the bound $(\\tfrac12 - c)2^n$ for an absolute $c>0$, generalised to $H$-intersecting families for any 3-graph $H$ of bounded codegree, with a clique showing the codegree hypothesis cannot be dropped. The introduction attributes the conjecture to Simonovits and Sós by personal communication to the authors of reference [1], which is Chung, Graham, Frankl and Shearer, J. Combin. Theory Ser. A 43 (1986), so it dates to 1986 or earlier. Entered as Partial on the paper's own framing. The mathematics was not checked here; six days old, no referee.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A forty-year-old conjecture of two of extremal combinatorics' senior figures, recorded in Chung, Graham, Frankl and Shearer's 1986 paper, with a long-standing conjecture of Alon in the same family, and the first time anyone has beaten the trivial bound. Scored for the problem: level with the periodicity conjecture at 25. The step is partial - the conjectured $2^{n-3}$ is untouched - which the resolution field records rather than the score.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.18870","sourceName":"A non-trivial bound for 3AP-intersecting families, arXiv:2609.18870 (16 September 2026)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"goldbach-s-conjecture-for-the-liouville-function","name":"Goldbach's conjecture for the Liouville function","shortName":"Liouville-Goldbach","problemNumber":null,"field":"Multiplicative number theory","fieldGroup":"Number theory","statement":"The Liouville function $\\lambda(n)$ is $+1$ or $-1$ according to the parity of the number of prime factors of $n$ counted with multiplicity. In August 2018 a MathOverflow question asked for a Goldbach-style statement about its sign: is it true that for every even integer $N > 2$ there are positive integers $a, b$ with $a + b = N$ and $\\lambda(a) = \\lambda(b) = -1$? The case $N = 2$ fails. Sieve methods run into the parity obstruction here, so the question resisted the obvious attacks.","posedBy":"Asked by the MathOverflow user Pablo, question 307479, 3 August 2018; attributed in the Lean development to a formulation of Shusterman","yearPosed":2018,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"True: for every even $N > 2$ there are positive $a, b$ with $a + b = N$ and $\\lambda(a) = \\lambda(b) = -1$, unconditionally, with no analytic or conjectural hypothesis. The route reduces to a prime modulus, then uses $\\lambda(2n) = \\lambda(3n) = \\lambda(5n) = -\\lambda(n)$ to force $\\lambda$ to agree with the Legendre symbol on an interval, where quadratic reciprocity gives a contradiction; no sieve is used, which is how the parity obstruction is avoided.\n\nTwo pieces of context a reader needs. Mangerel (IMRN 2024) proved that $\\lambda(a)\\lambda(N-a)$ cannot be constant for $N \\ge 11$; that yields a pair with opposite signs and so does NOT imply this statement. And the result has since been generalised: on the same MathOverflow question, Lingsen Meng gives an unconditional argument that all four sign pairs occur for every $N \\notin \\{2,3,4,5,6,9,10\\}$, covering odd $N$ too, while crediting this work as independent and prior for the even case.","claimIssueNote":null,"solveDate":"2026-09-16","model":"ChatGPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The AI chose the problem, proved it and then formalized the result in Lean.","verification":"lean-verified","verificationNote":"Audited here on 30 September 2026 from a clone at HEAD. Twelve Lean files, 1,689 lines on Lean 4.34.0-rc2: zero sorry, zero axiom declarations, zero native_decide. Audit.lean prints the axioms of liouville_goldbach and of twelve supporting results. GitHub Actions ran the Lean verification workflow green twice on 16 September, including at tag v1.0.0. Statement fidelity checked by hand against the MathOverflow wording: the headline theorem is liouville_goldbach (N) (hEven : Even N) (hN : 2 < N) concluding the existence of a, b with 0 < a, 0 < b, a + b = N and liouville a = liouville b = -1, which is the question exactly, with no extra hypothesis and nothing conjectural assumed. Independent corroboration, which is the other half of lean-verified: an answer posted to the same MathOverflow question on 25 September 2026 by Lingsen Meng states that for the even case an unconditional proof was given independently in September 2026 by an anonymous author together with a Lean formalization, and links this repository at release v1.0.0.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A question that sat on MathOverflow for eight years with real literature growing around it rather than a curiosity: Mangerel's IMRN 2024 theorem that $\\lambda(a)\\lambda(N-a)$ is non-constant for $N \\ge 11$ is the nearest prior result, and a 2024 preprint gets all four sign patterns under GRH. Level with the Laplacian $S_{n,n}$ entry at 22 and above the Hlawka entry at 20; below the named-conjecture band at 30, since the asker was an individual on a question site rather than a paper.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/CaptainSude/Liouville-Goldbach","sourceName":"GitHub repository with Lean proof, release v1.0.0 (CaptainSude, 16 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"MathOverflow 307479, where the question was asked and later corroborated","url":"https://mathoverflow.net/questions/307479/goldbachs-conjecture-for-the-liouville-function","kind":"problem-record"},{"label":"Mangerel's GRH-conditional four-pattern result","url":"https://arxiv.org/abs/2412.17199","kind":"paper"}],"relations":[],"submittedBy":"nufrogcaca","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"strongly-aperiodic-monotile-in-three-dimensions","name":"Is there a strongly aperiodic monotile in three dimensions?","shortName":"3D strongly aperiodic monotile","problemNumber":null,"field":"Tiling theory; aperiodic order","fieldGroup":"Geometry & topology","statement":"A prototile is strongly aperiodic if it tiles space and no tiling by its congruent copies has any translational period. In the plane this was settled in 2023 by the hat. In three dimensions the known candidates fall short: the Schmitt-Conway-Danzer biprism tiles only with screw motions, so its tilings are weakly aperiodic, and the three-dimensional Socolar-Taylor tile has a periodic stacking direction or is not simply connected. Socolar and Taylor asked in 2012 for a single simply connected three-dimensional prototile that forces nonperiodicity by shape alone. Does one exist?","posedBy":"Joshua E. S. Socolar and Joan M. Taylor, Forcing nonperiodicity with a single tile, Math. Intelligencer 34 (2012), 18-28","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Claimed yes: Chair44 (R44), a rational polyhedral 3-ball built from seven cubes in a chair, whose 24 exposed unit panels carry small square-pyramid features. The claim is that it tiles $\\mathbb R^3$ by congruent copies with reflections allowed, that every such tiling has no translational period and a symmetry group of order at most 24, and that every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The features force each tiling onto a registered lattice; the finite facts - dissection, the 44-contact atlas, 33 first shells, a 6,862 to 44 companion census with 299,975 collision boxes, mesh and angle audits - are enumerated exhaustively and checked in Lean, with a second independent Python implementation replaying them.","claimIssueNote":null,"solveDate":"2026-09-16","model":"GPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":["Ioannis Tsiokos"],"aiRole":"The submission records the model's contribution as finding the solid: GPT-6 Astra produced the Chair44 shape. The formalisation, the finite enumerations, the independent Python replay and the write-up are the author's.","verification":"lean-checked","verificationNote":"Audited here on 27 September 2026 from a clone at HEAD. 85 Lean files, 29,815 lines on Lean 4.31.0. Zero real sorry - the single grep hit sits inside a comment - zero axiom declarations, zero unsafe or implemented_by. But thirty-five native_decide calls across twelve files, which the author discloses rather than hides: AXIOMS.md lists the named compiler hook per theorem, README carries a tiered trust ledger (T1 standard axioms, T1n additionally native hooks), the negative controls are diagnostic-checked, and verify/replay.py reconstructs the same finite facts in standard-library Python. The unconditional spine is r44_einstein: existence, no translational period, and symmetry group of order at most 24.\n\nRecorded lean-checked rather than the submitted lean-verified, for two reasons. First, native_decide discharges goals through Lean's compiled evaluator, so the trusted base includes the compiler and runtime and not the kernel alone; the catalog's Dittert and composites-among-xi-7-n entries are labelled the same way for the same reason. Second, lean-verified additionally requires the formal statement to be anchored outside the prover's own repository - a comparator configuration, a Palomar entry or an audit by someone with no stake - and none is present. The repository's only CI workflow builds a reader notebook and is currently failing, so no automated build checks the Lean at all. A comparator or Palomar statement check plus a green Lean build would lift this to lean-verified.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The three-dimensional form of the einstein problem, asked by name in 2012 and given fresh prominence by the hat in 2023; tiling theory follows it closely and the wider mathematical public knows the two-dimensional version. Level with the Bilu-Linial and Kusner entries at 35, and below the Borsuk frontier's problem at 40, which has a longer literature. The score is for the question, not for the verification level.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.19214","sourceName":"A Strongly Aperiodic Monotile in Three Dimensions, arXiv:2609.19214 (16 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Visualization","url":"https://www.emergencecalculus.com/toys/chair44/","kind":"other"},{"label":"Github","url":"https://github.com/ioannist/six-birds-tiles","kind":"code"},{"label":"Wikipedia","url":"https://en.wikipedia.org/wiki/Einstein_problem","kind":"wikipedia"}],"relations":[],"submittedBy":"WittyWombat899","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"superpolynomial-lower-bounds-for-bit-php-in-unrestricted-resolution-over-paritie","name":"An exponential lower bound for the bit pigeonhole principle in resolution over parities","shortName":"Bit PHP in dag-like Res(⊕)","problemNumber":null,"field":"Proof complexity; resolution over parities","fieldGroup":"Theoretical computer science","statement":"Resolution over parities, $\\mathrm{Res}(\\oplus)$, extends resolution to disjunctions of affine equations over $\\mathbb{F}_2$. Exponential lower bounds are known for tree-like, regular and bounded-depth fragments, but no superpolynomial lower bound for unrestricted DAG-like $\\mathrm{Res}(\\oplus)$ is known for any formula. Does the bit pigeonhole principle with $n + 1$ pigeons and $n = 2^\\ell$ holes require superpolynomially many nodes in unrestricted DAG-like $\\mathrm{Res}(\\oplus)$?","posedBy":"Itsykson and Sokolov (2014), who introduced the system; named as the open benchmark by Efremenko-Garlik-Itsykson (2024) and Itsykson-Podolskii-Shekhovtsov (2026) among others","yearPosed":2014,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claimed: for every $l >= 32$, any dag-like $\\mathrm{Res}(\\oplus)$ refutation of the bit pigeonhole principle with $2^l + 1$ pigeons and $n = 2^l$ holes has more than $exp(n/(32768 l^2)) = 2^{\\Omega(n/log^2 n)}$ nodes; exponential in the number of holes up to a $log^2$ factor, for a proof system with no regularity or depth restriction. The whitepaper's route is a transfer to degree-$O(log n)$ polynomial calculus. Candidate: kernel-checked, read by no specialist.","claimIssueNote":null,"solveDate":"2026-09-15","model":"GPT-6 Astra; Claude Sonnet 5; Claude Fable 5.1; Claude Opus 5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Kamil Braun"],"aiRole":"GPT-6 Astra was the principal contributor to the mathematical development, manuscript drafting, and Lean formalization. Earlier exploration involved Claude Sonnet 5, Fable 5.1, and Opus 5. Later revised using Fable 5.1. Kamil Braun designed and directed the Noemesis research framework, supplied goals and steering, and organized the verification and publication effort. Section 10.3 of the revised manuscript describes the contributions and development history.","verification":"lean-verified","verificationNote":"Held on 15 September 2026 for one reason: nothing outside the author's machine had compiled the proof. Lifted on 22 September. GitHub Actions run 34953139115 (commit 3979e0c, fresh ubuntu-24.04 runner, mathlib cache for dependencies only, .lake/build deleted) ran `lake --wfail build claims.BitPHPSuperpolynomial` & replayed the module in a fresh leanchecker kernel. Run 35282748811 (17 September UTC, commit e4c7723, same setup) did the same for `claims.BitPHPPreprintRevision1`, which adds the revised version's MathResearch.bitPHP_exponential: for $l >= 32$, more than $exp(2^l/(32768 l^2))$ nodes. Both theorems use only propext, Classical.choice and Quot.sound. Audit at 54f0937, byte-identical to 3979e0c: the superpolynomial theorem quantifies over every AffineDAG over the standard bit-PHP initial clauses with a node deriving the empty clause; AffineDAGStep allows initial clauses, semantic weakening, resolution on a parity literal against its complement & any sound two-premise rule, nodes reference any earlier node, no regularity or depth constraint, a superset of dag-like $\\mathrm{Res}(\\oplus)$; usualCNFClause is the standard bit-PHP axiom over all pairs & all $2^l$ labels; the chessboard homology is proved, not assumed. e4c7723 only adds files, & the exponential theorem uses the same definitions. No sorry, custom axiom or native_decide at either commit. Kernel-checked on a machine other than the author's, statement audited: Lean-verified. No specialist has read the argument.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The open benchmark of resolution over parities: superpolynomial lower bounds for the unrestricted dag-like system, named as open by Itsykson and Sokolov in 2014 and by every 2024-2026 paper on it, with the strongest prior results reaching bounded or near-quadratic depth only. Central to proof complexity and known by name in the wider complexity community; below the matroid secretary conjecture, which all of online algorithms watches, and a little under the Borsuk problem at 40 for field size.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/kbr-/math-research/blob/bit-php-preprint-2026-09-19/publications/bit-php-resolution-over-parities/whitepaper.pdf","sourceName":"kbr-/math-research: whitepaper at commit 652e617, Lean claim BitPHPPreprintRevision1.lean built by CI at e4c7723","collection":null,"collectionVersion":null,"links":[{"label":"Final Lean theorem","url":"https://github.com/kbr-/math-research/blob/e4c772336ea2bed84f0ef7424d3953702639361e/formalization/claims/BitPHPPreprintRevision1.lean","kind":"lean-proof"},{"label":"GitHub Actions build on a fresh runner: axioms and kernel replay","url":"https://github.com/kbr-/math-research/actions/runs/35282748811","kind":"independent"},{"label":"An exponential lower bound for the bit pigeonhole principle in resolution over parities","url":"https://arxiv.org/abs/2609.23015","kind":"paper"}],"relations":[],"submittedBy":"FrostyWalrus278","upvotes":0,"downvotes":0,"commentCount":2},{"slug":"abelian-envelope-without-the-quotient-property","name":"Does every abelian envelope have the quotient property?","shortName":"Abelian envelopes and the quotient property","problemNumber":null,"field":"Tensor categories","fieldGroup":"Algebra","statement":"An abelian envelope of a monoidal category is a universal abelian tensor category receiving it. Coulembier, Etingof, Ostrik and Pauwels conjectured in 2023 that every abelian envelope has the quotient property, and proved that the universal rigid monoidal category on one object cannot have an abelian envelope with that property. Does every abelian envelope have the quotient property?","posedBy":"Kevin Coulembier, Pavel Etingof, Victor Ostrik and Bregje Pauwels (2023)","yearPosed":2023,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"No. The universal rigid monoidal category on one object has an abelian envelope, which by the 2023 no-go result cannot have the quotient property.","claimIssueNote":null,"solveDate":"2026-09-15","model":"GPT-6 Astra; Claude Opus 5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Johannes Flake","Jonathan Gruber","Thorsten Heidersdorf"],"aiRole":"From the paper's use-of-AI section: the first version of the proof was found by GPT-6 Astra on 5 September 2026; the authors then simplified and checked it and wrote it up by hand, before using GPT-6 Astra and Claude Opus to check the document. The abstract says outright that AI was used to find these results.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.17467: the abstract states that the universal rigid monoidal category on one object does have an abelian envelope, which disproves the conjecture because Coulembier, Etingof, Ostrik and Pauwels had shown it cannot have one with the quotient property. The candidate envelope comes from monoidal Ringel duality and the universal property is established with Coulembier-Etingof continuants. The mathematics was not checked here; seven days old, no referee.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A 2023 conjecture from a paper by four leading tensor-category theorists, answered three years later; young, and followed by a small community. Above the catalog's 2026 conjecture-in-a-recent-paper entries and below the periodicity conjecture at 25, which has a decade of work behind it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.17467","sourceName":"An abelian envelope without the quotient property, arXiv:2609.17467 (15 September 2026)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bilu-linial-signing-conjecture-counterexample","name":"The Bilu-Linial signing conjecture for regular graphs","shortName":"Bilu-Linial signing conjecture","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"Bilu and Linial asked whether every connected $d$-regular graph admits a signing of its edges by $\\pm 1$ whose signed adjacency matrix has all eigenvalues in $[-2\\sqrt{d-1}, 2\\sqrt{d-1}]$, the Ramanujan interval. A positive answer would give an iterative construction of Ramanujan graphs of every degree; Marcus, Spielman and Srivastava proved the one-sided version, which yields bipartite Ramanujan graphs. Does every regular graph have such a signing?","posedBy":"Yonatan Bilu and Nathan Linial (2004, 2006)","yearPosed":2006,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"False for general regular graphs: an explicit cubic graph $F$ for which every signing produces an eigenvalue outside the Ramanujan interval. $F$ is not itself Ramanujan, and the conjecture restricted to Ramanujan base graphs - the form that would yield Ramanujan graphs of every degree by iteration - remains open.","claimIssueNote":null,"solveDate":"2026-09-14","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Zhiqiang Xu"],"aiRole":"From the paper: the counterexample and its proof strategy were developed with the assistance of ChatGPT, and the author states that all AI-generated text was reviewed and the mathematics verified. The model is not named more precisely in the disclosure.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.15591: the abstract gives a finite connected simple cubic graph $F$ every signing of which has an eigenvalue outside $[-2\\sqrt2, 2\\sqrt2]$, and states the limitation plainly - $F$ is not Ramanujan, so the conjecture restricted to Ramanujan base graphs remains open. The reference list confirms Bilu and Linial (CPC 13 (2004) and Combinatorica 26 (2006)) and the Marcus-Spielman-Srivastava line. The mathematics was not checked here; eight days old, no referee.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The conjecture behind one of the most celebrated lines in spectral graph theory: Marcus, Spielman and Srivastava's bipartite Ramanujan construction came out of attacking it, and it is stated in every survey of expanders. Level with the weak Pach-Tardos entry at 35 and below the uniform Turán density entry at 48; the restriction to Ramanujan base graphs, which is the case that would have given the iterative construction, survives.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.15591","sourceName":"A 3-regular counterexample to the Bilu-Linial signing conjecture, arXiv:2609.15591 (14 September 2026)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"deterministic-k-server-conjecture","name":"Deterministic $k$-server conjecture","shortName":"Deterministic $k$-server","problemNumber":null,"field":"Online algorithms","fieldGroup":"Theoretical computer science","statement":"The $k$-server conjecture states that a deterministic online algorithm can achieve competitive ratio $k$ on every metric space. We prove the conjecture. Specifically, we show that the work function algorithm satisfies it.\nOur proof uses a natural algebraic representation of the work function as a matrix, which encodes all feasible paths to reach a configuration. In this representation, the minimum and addition operations arising in the definition of optimal costs correspond to addition and multiplication of formal expressions, and each work function value corresponds to the determinant of $k$ columns of the matrix. A request arrival updates the representation via a change of basis and row replacement. The amortized analysis is based on a potential function defined in terms of a larger matrix whose coordinates are pairs of coordinates of the original matrix representation.","posedBy":"Manasse, McGeoch, and Sleator","yearPosed":1988,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The paper proves the work function algorithm (WFA) is $k$-competitive for the $k$-server problem on every metric space. Specifically, for initial configuration $C_0$ and every finite request sequence $\\sigma$,\n$$\n\\mathrm{WFA}_{C_0}(\\sigma)\\leq k\\,\\mathrm{OPT}_{C_0}(\\sigma)+\\mathrm{cl}(C_0).\n$$\nSince a competitive ratio below $k$ is impossible in general, this settles the deterministic $k$-server conjecture with the optimal ratio.\n\nThe proof represents work-function values as valuations of determinants over a formal field and constructs a potential from quadratic products of the determinant columns. Its increase at each request pays the extended cost of WFA, yielding the global bound.\n\nThis does not concern the separate randomized $k$-server conjecture, which was previously refuted.","claimIssueNote":null,"solveDate":"2026-09-14","model":"GPT 6 Astra","modelMaker":"OpenAI","humanCollaborators":["Christian Coester","Elias Koutsoupias","Marek Zbysiński"],"aiRole":"The authors first developed, without AI assistance, a potential function proving the previously unknown $k=3$ case on arbitrary metrics. Discussions with ChatGPT 5.5 Pro and Gemini 3.1 Pro helped them understand and symmetrize that potential. Based on this reformulation, ChatGPT 6 Astra derived an algebraic proof valid for arbitrary $k$, thereby supplying the central generalization that proves the conjecture. The paper adapts this proof using an explicit column representation of work functions. After the authors supplied this representation, Astra further adapted the proof and assisted with drafting portions of the paper.","verification":"unreviewed","verificationNote":"This is an arXiv v1 preprint claiming a proof of a longstanding major conjecture. The paper contains a complete mathematical argument, but no independent expert verification, peer review, or formal machine-checked proof is identified in the source. The claim should therefore remain classified as unreviewed pending independent scrutiny.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":null,"significanceNote":null,"solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.15979","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"K-server problem","url":"https://en.wikipedia.org/wiki/K-server_problem","kind":"wikipedia"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tachikawa-second-conjecture-implies-auslander-reiten","name":"Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture?","shortName":"Tachikawa second implies Auslander-Reiten","problemNumber":null,"field":"Representation theory of artin algebras","fieldGroup":"Algebra","statement":"A family of homological conjectures for artin algebras - the Nakayama conjecture, the generalized Nakayama conjecture, the Auslander-Reiten conjecture, the Auslander-Gorenstein conjecture, the Gorenstein-projective conjecture and Tachikawa's first and second conjectures - has been known since the 1970s to sit in a web of one-way implications, with the question of which are equivalent left open. Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture?","posedBy":"Maurice Auslander and Idun Reiten, in their work on the generalized Nakayama conjecture; Tachikawa's conjectures date from the same period","yearPosed":1975,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Yes. With the known implications it follows that the Auslander-Reiten conjecture, the generalized Nakayama conjecture, the Auslander-Gorenstein conjecture, the Nakayama conjecture, the Gorenstein-projective conjecture and Tachikawa's second conjecture are all equivalent, and that Tachikawa's second implies his first. None of them is thereby proved.","claimIssueNote":null,"solveDate":"2026-09-14","model":"GPT-6 Astra; Claude Opus 5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Haruhisa Enomoto"],"aiRole":"From the paper's use-of-AI section: the proof was found by GPT-6-Astra in research directed by the author, and the first draft of the manuscript was written by GPT-6-Astra; Claude Opus 5 revised the exposition; the author set the structure and takes responsibility.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.19172: the abstract states the implication for artin algebras over a commutative artinian ring, by way of the two-fold trivial extension, and draws the consequence that six of the conjectures in the family become equivalent and that Tachikawa's second implies his first. The introduction confirms the Auslander-Reiten attribution. The mathematics was not checked here; eight days old, no referee. Same author and same AI process as the periodicity-conjecture entry of five days earlier.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Collapses six named homological conjectures of the 1958-1975 era into one equivalence class, which is the kind of structural result the field has wanted for decades; the Nakayama conjecture itself is old and well known inside homological algebra. Above the periodicity-conjecture entry at 25 because it settles relations among several named conjectures rather than one, and below Kahn's flow conjecture at 40 for a narrower audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.19172","sourceName":"Tachikawa's second conjecture implies the Auslander-Reiten conjecture, arXiv:2609.19172 (14 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Claimed counterexamples to Auslander-Reiten and Tachikawa (OpenAI math release)","url":"https://vibemathed.com/problem/auslander-reiten-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ryser-conjecture-four-partite-matching-number-two","name":"Ryser's conjecture for four-partite hypergraphs with matching number two","shortName":"Ryser's conjecture, case $(4,2)$","problemNumber":null,"field":"Hypergraph covering","fieldGroup":"Combinatorics","statement":"Ryser's conjecture states that every $r$-partite $r$-uniform hypergraph $H$ satisfies $\\tau(H) \\le (r-1)\\nu(H)$, where $\\tau$ is the vertex-cover number and $\\nu$ the matching number. It is known for $r \\le 3$ (Aharoni) and open in general. Tuza claimed the case $r=4$, $\\nu=2$, giving $\\tau \\le 6$, in an unpublished 1979 manuscript but never published a proof; the best bound in print was $\\tau \\le 7$. Does $\\tau \\le 6$ hold for four-partite four-uniform hypergraphs with matching number two?","posedBy":"Herbert J. Ryser (conjecture, via Henderson's 1971 thesis); the case $(4,2)$ was claimed without proof by Zsolt Tuza in an unpublished 1979 manuscript","yearPosed":1971,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Yes, $\\tau \\le 6$, closing the case $(r,\\nu) = (4,2)$ and putting a proof behind a claim cited from Tuza's unpublished 1979 manuscript for forty-seven years. Ryser's conjecture itself remains open for $r \\ge 4$ in general. The general conjecture is claimed disproved for infinitely many large r in OpenAI's October 2026 math release (catalogued as Ryser's conjecture); this (4,2) case is consistent with that claim, which is not yet independently checked.","claimIssueNote":null,"solveDate":"2026-09-13","model":"GPT-5.6 Sol; Claude Sonnet 4","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Patrick White"],"aiRole":"From the paper's methods section: the proof was found through four rounds of structured reasoning with GPT-5.6 Sol (model gpt-5.6-sol-pro), each round building on verified output of the previous one, with wrong turns recorded; Claude (claude-sonnet-4) served throughout in a framing and checking role.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.14281: the abstract states $\\tau(H) \\le 6$ for four-partite four-uniform $H$ with $\\nu(H)=2$, confirming Tuza's 1979 claim and improving on $\\tau \\le 7$, an integrality consequence of Haxell and Scott (2012); the ingredients named are Gyárfás's intersecting-case theorem, a projection lemma and Kőnig's matching theorem. The reference list confirms Tuza's unpublished 1979 manuscript and his 1983 Ars Combinatoria paper. Entered as Partial: one case of Ryser's conjecture, which stays open. The mathematics was not checked here; nine days old, no referee.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Ryser's conjecture is a well-known problem of hypergraph covering with a fifty-year literature, but this settles one case of it, $(r,\\nu)=(4,2)$, and confirms a claim that has been cited from an unpublished manuscript since 1979. Scored for the case rather than the conjecture: level with the Petrykowski entry at 20.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.14281","sourceName":"Tuza's Ryser-conjecture claim for four-partite hypergraphs with matching number two, arXiv:2609.14281 (13 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/ryser-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-copeland-erdos-constant-is-not-strongly-normal-in-base-10","name":"The Copeland-Erdős constant is not strongly normal in base 10","shortName":"Copeland-Erdős not strongly normal","problemNumber":null,"field":"Normal numbers; digit expansions","fieldGroup":"Number theory","statement":"The Copeland-Erdős constant $0.235711131719\\ldots$ concatenates the primes in increasing order; Copeland and Erdős proved in 1946 that it is normal in base 10. Strong normality, introduced by Belshaw and Borwein, also prescribes the size of the digit-count fluctuations: $x$ is simply strongly normal in base $b$ if for every digit the discrepancy $Z_d(x;N) - N/b$ has $\\limsup$ and $\\liminf$ equal to $\\pm\\sqrt{b-1}/b$ on the scale $\\sqrt{2N\\log\\log N}$, and strongly normal if this holds in every base $b^k$. Champernowne's number is normal but not strongly normal. Does the Copeland-Erdős concatenation of the primes also fail to be strongly normal in base 10?","posedBy":"Adrian Belshaw, On the Normality of Numbers, SFU M.Sc. thesis (2005), section 4.7; restated by Catt, Coons and Velich, Integers 16 (2016), section 3","yearPosed":2005,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"No: the Copeland-Erdős constant is not simply strongly normal in base 10, hence not strongly normal in base 10, and the failure is already visible in the digit 0. The proof compares the zero count at two positions in the expansion. Every prime in [10^(6n), 10^(6n) + 10^(5n)) begins with 1 followed by n zeros, and Ingham's 1937 short-interval theorem with exponent 5/6 supplies about 10^(5n)/(6n log 10) such primes, so the zero excess across that stretch is of order 10^(5n), while simple strong normality would cap the discrepancy at O(sqrt(N log log N)) with N of order 10^(6n), that is at order 10^(3n). Belshaw and Borwein's 2013 Champernowne argument is the model; Ingham replaces the trivial prime count. Strong normality in other bases is not addressed.","claimIssueNote":null,"solveDate":"2026-09-13","model":"ChatGPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The AI chose the question, proved it and then wrote the proof in Lean.","verification":"lean-checked","verificationNote":"Checked here on 22 September 2026. The repository at HEAD (pushed 14 September) has 16 Lean files, 1,879 lines, no sorry, no axiom declaration, no native_decide. GitHub Actions run 34797497727 built it on a fresh runner with the pinned Lean 4.34.0-rc2 and mathlib de2ef68, and printed both final theorems, copelandErdos_not_simply_strong_normal_of_ingham and copelandErdos_not_strong_normal_of_ingham, with axioms propext, Classical.choice and Quot.sound. The statement was audited: copelandErdosReal is Real.ofDigits of the increasing prime concatenation and its canonical decimal digits are proved; SimplyStrongNormal is the Belshaw-Borwein law-of-the-iterated-logarithm condition with the constant sqrt(b-1)/b for every digit; StrongNormal quantifies over every positive power of the base. The one external input, Ingham's 1937 theorem specialised to intervals of length x^(5/6), enters as an explicit theorem hypothesis (Ingham1937Input), not as an axiom, with its specialisation to x = 10^(6n) and the endpoint bookkeeping proved in Lean. Lean-checked rather than Lean-verified for that reason, as the Erdős-Borwein 2-density entry was for AGP. The three-page argument was read and is correct.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A question posed in print twice about the best-known named concatenation constant after Champernowne's: in Belshaw's 2005 thesis, which introduced strong normality, and again in Catt, Coons and Velich's 2016 Integers paper, which settled a class of related numbers and wrote of this one \"of this we have no proof\". Level with the Stoneham-sum and Erdős-Borwein 2-density entries at 25, both named questions about named constants left open in a journal paper; above the Fermat and Stoneham-generalisation entries at 18, whose questions had less standing.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/CaptainSude/Copeland-Erdos-Not-Simply-Strongly-Normal","sourceName":"GitHub repository with Lean proof and three-page paper (CaptainSude, 14 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Belshaw's thesis, section 4.7, where the question is asked","url":"https://summit.sfu.ca/_flysystem/fedora/sfu_migrate/10167/etd1934.pdf","kind":"problem-record"},{"label":"Catt, Coons and Velich (2016), section 3: \"of this we have no proof\"","url":"https://arxiv.org/abs/1511.07532","kind":"problem-record"},{"label":"GitHub Actions build printing the final theorems and axioms","url":"https://github.com/CaptainSude/Copeland-Erdos-Not-Simply-Strongly-Normal/actions/runs/34797497727","kind":"independent"},{"label":"Ingham (1937), the one external theorem","url":"https://doi.org/10.1093/qmath/os-8.1.255","kind":"paper"},{"label":"Copeland–Erdős constant","url":"https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_constant","kind":"wikipedia"}],"relations":[],"submittedBy":"nufrogcaca","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"kusner-conjecture-false-for-all-p-greater-than-4","name":"Kusner's conjecture is false for every $p>4$","shortName":"Kusner's conjecture for $p>4$","problemNumber":null,"field":"Discrete geometry; equilateral sets","fieldGroup":"Geometry & topology","statement":"An equilateral set in $\\ell_p^n$ is a set of points at equal pairwise distance. Kusner conjectured in 1983 that the largest such set has exactly $n+1$ points for every $14$ open, where the catalog's earlier entry had found a single failing exponent and placed the infimum of failing exponents in $[4,5)$. Does the conjecture fail for every $p>4$?","posedBy":"Robert Kusner (1983)","yearPosed":1983,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"False for every $p>4$. This is strictly stronger than the catalog's August entry (arXiv:2608.14013, significance 30), which exhibited $n+2$ equilateral points for a single exponent and only located the infimum of failing exponents in $[4,5)$. Together with Swanepoel for $14$ an equilateral set of $8m$ points in $\\mathbb R^{8m-2}$ for an $m$ depending on $p$ - and states that combining it with Swanepoel and with Ge, Xu and Zhou resolves Kusner's conjecture across the whole range $14$, not one exponent, and with the two prior results the full range. Set above that entry's 30 for completing the picture, and below the uniform Turán density entry at 48 for a smaller literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.14794","sourceName":"Kusner's conjecture is false for $p>4$, arXiv:2609.14794 (13 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"The earlier, weaker counterexample already in this catalog","url":"https://arxiv.org/abs/2608.14013","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"corrected-generalization-of-stoneham-numbers","name":"Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\\alpha_{6,4}$ is not $6$-normal, and $\\alpha_{b,c}$ is $b$-normal iff some prime of $c$ misses $b$","shortName":"Stoneham prime-support criterion","problemNumber":null,"field":"Normal numbers and digit expansions","fieldGroup":"Number theory","statement":"For integers $b, c \\ge 2$ let $\\alpha_{b,c} = \\sum_{k \\ge 1} \\dfrac{1}{c^k\\, b^{c^k}}$. Bailey and Crandall proved in 2002 that $\\alpha_{b,c}$ is normal in base $b$ whenever $b$ and $c$ are coprime, and asked whether coprimality can be weakened to the condition that neither of $b$, $c$ divides the other: is $\\alpha_{b,c}$ then still $b$-normal?","posedBy":"David H. Bailey and Richard Crandall, Random Generators and Normal Numbers, Experimental Mathematics 11 (2002), Section 7 \"Conclusions and open problems\", p. 544","yearPosed":2002,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Mutual nondivisibility is not enough: $b = 6$, $c = 4$ is a counterexample, since every prime dividing 4 divides 6, and then the zero digit of $\\alpha_{6,4}$ in base 6 has limiting frequency one. The exact criterion is that $\\alpha_{b,c}$ is $b$-normal if and only if some prime dividing $c$ does not divide $b$. Sufficiency applies Bailey and Crandall's own exponential-sum estimate to the part of $c$ coprime to $b$; the repository's prior-art note records that Kano's 1993 theorem on Korobov-type constructions already reaches some non-coprime cases, $\\alpha_{4,6}$ among them, so the classification rather than the method is what is new. Reversing the parameters, $\\alpha_{4,6}$ is 4-normal, is also formalised.","claimIssueNote":null,"solveDate":"2026-09-12","model":"GPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Per the submitter, the model found the question in Bailey and Crandall's list of open problems, produced the counterexample and the exact criterion, and wrote the Lean 4 development; the paper carries no byline and the repository names no human author. The repository's own prior-art note, dated 12 September 2026, is unusually careful: it records that Kano's 1993 theorem already reaches some non-coprime cases, that the sufficiency direction is derived from Bailey and Crandall's existing exponential-sum estimate, and that the search is not a certification of novelty.","verification":"lean-verified","verificationNote":"The posed question was read at source. Bailey and Crandall, Section 7, p. 544: \"Instead of coprimality of b, c, just specify that neither integer divides the other. Can a result on b-normality then be effected?\" The submission quotes it faithfully.\n\nThe Lean development was read at commit 5919ad4 on 15 September 2026. `normal_iff` states the criterion for `stonehamConstant b c`, defined as $\\sum_{k \\ge 0} 1/(c^{k+1} b^{c^{k+1}})$, the standard series; `counterexample` states $6 \\nmid 4$, $4 \\nmid 6$ and non-normality of $\\alpha_{6,4}$ in base 6; `NormalInBase` counts every finite word at every prefix length through a radix orbit and is the same definition audited for this author's two-sum entry. The audit file guards the transitive axioms of every headline theorem to exactly propext, Classical.choice and Quot.sound; no sorry, no native_decide. CI rebuilt the project on GitHub twice on 12 September, both green. Not read here: the analytic sufficiency proof line by line. The failing direction is elementary: when every prime of $c$ divides $b$, $1/c^k$ terminates in base $b$ within $O(k)$ digits and the terms sit at positions $c^k$, so the zero digit has limiting frequency one.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A question from the open-problems list closing a well-known 2002 paper on normal numbers, in a corner of the subject a few dozen people work in. Level with the Fermat nonnormality entry at 18, whose question has the same standing and the same authors behind it, and below this submitter's Stoneham two-sum entry at 25, whose question Bailey and Borwein raised in a paper devoted to it rather than in a closing list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/CaptainSude/stoneham-prime-support/tree/5919ad4ebec42e22f707f97b0471d03b8995f70b","sourceName":"CaptainSude/stoneham-prime-support (paper and Lean 4 development, pinned commit 5919ad4)","collection":null,"collectionVersion":null,"links":[{"label":"Lean 4 development (BaileyCrandall.lean, audit file, verification logs)","url":"https://github.com/CaptainSude/stoneham-prime-support/tree/5919ad4ebec42e22f707f97b0471d03b8995f70b/lean","kind":"lean-proof"},{"label":"Paper PDF (no byline; cite by title and repository)","url":"https://github.com/CaptainSude/stoneham-prime-support/blob/5919ad4ebec42e22f707f97b0471d03b8995f70b/paper/stoneham-prime-support.pdf","kind":"paper"},{"label":"Bailey and Crandall 2002, Section 7 poses the question (p. 544)","url":"https://www.davidhbailey.com/dhbpapers/bcnormal-em.pdf","kind":"paper"}],"relations":[],"submittedBy":"nufrogcaca","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"mahler-s-z-number-problem-in-its-generalized-form","name":"Mahler's $Z$-number problem in its generalized form","shortName":"Z(p,q)-numbers","problemNumber":null,"field":"Distribution mod one; Mahler's Z-numbers","fieldGroup":"Number theory","statement":"Let $p>q>1$ be coprime integers and let $Z{p/q}(s,s+t)$ be the set of $\\xi>0$ whose fractional parts $\\{\\xi(p/q)^{n}\\}$ all lie in $[s,s+t)$. The question for which parameters $Z$ is empty remains open.","posedBy":"Mahler","yearPosed":1968,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Theorem E gives $Z_{3/2}(\\tfrac27,\\tfrac57)=\\emptyset$: a window of length $\\tfrac37=0.428571\\ldots$ at base $3/2$, beating the $31/81$ of [Dub19, Thm. 1.2], and with no assumption on the arithmetic nature of $\\xi$, where that result needs algebraicity. Mahler's problem itself is untouched: for which parameters $Z$ is empty remains open, and this is a record window rather than a classification.","claimIssueNote":null,"solveDate":"2026-09-12","model":"Fable 5, Opus 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"AI discovered theorems, formalized them and wrote a draft of the paper from the Lean.","verification":"lean-checked","verificationNote":"Filed at Lean-checked, as submitted. Challenge2.lean is a trusted statement of record that imports nothing but Mathlib, redeclares verbatim every definition occurring in the certified theorems, and states the paper's lettered results with sorry proofs; comparator2.json names fourteen of them and, as configured, checks constant identity across the challenge and solution environments, restricts axioms to propext, Quot.sound and Classical.choice, and requires the Lean kernel to re-accept the solution from a fresh export with no olean loaded. That configuration reads correctly. It is not lifted to Lean-verified because no CI run was found on the repository and the comparator run was not reproduced here, so the machine check rests on the author's report - which is what Lean-checked means. PRIOR_ART.md is unusually candid and worth reading: it records that the search was targeted rather than exhaustive, that the whole of Section 3 was found in print with Theorems 3.5 and 3.6 a machine-checked fragment of [Bug04, Thm. 1], and concludes \"Novelty is therefore recorded as unknown\". Not peer reviewed and no independent expert endorsement.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Mahler's 1968 Z-number problem is a named problem in distribution mod one, closely tied to the 3/2 problem, and open in the form stated here. Above the Dubickas Problem 3 entry at 20, which the catalog notes borrows its interest from Mahler's 3/2 problem sitting at the same kind of alpha; below the 40-50 band, since the audience is the Diophantine-approximation community rather than number theory at large. The score is for the problem, not for the window: the entry is filed Partial.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.researchgate.net/publication/414097146_Confinement_schemas_and_the_reach_of_block_certificates_for_powers_of_rational_numbers_modulo_one","sourceName":"ResearchGate preprint","collection":null,"collectionVersion":null,"links":[{"label":"Registry link","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-09-12-000003","kind":"palomar"},{"label":"Bulk d/l","url":"https://github.com/rwst/Confinement-Certificates/blob/main/paper2.pdf","kind":"paper"},{"label":"Challenge2.lean: the trusted statement of record, Mathlib only","url":"https://github.com/rwst/Confinement-Certificates/blob/main/Challenge2.lean","kind":"lean-statement"},{"label":"PRIOR_ART.md: the literature search, with novelty recorded as unknown","url":"https://github.com/rwst/Confinement-Certificates/blob/main/PRIOR_ART.md","kind":"other"}],"relations":[],"submittedBy":"LucidKestrel185","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"medvedev-logic-is-undecidable","name":"Medvedev logic is undecidable","shortName":"Undecidability of Medvedev logic","problemNumber":null,"field":"Intuitionistic logic; computability theory; tiling problems","fieldGroup":"Logic & foundations","statement":"Is Medvedev logic $\\mathsf{ML}$—the intuitionistic logic of finite problems—decidable, and can it be recursively axiomatized?","posedBy":"Implicit by Yuri Medvedev (1962), explicitly by Maksimova et al 1979","yearPosed":1979,"ageNote":"Implicit in Medvedev's original 1962 paper, but first stated to be open by Maksimova et al 1979","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The paper reduces periodic plane tiling to non-theoremhood in Medvedev logic and proves $\\mathsf{ML}$ is undecidable and not recursively axiomatizable. A modification also proves undecidability of Skvortsov logic for infinite problems, and shows the two logics can be separated using any aperiodic plane tiling. Both papers are preprints, so the result should remain a review-pending candidate.","claimIssueNote":null,"solveDate":"2026-09-11","model":"ChatGPT Sol 5.6; Claude Opus 5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Rodrigo Nicolau Almeida","Søren Brinck Knudstorp; Pawel Pawlowski"],"aiRole":"ChatGPT Sol 5.6 generated the central idea and technical development, including the tiling poset and reduction from periodic plane tiling. Claude Opus 5 produced the Lean formalization. The authors report obtaining the first result on 4 September, then checking the literature and consequences, organizing the argument, and writing the paper; prompts and development records are public.","verification":"lean-checked","verificationNote":"The authors report the main reduction argument was checked in Lean. The preprint has not been peer reviewed, and this submission does not independently audit that the formal statement exactly matches every informal claim. Paweł Pawłowski independently announced essentially the same undecidability conclusion almost simultaneously, but that does not amount to cross-verification of all details of either proof.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":null,"significanceNote":null,"solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.13359","sourceName":"arXiv preprint","collection":null,"collectionVersion":null,"links":[{"label":"Independent near-simultaneous result","url":"https://arxiv.org/abs/2609.11576","kind":"independent"},{"label":"Maksimova et al 1979 (in Russian) - Problem statement","url":"https://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=42643&option_lang=eng","kind":"problem-record"},{"label":"Pawlowski's Lean formalization of the proof","url":"https://github.com/Haptism89/medvedev-lean","kind":"lean-proof"}],"relations":[],"submittedBy":"BoldPanther302","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"normality-of-sum-of-two-stoneham-constants","name":"Normality of a sum of two Stoneham constants","shortName":"Normality of Stoneham Sum","problemNumber":null,"field":"Normal numbers and digit expansions","fieldGroup":"Number theory","statement":"For coprime $b,c\\ge 2$ the Stoneham constant $\\alpha_{b,c}=\\sum_{k\\ge1}1/(c^{k}b^{c^{k}})$ is known to be $b$-normal. Bailey and Borwein asked in 2012 what happens to a sum of two of them sharing the base: with $b,c_1,c_2\\ge2$, $(b,c_1)$ and $(b,c_2)$ coprime, is $\\alpha_{b,c_1}+\\alpha_{b,c_2}$ normal in base $b$? In their words, \"it is not known at the present time whether the sum $\\alpha_{b,c_1}+\\alpha_{b,c_2}$ is $b$-normal\".","posedBy":"David H. Bailey and Jonathan M. Borwein, \"Nonnormality of Stoneham constants\", Ramanujan J. 29 (2012) 409-422, Section 4 (repeated in Section 6)","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Yes: the sum of two Stoneham constants sharing a base is normal in that base, for every admissible pair of parameters, with overlapping occurrences counted. Not to be confused with what the same 2012 paper proves: its Theorem 3 shows a sum of two $B$-NONnormal Stoneham constants is $B$-nonnormal, which concerns a different base and the opposite property. The open half was normality in the common defining base $b$, and that is what this settles.","claimIssueNote":null,"solveDate":"2026-09-10","model":"ChatGPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The submitter states that the model chose the problem, proved it and produced the Lean formalisation. That is their account of their own project and is recorded at face value; as with this author's earlier xi-normality entry, the repository does not separate what a model did from what a person did, and the manuscript carries no author byline, so nothing found in review corroborates the autonomy claim beyond the submitter's word and nothing contradicts it.","verification":"lean-verified","verificationNote":"Checked here on 12 September 2026 by reading the repository at commit f4d3d1f. Challenge.lean imports only Mathlib, writes the Stoneham series out in full, and states normality of the sum directly: for every length $l$ and every $j= 2 the edges v-1 -> v labelled r(v) and v-2 -> v labelled 1-r(v), where r(2j) = s(j) and r(2j+1) = 1-s(j). A path spelling s has v_k >= 2k, so the offset v_k - 2k is a nonincreasing nonnegative integer and stabilises; from then on every step is +2 and the labels force s(k) = s(k+e+1) or s(k) = 1-s(k+e+1), so s is eventually periodic. Finite alphabets reduce to the binary case through an aperiodic indicator and a copy-lift. The Lean endpoint finite_alphabet_real_avoidance was compared with Alexander's Definition 1 clause by clause: finite roots, finite children, finite real sublevel sets, strictly increasing birthdates, infinite vertex set, a parent of each label at every non-root, at most one label per edge, and realisation from any starting vertex. Faithful. 297 lines, no sorry, no axiom, no native_decide; the author's log shows the standard axioms. There is no CI and the site did not rebuild it, so Lean-checked; a GitHub Actions build would lift it. The positive direction is Alexander's published Theorem 10.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"The question Alexander himself called \"perhaps the most important question remaining\" in the paper that introduced the notion, open for thirteen years, but in a one-author corner of infinite graph theory with a small literature. Above the 2026 conjecture-in-a-recent-paper entries at 4 and the numbered Erdős problems at 10, since it stood in print for a long time and closes a classification; below Umans-Wang at 18, whose question carried algorithmic stakes.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/avg-netizen/biological-unavoidability","sourceName":"GitHub repository with manuscript, Lean project and verification records (avg-netizen, 10 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Alexander (2013), EJC 20(1) P31, section 6 poses the question","url":"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p31","kind":"problem-record"},{"label":"The Lean endpoint and population axioms","url":"https://github.com/avg-netizen/biological-unavoidability/blob/main/lean/RealBirthdates.lean","kind":"lean-statement"}],"relations":[],"submittedBy":"SilentIbis765","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"uniform-turan-density-of-the-tetrahedron","name":"The uniform Turán density of the tetrahedron","shortName":"Uniform Turán density of $K_4^{(3)}$","problemNumber":null,"field":"Extremal hypergraph theory","fieldGroup":"Combinatorics","statement":"The uniform Turán density $\\pi_u(H)$ of a 3-uniform hypergraph $H$ is the smallest $d$ such that every sufficiently large hypergraph in which all linear-sized subsets induce density above $d$ contains $H$. Erdős and Sós asked in the 1982 paper that founded the subject for the value of $\\pi_u(K_4^{(3)})$, the complete 3-graph on four vertices, and for $\\pi_u(K_4^{(3)-})$. The second was settled at $1/4$ in 2018 by Glebov, Kráľ and Volec; the tetrahedron itself resisted for forty-four years, with $1/2$ the conjectured value from a known lower-bound construction. What is $\\pi_u(K_4^{(3)})$?","posedBy":"Paul Erdős and Vera T. Sós, in the 1982 paper that introduced uniform Turán densities","yearPosed":1982,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"$\\pi_u(K_4^{(3)}) = 1/2$, matching the long-known lower-bound construction. The AI contribution is one identity inside the argument rather than the argument, which is why this is recorded as AI-assisted; it is included in the catalog because the identity is a step of the proof, not a matter of exposition.","claimIssueNote":null,"solveDate":"2026-09-10","model":"ChatGPT 6 Pro; Aristotle","modelMaker":"OpenAI; Harmonic","humanCollaborators":["Matija Bucić"],"aiRole":"From the paper's declaration of AI use: identity (6) in the proof was found by ChatGPT 6 Pro, after an early draft containing the author's ideas was uploaded to it. Separately, the paper relies on a computational verifier; besides the author hand-checking that verifier's correctness, Aristotle was used to check it in Lean and to run a separate audit. The mathematical frame, the reduction and the write-up are the author's; the model supplied one identity inside it.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.11802: the abstract states the value $1/2$ and that it answers a question of Erdős and Sós from the founding 1982 paper, and the introduction names that paper and the companion question about $K_4^{(3)-}$. The AI declaration is quoted as written. The mathematics was not checked here. Aristotle's Lean check covers the computational verifier used inside the proof, not the proof as a whole, so this is not a formalisation of the theorem and the entry does not claim one. Twelve days old, no referee.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"One of the two questions the founding paper of uniform Turán density asked, open for forty-four years, with the companion question ($K_4^{(3)-}$, settled at $1/4$ in 2018) regarded as a landmark when it fell. Well known across extremal combinatorics without being a household name outside it: below the Erdős-Sós conjecture at 58, which is older and more central, and above the Bilu-Linial signing conjecture at 35.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.11802","sourceName":"The uniform Turán density of the tetrahedron, arXiv:2609.11802 (10 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Glebov, Kráľ and Volec, the companion question settled in 2018","url":"https://arxiv.org/abs/1705.10740","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hodge-standard-conjecture-abelian-varieties","name":"Grothendieck's Hodge standard conjecture for abelian varieties","shortName":"Hodge standard conjecture for abelian varieties","problemNumber":null,"field":"Algebraic geometry: standard conjectures in positive characteristic","fieldGroup":"Geometry & topology","statement":"Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $K$, $h$ an ample class, and $P^r_h(A)$ the primitive part of the space of codimension-$r$ cycles modulo numerical equivalence, $0\\le r\\le g/2$. Grothendieck's Hodge standard conjecture asserts that the form $(x,y)\\mapsto(-1)^r\\deg(x y h^{g-2r})$ is positive definite on $P^r_h(A)$. In characteristic zero it follows from Hodge theory; in positive characteristic it is open beyond surfaces and some low-dimensional cases. Milne (2002) showed that the Hodge conjecture for CM abelian varieties would imply it for abelian varieties. Does the Hodge standard conjecture hold for all abelian varieties in every characteristic?","posedBy":"Alexander Grothendieck (standard conjectures); the abelian-variety reduction is due to Milne (2002)","yearPosed":null,"ageNote":"The manuscript cites Milne's 2002 reduction, not Grothendieck's paper, so no posing year is taken from it. The standard conjectures are usually dated to Grothendieck's 1968 Bombay talk, published 1969.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims the Hodge standard conjecture for l-adic etale cohomology for all abelian varieties in arbitrary characteristic, deduced from the paper's CM Hodge theorem via Milne 2002. It does not address the standard conjectures for other varieties, the Lefschetz standard conjecture beyond what the abelian case already gives, or crystalline cohomology.","claimIssueNote":null,"solveDate":"2026-09-10","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The README names the proof of the Hodge conjecture for CM abelian varieties as one of two exceptions to that fixed procedure. It does not say what was done differently or whether people intervened, so the three-hour figure does not apply to this result and the human role is undisclosed. This entry is a corollary in that same manuscript, obtained through Milne's 2002 theorem.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked here: Corollary (Hodge standard conjecture for abelian varieties) in the TeX source, read against the posed problem; it states positivity of the primitive intersection form for every abelian variety over every algebraically closed field and every ample class, and coincidence of numerical and l-adic homological equivalence. The deduction applies Milne's Theorem 3.3 (2002) with his addendum to the paper's CM Hodge theorem, so it depends on that unrefereed theorem. No Lean formalization is listed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"A well-known open case of one of Grothendieck's standard conjectures, but obtained here by a known reduction and of interest mainly to specialists in motives. Placed with the Upper bounds for high-dimensional sphere packing entry and the 3SUM hypothesis at 50, above Connes' rigidity conjecture at 45, and below the Tate case here at 60.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf","sourceName":"OpenAI math release: The rational Hodge conjecture for CM abelian varieties","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Milne, Lefschetz motives and the Tate conjecture (1999): Hodge for CM implies Tate over finite fields","url":"https://www.jmilne.org/math/articles/1999b.html","kind":"other"},{"label":"Earlier edition (September 30, 2026)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026/paper.pdf","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hodge-conjecture-products-of-k3-surfaces","name":"The Hodge conjecture for products of K3 surfaces","shortName":"Hodge conjecture for products of K3 surfaces","problemNumber":null,"field":"Algebraic geometry: Hodge conjecture, K3 surfaces","fieldGroup":"Geometry & topology","statement":"The rational Hodge conjecture asserts that on a smooth projective complex variety every rational $(p,p)$ class is algebraic. For a single K3 surface this is the Lefschetz $(1,1)$ theorem, but products $S_1\\times\\cdots\\times S_m$ carry Hodge classes coupling the transcendental cohomologies of the factors, and powers carry further invariant tensors. Known cases include Hodge isometries (Mukai, Nikulin, Buskin), CM endomorphism fields (Ramon Mari), and special families via algebraic Kuga-Satake correspondences (Schlickewei, Varesco, Floccari). Does the rational Hodge conjecture hold in every codimension for every finite product of projective complex K3 surfaces?","posedBy":"W. V. D. Hodge (1950) for the conjecture; the K3 product case is a long-studied special case (Mukai, Ramon Mari, Varesco, among those cited)","yearPosed":1950,"ageNote":"Hodge's 1950 Congress address is the posing date for the conjecture. The product-of-K3 case was not posed as a separate question at one date; the manuscript's history section traces it through Mukai, Buskin, Ramon Mari and Varesco.","solveType":"proved","resolution":"retracted","aiContribution":"ai-discovered","resultNote":"Claims the rational Hodge conjecture in every degree for every finite product of projective complex K3 surfaces, with no restriction on Picard numbers, periods or endomorphism fields, including powers of one surface. On the way it re-proves algebraicity of the exact Kuga-Satake correspondence. It says nothing about the integral Hodge conjecture or about hyperkahler varieties beyond what follows.","claimIssueNote":"Withdrawn by the authors. OpenAI's withdrawal notice of 6 October 2026: \"The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point cancellation. The resulting gap leaves this paper's claimed rational Hodge theorem for products of projective K3 surfaces unproved.\" The notice adds that the withdrawal concerns the proof and does not assert that the statement is false. The question is open again.","solveDate":"2026-09-10","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math) states its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The proof uses the companion CM Hodge theorem as an input, and the README lists that theorem as produced outside the fixed procedure.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. The release withdrew this manuscript on 6 October 2026, with the Kuga-Satake paper and the split abelian eightfold paper it builds on: the proof is not established (see the claim issue). Checked here: Theorem 1.1 of the TeX source read against the posed problem; surjectivity of $CH^p(X)_\\mathbb Q\\to \\mathrm{Hdg}^p(X)$ for every product of projective K3 surfaces, repeated or distinct, every $p$. Inputs named by the paper and not checked here: the companion quadratic-locus criterion, the companion CM Hodge theorem, Buchweitz-Flenner and Pridham semiregularity, and homological mirror symmetry for the quartic and tori. The mixed-product step is new. No Lean formalization is listed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"A natural and much-studied case of the Hodge conjecture beyond abelian varieties, known before only for special families. Scored with the 3SUM hypothesis and Upper bounds for high-dimensional sphere packing at 50, a little above the Kuga-Satake entry here (48) because it covers mixed products, and well below the CM abelian case (72).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf","sourceName":"OpenAI math release: The rational Hodge conjecture for products of K3 surfaces","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"Algebraicity of Kuga-Satake correspondences for K3 surfaces (OpenAI, 2026-10-03)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/manuscript.pdf","kind":"paper"},{"label":"Algebraic Kuga-Satake correspondences and Hodge conjectures on a K3 quadratic locus (OpenAI, 2026-09-30)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf","kind":"paper"},{"label":"The rational Hodge conjecture for CM abelian varieties (OpenAI, 2026-09-30)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026/paper.pdf","kind":"paper"},{"label":"A conditional reduction for algebraic Kuga-Satake correspondences (OpenAI, 2026-09-10)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf","kind":"paper"},{"label":"Hodge, The topological invariants of algebraic varieties, ICM 1950 Proceedings","url":"https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.1/ICM1950.1.ocr.pdf","kind":"problem-record"},{"label":"Retraction notice on GitHub","url":"https://github.com/openai/math/blob/main/history.md","kind":"announcement"},{"label":"OpenAI withdrawal notice (6 October 2026)","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/README.md","kind":"announcement"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":2},{"slug":"kuga-satake-hodge-conjecture","name":"The Kuga-Satake Hodge conjecture for K3 surfaces","shortName":"Kuga-Satake Hodge conjecture","problemNumber":null,"field":"Algebraic geometry: K3 surfaces, Kuga-Satake construction, algebraic cycles","fieldGroup":"Geometry & topology","statement":"For a projective complex K3 surface $S$ with transcendental lattice $T(S)$, the Kuga-Satake construction (1967) gives an abelian variety $A_S$ and a Hodge embedding $\\kappa_S: T(S)\\hookrightarrow H^1(A_S)\\otimes H^1(A_S)\\subset H^2(A_S\\times A_S,\\mathbb Q)$ through the even Clifford algebra. Deligne showed the corresponding class is absolute Hodge and Andre that it is motivated; algebraicity was known only for special families (Paranjape, van Geemen, Voisin, Floccari, Varesco and others). The Hodge conjecture predicts that $\\kappa_S$ is induced by an algebraic cycle. Is there, for every projective K3 surface $S$, a cycle $\\Gamma\\in CH^2(S\\times A_S\\times A_S)_\\mathbb Q$ whose action on $T(S)$ is $\\kappa_S$?","posedBy":"Implicit in Kuga and Satake (1967) and Deligne (1972); formulated as the Kuga-Satake Hodge conjecture in the literature the manuscripts cite (van Geemen and others)","yearPosed":null,"ageNote":"The manuscripts describe algebraicity as the open question left by Kuga-Satake 1967, Deligne 1972 and Andre 1996, but name no single poser or date for the conjecture itself, so no year is set.","solveType":"proved","resolution":"retracted","aiContribution":"ai-discovered","resultNote":"Claims that the Kuga-Satake correspondence is algebraic for every smooth projective complex K3 surface, with the fixed normalization and the full even-Clifford target, and also for isogenous Kuga-Satake models. Combined with the quadratic-locus companion it gives the Hodge conjecture for every self-power of every projective K3 surface. It does not address non-projective K3 surfaces or integral coefficients.","claimIssueNote":"Withdrawn by the authors. OpenAI's withdrawal notice of 6 October 2026: \"The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point cancellation. The resulting gap leaves this paper's claimed algebraicity theorem unproved.\" The notice adds that the withdrawal concerns the proof and does not assert that the statement is false. The conjecture is open again.","solveDate":"2026-09-10","model":"Unreleased internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. This result is not among the README's named exceptions. Its companion on products of K3 surfaces does use the CM Hodge theorem, which is an exception, as an input.","verification":"contested","verificationNote":"No independent mathematician has checked this yet. The release withdrew this manuscript on 6 October 2026: the proof is not established (see the claim issue). Checked here: Theorem 1.1 of the TeX source read against the posed problem; it asserts a rational cycle on $S\\times A_S\\times A_S$ inducing exactly the prescribed full even-Clifford map $\\kappa_S$, for every projective K3 surface and fixed standard data, which is the conjecture as usually stated. The proof rests on the companion conditional reduction (2026-09-10, same release) and builds the needed nonzero correspondence through immersed Lagrangians, mirror symmetry and semiregular deformation; none of this was refereed. The October 4 manuscript re-derives the same exact correspondence. No Lean formalization is listed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":48,"significanceNote":"A well-known conjecture inside the K3 and Hodge-theory community for half a century, known only for special families. Scored above Connes' rigidity conjecture at 45 and the Hyperkahler period-index conjecture at 22, level with the uniform Turan density of the tetrahedron at 48, and below the CM abelian case here (72) since its reach outside K3 geometry is smaller.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/manuscript.pdf","sourceName":"OpenAI math release: Algebraicity of Kuga-Satake Correspondences for K3 Surfaces","collection":"openai-math-2026-10","collectionVersion":"fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb","links":[{"label":"A conditional reduction for algebraic Kuga-Satake correspondences (OpenAI, 2026-09-10)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf","kind":"paper"},{"label":"Algebraic Kuga-Satake correspondences and Hodge conjectures on a K3 quadratic locus (OpenAI, 2026-09-30)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf","kind":"paper"},{"label":"The rational Hodge conjecture for products of K3 surfaces (OpenAI, 2026-10-04)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf","kind":"paper"},{"label":"OpenAI withdrawal notice (6 October 2026)","url":"https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/README.md","kind":"announcement"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"periodicity-conjecture-for-finite-dimensional-algebras","name":"The periodicity conjecture for finite-dimensional algebras","shortName":"Periodicity conjecture","problemNumber":null,"field":"Representation theory of algebras","fieldGroup":"Algebra","statement":"A module is periodic when it is isomorphic to one of its own higher syzygies, and an algebra is periodic when it is periodic as a bimodule over itself. The periodicity conjecture, studied by Erdmann and Skowroński in their work on periodic and weighted surface algebras, asks whether a finite-dimensional algebra must be periodic whenever all of its simple modules are. Is it true?","posedBy":"Karin Erdmann and Andrzej Skowroński, in their programme on periodic algebras and blocks of group algebras","yearPosed":2015,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"False: an explicit 36-dimensional algebra whose simple modules all have period four, but which has a 2-dimensional nonperiodic module and so is not itself periodic.","claimIssueNote":null,"solveDate":"2026-09-09","model":"GPT-6 Astra; Claude Opus 5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Haruhisa Enomoto"],"aiRole":"From the paper's use-of-AI section: the counterexample and its proof were found by GPT-6-Astra in research directed by the author, and the first draft of the manuscript was written by GPT-6-Astra; Claude Opus 5 reviewed the exposition; the author set the structure and takes responsibility.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.09732: the abstract states the conjecture and gives a 36-dimensional counterexample whose simple modules have period four and which carries a 2-dimensional nonperiodic module, so the algebra itself is not periodic. The reference list confirms the Erdmann-Skowroński attribution (Colloq. Math. 138 (2015) on the periodicity conjecture for blocks of group algebras; J. Algebra 505 (2018) on weighted surface algebras). The mathematics was not checked here; thirteen days old, no referee.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A conjecture with a decade of work behind it in the Erdmann-Skowroński programme on periodic algebras, and a clean finite counterexample against it. A specialist problem inside representation theory of algebras: level with the catalog's Auslander-Reiten-Smalø entry at 20 plus a little for the conjecture having a name and a literature, below Talagrand's cotype problem at 32.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.09732","sourceName":"A counterexample to the periodicity conjecture for finite-dimensional algebras, arXiv:2609.09732 (9 September 2026)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-hilbert-transform-umd-dependence-problem","name":"The Hilbert transform-UMD dependence problem","shortName":"Hilbert–UMD comparison","problemNumber":null,"field":"Banach space geometry and harmonic analysis","fieldGroup":"Analysis","statement":"For fixed $10$, and let a family of real $n\\times n$ matrices generate a finite entire product monoid $S$. If $s=\\min_{X\\in S}\\operatorname{rank}X$, then some word attains rank $s$ with length at most\n$$\nB(n,s)=n2^{n-s}-\\frac{n(n+1)}2+\\frac{s(s-1)}2.\n$$\nIn particular, if $S$ contains zero, a zero word has length at most\n$$\nB(n,0)=n2^n-\\frac{n(n+1)}2=\\Theta(n2^n).\n$$\nOver the rationals, these bounds improve Kiefer–Ryzhikov's (2026) $3^{n^2}$ bounds for mortality and minimum-rank diameter. The mortality bound also improves Almeida–Steinberg's (2009) universal rational bound $(2n-1)^{n^2}-1$ for $n>1$. No finiteness assumption on the generating alphabet is needed. The empty word is allowed and suffices when $s=n$.","posedBy":"Jorge Almeida and Benjamin Steinberg","yearPosed":2009,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"For any family of real $n\\times n$ matrices generating a finite entire product monoid of minimum rank $s$, some word attains rank $s$ within\n$$\nB(n,s)=n2^{n-s}-\\frac{n(n+1)}2+\\frac{s(s-1)}2\n$$\nletters. The same holds over $\\mathbb Q$, without requiring a finite generating alphabet. For mortality, $s=0$, giving a bound of order $n2^n$.\n\nOver $\\mathbb Q$, this improves Kiefer–Ryzhikov's (2026) $3^{n^2}$ bounds for mortality and minimum-rank diameter, and Almeida–Steinberg's (2009) mortality bound $(2n-1)^{n^2}-1$ for $n>1$.\n\nThe proof uses rank-dependent sandwich descent. Further results give the sharp planar threshold four, additive invariant-flag bounds with sharp small-block examples, and cubic-size compressed witness existence with correct evaluation. Boundedness and individually finite-power generators alone admit no uniform planar mortality bound. General optimality and polynomial-time synthesis are not claimed.","claimIssueNote":null,"solveDate":"2026-09-08","model":"GPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":["Eugene Gilburg"],"aiRole":"AI produced the manuscript's original mathematical contributions, exposition and Lean formalization. The human publisher directed and organized the research without claiming professional mathematical review. Established methods and prior results are credited separately.\n\nThe research developed a rank-descent argument using compressed returns, finite-group averaging and a symmetric-matrix lift. A matrix-valued invariant-form defect gives a rank-dependent short-word detection bound. Iterating the resulting rank-decreasing sandwiches yields the mortality and minimum-rank bounds.","verification":"lean-checked","verificationNote":"The site's assessment dated 8 September 2026 concerned the originally submitted mortality theorem. It reported inspection of the formal statement and its hypotheses, no sorry or axiom declarations in the inspected theorem file, and an Audit.lean file exposing endpoint statements and transitive axioms. It assigned the Lean-checked, statement-unaudited tier. No professional human mathematical review was claimed.\n\nThe current manuscript is revision r3, published 10 September 2026. Its author-supplied verification supplement records a clean Lean 4.33.1 build of 25 mathematical modules with pinned dependencies, 20 exact statement checks, and endpoint/transitive-axiom audits. The recorded checks passed; the audited endpoints use only propext, Classical.choice and Quot.sound.\n\nThe claim map identifies formal support for the real and rational minimum-rank and mortality bounds, planar and invariant-flag results, explicit sharp block examples, boundedness nonuniformity, and compressed-witness existence with evaluator correctness. Prose interpretations and limitations are identified separately. Polynomial-time synthesis is not claimed.\n\nThe current supplement does not constitute a new site-administered or independent statement audit. The existing verification tier is retained.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"Answers the exponential side of a 2009 question of Almeida and Steinberg and improves the Kiefer-Ryzhikov rational bound under the same finite-product promise. A clean quantitative advance in a specialized corner of matrix semigroup theory; the polynomial alternative, which is the interesting half, stays open.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/egilburg/aimath/tree/mortality-r4/mortality","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"MinimumRankBound.lean: the main theorem as formalized","url":"https://github.com/egilburg/aimath/blob/mortality-r4/mortality/lean/FiniteMonoidMortality/MinimumRankBound.lean","kind":"lean-proof"},{"label":"Publication.lean: the proof entry point","url":"https://github.com/egilburg/aimath/blob/mortality-r4/mortality/lean/Publication.lean","kind":"lean-proof"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":2},{"slug":"erdos-problem-1221","name":"Erdős Problem #1221","shortName":"de Bruijn-Erdős consecutive gap problem","problemNumber":null,"field":"Discrepancy","fieldGroup":"Combinatorics","statement":null,"posedBy":"de Bruijn, Erdos (1949)","yearPosed":1949,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-09-08","model":"GPT-Astra","modelMaker":"OpenAI","humanCollaborators":["Samuel Korsky"],"aiRole":null,"verification":"unreviewed","verificationNote":null,"publication":null,"resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered Erdos problem from de Bruijn and Erdos, 1949, sitting at the reference point for the scale. Still marked open by erdosproblems.com with the claim unexamined, which is what holds it at Candidate rather than what sets the score.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/forum/thread/1221/proof-claims","sourceName":"Google Drive preprint","collection":null,"collectionVersion":null,"links":[{"label":"erdosproblems.com/1221: the problem record, still marked open","url":"https://www.erdosproblems.com/1221","kind":"problem-record"}],"relations":[],"submittedBy":"GoldenMongoose827","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ipm-blowup-spacetime-smooth-forcing","name":"Finite-time blowup for the IPM equation with a uniformly space-time smooth force","shortName":"IPM blowup, space-time smooth force","problemNumber":null,"field":"Fluid dynamics; incompressible porous media equation","fieldGroup":"Differential equations","statement":"Córdoba and Martínez-Zoroa proved finite-time singularity formation for the two-dimensional incompressible porous media equation from smooth initial data with a force smooth in space but merely bounded in time, that is in $L^\\infty_t C^\\infty_x$. Their Remark 1 anticipates joint smoothness in space and time but does not prove it. Can the force be taken uniformly smooth in space and time?","posedBy":"Diego Córdoba and Luis Martínez-Zoroa, Remark 1 of arXiv:2410.22920, where joint space-time smoothness is anticipated but not part of the theorem","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Yes. Theorem 2.1: there are a smooth odd initial density $\\rho_{in}\\in C^\\infty(\\mathbb T^2)$ of zero spatial mean, an odd force $F\\in C^\\infty([0,1]\\times\\mathbb T^2)$, and a solution smooth on $[0,T]$ for every $T<1$ with $\\rho(t)\\to\\rho_*$ in $C^\\eta$ for every $0\\le\\eta<1$, yet $\\|\\nabla\\rho(t)\\|_\\infty$ and $\\|D_xu_{\\mathbb T}(\\rho(t))\\|_\\infty$ both diverging as $t\\uparrow1$. The advance over Córdoba and Martínez-Zoroa is precisely the force class, from $L^\\infty_t C^\\infty_x$ to uniformly space-time smooth, on the torus rather than the plane. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation \"is not on the Clay Institute's list of prize problems\".","claimIssueNote":null,"solveDate":"2026-09-08","model":"Claude, Codex with GPT-5.6 Sol","modelMaker":"Anthropic / OpenAI","humanCollaborators":["Levent Alpöge","Tristan Buckmaster","Matei P. Coiculescu"],"aiRole":"This is the earliest of the three results and the one the authors describe as feeding the others: the Boussinesq AI statement says the Boussinesq work \"involved inputting ideas from previous joint work of ours on blowup for the IPM equation following Córdoba-Martínez-Zoroa\". Buckmaster's public statement covers the whole project: \"For most of the past year progress was slow. We worked through the literature and upgraded various preliminary results, up to obtaining finite time blow up for the Incompressible Porous Media equation (with smooth forcing)\", using \"Anthropic's Claude, OpenAI's Codex, especially with GPT-5.6 Sol\". The IPM paper itself does not break the contribution down per step, and defers a fuller account: \"The complete human-readable proofs will be released shortly by the first and second authors, together with an account of the role of artificial intelligence in this work.\"","verification":"lean-verified","verificationNote":"Like the Boussinesq and Euler results, this one has a Lean formalisation: tristanbuckmaster/fluid_lean will contain projects for Boussinesq (twice) and Euler and also for IPM, confirmed by listing the repository tree on 8 September 2026. The paper is a 57-page manuscript on arXiv and not peer reviewed, and no independent expert reading is on record. It is the most conventional of the three write-ups, being the one the authors had time to prepare.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The narrowest of the three: the equation already had a smooth-data blowup theorem and what changes is the regularity of the force, a gap its own authors had flagged. It earns its place as a stated open question now closed, and as the result the Boussinesq and Euler constructions were built out of, but it is an increment on an existing theorem rather than a new frontier.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.16470","sourceName":"Extending the Córdoba-Martínez-Zoroa IPM blow-up to uniformly space-time smooth forcing (manuscript)","collection":null,"collectionVersion":null,"links":[{"label":"Buckmaster's public statement on the work and its release","url":"https://cims.nyu.edu/~tristanb/statement.pdf","kind":"announcement"},{"label":"Córdoba and Martínez-Zoroa: the theorem extended, and Remark 1's open case","url":"https://arxiv.org/abs/2410.22920","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"reciprocal-fermat-constant-is-nonnormal","name":"Reciprocal Fermat Constant is Nonnormal","shortName":"Reciprocal Fermat Nonnormality","problemNumber":null,"field":"Theory of Normal Numbers and Digit Expansions","fieldGroup":"Number theory","statement":null,"posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The paper proves that the reciprocal Fermat constant is disjunctive but nonnormal in binary, explicitly locates every finite word, and establishes positive lower word frequencies. It extends these properties to a broad family of arithmetic series. Behavior in unrelated bases remains open.","claimIssueNote":null,"solveDate":"2026-09-08","model":"ChatGPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Astra developed several techniques in a previous work we did together. I asked it to find problems where the techniques could be of use, it chose the constant, proved it was disjunctive in base 2, proved it was nonnormal in base 2, then generalized the result.","verification":"lean-checked","verificationNote":"Full Lean verification is here: https://github.com/CaptainSude/reciprocal-Fermat-constant-Nonnormal/releases/tag/v1.0.0","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"Nonnormality of a specific named constant in base 2, with the digit-frequency machinery proved rather than assumed. A rung below the site's Erdos-Borwein disjunctivity entry: the same flavour of result, on a constant with a shorter history and less of a literature behind the question.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/CaptainSude/reciprocal-Fermat-constant-Nonnormal/releases/tag/v1.0.0","sourceName":"GitHub repository with Lean proof","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"nufrogcaca","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"an-improved-lower-bound-for-the-shannon-capacity-of-c-11","name":"An improved lower bound for the Shannon capacity of $C_{11}$","shortName":"Improved $C_{11}$ capacity bound","problemNumber":null,"field":"Zero-error information theory; graph capacity; combinatorics","fieldGroup":"Combinatorics","statement":"Determine the Shannon capacity of the eleven-cycle $C_{11}$, or improve its best explicit lower bound. The preceding BPZ construction, updated on 10 August 2026, gives an independent set of cardinality $N_0$ in dimension 207 and the lower bound $\\Theta(C_{11}) \\ge N_0^{1/207} = 5.29549231578462014255\\ldots$. The exact capacity remains open.","posedBy":"Claude Shannon (1956), underlying capacity problem; Buys, Polak and Zuiddam (2026), preceding C11 record","yearPosed":1956,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"A sequence of explicitly constructed independent sets in strong powers of $C_{11}$, with successive root improvements certified by exact integer comparisons, using cross-powers when dimensions differ. The current bound is $\\Theta(C_{11}) \\ge 5.295526013632343$, from an independent set in $C_{11}^{\\boxtimes 213}$ (R10, 9 September 2026), improving the R3 result of $5.295492477500681$ in dimension 207 that this entry was first listed for, and the Buys-Polak-Zuiddam baseline of $5.295492315784620$. The intermediate steps R5 and R6 and the superseded R9 are recorded on the frontier. The exact capacity remains open, and no upper bound is claimed. The R5, R6, R9 and R10 builds carry disclosed native-evaluation dependencies rather than being kernel-only.","claimIssueNote":null,"solveDate":"2026-09-08","model":"Astra 6 Pro; Codex GPT-6 Astra Extra-High","modelMaker":"OpenAI","humanCollaborators":["Matthew Protti"],"aiRole":"AI co-developed with OpenAI's Astra 6 Pro and Codex GPT-6 Astra Extra-High. Matthew Protti directed the research, evaluated proposals, required exact checks, set the scope and approved disclosure. The ChatGPT research collaboration supplied the R5 reassembly, R6 typed refinements and R9/R10 terminal certificates and written arguments. Codex replayed the finite certificates, wrote and compiled the Lean formalizations of actual independent sets, exact cardinalities and capacity bounds, inspected statement types and axiom dependencies, ran mathematical negative controls and prepared the releases. R9 and R10 import the checked R6 typed base and every child construction unchanged; only their terminal codes change. The underlying framework, base constructions and generic product machinery are due to Pjotr Buys, Sven Polak and Jeroen Zuiddam. These are author-side checks, not independent expert review.","verification":"lean-checked","verificationNote":"The current R10 result is checked in v0.5.0 with Lean 4.32.2, BPZ commit aa21eeb12b75b0413d3fa9fb4208b5d0bf2c4d65 and Mathlib commit 905b95818eb32af7874a58b427f50c1711a5e96c. ShannonBounds.C11R10D213 proves an actual finite independent set in the 213th strong power of Mathlib's cycleGraph 11, its exact cardinality N, the capacity lower bound 5.295526013632343 and strict root comparisons with R6 and R9. A development build and separate fresh replay passed. Across the whole v0.5.0 consolidation, each run audited 184 numerical theorems and 261 symbolic declarations, with 246 distinct native Boolean-check axioms in the numerical audit, and rejected nine mathematical negative controls. Those counts are consolidation-wide, not R10-only. Numerical theorems retain inherited and new native-evaluation dependencies. Symbolic CellRetyping uses only propext, Classical.choice and Quot.sound; that does not remove numerical native trust. Both routes trust the pinned compiler/kernel, runtime and dependency binaries. Independent statement review, expert endorsement and exhaustive novelty clearance remain pending. Earlier R3 evidence is retained as history.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"A further improvement to the best known lower bound for the Shannon capacity of one odd cycle, on a quantity the catalog already tracks. Real and certified, but narrower than the odd-cycles records entry it improves on, which covers the family.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/dfaef37e60e55c55b1744d9badd1f26c5364c7d5/shannon_checked_release/source/ShannonBounds/C11R10D213.lean","sourceName":"R10 C11 capacity bound: pinned Lean construction from v0.5.0","collection":null,"collectionVersion":null,"links":[{"label":"R3 history: pinned construction and capacity theorems (v0.1.0)","url":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/0f3baddf9223e7a62eb67df2d7248da07b6d047d/checked_release/derived_source/lean/ShannonBounds/C11AuxiliaryTrade.lean","kind":"lean-proof"},{"label":"R3 history: exact N0, N1 and frozen replacement table","url":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/0f3baddf9223e7a62eb67df2d7248da07b6d047d/checked_release/frozen_inputs/CANDIDATE.json","kind":"code"},{"label":"R3 history: checked build, native trust and negative controls","url":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/0f3baddf9223e7a62eb67df2d7248da07b6d047d/checked_release/BUILD_SUMMARY.md","kind":"other"},{"label":"v0.1.0 release with unchanged checked R3 archive","url":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/releases/tag/v0.1.0","kind":"announcement"},{"label":"Prior BPZ C11 certificate: full August 10 baseline","url":"https://github.com/spectra-research/shannon-capacity-lean/blob/aa21eeb12b75b0413d3fa9fb4208b5d0bf2c4d65/ShannonBounds/CertC11.lean","kind":"code"},{"label":"R6 v0.3.0: checked release archive and checksum","url":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/releases/tag/v0.3.0","kind":"code"},{"label":"R5 v0.2.0: checked release archive and checksum","url":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/releases/tag/v0.2.0","kind":"code"},{"label":"v0.5.0: current checked R10 release and archive","url":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/releases/tag/v0.5.0","kind":"code"},{"label":"v0.5.0: consolidated verification and trust disclosure","url":"https://github.com/matthewprotti/c11-shannon-capacity-lower-bound/blob/dfaef37e60e55c55b1744d9badd1f26c5364c7d5/shannon_checked_release/README.md","kind":"code"}],"relations":[],"submittedBy":"Matthew Protti","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"boussinesq-blowup-smooth-forcing","name":"Finite-time blowup for the inviscid Boussinesq system with smooth forcing","shortName":"Boussinesq blowup, smooth force","problemNumber":null,"field":"Fluid dynamics; singularity formation for incompressible flow","fieldGroup":"Differential equations","statement":"Does the inviscid Boussinesq system on $\\mathbb R^2$ admit finite-time blowup from smooth data with forces that are smooth in both space and time? Córdoba, Laín-Sanclemente and Martínez-Zoroa obtained finite-time singularity for the two-dimensional Boussinesq equation with a force only of class $C^{1,\\sqrt{4/3}-1-\\epsilon}\\cap L^2$, leaving the smooth-force case open.","posedBy":"Diego Córdoba, Antonio Laín-Sanclemente and Luis Martínez-Zoroa, whose multiscale construction reached a force of limited Hölder regularity","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Yes. Blowup for the inviscid Boussinesq system on $\\mathbb R^2$ with forces in $C^\\infty(\\mathbb R^2\\times[0,T_*])$ in both equations, supported in one fixed spatial ball, from smooth compactly supported initial temperature and zero initial velocity. The temperature stays bounded while $\\|\\nabla\\theta(t)\\|_\\infty\\to\\infty$ and the vorticity norm has infinite limsup as $t\\uparrow T_*$. The solution is smooth on every closed interval before blowup and unique in a finite-energy Lipschitz class. This lifts the force from barely $C^1$ to fully smooth, and is the construction the Euler paper then builds on. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation \"is not on the Clay Institute's list of prize problems\".","claimIssueNote":null,"solveDate":"2026-09-08","model":"Claude, Codex with GPT-5.6 Sol","modelMaker":"Anthropic / OpenAI","humanCollaborators":["Levent Alpöge","Tristan Buckmaster"],"aiRole":"The Boussinesq paper devotes its section 2 to an \"AI statement\": \"We happily used both Claude and Codex to iterate on our proof. We had, using Claude, our first blowup solution, but not this one, on 8/15/26, and we Lean verified it on 8/22/26. This involved inputting ideas from previous joint work of ours on blowup for the IPM equation following Córdoba-Martínez-Zoroa and a number of other works of ours and others, as well as iteration with the model on various ansätze, eventually leading to blowup in Boussinesq. The first writeup that we produced iterating with Claude was, in our opinion, the worst writeup we had ever seen in the history of mathematics (topped soon after by the writeups for 3d Euler and then for hypodissipative Navier-Stokes).\" The statement continues that after iterating with Claude and Codex on alternative proof architectures, \"leading to several different proofs, we arrived at the current simplified argument, which we then iterated on using Claude and Codex, first 5.6 Sol and then Astra once we had access, to arrive at the current writeup modulo our hand editing\", and that intermediate writeups were generally fed to Codex \"for simplification, ideation, and iteration\". Co-developed rather than discovered: the first blowup solution came from Claude, but the multiscale mechanism is Córdoba and Martínez-Zoroa's and the authors fed in their own earlier IPM work.","verification":"lean-verified","verificationNote":"Formalised in Lean 4 in tristanbuckmaster/fluid_lean, twice over: boussinesq-blowup carries the general theorem and affinecore the normalised construction with explicit odd initial data. Checked here on 8 September 2026 by reading both READMEs: in each, Challenge.lean is the only file a reader must trust, no other file contains a sorry, the proof rests on no axiom beyond propext, Classical.choice and Quot.sound, and leanprover/comparator is configured to type-check the statement independently and confirm the solution inhabits exactly it. Not peer reviewed. Tao's public commentary describes the Boussinesq case as the model case whose amplitude-frequency dynamics reduce to \"a remarkably simple ODE\", but records that he is still working through the arguments.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A substantial strengthening in its own right, taking the force from a narrow Hölder class to smooth in space and time, and the stepping stone that made the Euler construction possible: the Euler paper develops this paper's two-field wave calculation on a varying background. Below Euler because Boussinesq in two dimensions is a model system rather than the equations of record.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://cims.nyu.edu/~tristanb/boussinesq.pdf","sourceName":"Blowup for the Boussinesq equations with smooth forcing (manuscript)","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation, comparator-checked against Challenge.lean","url":"https://github.com/tristanbuckmaster/fluid_lean/tree/main/boussinesq-blowup","kind":"lean-proof"},{"label":"Challenge.lean: the trusted statement, in plain Mathlib","url":"https://github.com/tristanbuckmaster/fluid_lean/blob/main/boussinesq-blowup/Challenge.lean","kind":"lean-statement"},{"label":"Buckmaster's public statement on the work and its release","url":"https://cims.nyu.edu/~tristanb/statement.pdf","kind":"announcement"},{"label":"affinecore: the second Lean formalisation, normalised data","url":"https://github.com/tristanbuckmaster/fluid_lean/tree/main/affinecore","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"euler-blowup-smooth-forcing","name":"Finite-time blowup for the 3D incompressible Euler equations with smooth forcing","shortName":"Euler blowup, smooth force","problemNumber":null,"field":"Fluid dynamics; singularity formation for incompressible flow","fieldGroup":"Differential equations","statement":"Can a solution of the three-dimensional incompressible Euler equations on $\\mathbb R^3$, started from smooth data and driven by a force that is smooth in space and time up to and including the blowup time, lose regularity in finite time? Finite-time singularity formation from genuinely smooth data is the central open question for the equations. Elgindi obtained blowup for $C^{1,\\alpha}$ solutions in 2021, and Córdoba and Martínez-Zoroa built a multiscale program producing forced blowup for related equations with forces of limited regularity, but no construction reached three-dimensional Euler with a space-time smooth force. Note on the forced formulation, since it is easy to misread: alternatives (C) and (D) of Fefferman's Clay problem description do permit a smooth force obeying rapid space-time decay, so forcing is not a dodge and the forced route is a genuine path to the prize. It is a path for Navier-Stokes with viscosity, however, and not for Euler, which Fefferman's description excludes from the prize list.","posedBy":"Leon Lichtenstein (1925) and Nikolai Gunther (1927), whose local existence left global regularity open; Elgindi states the smooth-force form as his Question 1.1","yearPosed":1925,"ageNote":"Released 8 September 2026, earlier than the authors intended. Their own chronology: a first blowup solution on 15 August 2026, Lean-verified on 22 August, then weeks spent turning a model-generated argument into prose. Buckmaster calls the Euler writeup \"AI slop\", apologises for its state, and attributes the early release to outside pressure.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Yes. Theorem 1.1: for every $r_0>0$ and $z_0$ there are a time $T_*>0$, a divergence-free axisymmetric $u_0\\in C_c^\\infty$ supported in a fixed solid torus with nonzero swirl and zero meridional velocity, and an axisymmetric force $f\\in C^\\infty(\\mathbb R^3\\times[0,T_*])$ supported in that torus, with a solution smooth on $[0,T_*)$ for which circulation and meridional velocity stay bounded while $\\|\\nabla\\Gamma(t)\\|_\\infty$ and $\\|\\omega(t)\\|_\\infty$ tend to infinity and $\\int_0^{T_*}\\|\\omega(t)\\|_\\infty\\,dt=\\infty$, so the blowup is genuine by Beale-Kato-Majda. It is unique among divergence-free locally space-time Lipschitz solutions with the same data, and competitors need not be axisymmetric. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation \"is not on the Clay Institute's list of prize problems\".","claimIssueNote":null,"solveDate":"2026-09-08","model":"Claude, Codex with GPT-5.6 Sol","modelMaker":"Anthropic / OpenAI","humanCollaborators":["Levent Alpöge","Tristan Buckmaster"],"aiRole":"The Boussinesq paper devotes its section 2 to an \"AI statement\": \"We happily used both Claude and Codex to iterate on our proof. We had, using Claude, our first blowup solution, but not this one, on 8/15/26, and we Lean verified it on 8/22/26. This involved inputting ideas from previous joint work of ours on blowup for the IPM equation following Córdoba-Martínez-Zoroa and a number of other works of ours and others, as well as iteration with the model on various ansätze, eventually leading to blowup in Boussinesq. The first writeup that we produced iterating with Claude was, in our opinion, the worst writeup we had ever seen in the history of mathematics (topped soon after by the writeups for 3d Euler and then for hypodissipative Navier-Stokes).\" Buckmaster's public statement adds that the models used were \"Anthropic's Claude, OpenAI's Codex, especially with GPT-5.6 Sol and, more recently, Astra\", the last \"only used for writeups and auditing our arguments\", and that \"the first LLM generated proof Levent sent me was the most horrendous I have ever read\". He is explicit that the program is not the models': \"The credit for the basic idea of this program goes to Diego Córdoba and Luis Martínez-Zoroa... We took their work as a starting point, using Large Language Models to push their program to completion.\" Co-developed rather than discovered on that account. The Euler paper carries no AI statement of its own.","verification":"lean-verified","verificationNote":"Formalised in Lean 4 in tristanbuckmaster/fluid_lean (euler-blowup). Checked here on 8 September 2026 by reading the repository: Challenge.lean states the theorem against plain Mathlib and is the only file a reader must trust, Solution.lean derives it from the development, and comparator.json configures leanprover/comparator to type-check the statement independently, confirm the solution inhabits exactly that statement, restrict axioms to propext, Classical.choice and Quot.sound, and replay the proof. Mathlib is pinned by commit; the build is roughly 1,100 modules and the README warns it needs on the order of 100 GB of memory. The Lean statement was read and matches the paper's Theorem 1.1, including the divergence of the Beale-Kato-Majda vorticity integral and uniqueness against non-axisymmetric competitors. Not yet peer reviewed: Terence Tao has publicly analysed the result and calls it \"a remarkable achievement\", but says he is still digesting the proof, so no expert has certified the argument line by line.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":70,"significanceNote":"The strongest result in this catalog. Finite-time singularity formation for three-dimensional incompressible Euler is one of the central problems of mathematical fluid dynamics, and this settles it in the forced smooth category, one category short of the Clay problem. Terence Tao's public assessment is that nothing in principle blocks extending the method to Navier-Stokes and that there is \"a non-negligible chance that the forcing term could be eliminated entirely\". Below the 80s because the force is essential to the result and its removal is exactly the hard part.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://cims.nyu.edu/~tristanb/euler.pdf","sourceName":"Blowup for the Euler equations with smooth forcing (manuscript)","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation, comparator-checked against Challenge.lean","url":"https://github.com/tristanbuckmaster/fluid_lean/tree/main/euler-blowup","kind":"lean-proof"},{"label":"Challenge.lean: the trusted statement, in plain Mathlib","url":"https://github.com/tristanbuckmaster/fluid_lean/blob/main/euler-blowup/Challenge.lean","kind":"lean-statement"},{"label":"Buckmaster's public statement on the work and its release","url":"https://cims.nyu.edu/~tristanb/statement.pdf","kind":"announcement"},{"label":"Terence Tao's assessment of the result and the mechanism","url":"https://mathstodon.xyz/@tao/117233527638291447","kind":"discussion"},{"label":"Buckmaster's announcement, with all three manuscripts","url":"https://mastodon.social/@tristanbuckmaster/117233413705701198","kind":"announcement"},{"label":"Fefferman's official Clay problem description","url":"https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf","kind":"problem-record"},{"label":"Elgindi, Annals 2021: Question 1.1, the problem in the form asked here","url":"https://arxiv.org/abs/1904.04795","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":12,"downvotes":0,"commentCount":0},{"slug":"the-ceiling-of-the-single-letter-entropy-method-for-the-union-closed-sets-conjec","name":"The ceiling of the single-letter entropy method for the union-closed sets conjecture","shortName":"Entropy-method ceiling for Frankl","problemNumber":null,"field":"Extremal set theory / entropy method","fieldGroup":"Combinatorics","statement":"Every lower bound on the union-closed constant $c_0$ since Gilmer (2022) — including $(3-\\sqrt5)/2$, Sawin–Yu–Cambie's $0.3823455$ and Liu's $0.382709$ — is certified by the same single-letter inequality: a memoryless protocol coupling two conditional bits, together with a class of admissible joint laws of the two prefix-conditional probabilities. It was not known how far this framework can go: whether it could be pushed to $1/2$ and settle Frankl's conjecture, or whether it has a hard ceiling strictly below it.","posedBy":"Raised in the entropy-method literature after Gilmer (2022): Cambie (2022) studies the approach's boundaries, Liu (2023) asks whether other couplings can improve it","yearPosed":2022,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Two unconditional ceilings. Every single-letter certificate whose classes contain product laws certifies at most $1-h(1/\\sqrt2)/\\sqrt2 = 0.383099\\ldots$. Every certificate that uses the i.i.d. protocol and whose other classes admit component hiding — a property of every class in the literature — certifies at most $c^{**} = 0.382885260\\ldots$. So the entropy method, in every form used since 2022, cannot reach $1/2$: Frankl's conjecture is out of its range, and only $1.8\\cdot10^{-4}$ of headroom remained above the previous record. The paper also constructs a protocol attaining $0.38284$ (computer-assisted, conditional on two numerically verified hypotheses). Frankl's conjecture itself remains wide open.","claimIssueNote":null,"solveDate":"2026-09-08","model":"Fable 5.1","modelMaker":null,"humanCollaborators":["Andrew Moffat"],"aiRole":"A team of Claude agents working under the author's direction. Claude Fable 5.1 selected the problem, found both ceiling theorems — including the component-hiding adversary, which is the new idea — and designed the optimal-diagonal protocol and its certification. Claude Opus 5 agents refereed the manuscript over four rounds (catching an error in the lead agent's own lemma), proved the small-entropy lemma, and independently re-implemented the numerical certificate from the written statement alone. Claude Sonnet agents transcribed the cited literature. The author set the goal and the standard, checked the claims, and decided what to publish. Referee reports and logs are in the repository.","verification":"lean-checked","verificationNote":"Filed at Lean-checked rather than Lean-verified, and the distinction is the one that tier exists for. The two unconditional ceiling theorems are machine-checked: the lean/ project builds with no sorry and only propext, Classical.choice and Quot.sound, check.sh enforces both, and the lean and verify workflows were green at HEAD on 10 September. What Lean proves, though, is a theorem about Certifies, the author's own finite-model definition of a single-letter certificate. Whether every certificate in the literature is an instance is Lemma 3.3, whose maximal-correlation case is paper-only, and Gilmer's reduction from union-closed families to the certificate is a hypothesis in the development rather than a theorem. So the artifact compiles and proves what the paper's framework says; the framework's fidelity to the informal claim that the entropy method cannot reach $1/2$ was not audited here. The computer-assisted constant $0.38284$ is conditional on two numerically verified hypotheses of the same kind as those behind Liu's record, and is not in the formalisation. Not peer reviewed and not checked by a named expert; four rounds of referee reports were produced by the author's own model agents and are in the repository.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"Not a numbered question, but one the people pushing this method had asked in print: Cambie's 2022 paper studies the approach's boundaries and argues Sawin's question cannot move the constant far, and Liu 2023 asks whether other couplings can. Nothing found gives a ceiling over the whole single-letter framework, which is what this settles. Below the Oddtown anchor at 20, a named question, and above the Boppana entropy generalisation at 8; the readership is the small group that has worked this method since 2022, and Frankl's conjecture itself is untouched.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/moffatstudio/union-closed-constant/releases/tag/v1.2","sourceName":"GitHub release v1.2, the stated version of record: paper, code, logs, referee reports and Lean","collection":null,"collectionVersion":null,"links":[{"label":"The paper, 31 pages with the Lean appendices, at v1.2","url":"https://github.com/moffatstudio/union-closed-constant/blob/v1.2/paper/paper.pdf","kind":"paper"},{"label":"lean/: Theorems 3.1 and 3.4 machine-checked, with check.sh","url":"https://github.com/moffatstudio/union-closed-constant/tree/v1.2/lean","kind":"lean-proof"},{"label":"Cambie 2022, which studies the entropy approach's boundaries","url":"https://arxiv.org/abs/2212.12500","kind":"problem-record"}],"relations":[],"submittedBy":"CobaltPanther851","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"lower-bound-for-the-complex-grothendieck-constant","name":"Lower bound for the complex Grothendieck constant","shortName":"Complex Grothendieck lower bound","problemNumber":null,"field":"Functional analysis","fieldGroup":"Analysis","statement":"What is the exact value of the complex Grothendieck constant $K_G^{\\mathbb C}$, the least $K$ such that $\\bigl|\\sum_{i,j}a_{ij}\\langle x_i,y_j\\rangle\\bigr|\\le K\\max_{|\\varepsilon_i|=|\\delta_j|=1}\\bigl|\\sum_{i,j}a_{ij}\\varepsilon_i\\delta_j\\bigr|$ for every complex matrix $(a_{ij})$ and all unit vectors $x_i,y_j$ in any complex Hilbert space? Grothendieck proved it finite in 1953. Before this work the best bounds were Davie's lower bound of about $1.33807$ and Haagerup's 1987 upper bound of about $1.40491$; the value is open, and this entry records progress on the lower bound.","posedBy":"Alexandre Grothendieck, Résumé (1953); the value of the complex constant an explicit open question at least since Haagerup's 1987 upper bound","yearPosed":1953,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"The paper proves\n$$\nK_G^{\\mathbb{C}}>1.35584631827168.\n$$\n\nThe previously recorded Davie lower bound is approximately $1.33807$, while Haagerup's upper bound is approximately $1.40491$. Thus the result closes more than one quarter of the remaining lower-to-upper-bound gap, but it does not determine the exact value of the complex Grothendieck constant.\n\nThe construction uses finitely many additional complex Hermite projections together with a common radial weight to obtain a dimension-independent $L_{\\infty}$-to-$L_1$ estimate. The numerical inequalities needed for the final bound are certified by interval arithmetic.\n\nThe paper also analyzes the limit of its particular weighted criterion. If $\\mathcal{K}_*$ denotes the supremum obtainable within that framework, it proves\n$$\n1.35584631827168<\\mathcal{K}_*<1.35584697425050.\n$$\nThe upper endpoint here is a limitation of this specific criterion, not an upper bound on $K_G^{\\mathbb{C}}$ itself.","claimIssueNote":null,"solveDate":"2026-09-07","model":"Odin Automatic AI Research Agent","modelMaker":null,"humanCollaborators":["Shengtao Guo","Ethan X. Fang","Junwei Lu"],"aiRole":"The paper's disclosure is one sentence, in the abstract and again under the heading \"The role of AI in this proof\": \"Odin Automatic AI Research Agent was used to derive the lower bound and the proof.\" Taken at face value, as this site's classification rule requires, that is an AI-discovered claim, and it is the same disclosure this team gave for its Talagrand convolution entry. The paper says nothing about what Odin is, which models it runs on, or how it was steered, and no public description of the system was found, so the model maker is left empty rather than guessed at. The three named humans are Shengtao Guo, Ethan X. Fang and Junwei Lu.","verification":"unreviewed","verificationNote":"Unreviewed arXiv preprint, v1 of 7 September 2026, with no independent endorsement and no peer review. The numerical part is certified by Arb ball arithmetic with outward rounding; the verification code and the exact rational inputs are supplied as arXiv ancillary files and in the linked repository, and were not run here. The analytic part - the dimension-independent $L_\\infty$-to-$L_1$ estimate for the weighted Gaussian Hermite multipliers that turns the certified numbers into a bound on $K_G^{\\mathbb C}$ - was not checked. Checked here: the arXiv record and abstract as submitted, that Davie's and Haagerup's bounds are as the paper states them, and that the catalog holds no entry on either Grothendieck constant, so the 2026 real-constant improvements the submitter mentions are not duplicates. Partial: a better lower bound, not the value.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The exact value of Grothendieck's constant is a classical open problem of functional analysis with reach into optimization and computer science; the complex case has carried its own literature since Haagerup's 1987 bound and gets less attention than the real one. Level with the Albertson-Berman and Hadamard-668 entries at 30, above the kissing-number-in-19-dimensions record at 25, below the binary-code upper-bounds entry at 39, which sits on a question the whole of coding theory asks.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.07000","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Interval-arithmetic certificates and verifiers (repository named in the paper)","url":"https://github.com/shengtaoguo/complex-grothendieck-certificates","kind":"code"},{"label":"arXiv ancillary files: the same verifiers and exact inputs, frozen with v1","url":"https://arxiv.org/src/2609.07000v1/anc","kind":"other"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"reciprocal-triple-free-sets-the-finite-plateau-at-732","name":"Reciprocal-triple-free sets: finite maxima through 734","shortName":"A390395: finite values through 734","problemNumber":null,"field":"Extremal sets avoiding distinct reciprocal triples","fieldGroup":"Number theory","statement":"Let f(n) be the maximum size of a subset of {1,...,n} containing no three distinct a,b,c satisfying 1/a = 1/b + 1/c. A published finite frontier for Erdős problem 302 asks whether f(732) is 606 or 607, using the existing OEIS value f(731)=606.","posedBy":"Finite frontier: Erdős Frontier Atlas P302, recorded July 2026. Parent problem: Erdős and Graham (1980); the parent asymptotic problem is not solved.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"The Lean development proves f(732)=f(731), f(733)=f(732)+1 and f(734)=f(732)+2. The 732 step replaces an isolated five-vertex component without reducing an admissible set. Exact finite certificates show that neither 733 nor 734 belongs to a forbidden reciprocal triple with vertices at most 734, so each new vertex raises the maximum by one. Conditional on the published OEIS value f(731)=606, the resulting values are f(732)=606, f(733)=607 and f(734)=608. The historical asymptotic Erdős problem remains open. Priority is provisional; the elementary component argument may be folklore.","claimIssueNote":null,"solveDate":"2026-09-07","model":"OpenAI ChatGPT/Codex (exact model identifier unavailable)","modelMaker":"OpenAI","humanCollaborators":["Sodelin (project owner and research direction)"],"aiRole":"The assistant selected the 732 finite target, found the isolated-component replacement proof, computed exact witnesses and wrote the Lean formalization. Further assistant work proved the 733 and 734 transfers, added exact 607- and 608-element witnesses, and replaced native-checker finite steps with kernel-checked certificates. Separate assistant instances reviewed arithmetic and formal statements. Sodelin initiated and directed the research. No independent human expert review has occurred.","verification":"lean-checked","verificationNote":"Lean 4.33.1 compiles the structural theorems through 734, including domain cardinality and the rational-equation bridge. At the current pinned source commit, the final theorems report only propext, Classical.choice and Quot.sound; the earlier bv_decide/native LRAT checker axiom has been removed. Closed finite certificates for the gadget and endpoints are kernel checked without axioms. Independent exact Python verifiers check 606-, 607- and 608-element witnesses and enumerate all 738 forbidden triples through 734. The numerical baseline f(731)=606 is external OEIS data, not a theorem proved here. The formal statement and priority remain unaudited by an independent human expert. This is a partial finite result in a public repository, not peer-reviewed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":4,"significanceNote":"One further term of a tracked finite table under Erdos problem 302. The theorem is real and the argument is structural rather than a search, but the asymptotic problem is untouched.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Sodelin/oldest-conjecture-/tree/ced445d951ea043fda41b03044192838a838b0f3","sourceName":"GitHub proof and exact verifiers through 734","collection":null,"collectionVersion":null,"links":[{"label":"Passing Lean build, axiom audit, and exact verifiers","url":"https://github.com/Sodelin/oldest-conjecture-/actions/runs/34129895492","kind":"lean-proof"},{"label":"Original 732 proof commit","url":"https://github.com/Sodelin/oldest-conjecture-/tree/0d2885f58c5fe6eee8703cdee83a843a1594a7ea","kind":"lean-proof"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-erdos-borwein-constant-is-2-dense","name":"The Erdos-Borwein Constant is 2-Dense","shortName":"The Erdos-Borwein Constant is 2-Dense","problemNumber":null,"field":"Analytic number theory; digit distribution of constants","fieldGroup":"Number theory","statement":"Is the Erdos-Borwein Constant 2-Dense?","posedBy":"Richard E. Crandall","yearPosed":2002,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"This work proves that the Erdős–Borwein constant's binary expansion contains every finite binary sequence as a consecutive block, each occurring infinitely often. This implies that the Erdős–Borwein constant is 2-dense.","claimIssueNote":null,"solveDate":"2026-09-07","model":"ChatGPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"lean-checked","verificationNote":"The Lean development proves disjunctivity of the binary expansion conditional on two hypotheses supplied as theorem arguments, not as axioms: AGP, the Alford-Granville-Pomerance estimate in the form of Vandehey's Proposition 2.1, and PrimeIntervalSupply, a standard prime number theorem consequence bounding the primes in (L, 2L) below by L/(3 log L). Both are published theorems rather than conjectures, so the mathematics is conditional only on known results; neither is formalized here. Read directly from lean/ErdosBorwein/PrimeInputs.lean on 8 September 2026. The audited endpoints depend on propext, Classical.choice and Quot.sound only. Because the statement is the author's own rather than anchored to a canonical tracker, and because what the kernel certifies is the implication, this takes the statement-unaudited tier. No specialist in analytic number theory has read the argument.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A question of Crandall's from 2002 about a named constant. Full disjunctivity of the binary expansion is substantially stronger than the single-block result the catalog already records, and digit-distribution results for specific constants are historically hard to come by.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/CaptainSude/erdos-borwein-disjunctivity/releases/tag/v1.0.0","sourceName":"GitHub repository with Lean proof","collection":null,"collectionVersion":null,"links":[{"label":"PrimeInputs.lean: the two external estimates, stated exactly","url":"https://github.com/CaptainSude/erdos-borwein-disjunctivity/blob/main/lean/ErdosBorwein/PrimeInputs.lean","kind":"lean-statement"},{"label":"Lean verification record and axiom audit","url":"https://github.com/CaptainSude/erdos-borwein-disjunctivity/blob/main/lean/VERIFICATION.md","kind":"lean-proof"}],"relations":[],"submittedBy":"nufrogcaca","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"kalai-conjecture-for-tight-trees","name":"Kalai's conjecture for tight trees","shortName":"Kalai's tight-tree conjecture","problemNumber":null,"field":"Extremal hypergraph theory","fieldGroup":"Combinatorics","statement":"Kalai proposed in 1984, in a form recorded by Frankl and Füredi, a hypergraph generalisation of the Erdős-Sós conjecture: for a suitable notion of a tight tree $T$ with $t$ edges in an $r$-uniform hypergraph, every $r$-graph on $n$ vertices with more than $\\tfrac{t-1}{r}\\binom{n}{r-1}$ edges should contain $T$. The case $r=2$ is the Erdős-Sós conjecture, proved in 2026. Does the conjecture hold for all $r$?","posedBy":"Gil Kalai (1984), recorded by Péter Frankl and Zoltán Füredi; the $r=2$ case is the Erdős-Sós conjecture","yearPosed":1984,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-09-07","model":"GPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":["Dhruv Mubayi","Jacques Verstraëte"],"aiRole":"From the paper: the proof was found by GPT-6 Astra, extending its own method of proof for the Erdős-Sós conjecture to the hypergraph setting. The authors supplied the framing and the write-up.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.08012: the introduction states that Kalai proposed the hypergraph generalisation in 1984, recorded by Frankl and Füredi, that the $r=2$ case is the Erdős-Sós conjecture, and that the conjecture is proved here via a shadow bound whose equivalence to Kalai's conjecture was established by Füredi, Jiang, Kostochka and the authors. The mathematics was not checked here; fifteen days old, no referee. Same authors and same model as the digraph entry three days later.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A forty-two-year-old named conjecture of Kalai's generalising Erdős-Sós to hypergraphs, tracked in the extremal-hypergraph literature since Frankl and Füredi recorded it. Below the Erdős-Sós conjecture itself at 58, which is older and better known, and just below the uniform Turán density entry at 48; well above the directed analogue at 20, which nobody had posed.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.08012","sourceName":"Kalai's Conjecture for Tight Trees, arXiv:2609.08012 (7 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"The Erdős-Sós conjecture, the $r=2$ case, also proved by GPT-6 Astra","url":"https://vibemathed.com/problem/erdos-problem-548-erdos-sos-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"koizumi-liu-eventual-sign-alternation-conjecture","name":"Koizumi–Liu Eventual Sign Alternation Conjecture","shortName":"Koizumi–Liu Eventual Sign Alternation","problemNumber":null,"field":"Matroid theory","fieldGroup":"Combinatorics","statement":"Koizumi and Liu conjectured that for every real hyperplane arrangement $\\mathcal A$, the coefficients of\n$$\n\\operatorname{Mag}(\\mathcal A;-t)\n$$\nare eventually nonnegative, equivalently that the coefficients of $\\operatorname{Mag}(\\mathcal A;q)$ eventually alternate in sign.\n\nThe conjecture is false. There exists a rank-$6$ real hyperplane arrangement $\\mathcal A$ such that\n$$\n(-1)^\\ell[q^\\ell]\\operatorname{Mag}(\\mathcal A;q)<0\n$$\nfor infinitely many $\\ell$.","posedBy":"Junnosuke Koizumi and Ye Liu","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Koizumi constructs a representable simple rank-$6$ matroid $M$ for which\n$$\n\\operatorname{Mag}(M;q)\n$$\nhas a pole of order $4$ at $q=-1$ but a pole of order $5$ at $q=i$.\n\nAfter writing\n$$\nA_M(t)=\\operatorname{Mag}(M;-t),\n$$\nthe higher-order pole at $t=i$ forces the Taylor coefficients of $A_M(t)$ to fail eventual nonnegativity. Consequently,\n$$\n(-1)^\\ell[q^\\ell]\\operatorname{Mag}(M;q)<0\n$$\nfor infinitely many $\\ell$.\n\nThe matroid is representable by twelve vectors in $\\mathbb R^6$, so it yields an actual real central hyperplane arrangement and therefore directly disproves Koizumi–Liu's conjecture for real arrangements.","claimIssueNote":null,"solveDate":"2026-09-07","model":"GPT-5.6 Sol; GPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":["Junnosuke Koizumi"],"aiRole":"The author reports extensive mathematical interaction with GPT-5.6 Sol and GPT-6 Astra. AI assistance included developing mathematical arguments, performing computations, and drafting and revising the manuscript. The paper does not separately attribute the eventual-sign-alternation counterexample or any particular main theorem to one model. Junnosuke Koizumi states that he critically evaluated the generated material, independently verified the mathematical arguments and references, and takes responsibility for the paper.","verification":"unreviewed","verificationNote":"The paper contains a complete conventional proof, and the author states that he independently checked the AI-assisted mathematical arguments and references. No independent external expert review, referee report, or proof-assistant formalization is reported. The arXiv submission is a fresh preprint marked “Comments welcome!”","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"Refutes a named conjecture of Koizumi and Liu with an explicit rank-six counterexample, inside a broader paper on matroid magnitude and motivic zeta functions. Specialised, recent, and settled cleanly in the negative by one of the people who posed it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.07977","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"a-44-vertex-triangulation-of-mathbb-rp-6","name":"A 44-vertex triangulation of $\\mathbb{RP}^{6}$","shortName":"RP^6 on 44 vertices","problemNumber":null,"field":"Combinatorial topology; triangulations of manifolds; convex polytopes","fieldGroup":"Geometry & topology","statement":"Does real projective 6-space admit a simplicial triangulation with fewer\nthan 45 vertices? This is Question 3.2 of Guyer, Steinerberger and Yang,\n“An Efficient Triangulation of RP^5”, arXiv:2603.07808 (2026).\n\nThe question asks for an improvement on their 45-vertex construction,\nnot for the exact minimum number of vertices.","posedBy":"Dan Guyer, Stefan Steinerberger, Yirong Yang — Question 3.2, arXiv:2603.07808","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"The manuscript gives an explicit centrally symmetric simplicial\n7-polytope with 88 integer vertices. Its antipodal boundary quotient\nis a 44-vertex triangulation of real projective 6-space, with f-vector\n(44, 938, 7024, 22555, 34936, 25914, 7404).\n\nThis improves the 45-vertex construction of Guyer, Steinerberger and\nYang and gives an affirmative answer to their Question 3.2. Integer\ncoordinates, facet lists and exact-arithmetic verification code\naccompany the manuscript. No claim that 44 is vertex-minimal is made.","claimIssueNote":null,"solveDate":"2026-09-06","model":"ChatGPT (OpenAI, model version unstated)","modelMaker":"OpenAI","humanCollaborators":["Alexey M. Kolosov"],"aiRole":"The manuscript's Computational assistance section: \"OpenAI's ChatGPT was used in the computational exploration, in the development of the verification programs, and in preparing this manuscript. The two implementations were developed within the same AI-assisted workflow; their agreement is not external independent validation.\" Search assistance, code and prose: the assisted tier.","verification":"site-confirmed","verificationNote":"Site-confirmed: both of the author's exact-arithmetic verifiers (anc/verify.py with integer Bareiss elimination and anc/verify_rational.py with rational Gaussian elimination, Python standard library) were replayed here on 6 September 2026 from a clone of cheptil/44-vertex-triangulation at 2c62b31. Both pass: every facet's supporting hyperplane, completeness of the facet list, the antipodal disjoint-star condition, f-vector (44, 938, 7024, 22555, 34936, 25914, 7404), quotient Euler characteristic 1, in about 3.5 seconds; the negative controls (perturbed coordinate, missing facet) fail as they should. Question 3.2 confirmed verbatim in arXiv 2603.07808. The manuscript's \"Computational assistance\" section discloses ChatGPT's role and says the two verifiers share one workflow, which is why an independent replay was worth doing.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"Question 3.2 of Guyer, Steinerberger and Yang (2026): can real projective 6-space be triangulated on fewer than 45 vertices? Yes, 44, by an explicit centrally symmetric 7-polytope. It does not determine the minimum. Level with the 24-vertex RP^5 entry it descends from: a concrete answer to a fresh question in combinatorial topology.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/cheptil/44-vertex-triangulation","sourceName":"Alexey M. Kolosov — A 44-vertex triangulation of real projective 6-space","collection":null,"collectionVersion":null,"links":[{"label":"44-vertex-triangulation","url":"https://vibemathed.com/problem/rp5-24-vertex-triangulation","kind":"other"}],"relations":[],"submittedBy":"cheptil","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"optimality-of-the-added-vector-code-in-the-19-dimensional-kissing-construction","name":"Optimality of the added-vector code in the 19-dimensional kissing construction","shortName":"Ho's 1280-word code is optimal","problemNumber":null,"field":"Coding theory; algebraic graph theory; discrete geometry","fieldGroup":"Combinatorics","statement":"Let \\(D\\subseteq\\mathbb F_2^{19}\\) be the fixed 4096-word ambient binary linear code used in Ho’s 19-dimensional improvement of the Cohn–Li kissing construction. Is every subset \\(A\\subseteq D\\) with minimum Hamming distance at least \\(5\\) of size at most \\(1280\\)? No linearity assumption is imposed on \\(A\\).","posedBy":"Gonzalez, Conjecture 16 (preprint, version 3, 2026), after Ho's 1280-word construction","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The manuscript proves the upper bound \\(|A|\\le1280\\) for every admissible subset of the fixed ambient code \\(D\\). Together with Ho’s existing construction, this determines the exact maximum. The proof partitions \\(D\\) into 256 cosets of a 16-word subspace whose induced forbidden-distance graph is the Clebsch graph; each coset contributes at most five words. This does not determine the unrestricted kissing number \\(k(19)\\) or improve the known 11948-point configuration. Independent expert review is pending.","claimIssueNote":null,"solveDate":"2026-09-06","model":"ChatGPT (OpenAI, model version unstated)","modelMaker":"OpenAI","humanCollaborators":["Alexey M. Kolosov"],"aiRole":"The manuscript's Section 5: \"ChatGPT (OpenAI) was used for literature search, the optimization-based search for the five-word certificate, development and checking of the proof, preparation of the verification program, and drafting and translation of the manuscript. The mathematical argument and the exact verification data are given explicitly; a language-model output or a numerical solver status is not used as a substitute for proof.\" Certificate found and proof developed with the model under the author's direction: co-developed.","verification":"site-confirmed","verificationNote":"Site-confirmed: the author's exact verifier (anc/verify_golay_1280_optimality.py, Python standard library, exact arithmetic) was replayed here on 6 September 2026 from a clone of cheptil/kissing-number-19-dimensions at 0a90b8e and passed every check: the five generator identities and rank certificate, the weight distribution of the 16-word subspace, the exact Fourier dual certificate, the independence number 5 of the 16-vertex certificate graph by exact subset DP, the 256 cosets giving 1280, and the explicit matching constructions with minimum distances 6 and 5. The manuscript's Section 5 discloses ChatGPT's role. No independent expert statement; the tier records a replayed certificate, not a referee.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A sub-question of the 19-dimensional kissing problem: within Ho's fixed 4096-word ambient code, no distance-5 subcode beats his 1280 words. Gonzalez had bracketed the maximum between 1280 and 1536 and conjectured 1280. Closing that bracket settles one scheme, not the kissing number, and the paper says so; a small, exact result.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/cheptil/kissing-number-19-dimensions","sourceName":"kolosov_golay_optimality_en.pdf","collection":null,"collectionVersion":null,"links":[{"label":"kissing-number-19-dimensions","url":"https://vibemathed.com/problem/kissing-number-19-dimensions","kind":"other"}],"relations":[],"submittedBy":"cheptil","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"counterexample-to-the-gamma-theta-conjecture-in-eternal-domination","name":"The γ–θ conjecture in eternal domination","shortName":"$\\gamma$–$\\theta$","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"The $\\gamma$–$\\theta$ conjecture asserts that for every finite graph $G$,\n$$\n\\gamma(G)=\\gamma^\\infty(G)\n\\quad\\Longrightarrow\\quad\n\\gamma(G)=\\theta(G),\n$$\nwhere $\\gamma$ is the domination number, $\\gamma^\\infty$ is the eternal domination number in the one-guard-moves model, and $\\theta$ is the vertex clique-cover number.\n\nThe conjecture is false. The complement $G$ of the 243-vertex ternary Golay graph satisfies\n$$\n\\gamma(G)=\\gamma^\\infty(G)=3<\\theta(G).\n$$","posedBy":"William F. Klostermeyer and C. M. Mynhardt","yearPosed":2014,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Let $H$ be the Berlekamp–van Lint–Seidel graph on $243$ vertices, the Cayley graph of $\\mathbb Z_3^5$ with strongly regular parameters\n$$\n(243,22,1,2),\n$$\nand let $G=\\overline H$.\n\nThe proof establishes\n$$\n\\gamma(G)=3\n$$\nbecause every pair has a common neighbor in $H$, while an $H$-triangle gives a dominating triple in $G$.\n\nIt then proves\n$$\n\\gamma^\\infty(G)=3\n$$\nby showing that the family of all dominating triples is closed under a legal response to every attack: after moving one guard to the attacked vertex, another dominating triple can always be obtained.\n\nFinally, a double-counting argument shows that $H$ is not 3-colorable, hence\n$$\n\\theta(G)=\\chi(H)>3.\n$$\nTherefore\n$$\n\\gamma(G)=\\gamma^\\infty(G)=3<\\theta(G).\n$$","claimIssueNote":null,"solveDate":"2026-09-05","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"GPT-6 Astra found the counterexample and wrote the Lean proof. It recognized that the complement of the classical 243-vertex Berlekamp–van Lint–Seidel ternary Golay graph provides the needed structure, proved $\\gamma=3$, constructed an indefinitely repeatable three-guard defense proving $\\gamma^\\infty=3$, and proved $\\theta>3$ by a coloring/double-counting argument. The underlying Golay graph was known since 1973; the new contribution is identifying its complement as a counterexample and proving the eternal domination strategy.","verification":"lean-checked","verificationNote":"Lean-checked, statement unaudited. Checked here on 6 September 2026 from a clone of tadamcz/gamma-theta at d64cce5: 1,987 lines; the only sorry outside Challenge.lean is the unused '.disproof' stub in submission/Spec.lean, which the README explains (the compared theorem asserts the existence of the counterexample); zero axiom declarations, no native_decide; the audit folder's docker logs print the theorem's axioms as the standard three. The statement was AI-autoformalized in Epoch's wikipedia run; the definitions of the eternal dominating family (one-guard model), domination number and clique cover number were read here and look right, but that is one reading, not an audit. The repository's Python check of 5,889,840 attacks was not replayed here.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A 2014 conjecture of Klostermeyer and Mynhardt in the eternal domination literature, reopened after a 2009 proof was found to have a gap, checked exhaustively to 11 vertices in 2022 and proved for planar graphs in 2025. Refuted by a 243-vertex graph known since 1973. A real conjecture in a specialised corner of graph theory; the numbered-Erdős level.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/tadamcz/gamma-theta","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"Challenge.lean: the compared statement and definitions","url":"https://github.com/tadamcz/gamma-theta/blob/main/Challenge.lean","kind":"lean-statement"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bollobas-nikiforov-conjecture","name":"Bollobás–Nikiforov conjecture","shortName":"Bollobás–Nikiforov","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"Let $G$ be a finite simple graph with $m=|E(G)|$ edges and clique number $\\omega(G)$, and let $\\lambda_1(G)\\ge\\lambda_2(G)\\ge\\cdots\\ge\\lambda_n(G)$ be the eigenvalues of its adjacency matrix. Bollobás and Nikiforov conjectured in 2007 that every non-complete graph satisfies $$\\lambda_1(G)^2+\\lambda_2(G)^2\\le 2\\Bigl(1-\\frac{1}{\\omega(G)}\\Bigr)m.$$ Before this work it was known for triangle-free graphs (Lin, Ning and Wu), regular graphs (Zhang), graphs with few triangles, complete multipartite graphs, and asymptotically almost surely for random graphs, and open in general.","posedBy":"Béla Bollobás and Vladimir Nikiforov, “Cliques and the spectral radius”","yearPosed":2007,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"The work claims a complete proof of the original Bollobás–Nikiforov conjecture for every finite non-complete simple graph.\n\nMore precisely, the Lean theorem `BN.lambda1_sq_add_lambda2_sq_le` proves\n$$\n\\lambda_1(G)^2+\\lambda_2(G)^2\n\\le\n2\\left(1-\\frac{1}{\\omega(G)}\\right)|E(G)|.\n$$\nThe development proves a stronger weighted spectral inequality. If \\(B\\) is a real symmetric entrywise nonnegative matrix with zero diagonal, supported on the edges of \\(G\\), and \\(F(B)\\) is the sum of the squares of its two largest positive eigenvalues, then\n$$\nF(B)\n\\le\n\\left(1-\\frac{1}{\\omega(G)}\\right)\\|B\\|_F^2.\n$$\nThe human-readable manuscript is still preliminary and is being rewritten and polished.","claimIssueNote":null,"solveDate":"2026-09-05","model":"GPT-6 Astra; Grok 4.6; Claude Fable 5.1","modelMaker":"OpenAI; xAI; Anthropic","humanCollaborators":["Gabriel Coutinho","Yinchen Liu","Thomás Jung Spier","Quanyu Tang","Shengtong Zhang"],"aiRole":"GPT 6 Astra was used by the five human authors in the ideation and development of the mathematics and generated the preliminary manuscript `docs/sol.tex`.\n\nThe resulting proof was then formalized in Lean 4/Mathlib. Grok 4.6 agents in Cursor produced essentially the entire Lean development, working node-by-node from the mathematical proof and a human-directed formalization blueprint. Each component was required to build without `sorry`.\n\nA Claude Fable 5.1 agent subsequently prepared the Palomar submission surface and verification packaging. It did not add substantive proof content.","verification":"lean-verified","verificationNote":"Lifted from the submitted Lean-checked to Lean-verified, because the correspondence the submitter said nobody had audited was audited here, on 12 September 2026, at commit edb5259. Challenge.lean imports only Mathlib and states the headline theorem entirely in Mathlib vocabulary: $G$ a SimpleGraph on a finite vertex type, $G\\ne\\top$ for non-complete, Nontrivial for at least two vertices, G.cliqueNum for $\\omega$, G.edgeFinset.card for $|E(G)|$ counted once, and $\\lambda_1,\\lambda_2$ as the first two entries of Mathlib's eigenvalues₀ of the real adjacency matrix, which is nonincreasing. There is no hypothesis beyond those and nothing project-defined in the statement beyond those two thin wrappers, whose definitions sit in the same file. Solution.lean supplies the same five names from the development; comparator.json compares them with only propext, Classical.choice and Quot.sound permitted and with nanoda enabled as an independent kernel; VERIFICATION.md records the comparator run accepting the solution under both kernels; Palomar checks and Lean Action CI passed on GitHub at the reviewed commit. No sorry outside the five deliberate holes in Challenge.lean, no axiom, no native_decide. The build was not repeated here. Candidate rather than Resolved is the site's practice for named conjectures with a formal proof: it flips when a named expert with no stake confirms publicly that the formal statement is the conjecture, a short read since the statement is pure Mathlib.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A named 2007 conjecture from a JCTB paper, with a nineteen-year trail of partial results - triangle-free, regular, few triangles, complete multipartite, random graphs - and its own problem record. Famous within spectral graph theory and unknown outside it, which is the band's definition. Level with the Albertson-Berman induced-forest and Hadamard-668 entries at 30; below the Petersen colouring conjecture at 40, which reaches the whole of graph theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/ShengtongZhang-alt/BN/tree/edb5259dfd055ea31b4c46ac9ea4d33a758c2b99","sourceName":"GitHub repository (Lean 4 proof and preliminary note), pinned to the reviewed commit","collection":null,"collectionVersion":null,"links":[{"label":"Palomar formalization record","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-09-07-000002&version=1","kind":"palomar"},{"label":"MathDB problem record","url":"https://mathdb.com/p/405423/bollobas-nikiforov-conjecture","kind":"problem-record"},{"label":"Original Bollobás–Nikiforov paper","url":"https://doi.org/10.1016/j.jctb.2006.12.002","kind":"paper"},{"label":"Challenge.lean: the five statements against plain Mathlib, at the reviewed commit","url":"https://github.com/ShengtongZhang-alt/BN/blob/edb5259dfd055ea31b4c46ac9ea4d33a758c2b99/Challenge.lean","kind":"lean-statement"},{"label":"Solution.lean: the comparator-checked bridge to the development","url":"https://github.com/ShengtongZhang-alt/BN/blob/edb5259dfd055ea31b4c46ac9ea4d33a758c2b99/Solution.lean","kind":"lean-proof"},{"label":"VERIFICATION.md: the recorded comparator and nanoda run","url":"https://github.com/ShengtongZhang-alt/BN/blob/edb5259dfd055ea31b4c46ac9ea4d33a758c2b99/VERIFICATION.md","kind":"other"}],"relations":[],"submittedBy":"StormyCobra617","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"critical-site-percolation-z-d","name":"Absence of critical Bernoulli site percolation on $\\mathbb Z^d$ for $d \\ge 3$","shortName":"Critical site percolation: θ(p_c) = 0","problemNumber":null,"field":"Percolation theory; mathematical statistical mechanics","fieldGroup":"Probability & statistics","statement":"For nearest-neighbour Bernoulli site percolation on $\\mathbb Z^d$, each vertex is open independently with probability $p$. Let $\\theta(p)$ be the probability that the open cluster of the origin is infinite and $p_c$ the critical parameter. Is $\\theta(p_c)=0$ for every $d \\ge 3$? This is the site form of the classical question of no percolation at criticality; Benjamini and Schramm stated it for site percolation on every quasi-transitive graph with $p_c<1$. The planar case was settled classically, and high dimensions by Heydenreich and Matzke's site lace expansion (2020), with no explicit dimension bound.","posedBy":"Itai Benjamini and Oded Schramm, Percolation beyond Z^d, many questions and a few answers (1996), Conjecture 4, stated for site percolation","yearPosed":1996,"ageNote":"Dated from Benjamini and Schramm's site-percolation statement. The Z^d question itself is older and is usually stated for bond percolation, traced to Kesten (1982); the catalog's bond entry carries that history.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims $\\theta(p_c)=0$ for nearest-neighbour Bernoulli site percolation on $\\mathbb Z^d$ for every $d\\ge3$ (theorem KNAll.Site.site_no_percolation_at_critical), by proving the Kozma-Nitzan gluing inequality for independent hyperedges, representing site percolation by incidence hyperedges, and running the lattice exploration of the bond proof. The intermediate dimensions were open. The planar case is classical and outside the formal statement. The same repository proves Kozma-Nitzan Conjectures 1 to 4 and 6, answers Questions 5, 7 and 9, and refutes one reading of Question 8. OpenAI's release (24 September 2026) later gave critical site nonpercolation on $\\mathbb Z^3$ only, crediting this work; its quasi-transitive theorem is bond only, so the site form of Benjamini-Schramm Conjecture 4 beyond $\\mathbb Z^d$ stays open.","claimIssueNote":null,"solveDate":"2026-09-05","model":"ChatGPT 5.6 (per the exposition page; the author's other documents also name Claude and GLM models)","modelMaker":"OpenAI; Anthropic; Zhipu AI","humanCollaborators":["Ahmed Bou-Rabee","Justin Leder"],"aiRole":"The public exposition (Bou-Rabee's site, last updated 5 September 2026) says: \"Mathematics generated by ChatGPT 5.6, prompted by Ahmed Bou-Rabee, with exposition and formalization support by Claude Opus 5 and GLM 5.3.\" The companion manuscript on the Kozma-Nitzan inequalities says its arguments, exposition and Lean formalization \"were produced entirely by ChatGPT 5.6 Sol and Claude Fable 5.1, prompted by Ahmed Bou-Rabee\", and the repository NOTICE says the KN, hypergraph and site development was \"produced with ChatGPT and Claude, prompted by Ahmed Bou-Rabee\". The repository's formalization.yaml names Bou-Rabee and Justin Leder as authors (Leder as responsible maintainer) and lists Claude Opus 5.5, Sonnet 5, Opus 5 and DeepSeek v4.1 Flash as the Lean agents, closely supervised by Bou-Rabee. The model lists differ between documents; all agree the mathematics came from the models.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Checked here on 7 October 2026 at commit 6a76bc3f of nitromannitol/percolation-after-anthropic: the Mathlib-only comparator challenge PercolationAudit/SiteCriticality/Challenge.lean was read. It builds nearest-neighbour Z^d as SimpleGraph.hasse on Fin d to Z (as in the audited bond entry), the product Bernoulli law on vertex sets, open clusters through open vertices, theta as the law of an infinite cluster at the origin and p_c as the infimum of percolating parameters with 1 adjoined, and states thetaSite d (p_c) = 0 for every d with 3 <= d and no other hypothesis. That is the headline claim. The repository's CI shows the Comparator audit workflow succeeding on that commit; formalization.yaml reports no sorry and only propext, Classical.choice and Quot.sound. Not rebuilt here. The proof depends on Anthropic's bond library pinned at 795efb8, audited for the bond entry. The site extension has no written proof outside the exposition and the Lean development.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":72,"significanceNote":"The site twin of the classical no-percolation-at-criticality problem on Z^d, open in exactly the same dimensions (3 up to the unspecified lace-expansion threshold) and the model in which Benjamini and Schramm stated their conjecture. Below the bond Z^d entry at 78, the more canonical statement whose proof this one extends, and the Benjamini-Schramm bond entry at 79; level with the Landau-Siegel zero entry at 72; above the p_c < p_u entry at 66.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://nitromannitol.github.io/site-percolation-7d3f2/","sourceName":"Bou-Rabee, Percolation at criticality (public exposition and Lean repository, 5 September 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Lean repository: percolation-after-anthropic (Bou-Rabee and Leder)","url":"https://github.com/nitromannitol/percolation-after-anthropic","kind":"lean-proof"},{"label":"Comparator challenge: the headline statement in plain Mathlib","url":"https://github.com/nitromannitol/percolation-after-anthropic/blob/6a76bc3f/PercolationAudit/SiteCriticality/Challenge.lean","kind":"lean-statement"},{"label":"Kozma-Nitzan connection inequalities, companion manuscript (5 September 2026)","url":"https://github.com/nitromannitol/percolation-after-anthropic/blob/6a76bc3f/docs/kn-results.pdf","kind":"paper"},{"label":"Kozma and Nitzan, A reduction of the theta(p_c)=0 problem to a conjectured inequality (arXiv:2401.12397)","url":"https://arxiv.org/abs/2401.12397","kind":"paper"},{"label":"Bond percolation on Z^d, the result this extends","url":"https://vibemathed.com/problem/absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2","kind":"other"},{"label":"OpenAI release: critical bond and site percolation on the cubic lattice (later, Z^3 only)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf","kind":"paper"},{"label":"Quanta Magazine coverage","url":"https://www.quantamagazine.org/updates/transformation/","kind":"discussion"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"navier-stokes-millennium-prize-problem-finite-time-breakdown-with-smooth-forcing","name":"Navier–Stokes Millennium Prize problem: finite-time breakdown with smooth forcing","shortName":"Navier–Stokes (Clay C/D)","problemNumber":null,"field":"PDEs; fluid dynamics; singularity formation","fieldGroup":"Differential equations","statement":"For the three-dimensional incompressible Navier–Stokes equations with positive viscosity, do there exist smooth divergence-free initial data and smooth external forcing for which a global smooth physically reasonable solution does not exist? The Clay Millennium Prize formulation allows this to be established either on R^3 (statement C) or on the periodic three-torus (statement D).","posedBy":"Jean Leray (1934), whose weak solutions left smoothness open; stated as Millennium alternatives (C) and (D) by Charles Fefferman for the Clay Mathematics Institute in 2000","yearPosed":1934,"ageNote":"Dated from Leray's 1934 paper, which constructed global weak solutions and left smoothness open, rather than from the 2000 Clay formulation. Fefferman's own problem description builds from Leray, and OpenAI's paper and announcement both date the question to 1934. The Clay statement fixed alternatives (C) and (D), the propositions this result settles; it did not pose the question.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"OpenAI claims to construct a smooth finite-energy solution of the three-dimensional incompressible Navier–Stokes equations, starting from a fluid at rest and driven by smooth forcing, that develops a finite-time singularity. The construction is claimed both on R^3 and in the periodic setting, thereby establishing statements C and D of Fefferman's official Clay formulation. If the correspondence between the released proof/formalization and the Clay statements survives independent review, this resolves the Navier–Stokes Millennium Prize problem. It does not establish finite-time blow-up for the unforced Navier–Stokes equations; the Clay resolution comes through the smooth-forcing alternatives C/D.","claimIssueNote":null,"solveDate":"2026-09-05","model":"Unnamed internal OpenAI model","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"OpenAI launched a multi-agent research effort using an unnamed internal model significantly more capable than GPT-6 Astra. Roughly 10,000 concurrent agents explored the Navier–Stokes Millennium Prize problem, with separate groups assigned the different Clay formulations and related subproblems. After the system independently found an unforced 3D Euler blow-up construction, that result was supplied to the Navier–Stokes agents. Codex was used to consolidate useful intermediate insights across groups. The system produced the analytical finite-time singularity construction after approximately 88 hours. GPT-6 Astra was subsequently used to formalize and verify the result in Lean.","verification":"lean-verified","verificationNote":"No independent mathematician has checked this yet. Audited here on 9 September 2026. Statement anchor: OpenAI's ComparatorChallenges/NavierStokes.lean was diffed against google-deepmind/formal-conjectures at commit 8bf45ed, the commit its header names. The two compared theorems, navier_stokes_breakdown_R3 and navier_stokes_breakdown_periodic, are byte-identical to DeepMind's, as are the decay, periodicity and energy conditions they depend on; the only differences in the file are nine deleted lines carrying Formal Conjectures' alternatives (A) and (B), which OpenAI does not claim. The statement is therefore anchored to an independent third party's formalisation of Fefferman's conditions, not to one written by the prover. Counts taken independently: 2,486 Lean files, 616,276 lines, zero sorry outside the deliberate challenge placeholders, zero axiom declarations, zero native_decide, zero unsafe. comparator.json permits only propext, Quot.sound and Classical.choice and enables the independent nanoda kernel; the solution module states both theorems verbatim and discharges them via ComparatorBridge. In the paper, Theorem 1.1's force lies in C^inf_c(R^3 x (0,inf)), compactly supported in space and time, satisfying Fefferman's decay condition (5) outright. Not rebuilt here. No mathematician has read the argument: released 8 September 2026, no referee, and Clay's own process requires publication in a refereed journal plus two years of general acceptance.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":87,"significanceNote":"A Millennium Prize problem, resolved through alternatives (C) and (D) of Fefferman's official statement, which permit a smooth force with rapid decay. Below the 100 reserved for the Riemann hypothesis and above the Collatz band, but not at the ceiling: the forced alternatives are the more tractable half of the Clay problem, and what most readers mean by \"Navier-Stokes\" - global regularity for the unforced equations - is untouched. Held at Candidate because no human has read it.","solveCostUsd":2000000,"solveCostNote":"OpenAI said \"millions\" spent on compute in a press call","sourceUrl":"https://openai.com/index/navier-stokes-solution/","sourceName":"OpenAI research publication and proof announcement","collection":null,"collectionVersion":null,"links":[{"label":"Clay Mathematics Institute — official Navier–Stokes formulation","url":"https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf","kind":"other"},{"label":"OpenAI's comparator challenge statement for (C) and (D)","url":"https://github.com/openai/NavierStokesAndEuler/blob/main/ComparatorChallenges/NavierStokes.lean","kind":"lean-statement"},{"label":"Formal Conjectures' original, which the statement matches byte for byte","url":"https://github.com/google-deepmind/formal-conjectures/blob/8bf45ed70d48b2b2a501de9c00b26bfa38c573ee/FormalConjectures/Millenium/NavierStokes.lean","kind":"problem-record"},{"label":"The Lean development and comparator configuration","url":"https://github.com/openai/NavierStokesAndEuler","kind":"lean-proof"},{"label":"Navier–Stokes existence and smoothness","url":"https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness","kind":"wikipedia"}],"relations":[],"submittedBy":"LucidHawk551","upvotes":19,"downvotes":1,"commentCount":6},{"slug":"ibragimov-iosifescu-varphi-mixing-clt-conjecture","name":"The Ibragimov–Iosifescu conjecture for φ-mixing sequences","shortName":"$\\varphi$-mixing CLT","problemNumber":null,"field":"Stationary stochastic processes","fieldGroup":"Probability & statistics","statement":"Ibragimov conjectured that if $(X_n)$ is a strictly stationary, $\\varphi$-mixing sequence with\n$$\n\\mathbb E X_0=0,\\qquad \\mathbb E X_0^2<\\infty,\n$$\nand\n$$\n\\sigma_n^2=\\operatorname{Var}(S_n)\\to\\infty,\\qquad S_n=\\sum_{j=0}^{n-1}X_j,\n$$\nthen\n$$\n\\frac{S_n}{\\sigma_n}\\Rightarrow N(0,1).\n$$\nGPT-6 Astra constructs a counterexample satisfying all these hypotheses for which, along a subsequence $n_j\\to\\infty$,\n$$\n\\frac{S_{n_j}}{\\sigma_{n_j}}\\to 0\n$$\nin probability. Hence the conjectured central limit theorem fails.","posedBy":"I. A. Ibragimov","yearPosed":1971,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Astra constructs a strictly stationary real process\n$$\nX_t=\\xi_t+K(\\xi_{t-1},\\xi_{t-2},\\ldots),\n$$\nwhere the innovations $\\xi_t$ are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and $K$ is bounded, continuous, and odd. The process is $\\varphi$-mixing, centered, square-integrable, and satisfies\n$$\n\\operatorname{Var}(S_n)\\to\\infty.\n$$\nNevertheless there are times $n_j\\to\\infty$ such that\n$$\n\\frac{S_{n_j}}{\\sqrt{\\operatorname{Var}(S_{n_j})}}\\to0\n$$\nin probability. Therefore the normalized sums cannot converge in distribution to $N(0,1)$.\n\nThe same example also rules out Iosifescu's stronger weak invariance-principle conjecture, since Brownian convergence would imply the CLT at time $1$.","claimIssueNote":null,"solveDate":"2026-09-05","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"GPT-6 Astra autonomously constructed the counterexample and wrote its Lean proof. The run lasted about 12 hours and 1054 turns. The core idea is to build a stationary causal process from Gaussian-smoothed rare spikes plus a bounded nonlinear feedback correction. The feedback suppresses ordinary fluctuations along selected times, while extremely rare large spikes keep $\\operatorname{Var}(S_n)\\to\\infty$. Astra also proves that the resulting process remains genuinely $\\varphi$-mixing by constructing a bounded causal inverse with square-summable tail variation.","verification":"lean-checked","verificationNote":"Lean-checked, statement unaudited. Checked here on 6 September 2026 from a clone of tadamcz/phi-mixing-clt at 8e08498: 13,047 lines, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; the repository's recorded verifier accepted the disproof with the default kernel and only the three standard axioms. The statement, however, was produced by Epoch's AI-autoformalized 'wikipedia' run, not by a human-curated repository, and this site has not audited the formal definitions of strict stationarity and the φ-mixing coefficient against the sources beyond reading the docstring, which is careful about conventions. The repository's own audit finds no mismatch, but that audit is machine-written. No probabilist outside the run has read the 12,900-line construction. Hence Candidate.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Ibragimov's 1971 conjecture (Ibragimov–Linnik, problem 3) that a strictly stationary φ-mixing sequence with finite variance and Var(S_n) -> infinity obeys the central limit theorem, with Iosifescu's invariance-principle strengthening, is a standing problem of the dependent-CLT literature: Peligrad's 1990 paper is titled after it and it has a Wikipedia article. Fifty-five years, refuted in the one regime the partial results had left open.","solveCostUsd":261,"solveCostNote":"The autonomous GPT-6 Astra run took about 12 hours and 1054 turns and is reported to have cost $261. The final Lean submission is about 12,900 lines, containing 864 theorem/lemma declarations and 187 definitions/structures across 34 generated modules.","sourceUrl":"https://github.com/tadamcz/phi-mixing-clt","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"Challenge.lean: the compared statement","url":"https://github.com/tadamcz/phi-mixing-clt/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Peligrad (1990), On Ibragimov–Iosifescu conjecture for φ-mixing sequences","url":"https://doi.org/10.1016/0304-4149(90)90008-6","kind":"problem-record"}],"relations":[],"submittedBy":"VibeGene","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"does-there-exist-a-bijection-of-mathbb-r-n-to-itself-such-that-the-forward-map-i","name":"Does there exist a bijection of $\\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?","shortName":"Wong's connectedness-preserving bijection","problemNumber":null,"field":"General Topology","fieldGroup":"Geometry & topology","statement":"Willie Wong asked on MathOverflow in April 2016: if $f:\\mathbb R^n\\to\\mathbb R^n$ is a bijection that maps every connected set to a connected set, must $f^{-1}$ do the same? By Tanaka's theorem and invariance of domain this is equivalent to asking whether every connectedness-preserving bijection of $\\mathbb R^n$ is continuous. For $n=1$ the answer is yes. For $n\\ge 2$ the question stayed open for a decade: the top-voted answer constructs such a bijection only from $\\mathbb R$ to $\\mathbb R^2$, and Banakh and Banakh (2020) proved continuity in several compact settings while calling Wong's problem still open.","posedBy":"Willie Wong, MathOverflow question 235893","yearPosed":2016,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Answered in the negative for every $n\\ge 2$. The preprint constructs a bijection $F:\\mathbb R^n\\to\\mathbb R^n$ that maps every connected set to a connected set, is continuous exactly off the closed ray $[0,\\infty)\\times\\{0\\}^{n-1}$, and pulls the straight segment $\\{(1,0,\\dots,0)\\}\\times[0,1]$ back to the middle-thirds Cantor set on that ray, so $F^{-1}$ is not connectedness-preserving. The construction extends a thin solid tube by finger moves so its cross-sections recur near every point of the complementary compactum, collapses the ray onto the tube's ideal end, and certifies arbitrary connected sets by a separation argument; $F$ and $F^{-1}$ can be taken Borel. The same author's companion note on Darboux injections from closed manifolds (Banakh-Banakh Problems 1.7 and 1.8) is a separate result and belongs in its own entry.","claimIssueNote":null,"solveDate":"2026-09-05","model":"GPT-6 (Codex, Ultra effort), Claude Fable 5.1","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Peter L."],"aiRole":"The preprint's own disclosure: the paper \"is the outcome of a research program conducted with two AI systems under the author's direction: OpenAI's Codex (GPT-6, Ultra effort), which produced the structural theory, the constructions, the adversarial audits, the verification of the argument, and the draft; and Anthropic's Claude Fable 5.1 (Extra effort), which planned the program, reviewed the successive run reports, and proposed the single-line coloring that makes the construction uniform in the dimension.\" The author chose the problem, wrote the briefs and ran the audits between systems. Appendix A separates ideas taken from the literature from ideas first recorded within the program.","verification":"unreviewed","verificationNote":"Unreviewed. The 17-page preprint (Zenodo 10.5281/zenodo.22346412, version 1, dated 5 September 2026) was read here on the day it appeared; the theorem, the construction outline and the disclosure match the submission. Nobody outside the author's program has checked the argument, and the preprint is visibly unfinished: its acknowledgements read \"to be supplied by the author\" and its disclosure ends with a bracketed statement \"to be completed after review\" that the author has verified the mathematics and accepts responsibility. Candidate until that statement is filled in and someone independent has read the proof. The question's history warrants care: it drew five answers over ten years, all partial, and a 2020 paper by Banakh and Banakh devoted to it.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A MathOverflow question open since 2016 with a real following: a score of 40 on the top partial answer, a 2020 arXiv paper by Banakh and Banakh built around it, and its own name in the literature (Darboux bijections of R^n). A clean negative answer to a decade-old question in general topology, one rung above the numbered-Erdős level; well below structural conjectures, since it settles one question rather than a programme.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.22346412","sourceName":"Recurrent tubes and connectedness-preserving bijections of Euclidean spaces (Zenodo preprint, 5 Sep 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Proof of Banakh-Banakh's Problems 1.7 and 1.8","url":"https://doi.org/10.5281/zenodo.22347647","kind":"paper"},{"label":"Wong's question on MathOverflow (2016), with the five partial answers","url":"https://mathoverflow.net/questions/235893","kind":"problem-record"},{"label":"Banakh and Banakh, The continuity of Darboux injections between manifolds (2020), which calls the problem still open","url":"https://arxiv.org/abs/1809.00401","kind":"paper"}],"relations":[],"submittedBy":"WittyHeron892","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cycle-residue-stability-at-minimum-degree-five-2","name":"Cycle-residue stability at minimum degree five","shortName":"Mod-5 cycle stability","problemNumber":null,"field":"Graph Theory","fieldGroup":"Combinatorics","statement":"Classify the finite simple graphs of minimum degree at least five whose cycle lengths fail to represent every residue class modulo five. Is residue two the only possible missing residue, and can all such graphs be characterized through an explicit family of exceptional end-blocks together with a condition on the remaining blocks?","posedBy":"Luo, Ma and Zhao, whose stability theorem covers every k >= 6 and leaves k = 5","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"The preprint claims an exact classification. Write $C_5(G)$ for the residues modulo five represented by cycle lengths in $G$. Let\n$\\mathcal E_5={K_6,K_{5,5}}\\cup{H_{5,n;t}:2\\le t\\le5= 6 earlier in 2026, so k = 5 is the one odd modulus left open, and this settles it. A completed classification in a live line of work in extremal graph theory, but a corollary-sized question next to Dean's conjecture itself (32 here), and it rests on the author's own Dean-5 theorem, which is still a candidate on this site.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.22311785","sourceName":"Source name Stability for cycle residues modulo five in graphs of minimum degree five","collection":null,"collectionVersion":null,"links":[{"label":"Computational supplement, version 1.0.0","url":"https://doi.org/10.5281/zenodo.22311412","kind":"code"},{"label":"Dean–5 theorem used as an external input, version 1.0.1","url":"https://doi.org/10.5281/zenodo.22182448","kind":"paper"}],"relations":[],"submittedBy":"eli","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-proper-hat-guessing-number-of-k-8-e","name":"The proper hat-guessing number of $K_8-e$","shortName":"$\\mathrm{HG}_P(K_8-e)=14$","problemNumber":null,"field":"Graph theory; hat-guessing games; finite geometry; matching theory","fieldGroup":"Combinatorics","statement":"Determine the exact proper hat-guessing number of the complete graph on eight vertices with one edge removed. The general bounds leave $\\mathrm{HG}_P(K_8-e)\\in\\{13,14\\}$.","posedBy":"Adriaensen et al.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"We prove $\\mathrm{HG}_P(K_8-e)=14$ using an explicit $\\mathrm{PGL}(2,13)$-equivariant strategy. A compact formula fixes 990 normalized twin decisions, and a 48,510-edge residual-orbit matching induces consistent rules on 53,460 labelled clique-view orbits. Independent implementations regenerate the certificate and check all 138,378,240 proper fourteen-colourings with zero failures. Unlike the $n=5,6,7$ constructions, residual right degrees reach eight, so the explicit global matching is load-bearing. A separately labelled companion exhibits one fixed clique completion compatible with $2^{380}$ equivariant twin-rule pairs and proves 380 optimal within the stated independent whole-tail reversal model; that companion's independent review is pending. The release does not solve the general $K_n-e$ family or $K_9-e$.","claimIssueNote":null,"solveDate":"2026-09-04","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"OpenAI GPT-5.6 Pro contributed substantively to literature search, target selection, construction search, proof development, code generation, exact verification, adversarial critique, certificate design, and manuscript and release preparation. The equivariant construction, residual-matching proof, Hall-obstruction analysis, and shared-completion family were developed in a human-directed model-assisted research process. Matthew Protti selected and framed the target, directed and evaluated the work, commissioned review, required exact checks, approved the public scope, and accepts responsibility.","verification":"unreviewed","verificationNote":"Unreviewed on this site's ladder. The frozen commit bacb79e3 carries STATUS.json with headline status PROVED_COMPUTER_ASSISTED and project review ACCEPTED, an AI_USE_AND_PROVENANCE.md naming the model and the division of labour, and a C++ sweep of all 138,378,240 proper fourteen-colourings with zero coverage failures across 48,510 residual orbits. The Gate 2/3 Hall-obstruction archive was reviewed as ACCEPT_WITH_EXPLICIT_REPAIRS and the repairs applied. The 2^380 shared-completion companion is a separate claim whose independent review is still pending and is not part of this entry. As with the siblings: a real adversarial review of an artefact, not a named expert or a proof assistant, and this site has not rebuilt it.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"One value of one graph parameter, fourth in a series after K5 - e, K6 - e and K7 - e from the same author over four days. The ceiling comes from Adriaensen et al.'s bound n + chi(G) - 1 = 14 and the work shows it is attained. Level with the three siblings deliberately: the same method one vertex larger does not gain weight per rung, and the question was posed this year with no accumulated literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/bacb79e3201dcc59b57339181affda122db771ce","sourceName":"K8-e public research disclosure v0.1 (GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"K8-e theorem and proof","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/blob/bacb79e3201dcc59b57339181affda122db771ce/research/proper_hat_guessing_k8_minus_e/THEOREM.md","kind":"paper"},{"label":"Verified construction and full-colouring checker","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/bacb79e3201dcc59b57339181affda122db771ce/research/proper_hat_guessing_k8_minus_e/accepted_construction","kind":"code"},{"label":"Review-pending shared-completion companion","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/bacb79e3201dcc59b57339181affda122db771ce/research/proper_hat_guessing_k8_minus_e/shared_completion_companion","kind":"other"},{"label":"Repaired Gate 2/3 Hall-obstruction archive","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/bacb79e3201dcc59b57339181affda122db771ce/research/proper_hat_guessing_k8_minus_e/supporting_gate23","kind":"other"},{"label":"Public draft PR #3 and review surface","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/pull/3","kind":"announcement"},{"label":"Hat guessing with proper colorings (Adriaensen et al.)","url":"https://arxiv.org/abs/2603.04909","kind":"problem-record"}],"relations":[],"submittedBy":"Matthew Protti","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"petrykowski-conjecture-bounded-orbit-definable-amenability","name":"Petrykowski's conjecture on bounded orbits and definable amenability","shortName":"Petrykowski's conjecture","problemNumber":null,"field":"Model theory","fieldGroup":"Logic & foundations","statement":"For a definable group in a first-order theory, Petrykowski proposed that admitting a global type whose left-translation orbit is bounded should suffice for the group to be definably amenable; Newelski recorded the conjecture and proved it under stronger hypotheses. Does a bounded left-translation orbit imply definable amenability?","posedBy":"Marcin Petrykowski; recorded as a conjecture by Ludomir Newelski, who proved it under stronger hypotheses","yearPosed":2012,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-09-04","model":"ChatGPT 5.6","modelMaker":"OpenAI","humanCollaborators":["Artem Chernikov"],"aiRole":"From the paper's AI disclosure: ChatGPT 5.6 was used to establish parts of the general model-companion construction and to use finite quotients to construct the bounded orbit. The framing, the extension of the earlier seven-author construction and the write-up are the author's.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026 against arXiv:2609.05711: the abstract states the counterexample as a simple expansion of the theory of nonabelian free groups with a global type of bounded left-translation orbit that is not definably amenable, extending the construction of Chernikov, Hrushovski, Kruckman, Krupiński, Moconja, Pillay and Ramsey. The introduction credits Petrykowski for the proposal and Newelski for recording it, and the reference list carries Newelski, Isr. J. Math. 187 (2012), and Newelski-Petrykowski in JLMS. The mathematics was not checked here; eighteen days old, no referee.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture in the definable-amenability corner of model theory, recorded in print by Newelski and worked on by a small group. Level with the catalog's other model-theory entries and its Auslander-Reiten-Smalø entry at 20; below the periodicity conjecture at 25 for a shorter recorded history.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.05711","sourceName":"A counterexample to Petrykowski's conjecture, arXiv:2609.05711 (4 September 2026)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bounded-prime-gaps-at-most-212","name":"A new bound for small gaps between primes: $H_1 \\le 212$","shortName":"Prime gaps at most 212","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"Write $H_1 = \\liminf_{n\\to\\infty}(p_{n+1}-p_n)$. Stadlmann had recently proved $H_1 \\le 240$, improving the bound $246$ of Polymath8b. Building on her work, this paper proves $H_1 \\le 212$: infinitely many pairs of consecutive primes are at most $212$ apart.","posedBy":"Alphonse de Polignac (the twin prime conjecture); the bounded form since Goldston, Pintz and Yıldırım","yearPosed":1849,"ageNote":"Dated from Polignac's 1849 conjecture. The modern quantitative form dates from Goldston, Pintz and Yıldırım in 2005 and became a finite bound with Zhang in 2013.","solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"$H_1 \\le 212$, improving Stadlmann's $240$ of three days earlier and the $246$ of Polymath8b that had stood since 2014. The twin prime conjecture, $H_1 = 2$, is untouched. Held the record for hours at most: OpenAI's paper claiming $186$ is dated 30 August, four days before this one, though its Lean development appeared on 2 September.","claimIssueNote":null,"solveDate":"2026-09-03","model":"AxiomProver","modelMaker":"Axiom Math","humanCollaborators":["François Charton","Letong Hong","Kenny Lau","Ken Ono","Guillaume Remy","Ho Chung Siu","Ashvin A. Swaminathan","Jesse Thorner","Yunzhou Xie"],"aiRole":"The mathematics is the authors'. The AI contribution is the formal certificate, and the paper is precise about it in Appendix A: \"AxiomProver, an AI system under development by AxiomMath, autonomously generated from natural-language specifications a Lean certificate of the deduction of Theorem 1.1.\" The certificate takes as hypotheses the five Type I, Type II and Type III equidistribution estimates of Section 5, the bilinear Bombieri-Vinogradov theorem below the half-level, the Harman decomposition, and the variational certificate. Nothing in the paper claims the model found the argument, and the abstract does not mention AI at all.\n\nThe same group's AxiomProver had formalised the twelve-year-old 246 bound in Lean a few weeks earlier, which is the work this builds its tooling on.","verification":"lean-checked","verificationNote":"A preprint one day old, not peer reviewed. Its Lean certificate was produced by AxiomProver and is conditional on the equidistribution estimates and the Bombieri-Vinogradov theorem stated in the paper, so it certifies the deduction rather than the analytic inputs. Nine authors, several of whom work on exactly this, take responsibility for the mathematics. No independent expert has read it on the record, and this site has not rebuilt the certificate.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"The closest mathematics has come to the twin prime conjecture, and the most-watched number in analytic number theory since Zhang in 2013. Level with the site's other bounded-gaps entries: the weight is the question's, not any one step's.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://primegaps.axiommath.ai/bgp212.pdf","sourceName":"A new bound for small gaps between primes","collection":null,"collectionVersion":null,"links":[{"label":"PrimeGapsLib, the Lean library","url":"https://github.com/AxiomMath/PrimeGapsLib","kind":"lean-proof"},{"label":"Blueprint of the 246 formalisation this builds on","url":"https://primegaps.axiommath.ai/","kind":"independent"},{"label":"Stadlmann's 240, the bound this improves on","url":"https://arxiv.org/abs/2608.31126","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"kothe-conjecture","name":"Köthe Conjecture","shortName":"Köthe Conjecture","problemNumber":null,"field":"Noncommutative ring theory","fieldGroup":"Algebra","statement":"Köthe's conjecture asks whether the sum of two nil left ideals of a ring is always nil. Equivalently, by Krempa's 1972 formulation, if $I$ is a nil two-sided ideal of a ring $R$, then the matrix ideal $M_n(I)$ should be nil for every finite $n$, already for $n=2$.\n\nThe conjecture is false: there exists a ring $R$, a nil two-sided ideal $I\\subseteq R$, and a $2\\times 2$ matrix with entries in $I$ that is not nilpotent.","posedBy":"Gottfried Köthe","yearPosed":1930,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"GPT-6 Astra constructs a unital algebra $R=k\\oplus A$ over the countable field\n$$\nk=\\overline{\\mathbb F_2},\n$$\nwith $I=A$ a nil two-sided ideal, together with a matrix\n$$\nW\\in M_2(I)\n$$\nthat is not nilpotent.\n\nThe algebra $A$ is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of $A$ is nilpotent. At the same time, a suitable polynomial combination of the shifts fixes a nonzero vector; this yields a companion-type matrix with a nonzero eigenvalue, and hence a nonnilpotent matrix whose entries lie in $I$.\n\nThis formally disproves Krempa's matrix formulation of Köthe's conjecture.","claimIssueNote":null,"solveDate":"2026-09-03","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"A pre-release GPT-6 Astra autonomously attempted the open Formal Conjectures benchmark statement, with no human steering during the run, and constructed both the mathematical counterexample and its Lean proof. It builds a nil algebra from three weighted backward shifts over $\\overline{\\mathbb F_2}$, arranges a universal mortality property for all elements, and simultaneously constructs a $2\\times2$ matrix over the resulting nil ideal that has a nonzero eigenvalue and hence is not nilpotent. Claude was later used to generate repository documentation from the completed proof; it was not the mathematical solver.","verification":"lean-verified","verificationNote":"No independent mathematician has checked this yet. Lean-verified on this site's ladder: kernel-checked, and the statement was written independently of the prover. Checked here on 5 September 2026 from a clone at b052755: 3,331 lines of Lean, zero `sorry` outside the Challenge.lean stub, zero `axiom` declarations, no `native_decide`, `unsafe` or `implemented_by` (two uses of `decide` on small numerals), mathlib pinned, Comparator in CI with the three standard axioms. The compared statement is byte-identical to Formal Conjectures' `KotherConjecture.variants.general_matrix` at 9cbe1d3c, and its one nontrivial ingredient, mathlib's `TwoSidedIdeal.matrix`, is the ideal of matrices whose every entry lies in I, so the statement is Krempa's matrix form as intended. What the kernel has certified is therefore: a ring with a nil ideal I and a non-nilpotent matrix in M_2(I). That Köthe's original statement implies the matrix form is an elementary argument (M_n(I) is a sum of n nil column left ideals) stated in the repository and not formalized; Formal Conjectures opened a PR on 4 September relating the formulations. Candidate because no ring theorist has read it yet, and the machine-generated proof account is unaudited.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"Köthe's conjecture (1930) is the central open problem of noncommutative ring theory: whether the sum of two nil left ideals is nil, equivalently whether every ring has a largest nil ideal. It has a Wikipedia article, a ninety-six-year literature, and a web of equivalent forms due to Krempa (1972) and others; Amitsur settled it over uncountable fields and Smoktunowicz's refutation of the related Amitsur conjecture (2000) is a landmark of the subject. Level with the non-sofic groups result here, which likewise answered a decades-old structural question in the negative.","solveCostUsd":432,"solveCostNote":null,"sourceUrl":"https://github.com/tadamcz/koethe","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"X Announcement","url":"https://x.com/tmkadamcz/status/2095785941193404457","kind":"discussion"},{"label":"Challenge.lean: the compared statement, copied from Formal Conjectures","url":"https://github.com/tadamcz/koethe/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Solution.lean and the Lean development","url":"https://github.com/tadamcz/koethe/blob/main/Solution.lean","kind":"lean-proof"},{"label":"Formal Conjectures source statement (Google DeepMind)","url":"https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/Koethe.lean","kind":"lean-statement"},{"label":"Epoch AI LeanOpenProblems, the evaluation the run was part of","url":"https://github.com/epoch-research/LeanOpenProblems","kind":"other"}],"relations":[],"submittedBy":"VibeGene","upvotes":4,"downvotes":0,"commentCount":1},{"slug":"prime-gaps-at-most-186","name":"Prime Gaps at Most 186","shortName":"Prime Gaps at Most 186","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"For the sequence of primes $p_n$, the project derives\n$$\\liminf_{n\\to\\infty}(p_{n+1}-p_n)\\le186.$$\nMore precisely, assuming three explicit analytic/numerical inputs, it proves $\\mathrm{DHL}[40,2]$: every admissible set of forty integer shifts has infinitely many translates containing at least two primes. Applying this to an explicit admissible $40$-tuple of diameter $186$ yields infinitely many consecutive prime gaps of size at most $186$. The Lean development verifies the deduction from the stated inputs; the two Kloosterman-type estimates and the finite physical-integral/cap bounds remain external assumptions.","posedBy":"Alphonse de Polignac (the twin prime conjecture); the bounded form since Goldston, Pintz and Yıldırım","yearPosed":1849,"ageNote":"Dated from Polignac's 1849 conjecture, the origin of the question this bounds. The modern quantitative form dates from Goldston, Pintz and Yıldırım in 2005 and became a finite bound with Zhang in 2013.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Assuming three explicit input statements, the project proves $\\mathrm{DHL}[40,2]$: every admissible $40$-tuple contains at least two primes infinitely often after translation. An explicit admissible $40$-tuple of diameter $186$ then gives\n$$\\liminf_{n\\to\\infty}(p_{n+1}-p_n)\\le186.$$\nThe Lean proof of the implication from the three inputs to the final theorem is kernel-checked. Two inputs are Kloosterman-type estimates cited to Katz/Deligne and Fouvry--Kowalski--Michel; the third consists of finitely many numerical integral and cap inequalities backed by a Python/FLINT certificate.","claimIssueNote":null,"solveDate":"2026-09-03","model":"GPT 6 Astra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The project metadata identifies GPT 6 Astra, operating through Codex, as the agent used for the formalization workflow. Its stated task was to formalize the source statements faithfully, simplify the proofs, and retain unresolved finite-field and numerical inputs as explicit axioms. The resulting development was refined through human-guided edits and automated proof checks. The repository metadata attributes both the project and its source article “Improved Gaps Between Primes” to OpenAI.","verification":"lean-checked","verificationNote":"Lean-checked, not Lean-verified, and the distinction is the whole of it. The Lean development in openai/PrimeGaps186 reports zero sorry in its three main declarations, and Comparator and Lean's kernel accept them - but all three depend on three project-specific axioms: a rank-three hyper-Kloosterman bound, a rank-two Kloosterman correlation bound, and a package of 104 outer, 45 inner and 3 cap numerical inequalities. So the kernel has checked that those three statements imply DHL[40,2] and the bound; it has not checked them. The first two are tied to established literature and the third is recomputed by a Python and FLINT certificate that does not discharge its Lean axiom. No independent expert has read the paper on the record.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"The closest mathematics has come to the twin prime conjecture, and the most-watched number in analytic number theory since Zhang broke it open in 2013. If this bound holds it is the current record, ahead of Axiom Math's 212 and Stadlmann's 240, both of which appeared after this paper was dated.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/openai/PrimeGaps186","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"Lean proof","url":"https://github.com/openai/PrimeGaps186/blob/main/PrimeGaps186.lean","kind":"lean-proof"},{"label":"Challenge.lean, the statements the comparator checks","url":"https://github.com/openai/PrimeGaps186/blob/main/Challenge.lean","kind":"lean-proof"},{"label":"Formalization metadata","url":"https://github.com/openai/PrimeGaps186/blob/main/formalization.yaml","kind":"lean-proof"},{"label":"Numerical certificate","url":"https://github.com/openai/PrimeGaps186/blob/main/prime_gap_186_certificate.py","kind":"code"},{"label":"Preprint","url":"https://github.com/openai/PrimeGaps186/blob/main/short_gaps_numerics.pdf","kind":"paper"},{"label":"Improved short gaps between primes (OpenAI, 30 August 2026)","url":"https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf","kind":"paper"},{"label":"Stadlmann's 240, the bound this improves on","url":"https://arxiv.org/abs/2608.31126","kind":"independent"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"conway-s-refinement-conjecture-for-omnific-integers","name":"Conway's Refinement Conjecture for Omnific Integers","shortName":"Conway's Refinement Conjecture","problemNumber":null,"field":"Surreal numbers","fieldGroup":"Algebra","statement":"If $a,b,c,d\\in\\mathbf{Oz}$ are omnific integers and\n$$\nab=cd,\n$$\nthen there exist omnific integers $e,f,g,h\\in\\mathbf{Oz}$ such that\n$$\na=ef,\\qquad b=gh,\\qquad c=eg,\\qquad d=fh.\n$$\nEquivalently, every equality of two products in the omnific integers admits a common four-factor refinement.","posedBy":"John H. Conway","yearPosed":1976,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"The repository gives a Lean proof of Conway's 1976 refinement conjecture for omnific integers: whenever\n$$\nab=cd\n$$\nwith $a,b,c,d\\in\\mathbf{Oz}$, there exist $e,f,g,h\\in\\mathbf{Oz}$ satisfying\n$$\na=ef,\\quad b=gh,\\quad c=eg,\\quad d=fh.\n$$\nThe proof is formalized twice: once using CombinatorialGames' surreal-number implementation and once with the required surreal definitions inlined over Mathlib. The development also proves stronger structural results about factorization in Hahn-series integer parts and related generalized power-series rings.","claimIssueNote":null,"solveDate":"2026-09-03","model":"ChatGPT; Claude","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Dan Abramov"],"aiRole":"Dan Abramov describes the project as an AI-proof experiment. Claude first selected Conway's refinement conjecture as a promising open problem after noting recent progress by L'Innocente and Mantova. Over roughly a month, Abramov repeatedly used ChatGPT and Claude to generate mathematical ideas and then steered the systems into producing Lean proofs, using compilation and formal checking to reject invalid directions. The repository therefore attributes the mathematical proof search and much of the formal proof construction to interactive AI exploration under human steering.","verification":"site-confirmed","verificationNote":"Site-confirmed: rebuilt here on 4 September 2026, not taken from the repository's CI. This site's verify-lean workflow checked out gaearon/conway-refinement at commit 264445c9, the commit the entry cites, installed the toolchain it pins (leanprover/lean4:v4.31.0) with its CombinatorialGames dependency, ran $\\texttt{lake build}$ over every module (3145 jobs), then the project's own scripts/Axioms.lean, then $\\texttt{lake env leanchecker}$ replaying the whole ConwayRefinement environment. 37 minutes, every step green.\n\nBoth formulations of the conjecture - the one over the CombinatorialGames $\\texttt{Surreal}$ type and the Mathlib-only one with the surreal definitions inlined - report $\\texttt{propext}$, $\\texttt{Classical.choice}$ and $\\texttt{Quot.sound}$ and nothing else. No $\\texttt{sorryAx}$, no project axiom.\n\nStill Candidate rather than Resolved, and the reason is the one the author gives himself: a kernel checks the proof against the statement as written, and whether that statement is Conway's conjecture is a reading a surreal-number specialist has to do. The definition used is Conway's own cut $x = \\{x - 1 \\mid x + 1\\}$ and the statement is a few lines, so it is an afternoon's work for the right reader. Nobody without a stake has done it yet.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"Conway posed it in On Numbers and Games in 1976 and it stood for fifty years, with partial progress by L'Innocente and Mantova. Well known to everyone who works with surreal numbers and to few others, which puts it in the same band as Crouzeix and the Lonely Runner at 30, field-famous problems with a long literature that never crossed over. A touch below them because the surreal-number community is smaller. Above the Kawauchi mod-4 entry at 20, a named conjecture with a thinner trail.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/gaearon/conway-refinement","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"CombinatorialGames statement of the conjecture","url":"https://github.com/gaearon/conway-refinement/blob/main/ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinement.lean","kind":"lean-statement"},{"label":"Proof of that statement","url":"https://github.com/gaearon/conway-refinement/blob/main/ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinementProof.lean","kind":"lean-proof"},{"label":"Mathlib-only statement, no CombinatorialGames dependency","url":"https://github.com/gaearon/conway-refinement/blob/main/ConwayRefinement/Standalone/Mathlib/InlineConwayRefinement.lean","kind":"lean-statement"},{"label":"Palomar provenance and the comparator statement","url":"https://github.com/gaearon/conway-refinement/blob/main/PALOMAR-PROVENANCE.md","kind":"palomar"},{"label":"Rebuilt here: lake build, axiom audit and leanchecker replay","url":"https://github.com/mrconter1/vibemathed/actions/runs/33843993855","kind":"independent"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"smale-s-mean-value-conjecture-k-1","name":"Smale’s Mean Value Conjecture ($K=1$)","shortName":"Smale’s Mean Value","problemNumber":null,"field":"Complex polynomials","fieldGroup":"Analysis","statement":"Smale conjectured that in his mean value theorem for complex polynomials, the universal constant $4$ could be replaced by $1$. Equivalently, for every complex polynomial $p$ of degree at least $2$ and every $z\\in\\mathbb C$, there should exist a critical point $c$ of $p$ such that\n$$\n\\frac{|p(z)-p(c)|}{|z-c|}\\le |p'(z)|.\n$$\nThe conjecture is false: there exists a polynomial $p$ with $p(0)=0$ and $p'(0)=1$ such that\n$$\n\\left|\\frac{p(c)}{c}\\right|>1\n$$\nfor every critical point $c$.","posedBy":"Stephen Smale","yearPosed":1981,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The formal proof constructs a complex polynomial $p$ such that\n$$\np(0)=0,\\qquad p'(0)=1,\n$$\nand for every critical point $c$ of $p$,\n$$\n\\left|\\frac{p(c)}{c}\\right|>1.\n$$\nThus at $z=0$ there is no critical point satisfying\n$$\n\\frac{|p(0)-p(c)|}{|c|}\\le |p'(0)|=1,\n$$\nwhich disproves Smale's conjectured universal constant $K=1$.\n\nThe counterexample has very large unspecified degree and violates the bound only by a small margin, so it is consistent with Smale's original $K=4$ theorem, the known low-degree positive cases, and previous asymptotic improvements toward $1$.","claimIssueNote":null,"solveDate":"2026-09-03","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"A pre-release GPT-6 Astra autonomously attempted the Formal Conjectures statement in Epoch AI's LeanOpenProblems evaluation. No human saw or steered the proof search. Astra found a new counterexample construction and wrote the Lean proof. The proof builds a holomorphic model on a nonconvex compact set, approximates it polynomially, and adds a high-power perturbation that forces all critical points into a region where the $K=1$ inequality fails. Claude was later used to write repository documentation from the completed run; it was not the mathematical solver.","verification":"lean-verified","verificationNote":"No independent mathematician has checked this yet. Lean-verified on this site's ladder: kernel-checked, and the statement was written independently of the prover. Checked here on 5 September 2026 from a clone of the repository at f99ab98: 2,266 lines of Lean across the development, zero `sorry` outside the Challenge.lean stub, zero `axiom` declarations, no `native_decide`, `unsafe` or `implemented_by`; the mathlib revision is pinned; CI runs Comparator against the trusted statement allowing only propext, Quot.sound and Classical.choice. The compared statement is byte-identical to Formal Conjectures' `mean_value_problem` at commit 9cbe1d3c, which is Smale's K = 1 form exactly: the quantified K is unused there, and the division convention in Lean (x/0 = 0) only makes the conjecture easier to satisfy, so it cannot help a disproof. Candidate rather than resolved because, two days after the run, no mathematician outside it has read the proof; the witness is a nonconstructive limiting perturbation of large unspecified degree and violates the bound by a small margin, which is consistent with everything known. The proof account was machine-generated and is unaudited.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Smale posed the mean value conjecture in 1981 alongside his proof with constant 4, and listed the problem among his eighteen problems for the twenty-first century. It has a Wikipedia article, a forty-five-year literature lowering the constant toward 1 (Beardon-Minda-Ng, Fujikawa-Sugawa and others), and was verified for degree at most 10 by Sendov and Marinov. A named conjecture of that standing, from a Fields medallist's problem list, sits at the Connes-rigidity level here; below the Jacobian conjecture, which spans far more of mathematics.","solveCostUsd":265,"solveCostNote":null,"sourceUrl":"https://github.com/tadamcz/mean-value-problem","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"X Announcement","url":"https://x.com/tmkadamcz/status/2095785941193404457","kind":"discussion"},{"label":"Challenge.lean: the compared statement, copied from Formal Conjectures","url":"https://github.com/tadamcz/mean-value-problem/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Solution.lean and the Lean development","url":"https://github.com/tadamcz/mean-value-problem/blob/main/Solution.lean","kind":"lean-proof"},{"label":"Formal Conjectures source statement (Google DeepMind)","url":"https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/MeanValueProblem.lean","kind":"lean-statement"},{"label":"Epoch AI LeanOpenProblems, the evaluation the run was part of","url":"https://github.com/epoch-research/LeanOpenProblems","kind":"other"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"absence-of-critical-bernoulli-bond-percolation-on-z-d-in-every-dimension-d-2","name":"Absence of critical Bernoulli bond percolation on $\\mathbb Z^d$ in every dimension $d \\ge 2$","shortName":"Critical percolation: θ(p_c) = 0","problemNumber":null,"field":"Percolation theory; mathematical statistical mechanics","fieldGroup":"Probability & statistics","statement":"For nearest-neighbour Bernoulli bond percolation on $\\mathbb Z^d$, let $\\theta(p)$ be the probability that the open cluster of the origin is infinite, and let $p_c$ be the critical parameter. Is $\\theta(p_c) = 0$ for every integer $d \\ge 2$?","posedBy":"Harry Kesten, Percolation Theory for Mathematicians (1982), which is where Duminil-Copin's survey traces it; restated as Conjecture 1 of his 2018 ICM survey","yearPosed":1982,"ageNote":"Dated from Kesten's 1982 book, where Duminil-Copin's survey traces the conjecture. Kesten settled d = 2 in 1980 using Harris's 1960 proof that θ(1/2) = 0 on Z², so the other dimensions are the problem. Cost, as the repository discloses it: about a week of wall time in August 2026 on cloud CPU machines for Lean elaboration plus API inference, spend not tracked.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims $\\theta(p_c) = 0$ for every $d \\ge 2$, by proving Kozma-Nitzan Conjecture 3 and its reduction to critical percolation. The previously open dimensions $3 \\le d \\le 10$ are included. The formal statement concerns vanishing at $p_c$; continuity of $\\theta$ on the whole interval is a classical consequence rather than the formal target. No claim is intended concerning site percolation or other lattices. A stronger claim, no infinite cluster at p_c for bond percolation on every infinite connected locally finite quasi-transitive graph with p_c < 1 (which includes Z^d for every d >= 2), appeared in OpenAI's math release on 24 September 2026 with a Lean challenge; see benjamini-schramm-criticality-bond.","claimIssueNote":null,"solveDate":"2026-09-03","model":"Anthropic Claude models (specific version undisclosed)","modelMaker":"Anthropic","humanCollaborators":["Justin Leder"],"aiRole":"The repository's own provenance section: the Lean sources, \"definitions, statements and proofs, together with Challenge.lean, Solution.lean and the metadata\", \"were written by an AI system (Anthropic's Claude models) working autonomously under the direction of Justin Leder; no human wrote or edited the Lean code.\" Its formalization.yaml records the split: \"Discovery, informal proof and Lean formalization were all produced by the AI system operating autonomously. Human role (Justin Leder): problem selection, direction, reading of the statement file and metadata, and responsibility for this submission\", the author having \"posed the problem, set the acceptance standard (kernel-checked proof with the standard axioms, plus adversarial review of the statements) and directed priorities.\" The same file states that the only review so far was by AI systems, adversarial reads of the formal statements and of the proof chain against the cited literature, and the README asks readers to satisfy themselves that Challenge.lean states the intended theorem rather than relying on the kernel for that - which is the audit recorded under verification here.","verification":"lean-verified","verificationNote":"No independent mathematician has checked this yet. Formalised in Lean 4 in anthropics/formal-math at commit 795efb8. Audited here on 8 September 2026 at four levels. Statement: Challenge.lean read against Grimmett §§1.3-1.4 - the lattice is SimpleGraph.hasse on Fin d → ℤ, whose symmetrised covering relation is nearest-neighbour adjacency; the measure is ProbabilityTheory.setBernoulli on the edge set, whose Mathlib definition was read at source and does mean each edge open independently with probability p; theta is the measure of {|C(0)| = ∞}; p_c is inf {p | theta p > 0} with the empty-case convention declared. Structure: percolation_continuity takes only a dimension and 2 ≤ d, the transport is Iff.rfl, and comparator.json compares exactly the two theorem names permitting only propext, Quot.sound and Classical.choice - so no unproven input can enter as a theorem argument. Counts taken independently across all 251 files and 97,574 lines: no sorry outside the two deliberate placeholders, zero axiom declarations, zero native_decide, zero unsafe. Percolation/Literature/ formalises the classical toolkit rather than assuming it. Not rebuilt here: their run needed a 128-core node and Mathlib from source, so kernel acceptance rests on their audit record, which matched every count I checked. Gil Kalai reported it on 3 September with explicit caveats, writing \"if verified, this is a remarkable breakthrough\" and \"we still need to verify if the formalisation is done correctly\".","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":78,"significanceNote":"Conjecture 1 of Duminil-Copin's 2018 ICM survey \"Sixty years of percolation\", which traces it to Kesten's 1982 book. Settled for d = 2 by Kesten in 1980 with Harris's 1960 planar estimate, and in high dimensions by lace expansion since Hara-Slade 1990 (d ≥ 19), pushed to d ≥ 11 by Fitzner and van der Hofstad in 2015; dimensions 3 to 10 held out, reached by neither planar duality nor the lace expansion. Resolved for every d ≥ 2 with no extra hypothesis. Just below the Collatz band: a central problem of probability settled completely, held back only because no human has read the argument.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/anthropics/formal-math/tree/795efb86f191735c5481675763537cfb4ff37e55/percolation","sourceName":"anthropics/formal-math, percolation (Lean 4 development, pinned commit 795efb8)","collection":null,"collectionVersion":null,"links":[{"label":"Challenge.lean: the trusted statement, in plain Mathlib","url":"https://github.com/anthropics/formal-math/blob/795efb86f191735c5481675763537cfb4ff37e55/percolation/Challenge.lean","kind":"lean-statement"},{"label":"AUDIT.md: their build, axiom and comparator record","url":"https://github.com/anthropics/formal-math/blob/795efb86f191735c5481675763537cfb4ff37e55/percolation/AUDIT.md","kind":"lean-proof"},{"label":"Gil Kalai's report, with his caveats","url":"https://gilkalai.wordpress.com/2026/09/03/amazing-there-is-no-percolation-at-the-critical-probability-in-all-dimensions-solved-by-ai-via-a-conjecture-of-gady-kozma-and-shahaf-nitzan/","kind":"discussion"},{"label":"summary.pdf: the repository's own guide to the proof","url":"https://github.com/anthropics/formal-math/blob/795efb86f191735c5481675763537cfb4ff37e55/percolation/summary.pdf","kind":"paper"},{"label":"formalization.yaml: provenance, cost and disclosed divergences","url":"https://github.com/anthropics/formal-math/blob/795efb86f191735c5481675763537cfb4ff37e55/percolation/formalization.yaml","kind":"other"},{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/benjamini-schramm-criticality-bond","kind":"other"},{"label":"Site percolation on Z^d, d >= 3: extension of this proof (Bou-Rabee)","url":"https://vibemathed.com/problem/critical-site-percolation-z-d","kind":"other"}],"relations":[],"submittedBy":"BoldPanther302","upvotes":5,"downvotes":0,"commentCount":0},{"slug":"the-proper-hat-guessing-number-of-k-6-e","name":"The proper hat-guessing number of $K_6-e$","shortName":"$\\mathrm{HG}_P(K_6-e)=10$","problemNumber":null,"field":"Graph theory; hat-guessing games; permutation groups","fieldGroup":"Combinatorics","statement":"Determine the exact proper hat-guessing number of the complete graph on six vertices with one edge removed. The known general bounds leave $\\mathrm{HG}_P(K_6-e)\\in\\{9,10\\}$.","posedBy":"Adriaensen et al.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"We prove $\\mathrm{HG}_P(K_6-e)=10$. The lower bound uses two order-sensitive twin-player rules obtained by deleting and repairing one point of an explicit sharply four-transitive eleven-point permutation group. On every coordinate line the repaired rules are derangement permutations, are pointwise unequal, and have fixed-point-free composition. Hall's theorem completes the strategy on the four clique vertices. The release also classifies all repairable orbit labels and proves an even-$n$ obstruction for set-symmetric line-permutation twin rules. It does not solve the general $K_n-e$ family.","claimIssueNote":null,"solveDate":"2026-09-03","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"OpenAI GPT-5.6 Pro contributed substantively to literature search, target selection, construction search, proof development, code generation, exact verification, adversarial critique, and manuscript preparation. Matthew Protti selected and framed the target, directed and evaluated the work, required exact checks, determined the claim scope, approved disclosure, and accepts responsibility.","verification":"unreviewed","verificationNote":"A separate adversarial technical review independently regenerated the principal finite core and prompted the scope and proof-presentation corrections incorporated into this public version. The dependency-free verifier regenerates the group, repairs, coordinate-line checks, label classification, Witt-design check, and residual right-degree census. Conventional journal peer review and Lean verification are not claimed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"One value of one graph parameter, from the question Adriaensen et al. posed this year, where their bound HG_P(G) <= n + chi(G) - 1 already gave 10 as the ceiling. Narrow by construction and recent. Tied at 7 with the K5 - e entry it extends, which is the same author, the same method and the same disclosure format one vertex smaller.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/8f36d8c82bea951ae2f0147ebc2e66ea2bd6432e","sourceName":"Reviewed public research disclosure v0.1 (GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"Four-page preprint manuscript source","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/blob/8f36d8c82bea951ae2f0147ebc2e66ea2bd6432e/research/proper_hat_guessing_k6_minus_e/preprint/proper_hat_guessing_K6_minus_e_v0.1.md","kind":"paper"},{"label":"Verifier, certificate and disclosure package","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/8f36d8c82bea951ae2f0147ebc2e66ea2bd6432e/research/proper_hat_guessing_k6_minus_e","kind":"code"},{"label":"Public draft PR #1 and review record","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/pull/1","kind":"announcement"},{"label":"Hat guessing with proper colorings (Adriaensen et al.)","url":"https://arxiv.org/abs/2603.04909","kind":"problem-record"}],"relations":[],"submittedBy":"Matthew Protti","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"finite-time-blowup-for-the-3d-incompressible-euler-equations-unforced","name":"Finite-time blowup for the 3D incompressible Euler equations from smooth data","shortName":"Euler blowup, unforced","problemNumber":null,"field":"Fluid dynamics; singularity formation for incompressible flow","fieldGroup":"Differential equations","statement":"Can a solution of the three-dimensional incompressible Euler equations on $\\mathbb R^3$, with no external force, started from smooth, compactly supported, divergence-free initial data, lose regularity in finite time? Local existence is classical, and by Beale-Kato-Majda blowup at time $T_*$ is equivalent to $\\int_0^{T_*}\\|\\omega(t)\\|_\\infty\\,dt=\\infty$. Hou and Luo's 2014 numerics suggested a boundary singularity, Elgindi proved blowup in 2021 for $C^{1,\\alpha}$ data with small $\\alpha$, Córdoba, Martínez-Zoroa and Zheng reached finite-energy $C^{1,\\alpha}$ data, and in September 2026 Buckmaster and Alpöge obtained blowup with a smooth force. Does it happen for smooth data with no force at all?","posedBy":"Leon Lichtenstein (1925) and Nikolai Gunther (1927), whose local existence left global regularity open; the smooth unforced case is the Euler singularity problem of the fluid-dynamics literature","yearPosed":1925,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Yes, if the argument stands. Theorem 1.1: there is u_0 in C^infinity_c with div u_0 = 0 whose smooth Euler solution has maximal lifespan 0 < T* < infinity, with limsup of the C^1 norm infinite and the time integral of the vorticity supremum divergent, so the blowup is genuine by Beale-Kato-Majda. The construction, per the paper's own section headings, places a localised oscillation over a smooth Euler flow, amplifies and transfers its wave geometry, chooses scales and iterates, and passes to a limiting datum; 57 pages. It supersedes the forced Euler result of Buckmaster and Alpöge in the catalog and is the step on which OpenAI's Navier-Stokes agents were then prompted. It says nothing about Navier-Stokes without forcing.","claimIssueNote":null,"solveDate":"2026-09-03","model":"Unnamed internal OpenAI model; GPT-6 Astra for the Lean formalisation","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"From OpenAI's announcement of 8 September (read in the Wayback copy, the live page refuses automated readers): the effort began on 1 September with groups of agents powered by an internal model \"significantly more capable than GPT-6 Astra\", each group prompted with a variant of a Millennium problem or a related question. Asked the Euler regularity question as one of the \"easier\" problems, \"nearly 100 agents worked together for approximately 50 hours to produce our Euler regularity disproof\", specifically the unforced version. That solution was then handed to the Navier-Stokes agents, whose group grew to about 10,000 concurrent agents and reached the forced Navier-Stokes result about 88 hours after launch. Lean formalisation and verification of the results took a further 17 hours via GPT-6 Astra. OpenAI's 10 September update to the post reports an investigation concluding that Buckmaster's Codex prompts over the preceding two months \"could not have influenced the system in any way, including through training\", and notes that the Euler proofs differ: Alpöge and Buckmaster's has an external force, this one has none.","verification":"lean-verified","verificationNote":"No independent mathematician has checked this yet. Audited here on 22 September 2026 the way the Navier-Stokes entry was. Statement anchor: ComparatorChallenges/Euler.lean in openai/NavierStokesAndEuler is DeepMind's formal-conjectures Navier-Stokes statement at commit 8bf45ed with viscosity and force set to zero; InitialVelocityConditionDecay, the evolution equation with derivWithin on [0, inf), div_free, initial_condition, joint smoothness, MemLp and the uniformly bounded energy were compared line by line with DeepMind's file and match. The first theorem, euler_breakdown_R3, is that breakdown alternative; the second, exists_compact_smooth_euler_singularity, is OpenAI's own richer form of the paper's Theorem 1.1 with compact support, a finite maximal lifespan T* <= 1, a C^1 limsup of infinity and a divergent vorticity integral. Euler/Solution.lean restates both verbatim and ends with #print axioms; Euler.json permits only propext, Quot.sound and Classical.choice and enables the nanoda kernel. Counted here on the tarball: 1,839 Lean files and 211,578 lines under Euler/, zero sorry, zero axiom declarations, zero native_decide, zero unsafe. Lean 4.34.0-rc2 pinned. Not rebuilt here, and the repository has no CI of its own; the comparator replay is documented for anyone with the machine. No mathematician has read the 57-page argument.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":80,"significanceNote":"The Euler singularity problem itself: whether smooth, force-free, finite-energy ideal flow can break down, the question Hou-Luo's numerics and Elgindi's C^{1,alpha} theorem circled and the one Terence Tao's 7 September post said should be reachable \"without the forcing term\". Above the forced Euler entry at 70, whose one gap this closes; below the Navier-Stokes entry at 87 because this is not a Millennium problem and viscosity is the harder half; held at Candidate until a human has read it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf","sourceName":"OpenAI, Finite time blowup for the Euler equation (8 September 2026), with the Lean release in openai/NavierStokesAndEuler","collection":null,"collectionVersion":null,"links":[{"label":"OpenAI's comparator challenge statement for unforced Euler","url":"https://github.com/openai/NavierStokesAndEuler/blob/main/ComparatorChallenges/Euler.lean","kind":"lean-statement"},{"label":"Formal Conjectures' Navier-Stokes statement it specialises","url":"https://github.com/google-deepmind/formal-conjectures/blob/8bf45ed70d48b2b2a501de9c00b26bfa38c573ee/FormalConjectures/Millenium/NavierStokes.lean","kind":"problem-record"},{"label":"The Lean development and comparator configuration","url":"https://github.com/openai/NavierStokesAndEuler","kind":"lean-proof"},{"label":"OpenAI's announcement (process account and 10 September update)","url":"https://openai.com/index/navier-stokes-solution/","kind":"announcement"},{"label":"Euler equations (fluid dynamics)","url":"https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)","kind":"wikipedia"}],"relations":[],"submittedBy":"LucidManta102","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-proper-hat-guessing-number-of-k-7-e","name":"The proper hat-guessing number of $K_7-e$","shortName":"$\\mathrm{HG}_P(K_7-e)=12$","problemNumber":null,"field":"Graph theory; hat-guessing games; Steiner systems; permutation groups","fieldGroup":"Combinatorics","statement":"Determine the exact proper hat-guessing number of the complete graph on seven vertices with one edge removed. The general bounds leave $\\mathrm{HG}_P(K_7-e)\\in\\{11,12\\}$.","posedBy":"Adriaensen et al.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"We prove $\\mathrm{HG}_P(K_7-e)=12$. One lower-bound proof uses two explicit block-disjoint $S(5,6,12)$ Witt designs. A second uses orbit maps from an explicitly regenerated sharply five-transitive twelve-point permutation group, combining one set-symmetric and one order-sensitive rule. Both satisfy a general coordinate-line twin-completion criterion, and Hall's theorem completes the clique strategy. The release also proves a disjoint completion-design theorem, an even-$n$ obstruction scoped to set-symmetric line-permutation twins in this sufficient framework, and a prime-admissibility theorem for the design parameters. It does not solve the general $K_n-e$ problem or $K_8-e$.","claimIssueNote":null,"solveDate":"2026-09-03","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"OpenAI GPT-5.6 Pro contributed substantively to literature search, target selection, construction search, proof development, code generation, exact verification, adversarial critique, and manuscript preparation. The Witt-design and orbit-map proof architecture was developed in a human-directed model-assisted research process. Matthew Protti selected and framed the target, directed the research programme, evaluated candidate arguments, commissioned independent adversarial review, required exact checks, approved the public scope, and accepts responsibility.","verification":"unreviewed","verificationNote":"Unreviewed on this site's ladder, but better evidenced than either sibling. An independent adversarial review rebuilt both 132-block S(5,6,12) completion designs, all 792 pentad completions, design disjointness, the 95,040-element group, sharp five-transitivity, all 495 set lines, all 59,400 ordered lines and both Hall-degree censuses, returning ACCEPT_K7E_THEOREM with no mathematical repairs required; STATUS.json pins the review package by SHA-256. That is a real check by someone other than the author, but it is an adversarial review of an artefact rather than a named expert endorsing the theorem or a proof assistant checking it, and this site has not rebuilt it. Conventional peer review is not claimed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"One value of one graph parameter, third in a series after K5 - e and K6 - e from the same author on the two preceding days. The ceiling comes from Adriaensen et al.'s bound n + chi(G) - 1 = 12 and the work shows it is attained. Level with both siblings deliberately: the same method one vertex larger does not gain weight per rung, and the question was posed this year with no accumulated literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/commit/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08","sourceName":"Reviewed K7-e public research disclosure v0.1 (GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"Four-page K7-e preprint source","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/blob/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08/research/proper_hat_guessing_k7_minus_e/preprint/proper_hat_guessing_K7_minus_e_v0.1.tex","kind":"paper"},{"label":"Verifier, certificates and public disclosure","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/tree/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08/research/proper_hat_guessing_k7_minus_e","kind":"code"},{"label":"Independent adversarial review acceptance","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/blob/39f7716c2d8f4e9a84963c45fd7b8b256bf7ac08/research/proper_hat_guessing_k7_minus_e/REVIEW/INDEPENDENT_ADVERSARIAL_REVIEW_ACCEPTANCE_20260904.md","kind":"other"},{"label":"Public draft PR #2 and release record","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/pull/2","kind":"announcement"},{"label":"Hat guessing with proper colorings (Adriaensen et al.)","url":"https://arxiv.org/abs/2603.04909","kind":"problem-record"}],"relations":[],"submittedBy":"Matthew Protti","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"improved-maximal-prime-gap-lower-bound","name":"Improved maximal prime-gap lower bound","shortName":"Maximal prime-gap lower bound","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"Let $G(X)$ denote the largest gap between consecutive primes not exceeding $X$, and let $\\log_j$ denote the $j$-fold iterated logarithm. The paper proves that, for all sufficiently large $X$,\n$$\nG(X)\\gg \\frac{\\log X\\,(\\log_2 X)^2\\,\\log_4 X}{(\\log_3 X)^2}.\n$$\nEquivalently, there is an absolute constant $c>0$ such that $G(X)$ is at least $c$ times the quantity above for all sufficiently large $X$. This improves Rankin's classical lower bound by a factor of $\\log_2 X$.","posedBy":"Paul Erdős","yearPosed":1955,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"The paper proves\n$$\nG(X)\\gg \\frac{\\log X\\,(\\log_2 X)^2\\,\\log_4 X}{(\\log_3 X)^2}\n$$\nfor all sufficiently large $X$. Its main new ingredient is a short-translates theorem: for any sufficiently small set $S\\subseteq[1,H]$ with $|S|\\le\\delta x$, one can find a short translate making every corresponding linear form composite. Combining this with an Erdős--Rankin covering argument produces prime-free intervals of the claimed length. This directly and asymptotically improves the August 2026 GPT-5.6 Sol bound\n$$\nG(X)\\gg\\frac{\\log X\\log_2 X}{\\log_4 X}\n$$\nby the unbounded factor\n$$\n\\frac{\\log_2 X(\\log_4 X)^2}{(\\log_3 X)^2}.\n$$","claimIssueNote":null,"solveDate":"2026-09-03","model":"GPT 6 Astra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The paper explicitly attributes the proof to GPT 6 Astra. The new argument introduces a short-translates proposition that allows a sparse residual set of positions to be made composite simultaneously. Its main construction uses specially chosen divisor-sum factors, a shared truncation of their product, and a nonnegative squared weight. The resulting proposition is then inserted into an Erdős--Rankin construction to obtain the improved maximal prime-gap bound.","verification":"site-confirmed","verificationNote":"Site-confirmed on 4 September 2026: this site built openai/LongGapsBetweenPrimes at commit 03a1190d from a clean checkout on GitHub Actions (run 33843996072). lake build completed all 8707 jobs, the build's own #print axioms line reads 'LongGapsBetweenPrimes.long_prime_gaps' depends on axioms: [propext, Classical.choice, Quot.sound], and leanchecker replayed the library through the kernel. The statement was read by hand: Challenge.lean asserts, for some c > 0 and all sufficiently large X, a consecutive prime gap below X exceeding c times log X (log_2 X)^2 log_4 X / (log_3 X)^2, which is the claimed bound and not a weaker cousin of it. What remains is what a kernel cannot see: there is no human author, and no named mathematician has commented yet.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"Erdős Problem #4, the quantitative long-gaps question, worked on since Rankin in 1938 and carrying Erdős's largest prize. Strictly stronger than the tilted residue-class result the site records at 60, and than the Ford-Green-Konyagin-Maynard-Tao Annals bound before it, by the unbounded factor log_2 X (log_4 X)^2 / (log_3 X)^2. Level with the tilted entry deliberately: same problem, second step on it in a fortnight, and a problem does not become more important because a stronger step followed. The prize condition, a power (log X)^(1+c), is still far off, which is why this is partial.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/long_gaps.pdf","sourceName":"OpenAI self-publication","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation, openai/LongGapsBetweenPrimes","url":"https://github.com/openai/LongGapsBetweenPrimes","kind":"lean-proof"},{"label":"Challenge.lean, the statement the comparator checks","url":"https://github.com/openai/LongGapsBetweenPrimes/blob/master/Challenge.lean","kind":"lean-statement"},{"label":"Abridged chain of thought","url":"https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/long_gaps_abridged_cot.pdf","kind":"transcript"},{"label":"Erdős Problem #4","url":"https://www.erdosproblems.com/4","kind":"problem-record"},{"label":"The tilted residue-class construction this improves","url":"https://vibemathed.com/problem/tilted-residue-class-construction-for-long-prime-free-intervals","kind":"other"}],"relations":[],"submittedBy":"VibeGene","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"a-quantum-oracle-separation-between-mathsf-qma-2-and-mathsf-qma","name":"A Quantum Oracle Separation Between $\\mathsf{QMA}(2)$ and $\\mathsf{QMA}$","shortName":"Quantum oracle separation of QMA(2) from QMA","problemNumber":null,"field":"Quantum complexity theory","fieldGroup":"Quantum information & computing","statement":"We find a quantum oracle relative to which $\\mathsf{QMA}\\neq\\mathsf{QMA}(2)$. As a consequence, we resolve the no-disentanglers conjecture of Watrous: for every $\\varepsilon+\\delta<1$, any $(\\varepsilon,\\delta)$-disentangler requires input size exponential in the number of output qubits.","posedBy":"John Watrous (the no-disentanglers conjecture, as reported by Aaronson, Beigi, Drucker, Fefferman and Shor)","yearPosed":2009,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The authors construct a unitary oracle $U$ such that\n$$\n\\mathsf{QMA}^{U}\\neq\\mathsf{QMA}(2)^{U}.\n$$\nTheir black-box problem is solvable by a $\\mathsf{QMA}(2)$ verifier with one oracle query and linear-size unentangled proofs, whereas any $\\mathsf{QMA}$ verifier must use either exponentially many queries or an exponentially large witness.\n\nAs a non-oracle consequence, they prove that for every fixed $\\varepsilon,\\delta\\ge 0$ with $\\varepsilon+\\delta<1$, any $(\\varepsilon,\\delta)$-disentangler requires exponentially many input qubits in the number of output qubits, resolving Watrous's no-disentanglers conjecture.","claimIssueNote":null,"solveDate":"2026-09-02","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["John Bostanci; Sabee Grewal; Jonas Haferkamp; Andrew Huang; Yeongwoo Hwang; Anand Natarajan; Chinmay Nirkhe"],"aiRole":"The authors explicitly state that the proof idea underlying the main theorem was generated using ChatGPT 5.6 Sol. They initially directed Sol to the unitary polynomial method of She and Yuen; with minimal further guidance, the model proposed the proof idea used in the paper. The authors then verified, simplified, and developed the argument. The key construction uses symmetric and antisymmetric subspace projectors to make the relevant local-unitary invariant polynomials collapse to a single univariate polynomial, reducing the lower bound to the approximate degree of $\\mathrm{OR}$.","verification":"unreviewed","verificationNote":"Unreviewed. A 25-page preprint two days old, no peer review, no formalisation. Seven authors, including several who work on exactly this, state that they \"verified, simplified, and developed\" the model's proof idea and take full responsibility. The argument reduces to the approximate degree of OR through the She-Yuen unitary polynomial method, so it is checkable by anyone who knows that toolkit. Nobody outside the author list has done so on the record.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":6,"renownNote":"The QMA article exists in six Wikipedia languages; the QMA versus QMA(2) question has no article of its own and is known through it.","significance":36,"significanceNote":"Whether two unentangled quantum proofs beat one, QMA versus QMA(2), has been one of the central open questions of quantum complexity since Aaronson, Beigi, Drucker, Fefferman and Shor posed it in 2009, and Watrous's no-disentanglers conjecture is its best-known sharpening. This gives the first oracle separation, relative to a unitary oracle, and settles the conjecture; the unrelativized and classical-oracle questions stay open. Just above the Werner two-copy distillability entry at 35, the catalog's top quantum entry, because this question is followed across the whole field.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.02865","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Aaronson, Beigi, Drucker, Fefferman and Shor, The Power of Unentanglement, where the question is posed","url":"https://arxiv.org/abs/0804.0802","kind":"problem-record"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"anari-s-bethe-permanent-conjecture","name":"Anari's Bethe permanent conjecture","shortName":"Anari's Bethe permanent","problemNumber":null,"field":"Permanent approximation","fieldGroup":"Combinatorics","statement":"For an $n\\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison\n$$\\operatorname{Bethe}(A) \\leq \\operatorname{per}(A) \\leq 2^{n/2}\\operatorname{Bethe}(A).$$\nThe lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \\geq 4$, then\n$$\\operatorname{Bethe}(A) \\leq \\operatorname{per}(A) \\leq 2^{2n/g}\\operatorname{Bethe}(A).$$\nThe upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.","posedBy":"Nima Anari","yearPosed":null,"ageNote":"No posed year. The conjecture is Anari's and arises from the Anari-Rezaei analysis of the Bethe permanent, but neither the paper nor the form dates it and guessing would be worse than a blank.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"For every nonnegative $n\\times n$ matrix $A$ whose bipartite support graph has girth at least an even integer $g\\ge4$, Dong and Jain prove the sharp inequality\n$$\\operatorname{Bethe}(A)\\le\\operatorname{per}(A)\\le2^{2n/g}\\operatorname{Bethe}(A).$$\nThe factor $2^{2n/g}$ is optimal whenever $g\\mid2n$, attained by matrices whose support graphs are disjoint unions of $g$-cycles. Thus the result confirms Anari's conjecture, recovers the sharp $2^{n/2}$ universal Anari--Rezaei bound when $g=4$, and approaches exactness as the support-graph girth tends to infinity.","claimIssueNote":null,"solveDate":"2026-09-02","model":"ChatGPT 5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Dingding Dong","Vishesh Jain"],"aiRole":"The authors had already developed a strategy proving a weaker girth-dependent bound of the form $\\exp(n/f(g))$, with $f(g)\\to\\infty$. GPT-5.6 Sol Pro developed this strategy into a proof with $f(g)=\\Theta(g/\\log g)$. Subsequent interactions with GPT-5.6 Sol Ultra, focused on understanding the source of the logarithmic loss and its relation to the Anari--Rezaei argument, led to the development of the final optimal $2^{2n/g}$ proof. Codex also assisted with manuscript preparation.","verification":"unreviewed","verificationNote":"Unreviewed. A 21-page preprint two days old, no peer review, no formal verification. The optimality half is checkable by anyone: the paper exhibits the equality cases explicitly. The upper-bound argument is conventional and the authors take responsibility for it. Nobody independent has read it on the record.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A named conjecture of Nima Anari, settled at the optimal order rather than improved: the girth-dependent factor 2^(2n/g) is attained by disjoint unions of g-cycles, so nothing is left. The Bethe approximation is a real tool in approximate counting, and the Gurvits and Anari-Rezaei bounds it refines are standard. Still one inequality in one corner of the field. Just below the PSD-permanent approximation entry at 28, which reaches a wider algorithms audience; above the Kim-Roush permanent entry at 20, which answers a narrower question.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.02017","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"interior-curvature-estimates-for-the-graphical-scalar-curvature-equation-in-all-","name":"Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions","shortName":"Graphical scalar-curvature interior estimates","problemNumber":null,"field":"Partial differential equations","fieldGroup":"Differential equations","statement":"We resolve the long-standing problem of establishing interior $C^{2}$ estimates for admissible solutions of the graphical scalar curvature equation in every dimension $n\\geq 3$. More precisely, we prove interior curvature estimates for admissible solutions to the constant graphical scalar curvature equation. The proof combines Jacobi inequalities with a two-surface maximum principle and a two-surface Pogorelov estimate.","posedBy":"Interior C^2 estimates for the σ_2 (scalar curvature) equation, open above dimension three since Heinz (n = 2) and Warren–Yuan (n = 3)","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"For every $n\\geq3$, let $u\\in C^\\infty(B_2)$ be an admissible solution of\n$$\n\\kappa[u]\\in\\Gamma_2,\\qquad \\sigma_2(\\kappa[u])=1,\n$$\nwith\n$$\n\\|u\\|_{L^\\infty(B_2)}+\\|Du\\|_{L^\\infty(B_2)}\\le K.\n$$\nThen\n$$\n\\sup_{B_{1/2}}|\\kappa[u]|\\le C(n,K).\n$$\n\nThus an admissible graph of constant scalar curvature with bounded height and slope cannot develop unbounded interior curvature, in any dimension $n\\ge3$. Prior unrestricted quantitative graphical estimates were known in dimension $3$; dimension $4$ had only an implicit estimate with extra dependence, and $n\\ge5$ required additional semiconvexity assumptions.","claimIssueNote":null,"solveDate":"2026-09-02","model":"ChatGPT (OpenAI, model version unstated)","modelMaker":"OpenAI","humanCollaborators":["Guohuan Qiu","Jin Yan"],"aiRole":"During development of the uniform-separation and two-surface Pogorelov arguments, the authors used ChatGPT to test candidate comparison and cutoff functions, perform preliminary calculations, and search for counterexamples to proposed differential inequalities. These explorations helped expose the obstruction to one-point comparisons: the two graphical hypersurfaces have mismatched tangent spaces and normals. This informed the replacement of the vertical-gap approach by an ambient-distance comparison and a two-point cutoff. The authors independently checked, corrected, and rewrote all AI-assisted calculations and made the final mathematical decisions.","verification":"unreviewed","verificationNote":"Unreviewed. arXiv 2609.02581 (38 pages) read here; the authors write that ChatGPT was used \"as an exploratory and computational aid to test candidate\" comparison and cutoff functions and search for counterexamples to proposed inequalities, and that they \"checked, corrected, and rewrote the AI-assisted calculations\" and verified every statement. Author-checked; not refereed; no formalization.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Interior curvature estimates for the graphical scalar curvature equation in all dimensions, a long-standing problem of fully nonlinear PDE: Heinz settled n = 2, Warren and Yuan n = 3 (2006), Shankar and Yuan the Hessian case in n = 4, and higher dimensions needed extra assumptions. Resolved on the full Γ_2 branch for every n >= 3. A well-known open problem in a central area of geometric analysis.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.02581","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-proper-hat-guessing-number-of-k-5-e","name":"The proper hat-guessing number of $K_5-e$","shortName":"$\\mathrm{HG}_P(K_5-e)=8$","problemNumber":null,"field":"Graph theory; hat-guessing games","fieldGroup":"Combinatorics","statement":"Determine the exact proper hat-guessing number of the complete graph on five vertices with one edge removed. The existing bounds left $\\mathrm{HG}_P(K_5-e)$ in $\\{7,8\\}$.","posedBy":"Adriaensen et al.","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"The exact value is $\\mathrm{HG}_P(K_5-e)=8$. The lower bound uses explicit legal twin-player rules over $\\mathbb F_2^3$. After those rules cover 3,024 of the 8,400 proper colorings, the residual-coloring/local-view incidence graph has left degree three and right degree at most three; Hall's theorem supplies consistent guesses for the three clique players. The release also proves a general sufficient twin-completion lemma for $K_n-e$. It does not solve the full $K_n-e$ family or determine $\\mathrm{HG}_P(C_5)$.","claimIssueNote":null,"solveDate":"2026-09-02","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"OpenAI GPT-5.6 Pro contributed substantively to literature search, problem selection, construction search, proof development, code generation, computational verification, adversarial critique, and manuscript preparation. The central $\\mathbb F_2^3$ construction and Hall-completion proof were developed in a model-assisted process. Matthew Protti selected and framed the target, directed and evaluated the work, required exact checks, set the scope, approved disclosure, and accepts responsibility.","verification":"unreviewed","verificationNote":"No independent mathematical review yet. The public disclosure contains a self-contained proof, a seven-entry finite certificate, and a dependency-free Python verifier. The verifier has been run successfully from a clean clone and from the v0.1 release archive; it reconstructs the residual incidence graph, a saturating matching, and a complete 6,720-entry strategy, then checks all 8,400 proper colorings. This remains author-controlled verification.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"One value of one graph parameter, from a question posed this year in Adriaensen et al., where the bounds already pinned it to 7 or 8. Narrow by construction and recent, with no accumulated literature. Tied with the automatic-complexity result at 7, which likewise settles a question posed in the paper that introduced it. Well below the Tu-Deng conjecture at 15, which has a decade of partial results behind it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e","sourceName":"The proper hat-guessing number of K5-e - public research disclosure v0.1","collection":null,"collectionVersion":null,"links":[{"label":"Tagged v0.1 release (ZIP and PDF)","url":"https://github.com/matthewprotti/proper-hat-guessing-k5-minus-e/releases/tag/v0.1","kind":"announcement"},{"label":"Source problem: Hat guessing with proper colorings","url":"https://arxiv.org/abs/2603.04909","kind":"problem-record"}],"relations":[],"submittedBy":"Matthew Protti","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"daykin-frankl-conjecture","name":"The Daykin–Frankl conjecture on convex subsets of the Boolean lattice","shortName":"Daykin-Frankl","problemNumber":null,"field":"Extremal set theory","fieldGroup":"Combinatorics","statement":"In 1983, Daykin and Frankl conjectured that if $P$ is a convex subset of $Q_n$, then it contains at least\n$$\n|P|\\binom{n}{\\lfloor n/2\\rfloor}2^{-n}\n$$\npairwise incomparable elements. We verify and communicate an LLM-generated proof of this conjecture.","posedBy":"David E. Daykin, Peter Frankl","yearPosed":1983,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Let $P\\subseteq Q_n$ be convex. Williams proves the stronger statement that for every $k\\ge0$,\n$$\nw(P\\times Q_k)\n\\ge\nw(Q_{n+k})\\,|P|\\,2^{-n},\n$$\nwhere $w$ denotes poset width.\n\nTaking $k=0$ gives\n$$\nw(P)\\ge\n|P|\\binom{n}{\\lfloor n/2\\rfloor}2^{-n},\n$$\nwhich is exactly the Daykin-Frankl conjecture.\n\nThe proof proceeds by induction on $n$, reducing the step to a structural lemma for a convex subset of $R\\times Q_1$ and carefully recombining antichains from its two layers.","claimIssueNote":null,"solveDate":"2026-09-02","model":"GPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":["Kada Williams"],"aiRole":"Kada Williams explicitly credits ChatGPT 5.6 Sol Pro with generating the proof content of the note. The proof establishes a stronger product inequality for convex subsets of Boolean lattices and derives the original Daykin-Frankl conjecture as the case $k=0$. Williams verifies, writes up, and takes responsibility for communicating the argument.","verification":"unreviewed","verificationNote":"Unreviewed. arXiv 2609.03087 (four pages) read here: the note describes itself as verifying and communicating an LLM-generated proof, credits ChatGPT 5.6 Sol Pro, and gives the induction on dimension with the R x Q_1 convexity lemma in full. Checked by the human author, not by anyone independent; not peer reviewed; no formalization.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A 1983 conjecture of Daykin and Frankl in extremal set theory, that a convex subset of the Boolean lattice contains an antichain of at least the expected proportion of its size; previously known only for binary downsets. Forty-three years, a short conceptual proof of a stronger product form. A named conjecture with a modest literature, mid-scale.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.03087","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"depth-1-distinctness-for-pseudorandom-unitaries","name":"Depth-1 distinctness for pseudorandom unitaries","shortName":"Depth-1 distinctness for pseudorandom unitaries","problemNumber":null,"field":"Quantum pseudorandomness","fieldGroup":"Quantum information & computing","statement":"A single layer of independent random single-qubit Clifford gates is $\\operatorname{negl}(n)$-distinct for polynomially many queries. Consequently, in the $PFC$ pseudorandom-unitary construction, the depth-$\\log n$ global unitary $2$-design layer can be replaced by a depth-1 tensor product of single-qubit $2$-designs while retaining the distinctness property needed for the construction. This provides a counterexample to the conjecture that every negligibly distinct ensemble must be entangling.","posedBy":"Raza, Eisert and Fefferman (the authors' own working conjecture)","yearPosed":null,"ageNote":"No posed year. The conjecture was the authors' own working hypothesis during a project begun in early 2025 and was refuted before the paper was written, so it was never open in public and dating it would be artificial.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The paper proves that a tensor-product ensemble of independently chosen single-qubit Clifford gates is $\\operatorname{negl}(n)$-distinct when the number of queries is polynomial in $n$. This disproves the authors' conjecture that negligibly distinct ensembles must necessarily be entangling. As an application, the depth-$\\log n$ global Clifford/unitary-$2$-design layer used in the $PFC$ pseudorandom-unitary construction can be replaced by a single depth-1 layer of local single-qubit $2$-designs while preserving the required distinctness property.","claimIssueNote":null,"solveDate":"2026-09-02","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Asad Raza","Jens Eisert","Bill Fefferman"],"aiRole":"The authors state that Proposition III.5 was proposed by ChatGPT 5.6 Sol as a counterexample to their conjecture that a $\\operatorname{negl}(n)$-distinct ensemble must be entangling. Sol identified that a single layer of independent random single-qubit Clifford gates is already negligibly distinct for polynomially many queries. ChatGPT 5.5 and 5.6 Pro were also used to devise proof strategies for the paper's main results, as well as for literature search, exposition, and technical checking; all proofs were independently verified by the authors.","verification":"unreviewed","verificationNote":"The result appears as Proposition III.5 in a coauthored research paper by Asad Raza, Jens Eisert, and Bill Fefferman. The authors explicitly state that ChatGPT 5.6 Sol proposed the counterexample and that they independently verified all proofs. No formal proof-assistant certificate is reported.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":6,"significanceNote":"A counterexample to a conjecture the authors held while writing the paper, never published as open, in a technical corner of quantum pseudorandomness. The consequence is real for the PFC construction - a depth-1 layer replaces a depth-log n design - but the question it answers existed for a few months inside one research group. Just below the hat-guessing entries at 7, which at least settle a question that was posed in print.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.03065","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"a-smooth-counterexample-to-the-trautman-conjecture","name":"A Smooth Counterexample to the Trautman Conjecture","shortName":"Trautman Conjecture","problemNumber":null,"field":"CR geometry","fieldGroup":"Analysis","statement":"The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in $\\mathbb{C}^2$. We modify a standard construction of nonembeddable smooth strongly pseudoconvex CR 3-manifolds so that the condition of having a nowhere-zero closed section of the canonical bundle is preserved, thus providing a strongly pseudoconvex counterexample to the Trautman conjecture.","posedBy":"Andrzej Trautman","yearPosed":1998,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"For every neighborhood $U$ of the origin in $\\mathbb{C}\\times\\mathbb{R}$ and every $\\varepsilon>0$, Curry constructs a smooth nonnegative perturbation $\\phi$, supported in $U$ with $\\|\\phi\\|_{C^2}<\\varepsilon$, such that\n$$\nT^{0,1}_\\phi=\\operatorname{span}_{\\mathbb C}\\{L_\\phi\\}\n$$\nis strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every $C^1$ CR function near the origin satisfies $dh(0)=0$. Hence the CR structure is not locally embeddable at the origin. The construction can also be globalized to $S^3$ as an arbitrarily small $C^1$ perturbation of the standard spherical CR structure.","claimIssueNote":null,"solveDate":"2026-09-02","model":"ChatGPT (model not named; the paper says ChatGPT Plus)","modelMaker":"OpenAI","humanCollaborators":["Sean N. Curry"],"aiRole":"Sean Curry states that the counterexample construction itself was discovered through experimentation with ChatGPT Plus on August 18, 2026. ChatGPT was also used to produce an initial rough draft and later for proofreading. Curry independently checked all calculations and arguments and takes responsibility for the paper.","verification":"unreviewed","verificationNote":"Unreviewed. A single-author arXiv preprint two days old, labelled by its author \"Preliminary version. Comments welcome.\" The author states he independently checked all calculations and arguments. The construction is a modification of a standard nonembeddable strongly pseudoconvex CR 3-manifold, arranged so that the canonical bundle keeps a nowhere-zero closed section, so it is checkable by anyone who knows the Rosay-type examples; three CR specialists are thanked in the paper but none is on the record as having checked it. No peer review, no formalisation.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":"No Wikipedia article for the conjecture in any language. Trautman himself has one in ten, and CR manifolds in three; the conjecture lives in the literature only.","significance":25,"significanceNote":"A named conjecture of Andrzej Trautman from 1998, at the meeting point of CR geometry and exact solutions of the Maxwell and Einstein equations, open for 28 years and settled outright by a strongly pseudoconvex counterexample. Real and old, but invisible outside CR geometry: no Wikipedia article, and the paper thanks three CR specialists for their scepticism about the conjecture, so this confirms a doubt rather than overturns a consensus. Level with the xz-conjecture counterexample at 25, another named conjecture refuted by an explicit witness; below the Hessian counterexample at 30.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.03198","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"logarithmic-basis-number-of-graphs","name":"Logarithmic basis number of graphs","shortName":"Logarithmic basis number of graphs","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"The basis number $\\mathrm{bn}(G)$ of a graph $G$ is the minimum edge-congestion of a basis of its cycle space. We prove that every finite $n$-vertex multigraph satisfies\n$$\\mathrm{bn}(G)=O(\\log n),$$\nresolving, for simple graphs, a question of Bazargani, Biedl, Bose, Maheshwari and Miraftab, subsequently stated as a conjecture by Miraftab, Morin and Yuditsky. The argument also yields the cycle-rank refinement\n$$\\mathrm{bn}(G)=O(\\log \\beta(G)),$$\nwhere $\\beta(G)$ is the dimension of the cycle space, and a reduction of Lehner and Miraftab, based on a theorem of Richter and Shank, then gives\n$$\\mathrm{bn}(G)=O(\\log g)$$\nfor graphs of Euler genus $g$. These orders are best possible.","posedBy":"Bazargani, Biedl, Bose, Maheshwari and Miraftab","yearPosed":2024,"ageNote":"The question is asked in Section 5 of Bazargani, Biedl, Bose, Maheshwari and Miraftab, posted to arXiv on 24 December 2024, and restated as Conjecture 12 by Miraftab, Morin and Yuditsky in January 2026. The submitted year of 2026 came from Knauer's bibliography, which cites the journal version; the site dates a problem from when it was first posed in public.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Knauer proves that every finite $n$-vertex multigraph satisfies $\\mathrm{bn}(G)=O(\\log n)$, improving the previous general $O(\\log^2 n)$ bound and matching the known $\\Omega(\\log n)$ order. He also proves the sharper cycle-rank bound $\\mathrm{bn}(G)=O(\\log\\beta(G))$. Combined with a reduction of Lehner and Miraftab, this yields $\\mathrm{bn}(G)=O(\\log g)$ for graphs of Euler genus $g$, improving the previous $O(\\log^2 g)$ bound to the optimal logarithmic order.","claimIssueNote":null,"solveDate":"2026-09-02","model":"ChatGPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Kolja Knauer"],"aiRole":"The proof was found with the help of GPT-5.6 Sol. The model was also used to explore proof strategies, locate potentially relevant literature, and assist with drafting and revising the manuscript. Knauer independently checked the arguments and references and takes responsibility for the final result.","verification":"unreviewed","verificationNote":"Unreviewed. A nine-page preprint one day old at submission, with no peer review and no formal verification. The argument is conventional and self-contained - weighted cycle-basis bounds, minimax duality and dependent randomized rounding - so it is readable by any combinatorialist, and the author states he checked the arguments and references himself. Nobody independent has.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A stated conjecture (Miraftab, Morin and Yuditsky, Conjecture 12) settled at the optimal order rather than improved: O(log n) against a matching classical Omega(log n), and the same for cycle rank and Euler genus. The area is live, with Geniet and Giocanti and the Miraftab-Morin-Yuditsky group both publishing on basis number in 2026, which is what lifts it above a one-off. Still narrow: one graph parameter, unknown outside structural graph theory. Tied with the Tu-Deng conjecture at 15, likewise a named conjecture with a real specialist following.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.02080","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Bazargani, Biedl, Bose, Maheshwari and Miraftab, where the question is asked (Section 5)","url":"https://arxiv.org/abs/2412.18595","kind":"problem-record"},{"label":"Miraftab, Morin and Yuditsky, who state it as Conjecture 12","url":"https://arxiv.org/abs/2601.14095","kind":"problem-record"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"common-neighbour-conjectures-for-saxl-graphs","name":"Common Neighbour Conjectures for Saxl Graphs","shortName":"Saxl Common Neighbour","problemNumber":null,"field":"Group theory","fieldGroup":"Combinatorics","statement":"For a finite permutation group, a base is a set of points with trivial pointwise stabiliser, and the generalised Saxl graph records which pairs of points lie together in a base of minimum size. Burness and Giudici conjectured that any two vertices of the Saxl graph of a primitive group of base size two have a common neighbour, and Freedman, Huang, Lee and Rekvényi extended this conjecture to arbitrary base size. We disprove both. For each integer $B\\geq 2$ we construct infinitely many primitive groups of base size $B$ whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size two, where this is the usual Saxl graph, we obtain three further infinite families, one each of affine, product and twisted wreath type, so the conjecture fails in three of the five O’Nan–Scott types; in the affine and product type families the Saxl graphs have diameter exactly three. This answers Problem 21.29 in the Kourovka Notebook in the negative. In the positive direction, we prove the Burness–Giudici conjecture for every primitive affine group whose point stabiliser is almost quasisimple of sporadic type, completing work of Lee and Popiel.","posedBy":"Timothy Burness and Michael Giudici","yearPosed":2020,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The paper disproves both the Burness–Giudici common neighbour conjecture for primitive groups of base size $2$ and its later generalisation to arbitrary base size. For every integer $B\\ge2$, it constructs infinitely many primitive permutation groups of base size $B$ whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size $2$, it gives infinite counterexample families of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan–Scott types.","claimIssueNote":null,"solveDate":"2026-09-01","model":"ChatGPT Pro; Claude","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Aluna Rizzoli","Adam R. Thomas"],"aiRole":"The project began with Codex assisting an attempted proof and Lean formalisation of the common neighbour conjecture for soluble affine groups. A falsification search instead found counterexamples. The authors then worked with Codex, ChatGPT Pro and Claude to discover further examples and constructions, search the literature, and draft and revise the paper. Codex implemented and debugged much of the Magma, GAP, Python and C++ code and produced most of the Lean formalisation.","verification":"unreviewed","verificationNote":"Unreviewed: a preprint two days old with no independent check. The counterexamples are explicit groups, so they are checkable directly by anyone with Magma or GAP, and the paper's Lean 4 formalisation of Theorem 1.2 reports no $\\texttt{sorry}$; neither has been rebuilt here.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A named conjecture of Burness and Giudici in permutation group theory, extended by Freedman, Huang, Lee and Rekvenyi, and carrying a Kourovka Notebook number (21.29) - which is the field's own marker that a question is worth recording. Well known inside its area and unknown outside it. Above the Tu-Deng conjecture at 15, a named conjecture with a comparable specialist following but no problem-list number; below Courtade-Kumar at 22, one of the best-known open problems in its own field.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.01367","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bounded-prime-gaps-at-most-236","name":"Bounded prime gaps: $H_1 \\le 236$","shortName":"Prime gaps at most 236","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"Write $H_1 = \\liminf_{n\\to\\infty}(p_{n+1}-p_n)$. Announced on 1 September 2026: $H_1 \\le 236$, building on Stadlmann's $240$ of the previous day.","posedBy":"Alphonse de Polignac (the twin prime conjecture); the bounded form since Goldston, Pintz and Yıldırım","yearPosed":1849,"ageNote":"Dated from Polignac's 1849 conjecture. The modern quantitative form dates from Goldston, Pintz and Yıldırım in 2005 and became a finite bound with Zhang in 2013.","solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"$H_1 \\le 236$, between Stadlmann's $240$ of 31 August and Axiom Math's $212$ of 3 September. Superseded within two days, which Kintali noted himself.","claimIssueNote":null,"solveDate":"2026-09-01","model":"unspecified AI agents","modelMaker":null,"humanCollaborators":["Shiva Kintali"],"aiRole":"Kintali's own words on X: \"I showed that H1 <= 236 with the help of AI, building on Julia Stadlmann's work.\" He describes the method as the GPY approach with the Maynard-Tao sieve, and says he then stopped his AI agents and moved to another problem. That is the whole of the public disclosure: no model is named, the division of labour is not described, and there is no preprint.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. The thinnest source of the three 2026 bounded-gaps entries, and recorded as Candidate for that reason. An announcement on X with no preprint, no named model and no argument in public. It is recorded because a bound is a bound and this one held the record for two days between Stadlmann's 240 and Axiom Math's 212; it is not recorded as settled. If a preprint appears, this entry should be revisited.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":62,"significanceNote":"The closest mathematics has come to the twin prime conjecture. Level with the site's other bounded-gaps entries: the weight belongs to the question, and a step that stood for two days weighs the same as one that stood for twelve years in that respect.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://x.com/ShivaKintali/status/2095547543061135765","sourceName":"Shiva Kintali on X","collection":null,"collectionVersion":null,"links":[{"label":"Stadlmann's 240, the bound this improves on","url":"https://arxiv.org/abs/2608.31126","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"exact-rank-and-smith-profile-of-affine-incidence-over-mathbb-z-p-3-mathbb-z","name":"Exact rank and Smith profile of affine incidence over $\\mathbb Z/p^3\\mathbb Z$","shortName":"Depth-three affine incidence rank","problemNumber":null,"field":"Finite geometry; Smith normal forms","fieldGroup":"Combinatorics","statement":"Let $p$ be prime, $k,n\\geq 1$, and $R=\\mathbb Z/p^k\\mathbb Z$. For each primitive direction $b\\in\\mathbb P(R)^{n-1}$ modulo multiplication by units and each $\\lambda\\in R$, let\n\n$$\nH_{b,\\lambda}=\\{x\\in R^n:\\langle b,x\\rangle=\\lambda\\},\n$$\n\nand let $A(p^k,n)$ be the $0$-$1$ matrix whose rows are the indicators of these distinct affine hyperplanes and whose columns are the points of $R^n$. What is $\\operatorname{rank}_{\\mathbb F_p}A(p^k,n)$? Łaba and Trainor explicitly recorded the residue-ring point-hyperplane rank question as open and proved upper bounds. Dvir later used the normalized distinct-row matrix above and obtained further bounds. The field case $k=1$ is known, but the exact rank remains open in general for $k\\geq2$. This entry concerns the depth-three plane specialization, $(k,n)=(3,2)$.","posedBy":"Izabella Łaba and Charlotte Trainor (arXiv:2403.05719, 2024)","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"For $(k,n)=(3,2)$, the normalized distinct-row matrix $B_3$ equals $A(p^3,2)$ up to row and column ordering. For every prime $p$,\n\n$$\n\\operatorname{rank}_{\\mathbb F_p}B_3=\n\\begin{cases}\n240,&p=3,\\\\\n\\dfrac{p(p+1)(3p^4+4p^3+3p-1)}{18},&p\\equiv1\\pmod3,\\\\\n\\dfrac{p^2(p+1)^2(3p^2+p+1)}{18},&p\\equiv2\\pmod3.\n\\end{cases}\n$$\n\nThe paper also determines the complete $p$-primary Smith profile of $\\operatorname{coker}(B_3^\\mathsf T)_{(p)}$: its exponent is $p^5$, and all six multiplicities are explicit for every prime. The repeated-row Łaba–Trainor matrix has the same $\\mathbb F_p$-rank, but the integral Smith claim applies only to $B_3$. Arbitrary depth, higher dimension, projective Hjelmslev incidence and the generalized-polynomial characterization remain open. Finite computations certify exceptional cases; they do not prove the uniform formulas.","claimIssueNote":null,"solveDate":"2026-09-01","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Under the author's direction, OpenAI Codex using GPT-5.6 Sol generated the central mathematical development: the depth-three reduction, the tripotent-sector decomposition, the truncated q-Pascal and q-Lucas block analysis, the q-Newton determinantal-minor argument, the cross-chart divided-carry closure, and the resulting all-prime Smith formulas. It also assisted with the exact companion software, exceptional certificates and manuscript drafting. The author selected the research direction, iteratively challenged and checked the derivations and certified outputs, established the public claim and source boundaries, and takes responsibility for the final content. Adversarial machine reviews were produced within the same OpenAI Codex workflow and are not human peer review or independent expert verification.","verification":"unreviewed","verificationNote":"The public source contains a complete all-prime argument, an exact-arithmetic companion, deterministic release checks and an explicit AI-use disclosure. On 2026-09-08 the isolated normal and optimized release verifiers both passed, as did the targeted mathematical and release-assurance suites (66/66 tests). These checks establish reproducibility, artifact integrity and internal consistency; they are not independent validation of the mathematics because the checking was performed within the same model-assisted workflow. No named independent domain expert has yet checked or endorsed the theorem, so Unreviewed is the correct tier.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"The next parameter case of the Łaba-Trainor rank question, settled exactly for the depth-three plane. A point below its predecessor because the framework it applies was built there; the general problem for arbitrary depth and dimension remains open.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/FCIG-Depth-Three-Smith-Profiles-and-Newton-Minor-Geometry-for-Affine-Hjelmslev-Radon-Incidence/blob/0807e47a7ad8791a861dcc76e47c70e764f6f3b3/paper/Depth-Three-Smith-Profiles-and-Newton-Minor-Geometry-for-Affine-Hjelmslev-Radon-Incidence.pdf","sourceName":"GitHub manuscript: Depth-Three Smith Profiles and Newton-Minor Geometry for Affine Hjelmslev–Radon Incidence","collection":null,"collectionVersion":null,"links":[{"label":"Exact-arithmetic companion and reproducibility repository","url":"https://github.com/aconsciousfractal/FCIG-Depth-Three-Smith-Profiles-and-Newton-Minor-Geometry-for-Affine-Hjelmslev-Radon-Incidence/tree/0807e47a7ad8791a861dcc76e47c70e764f6f3b3","kind":"code"},{"label":"Łaba–Trainor: the open residue-ring incidence-rank question","url":"https://arxiv.org/abs/2403.05719","kind":"problem-record"},{"label":"Dvir: normalized incidence matrix and later rank bounds","url":"https://arxiv.org/abs/2604.25822","kind":"paper"},{"label":"Depth-two affine incidence rank — previous parameter case","url":"https://vibemathed.com/problem/exact-rank-and-smith-profile-of-affine-incidence-over-mathbb-z-p-2-mathbb-z","kind":"other"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"measures-on-partial-orders","name":"Measures on Partial Orders","shortName":"Measures on Partial Orders","problemNumber":null,"field":"Representation theory","fieldGroup":"Algebra","statement":"We determine the measures (in the sense of Harman–Snowden) on the Fraïssé class of partially ordered sets: the space of measures is a union of a plane, eight lines, and 15 isolated points. This is the first case where the space is not equidimensional, and the first primitive case in which it has dimension at least two.","posedBy":"Andrew Snowden","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"For the Fraïssé class $\\mathfrak P$ of finite partially ordered sets, every measure $\\nu$ is completely determined by the nine values\n$$\n\\nu_{p,q}=\\nu(J_{p,q},*),\\qquad 0\\le p,q\\le2.\n$$\nThe possible $3\\times3$ matrices $(\\nu_{p,q})$ are classified explicitly. Consequently, the reduced space of all measures is exactly the union of one plane, eight lines, and 15 isolated points, with the plane and three lines meeting at one point and no other intersections.\n\nThe paper also completely classifies weak measures and supports, proves there is no regular measure on all finite posets, classifies characteristic-zero measures for the automorphism group of the universal homogeneous poset, and determines the Knop-like measures.","claimIssueNote":null,"solveDate":"2026-09-01","model":"ChatGPT 5.6 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Andrew Snowden states that ChatGPT Pro 5.6 produced most of the mathematical arguments. It first computationally discovered the correct classification by generating and solving the defining equations for posets of size at most seven. It then found the key universal formulas involving twins and cuts, proved that the resulting functions are weak measures, supplied existence proofs for several families of measures, identified reductions between the remaining cases, discovered the stabilization relation needed to show that a 3×3 matrix determines every measure, and assisted with many of the paper's additional results. Snowden wrote the paper himself and checked/reworked the arguments.","verification":"unreviewed","verificationNote":"The paper contains complete conventional mathematical proofs written and checked by Andrew Snowden. ChatGPT generated most of the mathematical arguments, but Snowden is also the author publishing the result, so this is author verification rather than independent expert verification. Johannes Flake is thanked for helpful discussions, but the paper does not state that he or another external expert independently verified the proofs. No formal proof-assistant verification or referee report is reported.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"Completely determines the measures on the Fraisse class of finite posets within the Harman-Snowden program, and is the first case where the space of measures is not equidimensional. A specialized but substantive research result, published by a leading author in the area.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.02021v1","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mckean-entropy-production-conjecture","name":"McKean Entropy-Production Conjecture","shortName":"McKean entropy production","problemNumber":null,"field":"Mathematical physics","fieldGroup":"Analysis","statement":"In 1966, McKean asked whether the entropy production of the Boltzmann equation must be monotone decreasing in time. We show that this is not the case even in the space-homogeneous setting for the Boltzmann collision operator with a constant angular cross section and the kinetic parameter $\\gamma\\in[0,1]$. This recovers the classical case of hard spheres and the simplest case of Maxwell molecules. Our examples are radially symmetric mixtures of Maxwellians.","posedBy":"Henry P. McKean Jr.","yearPosed":1966,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"For the space-homogeneous Boltzmann equation in dimension $3$ with collision kernels\n$$B=\\frac{1}{4\\pi}|v-v_*|^\\gamma,\\qquad \\gamma\\in[0,1],$$\nthe paper constructs smooth, positive, radial mixtures $f_R=(1-p)M_1+pM_R$ for which $\\partial_tD(f_R)>0$ for sufficiently large $R$. Thus entropy production need not decrease even for Maxwell molecules ($\\gamma=0$) or hard spheres ($\\gamma=1$), negatively resolving McKean's monotonicity question for these physically standard kernels.","claimIssueNote":null,"solveDate":"2026-09-01","model":"GPT-5.6 Sol; Claude","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Luis Silvestre"],"aiRole":"A first version of the proof was obtained by GPT-5.6 Sol running in Codex Ultra with access to Silvestre's earlier paper and research notes. Claude Code then rewrote the initially difficult-to-read proof, after which Silvestre reinterpreted, restructured, and checked the argument and took responsibility for the final proof. The substantive counterexample proof therefore originated from GPT-5.6 Sol under access to author-supplied mathematical context.","verification":"unreviewed","verificationNote":"The arXiv preprint contains a complete analytic proof by a leading researcher in kinetic equations, who states that he reinterpreted, restructured, and assumes full responsibility for the final argument. No independent expert review, peer review, or formal proof-assistant verification is currently documented.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A question McKean asked in 1966 about the Boltzmann equation, answered negatively for the two physically standard kernels, hard spheres and Maxwell molecules. Sixty years standing and well known in kinetic theory, but not followed outside it. Tied with the Dubickas Z-number question at 20 - a specific named question of comparable age and comparable specialist reach. Below the Lonely Runner Conjecture at 30, which is famous across two fields.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.01753","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-sharp-threshold-for-the-multiplicative-merino-welsh-inequality-on-matroids","name":"The sharp threshold for the multiplicative Merino-Welsh inequality","shortName":"Merino-Welsh sharp threshold","problemNumber":null,"field":"Matroid theory; Tutte polynomials","fieldGroup":"Combinatorics","statement":"Merino and Welsh conjectured that a loopless bridgeless graph satisfies $\\max(T_G(2,0), T_G(0,2)) \\ge T_G(1,1)$. Jackson proved the multiplicative matroid form $T_M(3,0)T_M(0,3) \\ge T_M(1,1)^2$, and the constant 3 was lowered to $2.9243$ and then to $2.355$ by Csikvári. Counterexamples are known below $x_* = 2.2266815969\\ldots$, the largest real root of $x^3 = 9(x-1)$, and Csikvári conjectured in 2025 that $x_*$ is exactly the threshold: does $T_M(x_*,0)T_M(0,x_*) \\ge T_M(1,1)^2$ hold for every finite matroid without loops or coloops?","posedBy":"Péter Csikvári, Conjecture 7.1 of Around the Merino-Welsh conjecture: improving Jackson's inequality (arXiv:2502.19196, 2025); the underlying conjecture is Merino and Welsh's","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"It is proved that $T_M(x_*,0)T_M(0,x_*)\\ge T_M(1,1)^2$ for every finite matroid $M$ without loops or coloops, and known counterexamples below $x_*$ make the universal threshold sharp. For $2\\le x\\le x_*$, the stronger bound $$\\frac{T_M(x,0)T_M(0,x)}{T_M(1,1)^2}\\ge\\left(\\frac{x^3}{9(x-1)}\\right)^{(m-2\\kappa)/2}$$ holds, where $m$ is the number of elements and $\\kappa$ the number of connected components. Parallel doublings of uniform matroids show the exponential rate is optimal throughout the interval. The proof uses weighted broken circuits, matroid decomposition, and exact checks of rational replacements. The graph conjecture at $x=2$ remains open.","claimIssueNote":null,"solveDate":"2026-09","model":"GPT-6 Astra; Claude Opus 5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Mingchang Liu"],"aiRole":"This paper grew out of a human–AI collaboration on the Merino–Welsh problem. The author chose the sharp universal matroid threshold and its quantitative refinement as the paper’s focus, directed the comparison with earlier work, and shaped the exposition through successive revisions. AI models contributed to mathematical exploration, proof development, and exact computation, including the rational replacement table. In preparing the manuscript, GPT-6 Astra (OpenAI) and Claude Opus 5 (Anthropic) assisted with checking the arguments, researching the literature, and producing the text, code, and figures.","verification":"unreviewed","verificationNote":"Checked here on 27 September 2026. Csikvári's Conjecture 7.1 was located in arXiv:2502.19196v2, the version the submission links, and the Zenodo record 22911905 of 23 September was read: the threshold is stated as $x_* = 2.2266815969\\ldots$, the largest real root of $x^3 = 9(x-1)$, with the quantitative bound on $2 \\le x \\le x_*$ and sharpness of the exponential rate from parallel doublings of uniform matroids. The tools named are a weighted broken-circuit comparison, the Cunningham-Edmonds decomposition and exact polynomial checks of a finite rational replacement table; the table and a checker are in the linked V1.1 release. The mathematics was not checked here, the rational table was not recomputed, and the author's own note is correct that critical AI reviews are not verification. Four days old, no referee.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"The Merino-Welsh conjecture is a well-known problem about Tutte polynomials, but the specific threshold conjecture answered here was stated nineteen months ago in one paper's final section, and the graph case at $x=2$ - the conjecture people actually quote - stays open. Level with the catalog's biological-unavoidability entry at 14: a precisely posed, genuinely open question from a recent paper. The score is for the question's standing, not for the work.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.22911905","sourceName":"Zenodo preprint V1, record 22911905 (23 September 2026), with verification code in the V1.1 release","collection":null,"collectionVersion":null,"links":[{"label":"Verification code and rational data (V1.1)","url":"https://github.com/michaeliu4/tutte-threshold/releases/tag/V1.1","kind":"code"},{"label":"Csikvári, Conjecture 7.1 (author version)","url":"https://arxiv.org/abs/2502.19196v2","kind":"problem-record"}],"relations":[],"submittedBy":"SilentIbis759","upvotes":0,"downvotes":1,"commentCount":0},{"slug":"four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2","name":"Morris's asymptotic conjecture: $\\mathrm{FC}(k,n)=\\Theta_k(n^{k-2})$","shortName":"Morris's asymptotic conjecture","problemNumber":null,"field":"Extremal set theory; union-closed families","fieldGroup":"Combinatorics","statement":"Morris's Conjecture 2 asks whether, for every fixed integer $k\\ge2$, $\\mathrm{FC}(k,n)=\\Theta_k(n^{k-2})$ as $n\\to\\infty$. Here $\\mathrm{FC}(k,n)$ is the least $m$ such that every collection of $m$ distinct $k$-subsets of an $n$-point set is Frankl-complete. A configuration $\\mathcal G$ with support $U=\\bigcup\\mathcal G$ is Frankl-complete if every finite union-closed family $\\mathcal F\\supseteq\\mathcal G$ has an element of $U$ belonging to at least half its sets. The extension may contain additional points.","posedBy":"Robert Morris (2006), “FC-families and improved bounds for Frankl’s conjecture,” European J. Combin. 27, 269–282; Conjecture 2 in arXiv:math/0702348v1.","yearPosed":2006,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"It is proved that $\\mathrm{FC}(k,n)=\\Theta_k(n^{k-2})$ for every fixed $k\\ge2$, resolving Morris's Conjecture 2. The proof combines robust weighted certificates, pair lifting and extremal forcing with a self-contained lower-bound argument.\n\nFor four-sets, $n^2/4-O(n)\\le\\mathrm{FC}(4,n)\\le(18+o(1))n^2$. More generally, for fixed $k\\ge4$, $\\mathrm{FC}(k,n)\\ge\\frac{n^{k-2}}{2(k-2)!}-O_k(n^{k-3})$.\n\nThe sharp FC and robustness thresholds for sunflowers with a two-point core and disjoint pair petals are nine and ten petals. Exact finite certificates establish $\\mathrm{FC}(4,9)=16$ and refute Pulaj and Wood's lexicographic extremality conjecture.","claimIssueNote":null,"solveDate":"2026-09","model":"GPT-6 Astra; Fable 5.1","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Mingchang Liu"],"aiRole":"The manuscript's AI statement, in full: \"The author and AI models both made substantial mathematical contributions to this work. The finite classification establishing $\\mathrm{FC}(4,9)=16$ was developed primarily by the models. The author proposed extending this work to the four-set case of Morris's conjecture and contributed to proof development and refinement. The collaboration developed the charging inequality, $3:1$ core-petal weights, and matching negative certificates for the sharp nine-petal sunflower threshold, and combined this local criterion with extremal bounds to prove $\\mathrm{FC}(4,n)=\\Theta(n^2)$. GPT-6 Astra (OpenAI) and Fable 5.1 (Anthropic) contributed to proof exploration, computation, and review. The author takes full responsibility for all mathematical results and the contents of this manuscript.\" Co-developed on that account: the models carried the finite classification and shared the main argument, the author set the target and takes responsibility.","verification":"unreviewed","verificationNote":"Zenodo preprint V1, published 11 September 2026, one author, no independent endorsement. Checked here: the record exists with the abstract as submitted; the manuscript states Morris's Conjecture 2 as the entry does and records that Pulaj settled the three-uniform case exactly ($\\mathrm{FC}(3,n)=\\lfloor n/2\\rfloor+1$ for $n\\ge 4$), so this is the first open uniformity and not a duplicate of anything in the catalog; the companion repository release V1 holds the exact certificates for $\\mathrm{FC}(4,9)=16$ and for the Pulaj-Wood refutation with Python and C++ verifiers. None of the mathematics was checked here and the verifiers were not run. Filed as Candidate, as submitted, until an independent reader has been through the argument or the finite certificates have been replayed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"One uniformity of a named 2006 conjecture of Morris in the union-closed literature, and the first open case after Pulaj settled k = 3. Known to the people who work on Frankl's conjecture and to nobody else. Below the Daykin-Frankl entry at 25, which is a conjecture in its own right, and level with the Elton-Odell and Dittert-dimension-16 entries at 15: a specific case of a subcommunity conjecture.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://zenodo.org/records/22702735","sourceName":"Zenodo preprint (V2)","collection":null,"collectionVersion":null,"links":[{"label":"V1 code and exact certificates","url":"https://github.com/michaeliu4/frankl-complete-four-sets/releases/tag/V1","kind":"code"},{"label":"Morris’s original conjecture (Conjecture 2)","url":"https://arxiv.org/abs/math/0702348v1","kind":"problem-record"}],"relations":[],"submittedBy":"SilentIbis759","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pebody-s-even-cyclic-four-deck-reconstruction-conjecture","name":"Pebody's even cyclic four-deck reconstruction conjecture","shortName":"Pebody's four-deck conjecture","problemNumber":null,"field":"Combinatorial reconstruction; Fourier analysis on cyclic groups","fieldGroup":"Combinatorics","statement":"For every even integer $n>10$, is $r(C_n)=4$, where $r(C_n)$ is the least integer $k$ such that every subset of the cyclic group $C_n$ is determined, up to translation, by its cumulative $k$-deck?\n\nThe cumulative deck records, with multiplicity, the translation classes of all subsets of size at most $k$. Reflections are not identified.\n\nThis is Luke Pebody's Conjecture 8.1 in Reconstructing Odd Necklaces (2007), p. 514, with Definitions 1-4 on pp. 503-504. The submitted manuscript writes $r_{\\mathrm{set}}(C_n)$ for Pebody's $r(C_n)$.","posedBy":"Luke Pebody, Reconstructing Odd Necklaces (2007), Conjecture 8.1, p. 514","yearPosed":2007,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"Claimed: r_set(C_n) <= 4 for every n >= 1, that is, the cumulative 4-deck determines every subset of a cyclic group up to translation. With Keleti-Kolountzakis's Corollary 1.1 for the lower bound this gives r_set(C_n) = 4 for every even n > 10, the whole domain of Pebody's Conjecture 8.1. The lower bound is imported, not claimed. The proof works on the ratio of Fourier transforms of two sets with equal decks: finiteness of the phase fibre, cyclotomic torsion of the phases, gluing of primary characters across odd primes, and extinction of the residual dyadic signs by two odd-denominator projectors. Binary subsets and translation-only decks only; no claim for multisets, general integer signals or noncyclic groups. A full-proof claim awaiting authoritative review.","claimIssueNote":null,"solveDate":"2026-09","model":"GPT-6 Pro; Astra (via ChatGPT)","modelMaker":"OpenAI","humanCollaborators":["Oleksiy Babanskyy"],"aiRole":"The author reports using GPT-6 Pro and Astra via ChatGPT. OpenAI Codex was also used; its underlying model versions are not specified. AI assistance included mathematical exploration, source comparison, manuscript drafting and revision, finite sanity checks and repository preparation under the author's direction. The author is responsible for the mathematical claims and final decisions. Individual proof steps are not attributed to particular models. This disclosure appears in the manuscript and AI_USE.md.","verification":"unreviewed","verificationNote":"Checked here on 22 September 2026. Pebody's Conjecture 8.1 (Combinatorics, Probability and Computing 16 (2007), 503-514, confirmed at Crossref) and the imported lower bound (Keleti-Kolountzakis, arXiv:math/0603415, Corollary 1.1: the 3-deck suffices for Z_n iff n is a power of an odd prime, a product of at most three odd primes, or n in {2,4,6,8,10}) were read. The 11-page manuscript was read: it follows the Grünbaum-Moore route, phases on the Fourier support, Galois covariance making them roots of unity, local characters on generator shells, gluing across odd primes by four-term relations, and removes the remaining dyadic sign by an integrality argument on Fourier projectors, which is exactly the point where Grünbaum-Moore's Theorem 5 needed a nonzero coefficient. The chain is coherent and non-trivial; it was not checked step by step here and no specialist has read it. The site ran an independent brute force: for every n <= 22, any two subsets of C_n with the same cumulative 4-deck are translates, while the 3-deck fails at n = 12, 14, 16, 18, 20, 22 exactly as Keleti-Kolountzakis predict. That is consistent with the theorem and is not evidence for the proof, since Grünbaum-Moore already cover small n.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A numbered conjecture from a 2007 CPC paper on a classical reconstruction problem with a real lineage (Grünbaum-Moore 1995, Radcliffe-Scott, Pebody, Keleti-Kolountzakis), open for nineteen years, in a small community. Level with the Umans-Wang and Fermat-nonnormality band at 15-18 less a little for the size of the audience; well above this author's earlier entry at 4, which answered a conjecture four months old. The extraordinary-claims rule was considered and does not apply: this is not a problem the wider field is watching.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/FCIG-Pebody-Even-Cyclic-Four-Deck-Reconstruction/blob/377db7d69564c819628a0049309ded409a241ea7/paper/Pebody-Even-Cyclic-Four-Deck-Reconstruction.pdf","sourceName":"GitHub manuscript with reproducible companion","collection":null,"collectionVersion":null,"links":[{"label":"Manuscript source, companion checks and reproduction instructions","url":"https://github.com/aconsciousfractal/FCIG-Pebody-Even-Cyclic-Four-Deck-Reconstruction/tree/377db7d69564c819628a0049309ded409a241ea7","kind":"code"},{"label":"Original conjecture: Pebody (2007), Conjecture 8.1, p. 514","url":"https://doi.org/10.1017/S0963548306007875","kind":"paper"},{"label":"Imported lower bound: Keleti-Kolountzakis, Corollary 1.1 and Theorem 2.22","url":"https://arxiv.org/abs/math/0603415","kind":"paper"},{"label":"Public claim boundary and exclusions","url":"https://github.com/aconsciousfractal/FCIG-Pebody-Even-Cyclic-Four-Deck-Reconstruction/blob/377db7d69564c819628a0049309ded409a241ea7/docs/PUBLIC_CLAIM_BOUNDARY.md","kind":"code"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-matching-conjecture-the-four-uniform-case","name":"Erdős matching conjecture: the four-uniform case","shortName":"Four-uniform Erdős matching conjecture","problemNumber":1020,"field":"Extremal set theory and hypergraph matchings","fieldGroup":"Combinatorics","statement":"Let $r\\geq2$, $s\\geq1$ and $n\\geq r(s+1)$ be integers. If $\\mathcal F\\subseteq\\binom{[n]}r$ contains no $s+1$ pairwise disjoint members, must\n\n$$\n|\\mathcal F|\\leq\\max\\left\\{\\binom{r(s+1)-1}{r},\\binom nr-\\binom{n-s}{r}\\right\\}?\n$$\n\nThe two candidate extremal families are all $r$-sets inside an $(r(s+1)-1)$-set and all $r$-sets meeting a fixed $s$-set. This is the Erdős matching conjecture (1965), recorded as Problem #1020. This entry concerns only the four-uniform specialization, $r=4$; it does not claim to settle arbitrary uniformity. The tracker's forbidden matching parameter is $k=s+1$.","posedBy":"Paul Erdős, A problem on independent r-tuples (1965); Erdős Problem #1020","yearPosed":1965,"ageNote":"1965 dates the general Erdős matching conjecture. This entry concerns only r=4; it does not close Problem #1020 in arbitrary uniformity.","solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"For $r=4$, the manuscript claims the sharp extremal value\n\n$$\n\\max\\left\\{\\binom{4s+3}{4},\\binom n4-\\binom{n-s}{4}\\right\\}\n$$\n\nfor every $s\\geq1$ and $n\\geq4s+4$. Both terms are attained by the standard clique and cover constructions. The claimed advance is an all-parameter four-uniform theorem, extending the sufficiently-large matching-number range of Hou–Hu–Liu ($s\\geq6961$) while retaining and crediting their framework and eight computations. The proof combines finite exact certificates and exhaustive searches with a written combinatorial reduction and analytic propagation.\n\nAdditional equality and stability conclusions have restricted domains; no classification of every extremizer or arbitrary-family stability is claimed. This is a claimed complete r=4 result and a partial result toward the general Erdős matching conjecture. Uniformity five and arbitrary uniformity are not settled by this paper.","claimIssueNote":null,"solveDate":"2026-09","model":"GPT-5.6 Sol; Astra; GPT-5 Pro; GPT-6 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The author reports using GPT-5.6 Sol, Astra, GPT-5 Pro and GPT-6 Pro through ChatGPT. Under the author's direction, AI systems explored proof approaches, proposed lemmas and counterchecks, developed exact certificates and verification code, drafted and revised the manuscript, and conducted adversarial reviews. The public manuscript explicitly discloses scientific and computational assistance, beyond language editing. The model names are supplied by the author; the manuscript does not identify their versions or assign individual results to particular models. AI-assisted is the conservative classification on this disclosure. The author selected the research scope and is responsible for the final claims. Reviews within this workflow are not independent expert verification.","verification":"unreviewed","verificationNote":"Checked here on 13 September 2026, at the sources rather than from the submission. erdosproblems.com/1020 carries the conjecture in the form stated and still marks it open; the tracker's forbidden-matching parameter is $k$ where this entry uses $s+1$, and substituting turns one formula into the other, so the transcription is faithful. Hou, Hu and Liu (arXiv:2605.26060) state in their abstract \"We prove the 4-uniform Erdos Matching Conjecture for every matching number $s\\ge 6961$\", so the standing state of the four-uniform case really was all sufficiently large $s$, and the claimed advance - closing the remaining finite range - is well defined.\n\nThe mathematics was not checked. The author's own audit of 12 September records replay of all 27 computational obligations including the eight retained Hou-Hu-Liu searches, 52 tests, 12 receipt-integrity mutation controls, a matching PDF rebuild and an anonymous clone matching all 97 reviewed files. That supports reproducibility and the declared finite computations; it does not validate the written reduction and induction, and no independent domain expert endorsement or formal proof is supplied. Announcement reflects repository-only publication.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Scores the Erdos matching conjecture itself, not the four-uniform case this entry settles. A named 1965 conjecture of Erdos with sixty years of partial results behind it - Erdos-Gallai for graphs, then Kleitman, Frankl, Frankl-Rodl-Rucinski and Frankl-Kupavskii across successive ranges of n and of the matching number. Tied at 40 with the planar unit-distance problem, Sendov's conjecture and Petersen colouring, all named problems famous inside their communities; below Erdos #1 on distinct subset sums at 50 and Erdos-Sos at 58.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/Four-Uniform-Erdos-Matching-Conjecture/blob/60ce10893953bcf3ac0642d992b18dc0ac41d5b9/paper/Four-Uniform-Erdos-Matching-Conjecture.pdf","sourceName":"GitHub manuscript: A computer-assisted proof of the four-uniform Erdős matching conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Paper source, certificates and reproduction instructions","url":"https://github.com/aconsciousfractal/Four-Uniform-Erdos-Matching-Conjecture/tree/60ce10893953bcf3ac0642d992b18dc0ac41d5b9","kind":"code"},{"label":"Erdős Problem #1020: original problem and current scope","url":"https://www.erdosproblems.com/1020","kind":"problem-record"},{"label":"Hou–Hu–Liu: the four-uniform bound for s >= 6961","url":"https://arxiv.org/abs/2605.26060v1","kind":"paper"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"unrestricted-multiplicative-complexity-mul4","name":"Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication","shortName":"Unrestricted multiplicative complexity of Mul4","problemNumber":null,"field":"Circuit complexity","fieldGroup":"Theoretical computer science","statement":"Let Mul4: F_2^8 -> F_2^7 output the seven coefficients of the product of two four-term binary polynomials. The manuscript proves that its unrestricted XOR--AND multiplicative complexity is exactly 9. The upper bound is a nine-AND Karatsuba--Ofman construction. The lower bound rules out every unrestricted eight-AND circuit, including circuits that reuse nonlinear intermediate wires and exploit Boolean idempotence. Thus, for this natural vector-valued quadratic function, allowing nonlinear feedback does not improve on the optimal quadratic circuit.","posedBy":"Joan Boyar and Magnus Gausdal Find","yearPosed":2014,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"The exact Mul4 instance is resolved: its unrestricted XOR--AND multiplicative complexity is 9. The restricted bilinear/quadratic value 9 was classical; the new theorem proves that arbitrary nonlinear Boolean reuse cannot lower it. This is a natural positive special case of the Boyar--Find question, not a solution of the general (n,m) problem. The lower bound is a conceptual structural argument. A complete Lean 4 formalization checks the unrestricted circuit semantics and the exact equality, while the development-time Python/C++ programs remain independent regression checks rather than logical premises. The corresponding unrestricted questions for five or more terms remain open. Independent statement-fidelity review remains pending.","claimIssueNote":null,"solveDate":"2026-08-31","model":"OpenAI GPT-5.6 Sol (extra-high); Anthropic Opus 5 (high, referee)","modelMaker":"OpenAI; Anthropic","humanCollaborators":[],"aiRole":"OpenAI GPT-5.6 Sol in extra-high thinking mode was used for research, proof exploration and development, computational checking, Lean 4 formalization and formal-proof engineering, literature and citation verification, manuscript drafting and revision, and release and submission preparation. Anthropic Opus 5 in high thinking mode was used as a critical referee, not as the discovery model. The public disclosure does not attribute a named essential lemma or the central informal proof solely to a model, so the conservative AI-assisted tier is retained rather than AI co-developed or AI-discovered. Gregory Morse directed and reviewed the mathematical claims, informal and formal proofs, computations, citations, code, and manuscript text and assumes full responsibility.","verification":"site-confirmed","verificationNote":"Site-confirmed: rebuilt here on 3 September 2026, not taken from the author's CI badge. This site's verify-lean workflow checked out release n4-arxiv-v2 (commit 2ebc0cf4) of GregoryMorse/unrestricted-boolean-mul, installed the toolchain the repository pins (leanprover/lean4:v4.32.1), ran $\\texttt{lake build}$ over every module, then the project's own AxiomAudit.lean, then $\\texttt{lake env leanchecker}$ replaying the whole UnrestrictedBooleanMul environment. 24m40s, every step green.\n\nAll six headline theorems - the exact values for $n = 0,1,2,3$, the eight-gate impossibility and $\\mathrm{MC}(\\mathrm{Mul}\\,4) = 9$ - report $\\texttt{propext}$, $\\texttt{Classical.choice}$ and $\\texttt{Quot.sound}$ and nothing else. No $\\texttt{sorryAx}$, no project axiom, no $\\texttt{native\\_decide}$, no $\\texttt{bv\\_decide}$.\n\nThe release tag is one commit ahead of 1533276b, which the author's own run used; that commit touches only READMEs, a citation file and checksum lists, no Lean source.\n\nNot Lean-verified. The kernel checks the proof against the statement the author wrote; whether that statement expresses the Boyar-Find question is the anchoring half, and nobody without a stake has audited it.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"The Boyar-Find question - whether quadratic vector-valued Boolean functions over F2 are always computed optimally by quadratic circuits - is a real open question in circuit complexity, and AND-gate count is the cost metric that matters in secure computation and FHE. But this settles one instance, the four-term case, not the classification. Tied with the (2,1)-gapped consecutive-ones case at 12, likewise a single open parameter case of a broader classification with a specialist audience. Below the Tu-Deng conjecture at 15, a named conjecture with its own trail of partial results.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.30238","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Boyar--Find finite-field question","url":"https://arxiv.org/abs/1407.6169v1","kind":"problem-record"},{"label":"Complete Lean proof and immutable verification release","url":"https://github.com/GregoryMorse/unrestricted-boolean-mul/releases/tag/n4-arxiv-v2","kind":"lean-proof"},{"label":"Headline theorem MC(Mul 4) = 9","url":"https://github.com/GregoryMorse/unrestricted-boolean-mul/blob/n4-arxiv-v2/UnrestrictedBooleanMul/N4/Main.lean","kind":"lean-statement"},{"label":"Rebuilt here: lake build, axiom audit and leanchecker replay","url":"https://github.com/mrconter1/vibemathed/actions/runs/33722071409","kind":"independent"}],"relations":[],"submittedBy":"SwiftFerret455","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"a-counterexample-to-the-stable-forking-conjecture","name":"A Counterexample to the Stable Forking Conjecture","shortName":"Stable Forking Conjecture","problemNumber":null,"field":"Model theory","fieldGroup":"Logic & foundations","statement":"Using ChatGPT 5.6, we find a counterexample to the stable forking conjecture. This answers a long-standing open question of Hart, Kim and Pillay (1996).","posedBy":"Bradd Hart, Byunghan Kim, and Anand Pillay","yearPosed":1996,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if $a \\not\\downarrow_C b$ then there is $\\varphi(x,\\bar b) \\in \\mathrm{tp}(a/Cb)$ forking over $C$ whose parameter-free form $\\varphi(x,y)$ is stable.\n\nThe counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph algebra of the random graph. Forking is characterised by an abstract independence relation, and the random graph is encoded into that relation so that it has the order property; stable formulas therefore cannot determine all forking in simple theories.","claimIssueNote":null,"solveDate":"2026-08-31","model":"GPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["James Freitag; Scott Mutchnik"],"aiRole":"GPT-5.6 Sol generated the counterexample through an interactive, human-guided search. Freitag and Mutchnik prompted it with structural restrictions any counterexample would have to satisfy, including using the Kim-Pillay abstract-independence criterion to prove simplicity and characterise forking, and directed the search toward infinite rank - earlier work having placed severe obstructions on a finite-rank example. They independently wrote the proofs and the manuscript.\n\nThe paper says it plainly, in the abstract and again in the text: \"This is an AI-generated result proven with the help of GPT-5.6 Sol. Specifically, we prompted ChatGPT to construct a counterexample to the stable forking conjecture with a detailed series of prompts which took into account the likely restrictions such a counterexample would have to satisfy.\" They add that \"in retrospect, it seems unlikely that this counterexample would have been found in the near term without the use of generative AI\".","verification":"unreviewed","verificationNote":"Unreviewed. A preprint three days old. The construction is explicit and the authors give the abstract-independence argument in full, so it is checkable by a model theorist, but none has done so on the record.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":26,"significanceNote":"The stable forking conjecture is a founding problem of neostability theory, posed by Hart, Kim and Pillay in 1996 and worked on for thirty years; simple theories and forking are the subject's core machinery, so this is close to field-famous within model theory. Just below the Lonely Runner Conjecture and the Polynomial-Time Low-Degree Conjecture at 30, both of which are known well outside their home fields; clearly above Courtade-Kumar at 22.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.00436","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":1,"downvotes":0,"commentCount":1},{"slug":"bugeaud-problem-10-61","name":"$(\\xi\\alpha^n)_{n\\ge1}$ is not uniformly distributed modulo one for Pisot $\\alpha$ and $\\xi$ in the Cantor set $C(\\alpha)$","shortName":"Bugeaud Problem 10.61","problemNumber":null,"field":"Equidistribution mod 1","fieldGroup":"Number theory","statement":"Bugeaud's Problem 10.61, due to Michel Mendès France in 1967: for a Pisot number $\\alpha > 2$ and the Cantor set $C(\\alpha) = \\{(\\alpha-1)\\sum_{k\\ge1}\\varepsilon_k\\alpha^{-k} : \\varepsilon_k \\in \\{0,1\\}\\}$, no $\\xi \\in C(\\alpha)$ has $(\\xi\\alpha^n)_{n\\ge1}$ uniformly distributed modulo one.\n\nThe problem itself remains open. What is proved is a set of criteria for it, and two instances. The criteria: a reduction to symbolic dynamics that is an equivalence; a pressure criterion; and a covering criterion which, for a quadratic setup of norm $b$, applies exactly when $(\\log_2\\alpha - 1)(\\log_2(\\alpha/|b|) - 1) > 1$, a condition that reduces to $\\alpha > 4$ for units. The two instances are both quadratic: at $\\alpha = 2+\\sqrt5$ in the strong form, an explicit interval that every orbit misses at every time, and at $\\alpha = 2+\\sqrt3$ by a confinement-gap certificate.","posedBy":"Michel Mendès France","yearPosed":1967,"ageNote":"Posed by Michel Mendès France in 1967, and restated as Problem 10.61 in Bugeaud's Distribution Modulo One and Diophantine Approximation (Cambridge, 2012), which is where it is usually cited from.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"The problem is open, and the repository says so: what is proved are criteria for it and two instances, not the general case.\n\nEvery compared statement that concludes Problem 10.61 does so for a quadratic setup - a real root $\\alpha > 1$ of $X^2 - aX - b$ whose conjugate has modulus below one - and both instances are quadratic. The arbitrary-degree material is conditional ingredients: for the family $X^d - aX^{d-1} - 1$ the real root exceeding $a$ is shown to be Pisot for $a \\ge 3$, with a conjugate-modulus bound and a numerical inequality. No compared statement carries those above degree two, because the covering criterion is proved only for quadratic setups.\n\nThe covering criterion also leaves quadratic $\\alpha$ with route-A exponent at least one undecided, about which nothing is claimed.","claimIssueNote":null,"solveDate":"2026-08-31","model":"Fable 5, Opus 5","modelMaker":"Anthropic","humanCollaborators":["Ralf Stephan"],"aiRole":"Directed by a human, the model planned and executed the discovery, the formalization and the write-up. The repository's formalization.yaml records the work as \"(C) 2026 Ralf Stephan, in collaboration with Claude Code\", and describes paper.pdf there as machine-written notes documenting the Lean development rather than a prior paper the formalization followed.","verification":"lean-checked","verificationNote":"Lean-checked, statement unaudited. Registered in the Palomar registry as PALOMAR-2026-08-31-000013: status registered, trust high, pinned to commit d61132ff, mirrored to PalomarArchive.\n\nSeventeen statements are compared with leanprover/comparator, which checks three things per theorem - that the statement in Solution is definitionally the same statement as in the trusted Challenge module, compared constant by constant; that the proof uses no axiom outside a permitted list; and that the resulting environment is re-accepted by the Lean kernel. All seventeen registered theorems sit in the axiom-free lane, permitting only propext, Quot.sound and Classical.choice.\n\nThe repository maintains a second lane permitting one cited literature input, LY.entropyRate_floor. Two theorems consume it and neither is among the registered seventeen; the $\\alpha = 2+\\sqrt3$ instance is registered in its axiom-free form.\n\nNot Lean-verified, on the anchoring half. A Palomar listing is a strong precondition rather than the anchoring itself: it makes the audit cheap, but nobody without a stake has checked that the formal statements say what Problem 10.61 says, and Palomar states plainly that a listing is not a certificate of novelty or relevance.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A numbered problem in Bugeaud's 2012 Cambridge tract on distribution modulo one, originally posed by Mendès France in 1967, so nearly sixty years standing in a standard reference for the field. Placed just above Problem 3 of Dubickas (2006) at 20, which sits in the same corner of distribution mod 1 but is twenty years old and lives in a journal problem list rather than a tract. Well below the Lonely Runner Conjecture at 30, which is famous across two fields and carries its own Wikipedia article.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/rwst/Pisot-Cantor-61/blob/main/paper.pdf","sourceName":"rwst/Pisot-Cantor-61 (paper.pdf)","collection":null,"collectionVersion":null,"links":[{"label":"Registry link","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-08-31-000013","kind":"palomar"},{"label":"Github repo","url":"https://github.com/rwst/Pisot-Cantor-61","kind":"code"},{"label":"Preprint","url":"https://www.researchgate.net/publication/414009170_Criteria_for_the_non-equidistribution_of_xan_on_the_Cantor_set_Ca","kind":"paper"}],"relations":[],"submittedBy":"LucidKestrel185","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"a-complex-structure-on-s-2-times-s-4","name":"A complex structure on $S^2\\times S^4$","shortName":"A complex structure on $S^2\\times S^4$","problemNumber":null,"field":"Differential geometry","fieldGroup":"Geometry & topology","statement":"On page 408 of Calabi's 1958 paper \"Construction and Properties of Some 6-Dimensional Almost Complex Manifolds\" we find: \n\n\"...it is still unknown whether the sphere $S^6$ or any product manifold $V^2 \\times S^4$ ($V^2 = $ any closed, orientable surface) admit any complex analytic structure.\"","posedBy":"Eugenio Calabi","yearPosed":1958,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The authors give an explicit construction of a complex manifold (called $X_+$) which is diffeomorphic to $S^2 \\times S^4$.","claimIssueNote":null,"solveDate":"2026-08-30","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Ruiming Liang","Chenhan Liu","Yang Zhang"],"aiRole":"The human authors write the following: \n\n\"Statement on the use of AI. At first, GPT 5.6 Sol used the fibration method similar to [7] to construct a complex manifold $X$ fibreed over $P^1$ and claimed that it is homeomorphic to $S^2 \\times S^4$. However, the proof provided by GPT 5.6 Sol is cumbersome, incomplete, and misses an essential lemma. Then by comparing the singular fibre of $X$ with the singular fibres of $X_{sph}$, we discovered that that $X$ is likely to be obtained by the blowup-quotient-flop process introduced in this paper, which led to the main theorem 1.2 of this article. Wall’s classification result is also introduced to the authors by GPT 5.6 Sol. \n\nThe proofs of lemma 2.3, lemma 3.1, proposition 4.6 are first generated by GPT 5.6 Sol, but it is checked and rewrote by the authors. \n\nThis article is also polished by GPT 5.6 Sol on grammars, language, notations, conventions, and some graphs.\"","verification":"unreviewed","verificationNote":null,"publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":null,"significanceNote":null,"solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.29651","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"Saul Schleimer","upvotes":1,"downvotes":0,"commentCount":1},{"slug":"dean-s-conjecture-for-k-5","name":"Dean's conjecture for $k=5$","shortName":"Dean $k=5$","problemNumber":null,"field":"Graph Theory","fieldGroup":"Combinatorics","statement":"Every finite simple graph $G$ with minimum degree $\\delta(G)\\ge 5$ contains a simple cycle $C$ whose length satisfies $|C|\\equiv 0\\pmod 5$.","posedBy":"Nathaniel Dean","yearPosed":1988,"ageNote":"Attributed to Nathaniel Dean, and 1988 is plausible but uncorroborated here: the paper cites nine references, none of them Dean's original, and Luo, Ma and Zhao date the conjecture only as \"three decades ago\", nearer the mid-nineties from 2026. The attribution is not in doubt - it is named for him throughout the literature - only the year, which wants a primary source.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claims the last open case of Dean's conjecture: every finite simple graph with minimum degree at least five contains a cycle of length divisible by five. Verified independently against Luo, Ma and Zhao (arXiv:2601.13552), whose abstract confirms the landscape - the conjecture was known for $k\\in\\{3,4\\}$ and they proved every $k\\ge6$, leaving $k=5$ open. If this proof stands, Dean's conjecture holds for all $k\\ge3$.\n\nNine finite proposition families in the bipartite and triangle-free branches are computer-assisted, with verifiers and certificates in a separately archived supplement. The reductions from arbitrary graphs to those finite state spaces are prose arguments in the paper and are not machine-checked.\n\nThe claim has not been refereed, and no independent mathematician has audited the graph-theoretic core. The author describes extensive model-assisted hostile auditing of his own argument, which is worth something but is not external review, and says so plainly.","claimIssueNote":null,"solveDate":"2026-08-29","model":"GPT-5.6 Sol (primary); Claude Opus 5; GLM 5.3 Flash","modelMaker":"OpenAI (primary); Anthropic; Z.ai","humanCollaborators":[],"aiRole":"GPT-5.6 Sol was the primary proof-development and hostile-audit system. It contributed substantial portions of the graph-theoretic reductions and terminal arguments, audited smoothing-edge scope, path and cycle simplicity, branch exhaustiveness, and graph-to-certificate coverage, and assisted in constructing and checking the finite verifiers. Claude Opus 5 was used as an ancillary prover for certain problems. GLM 5.3 Flash agents were used with the Danus framework earlier on in the problem-solving process.","verification":"site-confirmed","verificationNote":"Site-confirmed: the computational supplement was replayed here in full on 30 August 2026, and it passes.\n\nWhat was run. Downloaded the supplement from its own Zenodo record (10.5281/zenodo.22167084), checked all 86 files against the shipped $\\texttt{MANIFEST\\_SHA256.txt}$ - 86 match, 0 mismatch, 0 missing - then executed its $\\texttt{run\\_all.ps1}$ driver end to end. Result: \"All certificate runs passed: 47\", exit 0, in 33 min 53 s. Every per-run line reported $\\texttt{exit=0}$, and the only occurrences of \"fail\" in the fresh verification record are four instances of \"failures: 0\". Both toolchains ran, Python and JavaScript, including the bipartite verifiers, whose output reports \"checked rows: 580, failures: 0\" and \"verified rich pairs: 78\".\n\nWhat this does NOT establish, and it is the larger half. In the supplement's own words: \"The programs verify only the finite propositions listed in PROPOSITION_MAP.md. The reductions from arbitrary graphs to those finite state spaces, and the proofs that a reported forbidden object expands to a simple cycle or path in the original graph, are mathematical arguments in the paper. The programs do not replace those graph-to-state theorems.\"\n\nSo the nine computer-assisted proposition families are exactly what the author says they are, and reproducibly so. The argument that carries them to every graph of minimum degree five is unrefereed prose that no independent mathematician has read. That is why the entry remains a candidate.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"Dean's conjecture is a named problem in extremal graph theory that stood for roughly three decades, and it is live rather than dormant: Luo, Ma and Zhao settled every $k\\ge6$ earlier in 2026, which is what left $k=5$ as the last case. Completing a conjecture of that standing is a substantial result, and the score reflects the problem rather than the confidence - the candidate status and unreviewed tier carry the latter. Held below the top band because it is a single-author unrefereed preprint whose central reductions nobody independent has read.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://zenodo.org/records/22182448","sourceName":"Cycles of length divisible by five in graphs of minimum degree five - The k=5 case of Dean's conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Computational supplement: verifiers, certificates, manifest","url":"https://doi.org/10.5281/zenodo.22167084","kind":"paper"},{"label":"Luo, Ma and Zhao - Dean's conjecture for every k >= 6","url":"https://arxiv.org/abs/2601.13552","kind":"paper"}],"relations":[],"submittedBy":"eli","upvotes":1,"downvotes":0,"commentCount":1},{"slug":"improving-randomized-metric-distortion-to-2-3282","name":"Improving Randomized Metric Distortion to 2.3282","shortName":"Randomized metric distortion 2.3282","problemNumber":null,"field":"Game theory","fieldGroup":"Algorithms & optimization","statement":"In metric social choice, voters rank candidates by distance in an unknown metric space, while a randomized voting rule must use only these rankings. The paper introduces random-size stable lotteries and proves that, by mixing a suitably chosen random-size stable lottery with Integrated Veto, one obtains a randomized voting rule with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$. The proof combines infinite-dimensional conic linear-programming duality, heuristic nonlinear optimization, and exact rational verification using polynomial nonnegativity in the Bernstein basis.","posedBy":null,"yearPosed":null,"ageNote":"The paper gives no origin for the question of the optimal randomized distortion, so no year is recorded. The randomized side of metric distortion has been studied at least since Anshelevich and Postl's 2016 work on randomized social choice under metric preferences, which gave the first bounds.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"The paper proves that there exists a randomized voting rule using only ordinal rankings with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$ and closes about $44\\%$ of the gap to the known asymptotic lower bound of approximately $2.1126$. It does not determine the optimal randomized metric distortion: for $m\\ge4$, the exact optimum and its asymptotic limit remain open.","claimIssueNote":null,"solveDate":"2026-08-29","model":"GPT-5.6 Sol; Claude Opus 5.0","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Nisarg Shah"],"aiRole":"GPT-5.6 Sol derived all mathematical proofs in the paper from research directions, literature connections, proof and search strategies, and inspiration supplied by Nisarg Shah. It autonomously introduced stable-lottery ingredients, developed progressively stronger bounds, and derived the proofs leading to $2.3282$. Shah then generalized one proposed lottery to random-size stable lotteries, guided the search over distributions, verified all final mathematical details, and rewrote and simplified the exposition with GPT-5.6 Sol and Claude Opus 5.\n\nThe disclosure is in the paper itself, not only in this entry: \"All the proofs in this document were obtained using GPT-5.6-Sol with guidance from the author.\"","verification":"unreviewed","verificationNote":"Unreviewed. The author states that he verified all final mathematical details; by this site's ladder an author's own check does not move the tier, however expert, and Shah is among the leading researchers on metric distortion. The bound $11641/5000$ rests, per the abstract, on an exact rational verification via polynomial nonnegativity in the Bernstein basis, which is checkable in principle but has not been re-run here. No referee and no formalization.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"The optimal randomized metric distortion is the central remaining open question of the metric distortion programme in computational social choice; its deterministic half was a well-known conjecture settled at 3 in 2020. A tracked constant with a documented ladder, 3 to 2.753 to 2.5 and now 2.3282, against a lower bound near 2.1126. Above the Max-k-CSP approximation constant at 15, a similar tracked ladder with a narrower audience; below Courtade-Kumar at 22, one of the best-known open problems in analysis of Boolean functions.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.29308","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-problem-126-prime-divisors-of-pairwise-sums","name":"Erdős Problem #126: prime divisors of pairwise sums","shortName":"Erdős #126","problemNumber":126,"field":"Elementary number theory","fieldGroup":"Number theory","statement":"Let $f(n)$ be maximal such that for every $n$-element set $A\\subseteq\\mathbb N$,\n$$\n\\prod_{\\substack{a,b\\in A\\\\a\\ne b}}(a+b)\n$$\nhas at least $f(n)$ distinct prime factors. Erdős asked whether\n$$\n\\frac{f(n)}{\\log n}\\to\\infty.\n$$\nThe answer is yes.","posedBy":"Paul Erdős","yearPosed":1934,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"For\n$$\nf(n)=\\min_{|A|=n}\\left|\\left\\{p\\text{ prime}:p\\mid a+b\\text{ for some distinct }a,b\\in A\\right\\}\\right|,\n$$\nAstra formally proves\n$$\n\\frac{f(n)}{\\log n}\\to\\infty.\n$$\n\nThe repository contains substantially stronger proofs. In particular, one verified alternate resolution establishes\n$$\nn\\le 3r^2,\n$$\nwhere $r$ is the number of supporting primes, yielding\n$$\nf(n)\\gg n^{1/2}.\n$$\nOther independent resolutions give exponents $1/3$, $1/5$, and $1/8$. Thus the formal work goes well beyond the qualitative conjecture, although only the limit statement is the registered benchmark theorem.","claimIssueNote":null,"solveDate":"2026-08-28","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"GPT-6 Astra autonomously solved the Formal Conjectures benchmark statement and wrote the Lean proofs, with no human seeing or steering the proof search. Remarkably, four independent successful runs produced polynomial lower bounds by apparently three distinct elementary arguments. The primary proof gives $f(n)\\gg n^{1/8}$; alternate proofs give $f(n)\\gg n^{1/3}$, $f(n)\\gg n^{1/2}$, and $f(n)\\gg n^{1/5}$. Only the weaker limit $f(n)/\\log n\\to\\infty$ is advertised and Comparator-checked as the benchmark theorem.","verification":"lean-verified","verificationNote":"Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos126 at abd4239: 7,866 lines of Lean, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; Comparator configuration present and CI runs it with only propext, Quot.sound and Classical.choice. The statement is copied verbatim from Formal Conjectures' ErdosProblems/126.lean at commit 488aade2. Only the qualitative limit f(n)/log n -> infinity is the compared theorem; the polynomial bounds (exponents 1/8, 1/3, 1/2, 1/5 across four runs) are stronger internal results, and erdosproblems.com's page records the n^(1/2) one. erdosproblems.com, the field's own record, marks the problem PROVED (LEAN) with a proof exposition by Thomas Bloom, which is why this is Resolved rather than Candidate: the canonical tracker has accepted it.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"From the first Erdős–Turán paper (1934), with a $250 prize: does the number of primes dividing some pairwise sum of an n-set grow faster than log n? Ninety-two years open, and the answer is not just yes but polynomial, f(n) >> n^(1/2) in the strongest of four independent runs. A precise question rather than a structural conjecture, hence the middle of the scale.","solveCostUsd":247,"solveCostNote":null,"sourceUrl":"https://github.com/tadamcz/erdos126","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"erdosproblems.com/126: status and Thomas Bloom's proof exposition","url":"https://www.erdosproblems.com/126","kind":"problem-record"},{"label":"Challenge.lean: the compared statement","url":"https://github.com/tadamcz/erdos126/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Solution.lean and the Lean development","url":"https://github.com/tadamcz/erdos126/blob/main/Solution.lean","kind":"lean-proof"},{"label":"Epoch AI, Announcing FrontierMath Erdős (1 September 2026)","url":"https://epoch.ai/latest/announcing-frontiermath-erdos","kind":"announcement"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"non-mf-groups-and-non-finite-full-group-c-algebras","name":"Non-MF groups and non-finite full group C*-algebras","shortName":"Explicit non-MF groups","problemNumber":null,"field":"Operator algebra","fieldGroup":"Algebra","statement":"Let $\\Gamma$ be a property (T) group admitting an injective, non-surjective endomorphism and let $G$ be the associated ascending HNN-extension. Let\n$$\nW=\\left(\\bigoplus_{G/\\Gamma}\\mathbb Z/2\\mathbb Z\\right)\\rtimes G.\n$$\nWe show that $W$ is not an MF group and that $C^*(G)$ is not a finite $C^*$-algebra. The ideas and proofs were generated by ChatGPT 5.6 Sol, we have only refined their arguments in a hopefully more palatable form.","posedBy":"Whether a stably finite C*-algebra must be MF, and whether every full group C*-algebra is finite; questions of the C*-algebra literature the paper's introduction cites","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"If $\\Gamma$ is a non-coHopfian property-(T) group and $G$ is its ascending HNN extension, then the full group algebra $C^*(G)$ is not finite. Indeed, Kazhdan projections $p_H0$ such that every sum-distinct set $A\\subseteq\\{1,\\dots,N\\}$ satisfies\n$$\nN>C\\,2^{|A|}.\n$$\nThe conjecture is false: for every $\\varepsilon>0$ there are arbitrarily large $n$ and sum-distinct sets $A\\subseteq\\{1,\\dots,N\\}$ with\n$$\n|A|=n,\\qquad N\\le \\varepsilon 2^n.\n$$","posedBy":"Paul Erdős","yearPosed":1931,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The formal theorem proves that no universal constant $C>0$ can satisfy\n$$\nN>C\\,2^{|A|}\n$$\nfor every nonempty interval bound $N$ and every sum-distinct $A\\subseteq\\{1,\\dots,N\\}$.\n\nEquivalently, for every $\\varepsilon>0$ there are arbitrarily large $n$ and sum-distinct $n$-element sets contained in $\\{1,\\dots,N\\}$ with\n$$\nN\\le\\varepsilon 2^n.\n$$\n\nThe proof is ineffective: it establishes the existence of arbitrarily large such $n$ but gives no explicit bound for how large $n$ must be in terms of $\\varepsilon$.","claimIssueNote":null,"solveDate":"2026-08-28","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"A pre-release GPT-6 Astra autonomously solved the Formal Conjectures benchmark statement with no human steering during proof search. The primary run constructs, for large $n$, rational $n\\times n$ matrices with small determinant and additional admissibility properties, then uses integral changes of basis, saturated bidiagonal perturbations, and a binary-block construction to obtain sum-distinct sets with $N\\le\\varepsilon 2^n$. An independent larger-budget Astra run found an essentially equivalent lattice-based argument. Astra also wrote the Lean proofs; Claude was later used to prepare repository documentation from the completed runs.","verification":"lean-verified","verificationNote":"Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos1 at db6f909: 4,608 lines of Lean, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; Comparator configuration present and CI runs it with only propext, Quot.sound and Classical.choice. The statement is copied verbatim from Formal Conjectures' ErdosProblems/1.lean at commit 488aade2, the human-curated formalization the model was given, and the proved theorem is its negation. Two independent runs found essentially equivalent lattice-based arguments. The proof is ineffective: it gives no bound on how large n must be. erdosproblems.com, the field's own record, marks the problem DISPROVED (LEAN) with a proof exposition by Thomas Bloom, which is why this is Resolved rather than Candidate: the canonical tracker has accepted it.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"Erdős called it \"perhaps my first serious problem\" and dated it to 1931; it opens his problem collection and carried a $500 prize. Whether a sum-distinct set of n integers must reach about 2^n was believed true for ninety-five years; the answer is no, ineffectively. Level with Smale's mean value conjecture here rather than the great structural conjectures, since the question is sharp but narrow.","solveCostUsd":405,"solveCostNote":null,"sourceUrl":"https://github.com/tadamcz/erdos1","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"erdosproblems.com/1: status and Thomas Bloom's proof exposition","url":"https://www.erdosproblems.com/1","kind":"problem-record"},{"label":"Challenge.lean: the compared statement","url":"https://github.com/tadamcz/erdos1/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Solution.lean and the Lean development","url":"https://github.com/tadamcz/erdos1/blob/main/Solution.lean","kind":"lean-proof"},{"label":"Epoch AI, Announcing FrontierMath Erdős (1 September 2026)","url":"https://epoch.ai/latest/announcing-frontiermath-erdos","kind":"announcement"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"large-systoles-in-every-sufficiently-large-genus","name":"Large systoles in every sufficiently large genus","shortName":"Large systoles of hyperbolic surfaces","problemNumber":null,"field":"Hyperbolic geometry","fieldGroup":"Geometry & topology","statement":"We show that for every sufficiently large genus $g$, there exists a closed hyperbolic surface $S_g$ with systole $\\mathrm{sys}(S_g)\\geq \\log g-12\\log\\log g$. In particular,\n\n$$\n\\liminf_{g\\to \\infty}\\frac{\\max\\{\\mathrm{sys}(S):S\\in \\mathcal{M}_g\\}}{\\log g}\\geq 1,\n$$\n\nimproving the previously known bound $2/9$. This note is a continuation of our previous work on the diameter of finite covers arXiv:2608.12887, using the same framework of constant-twist pants decomposition to study systoles.\n\nThe proof was developed by GPT-5.6 Sol through an extended discussion with the author.","posedBy":null,"yearPosed":null,"ageNote":"No poser and no year recorded, because the target is a standing asymptotic question rather than a posed conjecture: the paper says only that \"It is still widely open to determine the asymptotic behavior of this maximum\". The modern form dates from the Brooks (1988) and Buser-Sarnak (1994) lower bounds, which is where the $\\limsup\\ge4/3$ and the positivity of the liminf come from.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"The theorem improves the best lower bound valid in *every* sufficiently large genus from asymptotic constant $2/9$ to $1$. The every-genus ladder it climbs is Katz-Sabourau's $19/120$ and then Liu-Petri's $2/9$, the latter also by a random construction. Constant $1$ was already reached by Petri-Walker along a subsequence of genera, following Erdos-Sachs, so the new contribution is achieving it uniformly rather than the constant itself.\n\nThe asymptotic problem stays open, and the remaining gap is wide: Brooks and Buser-Sarnak give $\\limsup\\ge4/3$, while the elementary area bound is $\\max\\mathrm{sys}(S)\\le2\\log(4g-2)$, asymptotically $2\\log g$. So this closes much of the liminf gap and determines no optimal constant.","claimIssueNote":null,"solveDate":"2026-08-27","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Yifei Cai"],"aiRole":"Disclosed twice: in the abstract, and in a dedicated section 1.3 \"Declaration on the use of AI\", which reads in full: \"Starting from the constant-twist pants decomposition approach described in this note, GPT-5.6 Sol (OpenAI) developed the first complete proof of the main theorem through an extended discussion with the author. The proof in this manuscript is checked, simplified and reorganized by the author. The author takes full responsibility for the content and correctness of this manuscript.\"\n\nAI-discovered on that wording: the model produced the proof and the human verified and wrote it up, which is what the tier means. The framework it started from was not the model's - the constant-twist pants decomposition comes from Cai and Luo's earlier work on the diameter of finite covers (arXiv:2608.12887), and the note is explicitly a continuation of it. So the human set the approach and the model built the proof inside it.","verification":"unreviewed","verificationNote":"Unreviewed: an arXiv preprint one day old (v1, 27 August 2026, math.GT), unrefereed, with no formalization and no computational certificate, so there was nothing mechanical to re-run and no mathematics was checked here. What was verified on 28 August 2026: the paper exists at arXiv:2608.26660 with this title and author; the theorem and the $\\liminf\\ge1$ corollary are its abstract and Theorem 1; the AI declaration is section 1.3, quoted in the AI-role note; and every prior-work claim is as the introduction states - Katz-Sabourau's $19/120$, Liu-Petri's $2/9$, Petri-Walker's constant $1$ along a subsequence, Brooks and Buser-Sarnak's $\\limsup\\ge4/3$, and the area bound $\\max\\mathrm{sys}\\le2\\log(4g-2)$.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"The maximal systole in genus $g$ is a classical quantity in hyperbolic geometry and its asymptotic constant is a well-known open question with a long ladder behind it. Moving the every-genus constant from $2/9$ to $1$ is a large step on that ladder and reaches the value previously available only along a subsequence. It is still a bound rather than a resolution, with $4/3$ and $2$ left above it. Placed at 22, in the band with the specialist geometry and mixing entries in the twenties and thirties, below results that settle their question outright.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.26660","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"failure-of-higher-order-truth-within-intuitionistic-propositional-logic","name":"Failure of Higher-Order Truth within Intuitionistic Propositional Logic","shortName":"Higher-order truth in IPL","problemNumber":null,"field":"Intuitionistic propositional logic","fieldGroup":"Logic & foundations","statement":"We answer the question whether all Heyting algebras can appear as the lattice of subterminal objects of an elementary topos in the negative. Concretely, we have shown that the free Heyting algebra on two generators cannot be such a Heyting algebra. The mathematical results in this document were obtained with the help of ChatGPT 5.6 Sol, although the document itself was written entirely by us and we take full responsibility for its contents.","posedBy":null,"yearPosed":null,"ageNote":"No poser and no year, because the paper names none: it introduces the target as \"a long-standing problem in categorical logic\", citing Pitts for a recent summary rather than an original source. The equivalent logical form - whether the truth values of a higher-order intuitionistic theory carry structure beyond a Heyting algebra - is the framing the paper itself gives.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"The paper proves that not every Heyting algebra can occur as the lattice of subterminal objects of an elementary topos. Specifically, the free Heyting algebra $F_2$ on two generators cannot occur.\n\nUsing Bellissima’s representation $F_2\\hookrightarrow\\mathcal O_\\uparrow(K_2)$, the authors construct an upward-closed subset $A\\subseteq K_2$ with $A\\notin F_2$. They show that if some elementary topos $\\mathcal E$ satisfied $\\operatorname{Sub}_{\\mathcal E}(1)\\cong F_2$, then higher-order internal logic would make $A$ definable as a global proposition, forcing $A$ to correspond to an element of $F_2$, a contradiction.\n\nThus no elementary topos has subterminal lattice isomorphic to $F_2$, disproving the claim that every Heyting algebra can arise this way. The paper does not classify which Heyting algebras are realizable.","claimIssueNote":null,"solveDate":"2026-08-27","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Lingyuan Ye","Yiqi Xu"],"aiRole":"The whole disclosure is one sentence of the abstract: \"The mathematical results in this document were obtained with the help of ChatGPT 5.6 Sol, although the document itself was written entirely by us and we take full responsibility for its contents.\" There is no acknowledgements section and no other mention of a model in the paper.\n\nThat sentence does credit the mathematics rather than tooling, which is what puts this in scope: it says the results were obtained with the model's help, not that a model wrote code or checked prose. But it identifies no lemma, construction or step, and a disclosure this general takes the lower tier, so the entry records AI-assisted rather than co-developed. If a later version attributes the obstruction term, the higher-order formula, or their identification to the model, the tier should move up.","verification":"unreviewed","verificationNote":"Unreviewed: an arXiv preprint one day old (v1, 27 August 2026, math.CT), unrefereed, with no formalization, and the argument runs through Bellissima's representation and higher-order internal logic, none of which was checked here. Verified on 28 August 2026: the paper exists at arXiv:2608.26874 with this title and both authors; the statement and the negative answer are its abstract; the target is described in its introduction as \"a long-standing problem in categorical logic\" with Pitts cited for a recent summary and a related positive result of Awodey et al. cited alongside; and the proof strategy is as the entry describes it, constructing $A\\in\\mathcal O_\\uparrow(K_2)\\setminus F_2$ and deriving a contradiction from its definability as a global proposition. The single AI sentence in the abstract is the paper's only mention of a model.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A long-standing problem in categorical logic, answered negatively and cleanly: not every Heyting algebra is the subterminal lattice of an elementary topos, with the free algebra on two generators as the witness. It is a definite settlement of a named standing question rather than a bound, which lifts it above the specialist entries in the low teens; the field is narrow and the result classifies nothing further, which keeps it below the tier at 30 and up. Placed at 18, beside the Auslander-Reiten-Smalo and Keisler-measure entries at 20.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.26874","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"supporting-affine-functionals-for-entanglement-of-formation","name":"Supporting affine functionals for Entanglement of Formation","shortName":"Entanglement of Formation supporting functional","problemNumber":null,"field":"Entanglement theory","fieldGroup":"Quantum information & computing","statement":"The paper disproves the assumption that finite-dimensionality and the convex-roof structure of Entanglement of Formation guarantee a global supporting affine functional at every bipartite state. It gives an explicit degenerate two-qubit state $\\rho$ for which no Hermitian $\\Lambda_\\rho$ satisfies both $E_F(\\rho)=\\mathrm{Tr}\\Lambda_\\rho\\rho$ and $E_F(\\sigma)\\geq\\mathrm{Tr}\\Lambda_\\rho\\sigma$ for every state $\\sigma$. The construction uses the equivalence between existence of such a functional and Lipschitz lower semicontinuity of $E_F$, together with Wootters’ formula.","posedBy":"A.S. Holevo and M.E. Shirokov","yearPosed":null,"ageNote":"No year recorded, and the attribution is unusual: this is a self-correction. The assumption being disproved is one Holevo and Shirokov say they themselves relied on in several of their own earlier articles, so the names in the poser field are the source of the claim rather than of a challenge to it. The abstract's own phrasing is \"In several articles, the authors assume that...\".","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"For two qubits, the paper considers\n\n$\\rho=\\frac12|\\Phi^+\\rangle\\langle\\Phi^+|+\\frac12|01\\rangle\\langle01|$,\n\nwhere $|\\Phi^+\\rangle=(|00\\rangle+|11\\rangle)/\\sqrt2$. This rank-2 state has no global supporting affine functional for Entanglement of Formation.\n\nSetting $\\rho_t=(1-t)\\rho+t|10\\rangle\\langle10|$, Wootters’ formula gives\n\n$C(\\rho_t)=\\frac12-\\sqrt{2t}+O(t)$.\n\nConsequently,\n\n$\\lim_{t\\to0^+}[E_F(\\rho)-E_F(\\rho_t)]/t=+\\infty$,\n\nso $E_F$ is not Lipschitz lower semicontinuous at $\\rho$. By the paper’s criterion, no global supporting affine functional exists there. Thus the claimed universal existence fails even for two qubits, although it remains true for nondegenerate finite-dimensional states.","claimIssueNote":null,"solveDate":"2026-08-27","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"From the abstract: \"We use Wootters' formula and the help of Claude Fable 5 to find a state $\\rho$ of the system $AB$ for which the latter property does not hold.\" The authors had already reduced the existence of a global supporting affine functional to Lipschitz lower semicontinuity, and knew in principle that Wootters' two-qubit formula could yield a counterexample; what they did not have was a practical way to construct one. They report the model found such a state very quickly, and it is the state of Proposition 3.\n\nAI-co-developed rather than AI-discovered, and the distinction is the tier's definition rather than a judgement call: this is a subproblem the authors formulated, inside a proof they set up, which the model solved. The surrounding theory - the equivalence criterion, the existence conditions, the Lipschitz bounds in finite and infinite dimensions - is the authors'.","verification":"site-confirmed","verificationNote":"Site-confirmed: the counterexample was re-derived here on 28 August 2026, in exact arithmetic, from the entry's statement rather than the paper's method. Wootters' concurrence was implemented from scratch and evaluated symbolically on $\\rho=\\frac12|\\Phi^+\\rangle\\langle\\Phi^+|+\\frac12|01\\rangle\\langle01|$ and on $\\rho_t$.\n\nResults. $C(\\rho)=1/2$ exactly. $C(\\rho_t)$ has exact closed form at each rational $t$ - at $t=10^{-6}$ it is $999999/2000000-3\\sqrt{222222}/10^6$, which differs from $\\frac12-\\sqrt{2t}$ by $-4.99\\times10^{-7}$, the $O(t)$ term with coefficient about $-\\frac12$, confirming the paper's expansion. The difference quotient $[E_F(\\rho)-E_F(\\rho_t)]/t$ evaluates to $14.81$, $154.7$, $1550.9$, $15512.7$ and $155131.3$ at $t=10^{-2},10^{-4},10^{-6},10^{-8},10^{-10}$: a factor of ten per two decades, so it grows like $t^{-1/2}$ and diverges. $E_F$ is therefore not Lipschitz lower semicontinuous at $\\rho$, which by the paper's own criterion is exactly the failure claimed.\n\nWhat this does not establish. The equivalence between a global supporting affine functional and Lipschitz lower semicontinuity is the paper's, and was not checked here; nor were the further existence conditions or the infinite-dimensional bounds. The paper is an unrefereed preprint (v1, 27 August 2026, quant-ph) with no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A precise technical correction inside entanglement theory: finite dimensionality plus the convex-roof structure of the Entanglement of Formation does not give a global supporting affine functional at every state, and the failure appears already for two qubits. It matters because the authors and others had used the assumption, and because the paper replaces it with usable conditions and Lipschitz bounds. It is narrow - degenerate boundary states only - and settles no headline question. Placed at 10, with the quantum conditional entropy continuity entry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.27363","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lyons-white-conjecture-rate-monotonicity","name":"The Lyons–White conjecture: rate-monotonicity of $\\ell^{2m}$ distances for random walks on dihedral groups","shortName":"Lyons–White conjecture","problemNumber":null,"field":"Random walks on finite groups; mixing; harmonic analysis on groups","fieldGroup":"Probability & statistics","statement":"Let $D_n$ be the dihedral group of order $2n$ and run a continuous-time random walk on it driven by symmetric jump rates whose support generates the group. Call the pair $(D_n,p)$ rate-monotonic if, at every fixed time, the $\\ell^p$ distance between the walk's distribution and the uniform distribution can only decrease when the rates are increased. Lyons and White (Ann. Probab. 51, 2023) proved this for $p=2$ and $p=\\infty$, found pairs $(D_n,p)$ that fail it for $p$ in $[1,1.997]\\cup[2.001,3.999]\\cup[4.001,5.995]$, and asked whether any pair fails for $p=4$ or $p=6$.","posedBy":"Russell Lyons and Graham White, Monotonicity for continuous-time random walks","yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"No such pair exists: for every positive integer $m$ and every $n$, $(D_n,2m)$ is rate-monotonic (Theorem A), and more generally so is every inversion extension of a finite abelian group by an involution, a family containing the generalized dihedral, dicyclic and generalized quaternion groups (Theorem B). The picture is completed in the other direction: for every real $p\\ge 1$ that is not an even integer there is an $n$ with $(D_n,p)$ not rate-monotonic (Theorem C), so the even integers are exactly the exponents for which monotonicity holds on the dihedral groups.","claimIssueNote":null,"solveDate":"2026-08-27","model":"AxiomProver","modelMaker":"Axiom Math","humanCollaborators":["Colin Defant","Ken Ono"],"aiRole":"The paper's own account: \"The proofs in this paper were generated through human-AI collaboration. In dialogue with AI, the human authors developed and formalized [Theorems A, B and C] with AxiomProver, an AI system currently under development by Axiom Math. In particular, this resulted in a formal Lean certificate for these three theorems.\" Both authors are at Axiom Math. Co-developed rather than assisted because the proofs themselves, not only the formalization, are described as produced in dialogue with the system; not discovered, because the humans directed the work and no autonomous run is claimed.","verification":"lean-checked","verificationNote":"Lean-checked, statement unaudited, as with the other AxiomProver entries here. The repository AxiomMath/LyonsWhite carries a Challenge/Basic.lean statement surface and a Comparator configuration, and its README says the development was verified locally against the challenge; the paper says the formalization \"assumes standard facts from analysis and group theory\" and lists none, so the certificate is conditional on those assumptions and this site has not enumerated them or rebuilt the development. A thirteen-page preprint ten days old, not peer reviewed, no independent reader on record.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A precise question from a 2023 Annals of Probability paper by a leading probabilist, answered completely, with the converse characterisation thrown in. Three years old and in a specialised corner of mixing theory, so a rung above a numbered Erdős problem rather than a named conjecture with a long literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.27708","sourceName":"Proof of the Lyons–White Conjecture (arXiv 2608.27708)","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalization (AxiomMath/LyonsWhite), Comparator-verified against Challenge/Basic.lean","url":"https://github.com/AxiomMath/LyonsWhite","kind":"lean-proof"},{"label":"Challenge/Basic.lean, the statement surface","url":"https://github.com/AxiomMath/LyonsWhite/blob/main/Challenge/Basic.lean","kind":"lean-statement"},{"label":"Lyons and White, Monotonicity for continuous-time random walks (the question)","url":"https://arxiv.org/abs/2109.10386","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"an-exact-all-width-plateau-for-the-three-summand-tu-deng-count-modulo-2-k-1","name":"An Exact All-Width Plateau for the Three-Summand Tu-Deng Count modulo $2^k-1$","shortName":"Three-summand two-zero plateau","problemNumber":null,"field":"Combinatorial number theory; binary digit sums and cyclic carries","fieldGroup":"Combinatorics","statement":"Put $N=2^k-1$ and write $\\mathrm{wt}$ for the binary Hamming weight. Let $F_k(t)$ count the ordered triples $(a,b,c)\\in\\{0,\\dots,N-1\\}^3$ with $a+b+c\\equiv t \\pmod N$ and $\\mathrm{wt}(a)+\\mathrm{wt}(b)+\\mathrm{wt}(c) 11.846$; the conjecture fails again for every $n$ from $786$ to $5000$, and the deficit $\\lambda_1 - D$ grows roughly linearly in $n$, so no additive correction $\\lambda_1 \\le D + C$ survives either.","posedBy":"Graffiti (Siemion Fajtlowicz's program)","yearPosed":1995,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The repository supplies an executable verifier in exact integer arithmetic; for $n = 51$ an integer vector $x$ satisfies $x^{T}Ax > 11\\,x^{T}x$, so $\\lambda_1 > 11 = D$. The deficit $\\lambda_1(n) - D(n)$ grows through $n = 5000$ in the repository's computations, but no asymptotic theorem proving divergence is claimed, so \"false for every constant $C$\" is a computed pattern, not a proved one.\n\nOne anomaly, which the repository records itself: $n = 51$ lies inside the range Graffiti is said to have tested when it made the conjecture.","claimIssueNote":null,"solveDate":"2026-08-26","model":"Claude Opus 5 (AI Village)","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"Claude Opus 5 identified and verified counterexamples to WOW conjecture 806 in the square-free non-coprimality graph family. It reconstructed the intended graph and degree-count invariant from the original Written on the Wall source, searched the family, found the smallest counterexample at $n=51$, and produced exact integer Rayleigh-quotient certificates showing that the largest adjacency eigenvalue exceeds the number of distinct degree values.","verification":"site-confirmed","verificationNote":"Site-confirmed: the repository's verifier, verify_wow1_806.py, was re-run here on 2 September 2026. Fast mode passed 160 assertions and full mode 175, in 21 seconds, with every counterexample certified without floating point - an explicit integer vector $x$ with $x^{T}Ax > D\\,x^{T}x$, which forces $\\lambda_1 > D$ by the Rayleigh principle. For $n = 51$ the certificate is $x^{T}Ax = 1117310362790 > 11 \\cdot 94319113125$.\n\nThe verifier also confirms the graph construction against direct gcd tests, scans every $n \\le 300$ for the least counterexample (it is $51$), and checks the neighbouring conjectures 802, 805, 807 and 808 of the same block as controls, which hold on Graffiti's stated range. Its check against the original Written on the Wall text was skipped here, since the text file is not in the repository; statement fidelity rests on the transcription in the script header and on those controls.\n\nNo independent specialist review.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Graffiti); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://gitlab.com/ai-village-agents/village/graffiti-verification/-/blob/main/verify/verify_wow1_806.py","sourceName":"AI Village graffiti-verification (GitLab)","collection":null,"collectionVersion":null,"links":[{"label":"Repository: 197 refutations, each with a verifier script","url":"https://gitlab.com/ai-village-agents/village/graffiti-verification","kind":"code"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-problem-571","name":"Erdős Problem #571: rational exponents for bipartite Turán numbers","shortName":"Erdős #571","problemNumber":571,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"For every rational $\\alpha\\in[1,2)$, there exists a finite bipartite graph $G$ such that\n$$\n\\operatorname{ex}(n;G)=\\Theta(n^\\alpha).\n$$\nEquivalently, every rational exponent between $1$ and $2$ occurs as the order of growth of the Turán number of a single bipartite graph.","posedBy":"Paul Erdős, Miklós Simonovits","yearPosed":1974,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"For every rational $\\alpha$ satisfying\n$$\n1\\le\\alpha<2,\n$$\nthe Lean theorem constructs some finite $q$ and a bipartite graph\n$$\nG:\\operatorname{SimpleGraph}(\\operatorname{Fin} q)\n$$\nsuch that\n$$\n\\operatorname{ex}(n;G)=\\Theta(n^\\alpha)\n$$\nas $n\\to\\infty$.\n\nThis resolves the single-graph rational-exponents conjecture. Earlier work of Bukh and Conlon proved the corresponding statement only for a finite family of forbidden bipartite graphs, and subsequent work realized many large classes of individual rational exponents. Astra's theorem covers every rational $\\alpha\\in[1,2)$ with one forbidden graph for each exponent.","claimIssueNote":null,"solveDate":"2026-08-26","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"A pre-release GPT-6 Astra autonomously found the proof and wrote its Lean formalization in the FrontierMath Erdős benchmark, with no human seeing or steering the proof search. The roughly 10,000-line development constructs balanced rooted graph models for every rational exponent and proves closure operations that preserve the required extremal-number behavior, including edge subdivision by paths of arbitrary length, adding hubs to the two colour classes, and positive rooted powers. These constructions are combined to realize every rational $\\alpha\\in[1,2)$ by a single finite bipartite graph.","verification":"lean-verified","verificationNote":"Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos571 at 661cc1d: 10,460 lines of Lean, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; Comparator configuration present and CI runs it with only propext, Quot.sound and Classical.choice. The statement was autoformalized for the benchmark and reviewed by Thomas Bloom; it uses mathlib's extremalNumber and IsBipartite and Asymptotics.IsTheta, and the repository's README compares it to the informal statement. One grep hit for the word 'externally' in a docstring is the only match for the risky-feature scan. erdosproblems.com, the field's own record, marks the problem PROVED (LEAN) with a proof exposition by Thomas Bloom, which is why this is Resolved rather than Candidate: the canonical tracker has accepted it.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":52,"significanceNote":"The rational exponents conjecture of Erdős and Simonovits (1974): every rational in [1,2) is the Turán exponent of some single bipartite graph. Bukh and Conlon proved it for finite families in 2018 and a decade of work realised large classes of exponents; the single-graph statement stayed open. Bloom calls the proof the hardest of the five to interpret. A named conjecture with fifty years of literature, a step below Erdős–Sós.","solveCostUsd":617,"solveCostNote":null,"sourceUrl":"https://github.com/tadamcz/erdos571","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"erdosproblems.com/571: status and Thomas Bloom's proof exposition","url":"https://www.erdosproblems.com/571","kind":"problem-record"},{"label":"Challenge.lean: the compared statement","url":"https://github.com/tadamcz/erdos571/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Solution.lean and the Lean development","url":"https://github.com/tadamcz/erdos571/blob/main/Solution.lean","kind":"lean-proof"},{"label":"Epoch AI, Announcing FrontierMath Erdős (1 September 2026)","url":"https://epoch.ai/latest/announcing-frontiermath-erdos","kind":"announcement"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-problem-548-erdos-sos-conjecture","name":"Erdős Problem #548: the Erdős–Sós conjecture","shortName":"Erdos #548 (Erdős–Sós)","problemNumber":548,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Let $n\\ge k+1$. Every graph $G$ on $n$ vertices with\n$$\n|E(G)|\\ge \\frac{k-1}{2}n+1\n$$\ncontains every tree on $k+1$ vertices as a not necessarily induced subgraph.","posedBy":"Paul Erdős and Vera T. Sós","yearPosed":1962,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"For every $n,k$ with $k+1\\le n$, every simple graph $G$ on $n$ vertices satisfying\n$$\n|E(G)|\\ge \\frac{k-1}{2}n+1\n$$\ncontains every tree on $k+1$ vertices.\n\nThe proof counts pairs $(\\pi,j)$ where $\\pi=(v_1,\\ldots,v_n)$ is an ordering of the host vertices and $v_1v_j$ is an edge. There are exactly\n$$\n2|E(G)|(n-1)!\n$$\nsuch pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus\n$$\n(k-1)n!.\n$$\nIf the target tree is absent, the rooted-copy term vanishes and one obtains\n$$\n2|E(G)|\\le (k-1)n,\n$$\ncontradicting the density hypothesis.\n\nIndependent human corroboration, September 2026: Riordan and Scott gave a short proof of the conjecture (arXiv:2609.15893) and Wood published an expository account of the argument (arXiv:2609.17877). Neither used AI. Mubayi and Verstraëte then extended the method, with GPT-6 Astra, to Kalai's 1984 hypergraph conjecture (arXiv:2609.08012) and to a sharp directed analogue (arXiv:2609.10987), both catalogued separately.","claimIssueNote":null,"solveDate":"2026-08-26","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"A pre-release GPT-6 Astra autonomously found the proof and wrote the Lean formalization in the FrontierMath Erdős benchmark, with no human seeing or steering the proof search. The proof uses a permutation-word counting argument: it counts ordered host-vertex configurations whose first vertex is adjacent to a later vertex, then uses two reversible word operations and induction on the target tree. If the tree is absent, the counting inequality yields\n$$\n2|E(G)|\\le (k-1)n,\n$$\ncontradicting the assumed edge density.","verification":"lean-verified","verificationNote":"Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos548 at 3766491: 1,311 lines of Lean, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; Comparator configuration present and CI runs it with only propext, Quot.sound and Classical.choice. The statement was autoformalized for the FrontierMath Erdős benchmark and reviewed by Thomas Bloom, and Challenge.lean is copied from that file; it follows erdosproblems.com's phrasing, which differs from the classical one by requiring one extra edge when (t-2)n is odd, a parity margin the repository's README discusses and the internal counting lemma closes. erdosproblems.com, the field's own record, marks the problem PROVED (LEAN) with a proof exposition by Thomas Bloom, which is why this is Resolved rather than Candidate: the canonical tracker has accepted it.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":58,"significanceNote":"The Erdős–Sós conjecture (1962) is one of the central conjectures of extremal graph theory: every graph with average degree above k-1 contains every tree on k+1 vertices. Sixty-four years, a large literature of special cases, and an announced-but-never-published asymptotic proof by Ajtai, Komlós, Simonovits and Szemerédi. erdosproblems.com now lists it proved. Level with Köthe and non-sofic groups here; below the Jacobian conjecture.","solveCostUsd":363,"solveCostNote":null,"sourceUrl":"https://github.com/tadamcz/erdos548","sourceName":"Github","collection":null,"collectionVersion":null,"links":[{"label":"erdosproblems.com/548: status and Thomas Bloom's proof exposition","url":"https://www.erdosproblems.com/548","kind":"problem-record"},{"label":"Challenge.lean: the compared statement","url":"https://github.com/tadamcz/erdos548/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Solution.lean and the Lean development","url":"https://github.com/tadamcz/erdos548/blob/main/Solution.lean","kind":"lean-proof"},{"label":"Epoch AI, Announcing FrontierMath Erdős (1 September 2026)","url":"https://epoch.ai/latest/announcing-frontiermath-erdos","kind":"announcement"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"rapid-mixing-for-spin-systems-on-graphs-of-girth-at-least-five","name":"Rapid mixing for spin systems on graphs of girth at least five","shortName":"Girth-5 spin-system rapid mixing","problemNumber":null,"field":"Glauber dynamics","fieldGroup":"Combinatorics","statement":"It is proved that, for every $\\delta\\in(0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q\\geq(1+\\delta)\\Delta$ and the underlying graph has girth at least $5$ and maximum degree $\\Delta=\\Omega_{\\delta}(1)$. This result also extends to general multi-spin systems satisfying a local spectral contraction condition, including the anti-ferromagnetic Potts model with $q\\geq(1+\\delta)(1-\\beta)\\Delta$.\n\nThese results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.","posedBy":"Mark Jerrum","yearPosed":1995,"ageNote":"Dated from Jerrum's 1995 paper, which is the standard citation and where the $q>2\\Delta$ bound was proved. Strictly, nobody posed the target as their own: the paper calls $q\\ge\\Delta+2$ \"a long-standing folklore conjecture\" and cites [Jer95] for it, so Jerrum is recorded here as the canonical reference rather than as an individual proposer.","solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"For fixed $\\delta\\in(0,1)$ and all sufficiently large $\\Delta$ depending only on $\\delta$, Glauber dynamics for proper $q$-colorings mixes rapidly on every graph of girth at least $5$ whenever $q\\ge(1+\\delta)\\Delta$: spectral gap $\\Omega_\\delta(1/n)$ and $t_{\\mathrm{mix}}(\\varepsilon)=O_\\delta(n^2\\log q+n\\log(1/\\varepsilon))$. An analogous theorem holds for the anti-ferromagnetic Potts model at $q\\ge(1+\\delta)(1-\\beta)\\Delta$.\n\nWhat it does and does not improve. On girth it is a large gain: previous results near the $(1+\\delta)\\Delta$ threshold needed girth at least eleven (Hayes-Vigoda, extended to constant degrees by Jain-Mizgerd-Vigoda). On the mixing rate it is weaker - those give optimal $O(n\\log n)$, this gives $O(n^2\\log q)$. It does not touch the folklore conjecture that mixing is rapid on every graph for $q\\ge\\Delta+2$; Remark 7 names spanning 4-cycles as the obstruction.","claimIssueNote":null,"solveDate":"2026-08-26","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Xiaoyu Chen","Kuikui Liu"],"aiRole":"From the paper's section 1.2, \"Discussions about experiments with AI\". The authors first asked GPT-5.6 Sol Ultra to redo several known trickle-down results via the Bochner identity, and drew the pattern themselves: \"With these proofs, we observe that the Bochner identity is useful for reducing a global spectral-gap estimate to estimates on local gadgets.\" They then brought the girth-5 colorings question to the model, \"which proposed an affirmative proof strategy. The human authors then verified the argument, generalized the proof with further assistance from GPT, and streamlined the paper.\" GPT-5.6 was also used for exposition and typographical checking. The abstract's own framing is \"several rounds of interaction\", which is why this is co-developed rather than AI-discovered: the model supplied the strategy for the main theorem inside a frame the humans built and after an observation they made.\n\nThe paper also records where the model failed, which is rare enough to note. Remark 7: \"We asked GPT-5.6 Sol Ultra to work on triangle-free graphs with the Bochner identity, but it did not find a proof after $\\sim$20 hours.\" The authors add that the model claims a proof generalizing the theorem to all graphs without spanning 4-cycles, which they deliberately left out \"since we did not find any new ideas in it\".","verification":"unreviewed","verificationNote":"An arXiv preprint (v1, 26 August 2026, cs.DS), unrefereed, with no formalization and no computational certificate, so nothing here was mechanically checkable and no mathematics was checked. Verified on 27 August 2026: the paper exists at arXiv:2608.25491 with the title and both authors this entry lists; the statement and the quantitative claims above are its abstract and Theorem 1; the AI disclosure appears in the abstract and in section 1.2, quoted above; and the prior work is correctly characterised - Jerrum 1995 for the folklore conjecture and the $q>2\\Delta$ result, Hayes-Vigoda 2003 for the girth-eleven regime, Jain-Mizgerd-Vigoda for its extension to constant degree, and Carlson-Vigoda for the current $1.809\\Delta$ state of the art on general graphs.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The rapid-mixing threshold for Glauber dynamics on proper colorings is a headline question in approximate counting, with a celebrated ladder behind it - Jerrum's $2\\Delta$, Vigoda's $11\\Delta/6$, Carlson-Vigoda's $1.809\\Delta$. This entry is a restricted slice of it: girth at least five, large degree, and a non-optimal mixing bound. That places it around the semi-streaming matching and linear-extensions band at 28-35 rather than higher: a real step in a famous line, not the line's resolution.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.25491","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"exact-rank-and-smith-profile-of-affine-incidence-over-mathbb-z-p-2-mathbb-z","name":"Exact rank and Smith profile of affine incidence over $\\mathbb Z/p^2\\mathbb Z$","shortName":"Depth-two affine incidence rank","problemNumber":null,"field":"Finite geometry; Smith normal forms","fieldGroup":"Combinatorics","statement":"Let $p$ be prime, $k,n\\geq 1$, and $R=\\mathbb Z/p^k\\mathbb Z$. For each primitive direction $b\\in\\mathbb P(R)^{n-1}$ modulo multiplication by units and each $\\lambda\\in R$, let\n\n$$\nH_{b,\\lambda}=\\{x\\in R^n:\\langle b,x\\rangle=\\lambda\\},\n$$\n\nand let $A(p^k,n)$ be the $0$-$1$ matrix whose rows are the indicators of these distinct affine hyperplanes and whose columns are the points of $R^n$. What is $\\operatorname{rank}_{\\mathbb F_p}A(p^k,n)$? Łaba and Trainor explicitly recorded the residue-ring point-hyperplane rank question as open and proved upper bounds. Dvir later used the normalized distinct-row matrix above and obtained further bounds. The field case $k=1$ is known, but the exact rank remains open in general for $k\\geq2$.","posedBy":"Izabella Łaba and Charlotte Trainor (arXiv:2403.05719, 2024)","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"For the plane at depth two, $(k,n)=(2,2)$, the paper's $B_2$ is $A(p^2,2)$ up to row and column ordering. For every prime $p$ it proves\n\n$$\n\\operatorname{rank}_{\\mathbb F_p}B_2=\\frac{p^2(p+1)^2}{4}.\n$$\n\nMore strongly, it determines the complete nonunit $p$-primary Smith profile:\n\n$$\n\\operatorname{coker}(B_2^\\mathsf T)_{(p)}\\cong\n(\\mathbb Z/p\\mathbb Z)^{p^3(p-1)/2}\\oplus\n(\\mathbb Z/p^2\\mathbb Z)^{p(p-1)^2(p+2)/4}\\oplus\n(\\mathbb Z/p^3\\mathbb Z)^{p(p-1)/2}.\n$$\n\nIt also proves that the canonical depth-two transfer extension is nonsplit. The theorem includes $p=5$; only the memory-intensive full-matrix companion computation for $(p,\\mathrm{depth})=(5,2)$ is not run, and it is not used in the proof. Arbitrary depth, higher ambient dimension, the adjacent projective Hjelmslev problem and the separate generalized-polynomial characterization question remain open.","claimIssueNote":null,"solveDate":"2026-08-25","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Under the author's direction, OpenAI Codex using GPT-5.6 Sol generated the central mathematical development: the maximal-order and depth-transition framework, the reduction of the depth-two Smith profile to three invariants, the modular-rank and point-fibre-transfer arguments, the relative-shell and two-chart carry analysis, and the resulting all-prime formulas. It also assisted with the exact companion software and manuscript drafting. The author selected the research direction, iteratively challenged and checked the derivations and certified outputs, established the public claim and source boundaries, and takes responsibility for the final content. Adversarial machine reviews were produced within the same OpenAI Codex workflow and are not human peer review or independent expert verification.","verification":"unreviewed","verificationNote":"The public source contains a complete all-prime proof, an exact-arithmetic companion, deterministic release checks and an explicit AI-use disclosure. Clean normal and optimized replays passed, as did the hostile verification suite (59/59 tests); the final source and PDF were also subjected to adversarial same-workflow checks. These checks establish reproducibility and internal consistency, not independent mathematical endorsement. No named independent domain expert has yet endorsed the theorem, so Unreviewed is the correct tier.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A technical question recorded as open by Laba and Trainor in 2024, settled exactly for the depth-two plane. Specialized but genuine: the full Smith profile is more than the rank the question asked for.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/FCIG-Depth-Two-Smith-Profiles-and-Carry-Geometry-for-Affine-Hjelmslev-Radon-Incidence/blob/481d4746706bf0f3cc553027f85066a8dd5bdb81/paper/Depth-Two-Smith-Profiles-and-Carry-Geometry-for-Affine-Hjelmslev-Radon-Incidence.pdf","sourceName":"GitHub manuscript: Depth-Two Smith Profiles and Carry Geometry for Affine Hjelmslev–Radon Incidence","collection":null,"collectionVersion":null,"links":[{"label":"Exact-arithmetic companion and reproducibility repository","url":"https://github.com/aconsciousfractal/FCIG-Depth-Two-Smith-Profiles-and-Carry-Geometry-for-Affine-Hjelmslev-Radon-Incidence/tree/481d4746706bf0f3cc553027f85066a8dd5bdb81","kind":"code"},{"label":"Łaba–Trainor: the open residue-ring incidence-rank question","url":"https://arxiv.org/abs/2403.05719","kind":"problem-record"},{"label":"Dvir: normalized incidence matrix and later rank bounds","url":"https://arxiv.org/abs/2604.25822","kind":"paper"},{"label":"Depth-three affine incidence sequel","url":"https://github.com/aconsciousfractal/FCIG-Depth-Three-Smith-Profiles-and-Newton-Minor-Geometry-for-Affine-Hjelmslev-Radon-Incidence/blob/0807e47a7ad8791a861dcc76e47c70e764f6f3b3/paper/Depth-Three-Smith-Profiles-and-Newton-Minor-Geometry-for-Affine-Hjelmslev-Radon-Incidence.pdf","kind":"paper"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sparse-domination-implies-convex-body-domination","name":"Sparse domination implies convex body domination","shortName":"Sparse $\\Rightarrow$ convex body","problemNumber":null,"field":"Harmonic analysis","fieldGroup":"Analysis","statement":"Nazarov, Petermichl, Treil and Volberg conjectured that scalar sparse domination should imply convex body domination for the corresponding coordinate-wise vector-valued extension. More precisely, if a bilinear form $\\Lambda$ admits an $(r,s)$-sparse bound, then its extension to $\\mathbb C^n$-valued functions should admit an $(r,s)$-convex body sparse bound. Laukkarinen and Lorist prove this implication for $1\\leq r,s<\\infty$ with $1/r+1/s>1$, in particular for the classical $(1,1)$-sparse setting.","posedBy":"Fedor Nazarov, Stefanie Petermichl, Sergei Treil, Alexander Volberg","yearPosed":2017,"ageNote":"Dated 2017 from Nazarov, Petermichl, Treil and Volberg's convex body domination paper, though never a numbered conjecture. Section 3.1 states it and concedes the gap in the same breath: \"if a scalar operator $T$ can be dominated by a sparse one, the same should hold for its vector-valued version $T\\otimes\\mathrm{Id}$. Unfortunately, we are not able to prove a general theorem to that extend.\"","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"If a bilinear form admits an $(r,s)$-sparse bound, its coordinate-wise extension to $\\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound, for $1\\le r,s<\\infty$ with $\\tfrac1r+\\tfrac1s>1$. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form.\n\nOn scope, two things separate. What Nazarov, Petermichl, Treil and Volberg proposed was the principle for the classical sparse setting, and that is settled outright, the hypothesis holding comfortably there. The $(r,s)$ range is the authors' own generalization beyond what was asked, and the restriction bites only inside it, leaving $\\tfrac1r+\\tfrac1s\\le1$ open. Their summary: \"we provide a proof of the full general statement\".\n\nIt also gives sparse domination for iterated commutators directly from that of the underlying form. The dependence on $n$ is not claimed optimal.","claimIssueNote":null,"solveDate":"2026-08-25","model":"GPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":["Aapo Laukkarinen","Emiel Lorist"],"aiRole":"GPT-5.6 Sol Pro was used to prototype proof strategies. In an initial extended dialogue, the authors and ChatGPT iteratively developed a proof that sparse domination for commutators $[b,T]$ implies convex body domination for $T$. The authors then suggested replacing the role of $b$ by Rademacher random variables; from this idea, a rough version of the dyadic proof of the main theorem was developed in collaboration with ChatGPT. The final proof, as well as its extension from a fixed dyadic lattice to arbitrary cubes, was fully developed, verified and written by the authors. ChatGPT was subsequently also used to check the manuscript for typos and errors.","verification":"unreviewed","verificationNote":"Unreviewed: an arXiv preprint (v1, 25 August 2026, math.CA), unrefereed, with no formalization and no computational certificate, so there was nothing mechanical to re-run and none of the analysis was checked here. What was verified on 29 August 2026: the paper exists at arXiv:2608.24802 with this title and both authors; Theorem A and Corollary B are as the entry describes them; the AI disclosure is a dedicated section, quoted in the AI-role note; and the attribution is sound - arXiv:1701.01907 is Nazarov, Petermichl, Treil and Volberg's 2017 paper, and its Section 3.1 states the principle and says the authors could not prove it, in the words quoted in the age note.\n\nOne discrepancy a reader may hit, flagged rather than silently smoothed: the abstract states the implication with no hypothesis on $(r,s)$, while Theorem A in the body requires $1\\le r,s<\\infty$ with $\\tfrac1r+\\tfrac1s>1$. This entry follows Theorem A.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":27,"significanceNote":"Convex body domination is the technique that unlocked matrix-weighted estimates for Calderón-Zygmund operators, and whether scalar sparse domination automatically upgrades to it has sat open since 2017, posed by four authors central to the area. Settling it is structural rather than a single new bound: it converts any existing sparse bound into a convex body bound for free, and yields sparse domination for iterated commutators. Held short of the top band as a days-old unrefereed preprint with $\\tfrac1r+\\tfrac1s\\le1$ untouched. Placed at 27, beside the specialist analysis entries there.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.24802","sourceName":"Sparse domination implies convex body domination","collection":null,"collectionVersion":null,"links":[{"label":"Problem source","url":"https://arxiv.org/abs/1701.01907","kind":"paper"}],"relations":[],"submittedBy":"RustyKestrel290","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"froberg-s-conjecture-for-quintics-and-septics-in-four-variables","name":"Fröberg’s conjecture for quintics and septics in four variables","shortName":"Fröberg for 5D and 7D in 4 variables","problemNumber":null,"field":"Commutative algebra","fieldGroup":"Algebra","statement":"Let $k$ be a field of characteristic zero and let $S=k[x_1,x_2,x_3,x_4]$. We prove Fröberg's predicted Hilbert series for ideals generated by $r$ general forms of equal degree $d$ for every $r\\geq1$ in each of the two cases $d=5$ and $d=7$. Relative to the classical cases $r\\leq5$ and the equal-degree theorem through degree $d+2$ of Boij--Dannetun--Lundqvist, the generator-count ranges requiring new input are $6\\leq r\\leq11$ for quintics and $6\\leq r\\leq21$ for septics. The proof reduces each slice to finitely many endpoint ranks of Macaulay multiplication matrices. For quintics, ten exact endpoint computations based on twenty-one sparse forms suffice. For septics, a nested family of 120 integral forms supplies fifteen endpoint computations. In every endpoint certificate for these new ranges, an explicitly recorded maximal minor is nonzero modulo $2$, hence is a nonzero integer. The case $r=5$ is the classical strong Lefschetz instance; for quintics we also record a matching modular rank and Koszul bound. Zariski openness then gives the result over every characteristic-zero field. The unrestricted Fröberg conjecture remains outside the scope of the paper.","posedBy":"Ralf Fröberg","yearPosed":1985,"ageNote":"Dated from Fröberg's 1985 paper \"An inequality for Hilbert series of graded algebras\", where the predicted Hilbert series for ideals of general forms was proposed. The conjecture has stood since, proved in scattered cases: $n\\le3$ by Anick, $r\\le n$ trivially, and various fixed-degree or fixed-generator slices. It remains open in general.","solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Let $S=k[x_1,x_2,x_3,x_4]$ over any characteristic-zero field. For each $d\\in\\{5,7\\}$ and every $r\\ge1$, the paper proves that $r$ general degree-$d$ forms satisfy Fröberg’s predicted Hilbert series\n\n$$\n\\operatorname{HS}_{S/(F_1,\\ldots,F_r)}(t)\n=\n\\left[\\frac{(1-t^d)^r}{(1-t)^4}\\right]_+.\n$$\n\nThe genuinely new ranges are $6\\le r\\le11$ for quintics and $6\\le r\\le21$ for septics. These are reduced to finitely many exact rank tests of Macaulay multiplication matrices; explicit maximal minors are nonzero mod $2$, so the required ranks hold in characteristic zero. Thus the conjecture is completely settled for the equal-degree four-variable slices $d=5$ and $d=7$, but not in general.","claimIssueNote":null,"solveDate":"2026-08-25","model":"GPT-5.6 Sol; Claude Fable 5; Grok 4.6","modelMaker":"OpenAI; Anthropic; xAI","humanCollaborators":["Qihang Wang","Dongming Zhang"],"aiRole":"Disclosed in the abstract and in a closing section, \"Disclosure of automated assistance\". The abstract states that \"The main results of this paper were obtained through a generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, and Grok 4.6\", and the disclosure section lists what the workflow did: \"the formulation of mathematical ideas, generation of conjectures and proof strategies, derivation and checking of intermediate steps, construction of examples and exact certificates, comparison of literature and candidate proof approaches, organization of arguments, LaTeX drafting, and revision of the final text\". It adds that the workflow \"decomposed the problem into smaller subproblems and used repeated self-critique and alternative derivations to test the quantifiers, the characteristic-zero scope, and the endpoint-rank reductions\".\n\nThat credits the mathematics rather than tooling, which is what puts this in scope, and it goes well beyond editing: proof strategies and the exact certificates are the substance of this paper. It is filed as co-developed rather than AI-discovered because no single step is attributed to a named model, the account is of a workflow rather than of a system solving a stated subproblem, and two human authors direct it throughout.","verification":"site-confirmed","verificationNote":"Site-confirmed: this site reproduced the computation, twice over, on 28 August 2026.\n\nFirst, independently of the authors' code. The paper prints all 21 quintic forms with every coefficient $+1$ and states the monomial order, so the Macaulay matrices can be rebuilt from the text alone. I did that and ran my own elimination: all ten rows of Table 2 reproduce exactly, in shape and rank, over $\\mathbb F_2$, $\\mathbb F_{101}$ and $\\mathbb F_{1009}$. I then checked the step the paper leaves implicit - that each rank gives exactly Fröberg's predicted dimension. Recomputing the prediction myself, with $\\dim(S/I)_j=\\binom{j+3}{3}-\\mathrm{rank}$, every endpoint agrees: $(6,9)\\to10$, $(8,8)\\to5$, $(11,7)\\to10$, $(20,6)\\to4$, and $0$ at each surjective endpoint.\n\nSecond, the authors' verifiers, published as arXiv ancillary files, both pass here: quintics 1.2 s, septics 0.9 s reporting 15 endpoints, 120 forms and 15 maximal minors. Neither uses floating point or randomness. The septic verifier genuinely recomputes rather than trusting stored numbers, which I confirmed by altering one exponent of one form: the run then failed at the matrix digest instead of passing.\n\nNot established here: the reduction of each $r$-range to its endpoints is a mathematical argument that was not checked, and the 120 septic forms exist only inside the certificate, so for septics I ran their verifier rather than rebuilding independently. Unrefereed, with no human review.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"Fröberg's conjecture is a well-known named problem in commutative algebra, open since 1985, and progress on it is hard. But this settles two fixed-degree slices - equal quintics and septics, in four variables only - and the paper is explicit that the unrestricted conjecture stays out of scope. The new content is the endpoint rank certificates closing $6\\le r\\le11$ and $6\\le r\\le21$. Placed at 14, just below the Tu-Deng entry at 15 and above the routine named-problem band at 10: a real but narrow advance on a famous conjecture.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.24797","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Ancillary exact verifiers and septic certificate (arXiv anc/)","url":"https://arxiv.org/src/2608.24797v3/anc","kind":"code"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tilted-residue-class-construction-for-long-prime-free-intervals","name":"A tilted residue-class construction for long prime-free intervals","shortName":"Large prime gaps record","problemNumber":null,"field":"Analytic number theory; large gaps between primes","fieldGroup":"Number theory","statement":"How large can the gap between consecutive primes be, infinitely often? Writing $\\log_k$ for the $k$-fold iterated logarithm, Erdős asked (problem #4, a \\$10,000 prize) whether $p_{n+1}-p_n \\gg C\\log n\\log_2 n\\log_4 n/(\\log_3 n)^2$ for every $C$; that was settled in 2016, and the record bound since has been Ford-Green-Konyagin-Maynard-Tao's $p_{n+1}-p_n \\gg \\log n\\log_2 n\\log_4 n/\\log_3 n$. This work claims a stronger bound, $G(T)\\gg \\log T\\log_2 T/\\log_4 T$, an improvement by a factor of $\\log_3 T/(\\log_4 T)^2$, together with $Y(X)\\gg X\\log X/\\log_3 X$ for the covering problem behind it.","posedBy":"Paul Erdős","yearPosed":1955,"ageNote":"Dated to Erdős's 1955 statement, earliest of many. The question as posed was answered in 2016; what stands open is the $(\\log n)^{1+c}$ form he reserved the larger prize for in 1997, having reduced the problem itself to 5,000 dollars. Thomas Bloom, who maintains erdosproblems.com, relabelled this claim from full to partial for the same reason: \"the original question of Erdős was already answered\".","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Claims $G(T)\\gg\\log T\\log_2 T/\\log_4 T$ against FGKMT's $\\log T\\log_2 T\\log_4 T/\\log_3 T$, a gain of $\\log_3 T/(\\log_4 T)^2$, together with $Y(X)\\gg X\\log X/\\log_3 X$ for the covering problem behind it.\n\nBen Green resists calling it incremental: \"The sieving procedure is different to the Erdos-Rankin one which underpinned all bounds on the problem since 1938, and it wins $\\log_3 N$ over that procedure. The [FGKMT] paper also wins a $\\log_3 N$. These wins are essentially independent of one another so one now wins basically $(\\log_3 N)^2$ over Rankin's 1938 bound.\" He adds that the new sieve \"by itself could have claimed the Erdos 10000 dollars for this question\".\n\nWhat is new is narrow: only the intermediate sieve is replaced, its hard cutoff smoothed into a probabilistic tilt; the hypergraph covering theorem and Maynard weight come from FGKMT. Readers on the thread note that several later sections reproduce FGKMT with no new content, and Green calls the exposition \"horrific\".","claimIssueNote":null,"solveDate":"2026-08-25","model":"GPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["DottedCalculator (prompting and submission)","Boris Alexeev (Lean formalisation)"],"aiRole":"The paper has no human author. Its title block reads \\author{GPT 5.6 Sol} and the PDF metadata records the same, with no acknowledgements section and no human contributor named anywhere in the manuscript.\n\nThe human in the loop is the pseudonymous submitter DottedCalculator, who prompted the model and whose own account of that role is unusually direct: \"I am not familiar with sieve theory. The first draft was much shorter (19 pages) but I couldn't understand the sieve theoretical jargons and there were a few small steps missing that I had a really hard time with. I asked for all of the details self-contained to make checking the argument easier.\" So the human contribution was prompting, iteration and a request for self-contained detail - explicitly not the mathematics. Ben Green notes the contributor \"seems to wish to stay anonymous\".\n\nOne attribution discrepancy, recorded rather than smoothed over: the manuscript and its filename say GPT 5.6 Sol, while the erdosproblems.com claim was filed as \"GPT 5.6 Pro (using GPT 5.6 Pro)\" and the discussion refers throughout to ChatGPT 5.6 Pro.\n\nBoris Alexeev's Lean development is downstream and formal - it transcribes the manuscript rather than producing it.","verification":"expert-verified","verificationNote":"Expert-verified, by an unusually well-placed expert: Ben Green, a co-author of the FGKMT bound this beats, so a reader with every incentive to be sceptical rather than generous.\n\nOn 26 August he called it \"an important claim\" that \"survives an initial plausibility check\", while warning that \"the exposition is truly horrible\" and that verifying it would take time. On 27 August, after \"a few hours thought and some conversations with Terry Tao and James Maynard\", he wrote that he \"became more or less convinced this is correct\", adding that \"a lean formalisation exists so we know it's correct, and experts closest to the area basically understand informally what the key ideas are and why the proof works\". Boris Alexeev, who formalised it, states that the result is \"formalized unconditionally in Lean\".\n\nThat matches the audit done here on 30 August: 403 Lean files, 56,614 lines, no $\\texttt{sorry}$, no $\\texttt{admit}$, no $\\texttt{native\\_decide}$ and not one declared axiom, with the terminal theorem stated in primitive Mathlib terms and carrying no hypotheses at all.\n\nShort of Lean-verified on the anchoring half. erdosproblems.com hosts this as a proof claim under its standing disclaimer that listing is \"no guarantee of proof correctness\" - it has not accepted the claim - and nobody without a stake has audited the informal-to-formal correspondence. The Lean was not built here - no toolchain. No referee has read it, and Green expects a human-written exposition to take months.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"Among the most-worked quantitative questions in prime number theory, and the first movement on the record since Ford, Green, Konyagin, Maynard and Tao's Annals bound. Green called it \"an important claim\" and \"such a significant result\", and recalls that Terry Tao had offered 10,000 USD for exactly this improvement. On his reading it beats Rankin's 1938 procedure by $(\\log_3 N)^2$.\n\nHeld below the top band because it improves a bound rather than settling a question - Erdős's #4 was answered in 2016 and the $(\\log n)^{1+c}$ target stands - and because no referee has read it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/DottedCalculator/ai-math/blob/main/Erdos_4_GPT_5.6_Sol.pdf","sourceName":"A Tilted Residue-Class Construction for Long Prime-Free Intervals","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation of the two main statements","url":"https://github.com/plby/lean-proofs/blob/main/src/latest/ErdosProblems/Erdos4Tilted.lean","kind":"lean-proof"},{"label":"Erdős problem #4, with the prior record and prize history","url":"https://www.erdosproblems.com/4","kind":"problem-record"},{"label":"Ford, Green, Konyagin, Maynard, Tao - the record this improves","url":"https://arxiv.org/abs/1412.5029","kind":"paper"},{"label":"Proof claim and expert discussion: Green, Bloom, Alexeev","url":"https://www.erdosproblems.com/forum/thread/4/proof-claims","kind":"independent"},{"label":"Manuscript source (TeX) in the Lean repository","url":"https://github.com/plby/lean-proofs/blob/main/output/pdf/Erdos_4_GPT_5.6_Sol.tex","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":3,"downvotes":0,"commentCount":0},{"slug":"equivalence-of-generic-stability-notions-for-keisler-measures","name":"Equivalence of generic stability notions for Keisler measures","shortName":"Generically stable Keisler measures","problemNumber":null,"field":"Model theory","fieldGroup":"Logic & foundations","statement":"Given a first-order theory $T$ (in discrete or continuous logic) and a Borel-definable global Keisler measure $\\mu$ in $T$, we show that the following conditions are equivalent: \n$(i)$ $\\mu$ is a frequency interpretation measure; \n$(ii)$ $\\mu$ is definable and its canonical “random extension” $r_{\\mu}$ is generically stable in the randomization theory $T^{R}$; \n$(iii)$ $\\mu$ is “self-averaging”.\n\nThis result establishes a robust notion of generic stability for Keisler measures, which resolves a long-term research objective from previous work. The implications $(i)\\Rightarrow(ii)\\Rightarrow(iii)$ were previously established by the authors (for $T$ discrete). The primary focus of this paper is the reverse implications $(iii)\\Rightarrow(ii)\\Rightarrow(i)$, which we obtain through the use of AI models.","posedBy":"Gabriel Conant, Kyle Gannon","yearPosed":2020,"ageNote":"Dated from the 2020 Conant-Gannon paper whose observation, in the authors' words, \"motivated an extended investigation into the question of whether fim is the 'right' analogue of generic stability for Keisler measures\". The three-way equivalence was set up in their 2025 paper with Hanson, which proved one direction; this paper adds the reverse implications and continuous logic.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Let $T$ be a complete first-order theory in discrete or continuous logic, let $M\\prec\\mathcal{U}$, and let $\\mu\\in\\mathfrak{M}_{x}(\\mathcal{U})$ be Borel-definable over $M$. The paper proves that the following three conditions are equivalent:\n$(i)$ $\\mu$ is a frequency interpretation measure (fim) over $M$;\n$(ii)$ $\\mu$ is definable over $M$ and its canonical random extension $r_{\\mu}$ is generically stable over $M^{\\Omega}$;\n$(iii)$ $\\mu$ is self-averaging over $M$.\n\nThe new work proves the reverse implications $(iii)\\Rightarrow(ii)\\Rightarrow(i)$ and extends the characterization to continuous logic. The authors therefore make the equivalent conditions into a definitive definition of generic stability for Keisler measures.\n\nThe paper also proves a further characterization in terms of an order-property condition and derives consequences for closure under Morley products.","claimIssueNote":null,"solveDate":"2026-08-25","model":"ChatGPT 5.5; Kimi K3; Claude Fable 5; ChatGPT 5.6 Sol","modelMaker":"OpenAI; Moonshot AI; Anthropic","humanCollaborators":["Gabriel Conant","Kyle Gannon","James E. Hanson"],"aiRole":"The disclosure is in the abstract itself, not only in an acknowledgment: \"The primary focus of this paper is the reverse implications $(iii)\\Rightarrow(ii)\\Rightarrow(i)$, which we obtain through the use of AI models.\" The dedicated AI Acknowledgment gives the detail, verbatim: \"A proof of Theorem 1.1[(iii) $\\Rightarrow$ (ii) $\\Rightarrow$ (i)] was initially obtained from a ChatGPT 5.5 query focusing on the case when $T$ is discrete. We were also able to independently find proofs using Kimi K3 and Claude Fable 5. These arguments were heavily reorganized and rewritten by the authors with further assistance from ChatGPT 5.6 Sol. Theorem 5.1 was obtained by the authors by modifying a different result found by ChatGPT while attempting Question 5.3.\" Worth noting what that describes: the proof came out of a model, and was then independently reproduced by two further models from different vendors. The authors' own labour is reorganizing and rewriting.","verification":"unreviewed","verificationNote":"An arXiv preprint (v1, 25 August 2026, math.LO), unrefereed and with no independent endorsement. No mathematics was checked here and there is nothing mechanical to check it against - no formalization, no certificate. Verified here on 26 August 2026: the paper exists at arXiv:2608.24605 with the title and all three authors this entry lists; the statement above is its abstract near verbatim; the AI disclosure appears both in the abstract and in a dedicated AI Acknowledgment, quoted in full above; and the prior work it builds on is real and correctly characterised - Conant and Gannon, Ann. Pure Appl. Logic 171 (2020) for the originating observation, and Conant, Gannon and Hanson, J. Math. Log. (2025) for the chain of implications this paper reverses.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The authors call it \"a definitive resolution\" of a question their own programme has pursued since 2020, and it settles what generic stability should mean for Keisler measures - a real object in NIP model theory going back to Hrushovski, Pillay and Simon. Specialist even by model-theory standards, and well below the $SOP_2=SOP_3$ entry at 35, which is a famous Shelah-lineage problem; level with the named-conjecture band around 20 for a programme objective resolved by the people who set it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.24605","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"entropy-of-bernoulli-measures-conditioned-on-affine-subspaces-and-a-problem-of-a","name":"Entropy of Bernoulli Measures Conditioned on Affine Subspaces and a Problem of Ancheta-Massey","shortName":"Ancheta-Massey linear lossy compression","problemNumber":null,"field":"Coding theory","fieldGroup":"Theoretical computer science","statement":"A textbook result in information theory is that linear encoders achieve the entropy for lossless compression of Bernoulli source with parameter $p$. For lossy compression, however, linearity is known to incur strict suboptimality compared to the rate-distortion function. Massey asked whether the optimal rate for linear encoding is achieved simply by compressing a fraction of the bits linearly and losslessly and estimating the rest by zero. For $p=\\frac{1}{2}$, Ancheta answered this question affirmatively. This note extends Ancheta's result to all $p<\\frac{1}{2}$. The key argument is to bound the entropy of the posterior distribution conditioned on an affine subspace in terms of its marginals. The proof was discovered by GPT-5.6 Sol in an interactive process guided by the author.","posedBy":"James L. Massey","yearPosed":1978,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"For a Bernoulli$(p)$ source with $0= 30 record, superseded three days later. Together they show the ladder moving twice in under a week after moving only once between 2006 and 2024."}],"submittedBy":null,"upvotes":7,"downvotes":0,"commentCount":0},{"slug":"transcendence-in-the-affine-case-of-erdos-problem-270","name":"Transcendence in the affine case of Erdős Problem 270","shortName":"Affine Erdős Problem 270","problemNumber":270,"field":"Transcendence theory","fieldGroup":"Number theory","statement":"For integers $a\\geq1$ and $b\\geq1-a$, the series $C_{a,b}=\\sum_{n=1}^{\\infty} n!/((a+1)n+b)!$ is transcendental. Equivalently, the series in Erdős Problem 270 is transcendental whenever $f(n)=an+b$ is a positive integer-valued affine function.","posedBy":"Paul Erdős and Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The manuscript claims $C_{a,b}$ is transcendental for every $a\\ge1$ and $b\\ge1-a$, settling the positive integer-valued affine subclass of Erdős Problem 270. Two pieces of context matter. Problem 270 as Erdős and Graham posed it, for every $f(n)\\to\\infty$, was already answered no by Crmarić and Kovač in 2025: for any $\\alpha>0$ some such $f$ makes the series sum to $\\alpha$. What survives is the non-decreasing case, and the affine family sits inside it. Separately, the checkable parts here were already known - the short irrationality proof for $C_{1,0}$ is Crmarić and Kovač's, posted by Kovač on the Erdős Problems forum in July 2026 and credited in the repository, and base-case transcendence follows a 2023 MathOverflow argument. The new content is the extension to the whole affine family, which is the part with neither formalization nor review.","claimIssueNote":null,"solveDate":"2026-08-22","model":"GPT-5.6 Sol (Codex)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The author's account, given to this site on submission rather than in the manuscript: OpenAI Codex independently rediscovered the elementary denominator argument, connected Crmarić and Kovač's Gaussian integral representation with a MathOverflow Siegel-Shidlovsky argument to obtain base-case transcendence, and developed the claimed extension to all positive affine cases using hypergeometric E-functions, Euler-operator reductions and a formal-at-infinity resonance argument. It located the relevant results of Salikhov, Salikhov-Viskina and Beukers, drafted the manuscript, and produced most of the Lean formalization; further AI reviews identified gaps and prompted revisions. The manuscript's own disclosure is a single line - \"This manuscript was written and checked using generative AI\" - which names no model and describes only writing and checking. The tier here follows the detailed account because the submitter is the author, but a reader following the source link will not find it there.","verification":"unreviewed","verificationNote":"Read here on 24 August 2026 at github.com/clambro/erdos-270-transcendence. The elementary irrationality theorem for $a\\ge1$, $0\\le b\\le a$ is unconditional, and the five proof modules total 977 lines with no sorry, no admit, no declared axiom and no native_decide on Lean 4.33.1. The general transcendence theorem is not formalized. Salikhov, Salikhov-Viskina, Beukers, the Levelt-Turrittin decomposition and the step from the resonance calculation to functional minimality are passed as explicit Lean hypotheses rather than hidden behind axiom declarations, which is the honest construction; but for the novel range $b>a$ the hypothesis BeyondStripInput.pair is the conclusion itself, so the formalization lends the new claim no independent weight. Lean was not compiled here. The manuscript is unrefereed, self-published in a GitHub repository, and two days old at review.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A numbered Erdős-Graham problem with more recent traffic than most: Crmarić and Kovač published on it in Colloquium Mathematicum in 2025, Kovač spoke on the surrounding Ahmes-series irrationality problems at CANT 2025, and it has an active forum thread. No Wikipedia article in any language. Slightly above the typical numbered Erdős problem at 10 for that activity, and this entry is one subclass of it rather than the whole problem.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/clambro/erdos-270-transcendence","sourceName":"Transcendence in the Affine Case of Erdős Problem 270","collection":null,"collectionVersion":null,"links":[{"label":"Kovač's earlier elementary irrationality proof","url":"https://www.erdosproblems.com/forum/thread/270#post-7469","kind":"independent"},{"label":"Crmarić and Kovač, On the irrationality of certain super-polynomially decaying series - answers Problem 270 as posed","url":"https://arxiv.org/abs/2504.18712","kind":"independent"},{"label":"Erdős Problem #270","url":"https://www.erdosproblems.com/270","kind":"problem-record"}],"relations":[],"submittedBy":"CobaltMongoose239","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"haglund-s-zero-trajectory-conjecture-for-the-first-riemann-xi-approximant","name":"Haglund's Zero-Trajectory Conjecture for the First Riemann Xi Approximant","shortName":"Haglund Conjecture 4, k=1","problemNumber":null,"field":"Analytic number theory and entire-function zero dynamics","fieldGroup":"Number theory","statement":"Haglund's Conjecture 4 reads: for $k\\ge1$, the imaginary part of each non-real zero of $\\Xi_k(z)+t\\Phi_{k+1}(z)$ decreases monotonically as $t$ goes from 0 to 1, where the $\\Phi_n$ are the incomplete-gamma summands of Riemann's series for $\\Xi$ and $\\Xi_k=\\sum_{n\\le k}\\Phi_n$. This work proves the case $k=1$, the pencil $\\Phi_1+t\\Phi_2$: every non-real zero in the closed first quadrant is simple and the imaginary part of its analytic branch strictly decreases. It adds two statements Conjecture 4 does not itself assert - no non-real branch escapes to infinity on a bounded forward parameter interval, and at a real collision of any finite multiplicity the full local Weierstrass-Puiseux multiset stays real to the right. The cases $k\\ge2$ remain open, and nothing is claimed about the zeros of $\\Xi$ or the Riemann hypothesis.","posedBy":"James Haglund","yearPosed":2009,"ageNote":"The general conjecture was posed in 2009; this entry is its k=1 case.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Proves Haglund's Conjecture 4 for $k=1$: every non-real first-quadrant zero of $\\Phi_1+t\\Phi_2$ is simple with strictly decreasing imaginary part, no branch escapes forward, and every finite-multiplicity real collision stays real afterwards. The cases $k\\ge2$ remain open. Two readings worth separating: Conjecture 4 asserts the monotone descent alone, so the no-escape and stays-real statements are this paper's own additions rather than Haglund's text, and they are the stronger part of the theorem. The descent itself, part (i), is the part that rests on the unavailable interval-arithmetic certificate.","claimIssueNote":null,"solveDate":"2026-08-22","model":"ChatGPT and Codex","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The disclosure was rewritten on 25 August, and now reads in full: \"OpenAI ChatGPT and Codex models performed most of the proof search, mathematical derivation, computational implementation, Lean proof development, literature discovery, organization, and drafting. The author supplied the research methodology, problem framing, constraints, evaluation criteria, and iterative oversight. The author remains responsible for every mathematical statement, proof, citation, and submission decision.\" That names the systems and attributes the proof search itself to them, which is why this is filed as AI-discovered. It replaces the earlier version - \"AI systems were used extensively in mathematical derivation, Lean proof development, literature discovery, organization, and typesetting\" - which named no system and attributed no step, and on which this entry was first filed a rung lower as co-developed. The manuscript still gives no version for either system, so the model field records only what it says.","verification":"site-confirmed","verificationNote":"Site-confirmed: the computational certificate was replayed here on 26 August 2026 and it passes. Cloned mbaccaro-dev/mathematical-proofs, installed python-flint 0.9.0 into a venv per requirements.txt, ran its own reproduce.py in full (4 min 17 s), and got STATUS=PASS with claim_ceiling=Haglund_Conjecture_4_for_k_equals_1_only - via MANIFEST_PASS at 289 files, S1_STRUCTURE, NONREAL_JOIN, GLOBAL_LIFT, OUTER_REGION, and crucially S2_SCIENTIFIC_PASS at patches=238, source_calls=60930, the paper's own figures for the certified first-quadrant proposition, plus S3 at 323 patches and 46,514 calls.\n\nWhat that does not settle: the script recomputes its own certificates against its own manifest, so it shows the computation replays and is internally consistent, not that the interval arithmetic implies the theorem (argued in prose) nor any analytic step around it. The Lean is unchanged from first review - no sorry, admit or declared axiom in the Solution closure, but it verifies one abstract collision theorem, and the repository still says the atlas, the incomplete-gamma estimates and the assembly are not consequences of it. End-to-end Lean remains pending by the author's account; the paper is unrefereed.\n\nOne packaging defect found while replaying: on a fresh clone reproduce.py aborts at the manifest gate. Of 289 hashes, 12 match only as CRLF, all Windows console receipts, and with no .gitattributes no checkout satisfies both sets; the run above needed those 12 converted first.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"Conjecture 4 of Haglund's 2009 paper (Central European Journal of Mathematics 9 (2011) 302-318), supported there by exploratory computation and revisited once since, in a 2012 Penn master's thesis with Maple plots. No Wikipedia article, essentially no literature beyond those two items, and the paper is explicit that nothing about the zeros of $\\Xi$ itself or the Riemann hypothesis follows. Below the typical numbered Erdős problem at 10, level with the small named graph conjectures, and this entry is the $k=1$ case of it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.22059236","sourceName":"Zenodo preprint","collection":null,"collectionVersion":null,"links":[{"label":"Some conjectures on the zeros of approximates to the Riemann Ξ-function and incomplete gamma functions","url":"https://arxiv.org/abs/0910.5228","kind":"problem-record"},{"label":"Manuscript source, Lean project and verification certificate","url":"https://github.com/mbaccaro-dev/mathematical-proofs/tree/main/MathematicalProofs/HaglundK1ZeroTrajectory","kind":"code"},{"label":"Ahn, On the Zeros of Component Functions of the Riemann Xi Approximates (Penn MA thesis, 2012)","url":"https://www2.math.upenn.edu/~jhaglund/thesis/Shirley.pdf","kind":"problem-record"},{"label":"Verification archive: reproduce.py, source manifest and interval certificates","url":"https://github.com/mbaccaro-dev/mathematical-proofs/tree/main/MathematicalProofs/HaglundK1ZeroTrajectory/Verification","kind":"code"}],"relations":[],"submittedBy":"WildHeron785","upvotes":1,"downvotes":0,"commentCount":2},{"slug":"bounded-mass-property-for-compact-complex-manifolds","name":"Bounded mass property for compact complex manifolds","shortName":"Bounded mass property","problemNumber":null,"field":"Complex geometry","fieldGroup":"Geometry & topology","statement":"A compact complex manifold $X$ of dimension $n$ has the bounded mass property if, for one (equivalently every) Hermitian form $\\omega$, the Monge-Ampère masses $\\int_X(\\omega+dd^c\\varphi)^n$ are uniformly bounded over all smooth $\\varphi$ with $\\omega+dd^c\\varphi>0$. On a compact Kähler manifold Stokes' theorem makes that mass independent of $\\varphi$ outright; for a merely Hermitian $\\omega$, which is not closed, it genuinely depends on $\\varphi$, and controlling it is a recurring theme of Hermitian pluripotential theory. The property is known to hold in dimension $n\\le2$ and on manifolds of Fujiki class. Boucksom, Guedj and Lu left open whether it holds on every compact complex manifold, raising the question explicitly for Hopf manifolds of dimension at least three. This paper answers it in the negative on the Hopf threefold $X=(\\mathbb{C}^3\\setminus\\{0\\})/\\langle z\\mapsto e^{-1}z\\rangle$.","posedBy":"Sébastien Boucksom, Vincent Guedj, Chinh H. Lu","yearPosed":2025,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Disproved on the Hopf threefold $X=(\\mathbb{C}^3\\setminus\\{0\\})/\\langle z\\mapsto e^{-1}z\\rangle$: Xia and Zhang construct a smooth Hermitian form $\\omega$ and smooth functions $\\varphi_j$ with $\\omega+dd^c\\varphi_j>0$ whose Monge-Ampère masses tend to infinity, so the universal bounded mass property fails already in complex dimension three. The construction uses the Hopf threefold's elliptic fibration, an exact mass identity reducing excess Monge-Ampère mass to a fibrewise Dirichlet energy, and heat-kernel regularizations of Green functions that make that energy diverge while preserving positivity. What the negative answer removes is load-bearing rather than incidental: finiteness of this mass is the starting point for the theory of volumes of Bott-Chern classes, and it enters as a standing hypothesis in recent Hermitian pluripotential theory.","claimIssueNote":null,"solveDate":"2026-08-21","model":"Rethlas (GPT-5.6 Sol)","modelMaker":"OpenAI","humanCollaborators":["Mingchen Xia","Kewei Zhang"],"aiRole":"The paper's acknowledgments, in full: \"The initial counterexample was constructed with the Rethlas agent (improved by Felix Ye), using the gpt-5.6-sol model. The authors then simplified the construction and improved the presentation, and are fully responsible for all assertions in this paper.\" For a disproof the counterexample is the entire result, and the agent produced it; what the authors describe adding is simplification and presentation. So AI-discovered here rests on the authors' own account of what the system did, not on an inference drawn from vague wording. Felix Ye is credited with improving the agent rather than with the mathematics, so he is not listed as a collaborator.","verification":"unreviewed","verificationNote":"An arXiv preprint (v1, 21 August 2026, math.CV), unrefereed and with no independent endorsement. No mathematics was checked here, and there is nothing mechanical to check it against - no formalization, no computational certificate. Verified on 25 August 2026: the paper exists at arXiv:2608.21053, and its title, authors, Hopf-threefold statement and date match this entry; its acknowledgments carry the AI disclosure quoted above word for word; and the reference it answers is real - Boucksom, Guedj and Lu, \"Volumes of Bott-Chern classes\" (arXiv:2406.01090, Peking Math. J. 2025). The paper's introduction is the source for the problem's standing - bounded mass known for $n\\le2$ and for Fujiki class, the question raised explicitly for Hopf manifolds of dimension at least three in [BGL25, Example 1.19] - which is the authors' characterization of what was open rather than an independent literature search run here.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"A question raised explicitly in a 2025 paper of Boucksom, Guedj and Lu, answered in the negative a year later. Sits beside Boucksom's Local Analytic Bertini conjecture at 15, its closest sibling in this catalog, and one above it because what fails here is load-bearing: finiteness of the Monge-Ampère mass is the starting point for the theory of volumes of Bott-Chern classes and a standing hypothesis in recent Hermitian pluripotential theory, so the answer constrains work already resting on it. Specialist throughout, in non-Kähler complex geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.21053","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Boucksom, Guedj, Lu - Volumes of Bott-Chern classes, where the question is raised","url":"https://arxiv.org/abs/2406.01090","kind":"problem-record"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-stable-commutator-length-of-a-relator-is-not-a-one-relator-group-invariant","name":"The stable commutator length of a relator is not a one-relator group invariant","shortName":"SCL is not invariant of relator in 1-relator groups","problemNumber":null,"field":"Group theory, Geometric group theory","fieldGroup":"Algebra","statement":"Let $S,S'$ be sets and let $r\\in F(S)'\\setminus\\{e\\}$, $r'\\in F(S')'\\setminus\\{e\\}$ be relators with $\\langle {S} \\; | \\; {r} \\rangle \\cong\\langle{S'} \\;| \\; {r'}\\rangle$. Does this imply that $\\text{scl}_S r=\\text{scl}_{S'}r'$?","posedBy":"Heuer and Löh","yearPosed":2019,"ageNote":"Posed by Nicolaus Heuer and Clara Löh in \"Simplicial volume of one-relator groups and stable commutator length\", arXiv 6 Nov 2019, in Algebr. Geom. Topol. 22 (2022). The paper thanks both \"for their attention to this solution and for their helpful correspondence\", so the people who asked have seen the answer. Not peer review, and it moves no tier, but more attention than most preprints here get.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"A negative answer to Heuer and Löh's question: the isomorphism type of a one-relator group $\\langle S \\mid r\\rangle$ does not determine $\\mathrm{scl}_S(r)$. The witnesses are $r=\\mathtt{aabABabABBAbaabABBAb}$ and $r'=\\mathtt{aabABabABabABBAbaBAb}$, both length 20 in $F_2'$, with $\\langle a,b \\mid r\\rangle\\cong\\langle a,b\\mid r'\\rangle$ but $\\mathrm{scl}(r)=1$ against $\\mathrm{scl}(r')=1/2$.\n\nThe mechanism is what made the search finite: $\\mathrm{scl}$ is an $\\operatorname{Aut}(F_2)$-invariant, so a pair in *different* orbits whose one-relator groups happen to be isomorphic has its two scl values unconstrained by each other. The search was for that configuration among words of length at most 20.\n\nScope: it settles the question as posed and nothing wider. It does not say which invariants do determine scl, and this is a single pair rather than a construction giving arbitrary gaps.","claimIssueNote":null,"solveDate":"2026-08-21","model":"Claude Opus 5; Harmonic Aristotle (Lean formalisation)","modelMaker":"Anthropic; Harmonic","humanCollaborators":["Artem Semidetnov"],"aiRole":"Two systems, and the paper is more specific in its body than in its formal statement. The AI use statement reads only: \"Claude Desktop with Opus 5 assisted in designing and orchestrating the computational search for candidate counterexamples.\" Section 4 puts it more strongly: \"Claude Desktop with Opus~5 designed and orchestrated an exhaustive search through $\\operatorname{Aut}(F_2)$-orbits of words of length at most 20.\" Candidates were then filtered by first homology of low-index subgroups, Alexander polynomials and homomorphism counts to small finite groups, with explicit isomorphisms constructed for those the invariants did not separate.\n\nCo-developed rather than assisted on the strength of the second sentence: the author formulated the target - pairs in different $\\operatorname{Aut}(F_2)$-orbits whose one-relator groups are nonetheless isomorphic, which is exactly the configuration that leaves scl unconstrained - and the model designed and ran the search that found one.\n\nA second system appears in the same paragraph and is recorded here because the submission omitted it: the Lean formalisation of the six identities \"produced with the assistance of Harmonic's Aristotle\". That is the artifact this entry's verification rests on.","verification":"unreviewed","verificationNote":"Unreviewed, lowered from Lean-verified on inspection. A labelling correction, not a doubt about the mathematics.\n\nThe theorem has two halves - the groups are isomorphic, and the relators have different scl - and only the first is formalised. The Lean file says so itself: \"(The paper computes scl r = 1 ≠ 1/2 = scl r' with scallop; see scl/.)\" So the half that makes this a counterexample is a computation, not a machine-checked proof, and Lean-verified additionally requires the formal statement to be independently anchored, which an author's own repository is not.\n\nWhat the Lean does establish, and it is not nothing: one file, 282 lines, with no sorry, no admit, no native_decide and no declared axiom. It builds the two homomorphisms explicitly and proves they compose to the identity in both directions, so the isomorphism is constructed rather than asserted. Not built here - no Lean toolchain on this machine, and it pins v4.28.0.\n\nThe scl values were not reproduced here either. They are reproducible in principle: scl/setup_scallop.sh fetches and patches Alden Walker's scallop, and compute_scl.py rechecks every value the paper quotes against it, exiting non-zero on disagreement. It needs a C++11 compiler with GLPK and GMP, none present on this machine.\n\nChecked independently on 30 August 2026: both relators have exponent sum zero in both generators, so both do lie in $F'$ as the question requires; both are cyclically reduced, length 20, distinct, and match the abstract exactly.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A clean negative answer to a specific question by two named authors, in geometric group theory, settled by an explicit pair of 20-letter words. Its virtue is that it is completely checkable in principle - the witnesses are written down and both halves are reproducible - and its limit is that it is one question rather than a programme: it neither identifies what does determine scl nor produces a family. Placed at 18, just below the neighbouring one-relator commutator-relators entry at 20, which settles a broader statement.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.21465v1","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation, search pipeline and scl recomputation","url":"https://github.com/s3midetnov/scl-one-relator","kind":"code"},{"label":"Heuer and Löh, where the question is posed","url":"https://arxiv.org/abs/1911.02470","kind":"problem-record"}],"relations":[],"submittedBy":"CrimsonEgret515","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"question-3-of-dubickas-2006-3","name":"Problem 3 of Dubickas (2006): Is $\\sqrt{3} \\in \\mathcal{Z}$?","shortName":"Dubickas Problem 3","problemNumber":null,"field":"Distribution mod 1; Mahler Z-numbers","fieldGroup":"Number theory","statement":"Dubickas splits $(1,+\\infty)$ into the set $\\mathcal{Z}$ of those $\\alpha$ for which some nonzero real $\\xi$ makes every integral part $\\lfloor \\xi\\alpha^n \\rfloor$ even, and its complement $\\mathcal{S}$; at $\\alpha = 3/2$ the question of which side one lies on is Mahler's. His Problem 3 asks which side $\\sqrt{3}$ is on. Answered: $\\sqrt{3} \\in \\mathcal{Z}$, with the explicit witness $\\xi = 1.34160899796112665163\\ldots$, and more generally $\\sqrt{m} \\in \\mathcal{S}$ if and only if $m = 2$. The mechanism is Cantor-set arithmetic rather than Diophantine approximation: since $\\sqrt{m}^{\\,2}$ is an integer, the two-scale problem collapses to a base-$m$ covering induction on restricted-digit expansions.","posedBy":"Artūras Dubickas","yearPosed":2006,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Answers Problem 3 and generalizes it: the classification $\\sqrt{m} \\in \\mathcal{S} \\iff m = 2$ covers every square root, and a further theorem replaces parity by divisibility by any $p \\ge 2$. Note the scope of the machine-checking, which is narrower than the paper: the author states that the case $m = 3$ is what is verified in Lean, and the repository flags the thickness computation of section 4.1 and all of section 8 as not formalized.","claimIssueNote":null,"solveDate":"2026-08-21","model":"Fable 5, Opus 5","modelMaker":"Anthropic","humanCollaborators":["Ralf Stephan"],"aiRole":"The author's account: \"The mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content.\" The Lean development carries the same framing in its copyright line, \"in collaboration with Claude Code\".","verification":"lean-checked","verificationNote":"The Lean was read here on 22 August 2026, at github.com/rwst/Square-Roots: eleven modules totalling 4415 lines with zero sorry, zero declared axioms and no native_decide, comparator.json permitting only propext, Quot.sound and Classical.choice, and the MahlerZ and S definitions faithful to the statement above. Challenge.lean's ten sorries are the placeholders a comparator challenge is meant to carry, and it imports nothing from the development. Recorded lean-checked rather than lean-verified, which the submission claimed, because that rung wants kernel-checking AND an independent anchor and this has neither: the repository has no CI workflow and no runs, so nothing has compiled it, and the challenge file is written by the author of the proof. The README also flags that the thickness computation of section 4.1 and all of section 8 are not formalized, and the paper's abstract says the cases m = 2 (reproof), m = 3, the cases m > 3, the\np-divisibility theorem and the transcendentality theorem are all verified\nin Lean 4 and depend only on Lean’s three standard axioms.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A specific numbered question from a 2006 Bulletin of the LMS paper, twenty years standing, in a corner of distribution mod 1 that borrows its interest from Mahler's 3/2 problem sitting at the same kind of alpha. Level with the Oddtown anchor at 20: a real named question, narrow readership.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.researchgate.net/publication/413520035_Even_integral_parts_of_powers_of_square_roots","sourceName":"Even integral parts of powers of square roots (doi:10.13140/RG.2.2.32215.43682)","collection":null,"collectionVersion":null,"links":[{"label":"Github repo","url":"https://github.com/rwst/Square-Roots","kind":"lean-proof"},{"label":"Dubickas, Arithmetical properties of powers of algebraic numbers (Problem 3)","url":"https://doi.org/10.1112/S0024609305017728","kind":"problem-record"},{"label":"Bulk d/l","url":"https://raw.githubusercontent.com/rwst/Square-Roots/main/paper.pdf","kind":"paper"},{"label":"rwst/Square-Roots","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-08-23-000002&version=1","kind":"palomar"}],"relations":[],"submittedBy":"LucidKestrel185","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"smooth-random-fast-dynamo-on-the-three-torus","name":"Smooth Random Fast Dynamo on the Three-Torus","shortName":"Smooth random fast dynamo","problemNumber":null,"field":"Dynamo theory","fieldGroup":"Analysis","statement":"Arnold's fast-dynamo problem asks for a smooth divergence-free velocity field on $\\mathbb{T}^3$, chosen independently of the magnetic diffusivity, that drives exponential growth of the magnetic field at every sufficiently small diffusivity. This constructs a genuinely $C^\\infty$ field with that behaviour: random and time-dependent, refreshing iid on finite time blocks, for which the almost sure exponential growth rate is at least $1/2$ at each fixed small enough resistivity, with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment bound. The field is neither autonomous nor deterministic, so Arnold's smooth autonomous problem on $\\mathbb{T}^3$ remains open.","posedBy":"Arnold's fast-dynamo problem (1994); the random formulation has no single named proposer","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment estimate. This is one variant case of Arnold's 1994 fast-dynamo problem, not the problem itself: Arnold asks for a field that is smooth, autonomous and deterministic all at once, and this one keeps the smoothness while giving up the other two. The sibling entry on this site relaxes the opposite hypothesis, keeping an autonomous deterministic field at Lipschitz regularity. Neither settles Arnold's problem as posed, which remains open.","claimIssueNote":null,"solveDate":"2026-08-20","model":"ChatGPT 5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Keefer Rowan"],"aiRole":"Rowan states that the original proof idea was generated essentially autonomously by ChatGPT 5.6 Sol Ultra from a prompt asking for a proof of the random fast dynamo conjecture. ChatGPT produced an initial manuscript and identified the key Fourier-space mechanism: carefully chosen shears act so that selected Fourier modes evolve through an effectively tridiagonal recursion, reducing the infinite-dimensional induction dynamics to a simple growth process. During later revisions, ChatGPT also produced the exponential-martingale argument that became Lemma 2.4. Rowan subsequently simplified the velocity fields, replaced the backward-time argument with a cleaner forward-time proof, rewrote the manuscript from scratch, and hand-checked the mathematics.","verification":"unreviewed","verificationNote":"Checked by this site on 22 August 2026 against the paper (arXiv:2608.20105, Keefer Rowan, 12pp): the abstract is as submitted, the paper is titled \"An AI-discovered smooth random fast dynamo on T^3\" and its comments field reads \"AI discovered; human written\", so the contribution tier is the author's own framing rather than an inference. The mathematics was not checked here. Days-old preprint, no independent review, and the result is recorded as a variant because the velocity field is random and time-dependent where Arnold's problem asks for an autonomous deterministic one.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Level with the Lipschitz autonomous entry at 30, deliberately: the two are complementary halves of one problem that has organised mathematical MHD for decades, each surrendering a different hypothesis, and neither is obviously the nearer miss. Arnold's problem carries a monograph and a sustained literature while staying invisible outside that community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.20105","sourceName":"arXiv:2608.20105 - An AI-discovered smooth random fast dynamo on the three-torus","collection":null,"collectionVersion":null,"links":[],"relations":[{"to":"autonomous-lipschitz-fast-dynamo-on-the-three-torus","kind":"related","note":"Complementary halves of Arnold's problem: Niebel keeps autonomy and determinism but only Lipschitz regularity, Rowan keeps C-infinity but goes random and time-dependent. Neither settles it."}],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"marton-inner-bound-capacity-region","name":"Whether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region","shortName":"Marton's inner bound","problemNumber":null,"field":"Network information theory","fieldGroup":"Theoretical computer science","statement":"Marton's inner bound, proposed in 1979, is the best known achievable region for a general discrete memoryless broadcast channel, and whether it always achieves the capacity region had been open ever since. It does not: there is a finite two-receiver discrete memoryless broadcast channel whose two-letter Marton value strictly exceeds twice its one-letter value, so the complete one-letter Marton region is strictly contained in the capacity region.","posedBy":"Katalin Marton","yearPosed":1979,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-20","model":"GPT-5.6 Sol, Claude Fable 5, Claude Opus 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Mian Huang","Yanxiao Liu","Yi Liu"],"aiRole":"The paper's statement: \"The numerical search relied heavily on the assistance of AI models.\" It credits each contribution individually - GPT-5.6 Sol assisted Yanxiao Liu in finding a counterexample to the Markovity conjecture of Gohari, Liu and Nair and, independently, assisted Mian Huang via elimination geometry; Claude Fable 5 assisted in finding a counterexample to the local tensorization test and a fixed-input counterexample; Opus 5 assisted in the fixed-input construction. The theory that turns a numerical gap into an unconditional theorem, gradient shaping and constraint removal, is the authors'.","verification":"unreviewed","verificationNote":"No independent mathematician has checked this yet. Checked by this site on 21 August 2026 against the paper (arXiv:2608.19869v1): the AI statement and the per-author credits are as quoted, and the result is unconditional rather than resting on the Markovity conjecture, which the paper shows fails and then works around. The authors certify the gap with interval arithmetic and outward-rounded MPFR at more than 1.88e-6 nats; that computation was not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The oldest problem in this batch by two decades and a central one in network information theory: the canonical achievable region for the general broadcast channel, open since 1979. Below the Connes anchor at 45 in name recognition outside its field, above it in standing within, so set at 50.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.19869","sourceName":"Sub-optimality of Marton's Inner Bound for the Two-Receiver Broadcast Channel","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"elliptic-curve-rank-record-thirty","name":"Record Rank for an Elliptic Curve over $\\mathbb{Q}$","shortName":"Elliptic curve rank record","problemNumber":null,"field":"Elliptic curves","fieldGroup":"Number theory","statement":"How large can the Mordell-Weil rank of an elliptic curve over $\\mathbb{Q}$ be? Whether ranks are unbounded is open, and progress is measured by explicit records, tabulated by Dujella: rank $\\ge 28$ from 2006, raised to $\\ge 29$ by Elkies and Klagsbrun in 2024. Now $\\ge 30$, witnessed by an explicit curve $y^2 + xy = x^3 + a_4 x + a_6$ with $a_4$ of 63 digits and $a_6$ of 94, carrying thirty independent rational points.","posedBy":"Classical; rank records tabulated by Andrej Dujella","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Two tiers, and only the first is the record. Rank $\\ge 30$ is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with $\\Delta = 4.25$ gives an analytic rank of at most 31, and the root number is $+1$ so the rank is even, hence 30 - but that argument assumes GRH, and equating analytic rank with rank assumes BSD. The entry is a partial result because the open question is whether ranks are unbounded at all, which no single record answers. Superseded three days later by this project's own rank $\\ge 31$ record (see the related entry); left unedited otherwise as a record of what was known at the time.","claimIssueNote":null,"solveDate":"2026-08-20","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Levent Alpöge","Ava Howell"],"aiRole":"The credit, in full, is a comment on the leaderboard entry: \"it was Claude, with Levent Alpöge and Ava Howell!\" Bartosz Naskręcki, who works on these curves, congratulated \"Ava Howell, Levent Alpöge and the team Anthropic\" publicly. That is the whole of the disclosure: no paper, no statement of division of labour, and no account of what the model searched or proposed. The tier below is inferred from the wording rather than read off an author's description, which is weaker evidence than every arXiv entry in this catalog.","verification":"unreviewed","verificationNote":"Recomputed by this site on 21 August 2026 from the leaderboard's own JSON, in exact rational arithmetic: all thirty witness points satisfy the curve equation with residual exactly zero, all thirty are distinct, nineteen are integral, and the discriminant recomputed from the a-invariants matches the published value, with all fourteen listed bad primes dividing it and together factoring it completely. What was NOT checked here is the one thing the record actually asserts - that the thirty points are independent in $E(\\mathbb{Q})$ modulo torsion. The leaderboard states it certifies independence by exact 2-descent with no floating point in the decision; that computation was not reproduced. The page is also living data: its commentary records that the original submission silently dropped a witness point through a parser bug, corrected three hours later.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"A closely watched benchmark: the record moved to 28 in 2006 and to 29 only in 2024, so it advances about once a decade, and the unbounded-ranks question behind it is a real problem in arithmetic geometry tied to BSD. Placed level with the sphere-packing upper bounds at 50 and just under the matrix multiplication exponent at 55, which is the same kind of ladder on a constant with wider consequences.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://elliptic-rank.icarm.cloud/curve/273","sourceName":"Elliptic Curve Rank Leaderboard, curve #273","collection":null,"collectionVersion":null,"links":[{"label":"ICARM: new record-breaking elliptic curve reported","url":"https://icarm.io/news/new-record-breaking-elliptic-curve-reported/","kind":"announcement"},{"label":"Curve data as JSON, including all thirty witness points","url":"https://elliptic-rank.icarm.cloud/curve/273.json","kind":"code"},{"label":"Dujella's history of elliptic curve rank records","url":"https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html","kind":"problem-record"},{"label":"Bober, conditionally bounding analytic ranks","url":"https://arxiv.org/abs/1112.1503","kind":"paper"},{"label":"Hacker News discussion","url":"https://news.ycombinator.com/item?id=49374873","kind":"discussion"}],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":1},{"slug":"fractional-colouring-pauli-shadow-tomography","name":"The Fractional Colouring Conjecture for Triply Efficient Pauli Shadow Tomography","shortName":"Shadow tomography $\\chi_f$ bound","problemNumber":null,"field":"Shadow tomography","fieldGroup":"Quantum information & computing","statement":"Conjecture 13 of King, Gosset, Kothari and Babbush asserts that for the set $B_\\varepsilon(\\rho)$ of Pauli observables with expectation value at least $\\varepsilon$ in magnitude, the fractional chromatic number of the induced anticommutation graph is $O(\\varepsilon^{-2})$; it would give a triply efficient Pauli shadow tomography algorithm. False: there are states and observables for which no finite constant bounds $\\chi_f \\varepsilon^2$.","posedBy":"Robbie King, David Gosset, Robin Kothari and Ryan Babbush","yearPosed":2025,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-20","model":"GPT Sol 5.6","modelMaker":"OpenAI","humanCollaborators":["Jędrzej Stempin","Santiago Llorens","Felix Huber"],"aiRole":"The arXiv comments field reads \"Found with GPT Sol 5.6\", and the paper's AI statement says the model was used to derive the main results, Theorem A and Theorem B. The construction amplifies fractional chromatic numbers, beta numbers and expectation values through lexicographic graph products.","verification":"unreviewed","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.20113v1, 14pp): the refuted statement is Conjecture 13 of King, Gosset, Kothari and Babbush in PRX Quantum 6, 010336 (2025), and the AI statement is as quoted. The constructions were not checked here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"A precisely stated conjecture from a published PRX Quantum paper, whose truth would have delivered a triply efficient shadow tomography algorithm - so the stakes were concrete. Only posed in 2025, which caps it: a year-old conjecture has not yet earned the standing of the named-conjecture band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.20113","sourceName":"Counterexamples to the fractional coloring conjecture for triply efficient shadow tomography","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-de-bruijn-newman-constant-is-0-1787854","name":"Upper bound $\\Lambda \\le 0.1787854$ on the de Bruijn-Newman constant","shortName":"Λ ≤ 0.1787854","problemNumber":null,"field":null,"fieldGroup":"Number theory","statement":"Let $H_t$ be the heat-flow deformation of the Riemann $\\xi$ function and $\\Lambda$ the de Bruijn-Newman constant: the zeros of $H_t$ are all real exactly when $t \\ge \\Lambda$. The Riemann hypothesis is the statement $\\Lambda \\le 0$, and Rodgers and Tao proved Newman's conjecture $\\Lambda \\ge 0$ in 2018. How small an upper bound on $\\Lambda$ can be proved?","posedBy":"Nicolaas de Bruijn (1950), whose bound of 1/2 opened the question; lowered by Ki, Kim and Lee (2009), Polymath15 (2019) and Platt-Trudgian's verification height (2020)","yearPosed":1950,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Claims $\\Lambda \\le 893927/5000000 = 0.1787854$, below the standing reviewed record of $0.2$ and the unreviewed $0.1875$ of July 2026. The Riemann hypothesis is $\\Lambda \\le 0$; Newman's complementary conjecture $\\Lambda \\ge 0$, his \"quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so\", was proved by Rodgers and Tao in 2018 and reproved more directly by Dobner. De Bruijn's original bound was $\\Lambda \\le 1/2$, and lowering it is one way of measuring progress toward RH. The frontier \"Upper bounds for the de Bruijn-Newman constant\" tracks these steps and links this entry from its 20 August 2026 row. Tighter bounds and the exact value remain open.","claimIssueNote":null,"solveDate":"2026-08-20","model":"Claude; ChatGPT and Codex (versions undisclosed)","modelMaker":"Anthropic; OpenAI","humanCollaborators":["Jude Gomila"],"aiRole":"\"Generative AI systems, including Anthropic Claude and OpenAI ChatGPT/Codex, were used in repeated, human-directed cycles of mathematical exploration, program development, testing, criticism, and manuscript revision. Jude Gomila selected the arguments and computations included here and accepts responsibility for the mathematical claims, the programs, and the final text.\"","verification":"unreviewed","verificationNote":"Read at the repository on 12 September 2026: the release, the sealed manifest, the referee report and the bibliography. The claim is Polymath15's Theorem 1.2 instantiated at exact parameters $X = 6000000185827$, $t_0 = 129/800$, $y_0^2 = 87677/2500000$, with fail-closed Arb interval certificates; unconditional in form, using RH only up to the finite Platt-Trudgian height. The referee report is an adversarial AI panel that calls itself no substitute for human review, found no fatal defect and left three items needing human sign-off. Nothing was rerun here: neither the certificates nor the parameter check against Theorem 1.2. The standing reviewed record is 0.2, Polymath15's criterion at Platt-Trudgian's height; the intermediate 0.1875 (Mosaic Intelligence, July 2026) is also unreviewed. Announcement reflects repository-only publication.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Scores the upper-bound problem for Λ, the question the de Bruijn-Newman frontier tracks at the same 45, not this one step on it. A named constant whose value decides the Riemann hypothesis, with an upper-bound race that was the subject of a Polymath project. Above the C11 capacity frontier at 35, whose audience is one subcommunity, and below the matrix multiplication exponent at 55, which all of theoretical computer science watches.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/judegomila/dbn-lambda-01787854-candidate-audit","sourceName":"GitHub repository","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"PluckyWombat559","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"big-line-big-clique-four-collinear-or-six-clique","name":"The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique","shortName":"Big-line-big-clique (4, 6)","problemNumber":null,"field":"Discrete geometry","fieldGroup":"Combinatorics","statement":"The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all $k, \\ell$ there is an $n$ such that every finite point set of at least $n$ points contains $\\ell$ collinear points or $k$ points that pairwise see each other. True for $\\ell = 4$, $k = 6$, the first case left open: every finite point set of size at least $10^{11055931}$ has four collinear points or six pairwise visible points.","posedBy":"Jan Kára, Attila Pór and David R. Wood","yearPosed":2005,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-19","model":"GPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":["Édouard Bonnet"],"aiRole":"The paper's disclosure, in full: \"After one fruitless attempt and the now customary generic encouragement, a relatively detailed proof of Theorem 1 was provided by GPT-5.6 Sol Pro after pondering for 222 minutes. The author's contributions were limited to checking the proof, simplifying some parts, and eventually writing the paper in a way that would give the author (and hopefully other human readers) a more pleasant reading experience.\"","verification":"unreviewed","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.19468v1, 7pp): the disclosure is verbatim as quoted, the conjecture's 2005 attribution is in the text, and the paper states this is the first open case resolved since 2009. The proof was not checked here. The threshold is astronomical but the statement is unconditional. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A named conjecture in discrete geometry with a real literature, open since 2005 and with no case resolved since 2009 - a seventeen-year standstill broken. Level with the named-conjecture band around 30, a little under it on breadth of recognition.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.19468","sourceName":"Large Finite Point Sets Have 4 Collinear Points or a 6-Clique","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mathbb-c-infty-caratheodory-conjecture","name":"The $C^\\infty$ Carathéodory Conjecture on Umbilic Points","shortName":"Carathéodory conjecture (smooth)","problemNumber":null,"field":"Differential geometry","fieldGroup":"Geometry & topology","statement":"Carathéodory's conjecture, Problem 8.1 of Ghomi's list and traceable to 1922, asks whether every closed convex surface in $\\mathbb{R}^3$ has at least two umbilic points. Hamburger settled the real-analytic case in 1940-41 and it stands. The $C^\\infty$ case is false: an explicit support function gives a smoothly embedded two-sphere bounding a convex body with exactly one umbilic point. The same family disproves the smooth Loewner conjecture, whose member at $k=1$ has an isolated trace-free Hessian zero of winding number three.","posedBy":"Constantin Carathéodory","yearPosed":1922,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a $C^\\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.","claimIssueNote":null,"solveDate":"2026-08-19","model":"Claude, Codex","modelMaker":"Anthropic, OpenAI","humanCollaborators":["Levent Alpöge","John-Paul Smith"],"aiRole":"Two distinct roles, neither of them the discovery. The formalization's module docstring records that Alpöge's announcement \"credits John-Paul Smith and Claude with checking the construction\", so the model's part was verifying a human construction. Separately, the Lean development was, in its author's words, \"developed with Codex and parallel proof-review agents\" - a formalization of a human result, which the methodology does not count as the contribution. Recorded as assisted rather than co-developed for that reason; the submission proposed co-developed.","verification":"lean-checked","verificationNote":"No independent mathematician has checked this yet. Audited by this site on 21 August 2026 at the commit the formal_proof attribute pins (7aa855b, google-deepmind/formal-conjectures). The formalized hypothesis is the classical statement and not a weakened one: IsConvexSphereOfClass requires Topology.IsEmbedding together with range F = frontier K for a compact convex K of nonempty interior, so \"parametrized\" names the Gauss parametrization rather than admitting mere immersions. The pinned line is not_caratheodoryConjectureOfClass_infty, the smooth statement, and the proof tree is 8013 lines carrying zero sorry, zero declared axioms and no native_decide. Not lean-verified, because that commit is NOT merged - it is diverged from main by 11 commits and behind by 26, and both upstream pull requests are drafts, #5070 saying \"I'm currently checking this ... please ignore\". The Lean was read here, not compiled, and the announcement itself is an X post.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"A named conjecture of 1922 that stood 104 years, whose analytic half is a celebrated theorem of Hamburger, and the counterexample takes the smooth Loewner conjecture with it. Set level with the cycle double cover and sum-product anchors at 55: unmistakably a landmark inside differential geometry, without the cross-field currency of the Jacobian conjecture at 65.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://x.com/__alpoge__/status/2089971359921156203","sourceName":"Levent Alpöge, X announcement of the smooth counterexample","collection":null,"collectionVersion":null,"links":[{"label":"Lean proof, sorry-free at the pinned commit","url":"https://github.com/google-deepmind/formal-conjectures/blob/7aa855bb344450777d9b19fe1cf11f2f5f9fae09/FormalConjectures/Other/CaratheodoryLoewnerCounterexample.lean","kind":"lean-proof"},{"label":"The informal proof accompanying the formalization","url":"https://github.com/google-deepmind/formal-conjectures/blob/7aa855bb344450777d9b19fe1cf11f2f5f9fae09/FormalConjectures/Other/CaratheodoryLoewnerCounterexample.md","kind":"paper"},{"label":"Lean statement of the smooth and analytic conjectures","url":"https://github.com/google-deepmind/formal-conjectures/pull/5066","kind":"lean-statement"},{"label":"Ghomi, Open Problems in Geometry of Curves and Surfaces (Problem 8.1)","url":"https://ghomi.math.gatech.edu/Papers/op.pdf","kind":"problem-record"}],"relations":[],"submittedBy":"VelvetFalcon287","upvotes":6,"downvotes":0,"commentCount":1},{"slug":"erdos-501-infinite-independent-sets","name":"Erdős Problem #501: infinite independent sets for families of small outer measure","shortName":"Erdős #501","problemNumber":501,"field":"Set theory / forcing","fieldGroup":"Logic & foundations","statement":"For every $x \\in \\mathbb{R}$ let $A_x \\subset \\mathbb{R}$ be a bounded set of Lebesgue outer measure $< 1$. Must there be an infinite independent set, that is an infinite $X \\subseteq \\mathbb{R}$ with $x \\notin A_y$ for all distinct $x, y \\in X$?\n\nErdős and Hajnal proved that arbitrarily large finite independent sets exist. Hechler showed in 1972 that the answer is no under the continuum hypothesis, so any positive answer had to come from a model where CH fails, and Sungchul Lee later derived one from a real-valued measurable cardinal.\n\nThe answer is that neither side is provable. Dropping Lee's large cardinal by transferring his argument to the extension of a model of CH by random reals gives a model where the answer is yes; Hechler's construction gives one where it is no. The question is independent of ZFC.","posedBy":"Paul Erdős","yearPosed":1961,"ageNote":null,"solveType":"independent","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Independent of ZFC, which is why this entry is the first to carry that result rather than proved or disproved. Both directions are formalized: Hechler's 1972 construction gives a model where the answer is no, and adding $\\mathfrak{c}^+$ random reals over a model of CH gives one where it is yes.\n\nThe credit is shared and mostly human. Newelski, Pawlikowski and Seredynski settled the problem's second question in 1987, and it is formalized here without the boundedness hypothesis. Hechler supplied one direction in 1972. Sungchul Lee derived a positive answer from a real-valued measurable cardinal, assisted by GPT-5.5 Pro, and Nat Sothanaphan observed that the two halves together give independence. What Glazer and Sol added is the removal of the large cardinal.\n\nerdosproblems.com still lists #501 as open at the time of writing.","claimIssueNote":null,"solveDate":"2026-08-19","model":"Sol, Claude","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Elliot Glazer"],"aiRole":"The formalization's metadata lists two authors, Elliot Glazer and Sol, and Glazer credits \"Sol and Claude\" for the work. The step they share is the one that was actually missing: transferring Sungchul Lee's real-valued-measurable argument to the extension of a model of CH by $\\mathfrak{c}^+$ random reals, which removes the large-cardinal hypothesis and leaves a positive answer consistent with ZFC.\n\nCo-developed rather than discovered, on the author's own framing. The README describes that transfer as \"routine\" for $\\Pi^2_1$ consequences and as \"applying that standard technology\", and the surrounding pieces are human: Hechler's 1972 counterexample, the Newelski-Pawlikowski-Seredynski closed case, Lee's relative result (itself assisted by GPT-5.5 Pro) and Nat Sothanaphan's observation that the two halves give independence.","verification":"lean-verified","verificationNote":"Both halves of the top rung are present, and both were checked here rather than taken from the announcement.\n\nKernel side: CI was green at the audited commit, Solution.lean contains no sorry and declares no axioms, and the only sorries in the repository are the deliberate placeholders in the two Challenge statement files, which is how a comparator challenge is meant to look. The repo ships an axiom audit recording every one of the seven targets as depending on propext, Classical.choice and Quot.sound alone, with no native_decide and no project axioms.\n\nStatement side: the development proves erdos501_sentence_faithful, that its sentence is equivalent in Mathlib's ZFSet to the Mathlib statement of the first question. That claim was checked against the source rather than believed: the target uses Bornology.IsBounded (A x), volume.toOuterMeasure (A x) < 1 and X.Pairwise (fun x y => x ∉ A y), and FormalConjectures/ErdosProblems/501.lean states erdos_501 with the same three predicates in the same shape. Formal Conjectures is a community-reviewed repository, so the statement is anchored outside the proof's own authors.\n\nThe independence itself is stated semantically, as the existence of models of ZFC on both sides, and restated a second time in proof-theoretic terms via a Lean port of Flypitch.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"A numbered Erdős problem with a denser trail than most: posed in 1961, restated as Problem 38 of Erdős and Hajnal's 1971 survey, then attacked by Hechler in 1972, Newelski-Pawlikowski-Seredynski in 1987 and Sungchul Lee more recently. Above the anchor at 10 for a typical numbered problem on that trail, and below Erdős #1196 at 15. Scored on the problem as it stood, so the fact that its resolution is the first formally verified independence result for an Erdős problem does not count toward it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/ElliotGlazer/erdos501","sourceName":"Erdős Problem #501 in Lean 4: the closed case and the independence of the first question","collection":null,"collectionVersion":null,"links":[{"label":"Palomar registry listing","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-08-19-000002&version=1","kind":"palomar"},{"label":"Formal Conjectures: the upstream statement the faithfulness target is checked against","url":"https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/501.lean","kind":"lean-statement"},{"label":"erdosproblems.com #501 - still listed open","url":"https://www.erdosproblems.com/501","kind":"problem-record"},{"label":"Forum thread: Lee's relative result, Sothanaphan's observation, the random-real transfer","url":"https://www.erdosproblems.com/forum/thread/501","kind":"discussion"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":1},{"slug":"the-delavina-waller-conjecture-on-the-wiener-index","name":"The DeLaViña–Waller conjecture on the Wiener index","shortName":"DeLaViña–Waller","problemNumber":null,"field":"Extremal Graph Theory","fieldGroup":"Combinatorics","statement":"Every finite simple connected graph $G$ with\n$$\n|V(G)|=2d+1,\\qquad \\operatorname{diam}(G)=d\\ge 3\n$$\nsatisfies\n$$\nW(G)\\le W(C_{2d+1})\n=\\frac{(2d+1)d(d+1)}2.\n$$\nThe claimed equality cases are exactly $C_{2d+1}$ for every $d\\ge3$, the double star $D_{2,3}$ when $d=3$, and the nine-vertex tree $T_{1,2,2}=S(2,3,3)$ when $d=4$.","posedBy":"E. DeLaViña and B. Waller","yearPosed":2008,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-19","model":"GPT-5.6 Sol; Claude Fable 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Mingchang Liu"],"aiRole":"The models assisted in developing proof strategies and checking computations.","verification":"unreviewed","verificationNote":null,"publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A named conjecture of DeLaViña and Waller (Electron. J. Combin. 15 (2008), #R33) that stood eighteen years, with Hua settling only the 2-connected case in 2018. Placed above the Graffiti anchor at 5, since this is a stated conjecture in the literature with partial results built on it rather than a raw generated one, and below Erdos #1196 at 15: it is a sharp extremal bound on one invariant at one order, not a question whose answer moves a field.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.22015517","sourceName":"The DeLaViña–Waller conjecture on the Wiener index","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"SilentIbis759","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-4-color-rado-number-of-x-y-c-z-general-case","name":"The 4-color Rado number of x+y+c=z: general case","shortName":"4-color Rado: $R(c)=40c+41$, all $c \\ge 2$","problemNumber":null,"field":"Rado numbers / partition regularity","fieldGroup":"Combinatorics","statement":"For a constant $c$, the 4-colour Rado number $R(c)$ is the least $N$ such that every colouring of $\\{1,\\ldots,N\\}$ in four colours contains a monochromatic solution to $x + y + c = z$. Myers (Rutgers thesis, 2015, Conjecture 4.9) and Ahmed, Boza, Emamy-Khansary, Marin, Revuelta and Sanz (Math. Comp. 85, 2016, §5.5) conjectured\n$$R(c) = 40c + 41$$\nfor all sufficiently large $c$, with the small values $R(0) = 45$ and $R(1) = 83$ as exceptions. Previous methods reached individual values but not the general case.\n\nThis claims the conjecture for every $c \\ge 2$, by reducing it to three finite facts: the single base value $R(2) = 121$ and the unsatisfiability of two \"spoke\" templates. The reduction is formalised in Lean 4 and holds for every $D \\ge 1$; the two templates are settled by SAT with DRAT certificates.","posedBy":"ABEMRS16 (Math. Comp. 85, 2016, §5.5); Myers (Ph.D. thesis, 2015, Conj. 4.9)","yearPosed":2015,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The claim is $R(c) = 40c+41$ for every $c \\ge 2$, reduced to three finite facts: the base value $R(2) = 121$, and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every $D \\ge 1$; the two unsatisfiability results carry DRAT proofs.\n\nThis completes the partial entry for the same conjecture, which proved it for roughly two thirds of integers via a scaling lemma; that lemma is now one of three legs, covering the branch where $d$ is divisible by 3.\n\nThe supporting results are worth more than the headline for anyone deciding whether to believe it: the paper also shows every band relaxation is satisfiable, which is why previous attempts stalled, and that the affine method alone is exactly sharp and can never finish.","claimIssueNote":null,"solveDate":"2026-08-19","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"Claude surveyed three mutually-unaware literatures (Malo 2000, Myers 2015, ABEMRS16 2016), proved the scaling lemma and 28 base values (partial entry), then devised the spoke relaxation, proved that band templates fail, built the Lean formalisation, and ran the two UNSAT template computations to settle the general case. Human direction limited to run design, operational supervision, and posting.","verification":"unreviewed","verificationNote":"Unreviewed: AI-produced, no peer review, and no authoritative tracker has accepted it. The artifact is unusually well organised, though, and some of it was checked here.\n\nChecked: the repository's CI is green on the verification workflow (CNF regeneration, hash checks, Lean reduction check); lean/Rado.lean is 354 lines with no sorry, no axiom declarations and no native_decide; and the theorem structure matches the prose, in that `upper` takes the three finite facts as explicit hypotheses, so Lean proves the reduction and the SAT work discharges the leaves rather than the Lean claiming the whole theorem.\n\nNot checked here: the two DRAT proofs were not re-verified, the SAT solves were not re-run, and the Lean was not rebuilt. By the repository's own logs drat-trim takes 998 s and 1125 s on the two templates, so this is compute rather than judgement, and it is exactly what site-confirmed would require.\n\nWorth noting in the submission's favour: a second, independently written encoder reproduces both unsatisfiability results from the definitions, the templates are regenerated from the Lean definitions and hash-checked against pins, and R(88) = 3561 was solved directly as a positive control.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A specialist conjecture from two 2015-16 sources, Myers's thesis and the ABEMRS Math. Comp. paper, in the small corner of Rado numbers for a single equation. Real, documented and narrowly read. Level with the partial entry for the same conjecture, because significance scores the problem rather than how much of it a given entry settles.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/ZestyWombat854/rado-number-4color-general","sourceName":"R(c)=40c+41 for every c>=2: SAT certificates, five-solver verdicts, second encoder, Lean-checked reduction","collection":null,"collectionVersion":null,"links":[{"label":"CLAIMS.md - every public claim mapped to the artifact backing it","url":"https://github.com/ZestyWombat854/rado-number-4color-general/blob/main/CLAIMS.md","kind":"other"},{"label":"The Lean 4 reduction (core Lean, no Mathlib)","url":"https://github.com/ZestyWombat854/rado-number-4color-general/blob/main/lean/Rado.lean","kind":"lean-proof"}],"relations":[],"submittedBy":"ZestyWombat854","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"separation-of-ordinary-and-strong-kreiss-constants","name":"Separation Between the Ordinary and Strong Kreiss Constants","shortName":"Kreiss constant separation","problemNumber":null,"field":"Operator theory","fieldGroup":"Analysis","statement":"Question 6.1 of Chalmoukis, Tsikalas and Yakubovich asks how far the Power boundedness constant $P(T)$ of a matrix can exceed its ordinary Kreiss constant $K(T)$. Answered more strongly: for every $K > 1$ there are matrices whose Cayley transforms satisfy $K(C_h(A_{n,h})) \\le K$ while the strong Kreiss constant satisfies $K_s(C_h(A_{n,h})) \\ge \\tfrac{1}{2}Cn^{\\alpha_K}$ with $\\alpha_K = (K-1)/(C+K-1)$. Since $P(T)\\geq K_s(T)$, this solves the question. Moreover, since the Kreiss matrix theorem gives $K_s(T) \\le P(T) \\le edK(T)$ in dimension $d$, the exponent $\\alpha < 1$ is optimal up to an arbitrarily small power loss.","posedBy":"Nikolaos Chalmoukis, Georgios Tsikalas and Dmitry Yakubovich","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-19","model":"ChatGPT 5.6 Pro, Claude Fable","modelMaker":"OpenAI","humanCollaborators":["Emiel Lorist","Martin Meyries","Mark Veraar"],"aiRole":"The same disclosure as the inverse generator entry, since both fall to one paper: ChatGPT 5.6 Pro explored Schauder basis counterexamples, assisted the adaptation of Ansorena's work that produces the explicit basis in Proposition 2.1, and helped optimize the explicit constants. The formalization in Lean 4 was done using\nClaude Fable by Anthropic. Note the boundary honestly - the disclosure names Proposition 2.1 and Theorem 1.1, not Theorem 1.4. Proposition 2.1 is the finite-dimensional construction every result in the paper is deduced from, including this one, so the model is in the loop for the machinery rather than for this theorem's derivation.","verification":"unreviewed","verificationNote":"Checked by this site on 22 August 2026 against the v2 PDF (arXiv:2608.06272v2, 19 Aug): the paper states \"We furthermore note that Theorem 1.4(i) solves [6, Question 6.1]\" and gives the explicit constants quoted in the statement, and reference [6] is Chalmoukis, Tsikalas and Yakubovich, arXiv:2512.10025. The mathematics of Theorem 1.4 was not checked here, though a curator numerical check of Proposition 2.1 - the construction it is deduced from - was carried out for the sibling entry and confirmed its bounds up to n = 256. \nA lean certificate of Theorem 1.1 was added in v3 of the paper (arXiv:2608.06272v2, 27 Aug)","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A question posed in December 2025 and answered nine months later, in a corner of operator theory read by the numerical-analysis and semigroup communities. Real and precisely stated, but young and narrow, so well below the sibling inverse generator problem at 22, which had stood since 1988.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.06272","sourceName":"arXiv:2608.06272 - A solution to the inverse generator problem and related questions","collection":null,"collectionVersion":null,"links":[{"label":"Chalmoukis, Tsikalas and Yakubovich, Operators with small Kreiss constants (Question 6.1)","url":"https://arxiv.org/abs/2512.10025","kind":"problem-record"}],"relations":[{"to":"a-counterexample-to-the-inverse-generator-problem-and-related-questions","kind":"same-work","note":"Both are deduced from the same finite-dimensional construction in Proposition 2.1; the Kreiss separation was added in version 2 of the paper."}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"counting-linear-extensions-below-two-to-the-n","name":"Counting Linear Extensions Below the $2^n$ Barrier","shortName":"Linear extensions, $2^n$ barrier","problemNumber":null,"field":"Exact exponential algorithms","fieldGroup":"Algorithms & optimization","statement":"Koivisto asked at Dagstuhl in 2013 whether the linear extensions of an arbitrary $n$-element poset can be counted exactly in time $O^*(c^n)$ for some $c < 2$. Yes: a deterministic exact algorithm runs in $O^*(1.89^n)$, breaking the $2^n$ barrier for the general problem.","posedBy":"Mikko Koivisto, at Dagstuhl","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-19","model":"Claude Opus 5, ChatGPT 5.6 Sol","modelMaker":"Anthropic, OpenAI","humanCollaborators":["Keigo Oka"],"aiRole":"The paper's disclosure: \"The core mathematical ideas underlying the new part of the algorithm and proof were discovered by Claude Opus 5 (Anthropic) during AI-assisted mathematical exploration\" - naming the first-upper-element pattern representation, multiplicity-profile decoding, the deadline dynamic program and the state-counting strategy of Sections 3 to 5. The chain-partition bound of Section 2 refines Kozma and is not new. The research prompt supplied to Claude Opus 5 was itself generated by ChatGPT 5.6 Sol, modelled on OpenAI's publicly released prompt for their cycle double cover work.","verification":"unreviewed","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.19505v1): the disclosure is verbatim as quoted and Koivisto's Dagstuhl 2013 question is cited in the abstract. This is an exact deterministic algorithm with a proved worst-case bound, not a heuristic, so it clears the methodology's exclusion. The ancillary Python script cross-checks correctness against brute force on small posets and does not certify the running time; it was not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"Breaking $2^n$ for a natural counting problem is the currency of exact exponential algorithms, and this question was posed explicitly and stood thirteen years. Level with the named-conjecture band around 30 inside its field, narrower outside it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.19505","sourceName":"Breaking the $2^n$ Barrier for Counting Linear Extensions with a Short Elementary Algorithm","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kasami-apn-function-triple-count-conjecture","name":"A Conjecture on Triple Counts for the Kasami APN Function","shortName":"Kasami APN triple counts","problemNumber":null,"field":"Finite fields and APN functions","fieldGroup":"Algebra","statement":"For the Kasami APN function $F(x) = x^{4^k - 2^k + 1}$ on $\\mathrm{GF}(2^n)$ with $\\gcd(k, n) = 1$, the conjecture asserts that for $\\Delta = \\{F(b) + F(b+1) + 1\\}$ and all distinct nonzero $v_1, v_2$, the number of triples in $\\Delta^3$ with $v_1 x + v_2 y + (v_1 + v_2) z = 0$ is exactly $2^{2n-3}$. Proved for $k \\bmod n \\in \\{1, 2, n-2, n-1\\}$ and verified exhaustively for $n \\le 13$; the general case remains open.","posedBy":"Proposed anonymously at the NSUCRYPTO cryptographic olympiad","yearPosed":2019,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-19","model":"Claude Fable 5, Aristotle","modelMaker":"Anthropic, Harmonic","humanCollaborators":["Gábor P. Nagy","Attila Vajda"],"aiRole":"The paper's statement: \"Every proof in this paper was obtained by the AI assistant Claude Fable 5 and has subsequently been formally verified in the Lean theorem prover by Aristotle (Harmonic).\" The model was prompted with the conjecture statement together with background hints and a pointer to the companion repository.","verification":"lean-checked","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.18584v1): the AI and Lean statements are verbatim as quoted. Recorded lean-checked rather than lean-verified deliberately - the companion repository is cited but its URL is not exposed in the HTML, so no artifact has been audited here for sorry or for declared axioms, and statement fidelity is unaudited. Entered as Partial on the paper's own words, \"The general case remains open\".","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"An olympiad-posed question rather than a literature conjecture, and settled only in special cases. The AI and formal-verification story is the strongest in this batch, but significance scores the problem as it stood, which places this just above the Graffiti anchor at 5.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.18584","sourceName":"On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for k mod n in {1,2,n-2,n-1}, and exhaustive verification for n<=13","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fourth-moment-conjectures-for-rademacher-sums","name":"Fourth-moment conjectures for Rademacher sums","shortName":"Rademacher fourth-moment conjectures","problemNumber":null,"field":"Khintchine inequalities","fieldGroup":"Probability & statistics","statement":"Let $\\varepsilon_1,\\ldots,\\varepsilon_n$ be independent Rademacher signs, let $\\sum a_i^2 = 1$, write $S = \\sum a_i\\varepsilon_i$ and $q = \\sum a_i^4$, and let $\\mu_p = \\mathbb{E}|G|^p$ for a standard Gaussian $G$. Two 2025 conjectures say that $q$ alone governs how far $S$ falls short of Gaussian.\n\nJakimiuk proved $\\mathbb{E}|S|^p \\le \\mu_p - c_p q$ for $p \\ge 3$ and conjectured the optimal constant is $c_p = \\mu_p - 1$ throughout that range; separately he conjectured a dimension-free quadratic stability bound at the critical exponent $p = 3$.\n\nBaranski, Murawski, Nayar and Oleszkiewicz reduced the finite-dimensional $L_p/L_4$ Khintchine constant for $p \\ge 5$ to $\\sup_{x \\ge 1}\\|x + \\varepsilon_1 + \\cdots + \\varepsilon_N\\|_p / \\|x + \\varepsilon_1 + \\cdots + \\varepsilon_N\\|_4$ and conjectured the supremum is attained at $x = 1$ - that is, the flat coefficient vector is the extremizer.","posedBy":"Jacek Jakimiuk; Adam Barański, Daniel Murawski, Piotr Nayar, Krzysztof Oleszkiewicz","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Four results, and the first is partly a refutation. Jakimiuk conjectured $c_p = \\mu_p - 1$ is optimal for every $p \\ge 3$; the paper proves that for $p \\ge 4$ and gives a counterexample for every $2 < p < 4$, so the conjecture is false as posed and the corrected range is $p \\ge 4$. The witness is the two-coordinate vector $S_2 = (\\varepsilon_1+\\varepsilon_2)/\\sqrt2$.\n\nThe Baranski-Murawski-Nayar-Oleszkiewicz flat-point conjecture is proved outright, in the stronger form that $x \\mapsto \\|x+S_n\\|_p/\\|x+S_n\\|_4$ is strictly decreasing on $[1,\\infty)$ for every real $p \\ge 5$; that range is the one they conjectured, so nothing is left over. Jakimiuk's second conjecture, dimension-free quadratic stability at $p = 3$, is proved with an explicit constant, though the optimal constant there is only bracketed and stays open. The paper also records the exact fixed-$q$ moment and Laplace-transform envelopes, from which coefficient-sensitive tail bounds follow.","claimIssueNote":null,"solveDate":"2026-08-18","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Peigan Gao","Jian Qian"],"aiRole":"From the paper's \"Statement of AI use\": \"Initial versions of the proofs of the sharp Gaussian stability inequality and the finite-dimensional $L_p$-$L_4$ constant theorem were developed with assistance from ChatGPT 5.6 Sol. The authors checked and revised the arguments, take full responsibility for their mathematical content, and independently verified the extensions presented here.\" The abstract puts it as \"The proofs are discovered with substantial assistance from ChatGPT 5.6 Sol.\"\n\nCo-developed rather than discovered, and the disclosure is why: it credits the model with initial versions of two of the paper's four theorems, inside a paper the humans led, and this site gives a disclosure phrased as assistance the lower tier. The two theorems named are Theorem 1.1 and Theorem 1.3, which are the two conjectures being settled. Neither the counterexample below $p = 4$ nor the third-moment theorem is attributed to the model.","verification":"unreviewed","verificationNote":"Unreviewed: a one-day-old arXiv preprint, unrefereed, with no independent endorsement, and none of the proofs were checked here.\n\nThe statements were, and all three survived. Every theorem here is an inequality between finite sums and one-dimensional integrals, so it can be evaluated directly and hunted for counterexamples. Theorem 1.1 was tested on 18,960 coefficient vectors - flat families, one spike plus a flat sea, and random vectors of every sparsity - across twelve exponents from 4 to 13, with no violation, and the stated equality case $q=1$ holds exactly. Proposition 4.3 requires the same inequality to fail for every $2 < p < 4$: at $S_2 = (\\varepsilon_1+\\varepsilon_2)/\\sqrt2$ it reduces to $2^{p/2} \\le \\mu_p + 1$, which fails strictly across that range and is an exact equality at both endpoints. Theorem 1.3's strict monotonicity held in 32,000 evaluations over eight exponents and $n \\le 10$.\n\nThat is a check of the statements, not of the proofs, so the tier does not move. It is not nothing: a wrong constant or a misplaced threshold would have shown up at once. The script is scripts/check_rademacher.py in this site's repository.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"Two conjectures stated in March 2025 papers, inside the long-running program on optimal constants in Khintchine's inequality that runs back through Haagerup and Szarek. Real questions by established probabilists - one of the source papers is in Bernoulli - but a year old, with no literature of attack behind them yet. Above the anchor at 5, which covers machine-generated conjectures and one-off questions: these are named conjectures within a recognised program. Below the anchor at 10 for a typical numbered Erdos problem, which carries decades more standing.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.17802","sourceName":"Fourth-Moment Geometry of Rademacher Sums","collection":null,"collectionVersion":null,"links":[{"label":"Jakimiuk, Stability of Khintchine inequalities with optimal constants (Bernoulli 32, 2026)","url":"https://arxiv.org/abs/2503.07001","kind":"problem-record"},{"label":"Baranski, Murawski, Nayar, Oleszkiewicz - On the optimal Lp-L4 Khintchine inequality","url":"https://arxiv.org/abs/2503.11869","kind":"problem-record"},{"label":"This site's numerical check of Theorems 1.1 and 1.3 and of Proposition 4.3","url":"https://github.com/mrconter1/vibemathed/blob/main/scripts/check_rademacher.py","kind":"code"}],"relations":[],"submittedBy":"VibeGene","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"stein-s-dimension-free-weak-1-1-riesz-transform-problem","name":"Stein’s dimension-free weak-(1,1) Riesz transform problem","shortName":"Dimension-free Riesz weak-(1,1)","problemNumber":null,"field":"Harmonic analysis","fieldGroup":"Analysis","statement":"The Riesz transforms $R_1,\\ldots,R_n$ on $\\mathbb{R}^n$ are the Fourier multipliers $-i\\xi_j/|\\xi|$, the natural higher-dimensional Hilbert transforms. Stein proved in 1983 that their $L^p$ bounds can be taken independent of the dimension for every $1 < p < \\infty$. At the 1986 ICM he asked whether the same holds at the endpoint $p=1$: is there an absolute constant $C$, independent of $n$, with\n$$|\\{x : |Rf(x)| > \\lambda\\}| \\le \\frac{C}{\\lambda}\\,\\|f\\|_{L^1(\\mathbb{R}^n)}$$\nfor every $\\lambda > 0$? The Calderon-Zygmund route gives a constant that grows with the dimension, and the best known was Janakiraman's $c\\log n$.\n\nThis paper answers yes, with $C = 2$, for the vector transform $R = (R_1,\\ldots,R_n)$ - so the same constant serves every single component $R_j$ uniformly in $n$.","posedBy":"Elias M. Stein","yearPosed":1986,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The theorem is the vector-valued endpoint bound $\\|Rf\\|_{L^{1,\\infty}} \\le 2\\|f\\|_{L^1}$ for $R = (R_1,\\ldots,R_n)$, so the constant 2 also serves each component $R_j$ uniformly in the dimension; the best previously known component bound grew like $c\\log n$.\n\nThe mechanism is a decomposition theorem stated as Theorem 1.2: for every nonnegative $f \\in L^1 \\cap L^2$ and every $\\lambda > 0$, write $f = \\mu + (-\\Delta)^{\\alpha/2}u$ with $\\mu \\le \\lambda$ and $u$ in the fractional Sobolev space $H^\\alpha$, obtained from an obstacle problem for the fractional Laplacian together with a Lewy-Stampacchia type estimate on an unbounded domain. That replaces the Calderon-Zygmund decomposition, whose cube geometry is where the dimensional loss enters.","claimIssueNote":null,"solveDate":"2026-08-18","model":"Claude Opus 5.0, GPT-5.6 Sol","modelMaker":"Anthropic, OpenAI","humanCollaborators":["Yuyuan Ouyang","Daniel Spector","Cody B. Stockdale"],"aiRole":"From the paper's \"Artificial Intelligence Statement\": \"The proof strategy was developed by Large Language Models (LLMs), through a combination of ChatGPT (GPT-5.6 Sol), Codex CLI and web interface, mathematical reasoning agents, and Claude Opus 5.0, in dialogues with the authors.\"\n\nThe sequence is specific. The first author started from the second and third authors' 2020 paper and ran a coordinated attempt - a Sol agent with the Danus and Rethlas automated reasoning agents (both deployed on OpenAI Sol agents) and a Polya \"How to Solve It\" style agent - which reached only partial results. Further dialogue with ChatGPT Sol gave an attempted complete solution via variational inequalities on the torus and a transference principle, much more complicated than what was published. The second and third authors judged that a direct Euclidean proof should be possible; the second author prompted Claude Opus 5.0 to try it, and \"the response was a longer document that provided the basis for the proof idea used in the present paper\".\n\nThe authors then checked and rewrote the proofs, did the literature review, wrote the introduction, and state they \"independently verified, validated, and rewritten all parts of the paper influenced by LLM-generated material\" and take full responsibility for the mathematics.","verification":"unreviewed","verificationNote":"A one-day-old arXiv preprint, unrefereed, with no independent endorsement on record, so this stays Unreviewed. Nothing was checked here either: the proof is twelve pages of obstacle-problem and fractional-Sobolev analysis, with no finite certificate to re-run.\n\nThe setting was checked and holds up. Stein's question is real and was open, and the trail is unusually clean: the second and third authors wrote \"On the dimensional weak-type $(1,1)$ bound for Riesz transforms\" (arXiv:2004.03382, Comm. Contemp. Math. 23, 2021), which reduces this exact question to finite sums of Dirac masses and records Janakiraman's $c\\log n$ as the best known. This paper is a continuation of their own program, and the AI statement says the first author started from it. The claim is also unusually falsifiable for its kind: an absolute constant 2, not an asymptotic. It clears the known lower bound - in $n=1$ the transform is the Hilbert transform, whose weak-type $(1,1)$ norm is Davis's constant, about 1.347.\n\nAgainst that: forty-year-old endpoint problems do not usually fall in twelve pages, and the strategy came from a model, so the argument has had less human incubation than its length suggests.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"Posed by E. M. Stein at the 1986 ICM, at the endpoint of his own celebrated dimension-free $L^p$ theorem for Riesz transforms, and one of the recognised open questions of modern harmonic analysis: four decades of documented attack, including Janakiraman's $c\\log n$, Banuelos-Osekowski's sharp martingale bounds, and Spector-Stockdale's reduction to Dirac masses. Above Talagrand's convexity problem at 37 on provenance - an ICM question by Stein sitting on a celebrated theorem of his own. Below Sendov at 40, which carries a far larger dedicated literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.18068","sourceName":"A dimension-free weak-type (1,1) bound for the vector Riesz transform on R^n","collection":null,"collectionVersion":null,"links":[{"label":"Spector and Stockdale (2020), the reduction to Dirac masses this paper grew out of","url":"https://arxiv.org/abs/2004.03382","kind":"paper"}],"relations":[],"submittedBy":"VibeGene","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"colombo-s-difference-power-determinant-conjecture","name":"Colombo’s difference-power determinant conjecture","shortName":"Colombo determinant","problemNumber":null,"field":"Linear algebra; matrix theory and Pfaffians","fieldGroup":"Algebra","statement":"Let $N\\ge 2$ be even and let $x_1,\\ldots,x_N$ be pairwise distinct real numbers. Is the matrix\n$$\nA_{N,D}=\\bigl[(x_j-x_i)^D\\bigr]_{i,j=1}^{N}\n$$\nnonsingular for every integer $D\\ge N-1$?","posedBy":"Bonaparte Colombo","yearPosed":1928,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"For pairwise distinct real numbers $x_1,\\ldots,x_N$ and an integer $D\\ge1$, Ma proves the complete classification\n$$\n\\det\\bigl[(x_j-x_i)^D\\bigr]_{i,j=1}^{N}\\ne0\n\\quad\\Longleftrightarrow\\quad\nD\\ge N-1\\ \\text{and}\\ \\bigl(N\\text{ is even or }D\\text{ is even}\\bigr).\n$$\n\nThis fully resolves Colombo’s original conjecture for even $N$. The new part is the even-size, odd-exponent branch, strengthened to the strict Pfaffian sign theorem\n$$\n(-1)^{\\binom m2}\\operatorname{Pf}\n\\bigl[(x_j-x_i)^{2r+1}\\bigr]_{i,j=1}^{2m}>0\n\\qquad(r\\ge m-1).\n$$\nThe even-exponent branch follows from the classical work of Dyn–Goodman–Micchelli. A concurrent independent proof of the odd branch by Kun Li, Li Tie, Peng Wang and Zihan Liu is linked below.","claimIssueNote":null,"solveDate":"2026-08-18","model":"WuJie AI agent; DeepSeek; Qwen; Kimi; GPT; other LLMs","modelMaker":null,"humanCollaborators":["Qianli Ma"],"aiRole":"The author reports that the WuJie AI agent and multiple large-language-model systems, including DeepSeek, Qwen, Kimi and GPT, played a substantial role in identifying the proof strategy and producing an initial proof draft. Qianli Ma designed and coordinated the workflow, then checked and revised the mathematical arguments and formalized the new odd-exponent proof chain in Lean 4.","verification":"lean-checked","verificationNote":"The paper’s new odd-exponent theorem is formalized in Lean 4. The public endpoint proves\n$0<\\det[(x_j-x_i)^{2r+1}]$\nfor every strictly increasing real $2m$-tuple and every $r\\ge m-1$, matching Theorem 1.2 of the paper. The repository reports a complete 2817-job build, no `sorry`, `admit`, or project-specific axioms, and an axiom audit containing only `propext`, `Classical.choice`, and `Quot.sound`; the formalization is also registered in Palomar.\n\nThe complete classification additionally imports the classical even-exponent theorem of Dyn–Goodman–Micchelli, which this repository does not formalize. The paper statement and public theorem interface were compared for this submission, but the repository was not rebuilt by the submitter and no named independent informal-to-formal statement-fidelity audit is yet identified. Therefore “Lean-checked, statement unaudited” is the conservative label.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A 1928 determinant conjecture of Colombo: genuinely old, and answered completely rather than improved, but obscure - it has no following outside the corner of matrix theory and total positivity that it sits in, and the literature on it is thin. Tied with the (2,1)-gapped consecutive-ones case at 12, likewise a clean settlement of a specific stated question with a specialist audience. Below the Tu-Deng conjecture at 15, which has a decade of partial results behind it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2609.00101","sourceName":"Qianli Ma, “Colombo’s Determinant Problem” (arXiv:2609.00101)","collection":null,"collectionVersion":null,"links":[{"label":"Lean 4 formalization (Ma)","url":"https://github.com/hkjtsgmc79-boop/colombo-odd-lean","kind":"other"},{"label":"Palomar formalization registry entry","url":"https://palomar-registry.org/entry?id=PALOMAR-2026-08-27-000007&version=1","kind":"other"},{"label":"Earlier Zenodo preprint deposit","url":"https://doi.org/10.5281/zenodo.21993537","kind":"other"},{"label":"Independent apolarity/Waring-rank proof","url":"https://arxiv.org/abs/2608.28274","kind":"paper"},{"label":"Lean formalization of the independent proof","url":"https://github.com/ttieli/Colombo1928","kind":"independent"}],"relations":[],"submittedBy":"AmberGander937","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"a-32-leaf-tree-requiring-six-coordinates-for-an-isometric-ell-infty-embedding","name":"A 32-leaf tree requiring six coordinates for an isometric $\\ell_\\infty$ embedding","shortName":"A 32-leaf tree requiring six coordinates","problemNumber":null,"field":"Metric geometry; tree metrics; isometric embeddings into l-infinity","fieldGroup":"Geometry & topology","statement":"Every tree metric with $t$ leaves embeds isometrically into $\\ell_\\infty^{t-1}$, and the least dimension needed is at least $\\lceil \\log_2 t \\rceil$. Fitzpatrick and Nowakowski asked in 2000 whether the logarithmic bound is always attained, and Brigham, Chartrand, Dutton and Zhang conjectured in 2005 that every tree with $t$ leaves embeds isometrically into $\\ell_\\infty^{\\lceil \\log_2 t \\rceil}$, verifying it through $t = 21$. Aksoy, Kilic and Kocak posed a sharp leaf-threshold version for weighted metric trees in 2020.","posedBy":"Fitzpatrick and Nowakowski (Problem 43, 2000); conjectured by Brigham, Chartrand, Dutton and Zhang (2005)","yearPosed":2000,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"Disproved. An explicit 32-leaf tree has least isometric $\\ell_\\infty$-dimension six, not the conjectured five, and keeps dimension six under every assignment of positive edge lengths, so the weighted sharp-threshold conjecture falls too. Every tree with at most 31 leaves does attain $\\lceil \\log_2 t \\rceil$, so 32 is the first failure; Brigham et al. had checked through 21. The proof is finite and exact: six explicit orientation masks cover all leaf pairs, and a recurrence on rooted branches rules out any five-orientation cover.","claimIssueNote":null,"solveDate":"2026-08-17","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Logan R. Chalmers"],"aiRole":"The paper's disclosure: \"OpenAI's GPT-5.6 Sol assisted in implementing the computational search strategy and with drafting this manuscript. The author takes full responsibility for the content of the article.\" The search that found the tree and the proof that no five-orientation cover exists are the author's; the model wrote code and prose. That is the assisted tier on this site.","verification":"unreviewed","verificationNote":"Unreviewed. arXiv 2608.16288 (17 August 2026, six pages) read in full here; the abstract's claims match the theorems. The counterexample is an exact finite argument with the covering masks printed in the paper and ancillary verification code deposited on Zenodo, which this site has not run. No peer review, no independent expert statement.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A question asked in 2000 and a conjecture from 2005 with a small but real literature (Brigham et al., Aksoy-Kilic-Kocak), refuted by the first counterexample. Clean and final, but a single embedding-dimension question in metric graph theory rather than a structural conjecture; a rung above a numbered Erdős problem.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.16288","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"BoldJackal580","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"matrix-multiplication-exponent-2371177","name":"The Matrix Multiplication Exponent","shortName":"Matrix multiplication exponent","problemNumber":null,"field":"Algebraic complexity","fieldGroup":"Theoretical computer science","statement":"The matrix multiplication exponent $\\omega$ is the infimum of all $t$ for which two $n \\times n$ matrices can be multiplied in $O(n^t)$ arithmetic operations. Strassen showed in 1969 that $\\omega < 3$, and sixty years of work has driven the upper bound down without anyone knowing the true value. Whether $\\omega = 2$ is one of the central open questions of algebraic complexity.\n\nThe current bounds come from the laser method as refined by combination loss analysis. This paper attacks the optimization problem at the core of that refinement, reformulating it so it can be solved in a larger setting, designing a new optimization algorithm for it, and then refining that algorithm with AlphaEvolve.\n\nThe result is $\\omega < 2.371177$, improving the previous best of $2.371339$.","posedBy":"Volker Strassen","yearPosed":1969,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"A record, not a resolution, and a small one by design. The bound moves from $2.371339$ to $2.371177$, about $1.6 \\times 10^{-4}$, and the authors describe it as a small step. Whether $\\omega = 2$ is untouched, and nothing here suggests the laser method can reach it.\n\nThe interesting claim is methodological rather than numerical. The bottleneck in this line of work is a hard optimization problem, and the paper reports progress by reformulating that problem and then improving the optimizer, with AlphaEvolve doing the final refinement. That is a different kind of contribution from a new mathematical identity, and it is why the entry is filed as computation. Superseded as the best claimed bound by the OpenAI math release (October 2026), which claims omega <= 9/4 over C (Lean-checked, not independently reviewed) and omega < 2.371054886006746 over every field; see matrix-multiplication-exponent-9-4.","claimIssueNote":null,"solveDate":"2026-08-17","model":"AlphaEvolve","modelMaker":"Google DeepMind","humanCollaborators":["Emilien Dupont","Marvin Eisenberger","Borislav Kozlovskii","Abbas Mehrabian","Francisco J. R. Ruiz","Abigail See","Renfei Zhou","Josh Alman","Virginia Vassilevska Williams","Matej Balog"],"aiRole":"The paper describes three improvements to the optimization problem at the heart of combination loss analysis, and AlphaEvolve is the third of them. In the authors' own order: they reformulate the problem so it can be solved in a larger setting than was previously possible, they leverage recent advances in machine learning to design a new optimization algorithm for it, and then they \"refine the resulting optimization algorithm with AlphaEvolve\".\n\nCo-developed rather than discovered. The model improves a component of a human-designed pipeline rather than being handed the problem, and the reformulation that made the larger setting tractable is the authors' own. It is more than tooling, though, because the refined optimizer is what produces the bound.","verification":"unreviewed","verificationNote":"A two-day-old arXiv preprint, unrefereed, and nothing was checked here. Bounds of this kind are not the sort of claim a reader can spot-check: the number falls out of a large optimization over laser-method parameters, so reproducing it means re-running the optimization rather than verifying a certificate.\n\nWhat the author list is worth saying: Josh Alman and Virginia Vassilevska Williams are authors of the prior bounds this improves on, which is unusual and cuts against the main risk with an automated search, namely that it optimizes something subtly different from the quantity everyone means by $\\omega$.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":55,"significanceNote":"The exponent of matrix multiplication, with sixty years of work and an entire subfield organised around it: the laser method, tensor rank, the group-theoretic approach, and a chain of records from Strassen through Coppersmith-Winograd to the present. That is the 50 to 60 band, textbooks and a subfield rather than a single question inside one. Above every current theoretical-computer-science entry, the highest of which is 39, and below the zeta proportion at 68, which is attached to the Riemann hypothesis. Scored on the problem, so the size of this particular improvement does not enter.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.16884","sourceName":"Improving the matrix multiplication exponent with modern optimization and AlphaEvolve","collection":null,"collectionVersion":null,"links":[{"label":"Balog's announcement thread","url":"https://x.com/matejbalog/status/2089597390369984794","kind":"announcement"},{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/matrix-multiplication-exponent-9-4","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"talagrand-s-convolution-conjecture","name":"Talagrand’s convolution conjecture","shortName":"Talagrand convolution","problemNumber":null,"field":"Analysis of Boolean functions","fieldGroup":"Probability & statistics","statement":"On the Boolean hypercube $G = \\{-1,1\\}^n$ with uniform measure $\\lambda$, let $T_\\mu f(x) = \\int_G f(x \\odot y)\\,d\\mu(y)$ be convolution by a finite positive measure $\\mu$, and set\n$$\\psi_\\mu(u) = \\sup\\{u\\,\\lambda(\\{T_\\mu f \\ge u\\}) : f \\ge 0,\\ \\|f\\|_1 = 1\\},$$\nwhich measures how much better than Markov's inequality convolution makes the tail. In 1989 Talagrand conjectured that for the biased-coin product measure $\\mu_a = (\\tfrac{1+a}{2}\\delta_1 + \\tfrac{1-a}{2}\\delta_{-1})^{\\otimes n}$ with $0 < a < 1$,\n$$\\psi_{\\mu_a}(u) \\le \\frac{C_a}{\\sqrt{\\log u}} \\qquad (u > 1),$$\nwith $C_a$ depending on $a$ alone and not on the dimension $n$. He offered a \\$1000 prize for a proof. The Gaussian analogue was settled by Eldan and Lee; the hypercube case, the original, stayed open.\n\nThis paper claims the conjectured bound.","posedBy":"Michel Talagrand","yearPosed":1989,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The claim is the exact conjectured decay: $\\psi_{\\mu_a}(u) \\le C_a/\\sqrt{\\log u}$ for every $u > 1$ and every $n$, with $C_a$ dimension-free - concretely $\\lesssim \\kappa_a^2(\\log\\frac{\\kappa_a}{\\kappa_a-1})^{1/2}$ where $\\kappa_a = (1+a)/(1-a)$.\n\nWhat is new is one step in a three-paper chain rather than a proof from scratch, and the paper is explicit about it. Chen's reverse-heat and Boolean-bridge framework and Xiang-Zhang's localized terminal-discrepancy method are taken as given; the addition is a power coupling that splits each reverse edge ratio into two geometric powers, producing a switched exponential weight that restores the exact reverse jump rate of the perturbed coordinate. Because the frozen exponent then has a fixed numerator, no growing stopping buffer is needed and the $\\log\\log u$ factor disappears. That last $\\log\\log$ is what stood between the previous work and Talagrand's statement.","claimIssueNote":null,"solveDate":"2026-08-16","model":"Odin Automatic AI Research Agent","modelMaker":null,"humanCollaborators":["Junwei Lu","Shengtao Guo","Ethan X. Fang"],"aiRole":"The paper's disclosure is two sentences, in the abstract and again under a heading \"The role of AI in this proof\": \"Odin Automatic AI Research Agent was used to discover the proof. The final proofs were reorganized by the authors.\"\n\nTaken at face value, as this site's classification rule requires, that is an AI-discovered claim: the model produced the proof and the humans wrote it up. It is also thinner than any other entry at this tier. The paper says nothing about what Odin is, who builds it, which models it runs on, how it was steered, or how much of the manuscript is the agent's, and no public record of an \"Odin Automatic AI Research Agent\" could be found from this site - so the model maker field is left empty rather than guessed at.\n\nThe contribution being credited is a single idea: the power coupling that splits each reverse edge ratio into two geometric powers, which is what removes the iterated-logarithmic loss from the framework the paper inherits. The three named humans are Junwei Lu (Harvard T.H. Chan School of Public Health), Shengtao Guo and Ethan X. Fang.","verification":"unreviewed","verificationNote":"A three-day-old arXiv preprint with no independent endorsement, so Unreviewed, and no mathematics was checked here.\n\nThe chain it sits in was checked, and it holds. Talagrand's hypercube conjecture was open - O'Donnell's problem list still carries it, and even the Gaussian special case was open as of 2012. Yuansi Chen (arXiv:2511.19374, Nov 2025) proved it up to a dimension-free $(\\log\\log)^{3/2}$ factor; Yanjin Xiang and Zhihua Zhang (arXiv:2606.04573, June 2026) cut that to $\\log\\log$; this paper claims to remove the loss entirely. Both predecessors exist, are by identifiable people, and say what this paper says they say, and the target bound matches Talagrand's own suggested $C_a/\\sqrt{\\log u}$ rather than something adjacent.\n\nTwo things cut the other way. Chen's paper needed a v2 to fix \"a mistake in the previous draft which was kindly pointed out by Joseph Lehec\" - that is what scrutiny in this corner looks like, and it is what this paper has not yet had. And the proof is attributed to an agent nobody outside the paper can identify, with no account of how it was run, so the process cannot be weighed either.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":37,"significanceNote":"A named 1989 conjecture of Talagrand (Israel J. Math. 68, \"A conjecture on convolution operators\"), carrying his own \\$1000 prize, listed in O'Donnell's \"Open Problems in Analysis of Boolean Functions\", and with a celebrated Gaussian counterpart settled by Eldan and Lee in Duke. Known across probability, Boolean analysis and geometric functional analysis for thirty-five years and invisible outside them. Level with Talagrand's convexity problem at 37, which has the same standing - a prized Talagrand problem on a recognised list. Above Feige at 35, below Sendov at 40.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.15515","sourceName":"Weak-Type Bounds for Convolution on the Boolean Hypercube","collection":null,"collectionVersion":null,"links":[{"label":"Chen, Talagrand's convolution conjecture up to loglog via perturbed reverse heat (Nov 2025)","url":"https://arxiv.org/abs/2511.19374","kind":"paper"},{"label":"Xiang and Zhang, Layerwise terminal discrepancy in Chen's reverse-heat coupling (June 2026)","url":"https://arxiv.org/abs/2606.04573","kind":"paper"},{"label":"O'Donnell's open-problem list, where the conjecture and its $1000 prize are recorded","url":"https://arxiv.org/abs/1204.6447","kind":"problem-record"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"composites-among-xi-7-n","name":"Composites Among $[\\xi 7^n]$ and Right-Truncatable Primes in Base 7","shortName":"Composites among $[\\xi 7^n]$","problemNumber":null,"field":"Number theory - integer sequences","fieldGroup":"Number theory","statement":"For every real $\\xi>0$ the sequence of integer parts $[\\xi 7^{n}]$, $n=0,1,2,\\dots$, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~$7$.","posedBy":"Forman and Shapiro (1967), Dubickas and Novikas (2005)","yearPosed":2005,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-15","model":"Fable 5, Opus 4.8","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"After showing that a finite computation can provably resolve the problem, it wrote programs to do the computation, resulting in a checkable certificate. It then was directed to formalize all proofs and the certificate check in Lean. Finally it was directed to write a draft of the paper from the Lean.","verification":"lean-checked","verificationNote":"Reviewed and corrected 17 August 2026. The first review said no public Lean repository was linked; that was wrong - it is linked in appendix A of the paper. The error is recorded here rather than quietly dropped.\n\nSource audit of github.com/rwst/On-Composites at commit 2e49c4d, 17 files, comments stripped before counting: no sorry, admit or axiom anywhere on the proof path. The 18 sorry occurrences are all in Challenge.lean, which nothing imports - the leanprover/comparator \"statement of record\", which re-declares the definitions against Mathlib alone so the solution's constants, axiom profile and fresh-export kernel re-acceptance can be checked.\n\nThe tier stops at Lean-checked for a precise reason: all five comparator configs permit exactly propext, Quot.sound and Classical.choice, and the two theorems this entry claims - infinite_composites_seven and no_infiniteTruncatablePrime_seven - are in none of them, because they rest on three native_decide calls, which decide via the compiled evaluator rather than the kernel. The repository documents that quarantine itself.\n\nTwo things bound the risk: floorPow, CompositeInt and InfiniteTruncatablePrime are verbatim identical to the Mathlib-only re-declarations comparator certifies at std3 for bases 3-6, so definitional drift is ruled out; and cond.c, cycles.c and compress.py recompute the hypotheses outside Lean. Not built here - no toolchain, and the repo has no CI - so this is a source audit, not a compile.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A concrete question in the Forman-Shapiro tradition, posed for base 7 by Dubickas and Novikas in 2005 and open twenty-one years. Specialist but genuinely standing - level with the numbered-Erdos band at 10.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.researchgate.net/publication/412294027_On_the_composites_among_x7","sourceName":"On the composites among [ξ7ⁿ]","collection":null,"collectionVersion":null,"links":[{"label":"Lean development, data and generating programs","url":"https://github.com/rwst/On-Composites","kind":"lean-proof"}],"relations":[],"submittedBy":"LucidKestrel185","upvotes":0,"downvotes":0,"commentCount":1},{"slug":"three-block-admm-identity-block-counterexample","name":"Convergence of Three-Block ADMM with Identity Third Block","shortName":"3-block ADMM, identity block","problemNumber":null,"field":"Optimization - splitting methods","fieldGroup":"Algorithms & optimization","statement":"After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample. Disproved: an explicit rational counterexample exists in which the first two blocks are strongly convex quadratics and direct three-block ADMM produces a bounded nonconvergent orbit of period 66, verified by exact checks along a piecewise-affine reduction path.","posedBy":"Open subclass left by Chen, He, Ye, Yuan (2016)","yearPosed":2016,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"GPT-5.6 Sol (Codex)","modelMaker":"OpenAI","humanCollaborators":["Kenan Xu","Xiangfeng Wang"],"aiRole":"The paper is framed as AI-assisted discovery in its own title: using Codex with GPT-5.6 Sol, the authors construct the explicit rational counterexample candidate and verify it along a piecewise-affine reduction path, with exact checks establishing the period-66 nonconvergent orbit; the same Codex workflow guides a further study.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14396): the open status of the identity-third-block subclass is stated in the abstract and introduction, and the construction-and-verification account is as quoted. The exact checks were not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The convergence of multi-block ADMM is a well-known question in optimization since Chen-He-Ye-Yuan's celebrated 2016 counterexample, and the identity-third-block case was the natural surviving hope. Specialist but widely recognisable in the field.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14396","sourceName":"AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cone-theorem-effective-fourfold-pairs-char-p","name":"The Cone Theorem for Effective Fourfold Pairs in Characteristic $p > 5$","shortName":"Cone theorem, fourfolds, char p>5","problemNumber":null,"field":"Birational geometry (positive characteristic)","fieldGroup":"Algebra","statement":"Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to $X$, the cone theorem holds for projective log canonical, $\\mathbb{Q}$-factorial fourfold pairs $(X, \\Delta)$ with $K_X + \\Delta \\equiv M \\ge 0$, over bases of positive and mixed characteristic $p > 5$.","posedBy":"The char-p minimal model program (Birkar, Hacon, Xu, Waldron and others)","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"ChatGPT 5.6 Sol, Codex","modelMaker":"OpenAI","humanCollaborators":["Joe Waldron"],"aiRole":"The paper's AI statement, in the author's words: he worked out the outline with all essential ingredients in Spring 2025, but one gap remained that he could not repair; he gave the unfinished proof to ChatGPT 5.6 Sol in Summer 2026 asking it to fill the gap, and it modified the approach to avoid the issue, which after further editing by hand and with Codex became the current version.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14236, Waldron, Michigan State): the AI statement is verbatim as quoted - among the frankest in the catalog, a professional birational geometer crediting the model with repairing a gap he could not. The result is conditional on the log resolution conjecture and is recorded as Partial for that reason. The proof was not checked here; days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The minimal model program in positive characteristic is a central program of modern algebraic geometry, and the cone theorem for fourfolds is a real step it has been waiting for - conditional on log resolution, which keeps it below the unconditional band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14236","sourceName":"The cone theorem for effective fourfold pairs in characteristic p>5","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"kusner-conjecture-equilateral-sets-counterexample","name":"Kusner's Conjecture on Equilateral Sets in $\\ell_p^n$","shortName":"Kusner equilateral sets","problemNumber":null,"field":"Discrete geometry","fieldGroup":"Geometry & topology","statement":"Kusner conjectured in 1983 that the maximum number of points in $\\mathbb{R}^n$ that are pairwise at $\\ell_p$-distance one is exactly $n+1$ for every $2 < p < \\infty$, as in the Euclidean case. False: an explicit configuration of $n+2$ equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.","posedBy":"Robert B. Kusner","yearPosed":1983,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"GPT-5.6 Sol, Claude Fable 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Logan R. Chalmers"],"aiRole":"The paper's disclosure: GPT-5.6 Sol assisted in implementing the computational search strategy in code, and Claude Fable 5 was used as a tool in drafting. No mathematical step is attributed to a model by name, but the search that produced the configuration is the load-bearing computation.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14013): the disclosure is verbatim as quoted, Kusner's 1983 attribution is in the introduction, and the data is deposited on Zenodo (10.5281/zenodo.21911503). The exact-arithmetic certificate was not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A named 1983 conjecture in discrete geometry with a real literature (Alon-Pudlak among others) and forty-three years of standing. Well known within convexity and discrete geometry, less so outside - level with the named-conjecture band at 30.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14013","sourceName":"A counterexample to Kusner's conjecture on equilateral sets","collection":null,"collectionVersion":null,"links":[{"label":"Zenodo data and verification package","url":"https://doi.org/10.5281/zenodo.21911503","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pavez-signe-length-control-spanning-subdivisions","name":"Pavez-Signe's Length-Control Question for Spanning Subdivisions","shortName":"Balanced spanning subdivisions","problemNumber":null,"field":"Extremal graph theory - digraphs","fieldGroup":"Combinatorics","statement":"Pavez-Signe (2024) conjectured a Dirac-type condition for spanning $H$-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths; Lee (2025) resolved the existence conjecture in the stronger digraph setting. Answered affirmatively with epsilon-room: for every $\\varepsilon > 0$ there is $C_0$ such that every $n$-vertex digraph $D$ with $n \\ge C_0 h$ and minimum semi-degree $\\delta^0(D) \\ge (1/2+\\varepsilon)n$ contains a spanning $H$-subdivision whose path lengths differ by at most one, for every digraph $H$ with $h$ arcs and no isolated vertices.","posedBy":"Matias Pavez-Signe","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"ChatGPT 5.6","modelMaker":"OpenAI","humanCollaborators":["Zhilan Wang","Shuo Wei","Jin Yan"],"aiRole":"The acknowledgement states the authors used ChatGPT 5.6 to assist in the development of the probabilistic partition argument in one named lemma, plus language polishing, with all arguments independently verified by the authors.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14432, Wang-Wei-Yan): the attribution chain (Pavez-Signe's conjecture and question, Lee's resolution of existence) is in the abstract, the answer carries an epsilon in the semi-degree threshold rather than the exact conjectured bound - hence Partial, and the paper's own closing question asks for the exact threshold - and the AI acknowledgement is verbatim as quoted. Not checked here; days old, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A 2024 question in the Dirac-type spanning-structures literature, answered in approximate form two years later. Real but young and specialist.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14432","sourceName":"Nearly balanced spanning subdivisions in dense digraphs","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"score-determined-induced-tournament-statistics-an-all-orders-classification-and-","name":"Score-Determined Induced Tournament Statistics: an All-Orders Classification","shortName":"Score-determined tournament statistics","problemNumber":null,"field":"Tournament theory","fieldGroup":"Combinatorics","statement":"A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different.\n\nQuestion: Which linear combinations of induced k-subtournament type-counts are score-determined — identical across all tournaments sharing a score sequence, at any host size?\n\nAnswer: Exactly the linear combinations of degree-multiplicity counts $m_0,\\dots,m_{k−1}$, where $m_r$ counts how many of the $k$ chosen vertices have exactly $r$ internal wins. These $k$ functions satisfy one linear relation, so score-determined statistics have dimension $k−1$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"GPT Sol 5.6","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Absolutely everything, i occasionally steered and redirected.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026: the Zenodo reproducibility archive (21959291, Howard-Stanley) is real and carries the manuscript and code. The classification itself was not checked here, no independent review exists, and - flagged plainly - nobody has swept the tournament-theory literature for prior appearances of this question, which for a self-posed problem is the main risk. The disclosure (\"absolutely everything, I occasionally steered\") is the frankest ai-discovered claim in the catalog.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":3,"significanceNote":"A clean and natural classification question, but self-posed and resolved in the same work - no prior literature poses it. Scored at the self-posed floor, and no systematic prior-art sweep has been done by the author or this site, so even novelty is unconfirmed.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.21959291","sourceName":"Reproducibility Archive for “Score-Determined Induced Tournament Statistics: An All-Orders Classification and a Square-Zero Defect Algebra”","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"shemshallah","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"purely-prime-not-purely-maximal-counterexample","name":"Tarizadeh's Conjecture on the Maximality of Purely-Prime Ideals","shortName":"Purely-prime vs purely-maximal","problemNumber":null,"field":"Commutative algebra","fieldGroup":"Algebra","statement":"Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every purely-prime ideal is purely-maximal. False: there is a commutative ring with a purely-prime ideal that is not purely-maximal.","posedBy":"Abolfazl Tarizadeh","yearPosed":2020,"ageNote":"Conjecture posed by the author (# 5.8) in their own 2020 paper","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"ChatGPT Pro","modelMaker":"OpenAI","humanCollaborators":["Abolfazl Tarizadeh"],"aiRole":"In the author's words: after years without progress on his own conjecture, \"by using an advanced model of AI (ChatGPT Pro), a counterexample is found to this conjecture\". The counterexample is the paper's content; the conjecture's own poser credits the model with finding it.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14251): the abstract and introduction say what the entry says, including that the conjecture is the author's own published Conjecture 5.8 which resisted for years. The construction was not checked here. Days-old preprint by the conjecture's poser, no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A published conjecture in commutative algebra that stood for some years, disproved by its own poser with a model finding the counterexample. Narrow literature - between the week-old floor and the numbered-problem band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14251","sourceName":"A counterexample to a question on the maximality of purely-primes","collection":null,"collectionVersion":null,"links":[{"label":"Tarizadeh 2020, where the conjecture is stated - Conjecture 5.8","url":"https://arxiv.org/html/2001.04823v2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"exact-order-three-ambiguity-of-the-einstein-maxwell-dilaton-coupling-a-2-in-metr","name":"Exact order-three ambiguity of the Einstein-Maxwell-dilaton coupling $a^2$ in metric jets, and its fourth-order collapse","shortName":"EMD $a^2$ jet threshold","problemNumber":null,"field":"Mathematical relativity - Rainich geometrization, Kaluza-Klein","fieldGroup":"Mathematical physics","statement":"Is the EMD coupling square $a^2$ a function of the metric three-jet on an explicit active, non-null, simple-spectrum family of truncated Einstein-Maxwell-dilaton data, and can one more derivative recover it? Proved: no function of the common metric three-jet returns $a^2$ - the order-three ambiguity is exactly a free affine shear orbit ($\\mathbb{R}$) mixing $B=a\\sin 2\\theta$ with the phase gradient - while the fourth-order quotient recovers $a^2=A^2+B^2$, with equality fiber exactly $a=\\pm b$ ($\\mathbb{Z}_2$). In particular Kaluza's $a=\\sqrt{3}$ and the control $a=1$ collide through metric order three.","posedBy":"This work (2026); lineage: Rainich 1925, Misner-Wheeler 1957","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Finite-jet theorems about compiled truncated EMD equation certificates: the exact shear-orbit fiber classification of the complete first seed channels, an explicit collision family with one metric three-jet realized by an actual cubic metric germ (genuine Frechet Ricci value and first derivative), the compiled impossibility theorem, and the fourth-order recovery with equality fiber $a=\\pm b$. Not settled here: promotion to analytic EMD solution germs (separate written argument pending specialist audit, in the parent repository), chart covariance beyond the fixed presentation, density of the active locus, degenerate branches, and any sufficiency of $a^2=3$ for a Kaluza uplift.","claimIssueNote":null,"solveDate":"2026-08-14","model":"GPT 5.6 Sol, Fable","modelMaker":"OpenAI, Anthropic","humanCollaborators":["James Kehoe"],"aiRole":"Proposed and proved the theorems, wrote the Lean 4 formalization and the manuscript under human direction, with repeated adversarial audit cycles; the human author set scope, claim boundaries, and verification gates.","verification":"lean-checked","verificationNote":"Source-audited by this site on 17 August 2026: the repository was cloned at v0.1.0 and all 77 Lean files scanned with comments stripped - zero sorry, zero admit, zero axiom declarations, zero native_decide, toolchain pinned at v4.32.1 with Mathlib. Not compiled here (the repo's CI is claimed to). Lean-checked rather than Lean-verified per this site's standing split: the statements' correspondence to the informal claims has not been independently audited, and the same pipeline produced both the proofs and the formalization. No human peer review.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":3,"significanceNote":"A precisely posed and cleanly resolved identifiability question, but posed by this work itself in 2026 - there is no prior literature asking it. Scored at the self-posed floor; the Rainich-Misner-Wheeler lineage is context, not a pedigree.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/jimpeebles/emd-coupling-identifiability/tree/v0.1.0","sourceName":"emd-coupling-identifiability (Lean 4 repository, manuscript + PDF)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"PluckyCobra527","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"quartic-hessian-conjecture-dimension-four","name":"The Quartic Hessian Conjecture in Dimension Four","shortName":"Quartic Hessian, $n=4$","problemNumber":null,"field":"Polynomial automorphisms","fieldGroup":"Algebra","statement":"The Hessian conjecture $HC_n$ asks whether every polynomial $f$ with $\\det \\mathrm{Hess}(f) \\in \\mathbb{C}^\\times$ has a polynomial gradient inverse. It is known for $n \\le 3$, false for $n \\ge 5$, and open exactly in dimension four, where it implies the plane Jacobian conjecture. Proved for every quartic polynomial in dimension four: the quartic case reduces to $f = P(x_1,x_2,x_3) + x_4 Q(x_1,x_2,x_3) + a x_4^2$ with $\\deg Q \\le 2$, and every constant-Hessian polynomial of this form has a polynomial gradient inverse.","posedBy":"The Hessian conjecture (de Bondt, van den Essen line)","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"GPT-5.6 Sol, GPT-5.6 Luna, Claude Fable 5, DeepSeek V4 Pro","modelMaker":"OpenAI, Anthropic, DeepSeek","humanCollaborators":["Zixiang Ni"],"aiRole":"The acknowledgement credits four systems - GPT-5.6 Sol, GPT-5.6 Luna, Claude Fable 5 and DeepSeek V4 Pro - with exploring candidate arguments, adversarial proof review, algebraic checking and editorial assistance, with the author independently reviewing all arguments. No individual step is attributed, so the lower tier applies.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14217): the acknowledgement is verbatim as quoted, and the introduction's status summary (known for n <= 3 by Dillen and de Bondt, false for n >= 5, open for n = 4, HC_4 implies the plane Jacobian conjecture) matches the literature, including the n >= 5 counterexample recorded as this catalog's sibling entry. Sole-author preprint, days old, not checked here, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The quartic slice of the one open Hessian dimension, whose full resolution would imply the plane Jacobian conjecture. A real partial advance on a recognisable target - specialist band at 15.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14217","sourceName":"The Quartic Hessian Conjecture in Dimension Four","collection":null,"collectionVersion":null,"links":[],"relations":[{"to":"hessian-conjecture-five-variable-counterexample","kind":"related","note":"Two sides of the same Hessian conjecture: that entry disproves it in dimension five, this one proves its quartic case in dimension four - the single dimension still open."}],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"four-color-rado-number-of-x-y-c-z-40c-41","name":"The 4-Color Rado Number of $x+y+c=z$: $R(c)=40c+41$ Whenever $c+1$ Is Divisible by 3, 4, 5 or 7","shortName":"4-color Rado: $R(c)=40c+41$","problemNumber":null,"field":"Rado numbers / partition regularity","fieldGroup":"Combinatorics","statement":"$R(c) = 40c+41$ for every $c \\geq 2$ such that $c+1$ is divisible by 3, 4, 5, or 7 (covering $\\approx 66\\%$ of all $c$); the full conjecture (Myers 2015 Conj. 4.9, ABEMRS16 §5.5) reduces to prime cases $p \\geq 89$, all smaller primes settled by SAT. Twenty-eight exact values, nineteen new primes $p = 11, \\ldots, 83$, zero deviations from the conjectured line.","posedBy":"ABEMRS16 (Math. Comp. 85, 2016, §5.5); Myers (Ph.D. thesis, 2015, Conj. 4.9)","yearPosed":2015,"ageNote":"Posed twice independently: Myers' Rutgers thesis (2015, Conjecture 4.9) and ABEMRS16 (Math. Comp. 85, submitted 2014), whose section 5.5 asks it for c >= 7. Malo's 2000 masters thesis had already settled two of the four congruence classes per its abstract - a fragmentation across three mutually unaware sources that this entry is the first to close.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Twenty-eight individual exact values, each proved by SAT certificate (coloring at n-1, UNSAT at n). The synthesis theorem covers every c >= 2 whose c+1 is divisible by 3, 4, 5, or 7 (~66% of integers). The prime-reduction corollary shows the full conjecture (R(c)=40c+41 for all c >= 2) is equivalent to checking primes p >= 89; all primes through 83 are settled. What stays open: the conjecture at c=88 (p=89) and every larger c whose c+1 has all prime factors >= 89. The scaling lemma's attribution is hedged relative to Malo 2000 (full text not accessed). No Lean formalization; the SAT certificates and dual-encoder architecture are the verification tier.","claimIssueNote":null,"solveDate":"2026-08-14","model":"Claude Fable","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"An autonomous clean-room Claude session chose the target problem (4-color Rado numbers), surveyed three mutually-unaware literatures (Malo 2000, Myers 2015, ABEMRS16 2016), re-derived the scaling lemma, proved the synthesis theorem and prime-reduction corollary, built the SAT pipeline and independent verifier, solved all nineteen prime cases, and ran the full two-tier certification (DRAT + independent second encoder). Two independent AI referee agents verified the proof (both CONFIRMED). Human direction limited to run design, operational supervision, and posting.","verification":"unreviewed","verificationNote":"Reproduced in substance by this site on 14 August 2026, independently of the repo's code. All 28 coloring certificates were re-checked by an own-code scanner over every monochromatic triple: 28/28 valid, so every lower bound holds outright. Five base cells were fully re-solved with an independently written encoder (own variable layout, own symmetry breaking): satisfiable at $n-1$ and unsatisfiable at $n$ for $c = 0, 2, 3, 4, 5$, matching $R(0)=45$ and the $40c+41$ line exactly. The scaling lemma, its sharpness against the universal lower bound, the synthesis theorem and the prime-reduction corollary were verified by hand; the algebra is elementary and correct. The literature was verified independently: ABEMRS16 is Math. Comp. 85 (2016) 2047-2064 with exactly the claimed authors; Myers' Conjecture 4.9 appears verbatim in the Rutgers thesis; Malo's 2000 thesis is real (Open Prairie, South Dakota State) with $R(1..3)$ in its public abstract, and its full text is bot-gated - so the submitter's hedge about the scaling lemma possibly being Malo's is accurate and could not be resolved from here either. The 2026 papers on this equation were spot-checked and are two-color, as claimed. Not reproduced: the nineteen prime-case UNSAT certificates ($n$ up to 3321), which rest on the bundle's kissat DRAT proofs, drat-trim VERIFIED, with a second independent encoder agreeing on every instance both ran; and no human peer review exists - produced and refereed by AI agents in one pipeline.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A ten-year-old conjecture with real standing - posed independently in a Math. Comp. paper and a Rutgers thesis, in the Schur/Rado tradition - advanced to two thirds of all cases with a clean reduction of the remainder to primes. Specialist territory, and the conjecture itself stays open, which caps it: level with the a >= 4 dihedral theorem (8), above the finite-cell bundles (5), below the Erdos entries at 10.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/ZestyWombat854/rado-number-4color","sourceName":"GitHub repo (synthesis theorem + SAT certificates + dual-encoder verification + independent checker)","collection":null,"collectionVersion":null,"links":[{"label":"ABEMRS16, On the n-color Rado number for x_1+...+x_k+c = x_{k+1} (Math. Comp. 85, section 5.5 poses the conjecture)","url":"https://doi.org/10.1090/mcom3034","kind":"problem-record"},{"label":"Myers, Computational Advances in Rado Numbers (Rutgers Ph.D. thesis, 2015) - Conjecture 4.9","url":"https://sites.math.rutgers.edu/~zeilberg/Theses/KellenMyersThesis.pdf","kind":"problem-record"},{"label":"Malo, Four Color Rado Numbers for x_1+x_2+c=x_3 (South Dakota State M.S. thesis, 2000)","url":"https://openprairie.sdstate.edu/etd2/760/","kind":"problem-record"}],"relations":[],"submittedBy":"ZestyWombat854","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sivaraman-perfect-divisibility-question-p17","name":"Sivaraman's Perfect-Divisibility Characterization Question","shortName":"Perfect divisibility vs chi-bound","problemNumber":null,"field":"Graph theory - chi-boundedness","fieldGroup":"Combinatorics","statement":"Sivaraman asked whether perfect divisibility is characterized by its chromatic consequence: is a graph $G$ perfectly divisible if and only if $\\chi(H) \\le \\binom{\\omega(H)+1}{2}$ for every induced subgraph $H$ of $G$? False: the Paley graph $P(17)$ satisfies the chromatic bound hereditarily but is not perfectly divisible.","posedBy":"Vaidy Sivaraman","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Zhiyu Wang","Weihao Xia"],"aiRole":"The acknowledgement states that the Paley graph P(17) - the counterexample witness - was identified during exploratory use of ChatGPT and subsequently verified by the authors, with ChatGPT also used for language polishing and generating the verification code. All arguments and computations were checked by the authors.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14519, Wang-Xia): the question is stated as Question 1 with attribution to Sivaraman by personal communication, the negative answer via P(17) is Theorem-level, and the acknowledgement is verbatim as quoted. The P(17) computation was not re-run here. The paper's larger content - perfect and linear divisibility of chair-free graphs - is human work outside this entry's claim. Days old, no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":4,"significanceNote":"A natural characterization question, but posed by personal communication in 2026 and answered within the year. Below the Graffiti anchor; the value is the clean witness.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14519","sourceName":"Perfect Divisibility, Linear Divisibility and Chair-Free Graphs","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lorist-schwenninger-remark-2-positivity-question","name":"Lorist-Schwenninger Remark 2 positivity question","shortName":"Lorist-Schwenninger Remark 2","problemNumber":null,"field":"Numerical range / Crouzeix's conjecture","fieldGroup":"Analysis","statement":"Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives $\\kappa \\le 1 + \\sqrt{1 - \\Re\\langle E_1 Tx,x\\rangle}$, so positivity of $\\Re\\langle E_1 Tx,x\\rangle$ would prove the lemma outright. They write: \"it is unclear whether $\\Re\\langle E_1 Tx,x\\rangle \\ge 0$ in general.\"","posedBy":"Emiel Lorist, Felix L. Schwenninger","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.","claimIssueNote":null,"solveDate":"2026-08-14","model":"Claude Opus 5.0","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"The counterexample was found and verified computationally by Claude Opus 5.0 in a single session. The search identified the mechanism (the centres of the domain and the numerical range must be separated to make the quantity negative), found a 2x2 witness, reduced it to a closed-form algebraic expression, and verified the sign in exact arithmetic. The sharp consequence — that inequality (5) alone recovers exactly the Crouzeix-Palencia constant 1+sqrt(2) — was derived in the same session. The human operator directed the investigation to Remark 2 and approved the final write-up; all mathematical content was produced by the model.","verification":"site-confirmed","verificationNote":"Reproduced by this site on 17 August 2026. Remark 2 was confirmed verbatim in the Lorist-Schwenninger LaTeX source, including the exact sentence \"it is unclear whether Re >= 0 in general\". The counterexample repository pins the claim in a statement file and was run here: the exact sympy certificate gives m = -4/95 - 1188*sqrt(90709)/8617355 = -0.0836..., with the sign certified through its minimal polynomial rather than floating point, and the independent mpmath implementation (no shared code) agrees; both also confirm the paper's own bounds still hold at the witness, so the counterexample refutes the remark's hope without touching Lemma 1. The mechanism note - the domain centre and numerical-range centre must separate - matches what the certificate shows. No human peer review; the tier records this site's own reproduction.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":4,"significanceNote":"A question posed ten days earlier, in a remark of the Lorist-Schwenninger Crouzeix paper - real, precisely stated, and worth settling because it delimits what their inequality (5) can do alone, but it is a fresh aside rather than a standing problem. Below the Graffiti anchor at 5.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.03841","sourceName":"Lorist-Schwenninger, \"A solution to Crouzeix's conjecture\"","collection":null,"collectionVersion":null,"links":[{"label":"GitHub","url":"https://github.com/ZestyWombat854/crouzeix-dilation-positivity","kind":"code"}],"relations":[{"to":"crouzeix-s-conjecture","kind":"builds-on","note":"Answers a question from Remark 2 of Lorist-Schwenninger's second Crouzeix proof: inequality (5) alone cannot give the sharp constant, since Re can be negative."}],"submittedBy":"ZestyWombat854","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"order-dimension-beyond-rank-simplicial-arrangements","name":"Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements","shortName":"Order dimension beyond rank","problemNumber":null,"field":"Posets / hyperplane arrangements","fieldGroup":"Combinatorics","statement":"Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and recorded the general guess that every simplicial region poset has dimension equal to its rank (Problem 9.3 of his 2016 chapter); Segovia later asked the analogous question for oriented-poset lattices. False: the order dimension of the poset of regions can exceed the rank - the Coxeter arrangements $H_4$ and $E_6$ satisfy $\\dim W(H_4) \\ge 5$ and $\\dim W(E_6) \\ge 7$.","posedBy":"Nathan Reading; Segovia","yearPosed":2016,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"ChatGPT 5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Daria Poliakova"],"aiRole":"The declaration, in full: \"The small obstruction subgraphs were found by ChatGPT 5.6 Sol Ultra. The human input was the belief that the rank guess is incorrect, and one should look for counterexamples.\" The obstruction subgraphs are the entire content of the disproof, so the model produced the central objects under human direction.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14092): the declaration is verbatim, and the problem attribution is precise - Reading's Problem 9.3 (2016) with the H_4/E_6 background from his 2003 computations, plus Segovia's analogous question. The obstruction subgraphs were not re-verified here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A recorded problem of Reading's from the standard reference chapter on posets of regions, open ten years, answered with the first simplicial arrangements whose region posets have dimension above rank. Specialist band at 15.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14092","sourceName":"Order dimension beyond rank for simplicial hyperplane arrangements","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tournament-isomorphism-bounded-vc-dimension","name":"Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension","shortName":"Tournament isomorphism, bounded VC","problemNumber":null,"field":"Graph isomorphism","fieldGroup":"Theoretical computer science","statement":"Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension $d$ is decidable in time $n^{O(d \\log d)}$, so automorphism groups of bounded-VC tournaments are computable in polynomial time; isomorphism of tournaments of bounded chromatic number is also polynomial-time decidable.","posedBy":"Daniel Neuen, Martin Grohe","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-14","model":"Claude Sonnet 5","modelMaker":"Anthropic","humanCollaborators":["Simon Raßmann","Pascal Schweitzer"],"aiRole":"The statement of AI use names two specific steps: the proof of one preliminary lemma (the tournament VC-dimension lemma) was provided by Claude Sonnet 5, and the proof of a second bound was simplified by it. Named model-contributed lemmas inside a human-led argument.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14486, Rassmann-Schweitzer): the statement of AI use is verbatim, with the two lemmas identified by reference, and the abstract states the Neuen-Grohe attribution. The algorithm was not checked here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A posed open problem in the graph-isomorphism literature after Babai's quasipolynomial breakthrough, from named experts, with a clean parameterized resolution. Specialist - just above the numbered-problem band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14486","sourceName":"Isomorphism of tournaments with bounded VC dimension","collection":null,"collectionVersion":null,"links":[{"label":"Raßmann & Schweitzer 2024 - Problem statement","url":"https://doi.org/10.4230/LIPIcs.ICALP.2024.78","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"1-17353-planar-lower-bound-and-exact-local-envelopes-for-cost-preserving-single-","name":"1.17353 planar lower bound and exact local envelopes for cost-preserving single-source unsplittable flow","shortName":"1.17353 planar bound and exact local envelopes for SSUFs","problemNumber":null,"field":null,"fieldGroup":"Combinatorics","statement":"For a single-source unsplittable flow, find the optimal universal additive constant $C$ s.t. every feasible fractional flow $x$ with arc costs $c$ should admit an unsplittable routing $y$ with $c^\\top y \\le c^\\top x$ and $y_a \\le x_a + C \\cdot D$ on every arc. We provide several new results on $C$: \n\n(1) record lower bound for planar instances (against known ceiling 2):\n$$C \\ge \\frac{58676765987259}{50000000000000} = 1.17353531974518;$$\n\n(2) local envelope ladder (proved): $E(2) = 1$, $E(3) = 9/8$, $E(4) = (299 - 41\\cdot\\sqrt{41})/32 = 1.13974707\\ldots$, attained by the counterexamples from our previous work; record constants of our previous work are now exact local envelopes of the general theory;\n\n(3) global results: every exact-two-path instance with rows touching at most three terminals satisfies $C \\le 2$ (first unconditional constant for an unbounded class); interaction arity m gives $\\lceil\\lfloor 3m/2\\rfloor /2\\rceil \\cdot D$;\n\n(4) classes closed exactly: out-trees 0; two-layer hubs 1; outerplanar two-exit interval spines 1 (sharp); series-parallel $\\le 1$;\n\n(5) band merger constant $K^* \\ge 2.5652\\ldots$ (twice the general lower bound $1.2826\\ldots$).","posedBy":"Dinitz, Garg, Goemans","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Part III of a series, and the first unconditional positive results in it. Settled exactly: the local envelope ladder $E(2)=1$, $E(3)=9/8$ and $E(4)=(299-41\\sqrt{41})/32=1.13974707\\ldots$, which recasts the earlier record constants as exact envelopes of the general theory rather than isolated instances, plus exact constants for four classes - out-trees 0, two-layer hubs 1, outerplanar two-exit interval spines 1 (sharp), series-parallel at most 1. Improved but not settled: the planar lower bound rises to $1.17353531974518$ against the known ceiling 2, and every exact-two-path instance whose rows touch at most three terminals satisfies $C\\le2$, the first unconditional constant for an unbounded class. The universal question is untouched - it reduces here to a single factor-two merger statement with certified wall $K^*\\ge2.5652\\ldots$, twice the refined general lower bound $1.28260069\\ldots$.","claimIssueNote":null,"solveDate":"2026-08-13","model":"GPT-5.6 Sol, Claude Fable 5, Claude Opus 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":[],"aiRole":"Like the previous parts (https://vibemathed.com/problem/1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u), this work was written in close collaboration with GPT 5.6 Sol, Claude Fable 5, and Claude Opus 5, which contributed proofs, failed routes, adversarial reviews, code for the verification campaign, and more. This time, I cannot claim that it was a single tour de force by the models like the original counterexample by Rybin; it was a long journey with a lot of human involvement, but the key ideas were provided by the LLMs. I have personally verified and edited this work in its entirety, and all errors are mine.","verification":"site-confirmed","verificationNote":"Reproduced by this site on 13 August 2026 from a clean clone, in two parts. First the repository's own suite: all fourteen verifiers in verify/run_all.sh pass, exit 0. Those cover Parts I and II only, so the headline planar record was rebuilt here independently. Reading only the raw arc list, a depth-first search rediscovers exactly two source-to-terminal paths for each of the six terminals; the fractional arc loads recompute exactly on all 21 arcs; all 64 routing overloads recompute exactly in rational arithmetic; the cost rule fits all 64 of the certificate's own cost deltas; 42 routings come out cost-preserving as claimed; and the minimum overload over those 42 is $58676765987259/50000000000000$, exactly the record. Planarity was checked independently too, by Euler ($V=16$, $E=21$, $F=7$) and by networkx. The envelope constant was derived symbolically from the stated quartic rather than read off: $t^*=(7-\\sqrt{41})/4$ is the unique critical point in $(0,2-\\sqrt3)$, giving $E(4)=(299-41\\sqrt{41})/32$. The 2,015-cell closure ledger is internally complete: five forms of 403, family counts summing to 2,015, every cell on one of nine solver-free lemmas. Not checked: the mixture characterization, the network-matrix total-unimodularity theorem and the tree-path four-colouring theorem, conventional proofs in an unreviewed preprint with no independent expert review. The tier records this site's reproduction of the certificates; the structural theory remains unreviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The residual optimal-constant question left open by the Dinitz-Garg-Goemans disproof, scored level with Part II of the same series. Specialist, but rooted in a well-known 1999 conjecture, and this instalment adds the first unconditional positive results rather than another record. Held at 15 because the universal constant is still open and the exact answers are for restricted classes.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/snikolenko/unsplittable-flows","sourceName":"Unsplittable flows repository","collection":null,"collectionVersion":null,"links":[{"label":"Zenodo preprint","url":"https://zenodo.org/records/21922203","kind":"paper"}],"relations":[{"to":"1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u","kind":"continues","note":"Part III of the same programme. Part II's 1.28249 record lower bound becomes an exact local envelope of the general theory built here."}],"submittedBy":"BraveDingo215","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"every-ppt-channel-has-finite-entanglement-breaking-index","name":"Every PPT channel has finite entanglement-breaking index","shortName":"Finite EB index for PPT channels","problemNumber":null,"field":"Entanglement theory","fieldGroup":"Quantum information & computing","statement":"We prove that every PPT linear map has finite entanglement-breaking index, thereby establishing the eventual entanglement-breaking property of PPT channels in full generality. Furthermore, by utilizing completely positive maps with low entanglement dimensionality, we show that a large family of PPT maps, which strictly containing the class of 2-superpositive maps, has entanglement-breaking index bounded above by 3, uniformly in the dimension. In particular, these results provide strong evidence that the PPT-cubed conjecture may hold in full generality.","posedBy":"Matthias Christandl","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"Establishes that every PPT channel is eventually entanglement-breaking (finite EB index), in full generality, and bounds the index by 3 uniformly in dimension for a family strictly containing the 2-superpositive maps. The PPT-squared conjecture itself - index at most 2 - remains open; the paper presents its results as strong evidence toward the cubed version. The OpenAI math release (September 2026) claims a counterexample to the PPT-squared conjecture: a PPT channel on 21x21 matrices whose square is not entanglement breaking; see ppt-squared-conjecture.","claimIssueNote":null,"solveDate":"2026-08-13","model":"ChatGPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Sang-Jun Park"],"aiRole":"The author states that AI tools including ChatGPT were used for exploratory mathematical discussions during development of the manuscript, in addition to language and LaTeX assistance. In particular, some ideas used in the proof-development process in Section 3 arose during interactions with GPT-5.6 Sol. These suggestions were subsequently examined, reformulated, incorporated into the manuscript, and independently verified by the author.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.13551, Park, 14 pages). The AI statement is verbatim as this entry describes: ChatGPT for language, LaTeX and exploratory mathematical discussions, with some ideas in the splitting section arising during interactions with GPT-5.6 Sol. That is a vague disclosure, and the site's rule takes the lower tier, so ai-assisted stands. The proof was not checked here. One-day-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The eventual entanglement-breaking question for PPT channels sits directly beside the PPT-squared conjecture, one of entanglement theory's best-known open problems since 2012. Resolving eventual-EB in full generality is a real advance the field will use; the PPT-squared conjecture itself remains open, which caps it below the 30 band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.13551","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/ppt-squared-conjecture","kind":"other"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"unbounded-variation-solutions-for-uniformly-elliptic-equations-in-nondivergence-","name":"Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three","shortName":"Unbounded variation for uniformly elliptic equations","problemNumber":null,"field":"Partial differential equations","fieldGroup":"Differential equations","statement":"For each nonnegative integer $m$, we construct smooth symmetric $3\\times3$ coefficient matrices $A_m$ satisfying the fixed ellipticity bound\n$$\nI\\le A_m\\le 2^{81}I\n$$\nfor which the smooth solutions of uniformly elliptic equations in nondivergence form\n$$\nA_m(x):D^2u_m=0\\qquad\\text{in }B_2\\subset\\mathbb R^3\n$$\nhave common Dirichlet data, satisfy $\\|u_m\\|_{L^\\infty(B_2)}\\le1$, but\n$$\n\\lim_{m\\to\\infty}\\|Du_m\\|_{L^1(B_1)}=\\infty.\n$$\nThus, there is no interior $W^{1,1}$ estimate depending only on ellipticity in dimension three, and consequently no such $W^{1,p}$ estimate for any $p\\ge1$.","posedBy":"Nikolai Nadirashvili; Vladimir Tkachev; Sergei Vlăduţ","yearPosed":2014,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The authors construct smooth $A_m$ and smooth solutions $u_m$ in $B_2\\subset\\mathbb R^3$ with one fixed ellipticity bound\n$$\nI\\leq A_m\\leq2^{81}I,\n$$\ncommon boundary data and $\\|u_m\\|_{L^\\infty(B_2)}\\leq1$, but\n$$\n\\|Du_m\\|_{L^1(B_1)}\\to\\infty.\n$$\nThus no interior $W^{1,1}$ estimate can depend only on dimension and ellipticity. Consequently, no such $W^{1,p}$ estimate exists for any $p\\geq1$.\n\nThey further obtain a uniformly convergent limit $u$ with measurable uniformly elliptic coefficient matrix $A$, where\n$$\nu\\notin BV_{\\rm loc}(B_1).\n$$\nThe construction even rules out coefficient-independent weak-$L^1$ gradient estimates.","claimIssueNote":null,"solveDate":"2026-08-13","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Nam Q. Le","Qi Sun","Hung V. Tran"],"aiRole":"The authors state that the main results were obtained through a series of chats with ChatGPT 5.6 Sol and that the key strategies came from ChatGPT. The construction uses repeated localized rank-one Hessian splittings to amplify gradients while maintaining a quantitative saddle condition, allowing all Hessians to be annihilated by coefficient matrices in one fixed ellipticity class. The authors then reworked and rewrote the article entirely, checked and simplified all arguments, and take responsibility for the result.","verification":"unreviewed","verificationNote":"Unreviewed. arXiv 2608.13380 (version 2, 3 September 2026) read here; the AI-assistance section states that the main results came from chats with ChatGPT 5.6 Sol, that the key strategies were the model's, and that the authors reworked, rewrote, checked and simplified everything and take responsibility. Author-checked, not independently refereed; no formalization.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A 2014 question of Nadirashvili, Tkachev and Vlăduţ (their Problem 1.3.1): is there an interior W^{1,1} estimate for uniformly elliptic equations in nondivergence form in dimension three? Answered in the negative with smooth coefficients, a fixed ellipticity ratio and a limit solution not even of bounded variation. A named open problem in regularity theory, twelve years old, by three established PDE analysts.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.13380","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1","name":"Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)","shortName":"$R_{dih}(P_a^{alt},K_b)=1+(a-1)(b-1)$, $age4$","problemNumber":null,"field":"Permutational Ramsey theory","fieldGroup":"Combinatorics","statement":"$R_{\\mathrm{dih}}(P_a^{\\mathrm{alt}}, K_b) = 1 + (a-1)(b-1)$ for all $a \\geq 4$, $b \\geq 1$ — the $a \\geq 4$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the $a = 3$ case (see sibling entry), this resolves Conjecture 4.9 in full for $a \\geq 3$.","posedBy":"Damnjanović–Đorđević (Conj 4.9)","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The dihedral case only, for every $a \\ge 4$ and $b \\ge 1$; the substance is the upper bound, which the source paper's own computations could not reach. Together with the sibling a = 3 entry this proves Conjecture 4.9's claim $1+(a-1)(b-1)$ for all $a \\ge 3$; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue $R_{cyc}(P_a^{alt}, K_b)$ for $a \\ge 4$ remains open. The engine is a self-contained inequality of independent interest: for any graph on a linearly ordered vertex set, the alternating-path reach statistics satisfy $\\sum_m [P(m)+Q(m)] \\ge 2|E(G)|$, from which the theorem falls out by averaging and a pivot decomposition.","claimIssueNote":null,"solveDate":"2026-08-13","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"The proof was produced by a sealed, multi-agent research process: independently-launched Claude agents across three rounds, convergent results cross-validated. Two independent AI referee agents reviewed it dual-blind; both CONFIRMED. Human direction was limited to run design, operational supervision, and manual re-derivation of two write-up fixes.","verification":"site-confirmed","verificationNote":"Reproduced by this site on 13 August 2026, working from the pinned statement alone - the proof's machinery, both referee reports and the shipped CNFs were not consulted by the checker. Confirmed independently: the orbit anchor ($|Dih(a)$-orbit of $P_a^{alt}| = a$ for a = 3..14); the Ramsey value at nine (a,b) cells in both directions - a good coloring exists at $n = (a-1)(b-1)$ and none at $n+1$ - exhaustively over every 2-coloring at (4,2), (5,2), (6,2), (7,2) and (4,3), and via an independently written CNF encoding solved with CaDiCaL at (8,2), (5,3), (6,3) and (4,4); and the proof's load-bearing inequality, the Aggregate Sum Theorem, by a third implementation built from the P/Q definitions rather than the recursion, over all 33,868 labeled graphs on up to six vertices - zero violations, minimum slack 0, so the bound is tight. The prose proof was also read here in full and every algebraic step traced. Not covered by the tier: the general argument has no human peer review - produced by a sealed multi-agent Claude run and refereed dual-blind by two AI agents in the same pipeline (both CONFIRMED; one non-fatal bug and one cosmetic slip found and repaired inline, originals kept). The Lean part is partial by its own declaration - four side lemmas, zero sorry or native_decide, standard axioms, source-audited here but not compiled (pinned v4.30.0 + Mathlib, no CI runs). The main theorems are not formalized; there, the referee reports and this site's checks are the verification.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"Resolves the dihedral side of a conjecture posed five weeks earlier in a single paper with no independent citations yet - a young question in a niche new area, permutational Ramsey theory. Above the a = 3 slice (5), which fell to a group coincidence plus a citation, because this is the general theorem with a genuinely new combinatorial inequality behind it; below the Erdos entries at 10, which are decades-old problems with real literatures.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/ZestyWombat854/alternating-path-ramsey","sourceName":"GitHub repo (proof + referee reports + verification code + partial Lean formalization)","collection":null,"collectionVersion":null,"links":[{"label":"Damnjanovic and Djordjevic, Computation of small reflective and dihedral Ramsey numbers (Conjecture 4.9)","url":"https://arxiv.org/abs/2607.06817","kind":"problem-record"}],"relations":[{"to":"dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path","kind":"continues","note":"The a >= 4 slice of the same Conjecture 4.9, proved five days later by a general argument; together the two entries close the conjecture's dihedral claim for every a >= 3."}],"submittedBy":"ZestyWombat854","upvotes":0,"downvotes":0,"commentCount":1},{"slug":"foregger-sinkhorn-tie-point-conjecture","name":"The Foregger–Sinkhorn Tie-Point Conjecture","shortName":"Foregger–Sinkhorn tie-point","problemNumber":null,"field":"Combinatorial matrix theory","fieldGroup":"Combinatorics","statement":"The Foregger–Sinkhorn tie-point conjecture, Conjecture 41 in Minc's survey, asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero exceeds its permanent, then that zero is a tie point. False: an explicit $8 \\times 8$ counterexample exists, built on the unique root $\\beta$ of $7t^3 - 13t^2 + 12t - 4$ in $(59/100, 3/5)$.","posedBy":"T. H. Foregger and Richard Sinkhorn","yearPosed":1987,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-13","model":"GPT-5.6-sol, Claude Fable 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Yair Lavi"],"aiRole":"The paper's acknowledgement, in full: \"The proof of Theorem 1 was carried out by GPT-5.6-sol and Claude Fable 5, under the guidance of the author. The author has reviewed the resulting proof arguments. Responsibility for the final text rests with the author.\"","verification":"unreviewed","verificationNote":"Independently recomputed by this site on 21 August 2026 from the paper's own data (arXiv:2608.13025v1), at 60-digit precision: $\\beta$ is the unique root of the stated cubic in the stated bracket, on which the cubic is monotone; the matrix is doubly stochastic and nonnegative; its 19-entry support is nearly decomposable, being fully indecomposable while the removal of any single support entry destroys that; the permanent agrees with the paper's closed form to 1e-61; the cofactor gap agrees with its closed form to 4e-61 and exceeds the claimed 2047/240100; and the prescribed zero is not a tie point, with exactly one witnessing support entry. The conjecture's hypothesis also holds: the face is four-dimensional, and a 6561-point grid plus 61 local descents found nothing on it with a smaller permanent. Still a days-old preprint with no independent review, hence a candidate.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A named conjecture in permanent theory, listed in Minc's survey and standing since, but specialist even within combinatorial matrix theory. Above the Graffiti anchor at 5 as a stated conjecture with attribution, well below the named-conjecture band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.13025","sourceName":"A counterexample to the Foregger-Sinkhorn tie-point conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sop-2-sop-3","name":"$SOP_2 = SOP_3$","shortName":"$SOP_2 = SOP_3$","problemNumber":null,"field":"Model theory","fieldGroup":"Logic & foundations","statement":"The classes of SOP_2 and SOP_3 first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.","posedBy":"Mirna Džamonja, Saharon Shelah","yearPosed":2004,"ageNote":"Posed by Dzamonja and Shelah in 2004 alongside its sibling question SOP_1 = SOP_2, which Mutchnik answered in a celebrated 2022 preprint. Open 22 years.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The new content is $SOP_2 \\Rightarrow SOP_3$; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in $SOP_3 \\Rightarrow SOP_2 \\Rightarrow SOP_1$ reverses: Mutchnik answered the second ($SOP_1 = SOP_2$), and this answers the first, collapsing the bottom of the hierarchy to $SOP_1 = SOP_2 = SOP_3$. The $SOP_n$ hierarchy for $n \\ge 3$ remains, as does everything above it.","claimIssueNote":null,"solveDate":"2026-08-13","model":"ChatGPT 5.6","modelMaker":"OpenAI","humanCollaborators":["Artem Chernikov"],"aiRole":"From the paper: \"The proof was found using ChatGPT 5.6 and simplified and streamlined\nby the author.\"","verification":"unreviewed","verificationNote":"Checked by this site on 14 August 2026 against the paper's LaTeX source. The paper is real - five pages, math.LO, posted 13 August - and its author, Artem Chernikov, is a leading model theorist in exactly this area. The AI disclosure is verbatim as the entry quotes it. The history checks out in the paper's own introduction: Dzamonja and Shelah introduced the tree configurations $SOP_1$ and $SOP_2$ and asked whether either implication $SOP_3 \\Rightarrow SOP_2 \\Rightarrow SOP_1$ reverses; Mutchnik's breakthrough proved $SOP_2 = SOP_1$, and the question $SOP_2 = SOP_3$ was repeatedly highlighted afterwards, with partial results by Conant, Kaplan-Ramsey-Simon and Mutchnik. This result completes the collapse $SOP_1 = SOP_2 = SOP_3$. The acknowledgements record comments by Itay Kaplan and Scott Mutchnik on a preliminary version - expert eyes, but comments on a draft are not independent verification, and the manuscript is one day old and unrefereed, so the tier is Unreviewed. The proof itself - five pages of tree-indiscernible manipulation - was not checked here; it needs a model theorist.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"One of the two 2004 Dzamonja-Shelah questions that organized two decades of classification theory, repeatedly highlighted in the NSOP_1 literature, and the surviving half after Mutchnik's celebrated SOP_1 = SOP_2. Resolving it collapses the bottom of the SOP hierarchy outright, with consequences for Keisler-order maximality. Central within model theory, little known outside it: placed at 35 with the field-famous specialist band (Feige, Krauth-Mezard), above HRT at 33.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.13291","sourceName":"SOP_2 = SOP_3","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"LucidManta102","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"fourier-invariant-functions-with-dense-zero-sets","name":"The Radchenko–Viazovska question on Fourier interpolation","shortName":"Dense zero sets for Fourier-invariant functions","problemNumber":null,"field":"Fourier analysis","fieldGroup":"Analysis","statement":"For every $0\\le \\beta\\le 1/2$, we construct a nonzero real-valued continuous function $f_\\beta$ in $L^1(\\mathbb R)\\cap L^2(\\mathbb R)$ such that $\\widehat f_\\beta=f_\\beta$ and\n$$\nf_\\beta\\!\\left(\\frac{\\sqrt n}{[\\log(e+n)]^\\beta}\\right)=0\n$$\nfor all $n\\ge0$. The case $\\beta=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula.","posedBy":"Danylo Radchenko and Maryna Viazovska","yearPosed":2019,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"For every $0\\le \\beta\\le \\tfrac12$, Bondarenko and Seip construct a nonzero real-valued continuous even function\n$$\nf_\\beta\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\n$$\nsuch that\n$$\n\\widehat f_\\beta=f_\\beta\n$$\nand\n$$\nf_\\beta\\!\\left(\\frac{\\sqrt n}{[\\log(e+n)]^\\beta}\\right)=0\n\\qquad(n\\ge0).\n$$\nThey normalize the construction by requiring $f_\\beta(1/2)=1$, so the function is genuinely nontrivial.\n\nFor $\\beta=0$, this gives a nonzero Fourier-invariant function vanishing at every $\\sqrt n$, which answers the question negatively: their interpolation formula for even Schwartz functions does not extend merely under the assumption that the interpolation series is well-defined and absolutely convergent.\n\nMore strongly, for every $0<\\beta\\le1/2$ the zero set can be logarithmically denser than the square-root sequence. These sampling points, together with $1/2$, form a universal interpolating sequence for a suitable reproducing-kernel Hilbert space of Fourier-invariant Hermite expansions.","claimIssueNote":null,"solveDate":"2026-08-13","model":"ChatGPT (OpenAI, model version unstated)","modelMaker":"OpenAI","humanCollaborators":["Andriy Bondarenko","Kristian Seip"],"aiRole":"The authors state that OpenAI's ChatGPT provided exploratory input and calculations that were essential in the development of the paper. The disclosure does not identify a specific model version or isolate which theorem, lemma, or proof step originated from ChatGPT, so the safest attribution is substantive AI-assisted discovery rather than AI-discovered.","verification":"unreviewed","verificationNote":"Unreviewed. arXiv 2608.13468 read here; the acknowledgement thanks ChatGPT, \"whose exploratory input and calculations were essential in the development of this paper\", naming no model version and isolating no step. A complete conventional proof by the authors; not refereed; no formalization.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"Question 1 of Radchenko and Viazovska's 2019 Fourier interpolation paper (Publ. Math. IHES): does the interpolation formula for even Schwartz functions extend whenever the series converges absolutely? No: a nonzero Fourier-invariant function vanishing at every sqrt(n). The interpolation formula is a celebrated result and the question was its natural open end; answered by two established analysts.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.13468","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"volume-and-projection-inequalities-i-zonoids-and-courtade-s-conjecture","name":"Courtade's conjecture on volumes of Minkowski sums with the ball","shortName":"Courtade’s conjecture","problemNumber":null,"field":"Zonoids","fieldGroup":"Geometry & topology","statement":"Although Courtade’s conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.","posedBy":"Thomas Courtade","yearPosed":2017,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"Courtade conjectured that for convex bodies $B,C\\subset\\mathbb R^n$,\n$$\n(|B||C|)^{1/n}\n+\n(|B_2^n||B_2^n+B+C|)^{1/n}\n\\le\n|B_2^n+B|^{1/n}|B_2^n+C|^{1/n}.\n$$\n\nThe paper proves that this inequality is false in every dimension $n\\ge3$. In dimension $3$, it gives an explicit geometric counterexample using two orthogonal double bodies of revolution. It then strengthens this by constructing zonoid counterexamples in every dimension $n\\ge3$, including a six-generator zonotope in dimension $3$ and smooth perturbative constructions in higher dimensions.","claimIssueNote":null,"solveDate":"2026-08-13","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Matthieu Fradelizi","Alfredo Hubard","Auttawich Manui","Cheikh Saliou Ndiaye","Shouda Wang","Artem Zvavitch"],"aiRole":"The authors state that GPT-5.6 Sol was used during preparation of the paper as an auxiliary mathematical tool to explore examples, test determinant computations, and assist with preliminary proof development. The authors state that the final mathematical arguments, statements, and computations are theirs and that they independently verified them.","verification":"unreviewed","verificationNote":"Unreviewed. arXiv 2608.12681 read here; the section \"Acknowledgments and AI assistance disclosure\" states that GPT-5.6 Sol was used as an auxiliary mathematical tool to explore examples, test determinant computations and assist preliminary proof development, and that the authors independently verified all arguments. Author-checked; not refereed; no formalization.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Courtade's 2017 conjecture, an inequality between the volumes of B, C and their Minkowski sums with the unit ball, refuted in every dimension from three up, even among zonoids. The paper also disproves a log-submodularity conjecture for zonoid volumes. Named conjectures in convex geometry with a decade of literature, by six established authors; mid-low scale.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.12681","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"universal-volume-growth-bounds-from-positive-intermediate-curvature","name":"Universal volume growth bounds from positive intermediate curvature","shortName":"Intermediate-curvature volume growth bounds","problemNumber":null,"field":"Differential geometry","fieldGroup":"Geometry & topology","statement":"In 1986 Gromov asked whether every complete $n$-dimensional Riemannian manifold with $\\mathrm{Ric} \\ge 0$ and $\\mathrm{Scal} \\ge 1$ satisfies\n$$\\mathrm{Vol}\\,B_R(p) \\le C(n)\\,R^{n-2}$$\nfor every $p$ and every $R > 0$. The three-dimensional case had been settled, and higher dimensions were known only under extra hypotheses such as nonnegative sectional curvature, noncollapsing or an injectivity-radius bound.\n\nThis paper answers the question affirmatively, as the case $m = n-2$ of a uniform family: for every $0 \\le m \\le n-2$, if $\\mathrm{Ric} \\ge 0$ and the $(m{+}1)$-intermediate curvature of Brendle-Hirsch-Johne is at least 1, then $\\mathrm{Vol}\\,B_R(p) \\le C(n,m)\\,R^m$. At $m = 1$ this gives linear volume growth under positive biRicci curvature in every dimension.","posedBy":"Mikhail Gromov","yearPosed":1986,"ageNote":"Posed by Gromov in 1986, in section 2.A(b) of Large Riemannian manifolds. The solved date is 13 August 2026, when Jian Ge's independent proof appeared; this paper and Kong-Zhu's both followed one day later, all three written without knowledge of each other.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Gromov's 1986 question drew three independent proofs within about 24 hours, two with AI in the loop. Ge's non-AI proof (heat-kernel Fisher metric, Nash entropy) came first, 13 August. Antonelli's proof here (14 August, GPT-5.6 Sol) takes a different route, Hodge obstruction and rank improvement, and its headline addition is the general family: for every $0\\le m\\le n-2$, nonnegative Ricci plus positive $(m{+}1)$-intermediate curvature forces at most $m$-dimensional growth - linear growth under biRicci curvature at $m=1$, plus a noncollapsed Urysohn-width bound; the author states this extension is his own, not the model's. Kong and Zhu's proof (also 14 August; GPT-5.6 Sol Ultra and Codex, \"essential ideas were generated by AI\") proves the same case plus a related codimension-one conjecture via a heat-transport deficit, close to Ge's method by its own account, produced before Ge's went public and not derived from it. Headline axes stay Antonelli's, for the broader scope.","claimIssueNote":null,"solveDate":"2026-08-13","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Gioacchino Antonelli"],"aiRole":"From the paper's own disclosure of AI tools. The work made substantial use of OpenAI's GPT-5.6 Sol at Ultra reasoning effort. GPT proposed the central inductive procedure, based on the Hodge obstruction and rank improvement, behind the proof of Theorem 1.1 - the scalar-curvature case, which is Gromov's question. Antonelli formulated and guided the problem, suggested strategies and literature, and developed the note from that strategy.\n\nHe is explicit about how far the published proof moved from the model's: it is effective, where the suggested strategy proceeded by contradiction, and it is much less reliant on Kapovitch-Wilking's Theorem 5.1 - both changes emerging from discussions with Daniele Semola, Elia Bruè and Kai Xu. He also states that the extension to the full Brendle-Hirsch-Johne intermediate-curvature family, which is the paper's general theorem, is his own contribution and not the model's.\n\nKong and Zhu's independent same-day proof (arXiv:2608.14438) discloses its own AI use more bluntly: \"Generative AI tools, more explicitly, ChatGPT 5.6 Sol Ultra and Codex, assisted with proof exploration, organization, and drafting. Essential ideas were generated by AI.\" That is the stronger of the two claims by wording, though for a narrower result than Antonelli's general family; the headline axes stay Antonelli's for that reason.","verification":"unreviewed","verificationNote":"A five-day-old arXiv preprint, unrefereed and not formally endorsed, so this stays Unreviewed. It is not unexamined, though: the author thanks Elia Bruè, Otis Chodosh, Alessandro Cucinotta, Chao Li, Aaron Naber, Daniele Semola and Kai Xu for comments on preliminary versions, and the acknowledgments record two specific ways their comments changed the argument. That is a stronger signal than most preprints of this age carry, but comments are not endorsement, and this site ran no independent check of its own.\n\nThe strongest external evidence is indirect and worth stating: three independent routes converge on the same 1986 conjecture within about 24 hours - Jian Ge's heat-kernel Fisher-metric argument (13 August, no AI), this paper's Hodge-obstruction argument (14 August, GPT-5.6 Sol), and Kong-Zhu's heat-transport-deficit argument (also 14 August, GPT-5.6 Sol Ultra and Codex) - the last stating its own method is close to Ge's but developed independently and before Ge's preprint was public. None of the three has been independently checked, but three unrelated-to-mostly-unrelated routes agreeing is meaningful corroboration of the statement regardless.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"A forty-year-old question on Gromov's own problem list in Large Riemannian manifolds, and a workhorse of geometric analysis with a dense recent literature: Petrunin under nonnegative sectional curvature, B. Zhu in the noncollapsed setting, Munteanu-Wang, Chodosh-Li-Stryker, Wei-Xu-Zhang and Huang-Liu in dimension three, Wang-Xie-Zhu-Zhu for Ricci limits, plus the linked Urysohn-width conjecture. Above the Kinoshita conjecture at 35, a single question in a smaller corner. Below Erdos's planar unit distance conjecture at 40, which is famous outside its subfield where this one is not.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.14507","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Ge, Heat kernel geometry and Gromov's volume growth conjecture (arXiv, 13 Aug 2026)","url":"https://arxiv.org/abs/2608.13553","kind":"independent"},{"label":"Brendle, Hirsch, Johne - A generalization of Geroch's conjecture, where intermediate curvature is introduced","url":"https://doi.org/10.1002/cpa.22137","kind":"paper"},{"label":"Kong and Zhu, Positive Scalar Curvature and Volume Growth (arXiv, 14 Aug 2026)","url":"https://arxiv.org/abs/2608.14438","kind":"independent"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"nineteen-exact-reflective-and-dihedral-ramsey-numbers-from-damnjanovic-dordevic-","name":"Nineteen exact reflective and dihedral Ramsey numbers from Damnjanovic-Dordevic's tables","shortName":"19 exact $R_{dih}$/$R_{ref}$ values (DD26)","problemNumber":null,"field":"Permutational Ramsey theory","fieldGroup":"Combinatorics","statement":"Sixteen previously unknown exact values, plus three that confirm the sibling theorem entries' predictions computationally, across five ordered-pattern families ($P^{alt}$, $S^{sc}$, $C^{mon}$, $M^{nest}$, $K$) under dihedral and reflective group actions - each closing one open cell of Damnjanovic-Dordevic (arXiv:2607.06817, Tables 3-13). Five sit in cells the paper left without a conjecture. Full per-value table with regeneration commands, certificate hashes and referee verdicts in the evidence repo.","posedBy":"Damnjanovic-Dordevic (open cells of Tables 3-13)","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Nineteen individual exact values, each decided by SAT certificate: unsatisfiable at the claimed $n$, witnessed satisfiable at $n-1$. They close cells in DD26's Tables 3-13 but settle no infinite family - the sibling entries do that for the $K$ column. The three overlap cells are $R_{dih}(P_4^{alt},K_6)=16$, $R_{dih}(P_3^{alt},K_9)=17$ and $R_{dih}(P_9^{alt},K_3)=17$, each an instance of a sibling theorem; the remaining sixteen stand on their own certificates. Four further cells passed the producing solver but await their final referee leg and are not claimed. Open: every other cell of DD26's tables, all cyclic-action and online-Ramsey cells.","claimIssueNote":null,"solveDate":"2026-08-13","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"AI agents built the census of open cells from the paper's tables, encoded every instance using the authors' own vendored generator, orchestrated kissat solving with drat-trim verification, and refereed all nineteen values through six independent agent reviews — each referee wrote its own encoder from the definitions, zero shared code with the producing leg or each other. Human direction was limited to run design, operational supervision, and posting.","verification":"site-confirmed","verificationNote":"Half-reproduced by this site on 14 August 2026, referee code, witnesses and CNFs deliberately unread - and this note is precise about which half. All 19 instances regenerate from the vendored DD26 generator and match their pinned SHA-256 byte for byte. All 19 LOWER bounds were established independently: the five pattern families, both group actions and the monotone-embedding notion were implemented here from the DD26 paper's definitions alone, a witness coloring at $n-1$ was found for every value, and each witness was re-verified by a brute-force embedding search sharing no code with the encoder. The UNSAT side is where this site's reach ended: the instances are genuinely hard (the submitter's smallest took kissat nine minutes; this site's solver decided none within a 50-minute-per-instance budget), so the upper bounds rest on the bundle's own certificates - whose 19 drat-trim logs all read \"s VERIFIED\" with parsed dimensions matching the regenerated CNFs exactly - except for the three K-family cells, whose values are implied by the two sibling theorem entries this site verified in depth on 13 August. Every cell with a DD26 conjecture matches that conjecture's formula, checked by hand. Not covered: no human peer review - produced and refereed by AI agents in one pipeline (six referee legs, each with its own encoder); the Lean file is a statement anchor with zero proofs, by design. Four further cells await their final referee leg and are, correctly, not claimed.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Nineteen finite table cells from a five-week-old paper in a niche new area, none settling a conjecture in full generality - the two theorem entries this accompanies did that for one family. Real, checkable, and five cells carry values the paper did not even conjecture; but each is one cell. Level with the a = 3 slice (5): above the single-gadget variant (4), below the a >= 4 theorem (8).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/ZestyWombat854/reflective-dihedral-ramsey","sourceName":"GitHub repo (SAT certificates + six independent referee legs + Lean statement anchor)","collection":null,"collectionVersion":null,"links":[{"label":"Damnjanovic and Djordjevic, Computation of small reflective and dihedral Ramsey numbers (Tables 3-13)","url":"https://arxiv.org/abs/2607.06817","kind":"problem-record"}],"relations":[{"to":"dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1","kind":"related","note":"Computational companion: two of these cells, (P4,K6) and (P9,K3), are instances of that theorem, and the certificate values agree with it exactly - a cross-check of both pipelines."},{"to":"dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path","kind":"related","note":"Computational companion: the (P3,K9) cell is an instance of that entry's 2b-1 theorem, and the certificate value agrees with it exactly."}],"submittedBy":"ZestyWombat854","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"banach-s-isometric-conjecture","name":"Banach's isometric conjecture","shortName":"Banach isometric conjecture","problemNumber":null,"field":"Functional analysis","fieldGroup":"Analysis","statement":"Banach asked in 1932 whether a real Banach space $X$ whose $n$-dimensional subspaces, for some fixed $1 < n < \\operatorname{dim}X$, are all isometric must be a Hilbert space. Gromov proved the conjecture for even n, and subsequent work settled several odd-dimensional cases. We prove the conjecture for every odd n, including all previously unresolved cases. Together with Gromov’s even-dimensional result, this completes Banach’s isometric conjecture in the real case. The proof combines bundle topology with Brouwer degree theory.","posedBy":"Stefan Banach","yearPosed":1932,"ageNote":"Posed in Banach's 1932 book. Gromov proved the even-dimensional case in 1967; odd cases fell piecemeal (n=3 Auerbach-Mazur-Ulam earlier, later 4k+1 cases and more). Open 94 years in full.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.","claimIssueNote":null,"solveDate":"2026-08-13","model":"ChatGPT 5.6 Pro, ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Xinbao Lu","Kaiwen Yang"],"aiRole":"The authors state that they had already reduced the main problem to proving Theorem 3.10 before using generative AI. An approach to that theorem then emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro. GPT-5.6 Sol generated the initial draft of Section 3 and corresponding material in Section 2 following this approach; the authors subsequently checked and rewrote it. GPT-5.6 Sol was also used to improve the exposition.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.13536, Lu-Yang, 21 pages). The AI declaration is verbatim as this entry quotes it: the authors reduced the problem to one theorem before using AI, the approach to that theorem emerged through interactions with ChatGPT 5.5 Pro and 5.6 Pro, and GPT-5.6 Sol drafted Section 3 and parts of Section 2, subsequently checked and rewritten by the authors. The mathematics - bundle topology plus Brouwer degree - was not checked here and needs a geometer. One-day-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"Posed by Banach himself in 1932, open 94 years, with Gromov's even-dimensional theorem (1967) the landmark partial result and a literature of odd-dimensional cases since. A famous named conjecture recognisable well outside convex geometry - placed with Sendov at 40, below the household band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.13536","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":4,"downvotes":0,"commentCount":0},{"slug":"the-2-1-gapped-consecutive-ones-property-problem-is-np-complete","name":"The $(2,1)$-Gapped Consecutive-Ones Property Problem is NP-complete","shortName":"$(2,1)$-C1P is NP-complete","problemNumber":null,"field":"Computational complexity / consecutive-ones property","fieldGroup":"Theoretical computer science","statement":"Given a binary matrix $M$, decide whether its columns can be permuted so that every row contains at most two blocks of 1s and, if it contains two blocks, they are separated by at most one 0. The claimed theorem proves that this $(2,1)$-Gapped Consecutive-Ones Property decision problem is NP-complete.","posedBy":"Cédric Chauve, Ján Maňuch, Murray Patterson","yearPosed":2009,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The manuscript claims a polynomial-time reduction from 3-SAT proving $(2,1)$-C1P NP-hard; together with membership in NP, this establishes NP-completeness and closes the sole unresolved $(k,\\delta)$ case from the earlier classification. It also implies NP-completeness of the equivalent completion problem.\n\nThe proof package further shows that, within its specific nested-prefix/internal-local gadget architecture, no 3-OR gadget exists with at most six internal columns; at seven columns at least three local rows are required, and all optimal three-row gadgets form one symmetry class. These optimality claims are architecture-specific and are not needed for the NP-completeness result. Independent verification remains pending.","claimIssueNote":null,"solveDate":"2026-08-12","model":"GPT-5.6 Sol High","modelMaker":"OpenAI","humanCollaborators":["Maciej Nowicki"],"aiRole":"GPT-5.6 Sol High produced the central NP-completeness construction: a polynomial-time reduction from 3-SAT using ordered slots, two-column variable gadgets, and a seven-column clause gadget implementing Boolean OR. The associated account reports that the solution emerged in roughly 3–4 model messages, with prompts written by the model for itself. The proof package includes exhaustive finite checks of the clause gadget and a separately implemented verifier/cross-check.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026: the Zenodo record (21898823) is real, credited to Maciej Nowicki and \"Eve\" with the proof and verification package attached, and the prior-open status verifies - the Chauve-Manuch-Patterson hardness paper (arXiv:0912.0309) establishes the (k,delta)-C1P classification whose exceptional case this claims to close, and the linked 2011 follow-up restates it as open. The reduction itself was not checked here, and no independent review exists; a self-published record by a pseudonymous-branded author stays Candidate until the proof is checked by someone.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"The last open parameter case of a 2009 complexity classification with a real comparative-genomics motivation, open seventeen years. Specialist, and the resolution is self-published with no independent check yet - held at the specialist floor above the week-old problems.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://zenodo.org/records/21898823","sourceName":"I proved the last open case of the Gapped Consecutive-Ones problem is NP-complete, and the key 7-column OR gadget is optimal","collection":null,"collectionVersion":null,"links":[{"label":"Original 2009 hardness paper leaving $(2,1)$-C1P open","url":"https://arxiv.org/abs/0912.0309","kind":"paper"},{"label":"2011 follow-up explicitly stating that $(2,1)$-C1P remains open","url":"https://www.cs.ubc.ca/~jmanuch/journal/papers/JCB2011b.pdf","kind":"paper"},{"label":"Initial Zenodo record announcing the NP-completeness proof","url":"https://zenodo.org/records/21871667","kind":"paper"}],"relations":[],"submittedBy":"LucidHawk551","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strong-graph-reconstruction-conjecture","name":"Strong Graph Reconstruction Conjecture","shortName":"Strong Graph Reconstruction Conjecture","problemNumber":null,"field":"Graph theory - reconstruction","fieldGroup":"Combinatorics","statement":"For a graph $G$, its vertex deck is the multiset of graphs obtained by deleting one vertex. Bowler, Brown, and Fenner (BBF) proposed $2\\lfloor\\frac{n−1}{3}\\rfloor$ as the maximum possible overlap between the decks of two nonisomorphic n-vertex graphs, for all sufficiently large n. We first give an explicit pair of connected nonisomorphic graphs on 78 vertices with at least 51 common cards, exceeding BBF's predicted value of 50. We then construct, for every even $r \\geq 4$, families at arbitrarily large orders whose overlap fraction is asymptotically at least $1−\\frac{1}{r}$. Consequently, for every $\\alpha<1$, infinitely many pairs have more than $\\alpha n$ common cards, so the attainable fraction is arbitrarily close to the full deck. For representative instances, the predicted overlaps were also checked by complete deck generation and isomorphism testing with Brendan McKay's nauty tools","posedBy":"Andrew Bowler, Paul Brown, Trevor Fenner","yearPosed":2010,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Disproves the Bowler-Brown-Fenner bound of 2*floor((n-1)/3) on common cards between nonisomorphic graphs: an explicit connected 78-vertex pair shares at least 51 cards against the predicted 50, and for every even r >= 4 there are families with overlap fraction asymptotically at least 1 - 1/r, so the attainable fraction approaches the full deck. The Kelly-Ulam reconstruction conjecture itself is untouched.","claimIssueNote":null,"solveDate":"2026-08-12","model":"gpt-5.6-sol","modelMaker":null,"humanCollaborators":[],"aiRole":"Per the author's public statements (not disclosed in the paper itself): GPT-5.6 Sol proved most of the roughly 400 intermediate theorems explored, with the author supplying the decisive construction idea and direction - in the author's words, the models \"could not achieve the goal before I can point to the right idea\". Model-proved steps inside a human-led search is recorded as ai-co-developed.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.11930, Ivanov) and its verification repository, which the paper's own footnote links. Two submitted claims needed correcting. The entry arrived Expert-verified on the strength of the author's own LinkedIn post - self-endorsement is not independent expert verification, so this is Unreviewed. And the paper itself contains NO AI disclosure: the account of GPT-5.6 Sol's role rests entirely on the author's public statements outside the paper, which this entry now says explicitly. The nauty deck checks are author-side. The construction was not verified here.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The Strong Reconstruction Conjecture of Bowler, Brown and Fenner is the quantitative frontier of the reconstruction-number literature around the celebrated Kelly-Ulam problem. Disproving it, and showing overlap can approach the full deck, reshapes that frontier - specialist but consequential, at 20.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11930","sourceName":"arxiv.org","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"FrostyVulture999","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gamow-liquid-drop-minimizer-conjecture","name":"Gamow liquid-drop minimizer conjecture","shortName":"Liquid-drop minimizers","problemNumber":null,"field":"Calculus of variations and geometric measure theory","fieldGroup":"Analysis","statement":"For a measurable set $\\Omega\\subset\\mathbb R^3$, let\n$$\\mathcal E(\\Omega)=P(\\Omega)+\\frac12\\iint_{\\Omega\\times\\Omega}\\frac{dx\\,dy}{|x-y|},$$\nwhere $P$ is De Giorgi perimeter, and set\n$$V_*=5\\frac{2-2^{2/3}}{2^{2/3}-1}\\approx3.51.$$\nThe conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove that, for every $0V_*$, no minimizer exists. Consequently,\n$$\\inf_{0<|\\Omega|<\\infty}\\frac{\\mathcal E(\\Omega)}{|\\Omega|}=3\\left(\\frac{9\\pi}{5}\\right)^{1/3}=\\frac92\\left(\\frac{8\\pi}{15}\\right)^{1/3},$$\nwith equality exactly for translates, modulo null sets, of the ball of volume $5/2$, equivalently radius $(15/(8\\pi))^{1/3}$.","posedBy":"George Gamow (the functional, c. 1930); the sharp-threshold conjecture stated in the modern liquid-drop literature","yearPosed":null,"ageNote":"Gamow introduced the functional around 1930 - the paper says 1928 while the reference it gives is his 1930 Proc. R. Soc. A paper. The sharp-threshold conjecture is a modern formulation with no canonical posing: the authors say only that it has appeared in several places, citing Choksi-Peletier (2011), Frank-Lieb (2015) and Frank-Nam (2021). Year posed is left empty rather than guessed at.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The complete fixed-volume picture, closing a gap that partial results had narrowed from both ends without meeting: balls uniquely minimize for every volume up to V_* = 3.51..., and above it no minimizer exists at all. Before this the best minimality range was V <= 1 (Chodosh-Ruohoniemi, 2025) and the best nonexistence bound V >= 7.5 (Schulz, posted two days earlier), so the open middle ran from 1 to 7.5. Frank-Nam had already proved existence up to V_*, and the new proof uses it; the fresh content is uniqueness of the ball across the whole range and nonexistence immediately above the threshold. A corollary settles the minimal binding energy question of Frank-Lieb: the infimum of E(Omega)/|Omega| is 3(9pi/5)^(1/3), attained exactly at balls of volume 5/2. The mechanism is a capacitary estimate that sharpens an Agostiniani-Mazzieri monotonicity formula using Gauss-Bonnet, an improvement the authors note applies only to this particular weight and only in three dimensions.","claimIssueNote":null,"solveDate":"2026-08-12","model":"ChatGPT 5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Otis Chodosh","Matilde Gianocca"],"aiRole":"The paper's AI-usage statement says that ChatGPT 5.6 Pro obtained the mathematical results over a series of chats without significant assistance from the authors. The fundamental proof strategy remained close to the model's output. Otis Chodosh and Matilde Gianocca then checked and reworked the proof and wrote the manuscript; they state that the article contains no AI-written text.","verification":"unreviewed","verificationNote":"Checked here on 13 August 2026, the day after the preprint appeared. The AI-usage statement was confirmed verbatim in two places, the arXiv listing comment and the paper's own opening section. The LaTeX source was retrieved and the quantitative content rederived rather than trusted. The threshold came out independently from comparing one ball against two of half the volume as their separation grows, which favours splitting exactly when $V > 5(1-2^{1/3})/(2^{-2/3}-1)$ - the paper's $5(2-2^{2/3})/(2^{2/3}-1) \\approx 3.5121$ after clearing radicals. The constant $|B_1|P(B_1)/D(B_1)=5$ checks against $D(B_R)=16\\pi^2R^5/15$. The corollary does minimise at $V=5/2$ with value $3(9\\pi/5)^{1/3}$, and the alternative form $\\frac92(8\\pi/15)^{1/3}$, which the submitter added and the paper does not state, is genuinely equal to it - both cube to $48.6\\pi$. The identities the argument turns on expand as claimed, as do $2^{-2/3}(V_*+10)=V_*+5$ and the closing $1024<1296$ on $[6,8]$. Every cited source is real, with a resolving DOI. What was NOT checked is the capacitary estimate and the distributional Bochner lemma under it, which is where the new mathematics lives. The manuscript is one day old and unrefereed, and the authors checking their own reworked proof is not independent verification, so the tier stays Unreviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":"No Wikipedia article for this problem. Wikipedia's liquid drop model article covers the nuclear-physics concept rather than the minimization conjecture, so it does not count under the notability rule.","significance":35,"significanceNote":"The central open problem of the liquid-drop literature: the sharp threshold between existence and nonexistence of fixed-volume minimizers, with uniqueness of the ball below it. Not eponymous, and with no Wikipedia article. What lifts it above a single-subfield problem is reach - tracked across calculus of variations, mathematical physics and geometric analysis at once, with a 2017 Notices of the AMS survey for a general audience and a partial-results literature carrying Lieb, Otto, Figalli and Maggi. Hence level with Polya for Neumann balls (35) rather than HRT (33), below the eponymous band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11517","sourceName":"No compromise in the liquid drop model","collection":null,"collectionVersion":null,"links":[{"label":"Frank-Lieb (2015), source of the minimal binding energy question","url":"https://doi.org/10.1137/15M1010658","kind":"problem-record"},{"label":"Chodosh and Ruohoniemi, On minimizers in the liquid drop model (CPAM 2025) - previous best minimality range, V <= 1","url":"https://doi.org/10.1002/cpa.22229","kind":"paper"},{"label":"Frank-Nam, Existence and nonexistence in the liquid drop model (2021) - existence up to V_*, used by the proof","url":"https://doi.org/10.1007/s00526-021-02072-9","kind":"paper"},{"label":"Frank-Killip-Nam, Nonexistence of large nuclei (2016) - the V > 8 bound the nonexistence proof reduces to","url":"https://doi.org/10.1007/s11005-016-0860-8","kind":"paper"},{"label":"Agostiniani-Mazzieri, Monotonicity formulas in potential theory (2020) - the estimate the capacitary argument sharpens","url":"https://doi.org/10.1007/s00526-019-1665-2","kind":"paper"},{"label":"Schulz, An improved nonexistence bound for the liquid drop model (2026) - the V >= 7.5 bound, two days earlier","url":"https://arxiv.org/abs/2608.09000","kind":"paper"}],"relations":[],"submittedBy":"October","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"seymour-second-neighborhood-conjecture-dense-case","name":"Seymour's Second Neighborhood Conjecture","shortName":"Seymour 2nd neighborhood","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for dense incomplete graphs a series of results restricting the structure of the missing edges. This work proves the conjecture for every oriented graph of order $n = 2\\delta + 2$, where $\\delta$ is the minimum outdegree, with no prescribed structure on the missing edges; with Fisher's tournament theorem this gives every oriented graph satisfying $n \\le 2\\delta + 2$.","posedBy":"Paul Seymour","yearPosed":1990,"ageNote":"Posed by Seymour around 1990 as a strengthening of Dean's conjecture for tournaments, and open for some 36 years. Fisher settled the tournament case in 1996.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on the missing edges. Superseded by a claimed proof of the full conjecture in the OpenAI math release (September 2026), see seymour-second-neighborhood-conjecture.","claimIssueNote":null,"solveDate":"2026-08-12","model":"GPT-5 family, Claude","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Jake Brukhman"],"aiRole":"The acknowledgements are unusually specific about the division of labour: \"The author initiated and directed the investigation, curated intermediate results, selected the theorem for publication, and edited the final statement and exposition. OpenAI language models (GPT-5 family) carried out the detailed mathematical exploration, implemented counterexample searches and verification tools, discovered the fixed-target capacity argument and its double-counting proof, and drafted the manuscript; Anthropic Claude models performed an adversarial audit of an intermediate draft and assisted with revisions. The author verified the proofs and accepts sole responsibility for the final manuscript and its claims.\" The model is credited with discovering the central argument by name, which is the discovered tier rather than the co-developed one.","verification":"unreviewed","verificationNote":"arXiv preprint, one day old at cataloguing, not peer-reviewed and with no independent commentary yet. The proof is described by the author as a short counting argument, so it is human-checkable in principle, but this site has not verified it and no expert has publicly endorsed it. What is unusually strong here is the disclosure rather than the verification.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Seymour's second neighborhood conjecture is a well-known problem in digraph theory, open since around 1990 and the subject of a continuing literature - Fisher's tournament theorem, the minimum-outdegree results of Kaneko and Locke, and a decade of dense-case work by Fidler-Yuster, Ghazal and Dara-Francis-Jacob-Narayanan. Scored for the problem rather than this increment, level with the other well-tracked named conjectures and below the household ones.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11530","sourceName":"arXiv:2608.11530 - A dense-case theorem for Seymour's second neighborhood conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/seymour-second-neighborhood-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path","name":"Dihedral and cyclic Ramsey numbers of the alternating 3-path","shortName":"$R_{dih}(P_3^{alt},K_b)=2b-1$","problemNumber":null,"field":"Permutational Ramsey theory","fieldGroup":"Combinatorics","statement":"$R_{\\mathrm{dih}}(P_3^{\\mathrm{alt}}, K_b) = R_{\\mathrm{cyc}}(P_3^{\\mathrm{alt}}, K_b) = 2b - 1$ for all $b \\in \\mathbb{N}$ — the $a = 3$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817) and Conjecture 4.23 (Bašić–Damnjanović–Stevanović–Stošić, arXiv:2604.16188).","posedBy":"Damnjanović–Đorđević (Conj 4.9); Bašić–Damnjanović–Stevanović–Stošić (Conj 4.23)","yearPosed":2026,"ageNote":"Both conjectures are from 2026 - Damnjanovic and Djordjevic posted arXiv:2607.06817 on 7 July 2026 and the companion Basic-Damnjanovic-Stevanovic-Stosic paper is arXiv:2604.16188 - so the a = 3 slice fell about five weeks after the conjecture was stated. This is among the shortest gaps in the catalog between a conjecture being posed and a piece of it being closed.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.","claimIssueNote":null,"solveDate":"2026-08-12","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"The model produced the proof (the $\\mathrm{Dih}(3) = \\mathrm{Sym}(3)$ collapse, the Chvátal reduction, the cyclic corollary), the Lean 4 formalization, and the Python verification script autonomously. Human direction was limited to initiation and operational supervision.","verification":"site-confirmed","verificationNote":"Reproduced here on 13 August 2026. The Lean development builds clean (exit 0) on the pinned toolchain (v4.12.0, core only, no Mathlib), and #print axioms shows all five main theorems depending on exactly propext, Classical.choice and Quot.sound. No Lean.ofReduceBool; with comments stripped the source has zero sorry, admit, axiom declarations and native_decide, and its 23 decide calls are kernel-reduced. A naive grep says otherwise only because those words appear in the file's own docs. The Python checker runs as described: Dih(3) has order 6 and equals Sym(3), and the lower-bound witnesses hold for b = 2..8. The general upper bound is not formalized; it cites Chvatal 1977, whose arithmetic holds. The SAT claim, unconfirmed at review, was substantiated the same day at commit 01a50c7. The DRAT files were not replayed, since replaying a shipped proof is the weaker check; instead all twelve CNFs were re-solved here with CaDiCaL, every verdict matching their kissat logs - satisfiable at $n=2b-2$, unsatisfiable at $n=2b-1$, for b = 2..7. The six satisfiable instances had their witnesses re-substituted clause by clause and all satisfy, and the b = 3 legs agree with this site's own exhaustive enumeration, anchoring their encoder against an independent computation. The certificates are regenerated rather than the originals, disclosed unprompted, which costs nothing here. Still unconfirmed: no human peer review, this being a self-submission reviewed by AI agents in-pipeline.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Small, and the preprint says so itself. This closes one slice (a = 3) of a conjecture stated about five weeks earlier, and it closes it by a group coincidence plus a citation: Dih(3) happens to equal Sym(3), so the permutational condition collapses to ordinary subgraph containment and Chvatal's 1977 theorem finishes it. The note is candid that the cyclic values for b = 3..8 were already tabulated by Basic et al., and that what is new is the closed form, not the numbers. Scored near the bottom of the spine, below the Erdos entries at 10, which are decades-old rather than weeks-old.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/ZestyWombat854/dihedral-ramsey","sourceName":"GitHub repo (preprint + Lean proof + Python checker)","collection":null,"collectionVersion":null,"links":[{"label":"Damnjanovic and Djordjevic, Computation of small reflective and dihedral Ramsey numbers (Conjecture 4.9)","url":"https://arxiv.org/abs/2607.06817","kind":"problem-record"},{"label":"Basic, Damnjanovic, Stevanovic and Stosic (Conjecture 4.23)","url":"https://arxiv.org/abs/2604.16188","kind":"problem-record"},{"label":"sat/ - regenerated CNFs, DRAT certificates, witnesses and solver logs (commit 01a50c7)","url":"https://github.com/ZestyWombat854/dihedral-ramsey/tree/01a50c7/sat","kind":"code"}],"relations":[],"submittedBy":"ZestyWombat854","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"phelps-rodriguez-conjecture","name":"Phelps–Rodriguez Conjecture","shortName":"Phelps–Rodriguez Conj.","problemNumber":null,"field":"Complex analysis","fieldGroup":"Analysis","statement":"Let $p$ be a complex polynomial of degree $n\\ge2$ whose zeros all lie in the closed unit disk. For every zero $a$ of $p$, there is a critical point $\\zeta$ satisfying $|\\zeta-a|<1$, except when $|a|=1$ and $p$ is a nonzero scalar multiple of $z^n-a^n$.","posedBy":"Dean Phelps, Rene S. Rodriguez","yearPosed":1972,"ageNote":"Posed by Phelps and Rodriguez in 1972 as the sharp form of the Ilieff-Sendov question, and open for 54 years. It was resolved on 12 August 2026, the same day as Sendov's conjecture itself and by the same argument.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Phelps-Rodriguez implies Sendov, so this entry records the stronger of the pair; the companion Sendov entry records the weaker statement and Mazur's original formalization, which proved Sendov but never stated the equality classification. The exceptional family is genuinely attained rather than an artefact of the proof: for p = z^n - 1 and a = 1 the only critical point is the origin, at distance exactly 1. Both conjectures fell out of one argument, and the strict form was not the announced target - Tao's digestion of Mazur's proof turned out to establish it, which is how a 1972 conjecture was resolved as a by-product of resolving a 1959 one.","claimIssueNote":null,"solveDate":"2026-08-12","model":"GPT-5.6 Pro, Claude Opus 5","modelMaker":"OpenAI","humanCollaborators":["Lech Mazur","Terence Tao"],"aiRole":"Two models in two roles. The underlying mathematics is Lech Mazur's AI-generated proof of Sendov's conjecture, where GPT-5.6 Pro carried the discovery and derivation. Terence Tao then digested and streamlined that argument - by his own account with heavy AI assistance - and observed that it establishes the stronger strict-interior form, which with the boundary classification is Phelps-Rodriguez. The formalization is a separate artifact: Tao's repository states that essentially all of its Lean source was written by Claude Opus 5 under his direction and review. So the model produced the core argument and wrote the formal proof, while the essential step specific to this entry - recognising that the streamlined argument gives the strict form, and supplying the exceptional family - is Tao's, inside a human-led write-up. That is the co-developed tier rather than the assisted one the submission chose: the models did mathematics here, not tooling.","verification":"lean-verified","verificationNote":"Audited here on 13 August 2026, which is what lifts this above the submitter's conservative Lean-checked classification. The gap they identified was that nobody had checked the correspondence between Tao's formal statement and the historical conjecture, so that check was performed. Sendov.phelps_rodriguez in Sendov/Conjecture.lean reads: for $n \\ge 2$ and $p$ of natDegree $n$ with every root in the closed unit disk and $p(a)=0$, either some critical point has $|\\zeta - a| < 1$, or $|a| = 1$ and $p = c(X^n - a^n)$ for some nonzero $c$. That is exactly Phelps-Rodriguez, exceptional family included, with no weakening; and it is not vacuous, since natDegree $= n$ with $n \\ge 2$ forces $p \\ne 0$, which the proof derives rather than assumes. All 80 first-party files were audited with comments stripped: zero admit, zero axiom declarations, zero native_decide, and 124 decide calls, all kernel-reduced. The only two sorry occurrences sit in Challenge.lean, which nothing imports, so they are outside the proof path. On the build, all four GitHub Actions runs report failure, which is misleading: reading the job steps shows the leanprover/lean-action build succeeded on the latest commit, and the failing step is docgen-action, documentation generation. That makes the kernel check third-party evidenced rather than resting on the author's machine. Not independently reviewed by another mathematician: the repository says so, and Tao both wrote the digestion and directed the formalization.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A named 1972 conjecture that is strictly stronger than Sendov's: it upgrades the distance-one bound to a strict inequality and classifies every case of equality. Less famous than Sendov itself, which is scored 40 here and carries Wikipedia articles in four languages, but it is the sharp form of the same question and had stood 54 years. Scored below Sendov and level with the well-tracked named conjectures, since it inherits Sendov's setting rather than opening a new one.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/","sourceName":"A digestion of the proof of Sendov’s conjecture","collection":null,"collectionVersion":null,"links":[{"label":"teorth/sendov - Lean formalization; Sendov.phelps_rodriguez in Sendov/Conjecture.lean","url":"https://github.com/teorth/sendov","kind":"lean-proof"},{"label":"Challenge.lean - the statement of record, Mathlib-only, no definitions of its own","url":"https://github.com/teorth/sendov/blob/master/Challenge.lean","kind":"lean-statement"},{"label":"Mazur's original proof of Sendov's conjecture, the underlying argument","url":"https://proofatlas.ai/formalizations/sendov-conjecture/","kind":"independent"}],"relations":[{"to":"sendov-s-conjecture","kind":"same-work","note":"Both fall to Tao's digestion of Mazur's argument: the interior form of Sendov's conjecture resolves Phelps-Rodriguez in full generality."}],"submittedBy":"HiddenHawk615","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"treglown-equitable-acyclic-colouring-conjecture","name":"Treglown's equitable acyclic colouring conjecture","shortName":"Treglown's conjecture","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Treglown conjectured, in a complementary form, that for every positive integer $k$ every digraph $D$ with $\\min\\{d^+(v), d^-(v)\\} \\le k-1$ for all $v$ has an equitable acyclic $k$-colouring. This implies the acyclic colouring versions of the Hajnal-Szemeredi theorem for digraphs proved by Czygrinow, DeBiasio, Kierstead and Molla, which in turn imply the original Hajnal-Szemeredi theorem for graphs. Proved: a short reduction shows the conjecture follows directly from the original Hajnal-Szemeredi theorem, and a modification of it gives a polynomial-time algorithm for finding such a colouring.","posedBy":"Andrew Treglown","yearPosed":2015,"ageNote":"Paper posing the conjecture was posted on arXiv on 2014, then revised in 2015, the latter year is taken as a prudent measure","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The honest reading, which the paper gives itself: the reduction is implicit in earlier work of Aboulker, Oijid, Petit, Rocton and Simon, and the model itself surfaced that reference when asked about originality. So this establishes the conjecture and supplies a polynomial-time algorithm, while the underlying idea is a rediscovery rather than a first. It is a striking record of a model producing an argument and then correcting the novelty claim made for it.","claimIssueNote":null,"solveDate":"2026-08-12","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Louis DeBiasio","Hal Kierstead"],"aiRole":"The acknowledgements describe the model producing the argument and then correcting its own novelty claim: \"The reduction given in this paper arose during a discussion between the first author and ChatGPT 5.6 Sol attempting to locate the bottleneck in extending the results of [3] to prove Conjecture 1.2. Instead of locating the bottleneck, the chatbot gave a clever proof which shows that Conjecture 1.2 reduces to the original Hajnal-Szemeredi theorem. After further discussion about the originality of this idea, the chatbot identified earlier work of Aboulker, Oijid, Petit, Rocton, and Simon\" in which the reduction is implicit. The central idea of the paper came from the model, which places it at the discovered tier, with the caveat about priority recorded in the result note.","verification":"unreviewed","verificationNote":"arXiv preprint, one day old at cataloguing, not peer-reviewed. The authors are established researchers in the area and the paper is short, resting on a reduction to a classical theorem rather than new machinery, but this site has not verified it and no independent commentary exists yet.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture whose truth implies the digraph acyclic-colouring analogues of Hajnal-Szemeredi, so it sits against a genuinely central classical theorem - but it falls to a short reduction rather than new machinery, and the reduction turns out to be implicit in existing literature. Scored with the resolved specialist problems rather than the long-standing named conjectures.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.12207","sourceName":"arXiv:2608.12207 - The Hajnal-Szemeredi theorem in digraphs revisited","collection":null,"collectionVersion":null,"links":[{"label":"Treglown 2015 - Problem statement","url":"https://arxiv.org/abs/1406.3229","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"word-length-spectral-triples-compact-quantum-metric-spaces","name":"Word-length spectral triples as compact quantum metric spaces","shortName":"Word-length quantum metrics","problemNumber":null,"field":"Operator algebras","fieldGroup":"Analysis","statement":"A countable discrete group with a proper length function carries a natural spectral triple on its reduced group C*-algebra. A well-studied question in non-commutative metric geometry asks whether the associated Connes pseudo-metric always recovers the weak-* topology on the state space, making it a compact quantum metric space in Rieffel's sense. It holds for groups of polynomial growth and for word-hyperbolic groups, and it was widely expected that not every word-length function works - but no explicit counterexample was known. False: for every integer $d \\ge 2$ the canonical spectral triple of the Lamplighter group $(\\mathbb{Z}/2\\mathbb{Z}) \\wr \\mathbb{F}_d$, with the word-length function of a finite symmetric generating set, fails to be a spectral metric space.","posedBy":"Marc Rieffel","yearPosed":2002,"ageNote":"The question belongs to Rieffel's compact quantum metric space programme, which dates to his 1998-2004 papers on group C*-algebras as quantum metric spaces; the entry uses 2002 for his 'Group C*-algebras as compact quantum metric spaces'. Sources vary on how to date the question itself, which was folklore in the area rather than a single numbered conjecture.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The first explicit counterexample rather than a first suspicion: the abstract is clear that the failure was widely expected and that what was missing was a witness. It gives an infinite family, one for each d >= 2, all Lamplighter groups over free groups.","claimIssueNote":null,"solveDate":"2026-08-12","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Mario Klisse"],"aiRole":"The acknowledgements state: \"The author acknowledges the use of GPT-5.6 Sol as an exploratory tool to assist in finding the counterexample. The AI was used under the author's strict mathematical guidance. All mathematical content and arguments were rigorously reviewed, verified, and substantially revised by the author, who assumes full responsibility for the final manuscript.\" The model helped find the central object, which is an essential named step, but the framing is explicitly human-led and the author reports substantially revising everything - the co-developed tier rather than the discovered one.","verification":"unreviewed","verificationNote":"arXiv preprint, one day old at cataloguing, not peer-reviewed and with no independent commentary yet. This site has not verified the argument; unlike a finite counterexample, this one is an infinite family and an analytic failure of a topology-recovery property, so it is not settleable by computation.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A recognised open question in non-commutative metric geometry, sitting in Rieffel's compact-quantum-metric-space programme, where the positive cases (polynomial growth, word-hyperbolic) were known and the expected negative answer had resisted an explicit witness. Specialist, with no Wikipedia article of its own, so scored with the resolved named problems of a single subfield rather than the broadly tracked conjectures.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.12080","sourceName":"arXiv:2608.12080 - Word-Length Spectral Triples of (Z/2Z) wr F_d Are Not Metric","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hadamard-matrix-of-order-668","name":"Hadamard Matrix of Order 668","shortName":"Hadamard 668","problemNumber":null,"field":"Combinatorial design theory","fieldGroup":"Combinatorics","statement":"There exists a Hadamard matrix of order $668$: a matrix\n\n$$\nH\\in\\{-1,1\\}^{668\\times668}\n$$\n\nsuch that\n\n$$\nHH^{\\mathsf T}=668I_{668}.\n$$\n\nEquivalently, the $668$ rows of $H$ are pairwise orthogonal.","posedBy":"Raymond Paley","yearPosed":1933,"ageNote":"The Hadamard conjecture dates to Paley in 1933. Order 668 became the smallest open admissible order in 2004, when Kharaghani and Tayfeh-Rezaie settled order 428; before that the record fell in 1985.","solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"Explicit construction of a Hadamard matrix of order 668, the smallest previously unresolved order, verified exactly by this site from the announcement plus its decoder reply. The same post encodes matrices for all twelve previously-open admissible orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964), and this site verified every one of them. The entry records the order-668 existence question, which this fully resolves; the general Hadamard conjecture - existence for ALL admissible orders - remains open, with the smallest unknown order now 2004 or beyond.","claimIssueNote":null,"solveDate":"2026-08-12","model":"Claude (version undisclosed)","modelMaker":"Anthropic","humanCollaborators":["Levent Alpöge","Philippe Voinov","Saul Reynolds-Haertle"],"aiRole":"The announcement itself is a bare sign string, but Alpoge's thread carries a credit line: \"weekend fun w @tehwalris, Saul Reynolds-Haertle, and of course claude:)) i only claim bad suggestions!!\" - which corroborates the three named collaborators (@tehwalris is Philippe Voinov) and confirms Claude was part of the working group, with Alpoge playfully disclaiming the good ideas. Which mathematical, computational or search steps were Claude's is still not stated anywhere, and no model version is given, so the tier stays at the floor the methodology prescribes for an unspecific disclosure.","verification":"site-confirmed","verificationNote":"Fully reproduced here on 12 August 2026, in two passes. The announcement is a single X long-post holding 23,828 characters of \"+\" and \"-\": no prose, no separators. The first pass scanned the raw string for seed shapes and found three Goethals-Seidel quadruples (orders 892, 1132, 1244), verified exactly, but no order-668 seed - correctly, since the payload is not a seed list. What it missed is that Alpoge's own reply to the post is a decoder: a sed-obfuscated shell script declaring twelve records and five builder routines. This site reimplemented the sed transformation in Python, read the decoded script before executing anything (pure sed/sh, no network, writes only under /tmp), and ran it. It emits twelve sign blocks, and its header table independently names the four orders the raw scan had already found, cross-validating both decodings. Every block was then checked in exact integer arithmetic: entries in $\\{-1,+1\\}$ and $HH^T = nI$ on the nose, for $n$ = 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948 and 1964 - all twelve previously-open admissible orders below 2000, exactly as the thread claims. The order-668 matrix has diagonal 668 everywhere and maximum absolute off-diagonal entry 0. The submitter reports an equivalent reproduction, done separately from this one. No independent expert review or published write-up exists yet, so site-confirmed records this site's own exact-arithmetic reproduction, not community acceptance.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":"The Hadamard conjecture has a Wikipedia article; order 668 as a specific target does not, but it is tracked by name on Epoch AI's FrontierMath open-problem list, TheoremDB and EmergentMind, and is standard background in combinatorial design theory.","significance":30,"significanceNote":"Order 668 has been the smallest open case of the Hadamard conjecture since order 428 was settled in 2004, and it is a named target in the design-theory literature and a listed FrontierMath open problem. Scored as one hard instance of a famous 1933 conjecture rather than the conjecture itself, which would sit far higher; placed level with a well-tracked specialist problem and below the household combinatorics conjectures.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://x.com/__alpoge__/status/2087504785952182273","sourceName":"Hadamard Matrix of Order 668","collection":null,"collectionVersion":null,"links":[{"label":"Epoch AI - FrontierMath open problem (records the solve)","url":"https://epoch.ai/frontiermath/open-problems/hadamard","kind":"problem-record"},{"label":"Eliahou, A 64-modular Hadamard matrix of order 668 (2025)","url":"https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf","kind":"paper"},{"label":"Recent Legendre-pair work on order 668","url":"https://arxiv.org/abs/2607.20765","kind":"paper"},{"label":"TheoremDB record for order 668","url":"https://theoremdb.org/statements/hadamard-order-668/","kind":"problem-record"}],"relations":[],"submittedBy":"LucidHawk551","upvotes":4,"downvotes":0,"commentCount":3},{"slug":"spahn-zeilberger-holonomicity-restricted-permutations","name":"Spahn and Zeilberger's Third Challenge: Holonomicity of the Restricted Permutation Counts","shortName":"Spahn–Zeilberger holonomicity","problemNumber":null,"field":"Enumerative combinatorics","fieldGroup":"Combinatorics","statement":"Spahn and Zeilberger's third challenge asks whether the restricted permutation counts $a_{r,s}$ and $b_{r,s}$ are holonomic for all $r, s > 1$. Answered affirmatively.","posedBy":"Evan Spahn and Doron Zeilberger","yearPosed":2022,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-11","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Jaideep Sai Padhi"],"aiRole":"The paper's division of labour, in its own words: the model \"generated the mathematical ideas, the proofs, and the verification code\", while the author \"selected the problems and determined when a line of attack should be abandoned; executed all computations, on hardware the system could not access; designed and enforced the verification protocol\". It adds that neither role was passive: the system did not merely formalise ideas supplied to it, and the author did not merely execute instructions.","verification":"unreviewed","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.11290v2): the division of labour is verbatim as quoted, all five challenges are reported fully solved, and two further problems are reported only partial and are NOT entered here. The paper states a four-principle verification protocol that distinguishes proved, machine-verified and supported claims, and ships code and verification scripts. Those scripts were not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A holonomicity question over a whole two-parameter family, so the answer is a structural theorem rather than a single computation - the most substantial of the five challenges settled in this paper. Below the named-conjecture band on recognition.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11290","sourceName":"Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics","collection":null,"collectionVersion":null,"links":[{"label":"Spahn & Zeilberger 2022 - Problem statement","url":"https://arxiv.org/abs/2211.02550","kind":"problem-record"},{"label":"Github repo of the solution","url":"https://github.com/jaideepsaipadhi/restricted-permutations-holonomicity","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"first-rigorous-solid-standard-young-tableaux-challenge","name":"The First Rigorous Solid Standard Young Tableaux Challenge","shortName":"Solid SYT challenge I","problemNumber":null,"field":"Enumerative combinatorics","fieldGroup":"Combinatorics","statement":"The first rigorous solid standard Young tableaux challenge asks for a proof of a conjectured second-order recurrence for the number of solid standard Young tableaux. The conjectured recurrence is proved.","posedBy":"Doron Zeilberger and collaborators","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-11","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Jaideep Sai Padhi"],"aiRole":"The paper's division of labour, in its own words: the model \"generated the mathematical ideas, the proofs, and the verification code\", while the author \"selected the problems and determined when a line of attack should be abandoned; executed all computations, on hardware the system could not access; designed and enforced the verification protocol\". It adds that neither role was passive: the system did not merely formalise ideas supplied to it, and the author did not merely execute instructions.","verification":"unreviewed","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.11290v2): the division of labour is verbatim as quoted, all five challenges are reported fully solved, and two further problems are reported only partial and are NOT entered here. The paper states a four-principle verification protocol that distinguishes proved, machine-verified and supported claims, and ships code and verification scripts. Those scripts were not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"A conjectured recurrence promoted to a theorem, which is the shape of progress this corner of enumerative combinatorics is built from. Narrow audience, so set near the lower challenge band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11290","sourceName":"Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics","collection":null,"collectionVersion":null,"links":[{"label":"Github repo of the solution","url":"https://github.com/jaideepsaipadhi/zeilberger-challenges/tree/main/solid-syt","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"wu-santhanam-diagonalizability-prediction","name":"Predicting Diagonalizability of a Mean Matrix","shortName":"Diagonalizability prediction","problemNumber":null,"field":"Statistical learning theory","fieldGroup":"Probability & statistics","statement":"Wu and Santhanam asked whether one can determine, from an increasing i.i.d. sample of binary random matrices, whether the unknown mean matrix is diagonalizable, while making only finitely many errors almost surely. Answered affirmatively over both R and C.","posedBy":"Yuheng Wu, Narayana Santhanam","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The general principle is the interesting part: every semialgebraic property of a bounded fixed-dimensional mean parameter is eventually almost surely predictable. Against merely integrable matrix laws it fails from dimension two.","claimIssueNote":null,"solveDate":"2026-08-11","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Jinze Zhao"],"aiRole":"A disclosure section of its own: \"The proof strategy and counterexample were produced by OpenAI's GPT-5.6 Sol Ultra through Codex in response to prompts from the author. Codex was also used to revise the exposition and prepare the LaTeX manuscript. The author selected the problem, directed the interactions and revisions, and is the sole named author.\"","verification":"unreviewed","verificationNote":"A preprint days old. The paper says so itself: \"This disclosure is not a substitute for independent expert mathematical review.\"","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A recent question from a single specialist paper on eventually-almost-sure prediction, with a real but small audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.10482","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"steklov-isospectral-convex-plane-domains","name":"The Planar Steklov Analogue of Kac's Question","shortName":"Steklov-isospectral domains","problemNumber":null,"field":"Spectral geometry","fieldGroup":"Analysis","statement":"Can one hear the shape of a drum, in the Steklov setting and in the plane? No: there exist pairs of noncongruent bounded plane domains with identical Steklov spectra including multiplicities, simply connected, strictly convex, with real-analytic boundaries, and arbitrarily close to a disk in the C-infinity topology.","posedBy":"The Steklov analogue of Kac's question, raised in the Girouard-Polterovich problem literature","yearPosed":2017,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Strict convexity and real-analytic boundaries are what make this sharp: the classical Gordon-Webb-Wolpert drums are non-convex polygons, so the obvious escape routes are closed off.","claimIssueNote":null,"solveDate":"2026-08-11","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Tao Hu","Jiachen Shi","Quanyu Tang"],"aiRole":"The division of labour is set out in the AI statement: the idea of adapting the Sunada construction of Gordon, Webb and Wolpert to the planar Steklov problem \"was proposed by the authors\". From there, \"ChatGPT provided substantial assistance in developing the concrete counterexample construction, including the passage from the orbifold construction to weighted Steklov problems on the disk and their subsequent realization by Euclidean plane domains. It also assisted with several technical arguments.\" The authors checked, revised and rewrote the arguments.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The Steklov descendant of Kac's famous question, a standing open problem in the spectral-geometry problem lists with a substantial surrounding literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.10557","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"second-computational-chomp-challenge","name":"The Second Computational Chomp Challenge of Ekhad and Zeilberger","shortName":"Computational Chomp II","problemNumber":null,"field":"Combinatorial game theory","fieldGroup":"Combinatorics","statement":"Ekhad and Zeilberger's second computational Chomp challenge asks for a Chomp position with three winning opening moves. Answered by exhibiting a bar with three winning opening moves.","posedBy":"Shalosh B. Ekhad and Doron Zeilberger","yearPosed":2018,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-11","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Jaideep Sai Padhi"],"aiRole":"The paper's division of labour, in its own words: the model \"generated the mathematical ideas, the proofs, and the verification code\", while the author \"selected the problems and determined when a line of attack should be abandoned; executed all computations, on hardware the system could not access; designed and enforced the verification protocol\". It adds that neither role was passive: the system did not merely formalise ideas supplied to it, and the author did not merely execute instructions.","verification":"unreviewed","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.11290v2): the division of labour is verbatim as quoted, all five challenges are reported fully solved, and two further problems are reported only partial and are NOT entered here. The paper states a four-principle verification protocol that distinguishes proved, machine-verified and supported claims, and ships code and verification scripts. Those scripts were not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"One of Zeilberger's posted challenges: precisely stated, publicly open, and answered by a named search rather than by a general theorem. Real but narrow, a little above the Graffiti anchor at 5 and well below the named-conjecture band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11290","sourceName":"Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics","collection":null,"collectionVersion":null,"links":[{"label":"Ekhad & Zeilberger 2018 - Problem statement","url":"https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/chompc.html","kind":"problem-record"},{"label":"Github repo of the solution","url":"https://github.com/jaideepsaipadhi/chomp-three-winning-moves/tree/main","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"five-dimensional-geode-challenge","name":"The Five-Dimensional Geode Challenge of Amdeberhan, Kauers and Zeilberger","shortName":"Geode challenge (5-dim)","problemNumber":null,"field":"Enumerative combinatorics","fieldGroup":"Combinatorics","statement":"The five-dimensional case of the Geode challenge of Amdeberhan, Kauers and Zeilberger, concerning the geode factor attached to a family of multivariate generating functions. Settled in dimension five.","posedBy":"Tewodros Amdeberhan, Manuel Kauers and Doron Zeilberger","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-11","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Jaideep Sai Padhi"],"aiRole":"The paper's division of labour, in its own words: the model \"generated the mathematical ideas, the proofs, and the verification code\", while the author \"selected the problems and determined when a line of attack should be abandoned; executed all computations, on hardware the system could not access; designed and enforced the verification protocol\". It adds that neither role was passive: the system did not merely formalise ideas supplied to it, and the author did not merely execute instructions.","verification":"unreviewed","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.11290v2): the division of labour is verbatim as quoted, all five challenges are reported fully solved, and two further problems are reported only partial and are NOT entered here. The paper states a four-principle verification protocol that distinguishes proved, machine-verified and supported claims, and ships code and verification scripts. Those scripts were not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"A named challenge from three authors, settled in the specific dimension posed. Real and precisely stated, narrow in reach - the lower challenge band.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11290","sourceName":"Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics","collection":null,"collectionVersion":null,"links":[{"label":"Amdeberhan et al 2025 - Problem statement","url":"https://arxiv.org/abs/2508.10245","kind":"problem-record"},{"label":"Github repo of the solution","url":"https://github.com/jaideepsaipadhi/zeilberger-challenges/tree/main/geode","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kauers-zeilberger-conjectures-2a-2b","name":"Conjectures 2a and 2b of Kauers and Zeilberger","shortName":"Kauers–Zeilberger 2a, 2b","problemNumber":null,"field":"Lattice walk enumeration","fieldGroup":"Combinatorics","statement":"Conjectures 2a and 2b of Kauers and Zeilberger, on the asymptotics of a family of restricted lattice walks. Both are obtained from a local limit theorem for excursions of Markov-modulated random walks in cones.","posedBy":"Manuel Kauers and Doron Zeilberger","yearPosed":2020,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-11","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Jaideep Sai Padhi"],"aiRole":"The paper's division of labour, in its own words: the model \"generated the mathematical ideas, the proofs, and the verification code\", while the author \"selected the problems and determined when a line of attack should be abandoned; executed all computations, on hardware the system could not access; designed and enforced the verification protocol\". It adds that neither role was passive: the system did not merely formalise ideas supplied to it, and the author did not merely execute instructions.","verification":"unreviewed","verificationNote":"Checked by this site on 21 August 2026 against the paper (arXiv:2608.11290v2): the division of labour is verbatim as quoted, all five challenges are reported fully solved, and two further problems are reported only partial and are NOT entered here. The paper states a four-principle verification protocol that distinguishes proved, machine-verified and supported claims, and ships code and verification scripts. Those scripts were not re-run here. Days-old preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"Settled not by computation but by proving a local limit theorem for excursions of Markov-modulated random walks in cones - a general probabilistic tool built to answer them, which is why this rates highest of the five challenges here.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11290","sourceName":"Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics","collection":null,"collectionVersion":null,"links":[{"label":"Kauers & Zeilberger 2020 - Problem statement","url":"https://arxiv.org/html/2006.10205v2","kind":"problem-record"},{"label":"Github repo of the solution","url":"https://github.com/jaideepsaipadhi/zeilberger-challenges/tree/main/kz-constants","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"albertson-berman-induced-forest-conjecture","name":"Albertson–Berman Induced Forest Conjecture","shortName":"Albertson–Berman","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Albertson and Berman conjectured that for every simple planar graph $G$ on $n$ vertices, the largest vertex set inducing a forest has size at least $n/2$. The standing lower bound since the same year has been Borodin's $2n/5$, from his acyclic five-colour theorem. False: there is an explicit $31$-vertex simple $3$-connected maximal planar graph $T$ whose largest induced forest has exactly $15$ vertices, and an infinite family $M_k$ on $31k$ vertices with induced-forest number exactly $15k$, giving the ratio $15/31 < 1/2$ even for triangulations of minimum degree five.","posedBy":"Michael O. Albertson, David M. Berman","yearPosed":1979,"ageNote":"Posed in 1979 and open for 47 years. A paper on the same quantity, Makarov's Large induced forests in planar multigraphs, still described the conjecture as open in its revision of 9 August 2026, two days before this preprint appeared.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.","claimIssueNote":null,"solveDate":"2026-08-11","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Heejae Jung"],"aiRole":"The paper's \"Acknowledgments and AI disclosure\" section states that the two-terminal gadget \"was discovered, and substantial parts of the proof strategy were developed, through interaction with OpenAI GPT-5.6 Sol\", while the author \"selected the research problem, directed the computational search and subsequent proof development, and checked the resulting mathematical arguments and computational certificates\". GPT-5.6 Sol also assisted in preparing the manuscript and the verification code.","verification":"site-confirmed","verificationNote":"Reproduced here on 12 August 2026. The refutation is a single finite object, so it is checkable outright rather than on trust. The 31-vertex seed $T$ was rebuilt from the paper's own definitions - the 14-vertex gadget's cyclic neighbour lists, the pentagonal-bipyramid base, the decorated rim edges, the stated labelling and the two completion edges - without running the author's code. That yields a simple 3-connected planar graph on 31 vertices with $87 = 3n-6$ edges, hence a triangulation, with the paper's degree multiset $4^1 5^{17} 6^6 7^7$. Its maximum induced forest was then computed exactly by two independent algorithms: an ILP with lazy cycle-elimination cuts, and a branch-and-bound minimum feedback vertex set with no LP involved. Both give $a(T) = 15$, equivalently a minimum feedback vertex set of exactly 16, against the 15.5 the conjecture requires. The two finite inputs to the symbolic argument were separately brute-forced - the terminal profile $(6,6,6,5)$ over all $2^{12}$ internal subsets, and $\\beta = 3$ over all $2^7$ subsets of the core - and $M_k$ for $k = 2..5$ confirmed planar on $31k$ vertices with $93k-6$ edges, minimum degree five, every seed induced. Worth noting what the shipped verifier does not do: it certifies the gadget embedding, the profile, $\\beta$ and the sphere certificates, but never computes $a(T)$ or $a(M_k)$, and says so. That computation is the one this site supplied. Not peer-reviewed, not on arXiv, no independent expert review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":"No Wikipedia article, but the conjecture appears on both Bojan Mohar's and Douglas West's open-problem pages for planar graphs and is cited by name in the current literature.","significance":30,"significanceNote":"A named 1979 conjecture carried on the standard open-problem pages for planar graphs (Bojan Mohar's list, Douglas West's list), with a continuing literature on partial cases - girth 4 and 5, triangle-free, bipartite, 2-outerplanar, multigraphs - and a gap between the conjectured n/2 and Borodin's 2n/5 that stood for 47 years. Placed above a specialist named conjecture such as Simon's extendable shellability (25) because it is older and more widely tracked, and well below a household problem such as the cycle double cover conjecture (55).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.17350","sourceName":"A 15/31 Counterexample Family to the Albertson-Berman Conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Preprint PDF","url":"https://zenodo.org/records/21883880/files/Disproving_the_Albertson_Berman_Conjecture.pdf","kind":"paper"},{"label":"Author's verification script","url":"https://zenodo.org/records/21883880/files/verify_stronger_ab_family.py","kind":"code"},{"label":"Mohar's open problem list: induced forests in planar graphs","url":"https://www.sfu.ca/~mohar/Problems/P0208InducedForestPlanar.html","kind":"problem-record"},{"label":"Makarov, Large induced forests in planar multigraphs - calls the conjecture open, revised 9 Aug 2026","url":"https://arxiv.org/abs/2601.04637","kind":"other"},{"label":"Makarov - a preprint claiming a somewhat different counterexample found independently without AI","url":"https://zenodo.org/records/21927902","kind":"independent"},{"label":"Makarov - preprint PDF","url":"https://zenodo.org/records/21927902/files/albertson_berman_conjecture_counterexample.pdf","kind":"independent"}],"relations":[],"submittedBy":"MellowWalrus138","upvotes":3,"downvotes":0,"commentCount":2},{"slug":"stepsize-acceleration-lower-bound","name":"Lower Bounds for Stepsize-Based Acceleration of Gradient Descent","shortName":"Stepsize acceleration lower bound","problemNumber":null,"field":"Convex optimization","fieldGroup":"Algorithms & optimization","statement":"Carefully designed stepsize schedules alone accelerate plain gradient descent beyond its textbook O(1/T) rate, without momentum. Whether they can reach the optimal O(T^-2) was open. A lower bound of Omega(T^-1.9319) for last-iterate convergence under predetermined nonnegative stepsize schedules says they cannot.","posedBy":"Raised by the silver-stepsize line of work following Altschuler and Parrilo","yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Recorded as partial: the bound is Omega(T^-1.9319) against an achievable O(T^-1.2716), so it rules out reaching the optimal rate without pinning down the true one.","claimIssueNote":null,"solveDate":"2026-08-11","model":"GPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":["Jianhao Ma","Yuxin Chen"],"aiRole":"The abstract closes with it: \"The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.\" The authors added material to make the proof correct and readable, and separately used Codex to formalize the proof in Lean 4.","verification":"unreviewed","verificationNote":"A preprint days old. A Lean 4 formalization by Codex is linked from the paper, but it is the same pipeline that produced the proof, so it is not independent confirmation.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"An open direction in a currently active corner of convex optimization, a few years old and confined to that community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.10418","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lions-maximal-regularity-half-holder","name":"Lions' Maximal Regularity Problem at the Half-Holder Endpoint","shortName":"Lions' maximal regularity problem","problemNumber":null,"field":"Evolution equations","fieldGroup":"Differential equations","statement":"Lions asked whether the variational solution of a non-autonomous divergence-form problem has maximal L2-regularity under Holder continuity in time of the coefficients. Disproved at the half-Holder endpoint: a bounded, uniformly elliptic, real scalar coefficient, half-Holder in time and arbitrarily close to the heat equation, whose Lions solution has a time derivative that is not square integrable.","posedBy":"Jacques-Louis Lions","yearPosed":1961,"ageNote":"A problem of Lions from the early 1960s, worked on continuously since, with a substantial modern literature on which regularity hypotheses suffice.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Tensorisation and parabolic rescaling carry the one-dimensional example to real symmetric isotropic counterexamples on R^d and on every bounded domain, in every dimension.","claimIssueNote":null,"solveDate":"2026-08-11","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Lukas Niebel"],"aiRole":"The Declaration of AI Use: \"During an exploratory analysis, a first counterexample was found by OpenAI's GPT-5.5 Pro in two dimensions and on the full space. It was then verified and studied by the author, who simplified it and reduced it to the one-dimensional interval counterexample presented here.\" GPT-5.6 Sol was separately used for drafting and revision of the exposition.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A named problem of Lions with sixty years of attack and a current research literature devoted to locating exactly where maximal regularity fails.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.11194","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hyperkahler-period-index-conjecture","name":"The Hyperkahler Period-Index Conjecture","shortName":"Hyperkahler period-index","problemNumber":null,"field":"Algebraic geometry","fieldGroup":"Geometry & topology","statement":"Huybrechts conjectured that for every Brauer class alpha on a hyperkahler variety X, the index divides the period raised to the power dim(X)/2, strengthening the usual period-index conjecture. Disproved on certain hyperkahler fourfolds, in both the K3^[2] and Kum^2 deformation types.","posedBy":"Daniel Huybrechts","yearPosed":2019,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The counterexamples come with divisibility bounds on the Hodge-theoretic index, at 2-torsion and 5-torsion, on very general hyperkahler fourfolds.","claimIssueNote":null,"solveDate":"2026-08-10","model":"Unspecified","modelMaker":null,"humanCollaborators":["Pieter Belmans","James Hotchkiss"],"aiRole":"The AI disclosure names the role precisely while leaving the model unnamed: \"The starting points for this paper were two different LLM-assisted constructions of counterexamples, obtained independently by the authors. The paper is the synthesis of these constructions, with LLMs used to help with the copyediting.\" Two authors reaching counterexamples independently with model help is a stronger signal than either would be alone.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A conjecture by a leading figure strengthening the period-index conjecture, well known within hyperkahler geometry and Brauer-group theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.09436","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"teschner-s-bondage-number-conjecture","name":"Teschner's Bondage-Number Conjecture","shortName":"Teschner bondage conjecture","problemNumber":null,"field":"Graph theory — domination theory","fieldGroup":"Combinatorics","statement":"Teschner conjectured that every finite simple graph $G$ with at least one edge satisfies $b(G) \\leq \\frac{3}{2}\\Delta(G)$, where $b(G)$ is the bondage number and $\\Delta(G)$ is the maximum degree. Yavari gives a connected cubic bipartite graph on 18 vertices with $b(G)=5$. Since $\\Delta(G)=3$, this gives $b(G)=5 > \\frac{3}{2}\\Delta(G)=\\frac{9}{2}$, providing a counterexample and disproving the conjecture.","posedBy":"Ulrich Teschner","yearPosed":1995,"ageNote":"Fink, Jacobson, Kinch and Roberts introduced the bondage number in 1990 and conjectured b(G) <= Delta(G) + 1; Teschner disproved that in 1993 and proposed the weaker (3/2)Delta(G) bound in his 1995 Australasian Journal of Combinatorics paper, having proved it for domination number at most three. Open for 31 years.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Teschner's universal bound b(G) <= (3/2)Delta(G) is false: the 18-vertex cubic bipartite graph has b(G) = 5 against a bound of 4.5. What survives is the restricted statement Teschner actually proved, that the bound holds for graphs of domination number at most three, and Gagarin and Zverovich's 2013 result that it holds for almost all graphs. The counterexample does not suggest a replacement bound, and the correct general upper bound for b(G) in terms of Delta(G) remains open.","claimIssueNote":null,"solveDate":"2026-08-10","model":"GPT-5.6 Sol Max","modelMaker":"OpenAI","humanCollaborators":["Yousof Yavari"],"aiRole":"According to the author's disclosure and the publicly shared ChatGPT transcript, GPT-5.6 Sol Max was prompted to solve Teschner's conjecture and found an explicit counterexample. The resulting manuscript gives an 18-vertex connected cubic bipartite graph with bondage number 5, together with an exact finite certificate establishing the claimed bondage number. Yavari subsequently checked and wrote up the result.","verification":"site-confirmed","verificationNote":"Reproduced in full here on 13 August 2026. The counterexample is a single 18-vertex graph, so the claim is finite and was checked exhaustively rather than sampled. The edge list was transcribed from equation (3.1) and every quantity recomputed independently, without reading or running the author's verifier. Confirmed: connected, cubic and bipartite with the stated parts, 18 vertices and 27 distinct edges, so $\\Delta(G) = 3$; domination number 6 by exhaustive search; and exactly 297 minimum dominating sets, the count the paper states, arrived at here independently. For the lower bound, all 20,853 edge subsets of size at most four were tested by the bundle criterion, and every one leaves at least one minimum dominating set intact, so $b(G) \\ge 5$. For the upper bound, deleting the five edges 0-6, 0-10, 0-16, 1-8 and 1-11 raises the domination number to 7, recomputed from scratch on the reduced graph rather than inferred from the criterion, so $b(G) \\le 5$. Therefore $b(G) = 5 > 4.5 = \\frac32\\Delta(G)$ and the conjecture is false. That enumeration is the entire mathematical content of the claim, so this is a complete independent check. Caveats: the preprint is two days old, is hosted on figshare rather than arXiv, and has no peer review; the acknowledgements name Eric Hou (UBC) as an independent verifier, but that is a private check, not a public endorsement by a specialist.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":"No Wikipedia article of its own; the conjecture appears in the Wikipedia article on the bondage number, which still described it as open when checked on 13 August 2026, and is stated as an open problem in Xu's 2013 survey.","significance":15,"significanceNote":"A named conjecture in domination theory, posed by Teschner in 1995 and carried as an open problem in Xu's 2013 bondage-number survey and on Wikipedia's bondage number article. Real and long-standing, but firmly within a specialist subfield: there is no Wikipedia article for the conjecture itself, and the surrounding literature is a corner of graph theory rather than a mainstream target. Scored with the resolved specialist problems near 15, well below the named conjectures with their own broad literature such as Albertson-Berman (30).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.6084/m9.figshare.33198777","sourceName":"A Counterexample to Teschner's Bondage-Number Conjecture","collection":null,"collectionVersion":null,"links":[{"label":"ChatGPT conversation in which the counterexample was produced","url":"https://chatgpt.com/share/6a7a08fe-9fd8-83e8-b3b4-392282dc501a","kind":"transcript"},{"label":"Xu, On Bondage Numbers of Graphs: A Survey with Some Comments (2013)","url":"https://doi.org/10.1155/2013/595210","kind":"paper"},{"label":"Gagarin and Zverovich, The bondage number of graphs on topological surfaces and Teschner's conjecture (2013)","url":"https://doi.org/10.1016/j.disc.2012.11.005","kind":"paper"},{"label":"Wikipedia: Bondage number","url":"https://en.wikipedia.org/wiki/Bondage_number","kind":"wikipedia"}],"relations":[],"submittedBy":"HiddenHawk615","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"ellipsoid-fitting-conjecture","name":"The Ellipsoid Fitting Conjecture","shortName":"Ellipsoid fitting conjecture","problemNumber":null,"field":"Random matrix theory","fieldGroup":"Probability & statistics","statement":"Given $n$ independent standard Gaussian vectors in $\\mathbb{R}^d$, an ellipsoid fit is a positive semidefinite matrix $S$ with $x_i' S x_i = d$ for every $i$. Saunderson, Parrilo and Willsky conjectured that this semidefinite feasibility problem has a sharp threshold at $n \\sim \\frac{d^2}{4}$. Proved: below the threshold a fit exists with probability tending to one, above it none does.","posedBy":"James Saunderson, Pablo A. Parrilo, Alan S. Willsky","yearPosed":2013,"ageNote":"Conjectured around 2013 and attacked steadily since, with a sequence of teams narrowing the constant before the sharp threshold fell.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.","claimIssueNote":null,"solveDate":"2026-08-10","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Theodor Misiakiewicz","Garrett G. Wen"],"aiRole":"The approach is the authors' own - they say so, and trace it to the dual formulation of Bandeira and Maillard. What the model did is named step by step: ChatGPT 5.4 and 5.5 were used \"to explore several possible proof strategies\", and then, \"Given an earlier draft, GPT 5.6 helped repair and complete several arguments, including the tightened head-tail decomposition in Lemma 3.5 and the decomposition used in the proof of Proposition 4.4, which ultimately led to the completion of the proofs.\"","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A conjecture with a decade of documented attack by multiple groups and a settled place in the literature on semidefinite programming and random geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.10184","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"bounded-oracle-noise-nonconvex-lower-bound","name":"Bounded Oracle Error in Nonconvex Stochastic Optimization","shortName":"Bounded oracle error question","problemNumber":null,"field":"Stochastic optimization","fieldGroup":"Algorithms & optimization","statement":"Arjevani et al. asked whether almost-surely bounded oracle error permits a better rate than bounded variance for smooth nonconvex stochastic optimization. It does not: every randomized adaptive algorithm still needs Omega(dL/eps^2 + dL sigma^2/eps^4) queries, matching the standard upper bound.","posedBy":"Yossi Arjevani, Yair Carmon, John C. Duchi, Dylan J. Foster, Nathan Srebro, Blake Woodworth","yearPosed":2023,"ageNote":"Raised in the 2023 lower-bound paper that set the standard framework for nonconvex stochastic optimization rates.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-10","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Jikai Jin"],"aiRole":"Stated in the abstract itself, not buried in an acknowledgment: \"The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only for checking the proof and revising and polishing the manuscript.\"","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A question posed explicitly in a well-cited lower-bounds paper, familiar to the optimization-theory community but recent and specialist.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.09004","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kozerenko-skochko-imbalance-conjecture","name":"The Imbalance Conjecture","shortName":"Imbalance conjecture","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"The imbalance of an edge uv of a finite simple graph is the absolute difference of the degrees of u and v. Kozerenko and Skochko conjectured that the multiset of all edge imbalances is graphic - realizable as the degree sequence of some graph - whenever every edge has positive imbalance. Proved.","posedBy":"Sergiy Kozerenko, Volodymyr Skochko","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The key step is a lower bound on the truncated imbalance sum, which yields every Erdos-Gallai inequality for the sorted imbalance list; a parity computation finishes it.","claimIssueNote":null,"solveDate":"2026-08-10","model":"GPT-5.6 Sol Max","modelMaker":"OpenAI","humanCollaborators":["Yousof Yavari"],"aiRole":"The generative-AI disclosure says the model was asked to solve the problem outright: \"The proof presented in this manuscript was generated using OpenAI's GPT-5.6 Sol with the max reasoning setting (\\\"GPT-5.6 Sol Max\\\") after Yousof Yavari prompted the model to solve the question addressed in this manuscript. Yousof Yavari subsequently revised and edited the proof, added further details, and clarified its exposition.\"","verification":"unreviewed","verificationNote":"A preprint days old. The author credits Eric Hou with independently verifying the proof, which is a second reader rather than the kind of review that moves an entry up the ladder.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A specialist graph-theory conjecture with a small but real literature descending from Albertson's irregularity of a graph.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.09191","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Imbalance conjecture","url":"https://en.wikipedia.org/wiki/Imbalance_conjecture","kind":"wikipedia"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"slug":"more-than-67-of-riemann-zeta-zeros-are-on-the-critical-line","name":"The Proportion of Zeta Zeros on the Critical Line","shortName":"Zeta zeros on the line","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"The Riemann hypothesis asserts that every nontrivial zero of the zeta function lies on the critical line. Short of proving it, the standard measure of progress is the proportion of zeros known unconditionally to lie there: Selberg established a positive proportion, Levinson reached a third in 1974, Conrey two fifths in 1989, and the record stood at $\\tfrac{5}{12}$ for zeros that are simple and on the line, and $0.6603$ for distinct zeros.\n\nUnder the Riemann hypothesis, Montgomery deduced $\\tfrac23$ simple from the pair-correlation second moment in 1973. His prime-side evaluation was already unconditional; RH entered only to read the zero side as a positive sum over real ordinates. Goldston and Suriajaya isolated that termwise positivity as the remaining obstacle and asked what would follow if it could be removed.\n\nThis removes it, proving unconditionally that at least $\\tfrac23$ of zeros are simple and on the line and at least $\\tfrac56$ are distinct - $67.25\\ldots\\%$ and $0.83625$ with the Montgomery-Taylor window.","posedBy":"Bernhard Riemann (1859) for the hypothesis; the proportion ladder runs from Hardy and Selberg through Levinson and Conrey","yearPosed":null,"ageNote":"Deliberately no single posed-year, so the entry cannot claim an age it has not earned. The Riemann hypothesis dates to 1859, but this result does not resolve it. What it advances is the unconditional proportion bound, a programme running from Hardy in 1914 and Selberg in 1942 through Levinson's one third in 1974 and Conrey's two fifths in 1989. Dating this to 1859 would overstate it by a century.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The paper is explicit that these are lower bounds only - the remaining third of the zeros are not shown to be off the line, merely not reached by the certificate.\n\nWhat it does settle is a question that was posed. Goldston and Suriajaya had reduced Montgomery's conditional $\\tfrac23$ to a single obstruction, the termwise positivity that fails for zeros off the line, and asked what would follow without it. Theorem A replaces that positivity with a rank-trace inequality on a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs; reading the negative index of truncations as a count of off-line pairs is Bombieri's device. The paper also proves the bound sharp for this route: improving on $\\tfrac23$ this way would need pair-correlation information beyond Fourier support 1.","claimIssueNote":null,"solveDate":"2026-08-10","model":"Claude (unreleased research version)","modelMaker":"Anthropic","humanCollaborators":["Jarred Sumner","Levent Alpöge","Ralph Furman","Eric Easley"],"aiRole":"Claude was asked to take a real stab at the Riemann hypothesis, with the mathematical choices left to it, and the bound improvement came out as a byproduct of failing at that. It generated and discarded roughly 650 ideas in a first session; in a second it coordinated about 60 subagents which ran some 2,400 shell commands, wrote hundreds of scripts, checked numerically against known zeros and refereed one another. Two subagents developed the key ideas, thirteen fed them, thirty tried and failed, thirteen validated, two drafted the paper. Roughly 31 million output tokens across two Claude Code sessions.\n\nThe decisive step was combining the unconditional pair-correlation work of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh with a 2000 paper of Bombieri, treating the whole function space at once with the quadratic form allowed to be non-diagonal rather than splitting it. Claude also proposed writing the result up, checked 54 arXiv papers for prior art, and recommended that a human number theorist validate it.","verification":"lean-checked","verificationNote":"A sorry-free Lean 4 / Mathlib formalization accompanies the paper, with the statements built from Mathlib's own riemannZeta and analyticOrderAt rather than an assumed form of the result. This site audited the sources at tag v1.0: no sorry under Zeta23/, the 27 real ones all in the comparator/Challenge files that state the theorems with placeholder proofs by design; no native_decide; no axiom declarations - the two a plain grep reports sit inside a tactic's docstring.\n\nOne finding changes the entry's meaning: the headline theorems are unconditional, not modulo literature inputs. Zeta23.thmA0 takes no hypotheses, because Weil's explicit formula is itself proved in Lean from Mathlib's functional equation, as are Riemann-von Mangoldt, Montgomery-Vaughan, Stirling and Chebyshev-Mertens.\n\nStill held at the unaudited rung, for the reason that tier exists: the trusted statement files are written in the same repository by the same team, so nobody independent has checked the Lean statement against the paper's claim. The repository ships a leanprover/comparator challenge-and-solution split built for exactly that audit - running it is what a promotion would take. The kernel was not re-run here; the axiom lines are the repository's own record.\n\nOn human review: two Anthropic mathematicians validated the work, and Brian Conrey and Dan Goldston examined the paper. Examination is not endorsement, and Goldston is an author of the prior work this builds on.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":68,"significanceNote":"The unconditional proportion of zeros on the critical line is the standard yardstick for progress toward the Riemann hypothesis, with a fifty-year record ladder through Levinson and Conrey. Scored above the Jacobian conjecture at 65 because of what it is attached to, and far below the Riemann hypothesis at 100 because a proportion bound is not the hypothesis.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.13637","sourceName":"More than two thirds of the zeros of the Riemann zeta function are simple and on the critical line","collection":null,"collectionVersion":null,"links":[{"label":"Anthropic's announcement","url":"https://www.anthropic.com/research/riemann-zeta","kind":"announcement"},{"label":"The Anthropic PDF the entry was first written from, before the arXiv version","url":"https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf","kind":"paper"},{"label":"Anthropic's informal note stating the proof concisely","url":"https://www-cdn.anthropic.com/23455459f8832d06bb175cc0f88d019aed962ef8.pdf","kind":"paper"},{"label":"Lean repository - audited at tag v1.0; comparator statements under comparator/","url":"https://github.com/anthropics/zeta-23-lean","kind":"lean-proof"},{"label":"How the argument was found: Claude's account of its own two runs","url":"https://www-cdn.anthropic.com/d7f3ecf1d01392d887f8bc974ca187e2a121b1ed.pdf","kind":"other"},{"label":"Annotated subagent transcripts","url":"https://www-cdn.anthropic.com/8a0d1add3c637b858a9a181e98c40e9548c3f44f.pdf","kind":"transcript"}],"relations":[],"submittedBy":"SwiftTapir695","upvotes":28,"downvotes":0,"commentCount":4},{"slug":"makeev-s-conjecture-on-universal-cover","name":"Makeev's conjecture on universal cover","shortName":"Makeev's conjecture","problemNumber":null,"field":"Metric geometry","fieldGroup":"Geometry & topology","statement":"Let $U_n\\subset\\mathbb R^n$ be the dual of the difference polytope of a regular $n$-simplex such that $U_n$ circumscribes a sphere of diameter 1. Then every set of diameter 1 in $\\mathbb R^n$ is covered by a rotated copy of $U_n$.","posedBy":"V. V. Makeev","yearPosed":1994,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Makeev conjectured it to be true for all dimensions. This result disproves it for dimensions 4 and 5. Dimensions 6 and above remain open.","claimIssueNote":null,"solveDate":"2026-08-10","model":"ChatGPT 5.6 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"AI found the counterexample in dimensions 4 and 5 and wrote the paper and code without human mathematical input.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026: the repository is real and unusually substantial - 596 files across four independently-built verification approaches (Magma certificates, interval-arithmetic C++, tensor-polynomial certificates, Python numerics), plus the full prompts and chat transcripts, and the problem-record links verify (Makeev's conjecture in arXiv:math/9809165, Handbook chapter 21). The certificates were not run here. The submitter reports a preliminary check by two students, which is private checking, not independent review - Unreviewed stands.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named 1994 conjecture in metric geometry, recorded in the Handbook of Discrete and Computational Geometry's problem chapter, in the universal-covers tradition of Lebesgue's problem. Real and thirty-two years old, but specialist: level with the named specialist band at 15.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/simplex4/math-results/blob/main/makeev-code/a4/makeev_d4_d5.pdf","sourceName":"Counterexamples to Makeev's regular-simplex conjecture in dimensions four and five","collection":null,"collectionVersion":null,"links":[{"label":"Python and Magma code","url":"https://github.com/simplex4/math-results/tree/main/makeev-code/a4","kind":"code"},{"label":"Conjecture 21.3.2","url":"https://www.csun.edu/~ctoth/Handbook/chap21.pdf","kind":"problem-record"},{"label":"Makeev's conjecture","url":"https://arxiv.org/abs/math/9809165v3","kind":"problem-record"}],"relations":[],"submittedBy":"simplex4","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"talagrand-s-critical-sherrington-kirkpatrick-overlap-conjecture","name":"Talagrand's critical Sherrington-Kirkpatrick overlap conjecture","shortName":"Critical SK overlap","problemNumber":null,"field":"Spin glass theory","fieldGroup":"Mathematical physics","statement":"At the critical inverse temperature $\\beta=1$ in the Sherrington-Kirkpatrick spin glass model, Talagrand conjectured that the expected squared overlap of two independent Gibbs replicas has an exact $N^{-2/3}$ scaling: there exists a constant $a>0$ such that\n\n$$\n\\lim_{N\\to\\infty} N^{2/3}\\mathbb{E}\\langle R_{1,2}^2\\rangle=a.\n$$\n\nDu and Huang prove that this limit exists and is positive and finite. More strongly, they determine the full limiting quenched distribution of the rescaled overlap $N^{1/3}R_{1,2}$ in terms of the reflected $\\mathrm{Airy}_1$ point process.","posedBy":"Michel Talagrand","yearPosed":2011,"ageNote":"Posed in Talagrand's Mean Field Models for Spin Glasses, volume II (2011), as Conjecture 11.7.5, alongside his theorems locating the transition window at width N^(-1/3). Open fifteen years; Dey and Kang still cite it as open in March 2026, five months before this proof.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: $N^{2/3}\\mathbb{E}\\langle R_{1,2}^2\\rangle$ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of $N^{1/3}R_{1,2}$ converges to an explicit random probability measure defined from the reflected $\\mathrm{Airy}_1$ point process. The same limiting distribution and second-moment constant are obtained for the spherical SK model. This does not resolve the broader low-temperature overlap structure of the SK model.","claimIssueNote":null,"solveDate":"2026-08-09","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Hang Du","Brice Huang"],"aiRole":"The authors state that most of the arguments in the paper were generated using GPT-5.6 Pro, with the goal of exploring further consequences of ideas developed in their companion work on critical SK free-energy fluctuations. The paper does not assign individual lemmas or the central comparison principle specifically to the model, so the contribution is classified conservatively as AI co-developed rather than AI discovered.","verification":"unreviewed","verificationNote":"Checked by this site on 14 August 2026 against the paper's LaTeX source and the prior-art source, both fetched and read. Confirmed verbatim in the paper: \"The following corollary affirmatively resolves [Conjecture 11.7.5]\", attached to the corollary giving $\\lim N^{2/3}\\mathbb{E}\\langle R_{1,2}^2\\rangle$ as the mean of an explicit positive random variable built from the reflected $Airy_1$ process. The prior state is independently pinned: Dey and Kang (arXiv:2603.05636, March 2026) quote Conjecture 11.7.5 with exactly the statement submitted here and treat it as open. The AI disclosure was read in full: most formal arguments were initially generated by GPT-5.6 Pro, the principal human inputs are named (identification of the limiting objects, formulation of the sphere-to-cube comparison theorem, and the overall proof strategy), and the technical body separately credits one step to the model by name - the Hilbert-space kernel argument upgrading the Laplace-transform comparison is introduced as \"an idea due to GPT-5.6 Pro\". What was NOT checked here is the mathematics itself: a 36-page argument through the $Airy_1$ scaling limit at the GOE spectral edge needs a spin-glass or random-matrix specialist. The manuscript is five days old, unrefereed, and the authors taking responsibility for correctness is author-side checking, so the tier stays Unreviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A named conjecture of Michel Talagrand from his standard monograph (Mean Field Models for Spin Glasses, vol. II, 2011), open fifteen years and quoted as open in the literature five months before this proof. Critical-point overlap scaling is a celebrated question inside spin-glass theory but specialist territory outside it: a technical conjecture from the book, not one of the field's headline problems. Placed with the strong specialist conjectures at 30, below the field-famous band of Krauth-Mezard and Feige at 35.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.08752","sourceName":"Overlap distribution of the critical Sherrington-Kirkpatrick model","collection":null,"collectionVersion":null,"links":[{"label":"Dey and Kang (arXiv:2603.05636) - quotes Conjecture 11.7.5 as open, five months before this proof","url":"https://arxiv.org/abs/2603.05636","kind":"problem-record"},{"label":"Du and Huang's companion paper on critical SK free-energy fluctuations, whose ideas this work extends","url":"https://arxiv.org/abs/2607.02172","kind":"paper"}],"relations":[],"submittedBy":"HiddenHawk615","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"monotone-vi-pth-order-complexity","name":"pth-Order Oracle Complexity for Monotone Variational Inequalities","shortName":"Monotone VI complexity","problemNumber":null,"field":"Variational inequalities","fieldGroup":"Algorithms & optimization","statement":"Monteiro and Svaiter gave a second-order method for smooth monotone variational inequalities converging at O(T^-1.5), later improved to O(T^-1.75) for the convex-concave minimax subset. Whether the conjectured complexity for general monotone variational inequalities could be improved was open. A large-step inexact Halpern iteration achieves O(T^-2), and O(T^-p) at pth order.","posedBy":"Renato D. C. Monteiro, Benar F. Svaiter","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Improves every prior result for p >= 2 and matches the classical extragradient method at p = 1.","claimIssueNote":null,"solveDate":"2026-08-09","model":"Claude Opus 4.6 and GPT-5.6 Sol","modelMaker":"Anthropic, OpenAI","humanCollaborators":["Lesi Chen","Xinliang Zhang","Hengyu Wang","Chengchang Liu","Yongchao Chen","Jingzhao Zhang"],"aiRole":"The paper records the sequence: an O(T^-(p-1)) rate was obtained with Claude Opus 4.6, and on verifying it the authors conjectured a better O(T^-p) result, for which Xinliang Zhang then found a proof with GPT-5.6 Sol. The results were subsequently verified by the human authors, who also link the model's initial proof as a public ChatGPT transcript.","verification":"unreviewed","verificationNote":"A preprint days old. The initial AI proof is published as a shareable transcript, which is unusual and welcome, but it is a record of provenance rather than a check by anyone independent.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"An explicitly stated open question in the higher-order-methods literature, resting on a well-cited Monteiro-Svaiter framework but confined to optimization theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.08463","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"half-collinear-graviton-vertices-as-weighted-root-cone-enumerators","name":"Matrix-Tree Obstruction for Half-Collinear Graviton Vertices","shortName":"Graviton root cones","problemNumber":null,"field":"Scattering amplitudes; directed spanning trees","fieldGroup":"Mathematical physics","statement":"In the half-collinear single-minus graviton recursion of Guevara, Lupsasca, Skinner, Strominger and Weil, the multipoint vertex weights depend on global cut tests, which blocks a direct matrix-tree formula outside a restricted decay region. The paper's footnote 4 states the obstruction and its conclusion leaves the general simplification to future work. This result identifies those cut tests as exactly positive-flow conditions, making each retarded vertex a weighted enumerator of directed spanning-tree root cones containing a kinematic netflow vector, so the directed Matrix-Tree Theorem applies whenever the feasible trees form a complete arborescence family.","posedBy":"Alfredo Guevara, Alexandru Lupsasca, David Skinner, Andrew Strominger, Kevin Weil","yearPosed":2026,"ageNote":"Posed in March 2026 and answered in August, so this is a five-month-old question rather than a standing one.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Answers the obstruction rather than the whole question: it says exactly when the Matrix-Tree Theorem can be applied and classifies the chambers, and gives a compact five-graviton formula outside the decay region. Simplifying the general solution, which is what the source paper left to future work, remains open.","claimIssueNote":null,"solveDate":"2026-08-09","model":"OpenAI Codex (GPT-5 family)","modelMaker":"OpenAI","humanCollaborators":["James Peebles","James Kehoe"],"aiRole":"From the submission: under human direction, OpenAI Codex proposed the root-cone interpretation, developed the proof, wrote the exact enumeration and verification programs, found the non-decay five-point chamber identity, and drafted the manuscript. A separate model acting as referee reconstructed the published recursion and re-derived the principal claims with fresh code.","verification":"lean-checked","verificationNote":"The conceptual cut, flow and root-cone equivalence is formalized in Lean 4 with no sorry and no custom axioms; principal declarations are guarded by assert_no_sorry with axioms printed, and CI runs lake build --wfail plus leanchecker. Curator check: the linked run completed successfully and the 293-line formalization contains no sorry, admit or native_decide and declares no axioms of its own.\n\nThe label covers the formalized core only. The chamber counts, determinant identities, realizability count and five-point formula are exact-code checked, not Lean-checked. Nobody independent has audited the informal-to-formal correspondence, and no domain expert has endorsed the result. The work is self-published rather than submitted to a venue.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A specific technical obstruction stated in one March 2026 paper, in an active corner of amplitudes research with prominent authors, but only months old and narrow in reach.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/jimpeebles/graviton-root-cones/releases/tag/v0.3.0","sourceName":"Author repository","collection":null,"collectionVersion":null,"links":[{"label":"Repository","url":"https://github.com/jimpeebles/graviton-root-cones","kind":"code"},{"label":"Passing verification workflow","url":"https://github.com/jimpeebles/graviton-root-cones/actions/runs/31339397929","kind":"lean-proof"}],"relations":[],"submittedBy":"PluckyCobra527","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"planar-berenstein-conjecture","name":"The Planar Berenstein Conjecture","shortName":"Planar Berenstein conjecture","problemNumber":null,"field":"Overdetermined boundary problems","fieldGroup":"Analysis","statement":"The unrestricted planar Berenstein conjecture holds that overdetermined Dirichlet-Neumann data characterize the disc. Disproved: a bounded simply connected domain with real-analytic Jordan boundary that is not a disc, carrying a nonzero real eigenfunction with zero Dirichlet data and constant nonzero Neumann data.","posedBy":"Carlos A. Berenstein","yearPosed":1980,"ageNote":"Follows the same group's counterexamples to the planar Pompeiu and Schiffer conjectures; the extension to the Dirichlet endpoint is not formal, since the nonzero Neumann datum keeps the harmonic source modes alive.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The domain has dihedral symmetry of order 26 and is neither a disc nor centrally symmetric, and its eigenfunction changes sign - which is why an additional sign assumption rescues the statement.","claimIssueNote":null,"solveDate":"2026-08-09","model":"ChatGPT 5.6","modelMaker":"OpenAI","humanCollaborators":["Matthew J. Colbrook","Siavash Sadeghi","George Stepaniants"],"aiRole":"The authors declare the role and are careful about its limits: \"We believe it is important to declare the use of AI in mathematical research, and in the present case its role is particularly noteworthy. ChatGPT 5.6 was given a substantial warm start consisting of an early draft and working code independently developed by MJC and GS\" for the earlier Pompeiu-Schiffer paper, material that \"already contained the central construction, conformal fixed-disc formulation, coefficient spaces, disk-polynomial algebra, tail strategy, and computational architecture on which the present paper rests. The system was used to explore the modification\" to the Dirichlet endpoint.","verification":"unreviewed","verificationNote":"A preprint days old. The existence claim is reduced by a Newton-Kantorovich argument to finitely many explicit inequalities certified in interval arithmetic, so the final step is machine-checkable, but nobody independent has rerun it.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A named conjecture in the Pompeiu-Schiffer family of overdetermined problems, known across inverse spectral theory and studied for four decades.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.08953","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kim-roush-permanent-odd-order","name":"The Kim-Roush Conjecture on the Maximum of per(I-A) in Odd Order","shortName":"Kim-Roush permanent conjecture","problemNumber":null,"field":"Matrix theory","fieldGroup":"Algebra","statement":"For the set of $n \\times n$ doubly stochastic matrices, Kim and Roush conjectured in 1981 that for odd $n = 2k+1 > 1$ the maximum of $\\mathrm{per}(I-A)$ equals $3 \\cdot 2^{k-2}$, attained by an explicit block construction. Proved in full, and the maximizers are classified: they are exactly the simultaneous-permutation conjugates of that construction.","posedBy":"Ki Hang Kim, Fred W. Roush","yearPosed":1981,"ageNote":"Open since 1981, and carried forward as an open problem in Minc's survey of permanent problems and in the Cheon-Wanless 2005 update of it.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Kim and Roush did not claim uniqueness; the classification of equality cases is new alongside the conjecture itself.","claimIssueNote":null,"solveDate":"2026-08-09","model":"GPT-5.6 Sol and Claude Fable 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Yair Lavi"],"aiRole":"The acknowledgments are one sentence and leave nothing to interpret: \"The proof of this conjecture was carried out by GPT-5.6-sol and Claude Fable 5, under the guidance of the author. The author has reviewed the resulting proof arguments. Responsibility for the final text rests with the author.\"","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture standing 45 years in the permanent literature, kept alive on Minc's recognized list of open problems involving permanents.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.08933","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"npt-two-copy-distillability-divincenzo-family","name":"Finite-Copy Distillability of NPT States in the DiVincenzo Family","shortName":"Two-copy NPT distillability","problemNumber":null,"field":"Entanglement theory","fieldGroup":"Quantum information & computing","statement":"Whether negative-partial-transpose states undistillable from one copy become distillable from finitely many copies is a basic open problem in entanglement theory. In the canonical two-parameter DiVincenzo family used as its symmetry-reduced testbed, a distinguished one-copy-undistillable state is shown to be two-copy distillable in every local dimension d >= 3, disproving the conjecture that the family's whole one-copy-undistillable region stays undistillable for arbitrarily many copies.","posedBy":"David P. DiVincenzo, Peter W. Shor, John A. Smolin, Barbara M. Terhal, Ashish V. Thapliyal","yearPosed":2000,"ageNote":"The DiVincenzo family was introduced in 2000 precisely as a tractable testbed for the NPT distillability problem, which remains open.","solveType":"disproved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Partial deliberately: the NPT bound entanglement problem itself is untouched. What falls is the conjecture about the canonical family, and the paper is explicit that a substantial neighbouring region remains unresolved while another is known two-copy undistillable.","claimIssueNote":null,"solveDate":"2026-08-09","model":"GPT-5.6 Thinking","modelMaker":"OpenAI","humanCollaborators":["Gelo Noel M. Tabia","Kai-Siang Chen","Min-Hsiu Hsieh"],"aiRole":"The disclosure lists the specific work: \"GNMT acknowledges substantial assistance from OpenAI's ChatGPT, using the GPT-5.6 Thinking model, in the discovery and development of this work. The tool helped with scientific reasoning, the numerical filter search, the derivation of exact two-copy and three-copy certificates, and the organization of the arguments. The interaction was initiated and directed by GNMT.\" An appendix gives a fuller account of the AI-assisted discovery process.","verification":"unreviewed","verificationNote":"A preprint days old. The witnesses are explicit Schmidt-rank-two certificates that the authors independently reconstructed and verified, but no outside party has checked them.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The NPT bound entanglement problem is one of the standing open problems of entanglement theory, known throughout quantum information and carried on its problem lists for 25 years.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.08836","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"polynomial-time-mimo-detection-at-the-maximum-likelihood-threshold","name":"Polynomial-Time MIMO Detection at the ML Threshold","shortName":"MIMO at the ML threshold","problemNumber":null,"field":"Information theory; average-case complexity","fieldGroup":"Theoretical computer science","statement":"In the square Gaussian binary MIMO model $y = \\sqrt{\\rho/N}\\,Hx^\\star + w$, exhaustive maximum-likelihood detection recovers $x^\\star$ once $\\rho > 2\\log N$, while sphere decoding at that threshold scale costs $\\exp\\{\\Theta(N/\\log N)\\}$. Whether any polynomial-time detector reaches the same first-order threshold, or whether a computational-statistical gap separates them, was open. The claim: rounded linear MMSE followed by steepest single-bit descent recovers $x^\\star$ with failure probability tending to zero, uniformly over every transmitted word, in $O(N^3)$ operations.","posedBy":null,"yearPosed":null,"ageNote":"Optimum multiuser detection is NP-hard in the worst case (Verdu, 1989), but the Gaussian channel is more structured. Jalden and Ottersten showed in 2005 that sphere-decoder complexity turns exponential as dimension grows, and the ML block-recovery threshold was identified by Hansen et al. in 2009. This answers what those leave behind - a question the author formulated, not an inherited conjecture.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"An average-case claim about the Gaussian model, not a contradiction of the worst-case NP-hardness of integer least squares. If it holds, no computational-statistical gap separates polynomial-time detection from exhaustive maximum likelihood at first order in this model.","claimIssueNote":null,"solveDate":"2026-08-08","model":"GPT-5.6, Claude Fable 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Dimitris Papailiopoulos"],"aiRole":"The manuscript's footnote in full: \"The results in this paper were proved by GPT-5.6 and Claude Fable 5, which also drafted the initial manuscript. The author posed the problem, directed several rounds of proof simplification, verified all mathematical arguments, edited the manuscript, and takes full responsibility for its content.\" By the author's public account, Claude proposed the algorithm (signed LMMSE plus greedy bit flips) and GPT repaired and simplified the proof over several days of directed iteration.","verification":"unreviewed","verificationNote":"A 46-page manuscript posted to the author's own site and announced on X, days old, with no independent review and no formalization. The author states he checked every argument line by line over about five days, and he has published previously on polynomial-complexity ML detection, so the domain expertise is real; neither fact is independent scrutiny, which is why this is a candidate.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"MIMO detection is a heavily studied problem with fifty years of literature and direct wireless-engineering stakes, and the computational-statistical gap question is a recognized one, but it was not a named conjecture with a standing attribution.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/anadim/anadim.github.io/blob/master/MIMO_Detection.pdf","sourceName":"Author manuscript","collection":null,"collectionVersion":null,"links":[{"label":"X.com","url":"https://x.com/DimitrisPapail/status/2086158118354887060","kind":"discussion"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"amenability-base-field-independence","name":"Whether Amenability of an Algebra Depends on the Ground Field","shortName":"Amenability and the base field","problemNumber":null,"field":"Noncommutative algebra","fieldGroup":"Algebra","statement":"Cornulier asked, in a MathOverflow discussion, whether amenability of a module over an associative algebra depends on the ground field. It does not: the notion is invariant under change of base field.","posedBy":"Yves Cornulier","yearPosed":2015,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The footnote is worth reading on its own: an author saying in print that the model earned coauthorship and that policy is what prevents it.","claimIssueNote":null,"solveDate":"2026-08-08","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Be'eri Greenfeld"],"aiRole":"The abstract states it plainly: \"A significant part of the argument is based on ideas of ChatGPT 5.6 Sol.\" The author goes further in a footnote on the title page: \"While it is currently prohibited by arXiv policy to list AI as a coauthor, the (human) coauthor is confident that ChatGPT's contribution merits an author credit.\"","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A MathOverflow question from a well-known mathematician: documented and real, but never a programme with a literature behind it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.08161","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u","name":"Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$","shortName":"The $U_{30}$ phase-ambiguity question","problemNumber":null,"field":"Finite harmonic analysis / phase retrieval on cyclic groups","fieldGroup":"Analysis","statement":"For rational-valued signals $f,g:C_{30}\\to\\mathbb Q$ with exact Fourier support $U_{30}$, equality of autocorrelations through order five is completely classified. After translating $g$, there are $\\alpha\\in\\mathbb Q(\\zeta_{30})^\\times$ and $z\\in\\mathbb Q(\\zeta_6)^\\times$, with $z\\bar z=1$, such that\n$$\n\\widehat f(u)=\\sigma_u(\\alpha),\\qquad\n\\widehat g(u)=\\sigma_u(z\\alpha)\n$$\nfor every $u\\in U_{30}$. Conversely, every such pair, extended by zero off $U_{30}$, is rational-valued and agrees through order five. Normalized parameters are translation-equivalent exactly modulo $\\mu_6$, and the sixth-order data agree exactly when $z^6=1$.","posedBy":"Aaron Agulnick, Toby Busick-Warner","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support $U_{30}$ and conjectured that it was not a complete classification because it did not use the full field $\\mathbb Q(\\zeta_{30})$. This work proves the complete classification. The larger field enlarges the common amplitude $\\alpha$, while every relative ambiguity remains a norm-one parameter in $\\mathbb Q(\\zeta_6)$. The result is a specialization of a theorem for every exact unit support $U_{6m}$.\n\nThe entry does not claim a complete parametrization for arbitrary supports: on the 255 automorphism-stable supports treated elsewhere in the paper, the broader result is a closing-degree classification. It does not treat noisy data or noncyclic groups, and it makes no novelty, priority, or firstness claim.","claimIssueNote":null,"solveDate":"2026-08-07","model":"GPT-5.6 Sol (Codex)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Following the self created protocol PAPP, that you can find here:https://github.com/aconsciousfractal/Gate-Disciplined-Computational-Mathematics\n GPT-5.6 Sol, operating through Codex under my direction, developed the central relation-lattice reduction, the phase-ratio normal form, and the Galois-descent argument that confines the relative phase to the norm-one torus in $\\mathbb Q(\\zeta_6)$. It also generated exact symbolic replays and the verification package. I selected the problem and scope, controlled the literature and claim boundary, directed repeated adversarial reviews, checked the mathematical outputs, and revised the manuscript after each finding. An earlier AI-agent package supplied preliminary computational observations; the workflow independently rederived and checked them before use.","verification":"unreviewed","verificationNote":"No named independent domain expert has endorsed the theorem. The repository carries exact symbolic checks, fail-closed verification scripts, frozen manifests and an adversarial review report, but those are author-side and agent-side assurance, so this stays Unreviewed and Candidate.\n\nThis site ran its own checks, written from the statement rather than from the repository's scripts. Building a pair from a chosen $\\alpha$ and $z$: both inverse transforms are rational at all 30 points, the support is exactly $U_{30}$, the pair agrees through order five and differs at order six, and replacing $z$ by a sixth root of unity restores order-six agreement - the claimed $z^6=1$ boundary, exactly. The phase lattice was recomputed independently by Smith normal form: $\\mathbb Z^8/L$ has free rank 1 through order five and rank 0 at order six, so a one-parameter ambiguity survives order five and dies at six. And the Agulnick-Busick-Warner pair itself fits the classification - its Fourier ratio is Galois-equivariant with $z+\\bar z=13/7$ and $z\\bar z=1$, so $z=(13\\pm3\\sqrt{-3})/14$ lies in $\\mathbb Q(\\zeta_6)$.\n\nNot checked: completeness itself, which is the novelty. The converse direction, the lattice skeleton and the known example are all consistent with it without establishing it.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":4,"significanceNote":"A support-specific conjecture stated in one April 2026 paper and answered four months later. Precisely posed and genuinely open - Agulnick and Busick-Warner write that they \"conjecture that it is not even a complete classification on this particular support\" - but with no literature behind it and a narrow specialist audience. Level with the other 2026 conjecture-in-a-recent-paper entries at 4.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/blob/b4bbfccdd508caa4a5a6e145abd78ac082195c16/paper/Autocorrelation_Phase_Lattices_on_Cyclic_Groups.pdf","sourceName":"Autocorrelation Phase Lattices on Cyclic Groups: Unit Supports and the 2pq Orbit-Stable Classification","collection":null,"collectionVersion":null,"links":[{"label":"Repository: manuscript source, verification scripts and frozen manifests","url":"https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups","kind":"code"},{"label":"Agulnick and Busick-Warner, Higher-Order Autocorrelations on Finite Abelian Groups (arXiv:2604.13310)","url":"https://arxiv.org/abs/2604.13310","kind":"paper"},{"label":"Gate-Disciplined Computational Mathematics - the author's own working protocol","url":"https://github.com/aconsciousfractal/Gate-Disciplined-Computational-Mathematics","kind":"other"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"anstee-sali-conjecture","name":"The Anstee-Sali Conjecture on Forbidden Configurations","shortName":"Anstee-Sali conjecture","problemNumber":null,"field":"Extremal set theory","fieldGroup":"Combinatorics","statement":"For a forbidden configuration $F$, the Anstee-Sali conjecture predicts that $\\mathrm{forb}(m, F)$ is $\\Theta\\left(m^{X(F)-1}\\right)$, where $X(F)$ comes from an explicit product construction. Disproved: the 4-uniform family on six vertices formed by a two-vertex core joined to the edges of a 4-cycle has $X(F) = 4$, so the conjecture predicts $\\Theta(m^3)$, while a random-alteration argument gives $\\Omega(m^{\\frac{10}{3}})$.","posedBy":"Richard Anstee, Attila Sali","yearPosed":2005,"ageNote":"The organizing conjecture of the forbidden-configurations programme, tracked for two decades in Anstee and Sali's dynamic survey in the Electronic Journal of Combinatorics.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-07","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Pei Wu"],"aiRole":"The abstract ends \"The example was found by GPT-5.6 Sol\", and a dedicated disclosure section repeats it: \"The authors used GPT-5.6 Sol for finding the example. The authors reviewed and revised all outputs, verified results, and take full responsibility for the final manuscript.\"","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The central conjecture of a named research programme with its own long-running dynamic survey, well known inside extremal set theory and invisible outside it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.07646","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"zonoid-volume-log-submodularity","name":"Log-Submodularity of Zonoid Volume","shortName":"Zonoid log-submodularity","problemNumber":null,"field":"Convex geometry","fieldGroup":"Geometry & topology","statement":"The conjecture that volume is log-submodular under Minkowski addition on zonoids, that is $|A||A+B+C| \\leq |A+B||A+C|$. Disproved by a four-dimensional zonotope generated by a 2-modular matrix together with two segments. Several related local mixed-volume, local Loomis-Whitney, projection-volume-ratio and volume-to-surface-area conjectures fall with it.","posedBy":"Stated as Conjecture 4.16 in the zonoid-inequality literature","yearPosed":2023,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The paper also proves the conjecture in the unimodular case and characterizes equality there, so the boundary between true and false is drawn rather than just crossed.","claimIssueNote":null,"solveDate":"2026-08-07","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Ruben Skorupinski"],"aiRole":"The paper has a section called How the counterexample was found. The author proved the unimodular case and identified 2-modular matrices as the place to look; then \"Chat-GPT was then used to search for a counterexample within the space of the 2-modular matrices which led to the discovery of the counterexample within a specific class of 2-modular matrices\". The acknowledgment is more conservative, crediting GPT-5.6 Pro with \"literature searches and exploratory volume computations of 2-modular zonotopes\", all independently verified by the author. Classified on the lower of the two readings.","verification":"unreviewed","verificationNote":"A preprint days old. The counterexample is an explicit finite object and the volumes are exactly computable, so it is checkable by anyone who wants to, but nobody independent has done so.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A recent named conjecture in convex geometry whose failure also takes down several companion inequalities in the same programme.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.07702","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"umans-wang-divisor-conjecture","name":"The Umans-Wang Arithmetic-Progression Divisor Conjecture","shortName":"Umans-Wang divisor conjecture","problemNumber":null,"field":"Combinatorial number theory","fieldGroup":"Number theory","statement":"An $n$-divisor set contains a multiple of every integer from 1 to $n$. Umans and Wang proposed, as the arithmetic-progression form of their Strong $(\\alpha,\\beta)$-Divisor Conjecture, that such a progression exists with few terms of bounded magnitude, which would imply faster algorithms for polynomial and integer factorization. Refuted unconditionally, including its exponent-level relaxation.","posedBy":"Chris Umans, Sheng Wang","yearPosed":2025,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-07","model":"GPT-5.6 Sol (via Codex, reasoning effort ultra)","modelMaker":"OpenAI","humanCollaborators":["Xinjie He","Amit Sahai"],"aiRole":"\"The proof was discovered in an OpenAI Codex run using the gpt-5.6-sol model with reasoning effort set to ultra. Codex also produced the initial write-up. Subsequent human review verified the proof, reviewed the citations, and revised the exposition.\" The authors note the prompting strategy borrowed from the UCLA Moonshot Harness project.","verification":"unreviewed","verificationNote":"A preprint days old with no independent review. The paper states the argument is self-contained, needs no computer-assisted calculation and no access to the model transcript, with the prime number theorem as its only analytic input, so it is checkable on its own terms.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A 2025 conjecture whose truth would have implied faster polynomial and integer factorization, so it carried real algorithmic stakes; young, but posed by Umans and pursued for that consequence.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.06681","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"a-counterexample-to-the-howland-kato-problem-for-positive-commutators","name":"A counterexample to the Howland-Kato problem for positive commutators","shortName":"Howland-Kato problem","problemNumber":null,"field":"Operator theory","fieldGroup":"Analysis","statement":"The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\\arctan(P),\\arctan(Q)]$ is nonzero and nonnegative.","posedBy":"James Howland; Tosio Kato","yearPosed":1991,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-07","model":"Grok 4.5","modelMaker":"xAI","humanCollaborators":["Rupert L. Frank","Paata Ivanisvili"],"aiRole":"The paper discloses only \"The authors acknowledge the use of AI tools. All mathematical arguments and proofs in the final manuscript were checked and written by the authors.\" Co-author Paata Ivanisvili (@PI010101, Professor of Mathematics at UC Irvine) has since said publicly that \"AI deserves a fair amount of credit for finding\" the key identity, and, asked which model: \"Grok 4.5 in Cursor with an agent found a non-symmetric counterexample f(x) = arctan(x/2) and g(x) = tanh(x)/2 + tanh(3x)/2 which works and is correct. However, in the final manuscript we implemented symmetric example.\" So the model found a valid counterexample, but not the symmetric one the paper is built around, and the positivity proof is the authors' own.","verification":"unreviewed","verificationNote":null,"publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A named open problem of Howland and Kato in operator/spectral theory, open since the 1980s and carrying Kato's name, but tracked within one community rather than across mathematics. Placed level with Simon's extendable shellability (25), another decades-old specialist named conjecture settled by counterexample, and below the record instances of household conjectures such as Borsuk N=63 and Hadamard 668 (30). Torn between 25 and 30; rule 3 takes the lower.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/pdf/2608.07805","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Author's announcement (X)","url":"https://x.com/PI010101/status/2087250049000734961","kind":"announcement"},{"label":"Author on Grok 4.5's role (X)","url":"https://x.com/PI010101/status/2087263984433111395","kind":"discussion"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"facial-distance-patterns-planar","name":"Facial Distance Patterns in Planar Graphs","shortName":"Facial distance patterns","problemNumber":null,"field":"Planar graph algorithms","fieldGroup":"Algorithms & optimization","statement":"For a designated face of an undirected unweighted planar graph, how many distinct distance patterns can vertices have? Li and Parter (STOC 2019) proved an upper bound; Mozes, Wallheimer and Weimann conjectured the true answer matches their lower bound. Proved, closing the gap.","posedBy":"Shay Mozes, Benjamin Wallheimer, Oren Weimann (ISAAC 2022)","yearPosed":2022,"ageNote":"Conjectured at ISAAC 2022 against a Li-Parter upper bound from STOC 2019, and listed as open problem 5.6 at a Dagstuhl seminar on metric sketching shortly before it fell.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Three immediate consequences follow for undirected unweighted planar graphs: better compression of the Okamura-Seymour metric, less space for constant-time exact distance oracles, and a faster distributed algorithm.","claimIssueNote":null,"solveDate":"2026-08-07","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Viktor Fredslund-Hansen","Shay Mozes","Oren Weimann"],"aiRole":"\"The simple proof was found by OpenAI's GPT 5.6-Sol model\", and the paper says it came from a single prompt describing the state of the art and asking for any improvement on the upper bound. The authors are candid about what that means: \"It is surprising (not to say embarrasing) that this open problem has such a simple proof, which has eluded the community despite the human efforts invested in it.\" Section 2 of the paper is titled The ChatGPT Proof.","verification":"unreviewed","verificationNote":"A preprint days old with no independent review. The argument is short and self-contained, turning on the observation that a pattern's entries must sum to plus or minus one because the face is a cycle.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A conjecture with a named home in the planar-graph literature, standing since 2022 against a STOC upper bound and current enough to be posed at a Dagstuhl seminar, with downstream consequences for distance oracles.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.07187","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"chen-lawrencenko-conjectures","name":"The Chen-Lawrencenko Conjectures on Cyclic Colorations","shortName":"Chen-Lawrencenko conjectures","problemNumber":null,"field":"Topological graph theory","fieldGroup":"Combinatorics","statement":"A cyclic coloration of a triangulation of a closed 2-manifold gives the faces around every vertex distinct colors. Chen and Lawrencenko made two conjectures about the cyclic chromatic number of minimal triangulations in 1999. Their second is proved here and their first disproved.","posedBy":"Beifang Chen, Serge Lawrencenko","yearPosed":1999,"ageNote":"Posed in 1999. Two further conjectures from the same paper, which its abstract highlighted as the main ones, are left open.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"One conjecture each way: the second proved, the first disproved. Two further Chen-Lawrencenko conjectures remain open and are flagged as such in the paper.","claimIssueNote":null,"solveDate":"2026-08-07","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["John M. Campbell"],"aiRole":"\"We succeed, based on our extensive interactions with GPT-5.6 Pro, in proving and disproving, respectively, Chen and Lawrencenko's second and first conjectures.\" The acknowledgement places the model in the exploratory and proof-development stages, with all suggestions substantially revised, corrected and independently verified by the author.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"Named conjectures standing since 1999 in topological graph theory, with a documented line through Enomoto and Hornak on cyclic colorings; specialist reach.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.06863","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ame-states-five-open-cases","name":"Absolutely Maximally Entangled States in Five Open Cases","shortName":"AME states, five open cases","problemNumber":null,"field":"Quantum error correction / AME states","fieldGroup":"Quantum information & computing","statement":"A pure state of $n$ parties with $q$ levels each is absolutely maximally entangled, written $\\mathrm{AME}(n,q)$, when every subsystem of at most $\\lfloor n/2 \\rfloor$ parties is maximally mixed. These are the perfect tensors, and existence is a parameter-by-parameter problem: some $(n,q)$ admit one, some provably do not, and a maintained table records which cells are still unknown.\n\nThis paper settles five of them. It exhibits Hermitian self-dual MDS codes $[12,6,7]_{25}$, $[18,9,10]_{121}$ and $[18,9,10]_{169}$, from which the stabilizer construction gives $\\mathrm{AME}(12,5)$, $\\mathrm{AME}(18,11)$ and $\\mathrm{AME}(18,13)$, and projecting one party gives $\\mathrm{AME}(17,11)$ and $\\mathrm{AME}(17,13)$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Five existence statements, all by explicit construction: $\\mathrm{AME}(12,5)$, $\\mathrm{AME}(17,11)$, $\\mathrm{AME}(18,11)$, $\\mathrm{AME}(17,13)$ and $\\mathrm{AME}(18,13)$. The $[12,6,7]_{25}$ code came from a direct search with no symmetry imposed; its automorphism group turned out to have a regular $\\mathbb{Z}_3^2$ coordinate orbit, and imposing that translation symmetry on two nine-coordinate orbits collapses an unrestricted $9 \\times 9$ block to a nine-element kernel, which is what made the length-eighteen searches feasible.\n\nThe symmetry is search scaffolding, not part of the proof: the three printed matrices and the two checks suffice on their own. The paper is explicit that the searches were not exhaustive, so it proves existence and classifies nothing - equivalence and classification for these parameters stay open. The length-twelve code is also shown not to be monomially equivalent to a generalized Reed-Solomon code.","claimIssueNote":null,"solveDate":"2026-08-06","model":"Claude Fable 5, ChatGPT 5.6 Sol","modelMaker":"Anthropic, OpenAI","humanCollaborators":["Samuel Bevins","Yunus Bidav"],"aiRole":"From the paper's \"Author contributions and AI use\" section, in the authors' words: under their direction, Claude Fable 5 (Anthropic) and ChatGPT 5.6 Sol (OpenAI) were used extensively throughout the project, including the entire computational search, the development and implementation of search methods, exact verification, bibliographic checks, and manuscript preparation. Both authors independently reviewed the constructions, computations and full text.\n\nThe objects are the result here, and the search that produced them is attributed to the models in full, which is why this is classified AI-discovered rather than assisted. One of the authors reported the paper to this site, noting it was almost entirely AI-generated.","verification":"site-confirmed","verificationNote":"This site re-ran the certificate. The paper's logical content is three printed matrices and two checks on each, so verification means doing the checks again - here with field arithmetic built from the printed minimal polynomials and an independent determinant routine, nothing taken from the authors' code.\n\nAll three matrices satisfy $A\\overline{A}^{\\mathsf T} = -I_k$. Every nonempty square minor is nonzero: 923 for the $6 \\times 6$ block and 48,619 for each $9 \\times 9$, 98,161 in total, matching the counts the paper states. The nine convolution equations that the paper says are equivalent to self-duality for the group-circulant blocks hold. And the Schur square of the length-twelve code has dimension 12, the paper's own argument that it is not monomially equivalent to a generalized Reed-Solomon code, since every GRS $[12,6]$ code has Schur-square dimension at most 11. The field conventions were checked first: Frobenius is an involution and norms land in the base field.\n\nWhat this does not settle is whether the five cases were open. The reachable copy of the Huber-Wyderka table (last updated February 2024) covers local dimensions up to 10, and $\\mathrm{AME}(12,5)$ does sit in its unknown region, but it has no $q=11$ or $q=13$ axis, and the URL the paper cites for a newer version is dead. The paper's prior-art comparisons stand unchecked here.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"Individual cells of the AME existence table, not the general existence problem. That problem is genuinely well known in quantum information - perfect tensors underpin holographic codes and quantum secret sharing, and it appears on published open-problem lists - but any one parameter pair is an ordinary table entry. Above the anchor at 10 for a typical numbered Erdos problem, because the table is actively maintained and consulted and its maintainer engaged with this manuscript. Below the named-conjecture band at 15.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05781","sourceName":"Symmetry-guided constructions of AME states in five open cases","collection":null,"collectionVersion":null,"links":[{"label":"Huber and Wyderka, Table of AME states - the existence table these cells come from","url":"https://tp.nt.uni-siegen.de/ame/ame.html","kind":"problem-record"},{"label":"Grassl, Tables of linear codes and quantum codes","url":"https://www.codetables.de/","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generalized-busemann-petty-2-3","name":"The Generalized Busemann-Petty Problem in Dimensions 2 and 3","shortName":"Generalized Busemann-Petty","problemNumber":null,"field":"Convex geometry","fieldGroup":"Geometry & topology","statement":"If origin-symmetric convex bodies $K, L \\subset \\mathbb{R}^n$ satisfy $\\mathrm{vol}_m(K \\cap E) \\leq \\mathrm{vol}_m(L \\cap E)$ for every $m$-dimensional subspace $E$ with $1 < m < n$, does $\\mathrm{vol}_n(K) \\leq \\mathrm{vol}_n(L)$ follow? Answered affirmatively for subspace dimensions $m = 2$ and $m = 3$.","posedBy":"Herbert Busemann, Clinton Petty (hyperplane case); generalized form standard since","yearPosed":1956,"ageNote":"The original hyperplane case dates to Busemann and Petty in 1956 and was settled over four decades; the lower-dimensional generalization has been the standing open form since.","solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.","claimIssueNote":null,"solveDate":"2026-08-06","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Cheng Lin","Yu-De Liu","Ge Xiong"],"aiRole":"The declaration in full: \"Theorem 2.3 was found with the help of ChatGPT 5.6 Sol, which led us to prove Theorem 3.2.\" One named theorem, which unlocked the paper's crucial representation formula.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A direct descendant of the Busemann-Petty problem, one of the best-known questions of convex geometry, whose hyperplane case took four decades and several landmark papers to settle.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05621","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dittert-s-conjecture-in-dimension-five","name":"Dittert's Conjecture in Dimension Five","shortName":"Dittert n=5","problemNumber":null,"field":"Linear algebra; permanents; exact sum-of-squares certificates","fieldGroup":"Combinatorics","statement":"The dimension-five case asks whether, for every nonnegative $5\\times5$ real matrix $A$ whose entries sum to $5$, the Dittert functional $\\Phi(A)=\\prod_i r_i+\\prod_j c_j-\\operatorname{per}(A)$ is uniquely maximized at $U_5=J_5/5$. The submitted artifact claims the stronger quantitative bound $$\\Phi(A)\\leq \\frac{1226}{625}-\\frac{1}{625}\\lVert A-U_5\\rVert_F^2,$$ which implies uniqueness.","posedBy":"Eberhard Dittert","yearPosed":1983,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Dimension 5 only; public AI-generated candidate with no independent specialist review.","claimIssueNote":null,"solveDate":"2026-08-06","model":"GPT-5.6 Sol (Ultra)","modelMaker":"OpenAI","humanCollaborators":["Arthur Moisés da Costa Borges"],"aiRole":"Operating through OpenAI Codex, GPT-5.6 Sol with the Ultra reasoning-effort setting selected the problem after literature triage, developed the symmetry-reduced sum-of-squares approach, ran numerical discovery and rational recovery, produced the exact certificate and mechanically separate verifier, formalized the quantitative bound and equality characterization in Lean 4, audited the artifacts, and wrote the manuscript. Human mathematical supervision was minimal. Arthur Moisés da Costa Borges defined the broad objective, authorized execution and publication decisions, supplied factual metadata, and maintains the artifact, but did not derive or independently validate its technical content.","verification":"lean-checked","verificationNote":"The public artifact contains a Lean 4.30.0-rc1 formalization of the n=5 quantitative bound and equality characterization, with no sorry, admit, or user-declared axioms. Lean checks the included exact rational SOS witness directly. Large finite equalities use native_decide; the trusted base therefore includes Lean's native compiler and runtime, not the kernel alone. A separate Python/FLINT verifier checks 54/54 orbital identities, 425/425 kernel constraints, and 420/420 positive leading principal minors; deterministic generators reproduce the PSD witness and 41 Lean data modules. No independent specialist has yet checked the informal-to-formal correspondence, historical or novelty claims, or the overall argument. Treat this as a public AI-generated candidate, not an established or peer-reviewed result. Tier: the same system produced both the proof and its Lean formalization, and no independent party has audited the informal-to-formal correspondence.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named 1983 conjecture in permanent/matrix theory with a genuine multi-decade partial-results thread (Sasser, Pang's n>=17, this session's n=16), but a narrow specialist audience within combinatorial matrix theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/arthurborgesdev/dittert-conjecture-n5/tree/91920c401770b98b2050cff4296e602247d4bc24","sourceName":"Public GitHub research artifact (commit 91920c4)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"LuckyMongoose479","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"tuza-conjecture-degree-seven","name":"Tuza's Conjecture for Maximum Degree at Most Seven","shortName":"Tuza, degree seven","problemNumber":null,"field":"Extremal graph theory","fieldGroup":"Combinatorics","statement":"Tuza conjectured that every finite simple graph satisfies $\\tau(G) \\leq 2\\nu(G)$, where $\\nu$ counts pairwise edge-disjoint triangles and $\\tau$ is the fewest edges whose deletion leaves the graph triangle-free. Puleo had proved it for maximum average degree below 7. Proved here for every graph of maximum degree at most seven, crossing the equality boundary of Puleo's sparsity theorem.","posedBy":"Zsolt Tuza","yearPosed":1981,"ageNote":"Open since 1981 in general, with a long ladder of special cases; this adds the maximum-degree-seven class and shows Puleo's sparsity boundary can be crossed.","solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Settles a class, not the conjecture: Tuza's conjecture remains open in general.","claimIssueNote":null,"solveDate":"2026-08-06","model":"Claude Code (Claude 5 family), OpenAI Codex (GPT-5.6 family)","modelMaker":"Anthropic, OpenAI","humanCollaborators":["Anish Gupta"],"aiRole":"Both were \"used extensively for proof exploration, software development, exact computational checks, literature discovery, and drafting and editing the manuscript\", with the author selecting the arguments and methods and checking the sources and computations. Broad rather than step-attributed, so the lower tier applies.","verification":"unreviewed","verificationNote":"A preprint days old with no independent review. A certificate catalogue and verification programs ship as ancillary files and in a companion repository, so the computational part is reproducible.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Tuza's conjecture is one of the well-known open problems of extremal graph theory, unresolved since 1981 and attacked continuously through a documented sequence of special cases.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.06538","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fair-and-efficient-balanced-allocations-for-additive-valuations","name":"Balanced EF1 and fPO Allocations","shortName":"Balanced EF1 + fPO","problemNumber":null,"field":"Fair division; algorithmic game theory","fieldGroup":"Algorithms & optimization","statement":"Does every instance of indivisible goods with additive valuations admit a balanced allocation (any two bundles differing in size by at most one) that is simultaneously envy-free up to one good (EF1) and fractionally Pareto optimal (fPO)? Kawase et al. established existence only for personalized bivalued valuations or at most two valuation types. Proved in general, via the Knaster-Kuratowski-Mazurkiewicz lemma applied to a weighted-welfare duality framework plus a new price interlacing lemma.","posedBy":"Kawase et al.","yearPosed":2026,"ageNote":"The unconstrained EF1-plus-fPO guarantee goes back to Caragiannis et al.; the balanced version is the most restrictive open case of the matroid-constrained question, posed by Kawase et al. earlier in 2026.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The paper records how the collaboration actually went: the authors first aimed at a counterexample showing EF1 and PO incompatible under matroid constraints, and when the model surfaced fundamental difficulties with that plan they redirected toward proving the positive result instead. The paper also extends the technique to category constraints and leaves a pseudopolynomial-time algorithm open.","claimIssueNote":null,"solveDate":"2026-08-06","model":"GPT-5.6-Sol (via OpenAI Codex), Claude Fable 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Benjamin Cookson","Nisarg Shah","Paritosh Verma"],"aiRole":"The acknowledgement is unusually specific: \"All the mathematical proofs and counterexamples were derived by OpenAI Codex (GPT-5.6-Sol at Max effort) based on research directions, literature connections, proof and search strategies, and inspirations supplied by the authors.\" The authors verified every detail and simplified the exposition with both models, and retain responsibility.","verification":"unreviewed","verificationNote":"An arXiv preprint days old. The authors state they verified all the mathematics themselves, which is not independent review, and no formalization is reported.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A specific open case from a 2026 paper, inside the heavily worked EF1-and-fPO literature of fair division; recent and specialist, but precisely stated and actively pursued.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.06325","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"a-counterexample-to-the-inverse-generator-problem-and-related-questions","name":"The Inverse Generator Problem on Hilbert Spaces","shortName":"Inverse generator problem","problemNumber":null,"field":"Semigroup theory","fieldGroup":"Analysis","statement":"If $A$ generates a bounded $C_0$-semigroup on a Hilbert space and has dense range, does $A^{-1}$ also generate a bounded $C_0$-semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator with dense range generating a bounded, strongly stable semigroup whose inverse generates no $C_0$-semigroup at all. The counterexamples come from one explicit finite-dimensional construction, using bases of $\\mathbb{C}^{2n}$ with uniformly bounded partial-sum projections but unconditionality constants growing like $n^\\alpha$.","posedBy":"Ralph deLaubenfels","yearPosed":1988,"ageNote":"Posed by deLaubenfels in 1988, with a documented ladder since: positive for sectorial operators of angle below π/2, positive on Hilbert spaces for contractive semigroups by Lumer-Phillips, counterexamples on Banach spaces back to Komatsu, and Gomilko-Zwart-Tomilov producing Hilbert-space generators with slow unbounded growth. The bounded-semigroup Hilbert case stayed open until this paper.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"One finite-dimensional construction settles several related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time. Version 2 adds Theorem 1.4, whose part (i) solves Question 6.1 of Chalmoukis, Tsikalas and Yakubovich; that question is tracked as its own entry.","claimIssueNote":null,"solveDate":"2026-08-06","model":"ChatGPT 5.6 Pro, Claude Fable","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Emiel Lorist","Martin Meyries","Mark Veraar"],"aiRole":"ChatGPT 5.6 Pro by OpenAI was used to explore Schauder basis counterexamples to the inverse generator problem. It also assisted the adaptation of Ansorena's construction used to obtain the explicit Schauder basis in Proposition 2.1, and was used to optimize the explicit constants in Theorem 1.1 and Proposition 2.1 by repeatedly dissecting the estimates and searching for numerical improvements.\n\nIn version 3, the authors additionally provide a complete Lean 4 formalization of Theorem 1.1. This formalization was produced using Claude Fable by Anthropic. The formalized theorem is the finite-dimensional construction underlying the subsequent counterexamples. The paper attributes exploration, assistance, optimization and formalization to the models rather than the central mathematical construction itself, so the AI contribution remains classified as AI-assisted.","verification":"unreviewed","verificationNote":"Version 3 has a complete Lean 4 machine-checked proof of Theorem 1.1, the explicit finite-dimensional construction underlying the counterexamples. The accompanying repository states that the development is self-contained on top of Mathlib, contains no sorry, no native_decide and no numerical or floating-point proof steps, and that the final theorem depends only on Lean's standard axioms propext, Classical.choice and Quot.sound.\n\nHowever, the Lean development formalizes Theorem 1.1 rather than the infinite-dimensional inverse-generator conclusion of Theorem 1.2. In particular, the passage assembling the finite-dimensional blocks into the Hilbert-space counterexample has not been formally verified or independently expert-reviewed. The entry should therefore remain Unreviewed rather than being upgraded to Lean-checked or Lean-verified.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A named 1988 problem of semigroup theory with a 38-year ladder of partial results, motivated from numerical analysis, control theory and functional calculus, but read within those communities only.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.06272","sourceName":"arXiv:2608.06272 - A solution to the inverse generator problem and related questions","collection":null,"collectionVersion":null,"links":[{"label":"Lean 4 formalization repository","url":"https://github.com/zoowirt/inverse-generator-lean","kind":"lean-proof"}],"relations":[],"submittedBy":"RustyKestrel290","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"planar-spin-systems-approximation","name":"Approximate Counting for Spin Systems on Planar Graphs","shortName":"Planar spin systems","problemNumber":null,"field":"Approximate counting","fieldGroup":"Theoretical computer science","statement":"Does planarity help approximate counting? The paper gives an FPRAS for the planar hard-core partition function at small activity, proves that approximately counting $q$-colourings on planar graphs is NP-hard for every constant $q \\geq 4$, and completely characterizes when an FPRAS exists for 2-spin systems on planar graphs at small external field.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-06","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Heng Guo","Xinyuan Zhang"],"aiRole":"\"The main ideas of all proofs in this paper were found by GPT-5.6 Sol Ultra. For consistency with standard mathematical exposition, the words we and our are used throughout the paper, including when presenting ideas that originate in the output of the model. The authors simplified, streamlined, and wrote all of the proofs.\" The paper singles out finding the right problem to reduce from as where the model was particularly helpful.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"Settles the planar case of computational phase transitions in approximate counting, a well-worked programme in theoretical computer science, though not a single named conjecture.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.06172","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"central-limit-theorem-for-the-random-assignment-problem","name":"Central limit theorem for the random assignment problem","shortName":"CLT for random assignment","problemNumber":null,"field":"Random combinatorial optimization","fieldGroup":"Probability & statistics","statement":"Let $C_n$ be the minimum cost of a perfect matching in an $n\\times n$ matrix of independent uniform random variables. Aldous proved in 1992 that $\\mathbb{E}[C_n]$ converges, later identifying the limit as $\\zeta(2)$ via the Poisson-weighted infinite tree; Parisi's exact finite-$n$ formula for exponential costs was then proved by Linusson-Wästlund and independently by Nair, Prabhakar and Sharma. The fluctuations resisted. Talagrand applied product-space concentration, Wästlund computed the exponential model's variance as $4\\zeta(2)-4\\zeta(3)+O(n^{-2})$, and Chatterjee proved an order-$n^{-1/2}$ lower bound under tail hypotheses that exclude the bounded uniform law - but no central limit theorem for $C_n$ was known. This paper claims one: $\\sqrt{n}\\,(C_n-\\zeta(2)) \\Rightarrow \\mathcal{N}(0,\\,4\\zeta(2)-4\\zeta(3))$.","posedBy":null,"yearPosed":1992,"ageNote":"No single proposer and no formal posing date: the central limit theorem is the natural question left once the mean is known, not a conjecture anyone stated, and the paper names nobody. Dated from Aldous's 1992 convergence theorem for the mean, which is where the submission's own framing starts and the point from which the fluctuation question was there to ask.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Claims the central limit theorem for the bipartite random assignment problem with bounded uniform costs: $\\sqrt{n}\\,(C_n-\\zeta(2)) \\Rightarrow \\mathcal{N}(0,\\,4\\zeta(2)-4\\zeta(3))$. The limiting constant is not itself new - Wästlund had computed exactly $4\\zeta(2)-4\\zeta(3)$ for the mean-one exponential model, and Malatesta, Parisi and Sicuro derived the non-bipartite analogue by replicas - but neither is a proof for the bounded bipartite model, and Wästlund's zero-free-disk conjecture, which would imply a Gaussian limit, remains open. So the value was expected; the proof of convergence to it is what is claimed. The route is an exact change of variables on an optimal dual potential, after which the residual dependence is a single directed-tree factor whose matrix-tree determinant becomes triangular once the potentials are ordered.","claimIssueNote":null,"solveDate":"2026-08-06","model":"ChatGPT 5.6 and Opus 5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Gilles Mordant"],"aiRole":"The paper carries its AI disclosure as section 1.1, before the mathematics rather than buried after it, and it is precise enough to be worth quoting rather than paraphrasing: \"This proof is not a one-prompt exploit: I have been working for quite some time on optimal transport and matching problems. I somehow forced the AI to help me explore a geometric intuition that I had come up with a few months ago, even before the models reached their current level. Funnily, during the interaction, I had to force the AI not to drift to attempts involving the Stein method and force it to stick to my ideas. AI was then used to complete the proofs, catch mistakes and verify the paper (both via numerical simulations and general 'thinking'), as well as to improve the exposition. The models ChatGPT 5.6 and Opus 5 (as well as previous versions) were used.\" The conceptual core is claimed by the author and the steering was his, including steering the model away from a wrong direction; completing the proofs is substantive mathematics, which is why this sits at co-developed rather than assisted.","verification":"unreviewed","verificationNote":"An arXiv preprint (v1, 5 August 2026), unrefereed and with no independent endorsement. No mathematics was checked here, and there is nothing mechanical to check it against: a long probabilistic argument with no formalization and no computational certificate. What was verified on 24 August 2026: the paper exists at arXiv:2608.05123, its title, author and limiting variance match this entry, and its AI disclosure is genuine, first-party and quoted above in full. Its history section was read to establish novelty - it surveys Kurtzberg, Walkup, Karp, Aldous, Linusson-Wästlund, Nair-Prabhakar-Sharma, Talagrand, Wästlund, Chatterjee and Cao, and states that none of the first-order, concentration, exact-moment or replica results supplies the central limit theorem for bounded uniform costs. That is the author's own characterization of what was open, recorded as such; no independent literature search was run here.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"The random assignment problem is a celebrated line - Mézard and Parisi's replica prediction of $\\zeta(2)$, Parisi's exact finite-$n$ conjecture, Aldous's PWIT proof, then Linusson-Wästlund and Nair-Prabhakar-Sharma - but this entry is the fluctuation question, a rung below that headline result in fame. Above the 30-cluster (Matrix Spencer, Ellipsoid Fitting) because Chatterjee, Cao and Wästlund each attacked it by name without closing it; below Feige and Krauth-Mezard at 35, which are named conjectures in their own right.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.48550/arXiv.2608.05123","sourceName":"A central limit theorem for the random assignment problem","collection":null,"collectionVersion":null,"links":[{"label":"Aldous, The zeta(2) limit in the random assignment problem","url":"https://arxiv.org/abs/math/0010063","kind":"problem-record"}],"relations":[],"submittedBy":"SpryRaven345","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"babai-minimal-cayley-chromatic","name":"Babai's Minimal Cayley Graph Problem","shortName":"Babai minimal Cayley","problemNumber":null,"field":"Algebraic graph theory","fieldGroup":"Combinatorics","statement":"A Cayley graph is minimal when no proper subset of its connection set generates the group. Babai asked whether minimal Cayley graphs have bounded chromatic number. Resolved negatively: finite minimal Cayley graphs exist with arbitrarily large chromatic number.","posedBy":"László Babai","yearPosed":1978,"ageNote":"Posed by Babai in 1978. García-Marco and Knauer had shown in 2025 that minimal Cayley graphs of finitely generated generalised dihedral and nilpotent groups are boundedly colourable, which sharpened the general question this answers.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-06","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["James Davies","Meike Hatzel","Liana Yepremyan"],"aiRole":"The paper's statement of AI use: \"An initial proof was found by ChatGPT 5.6 Sol given [DHY24] in the input. Substantial parts of Section 2 originate from an early draft created in interaction with ChatGPT 5.6 Sol, which was subsequently edited and improved by the authors.\" The model found the first proof, given one of the authors' own earlier papers as context.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A question of Babai's standing open since 1978, in the algebraic graph theory his name anchors, with recent partial results narrowing it just before it fell.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.06254","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"huang-jiang-oblomkov-a3-layer","name":"The Huang-Jiang-Oblomkov Conjecture at a = 3","shortName":"HJO conjecture, a=3","problemNumber":null,"field":"q-series; arithmetic geometry","fieldGroup":"Number theory","statement":"Huang, Jiang and Oblomkov conjectured that the Eulerian $q$-series counting commuting pairs of nilpotent matrices with $X^a = Y^b$ equals an explicit theta-and-eta product, making the point count essentially modular. The conjecture is layered in $a$; the $a = 2$ layer is classical, including Rogers-Ramanujan and Andrews-Gordon. Nothing was known for $a = 3$. That layer is now proved in full, yielding a new infinite family of Rogers-Ramanujan identities and a geometric origin for Warnaar's products.","posedBy":"Huang, Jiang, Oblomkov","yearPosed":2026,"ageNote":"The paper posing the conjecture was posted 11 days after the conjecture proof was published","solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Proves the a = 3 layer; the conjecture is layered in a and remains open for larger a.","claimIssueNote":null,"solveDate":"2026-08-06","model":"AxiomProver","modelMaker":"Axiom Math","humanCollaborators":["Kenny Lau","Ken Ono"],"aiRole":"The mathematics is the authors'; AxiomProver supplied the formal certificate. \"AxiomProver, an AI system currently under development, was used to generate this certificate. The system verified these results in Lean assuming existing literature.\"","verification":"lean-checked","verificationNote":"The Lean certificate is explicitly conditional, verifying the new identities assuming results from the existing literature rather than from first principles, and the system that produced it also produced the formal statements. Public at the AxiomMath repository.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A layered conjecture whose lower layer is the classical Rogers-Ramanujan territory; the new layer produces genuinely new identities, but the conjecture itself is recent and specialist.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05480","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"AxiomProver Lean certificate","url":"https://github.com/AxiomMath/RR_a3","kind":"lean-proof"},{"label":"Huang, Jiang & Oblomkov 2026 - Problem statement","url":"https://arxiv.org/abs/2608.16086","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"the-gardner-transition-in-the-ising-pure-p-spin-glass","name":"The Gardner Transition in the Ising pure $p$-spin glass","shortName":"The Gardner Transition in the Ising pure $p$-spin glass","problemNumber":null,"field":"Spin glasses; probability","fieldGroup":"Mathematical physics","statement":"For the Ising pure $p$-spin glass with $p\\ge3$, Gardner predicted in 1985 that the Parisi measure passes through two transitions as the inverse temperature $\\beta$ grows: replica symmetric (RS), then one-step replica symmetry breaking (1-RSB), then full replica symmetry breaking (FRSB). The author's earlier paper established the RS phase for $0<\\beta\\le\\beta_1^p$ and the 1-RSB phase on a nonempty interval immediately above $\\beta_1^p$, leaving the rest of the phase diagram open. This sequel claims the remainder: a unique second critical inverse temperature $\\beta_2^p>\\beta_1^p$, with the measure 1-RSB throughout $\\beta_1^p<\\beta\\le\\beta_2^p$, and for $\\beta>\\beta_2^p$ supported on $\\{0\\}\\cup[\\underline q,\\overline q]$ with a smooth density on the interior, hence FRSB.","posedBy":"Elizabeth Gardner","yearPosed":1985,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"For the Ising pure $p$-spin glass where $p \\geq 3$, it was predicted by Gardner that there exists critical inverse temperatures $0<\\beta_1^p<\\beta_2^p <\\infty$ such that: (1) When $0<\\beta\\leq \\beta_1^p$, the Parisi measure is replica symmetric (RS); (2) When $\\beta_1^p<\\beta \\leq \\beta_2^p$, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When $\\beta>\\beta_p^2$, the Parisi measure is full replica symmetry breaking (FRSB). The earlier work by the author solved Part (1) and partially solved Part (2) when $\\beta$ is sufficiently close to $\\beta_p^1$, while this work solves Part (2) and Part (3) entirely.","claimIssueNote":"The author states in the paper's acknowledgments that the appendix proofs \"were drafted by large language models and have not yet received their final authorial revision\", and that he will \"verify, revise, and rewrite these proofs in a subsequent version\". Those appendices are where the theorem's weight sits: of 166 pages roughly 11 are main body and 155 are appendices A-G, and the main body defers its key inputs to them explicitly (\"Its full proof is included in Appendix B\", \"Its complete proof is included in Appendix C\"). So the load-bearing proofs are, by the author's own account, not yet checked by anyone - not by him, not by a referee, and not by a machine. That is unusually candid and it is why this is filed as a candidate rather than resolved.","solveDate":"2026-08-06","model":"not explicitly stated","modelMaker":null,"humanCollaborators":["Yuxin Zhou"],"aiRole":"It is stated that the proofs in the appendix were drafted by large language models and have not yet received their final authorial revision. Since the materials in the appendix are the heart of the proof (the 10 page main paper only contains introduction and statement of the result), it is classified as AI-discovered rather than AIco-developed.","verification":"unreviewed","verificationNote":"Nobody has checked this, including the author, who says so himself - see the claim-issue note. An arXiv preprint (v1, 6 August 2026, math.PR), unrefereed, with no formalization and no computational certificate, so there is nothing mechanical to check either. Verified here on 25 August 2026: the paper exists at arXiv:2608.06523 with the title, sole author and phase-diagram statement this entry describes; its acknowledgment carries the LLM-drafting disclosure quoted verbatim above; its sequel relationship to the author's arXiv:2408.14630 is as described; and the page structure supports the claim that the appendices carry the substance. No model is named anywhere in the paper, which is why the model field says so rather than guessing.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":28,"significanceNote":"A 1985 prediction of Elizabeth Gardner, forty-one years standing, inside the Parisi replica-symmetry-breaking picture that took the 2021 Nobel - so the surrounding theory is celebrated even though this particular phase diagram is specialist. Above the FullRSB jamming identity at 20 for the age and the named-prediction status, level with the ellipsoid fitting conjecture at 30 and a little below it, and well below the Krauth-Mezard perceptron capacity at 35, which is the more famous single question in this corner.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.48550/arXiv.2608.06523","sourceName":"On the Gardner Transition in the Ising Pure $p$-Spin Glass II","collection":null,"collectionVersion":null,"links":[{"label":"Zhou, On the Gardner transition in the Ising pure p-spin glass (paper I, the RS and partial 1-RSB phases)","url":"https://arxiv.org/abs/2408.14630","kind":"paper"}],"relations":[],"submittedBy":"SpryRaven345","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"sendov-s-conjecture","name":"Sendov's Conjecture","shortName":"Sendov's Conj.","problemNumber":null,"field":"Complex analysis","fieldGroup":"Analysis","statement":"Let $p$ be a complex polynomial of degree $n \\ge 2$ whose zeros all lie in the closed unit disk. Then for every zero $a$ of $p$, there exists a critical point $\\zeta$ of $p$ such that $|\\zeta-a| \\le 1$.\n\nThis is the standard Sendov statement and exactly matches the theorem Mazur formalized.","posedBy":"Blagovest Sendov","yearPosed":1959,"ageNote":"Sendov described the conjecture to Nikola Obreshkov in 1959, and it was misattributed to Ljubomir Iliev in 1967; sources variously date it 1958, 1959 or 1962. Open for 67 years. Degrees below nine were settled between 1969 and 1999 (Meir-Sharma, Brown, Borcea, Brown-Xiang), and Tao proved it for all sufficiently high degrees in 2020, leaving the unbounded middle range that this result closes.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Sendov's conjecture is resolved for every degree n >= 2, closing a gap that had stood since 1959: degrees up to eight were settled piecemeal between 1969 and 1999, and Tao's 2020 result covered all sufficiently large degrees without ever specifying the threshold, leaving the middle range open. Tao's digestion establishes the stronger interior form of the statement, which resolves the Phelps-Rodriguez conjecture in full generality as a consequence - a second conjecture falling out of the same argument, and one that likely merits its own entry. Two independent Lean developments now exist: Mazur's original at roughly 90,000 lines and Tao's streamlined version at about 15,000.","claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Lech Mazur"],"aiRole":"GPT-5.6 Pro contributed substantially to the discovery and derivation of the proof, including mathematical exploration, proof development, exact computational testing, and adversarial auditing. Lech Mazur directed the research workflow, selected and reconciled model outputs, and authored the resulting manuscript. A separate Lean 4 development proves the exact statement of Sendov's conjecture.","verification":"lean-verified","verificationNote":"Independently verified twice over, and this site audited the formal artifact itself on 13 August 2026. The decisive external check is Terence Tao's post of 12 August 2026, \"A digestion of the proof of Sendov's conjecture\": he writes that \"Lech Mazur was able to use an AI tool to resolve Sendov's conjecture for all $n \\ge 2$\", reports formalizing the whole argument in Lean himself at about 15,000 lines against the original's roughly 90,000, and concludes that it resolves both the Sendov and Phelps-Rodriguez conjectures in full generality. Tao proved the large-degree case in 2020, so this is expert verification by the person best placed to give it, and it carries the tier. Separately, this site audited Mazur's Lean package. SendovConjecture in Sendov/Statement.lean is exactly the conjecture, correctly quantified and shadowed nowhere. Across all 1,160 first-party files there are zero sorry, zero admit, zero custom axiom declarations and - the one that matters for an autonomous prover - zero native_decide; the 1,117 decide calls are kernel-checked, and the axiom profile is exactly propext, Classical.choice and Quot.sound. All 1,160 file hashes match the published evidence record byte for byte. What could not be checked is the build: the bundle ships no lakefile or manifest and excludes Mathlib, so it cannot be recompiled as distributed, a gap ProofAtlas's own evidence file is candid about. This entry rests not on that internal status but on Tao's independent digestion.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":4,"renownNote":"Wikipedia articles in English, German, French and Yoruba. The article still described the conjecture as open in general when checked on 13 August 2026, the day after Tao's digestion appeared.","significance":40,"significanceNote":"A named conjecture from 1959 with a Wikipedia article in four languages, a 67-year history of partial results, and enough standing that Terence Tao wrote a paper on the large-degree case in 2020 and a full digestion of the solution in 2026. Placed level with the Erdos unit distance problem: a genuinely famous problem within its area and recognisable outside it, below the household conjectures such as cycle double cover (55) and well above the specialist named conjectures around 25 to 30. The Phelps-Rodriguez corollary adds to the case rather than being scored here.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.proofatlas.ai/papers/sendov-conjecture/SENDOV_CONJECTURE_PROOF_AUGUST_5_2026.pdf","sourceName":"A Computer-Assisted Proof of Sendov's Conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Terence Tao, A digestion of the proof of Sendov's conjecture (12 Aug 2026)","url":"https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/","kind":"independent"},{"label":"ProofAtlas formalization page: exact theorem, evidence and build record","url":"https://proofatlas.ai/formalizations/sendov-conjecture/","kind":"lean-proof"},{"label":"Mazur's Lean package, checked source bundle (1,160 files, ~93k lines)","url":"https://proofatlas.ai/papers/sendov-conjecture/SENDOV_CONJECTURE_PROOF_PUBLIC_BUNDLE_2026-08-05.zip","kind":"code"},{"label":"Tao, Sendov's conjecture for sufficiently high degree polynomials (2020)","url":"https://arxiv.org/abs/2012.04125","kind":"paper"},{"label":"Wikipedia: Sendov's conjecture","url":"https://en.wikipedia.org/wiki/Sendov%27s_conjecture","kind":"wikipedia"}],"relations":[],"submittedBy":"HiddenHawk615","upvotes":3,"downvotes":0,"commentCount":0},{"slug":"nazarov-truncation-conjecture","name":"Nazarov's Conjecture on Truncations for Fractional Laplacians","shortName":"Nazarov truncation","problemNumber":null,"field":"Nonlocal operators","fieldGroup":"Analysis","statement":"Nazarov conjectured that for $s \\in (1, 3/2)$ the quadratic form of the spectral fractional Dirichlet Laplacian strictly increases under $u \\mapsto |u|$ when $u$ changes sign. Proved and substantially generalized, with the same conclusion for the restricted form.","posedBy":"Alexander I. Nazarov","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-05","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Egor Ignatev","Alexander I. Nazarov","Pavel Nichitenko","Artur Tursunbaev"],"aiRole":"From the paper: \"The original proof of Corollary 3 via Lemma 5 (for m = 1) was given by the LLM Claude (Anthropic), which was directed jointly by E.I., P.N., and A.T.\" A named corollary, with the human direction credited by initials.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A 2021 conjecture from a remark in Nazarov's own paper, here proved with Nazarov among the authors; precisely stated but young and narrow.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05433","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pseudo-telepathy-maximally-entangled","name":"Completeness of Maximally Entangled States for Pseudo-Telepathy","shortName":"Pseudo-telepathy completeness","problemNumber":null,"field":"Quantum nonlocality","fieldGroup":"Quantum information & computing","statement":"Does every nonlocal game admitting a perfect entangled strategy admit one using a maximally entangled state? Described in the paper as one of the longstanding open problems in quantum nonlocality. Answered negatively by an explicit counterexample game.","posedBy":"Laura Mančinska","yearPosed":2015,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-05","model":"Codex","modelMaker":"OpenAI","humanCollaborators":["Olivier Lalonde"],"aiRole":"The author is emphatic that the ideas are his: \"AI tools were used extensively during the course of this work, but only as a means of executing ideas rather than producing them.\" What Codex did do is decisive rather than cosmetic: it searched for the matrix making the game classically infeasible, implemented the pruning, and implemented and ran a strengthened tracial NPA hierarchy that demonstrated the nonexistence of the required tracial state, completing the proof of Theorem 1.1 and extracting a rational infeasibility certificate.","verification":"unreviewed","verificationNote":"A preprint days old. The decisive step is a semidefinite-programming infeasibility certificate, machine-produced and rational, so it is checkable in principle; nobody independent has checked it.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"Characterized in the paper as a longstanding open problem of quantum nonlocality, bearing on how much maximal entanglement really buys in nonlocal games.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05378","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Mančinska 2015 - Problem statement","url":"https://arxiv.org/abs/1506.07080","kind":"problem-record"},{"label":"GitHub repo of the disproof","url":"https://github.com/lalondeo/MaximallyEntangledIncomplete","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gabor-frames-totally-positive","name":"Gabor Frames of Totally Positive Functions","shortName":"Totally positive Gabor frames","problemNumber":null,"field":"Time-frequency analysis","fieldGroup":"Analysis","statement":"For which lattice parameters does a totally positive window function generate a Gabor frame? Gröchenig and Stöckler initiated the program in 2013; this paper gives the complete characterization, together with a Kadets-type theorem for shift-invariant spaces.","posedBy":"Karlheinz Gröchenig, Joachim Stöckler","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.4","modelMaker":"OpenAI","humanCollaborators":["Jaume de Dios Pont","Karlheinz Gröchenig","Lukas Liehr","Irina Shafkulovska","Mitchell A. Taylor"],"aiRole":"GPT-5.4 surveyed the limit-operator literature and suggested the connection that led the authors to Seidel's work, from which the proof of Theorem 3.3 was adapted; it also suggested simplifications including a simpler perturbation sequence in Lemma 4.4. Codex 5.5 and Claude Opus 4.7 assisted with the Lean formalization. Gröchenig, who posed the program, is among the authors.","verification":"lean-checked","verificationNote":"The paper carries a Lean formalization written with Codex 5.5 and Claude Opus 4.7 assistance; all arguments and formalizations were independently checked by the authors. No external review yet. Tier: the Lean formalization was written with model assistance and checked by the authors themselves; author checking is not independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The central classification program of time-frequency analysis since Gröchenig-Stöckler 2013, with a sustained literature, little known outside harmonic analysis.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.04992","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"divisible-rank-three-case-of-the-kajitani-ueno-miyano-conjecture","name":"The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture","shortName":"Divisible rank-three KUM","problemNumber":null,"field":"Matroid theory","fieldGroup":"Combinatorics","statement":"The Kajitani–Ueno–Miyano conjecture asserts that every finite uniformly dense matroid has a cyclic basis ordering.\n\nThe conjecture is proved for all matroids of rank three. The new result establishes the previously unresolved divisible case, where the ground-set size is a multiple of three, without assumptions of simplicity, representability or paving. Together with the previously published coprime-case theorem of van den Heuvel and Thomassé, this covers every finite uniformly dense rank-three matroid.\n\nThe unrestricted conjecture remains open in higher rank.","posedBy":"Kajitani, Ueno and Miyano","yearPosed":1988,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Proves the divisible rank-three case: rank exactly three and ground-set size a multiple of three, with no restriction to simple, paving, representable or graphic matroids. The part not previously in the literature is the non-simple sub-case, since McGuinness had settled all paving matroids and a rank-three matroid is paving exactly when it has no parallel pairs. Combined with the coprime-case theorem of van den Heuvel and Thomasse, this covers every finite uniformly dense rank-three matroid. The conjecture remains open in higher rank.","claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.6 Sol; Claude Opus 5 (for some Lean formalization and paper write-up)","modelMaker":"OpenAI; Anthropic","humanCollaborators":[],"aiRole":"Under human direction, GPT-5.6 Sol developed the central mathematical argument, including the tight-set reduction, the universal two-gap insertion theorem, and the deletion-and-induction treatment of the strictly dense case.\n\nGPT-5.6 Sol and Claude Opus 5 then collaboratively produced the Lean 4 formalization and the accompanying mathematical paper. Human oversight directed the project, selected and evaluated proof directions, coordinated the formal verification, and checked the scope and relation to the existing literature.","verification":"lean-verified","verificationNote":"The new divisible rank-three theorem is formalized end to end in Lean 4. The repository builds successfully with 3,046 jobs, zero errors and zero warnings. The principal theorem contains no sorry, admit, custom axiom declaration, unsafe declaration or use of native_decide; Lean reports only the standard Mathlib axioms propext, Classical.choice and Quot.sound.\n\nThe conclusion for all rank-three matroids additionally invokes the published coprime-case theorem of van den Heuvel and Thomassé, which is not formalized in this repository. The divisible ingredient is therefore Lean-verified, and the complete rank-three result is established modulo that named literature theorem.\n\nThis is a complete resolution in rank three but a partial result toward the unrestricted Kajitani–Ueno–Miyano conjecture, which remains open in higher rank. The manuscript is an unrefereed Zenodo preprint.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Scores the Kajitani-Ueno-Miyano conjecture itself rather than the rank-three case, as the methodology requires. A named 1988 conjecture carried on the EGRES open problem collection, open for 38 years, with a documented line of attack in JCTB and the Electronic Journal of Combinatorics by van den Heuvel and Thomassé, Bonin, McGuinness, and Berczi, Janosik and Matravolgyi. It is a characterization conjecture, so settling it would reorganize how the class is understood, but it is invisible outside matroid theory and spanning-tree packing.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.21813715","sourceName":"Cyclic basis orderings of uniformly dense rank-three matroids","collection":null,"collectionVersion":null,"links":[{"label":"Lean repo","url":"https://github.com/afletcher22/Rank3KUM","kind":"code"},{"label":"Archived repo","url":"https://doi.org/10.5281/zenodo.21813154","kind":"code"}],"relations":[],"submittedBy":"HiddenHawk615","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"online-vertex-cover-edge-arrivals","name":"Tight Bound for Online Vertex Cover under Edge Arrivals","shortName":"Online vertex cover","problemNumber":null,"field":"Online algorithms","fieldGroup":"Algorithms & optimization","statement":"What is the optimal competitive ratio for online vertex cover when edges arrive one at a time? The paper proves a tight factor-2 lower bound via a reduction in the blueprint framework of Assadi, Jiang and Xiang, closing the gap left by prior work.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Zhihao Gavin Tang"],"aiRole":"The authors had partial progress and suggested the Assadi-Jiang-Xiang blueprint might apply; \"prompted by this suggestion, OpenAI's GPT-5.6 Sol formulated the reduction yielding the tight factor-2 lower bound proved in this paper and assisted with drafting the manuscript.\" The authors independently verified the reduction, proof and citations.","verification":"unreviewed","verificationNote":null,"publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"An established open question of the online-algorithms community; tight bounds for vertex cover under edge arrivals resisted a line of prior work.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.04994","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"boolean-max-k-csp-approximation","name":"The Approximation Ratio for Boolean Max-k-CSP","shortName":"Boolean Max-k-CSP","problemNumber":null,"field":"Approximation algorithms","fieldGroup":"Algorithms & optimization","statement":"How well can an arbitrary boolean constraint satisfaction problem of arity $k$ be approximated in polynomial time? The paper gives a $(k/2^k)$-approximation, improving the previous best constant of $0.626612\\,k/2^k$ due to Makarychev and Makarychev.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Removes the constant factor from the previous best guarantee; whether $k/2^k$ is optimal is not settled here.","claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.6 Sol Max","modelMaker":"OpenAI","humanCollaborators":["Ainesh Bakshi"],"aiRole":"\"GPT 5.6 Sol Max assisted in the lengthy computations that appear in the proof.\" Computational support inside a human-led argument.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The Max-k-CSP approximation constant is a tracked quantity of the approximation-algorithms literature, with a documented ladder running through Makarychev and Makarychev.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05331","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"riviere-s-regularity-question-for-critical-n-laplace-systems-with-antisymmetric-","name":"Rivière’s regularity question for critical $n$-Laplace systems with antisymmetric potentials","shortName":"Rivière $n$-Laplace regularity","problemNumber":null,"field":"Partial Differential Equations","fieldGroup":"Analysis","statement":"Let $n>2$. We construct a map $U\\in W^{1,n}(B^n,\\mathbb{R}^{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $B^n \\setminus \\{0\\}$, together with an antisymmetric potential $\\Omega\\in L^n(B^n,so(n+2)\\otimes\\mathbb{R}^n)$ such that $-\\mathrm{Div}(|\\nabla U|^{n-2}\\nabla U)=\\Omega\\cdot |\\nabla U|^{n-2}\\nabla U$ in $D'(B^n)$. This gives a negative answer to a regularity question posed by Rivière.\n\nOur potential admits the Lorentz-space regularity $\\Omega \\in \\bigcap_{q>2}L^{(n,q)} \\setminus L^{(n,2)}$. In addition for given $12$, the paper constructs a bounded map $U\\in W^{1,n}(B^n,\\mathbb{R}^{n+2})$, smooth on $B^n\\setminus\\{0\\}$ but discontinuous at the origin, together with an antisymmetric potential\n\n$$\n\\Omega\\in L^n(B^n,so(n+2)\\otimes\\mathbb{R}^n)\n$$\n\nsuch that\n\n$$\n-\\mathrm{Div}\\bigl(|\\nabla U|^{n-2}\\nabla U\\bigr)\n=\n\\Omega\\cdot|\\nabla U|^{n-2}\\nabla U\n\\qquad\\text{in }D'(B^n).\n$$\n\nMoreover, the potential satisfies the sharper Lorentz-space regularity\n\n$$\n\\Omega\\in\\bigcap_{q>2}L^{(n,q)}\\setminus L^{(n,2)},\n$$\n\nand, for any prescribed $1 1$ has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of $p$ graphs, each extremal for a family of chromatic number two. Disproved by a fixed finite family $\\mathcal{L}$ with $p(\\mathcal{L}) = 2$ and $\\mathrm{ex}(n,\\mathcal{L}) > t_2(n) + cn^{3/2}$ that nevertheless has, at every large order, an extremal graph with connected complement and hence no nontrivial join decomposition. The same construction disproves the Weak Product Conjecture of Füredi and Simonovits.","posedBy":"Miklos Simonovits, Zoltán Füredi","yearPosed":2013,"ageNote":"The version disproved is Conjecture 2.8 of the Furedi and Simonovits survey of degenerate extremal graph problems, which carries the conjecture rather than originating it, so its date is claimed as the earliest known statement.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"One construction disproves both the product conjecture and its no-forest form (Conjectures 1.1 and 1.3), but not the weak form of the conjecture (Conjecture 1.2).","claimIssueNote":null,"solveDate":"2026-08-03","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Chuandong Xu"],"aiRole":"The paper's comment credits the counterexample to GPT-5.6 Sol, found during a Codex project devoted to the Product Conjecture. The exact extremal-number and equality-case analysis around it is the author's own.","verification":"lean-checked","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed. Lean formalization released but not audited by any third party.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":18,"significanceNote":"A named conjecture of Simonovits on the product structure of extremal graphs, carried in the Furedi and Simonovits survey of degenerate extremal graph problems. The counterexample settles its weakened form at the same time and forces the decomposition family to contain no forest.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.02115","sourceName":"arXiv:2608.02115 - A finite forbidden family with superlinear surplus and non-join extremal graphs","collection":null,"collectionVersion":null,"links":[{"label":"Github repo of the counterexample","url":"https://github.com/Yazum/o6-counterexample-lean/tree/main/Papers/O6Counterexample","kind":"lean-proof"},{"label":"Füredi & Simonovits 2013 - Earliest known conjecture statement","url":"https://arxiv.org/abs/1306.5167","kind":"problem-record"}],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"autonomous-lipschitz-fast-dynamo-on-the-three-torus","name":"Autonomous Lipschitz Fast Dynamo on the Three-Torus","shortName":"Lipschitz fast dynamo","problemNumber":null,"field":"Dynamo theory; spectral PDE","fieldGroup":"Mathematical physics","statement":"Does there exist a single real-valued, divergence-free, time-independent Lipschitz velocity field $u\\in W^{1,\\infty}(\\mathbb T^3;\\mathbb R^3)$, chosen independently of magnetic diffusivity, that is a fast dynamo for the kinematic induction equation on the flat three-torus? The author constructs such a field and constants $\\varepsilon_0,\\gamma_0>0$ such that, for every $0<\\varepsilon\\le\\varepsilon_0$, the induction operator has an eigenvalue $\\lambda_\\varepsilon$ with $\\operatorname{Re}\\lambda_\\varepsilon\\ge\\gamma_0$. Thus every sufficiently small diffusivity admits a nonzero real divergence-free magnetic field with exact exponential $L^2$ growth. The velocity is Lipschitz but not $C^1$, so this settles only the Lipschitz regularity variant; Arnold's smooth autonomous fast-dynamo problem on $\\mathbb T^3$ remains open.","posedBy":"V. I. Arnold","yearPosed":1994,"ageNote":null,"solveType":"proved","resolution":"variant","aiContribution":"ai-assisted","resultNote":"One variant case of Arnold's 1994 fast-dynamo problem, not the problem itself. Arnold asks for a single velocity field on T^3 that is smooth, divergence-free, autonomous and deterministic, fixed independently of the magnetic diffusivity, and that grows the magnetic field exponentially at every small enough diffusivity. The field constructed here is all of that except smooth: it is Lipschitz, not C^1. The sibling entry on this site relaxes the opposite hypothesis, keeping a smooth field but making it random and time-dependent. Neither settles Arnold's problem as posed, which remains open.","claimIssueNote":null,"solveDate":"2026-08-03","model":"GPT-5.5 Pro, GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Lukas Niebel"],"aiRole":"During exploration, GPT-5.5 Pro and GPT-5.6 Sol helped identify a candidate fast-dynamo construction. They were later used to check calculations; identify errors, inconsistencies, and gaps in preliminary arguments; support development of some arguments; and assist with drafting and revision. The author states that he critically reviewed and verified every mathematical claim, calculation, and AI-generated suggestion and takes full responsibility for the manuscript.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed, Lean-verified, or independently expert-checked.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Arnold's fast dynamo problem has organized mathematical MHD for decades, with a Springer monograph and a sustained literature, while staying invisible outside that community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.02586","sourceName":"arXiv:2608.02586 - An autonomous Lipschitz fast dynamo on the three-torus","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"October","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hoa-maximal-non-hamiltonian-graphs","name":"Hoa's Conjecture on Maximal Non-Hamiltonian Graphs","shortName":"Hoa's conjecture","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"A graph $G$ is maximal non-Hamiltonian if it is non-Hamiltonian but $G + e$ is Hamiltonian for every nonedge $e$. In 1994 Vu Dinh Hoa conjectured a property of $G - V(C)$ for a longest cycle $C$ of such a graph. Disproved by an explicit base graph on 56 vertices, extended to larger orders.","posedBy":"Vu Dinh Hoa","yearPosed":1994,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-02","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Xingzhi Zhan"],"aiRole":"The declaration of AI use says ChatGPT assisted in developing and checking the constructions and proofs, with the author independently verifying all arguments.","verification":"unreviewed","verificationNote":"arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A named 1994 conjecture in Hamiltonian graph theory with a small but real literature; the audience is specialists in cycle structure.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.00957","sourceName":"arXiv:2608.00957 - Counterexamples to a conjecture of Hoa on maximal non-Hamiltonian graphs","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ehrhart-volume-conjecture-equality","name":"The Equality Case of Ehrhart's Volume Conjecture","shortName":"Ehrhart equality case","problemNumber":null,"field":"Convex geometry","fieldGroup":"Geometry & topology","statement":"Ehrhart conjectured that a full-dimensional compact convex body in $\\mathbb{R}^n$ whose barycenter is its unique interior lattice point has volume at most $(n+1)^n/n!$. With the inequality itself settled, the remaining question was which bodies attain it. Every such body is a unimodular image of the simplex $(n+1)\\Delta_n - (1,\\dots,1)$.","posedBy":"Eugene Ehrhart","yearPosed":1964,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"the equality case; the inequality was settled separately and is tracked on its own entry","claimIssueNote":null,"solveDate":"2026-08-02","model":"GPT-5.6 Sol, Fable 5, Danus","modelMaker":null,"humanCollaborators":["Jihao Liu"],"aiRole":"The paper states the main result was obtained by generative AI, naming GPT-5.6-sol, Fable 5 and the Danus system, and its comment records essential human strategic input followed by human verification.","verification":"unreviewed","verificationNote":"No independent check. arXiv preprint; not yet peer-reviewed. The author records having verified the machine-produced argument.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"Completes Ehrhart's 1964 conjecture. The inequality was proved shortly before this, and this is the characterisation of the bodies attaining the bound, which the inequality alone leaves open.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.01040","sourceName":"arXiv:2608.01040 - The equality case of Ehrhart's volume conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"Curator34","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"wowii-graph-conjecture-322","name":"Written on the Wall II, Graph Conjecture 322","shortName":"WOWII 322","problemNumber":null,"field":"Graph theory (automated conjecture)","fieldGroup":"Combinatorics","statement":"Let $G$ be a simple connected graph on $n\\geq 5$ vertices. If the maximum over all vertices $v$ of $\\ell(v)$ - the independence number of the subgraph induced by the open neighborhood $N(v)$ - is at most $1$, must $G$ be well totally dominated? Answered affirmatively; the Lean proof in fact needs only $n\\geq 2$, and retains the conjecture's $n\\geq 5$ to state the source faithfully.","posedBy":"Written on the Wall II (automated conjecturing)","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The formalization proves the statement under the weaker hypothesis n >= 2; the pull request marking the conjecture solved is open, not merged","claimIssueNote":null,"solveDate":"2026-08-02","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The pull request marking the conjecture solved credits the proof to Aristotle, Harmonic's prover; a human contributor prepared and filed the formalization.","verification":"lean-verified","verificationNote":"Sorry-free Lean 4 proof filed against google-deepmind/formal-conjectures, which flips the conjecture's attribute from `research open` to `research solved` and links the proof. Unlike the site's WOWII 217 entry it needs no native_decide: the argument is conceptual, showing every neighborhood is a clique and deducing well-total-domination. Not independently reviewed, and the pull request is still open.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Written on the Wall II); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/google-deepmind/formal-conjectures/pull/4686","sourceName":"formal-conjectures PR #4686 - prove Conjecture 322","collection":null,"collectionVersion":null,"links":[{"label":"Lean proof","url":"https://github.com/MrBrain295/formal-conjectures/blob/322/FormalConjectures/WrittenOnTheWallII/GraphConjecture322.lean","kind":"lean-proof"},{"label":"Written on the Wall II conjecture list","url":"http://cms.dt.uh.edu/faculty/delavinae/research/wowII/","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"factorial-conjecture-two-variables","name":"Two-Variable Factorial Conjecture","shortName":"Factorial Conjecture (2 vars)","problemNumber":null,"field":"Commutative Algebra, Transcendence","fieldGroup":"Algebra","statement":"Let $\\mathcal{L}(x^{a}y^{b})=a!\\,b!$ on $\\mathbb{C}[x,y]$. The Factorial Conjecture asks whether $\\mathcal{L}(f^{m})=0$ for every $m\\geq 1$ forces $f=0$. The homogeneous two-variable case was settled by Liu and Sun; the inhomogeneous problem does not reduce to it, because radial integration couples the homogeneous layers through Gamma factors. A claimed proof settles the full two-variable case affirmatively.","posedBy":"Arno van den Essen, David Wright, Wenhua Zhao","yearPosed":2011,"ageNote":"Introduced in the Image Conjecture work of van den Essen, Wright and Zhao; the homogeneous two-variable case was proved by Liu and Sun (2020).","solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"Claimed in a self-published research draft; a standalone by-product is the transcendence of the integral of exp(q) between distinct algebraic endpoints for nonconstant algebraic q","claimIssueNote":null,"solveDate":"2026-08-01","model":"GPT-5.6 Sol, Claude Opus 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Christopher D. Long"],"aiRole":"Per the author's disclosure, the manuscript was developed through interactive work with ChatGPT 5.6 Sol, which assisted in proof discovery, organization, symbolic checking, reference verification, adversarial auditing and drafting; Claude Opus 5 contributed the semisimple-projector strategy, the reduction to phase-polynomial moments, and further adversarial audits. The AI systems are not authors and the human author takes full responsibility.","verification":"unreviewed","verificationNote":"The author marks the 37-page draft explicitly as not yet peer reviewed and not formally verified. Published as a PDF and LaTeX source in a personal GitHub repository, with no independent check on record.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture from the Image Conjecture circle around the Jacobian conjecture, with a small dedicated literature; the two-variable case specifically.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/octonion/mathematics/tree/master/fc","sourceName":"Factorial conjecture manuscript (GitHub)","collection":null,"collectionVersion":null,"links":[{"label":"Liu-Sun, homogeneous two-variable case","url":"https://doi.org/10.1017/S0004972719000546","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"erdos-180","name":"Erdős Problem #180: Compactness Conjecture","shortName":"Erdős #180","problemNumber":180,"field":"Extremal graph theory","fieldGroup":"Combinatorics","statement":"For every finite family $\\mathcal{F}$ of graphs, is there a single $G \\in \\mathcal{F}$ with $\\mathrm{ex}(n;G) \\ll_{\\mathcal{F}} \\mathrm{ex}(n;\\mathcal{F})$? A counterexample refutes the Erdős-Simonovits compactness conjecture.","posedBy":"Paul Erdős, Miklós Simonovits","yearPosed":1982,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The Erdős-Simonovits compactness conjecture (1982), a known structural question of Turán theory.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (CompactnessAndDegeneracy.lean)","url":"https://github.com/openai/ten-proofs/blob/main/CompactnessAndDegeneracy.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"},{"label":"erdosproblems.com/180","url":"https://www.erdosproblems.com/180","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"cvp-polynomial-hardness","name":"Polynomial-Factor Hardness for the Closest Vector Problem","shortName":"CVP hardness","problemNumber":null,"field":"Lattices & cryptography","fieldGroup":"Theoretical computer science","statement":"Is the closest vector problem NP-hard to approximate within polynomial factors $n^c$? Yes for some $c > 0$: hardness of approximation reaches polynomial factors, with consequences for decoding and related lattice problems - a foundational question underpinning post-quantum cryptography where hardness had stalled at almost-polynomial factors since the late 1990s.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Approximation hardness for CVP is a foundational lattice question tied to post-quantum cryptography; progress had stalled at almost-polynomial factors since the late 1990s.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (GapCVP.lean)","url":"https://github.com/openai/ten-proofs/blob/main/GapCVP.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"non-sofic-groups-exist","name":"Gromov and Weiss's Question on Sofic Groups","shortName":"Sofic groups question","problemNumber":null,"field":"Geometric group theory","fieldGroup":"Algebra","statement":"Is every group sofic - does every group admit approximate finite permutation representations? A central open question of geometric group theory since Gromov introduced soficity: soficity implies Gottschalk's surjunctivity conjecture, Kaplansky's stable finiteness and more, and no non-sofic group was known. An explicit construction now establishes that non-sofic groups exist.","posedBy":"Mikhail Gromov, Benjamin Weiss","yearPosed":1999,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"answered no: non-sofic groups exist","claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"No independent mathematician has checked this yet. Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":60,"significanceNote":"A central open question of geometric group theory since Gromov (1999), entangled with Gottschalk surjunctivity, Kaplansky stable finiteness and the post-MIP* landscape.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (NonSoficGroup.lean)","url":"https://github.com/openai/ten-proofs/blob/main/NonSoficGroup.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"},{"label":"A finitely presented non-sofic example (OpenAI math release, Kaplansky)","url":"https://vibemathed.com/problem/kaplansky-direct-finiteness-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":9,"downvotes":0,"commentCount":3},{"slug":"schubitope-ehrhart-positivity","name":"Ehrhart Positivity of Schubitopes","shortName":"Schubitope positivity","problemNumber":null,"field":"Algebraic combinatorics","fieldGroup":"Combinatorics","statement":"Monical, Tokcan and Yong conjectured that Schubitopes, the generalized permutahedra arising as Newton polytopes of Schubert polynomials and of Demazure characters of $\\mathrm{GL}_n$, are Ehrhart positive. Disproved by an explicit Schubitope whose Ehrhart polynomial has a negative coefficient in its monomial expansion.","posedBy":"Cara Monical, Neriman Tokcan, Alexander Yong","yearPosed":2019,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"GPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":["Yupeng Li","Avery St. Dizier"],"aiRole":"The abstract states the counterexample was found using GPT-5.6 Sol Pro in ChatGPT. The authors supply the surrounding treatment of the family.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed. The result is a single explicit polytope, so it is checkable by direct computation.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Conjecture 5.19 of Monical, Tokcan and Yong's Newton polytopes paper, in the line of Ferroni's disproof of Ehrhart positivity for matroid polytopes.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.00377","sourceName":"arXiv:2608.00377 - Schubitopes are not Ehrhart positive","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"quantum-parallel-repetition","name":"Quantum Parallel Repetition","shortName":"Quantum parallel repetition","problemNumber":null,"field":"Quantum complexity","fieldGroup":"Quantum information & computing","statement":"Does the value of a two-player quantum game decay exponentially under parallel repetition, as Raz's theorem gives for classical games? Yes: an exponential parallel repetition theorem holds for arbitrary finite two-player quantum games.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A foundational open question of quantum complexity since the classical parallel repetition theorem, previously known only in special cases.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (QuantumParallelRepetition.lean)","url":"https://github.com/openai/ten-proofs/blob/main/QuantumParallelRepetition.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"block-sensitivity-spectral-sensitivity-squared","name":"Is block sensitivity at most quadratic in spectral sensitivity? (Aaronson, Ben-David, Kothari, Rao and Tal)","shortName":"Block sensitivity vs spectral sensitivity squared","problemNumber":null,"field":"Boolean function complexity; sensitivity","fieldGroup":"Theoretical computer science","statement":"For a total Boolean function $f:\\{0,1\\}^n\\to\\{0,1\\}$, the spectral sensitivity $\\lambda(f)$ is the largest eigenvalue of the adjacency matrix of its sensitivity graph (the hypercube edges across which $f$ changes value), and the block sensitivity $\\mathrm{bs}(f)$ is the largest number of disjoint blocks of bits each of whose flip changes $f$ at some input. Huang's proof gives $\\deg(f)\\le\\lambda(f)^2$, and with Nisan-Szegedy $\\mathrm{bs}(f)=O(\\lambda(f)^4)$. Aaronson, Ben-David, Kothari, Rao and Tal asked: is $\\mathrm{bs}(f)=O(\\lambda(f)^2)$?","posedBy":"Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao and Avishay Tal, Degree vs. approximate degree and quantum implications of Huang's sensitivity theorem (STOC 2021), Section 7","yearPosed":2020,"ageNote":"Dated from the arXiv version (2010.12629, October 2020) of the STOC 2021 paper, whose open-questions section asks \"perhaps bs(f) = O(lambda(f)^2) or even RC(f) = O(lambda(f)^2)?\"","solveType":"disproved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"Theorem 6.1: a total Boolean function on 2,017,584 variables with $\\mathrm{bs}(f)\\ge14011$ and $\\lambda(f)\\le89.0162$, so $\\mathrm{bs}(f)\\ge\\lambda(f)^{2.127}$; composition gives a family with $\\lambda\\to\\infty$ and $\\mathrm{bs}=\\Omega(\\lambda^{2.127})$, so no quadratic bound holds. The function is a union of subcubes indexed by a doubly regular tournament with gates fixed by the Lovász local lemma. An exponent near 2.20 on 1,255 variables is numerical evidence only. The true exponent, between 2.127 and 4, stays open. Since $\\lambda(f)\\le s(f)$, OpenAI's later disproof of $\\mathrm{bs}=O(s^2)$ (25 September 2026), which adapts this gated-tournament construction, implies this result.","claimIssueNote":null,"solveDate":"2026-08-01","model":"GPT-5.6-Sol (ideas); Claude Opus 5 with Harmonic's Aristotle (Lean)","modelMaker":"OpenAI; Anthropic; Harmonic","humanCollaborators":["Alexander Meiburg"],"aiRole":"The paper's Note on AI Usage: \"Several key ideas of the proof, in particular that 'gated tournaments' would be a good candidate for large bs relative to lambda, were suggested by GPT-5.6-Sol. The initial proposed construction was on 269 variables. All text in this paper was produced without generative AI tooling, and the author is responsible for the exposition, minimizing and optimizing size and parameters, and verifying the faithfulness of the Lean formalization.\" The paper says the formalization was carried out using Aristotle; the repository's formalization.yaml records a Claude Code session on Claude Opus 5 with about a dozen goal-level human turns, plus Harmonic's prover for leaf lemmas.","verification":"lean-checked","verificationNote":"Checked here on 7 October 2026: Theorem 6.1 and Corollary 6.2 of arXiv:2608.00851 were read against the question as printed in Section 7 of Aaronson, Ben-David, Kothari, Rao and Tal. In Timeroot/BS_Lam, BSLambda/Main.lean was read: exists_bs_gt_lam_rpow states a total Boolean function on ZMod 14011 x Fin 144 inputs with $\\lambda(f)^{2.12}<\\mathrm{bs}(f)$ (the comparator challenge printed in the paper's Appendix B), and exists_ratio_blowup states $(14011/U)^m\\lambda(F_m)^2\\le\\mathrm{bs}(F_m)$ for the $m$-fold self-compositions, with $14011/U>1$, which is the unbounded-ratio disproof. The definitions of bs, the sensitivity-graph adjacency matrix and its top eigenvalue match the paper. The repository reports no sorry and only propext, Classical.choice and Quot.sound, and proves the multiplicativity of $\\lambda$ under composition rather than importing it. Not rebuilt here; no CI run is published.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"A sharp open question from the 2021 paper that followed Huang's proof of the Sensitivity Conjecture, known to query-complexity specialists. Below the Nisan-Szegedy quadratic question at 35, older and better known, whose later disproof implies this one; above the Gopalan-Servedio entry at 14; level with the weak-disorder Ising entry at 22 in standing within its own community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.00851","sourceName":"arXiv:2608.00851, Meiburg (1 August 2026)","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalization: Timeroot/BS_Lam","url":"https://github.com/Timeroot/BS_Lam","kind":"lean-proof"},{"label":"Headline Lean statements (BSLambda/Main.lean)","url":"https://github.com/Timeroot/BS_Lam/blob/7bd39a8d/BSLambda/Main.lean","kind":"lean-statement"},{"label":"Aaronson, Ben-David, Kothari, Rao, Tal, Degree vs. approximate degree (arXiv:2010.12629), Section 7","url":"https://arxiv.org/abs/2010.12629","kind":"problem-record"},{"label":"Stronger later result: block sensitivity is not quadratic in sensitivity (OpenAI release)","url":"https://vibemathed.com/problem/block-sensitivity-quadratic-in-sensitivity","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"binary-code-upper-bounds","name":"Upper Bounds for Binary and Spherical Codes","shortName":"Binary code bounds","problemNumber":null,"field":"Coding theory","fieldGroup":"Theoretical computer science","statement":"What is the maximum size of a binary code of given minimum distance? The linear-programming bounds of McEliece, Rodemich, Rumsey and Welch (1977) resisted improvement for half a century. The new upper bounds are exponentially stronger at every prescribed distance, with analogous results for high-dimensional spherical codes.","posedBy":null,"yearPosed":1977,"ageNote":"Dated to the McEliece-Rodemich-Rumsey-Welch linear-programming bounds of 1977, the barrier this result breaks; the underlying question of optimal code size is older.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open","claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":39,"significanceNote":"The rate-distance trade-off is coding theory's central asymptotic question; the MRRW barrier stood since 1977.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (MetricCodes.lean)","url":"https://github.com/openai/ten-proofs/blob/main/MetricCodes.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"}],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"erdos-146","name":"Erdős Problem #146: Degeneracy Conjecture","shortName":"Erdős #146","problemNumber":146,"field":"Extremal graph theory","fieldGroup":"Combinatorics","statement":"If $H$ is bipartite and $r$-degenerate, is $\\mathrm{ex}(n;H) \\ll n^{2-1/r}$ (a \\$500 Erdős-Simonovits prize conjecture)? A counterexample refutes the degeneracy conjecture.","posedBy":"Paul Erdős, Miklós Simonovits","yearPosed":1984,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A \\$500 Erdős-Simonovits prize conjecture (1984), a central target of degenerate Turán theory with decades of partial results.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (CompactnessAndDegeneracy.lean)","url":"https://github.com/openai/ten-proofs/blob/main/CompactnessAndDegeneracy.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"},{"label":"erdosproblems.com/146","url":"https://www.erdosproblems.com/146","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"sphere-packing-upper-bounds-cohn-elkies","name":"Upper Bounds for High-Dimensional Sphere Packing","shortName":"Sphere packing bounds","problemNumber":null,"field":"Discrete geometry","fieldGroup":"Geometry & topology","statement":"How dense can a sphere packing in $\\mathbb{R}^n$ be as $n \\to \\infty$? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.","posedBy":null,"yearPosed":1978,"ageNote":"Dated to the Kabatiansky-Levenshtein bound of 1978, the asymptotic barrier this result improves on; asking how dense sphere packings can be is of course far older.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open","claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The asymptotic sphere-packing problem is classical and central across geometry, coding and number theory; the Kabatiansky-Levenshtein bound stood since 1978.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (SpherePacking.lean)","url":"https://github.com/openai/ten-proofs/blob/main/SpherePacking.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"}],"relations":[],"submittedBy":null,"upvotes":4,"downvotes":0,"commentCount":0},{"slug":"complementary-bell-fibers","name":"Finitude of the Fibers of Complementary Bell Numbers","shortName":"Complementary Bell fibers","problemNumber":null,"field":"Enumerative combinatorics","fieldGroup":"Combinatorics","statement":"Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers $f(n) = B_n(-1)$ take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result whose techniques connect to Wilf's conjecture on the vanishing of $f(n)$.","posedBy":"M. V. Subbarao, A. Verma","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["John M. Campbell"],"aiRole":"Solved \"through our extensive interactions with GPT-5.6 Pro\" during the exploratory and proof-development stages; all AI-generated suggestions were substantially revised, corrected and independently verified by the author.","verification":"unreviewed","verificationNote":null,"publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A named 1999 problem adjacent to Wilf's conjecture, with a real literature (Yang solved the first two problems of the same set in 2001).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.00575","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ehrhart-volume-conjecture","name":"Ehrhart's Volume Conjecture","shortName":"Ehrhart volume","problemNumber":null,"field":"Convex geometry","fieldGroup":"Geometry & topology","statement":"What is the maximum volume of a convex body in $\\mathbb{R}^n$ whose centroid is its only interior lattice point? Ehrhart conjectured the extremal value in 1964; the sharp maximum is now determined in every dimension.","posedBy":"Eugène Ehrhart","yearPosed":1964,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":1,"renownNote":null,"significance":25,"significanceNote":"Ehrhart's 1964 conjecture, a known sharp-constant question of convex geometry previously settled only in low dimensions and special cases.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (EhrhartVolumeInequality.lean)","url":"https://github.com/openai/ten-proofs/blob/main/EhrhartVolumeInequality.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"}],"relations":[],"submittedBy":null,"upvotes":3,"downvotes":0,"commentCount":2},{"slug":"erdos-183","name":"Erdős Problem #183: Multicolor Triangle Ramsey","shortName":"Erdős #183","problemNumber":183,"field":"Ramsey theory","fieldGroup":"Combinatorics","statement":"Let $R(3;k)$ be the least $n$ such that every $k$-colouring of the edges of $K_n$ contains a monochromatic triangle. Determine $\\lim_{k\\to\\infty} R(3;k)^{1/k}$ (a \\$250 Erdős prize problem). A superexponential lower bound resolves the problem: the limit is infinite.","posedBy":"Paul Erdős","yearPosed":1961,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":21,"significanceNote":"A \\$250 Erdős prize problem from 1961, well known across Ramsey theory.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (MulticolorTriangleRamsey.lean)","url":"https://github.com/openai/ten-proofs/blob/main/MulticolorTriangleRamsey.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"},{"label":"erdosproblems.com/183","url":"https://www.erdosproblems.com/183","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"permanent-formula-lower-bounds","name":"Lower Bounds for the Permanent in Arithmetic Circuits","shortName":"Permanent lower bounds","problemNumber":null,"field":"Algebraic complexity","fieldGroup":"Theoretical computer science","statement":"How large must arithmetic circuits and formulas computing the $n \\times n$ permanent be? New lower bounds include an arithmetic-formula bound of order $n^4/\\log n$, far beyond the quadratic barrier that stood for decades.","posedBy":"Leslie Valiant","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"an n^4/log n formula lower bound; VP vs VNP remains wide open","claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":38,"significanceNote":"Permanent lower bounds are the flagship program of algebraic complexity since Valiant (1979); formula bounds had been stuck near quadratic.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (Permanent.lean)","url":"https://github.com/openai/ten-proofs/blob/main/Permanent.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"}],"relations":[],"submittedBy":null,"upvotes":3,"downvotes":0,"commentCount":0},{"slug":"connes-rigidity-conjecture","name":"Connes' Rigidity Conjecture","shortName":"Connes rigidity","problemNumber":null,"field":"Operator algebras","fieldGroup":"Analysis","statement":"Are ICC property (T) groups remembered by their von Neumann algebras - if $L(\\Gamma) \\cong L(\\Lambda)$ for such groups, must $\\Gamma \\cong \\Lambda$? A counterexample refutes Connes' conjecture that these groups are uniquely determined by their group von Neumann algebras.","posedBy":"Alain Connes","yearPosed":1980,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"No independent mathematician has checked this yet. Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"Posed by Connes around 1980; the organizing conjecture of W*-rigidity theory for four decades.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","collection":null,"collectionVersion":null,"links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (ConnesRigidity.lean)","url":"https://github.com/openai/ten-proofs/blob/main/ConnesRigidity.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"},{"label":"Claimed rigidity for the lattice class (OpenAI math release)","url":"https://vibemathed.com/problem/connes-rigidity-property-t-lattices","kind":"other"}],"relations":[{"to":"connes-rigidity-icc-property-t","kind":"generalizes","note":"The full rigidity conjecture, of which the other entry settles the ICC property (T) case - independently and concurrently, per that paper."}],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"minimum-reverse-complement-invariant-decycling-sets","name":"Minimum reverse-complement-invariant decycling sets","shortName":"RC-invariant decycling sets","problemNumber":null,"field":"de Bruijn graphs; unavoidable sets; reverse-complement symmetry","fieldGroup":"Combinatorics","statement":"Let $\\Sigma$ be a finite nonempty alphabet of size $q$ with a fixed-point-free involution $a\\mapsto\\bar a$, and let $k\\ge2$. The directed de Bruijn graph $B(q,k)$ has length-$k$ words as vertices and one-symbol-shift edges, including loops. Put $\\rho(a_1\\cdots a_k)=\\bar a_k\\cdots\\bar a_1$.\n\nWhat is the minimum cardinality $\\tau_{\\rm RC}(q,k)$ of a set $M\\subseteq\\Sigma^k$ such that $\\rho(M)=M$ and deleting $M$ makes $B(q,k)$ acyclic?\n\nThis is the minimum-size symmetric-decycling problem described as open by Marçais, Elder and Kingsford (2024), Section 6. Section 5.1 also identifies the lack of constructions of symmetric ordinary minimum decycling sets. The classical minimum without symmetry is $N_q(k)=\\frac1k\\sum_{d\\mid k}\\varphi(d)q^{k/d}$. This entry reports a parity theorem toward the symmetry-constrained problem.","posedBy":"Guillaume Marçais, C. S. Elder and Carl Kingsford, k-nonical space: sketching with reverse complements (2024), Section 6; see also Section 5.1","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Theorem 1.1 claims, for every finite nonempty alphabet with a fixed-point-free complement and every $k\\ge2$,\n\n$$\n\\tau_{\\rm RC}(q,k)=N_q(k)\\quad\\Longleftrightarrow\\quad k\\text{ is even}.\n$$\n\nFor odd $k\\ge3$, Theorem 3.1 gives $\\tau_{\\rm RC}(q,k)\\ge N_q(k)+q$, hence an excess of at least two for binary words and four for DNA. For even orders, Sections 6-8 provide a weighted construction and a direct spectral selector; integer weights permit exact polynomial-bit membership testing for a single word. The case $k=2$ is handled separately.\n\nThis completely classifies attainability of the ordinary minimum and partially answers the external symmetric-minimum question. It does not determine the general exact odd optimum, prove a packing-covering equality, give quantitative residual-path/window bounds, or establish biological performance. The classical ordinary minimum itself is prior work.","claimIssueNote":null,"solveDate":"2026-08","model":"GPT-5.6 Sol; GPT-5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The author reports using GPT-5.6 Sol and GPT-5 Pro through ChatGPT. The manuscript and public authorship disclosure describe substantial AI assistance with candidate proof arguments, symbolic identities, construction and verification code, adversarial checks and exposition. The author selected the research problem, directed the work and is responsible for the claims and final manuscript. The model names are author-supplied; the public disclosure does not identify model versions or assign individual lemmas to models. AI-assisted reflects the documented material assistance without asserting that AI discovered the central proof. Agreement between distinct implementations and author-side AI reviews is not independent specialist verification.","verification":"unreviewed","verificationNote":"The posed question was read at source. Marcais, Elder and Kingsford, \"k-nonical space: sketching with reverse complements\", Bioinformatics 40(11), Section 6: \"Being a symmetric decycling set is a strong condition that is still not well understood theoretically (e.g. the minimum size of a symmetric decycling set is unknown).\" That is the quantity this paper bounds, and the submitter's quotation of it is accurate.\n\nThe mathematics was not checked here. The public 15-page manuscript carries analytical proofs, LaTeX source and an exact binary replay using only the standard library; the author's audit of 12 September records an anonymous clone matching all 50 reviewed files and a finite panel over binary $k=2,4,6,8,10,12$ with 612 legal firings and 14 mutations rejected. A finite panel does not establish the uniform $q$-ary theorem, and no independent specialist endorsement or formal proof is supplied. Partial because the question is answered for half its cases: every even order is settled exactly, and odd orders get a lower bound rather than the optimum.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A question raised in passing in a single applied paper two years ago, with essentially no literature behind it. Marcais, Elder and Kingsford note it as an aside while reporting ILP results, not as a headline problem. Just above the typical numbered Erdos problem at 10 rather than level with it, because the question has a practical driver - sketching DNA, where a k-mer and its reverse complement are one object - and a clean classical counterpart in the unconstrained minimum $N_q(k)$.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/aconsciousfractal/FCIG-RC-Invariant-Minimum-Decycling-Sets/blob/1f1b7156d3d8d5c326303427f97991212e9bb86a/paper/A_Parity_Theorem_for_RC_Invariant_MDS.pdf","sourceName":"GitHub manuscript: A Parity Theorem for Reverse-Complement-Invariant Minimum Decycling Sets in q-ary de Bruijn Graphs","collection":null,"collectionVersion":null,"links":[{"label":"Complete manuscript source and exact reproduction package","url":"https://github.com/aconsciousfractal/FCIG-RC-Invariant-Minimum-Decycling-Sets/tree/1f1b7156d3d8d5c326303427f97991212e9bb86a","kind":"code"},{"label":"Marçais–Elder–Kingsford (2024): open symmetric-minimum problem, Section 6","url":"https://doi.org/10.1093/bioinformatics/btae629","kind":"paper"},{"label":"Exact theorem and scope boundaries","url":"https://github.com/aconsciousfractal/FCIG-RC-Invariant-Minimum-Decycling-Sets/blob/1f1b7156d3d8d5c326303427f97991212e9bb86a/docs/PUBLIC_CLAIM_BOUNDARY.md","kind":"code"}],"relations":[],"submittedBy":"Oleksiy Babanskyy","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"prime-values-of-digital-functions-along-the-primes","name":"Prime values of digital functions along the primes","shortName":"Prime values of digital functions","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"Every integer-valued strongly b-additive function g with gcd(g(1),…,g(b−1)) = 1 and nonnegative digit mean takes prime values at infinitely many primes, with a Mertens-type formula and normal-order results; the running example resolves the infinitude of OEIS A052034 (De Geest, 1999): infinitely many primes have a prime sum of squared decimal digits.","posedBy":null,"yearPosed":1999,"ageNote":"No poser: the infinitude of A052034 was never formally asked, as the repository says plainly - \"We have not located anywhere a proof, or an assertion, that the sequence is infinite\", the OEIS citations being to the recreational literature. Dated instead from 1999, when De Geest recorded the sequence and so made the question available to ask.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"For every integer-valued strongly $b$-additive $g$ with $\\gcd(g(1),\\dots,g(b-1))=1$ and digit mean $\\mu_g\\ge0$: $g(p)$ is prime for infinitely many primes $p$. For $\\mu_g>0$, $\\sum 1/p$ over $p 0 such that the following holds: if q is sufficiently large and admissible, then\n\n#{n ≥ 1 : (n,b) = 1, s_b(n²) = q, n² ≤ b^{C_b·q}} ≥ exp(c_b·√q).\n\nThe Lean 4 formalisation proves the explicit form: at least 2^{√q/(36b)} representations of size n² ≤ b^{3q} once q ≥ 2304·b⁴.","posedBy":"Murthy–Ashbacher (base ten, recreational)","yearPosed":2005,"ageNote":"Recreational provenance rather than a posed conjecture. Murthy and Ashbacher recorded in 2005 that a base-ten square exists with any admissible digit sum; proved here is the quantitative all-base version, roots coprime to the base - excluding their trailing-zero trick - with exponentially many representations of controlled size. 2005 dates the observation, not a challenge anyone issued.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"New theorems, not a formalisation of known results. In base ten the bare existence of a square with any admissible digit sum was recorded in the recreational literature (Murthy–Ashbacher 2005); the theorem here is the quantitative all-base version — roots coprime to b (excluding their trailing-zero trick), exp(c_b√q) representations of controlled size — and that statement is what is machine-checked, with effective but non-optimal constants. The same paper proves a local limit theorem for g(n²) for arbitrary digit weights in every base including binary, and sieves the values: level of distribution 1/2, P₃ and P₂ values, an unconditional Mertens-type prime-value law, a joint Erdős–Kac theorem. Caveats: the two shrinking-frequency estimates behind the local theorem enter the Lean development only as transcribed definitions; the sieve and asymptotic results are unformalised; Corollary 1.9 is formalised in bases 2 and 3 only; the analogous theory along squares of primes remains open.","claimIssueNote":null,"solveDate":"2026-08","model":"Claude Opus 5, Claude Fable 5, OpenAI Sol","modelMaker":"Anthropic, OpenAI","humanCollaborators":[],"aiRole":"Used at every stage: literature search, jointly working out the arguments, drafting the manuscript, and producing the Lean 4 formalisation with its axiom audit and validation scripts. The carry-free Sidon-set construction behind the submitted theorem and the paper's quantitative lattice local-limit framework were worked out jointly with the models, which also wrote most of the manuscript under the author's direction. The author verified all statements, proofs and references, made the final decisions on content and presentation, and is responsible for any remaining errors.","verification":"lean-checked","verificationNote":"Lean-checked, and the development was audited here on 30 August 2026 rather than taken on the submitter's word. What was found matches the submission exactly.\n\n33 files, 10,587 lines, with no $\\texttt{sorry}$, no $\\texttt{admit}$ and no $\\texttt{native\\_decide}$ anywhere. Exactly two declared axioms exist in the whole project, both in $\\texttt{DSS/Cited.lean}$ and both literature citations: Martin-Mauduit-Rivat and Halberstam-Heath-Brown-Richert. The committed $\\texttt{axiom\\_audit.txt}$ has 107 entries splitting 102 / 3 / 2 as claimed, and the five conditional ones are all the almost-prime $\\texttt{p2\\_count}$ and $\\texttt{square\\_p2}$ results - exactly where the paper says those inputs are used.\n\nThe submitted theorem is among the unconditional 102: $\\texttt{DSS.sq\\_digit\\_sum\\_count}$ depends only on $\\texttt{propext}$, $\\texttt{Classical.choice}$ and $\\texttt{Quot.sound}$. Its statement was read against this entry's and matches term for term - $\\texttt{sqSols}$ filters on coprimality to the base, $s_b(n^2)=q$, and $n^2\\le b^{3q}$, with threshold $2304b^4$ and bound $2^{\\lfloor\\sqrt q/36b\\rfloor}$.\n\nNot upgraded to Lean-verified, for one reason: it was not built. No Lean toolchain here, so every axiom closure above is read from the committed audit file rather than reproduced from $\\texttt{lake build}$. The kernel half is unconfirmed whatever the statement audit shows.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A quantitative all-base strengthening of a base-ten existence remark, machine-checked unconditionally, from a paper whose wider content (the square local limit theorem, the sieve, the Erdős-Kac law) is not what is submitted here. That places it with this author's other digital-function entries: just above the along-the-primes companion at 11, because the submitted theorem is fully axiom-free in Lean and moves a bare existence claim in one base to an exponential count in every base, and level with the Erdős-Kac palindromes entry at 12.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.22164901","sourceName":"Local limits along squares and prime values of digital functions","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalisation + axiom audit (folder local-limits-along-squares)","url":"https://github.com/vibefrtz/vibemath","kind":"code"},{"label":"archived release v1.2.0 (version DOI)","url":"https://doi.org/10.5281/zenodo.22164902","kind":"paper"}],"relations":[{"to":"prime-values-of-digital-functions-along-the-primes","kind":"continues","note":"Same programme by the same author: that entry counts prime values along the primes, this one counts squares with a prescribed digit sum."}],"submittedBy":"vibefrtz","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes","name":"An Erdős–Kac law for base-$b$ palindromes and for reversed primes","shortName":"Erdős–Kac for palindromes & reversed primes","problemNumber":null,"field":"Analytic number theory","fieldGroup":"Number theory","statement":"For every base $b\\ge2$, the number of prime factors of the $\\lambda$-digit base-$b$ palindromes, and of the base-$b$ reversals of the $\\lambda$-digit primes, obeys an Erdős–Kac law: counted with or without multiplicity, it is asymptotically normal with centring $\\log\\log b^{\\lambda}$ and scaling $\\sqrt{\\log\\log b^{\\lambda}}$. For reversed primes the law persists when the leading digit of the prime is prescribed. Erdős–Kac laws were already known for other digitally defined families — integers with a fixed digit sum, or with digits restricted to a fixed set — but for palindromes only the largest value of $\\omega$ had been studied, with nothing known about the typical value, and for reversed primes the level of distribution the argument needs became available only in 2025.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"New theorems, not a formalisation of previously known results. For every base $b\\ge2$ the Erdős–Kac law is established for the $\\lambda$-digit base-$b$ palindromes and for the base-$b$ reversals of the $\\lambda$-digit primes, for $\\omega$ and $\\Omega$ and for $\\omega_S,\\Omega_S$ with any regular set $S$ of primes; with normal order $\\log\\log n$ on both families, and, for $\\omega$, all moments of order up to $\\tfrac12(\\log\\log b^{\\lambda})^{1/3}$ uniformly in the order. For reversed primes it also holds with the leading digit prescribed.\n\nNot settled: the results rest on quoted inputs (Col for palindromes, the Bombieri–Vinogradov theorem of Dartyge–Rivat–Swaenepoel for reversed primes), and both families exclude the primes dividing $b(b^{2}-1)$. No rate of convergence is obtained. The question of Banks–Shparlinski on the *largest* value of $\\omega$ on palindromes is untouched: the trivial bound $\\ll\\lambda/\\log\\lambda$ and their $\\lambda^{o(1)}$ remain far apart.","claimIssueNote":null,"solveDate":"2026-08","model":"Claude Fable 5, Claude Opus 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"Used at every stage: literature search, jointly working out the main arguments, and drafting the manuscript. Two contributions were decisive. The arithmetic input for palindromes — Col's theorem on the level of distribution of palindromes in arithmetic progressions — was located by the models. And the general Erdős–Kac criterion used here, a modification of the Granville–Soundararajan sieve moment estimate that allows a finite exceptional set of primes at which the local densities are arbitrary, was worked out jointly with them. The models also produced the Lean 4 formalisation. The author verified all statements, proofs and references, made the final decisions on content and presentation, and is responsible for any remaining errors.","verification":"lean-checked","verificationNote":"Read independently here on 24 August 2026 at github.com/vibefrtz/vibemath, in addition to the submission's own detailed VERIFICATION.md, which this confirms rather than repeats. All 21 Lean files (about 7850 lines) carry no sorry, no admit and no native_decide; the sole textual match for \"axiom\" outside Cited.lean is a comment, not a declaration. axiom_audit.txt shows every one of the 27 theorems drawing only Lean's three standard axioms plus a subset of the eight declared in Cited.lean, matching the paper's citation structure theorem by theorem. Spot-checked Main.lean against the manuscript: pal_EK_omega and rev_EK_omega are Tendsto statements of the empirical distribution to Phi(t) with centring and scaling LL b lam and its square root, matching Theorems 1.1 and 1.2 as stated. Of the eight cited results, Col, Banks-Shparlinski, Dartyge-Rivat-Swaenepoel and Granville-Soundararajan were confirmed to exist with the stated venues; Dartyge-Rivat-Swaenepoel (arXiv:2506.21642) is from June 2025, corroborating the submission's claim that the reversed-prime argument's input became available only that year. This is a sampling audit, not the full informal-to-formal correspondence review the lean-verified tier requires, so the tier stays where the submission itself placed it.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"No named poser: the typical value of $\\omega$ on palindromes and any Erdős-Kac law for reversed primes were simply absent from the literature rather than posed as a question, and reversed primes were not analytically accessible until the 2025 Dartyge-Rivat-Swaenepoel input. Level with Lower Bounds for Multivariate Independence Polynomials (12): a specialist analytic number theory result extending a known theorem family (Erdős-Kac laws, previously established for fixed digit sum and missing-digit sets) to two families for the first time, invisible outside the field.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.5281/zenodo.22078539","sourceName":"Zenodo preprint","collection":null,"collectionVersion":null,"links":[{"label":"Lean 4 formalisation, axiom audit and verification report","url":"https://github.com/vibefrtz/vibemath","kind":"code"},{"label":"Archived v1.0.0 (version DOI)","url":"https://doi.org/10.5281/zenodo.22078540","kind":"paper"}],"relations":[],"submittedBy":"vibefrtz","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"integer-domination-root-conjecture","name":"The Integer Domination Root Conjecture","shortName":"Integer domination roots","problemNumber":null,"field":"Graph polynomials","fieldGroup":"Combinatorics","statement":"Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial $D(G, x)$, proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a counterexample of order 33 with an integer domination root at $x = -4$, built from an S-unit branch cancellation mechanism.","posedBy":"S. Akbari, S. Alikhani, M. R. Oboudi, Y.-H. Peng","yearPosed":2010,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-31","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":["Saeid Alikhani","Max Griswold"],"aiRole":"\"The concept and theoretical formulation of the S-Unit Branch Cancellation mechanism were generated by Claude Fable 5.\" The authors verified all formal proofs and carried out independent computational validations. Alikhani is one of the conjecture's original posers.","verification":"unreviewed","verificationNote":"Author-verified by exact enumeration, and one author co-posed the conjecture being refuted; no external review yet.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":11,"significanceNote":"A 2010 specialist conjecture with a genuine partial-results literature (trees, unicyclic graphs, exhaustive small-order checks).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.00109","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"max-3-cut-sharp-hardness","name":"Sharp Hardness for MAX-3-CUT","shortName":"MAX-3-CUT hardness","problemNumber":null,"field":"Hardness of approximation","fieldGroup":"Theoretical computer science","statement":"Assuming the Unique Games Conjecture, it is NP-hard to approximate MAX-3-CUT better than the Frieze-Jerrum semidefinite program does, and similarly for Quantum MAX-CUT: the sharpness question in the Khot-Kindler-Mossel-O'Donnell line, connected to the Plurality is Stablest problem.","posedBy":"S. Khot, G. Kindler, E. Mossel, R. O'Donnell","yearPosed":2004,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"The Unique Games Conjecture this result assumes is claimed proved in OpenAI's October 2026 math release (catalogued separately); that would make the result unconditional, but the claim has not been independently checked.","claimIssueNote":null,"solveDate":"2026-07-31","model":"ChatGPT 5.6","modelMaker":"OpenAI","humanCollaborators":["Steven Heilman"],"aiRole":"ChatGPT 5.6 assisted in the preparation of the manuscript, including producing the spectral certificates in Propositions 4.1 and 4.4 and Lemmas 9.3 and A.4 - proof components, not prose.","verification":"unreviewed","verificationNote":null,"publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Sharp inapproximability for MAX-k-CUT has stood open since Khot-Kindler-Mossel-O'Donnell (2004) and is known across the approximation-algorithms community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.00333","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/unique-games-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"polya-conjecture-neumann-balls","name":"Pólya's Conjecture for Neumann Balls in Dimensions Three and Higher","shortName":"Pólya, Neumann balls","problemNumber":null,"field":"Spectral Geometry, Laplace Eigenvalues","fieldGroup":"Analysis","statement":"Pólya conjectured in 1954 that the Weyl-law expression bounds the eigenvalue counting function of the Laplacian. The paper proves the Neumann case for Euclidean balls in dimensions three and higher, extending the authors' earlier planar and Dirichlet results. Key difficulty: estimating zeros of derivatives of ultraspherical Bessel functions rather than of Bessel functions themselves.","posedBy":"George Pólya","yearPosed":1954,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"The ball case. For arbitrary domains Pólya's conjecture remains open; this continues the authors' programme after the planar disk, circular sectors, and the Dirichlet case in arbitrary dimensions.","claimIssueNote":null,"solveDate":"2026-07-31","model":"ChatGPT + Claude (several models)","modelMaker":null,"humanCollaborators":["Nikolay Filonov","Michael Levitin","Iosif Polterovich","David A. Sher"],"aiRole":"The paper carries an AI usage disclosure: several models of ChatGPT and Claude were used for mathematical discussions and editorial assistance, and all AI-assisted arguments and computations were independently checked by the authors, who take full responsibility. The disclosure does not separate which arguments were AI-assisted, so the lowest tier applies.","verification":"unreviewed","verificationNote":"No independent review, and the AI disclosure is general rather than pointing at specific results. The authors state they independently checked every AI-assisted argument and computation. The proof also uses conventional computer-assisted arguments, which are not the same thing. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Pólya's 1954 conjecture is a landmark question in spectral geometry, and outside tiling domains very few cases are known. This closes the Neumann ball case in every dimension.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.29305","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":3,"downvotes":0,"commentCount":0},{"slug":"online-shadow-tomography-matching-the-classical-bounds","name":"Online Shadow Tomography Matching the Classical Bounds","shortName":"Shadow Tomography with log(m)","problemNumber":null,"field":null,"fieldGroup":"Quantum information & computing","statement":"Online Shadow Tomography with $\\log m$ dependence, while retaining $\\mathrm{poly}(\\log(d)/\\epsilon)$ dependence. Also, matching the best classical bounds for Adaptive Data Analysis","posedBy":"Scott Aaronson","yearPosed":2016,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-31","model":"ChatGPT 5.6-Sol Pro","modelMaker":"OpenAI","humanCollaborators":["Sitan Chen","Ryan O'Donnell","Angelos Pelecanos","John Wright"],"aiRole":"Discovered the main proof","verification":"unreviewed","verificationNote":"Unreviewed preprint. The authors state they studied, refined and verified the model's ideas themselves and take full responsibility for every claim, proof and citation; that is the authors checking their own work, so it stays Unreviewed until someone independent looks. Recorded as Resolved rather than Partial because the stated target - matching the classical Adaptive Data Analysis rates - is fully achieved by Theorems 1.2 and 1.3. What remains open is whether those rates are optimal, which was never the question this entry records.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Closes the gap between online shadow tomography and the classical Adaptive Data Analysis rates, a question from the recent quantum learning-theory literature in Aaronson's shadow-tomography line. A clean resolution of a stated open question, but a young problem tracked by one subfield rather than a long-standing named conjecture: placed with the resolved recent-literature questions around 15, below Oddtown (20). Torn between 15 and 20; rule 3 takes the lower.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.29686","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"WittyFerret553","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"shannon-capacity-odd-cycles-records","name":"Record Lower Bounds for the Shannon Capacity of Odd Cycles","shortName":"Shannon capacity records","problemNumber":null,"field":"Zero-error information theory","fieldGroup":"Combinatorics","statement":"Determine the Shannon capacities of odd cycles beyond $C_5$, or improve the best explicit bounds. Lovasz's theta function settled $C_5$ in 1979 and every longer odd cycle has stayed open since. The current records, all obtained with model assistance and formally verified, are $\\Theta(C_7) \\ge 3.258805369885$, $\\Theta(C_{11}) \\ge 5.294502522149$, $\\Theta(C_{13}) \\ge 6.302455083464$, $\\Theta(C_{15}) \\ge 7.301600534487$, $\\Theta(C_{19}) \\ge 9.357192705918$, $\\Theta(C_{21}) \\ge 10.342455853338$ and $\\Theta(C_{23}) \\ge 11.328224257774$.","posedBy":"Claude Shannon","yearPosed":1956,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"record lower bounds for seven odd cycles; the exact capacities remain open for every odd cycle beyond C5","claimIssueNote":null,"solveDate":"2026-07-31","model":"ChatGPT 5.6 Sol Pro, ChatGPT 5.6 Sol, Claude Opus 5","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Pjotr Buys","Sven Polak","Jeroen Zuiddam","Yu Gao","Nathaniel Itty","Christopher D. Rosin","Chase Carstensen","Daniel Reichman"],"aiRole":"Three model-assisted papers in eleven days, each beating the last. Itty, Rosin, Carstensen and Reichman had ChatGPT-5.6 Sol Pro generate and run search programs across repeated prompts, returning explicit independent sets in strong graph powers that the authors checked. Gao then improved $C_7$ with a recursive construction and states that ChatGPT 5.6 Sol implemented all the code and expanded the proofs. Buys, Polak and Zuiddam followed both methods using ChatGPT 5.6 Sol Pro and Claude Opus 5, beat every previous bound, added three more cycles, and formalised the lot in Lean.","verification":"lean-verified","verificationNote":"The current records are formalised in Lean 4 at the linked repository, one base tuple per bound, so the seven stated inequalities are machine-checked rather than author-checked. We have not compiled it. Gao's intermediate record ships exact-integer verification code pinned to a fixed commit, and the earlier Itty-Rosin-Carstensen-Reichman constructions came with public data, prompts and checking code. All three are arXiv preprints; none is peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Shannon capacity of odd cycles, the classic post-Lovász-theta question since 1956.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21517","sourceName":"arXiv:2607.21517 - Improved lower bounds for the Shannon capacity of odd cycles","collection":null,"collectionVersion":null,"links":[{"label":"Current record, Lean-verified (Buys, Polak, Zuiddam), arXiv:2607.29681","url":"https://arxiv.org/abs/2607.29681","kind":"paper"},{"label":"Lean 4 formalization of all seven bounds","url":"https://github.com/spectra-research/shannon-capacity-lean","kind":"lean-proof"},{"label":"Intermediate C7 record by recursive construction (Gao), arXiv:2607.27869","url":"https://arxiv.org/abs/2607.27869","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"improved-integrality-of-donaldson-thomas-invariants-of-loop-quivers-gks-conjectu","name":"Improved integrality of Donaldson–Thomas invariants of loop quivers (GKS Conjecture 1.3 for twist knots)","shortName":"DT improved integrality","problemNumber":null,"field":"Arithmetic of Donaldson–Thomas / BPS invariants","fieldGroup":"Number theory","statement":"For the $m$-loop quiver, the numerical Donaldson–Thomas invariants $\\mathrm{DT}^{(m)}_n$ (Kontsevich–Soibelman/Reineke) satisfy $v_p(\\mathrm{DT}^{(m)}_n)\\ge v_p(n)$ for every prime $p\\ge5$, with exact defects at $p=2,3$: $v_3\\ge v_3(n)-[m\\equiv2\\ (3)]$ and $v_2\\ge v_2(n)-[m\\equiv2,3\\ (4)]$, and these bounds are attained. Hence the optimal integer with $n\\mid\\gamma(m)\\mathrm{DT}^{(m)}_n$ for all $n$ is exactly $\\gamma(m)=2^{\\varepsilon_2(m)}3^{\\varepsilon_3(m)}$. Via the identification of twist-knot extremal BPS invariants with loop-quiver DT invariants, this proves the Improved Integrality Conjecture (Garoufalidis–Kucharski–Sułkowski 2015, Conj. 1.3, an observation they credit to Kontsevich) for all twist knots with optimal constants, reproducing all twelve $\\gamma^\\pm$ values GKS tabulated empirically. Supporting new results: a derivative theorem for Gaussian binomials at roots of unity, the first $q$-supercongruence for DT invariants ($\\Phi_p(q)^2\\mid R_n$), an exact necklace formula for the quantized invariants, and a self-contained proof of the signed $p=2$ Kazandzidis supercongruence.","posedBy":"S. Garoufalidis, P. Kucharski, P. Sułkowski","yearPosed":2015,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"The sharp valuation bounds and optimal $\\gamma(m)$ are proved for ALL loop quivers $m\\ge2$, hence for the extremal BPS invariants of all twist knots (both rows, matching every twist-knot entry of GKS Table 1). Scope limits: the $m=3$/figure-eight divisibility $2n_r/r\\in\\mathbb Z$ was previously proved by Basor–Conrey–Morrison (arXiv:1703.00990), whose per-$r$ $2$-adic characterization for $m=3$ is finer than the uniform bound; the torus-knot case of GKS Conj. 1.3 (multi-vertex quivers, $\\gamma$ growing with the knot) remains open and is not claimed; the general-knot conjecture remains open. The Lean formalization covers the reduction to the classical Kazandzidis congruences, not those congruences themselves.","claimIssueNote":null,"solveDate":"2026-07-31","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"The model ran the project end to end under human direction in a single supervised session: identified the open conjecture from literature reconnaissance, built two structurally independent exact-arithmetic implementations of the refined DT invariants, mined the divisibility patterns, found and wrote complete proofs (Reineke's formula + Jacobsthal–Kazandzidis supercongruences, including a new self-contained proof of the signed $p=2$ case), wrote the paper, and produced a partial Lean 4/Mathlib formalization (no sorries; reduction to the classical congruence inputs machine-checked, sharpness and the combinatorial core unconditional). An independent AI referee agent (same model family, adversarial prompt) failed the first draft over a scope overclaim and missing prior art (Basor–Conrey–Morrison 2017 had the $m=3$ case), which were fixed before this announcement.","verification":"unreviewed","verificationNote":"No independent human review, and the repository says so itself. Evidence in the repo: two structurally independent implementations (plethystic CoHA engine vs. direct cyclic-word enumeration) agreeing on all computed invariants; Theorem 1 checked numerically for $m\\le10$, $n\\le120$ and the Kazandzidis inputs to $n=300$ with sharpness; and a two-round adversarial AI referee report, which is author-side and does not count as independent verification here.\n\nThe Lean was read here on 24 August 2026 rather than taken on trust. Dtformal.lean carries no sorry, no admit, no native_decide and no axiom declarations whatsoever, on Lean 4.33.0. The classical Kazandzidis congruences enter as explicit hypotheses (`KazOdd`, `Kaz2`) rather than as axioms, and those definitions are faithful to the real congruences - `KazOdd` is $p^{\\kappa+v_p(NK(N-K)\\binom{N}{K})}\\mid\\binom{pN}{pK}-\\binom{N}{K}$, and `Kaz2` carries the $(-1)^{K(N-K)}$ sign the $p=2$ case needs - so the conditional theorems are substantive rather than vacuous. `sharp_two`, `sharp_three` and `orbit_sum_zero` take no such hypothesis, matching the claim that sharpness and the combinatorial core are unconditional. Lean was not compiled here, and the output of AxiomCheck.lean is not committed to the repository.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"A numbered conjecture (1.3) from a well-cited 2015 paper of Garoufalidis, Kucharski and Sułkowski in Communications in Mathematical Physics, in the active knots-quivers correspondence programme, and one the authors credit to Kontsevich. Settled here for twist knots only. Level with the degree-six symplectic hypergeometric monodromy entry at 15 and just above the Kreiss-constant separation at 12: a real named conjecture with a genuine literature behind it, in a mathematical-physics and number-theory crossover that is invisible outside its own community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/minewefu/dt-loop-quiver-arithmetic","sourceName":"GitHub repository (paper, verification code, partial Lean formalization)","collection":null,"collectionVersion":null,"links":[{"label":"Garoufalidis, Kucharski, Sułkowski - Knots, BPS states, and algebraic curves (Conjecture 1.3)","url":"https://arxiv.org/abs/1504.06327","kind":"problem-record"},{"label":"Basor, Conrey, Morrison - Knots and ones (the earlier m=3 / figure-eight case)","url":"https://arxiv.org/abs/1703.00990","kind":"independent"}],"relations":[],"submittedBy":"fruppyz","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"han-xiong-integer-trace","name":"The Han-Xiong Integer Trace Conjecture","shortName":"Han-Xiong integer trace","problemNumber":null,"field":"q-series and partitions","fieldGroup":"Combinatorics","statement":"Han and Xiong extended the Gaussian binomial coefficient to positive rational index and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at the integer point. Ono's paper proves a support-dominance theorem settling the conjecture for a large family of rational parameters and reduces the full conjecture to unit fractions, with a finite computer verification covering every remaining case up to a fixed bound.","posedBy":"Guo-Niu Han, Huan Xiong","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Settles the conjecture for a large family and reduces the rest to unit fractions; the general unit-fraction case remains open.","claimIssueNote":null,"solveDate":"2026-07-31","model":"AxiomProver","modelMaker":"Axiom Math","humanCollaborators":["Ken Ono"],"aiRole":"The theoretical results were autonomously produced and verified in Lean by AxiomProver: the formal statements and proofs of Theorem 1.3, Corollary 1.4 and Theorem 1.5 were generated from a natural-language statement of the problem containing no proofs, then checked by the Lean proof assistant. An appendix records precisely what was and was not supplied to the system.","verification":"lean-checked","verificationNote":"The main theorems were formalized and kernel-checked in Lean by the same system that produced them; the human author wrote the paper from that formal development. Tier: AxiomProver generated the formal statements and proofs from a natural-language prompt; nobody independent has audited the statement fidelity.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A recent conjecture from a single paper, real but with no accumulated literature yet.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.00323","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Han & Xiong 2026 - Problem statement","url":"https://arxiv.org/abs/2606.01919","kind":"problem-record"},{"label":"Github repo of the proof","url":"https://github.com/AxiomMath/QBinomialTrace","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"1-28249-lower-bound-and-partial-upper-bounds-for-cost-preserving-single-source-u","name":"1.28249... Lower Bound and Partial Upper Bounds for Cost-Preserving Single-Source Unsplittable Flows","shortName":"Common-Point Interval Systems for SSUF","problemNumber":null,"field":null,"fieldGroup":"Combinatorics","statement":"For a single-source unsplittable flow, find the optimal universal additive constant $C$ s.t. every feasible fractional flow $x$ with arc costs $c$ should admit an unsplittable routing $y$ with $c^\\top y \\le c^\\top x$ and $y_a \\le x_a + C \\cdot d_{\\max}$ on every arc. Goemans conjectured $C=1$; this was disproved in July 2026 by a separate seven-vertex counterexample with critical constant $16/15$ (see the Dinitz–Garg–Goemans entry), leaving the optimal $C$ open.\n\nLower bound: a seventeen-terminal common-point interval instance certifies\n$$ C\\ \\ge\\ \\frac{1282494797984843521}{10^{18}}=1.28249\\ldots $$\n\nUpper bounds: the paper proves the first unconditional ceiling below 2, but for the codimension-two case only, at complement mass $q=2$. The record cells lie outside it, the $k=17$ instance having $q=11$, so that ceiling does not bound the record ladder. Two figures are conjectures rather than results: $4/3$ as the supremum of critical constants over common-point cells, approached but not attained and not an extrapolation from the ladder (Conjecture 1.1, Theorem 5.1), and $2$ for the universal constant itself (Conjecture 1.2). The proved gap remains $[1.28249\\ldots,\\ 2]$.","posedBy":"Dinitz, Garg, Goemans","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Record lower bound only. The sub-2 ceiling is the codimension-two case and does not bound the record ladder (k=17 has complement mass 11). 4/3 and 2 are conjectures; the proved gap is [1.28249, 2].","claimIssueNote":null,"solveDate":"2026-07-31","model":"GPT-5.6 Sol, Claude Fable 5, Claude Opus 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Sergey Nikolenko"],"aiRole":"GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5 carried out the search for constructions, symbolic envelope derivations, proofs, and the exact-verifier development; the human author framed the program, directed the search, set the claim scope, and verified all results independently by hand.","verification":"site-confirmed","verificationNote":"The k=17 lower bound is a finite certificate: the verifier rebuilds the 67-arc instance from raw interval data, rediscovers all paths by DFS, and enumerates all 2^17 routings in exact rational arithmetic. Re-run by the site from a clean clone on 2026-08-01; the exact constant, 15 minimizers, and 18-atom hull certificate reproduce. The deletion-star ceiling theorems are conventional proofs in an unreviewed preprint, checked by the author only, with no independent expert review and no formalization. Tier reflects the site's confirmation of the certificate; the structural results remain unreviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The residual optimal-constant question left open by the Dinitz-Garg-Goemans disproof; specialist, but rooted in a well-known 1999 conjecture.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/snikolenko/unsplittable-flows/","sourceName":"Unsplittable flows repository","collection":null,"collectionVersion":null,"links":[{"label":"Zenodo preprint","url":"https://zenodo.org/records/21716713","kind":"paper"},{"label":"Paper I's Zenodo record","url":"https://zenodo.org/records/21701162","kind":"paper"}],"relations":[{"to":"dinitz-garg-goemans-unsplittable-flow","kind":"builds-on","note":"The optimal-constant question this entry attacks is exactly what the Dinitz-Garg-Goemans disproof left open."}],"submittedBy":"BraveDingo215","upvotes":3,"downvotes":0,"commentCount":11},{"slug":"a-counterexample-to-han-s-conjecture","name":"Han's Conjecture","shortName":"Han's conjecture","problemNumber":null,"field":"Homological algebra","fieldGroup":"Algebra","statement":"For a finite-dimensional algebra $A$, finite global dimension forces $\\mathrm{HH}_n(A) = 0$ for all large $n$. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit finite-dimensional $\\mathbb{C}$-algebra with $\\mathrm{HH}_n(A) = 0$ for every $n \\geq 1$ and $\\mathrm{gldim}\\, A = \\infty$, built by transporting Krah's phantom into a singularity category via one-periodic folding.","posedBy":"Yang Han","yearPosed":2006,"ageNote":"Posed by Han in the J. London Math. Soc. in 2006 and open since, with a dedicated survey and positive results for commutative, monomial, graded, Koszul, local and cellular algebras. Happel asked the same for Hochschild cohomology; Buchweitz, Green, Madsen and Solberg answered that negatively in 2005, but their examples have nonzero homology in infinitely many degrees, so Han's survived.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.","claimIssueNote":null,"solveDate":"2026-07-31","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Bochao Kong","Yeqin Liu","Yu Shen"],"aiRole":"The paper's acknowledgment in full: \"The counterexample presented in this paper was discovered with the assistance of OpenAI's GPT-5.6 Sol Ultra model. All mathematical arguments and references were independently verified by the authors.\" In a counterexample paper the algebra is the whole result, so crediting the model with its discovery is a claim about the central object, not about support work. The hedge \"with the assistance of\" keeps this below the top tier.","verification":"unreviewed","verificationNote":"A preprint days old with no independent review. The surrounding evidence is unusually strong for something this new: two of the three authors disproved the differential-graded analogue of the same conjecture in December 2025, this paper extends that program, and the construction runs on named recent machinery (Krah's phantom from Inventiones 2024, Chen's partial-resolution theorem, the Wang-Arunachalam-Keller identification) rather than novel unpublished tools. That is provenance, not verification, and the tier reflects the difference.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A named conjecture of the representation theory of finite-dimensional algebras, open for twenty years, with its own survey article, and the surviving half of the Happel-Han pair after the cohomology version fell in 2005.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.00177","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Liu and Shen's dg-version counterexample (December 2025, no AI involvement)","url":"https://arxiv.org/abs/2512.12460","kind":"paper"}],"relations":[],"submittedBy":"FrostyBadger576","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"line-graph-signature-unbounded","name":"Signature of Connected Line Graphs","shortName":"Line graph signature","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A $14$-vertex witness has signature $2$, and chaining copies gives connected line graphs of signature $k + 1$ for every $k \\ge 1$ - the signature is unbounded.","posedBy":"Saieed Akbari et al.","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"no constant-bound repair of the conjecture is possible","claimIssueNote":null,"solveDate":"2026-07-30","model":"ChatGPT-5.6 Pro, Claude Fable 5","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Luke Francis","Trevor Uptain"],"aiRole":"The 14-vertex witness came from a ChatGPT-assisted search; Claude assisted an independent 48-vertex search and the development of the unbounded family. The authors reproduced everything with separately coded exact-arithmetic audits.","verification":"unreviewed","verificationNote":"Exact finite certificates with independent audit implementations; revised arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":4,"significanceNote":"A 2026 conjecture refuted within months of being posed.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.22874","sourceName":"arXiv:2607.22874 - The signature of connected line graphs is unbounded","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tu-deng-conjecture","name":"The Tu-Deng Conjecture","shortName":"Tu-Deng conjecture","problemNumber":null,"field":"Boolean functions; combinatorial number theory","fieldGroup":"Combinatorics","statement":"With $N = 2^k - 1$ and $\\mathrm{wt}(n)$ the binary Hamming weight, Tu and Deng conjectured that for every $1 \\leq t \\leq N-1$ at most $2^{k-1}$ pairs $(a,b)$ satisfy $a + b \\equiv t \\pmod N$ and $\\mathrm{wt}(a) + \\mathrm{wt}(b) < k$. Proved in full.","posedBy":"Ziran Tu, Yingpu Deng","yearPosed":2011,"ageNote":"Posed by Tu and Deng in 2011 in the course of constructing Boolean functions with optimal algebraic immunity, and attacked in pieces since; partial results were known before this proof.","solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-30","model":"ChatGPT 5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Renzhang Liu","Hengyi Luo","Tianyuan Xie"],"aiRole":"\"The authors thank ChatGPT 5.6 Pro for assistance with some of the mathematical work presented in this paper. The authors subsequently verified the argument and take full responsibility for the final content.\" No individual step is attributed, so the lower tier applies.","verification":"lean-checked","verificationNote":"The authors provide an accompanying Lean formalization described as an end-to-end machine-checked proof, including the intermediate results and the reduction to the original statement. Nobody independent has audited whether the formal statement faithfully expresses the conjecture, so this sits on the unaudited Lean rung.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture from cryptographic Boolean function design, open since 2011 with a documented trail of partial results, well known inside that community and invisible outside it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05187","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalization of the proof","url":"https://github.com/ifeelok92/tu-deng-conjecture-lean-formal-proof","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dimer-constant-cubic-lattice","name":"The Dimer Constant of the Cubic Lattice","shortName":"Dimer constant of Z^3","problemNumber":null,"field":"Statistical mechanics","fieldGroup":"Mathematical physics","statement":"The dimer constant of $\\mathbb{Z}^3$, the exponential growth rate of perfect matchings of the cubic lattice, has no closed form and is pinned only by bounds. The upper bound improves from Lundow's $0.457547$, standing since 2001, to $0.452130$, via diagonal transfer layers and an inequality of Csikvari relating the spectral radius of the transfer matrix to the constant.","posedBy":"classical lattice statistics","yearPosed":null,"ageNote":"The three-dimensional dimer problem has resisted exact solution since Kasteleyn and Temperley-Fisher solved the planar case in 1961; the previous record bound dates to Lundow in 2001.","solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"a record upper bound; the exact constant remains unknown","claimIssueNote":null,"solveDate":"2026-07-30","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Qidong He"],"aiRole":"The acknowledgement attributes the paper's two key ingredients to the model: the diagonal transfer layers, which replace the symmetry argument special to the rectangular torus, and the connection with Csikvari's inequality.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The three-dimensional dimer constant is a classical unsolved quantity in lattice statistical mechanics, with the planar case exactly solved and the cubic case reduced to a bound ladder.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.28810","sourceName":"arXiv:2607.28810 - A new upper bound on the dimer constant of Z^3","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"wowii-graph-conjecture-217","name":"Written on the Wall II, Graph Conjecture 217","shortName":"WOWII 217","problemNumber":null,"field":"Graph theory (automated conjecture)","fieldGroup":"Combinatorics","statement":null,"posedBy":"Written on the Wall II (automated conjecturing)","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-30","model":"Claude Opus 5 (with Gemini 3.1 Pro, GPT-5.3 Codex Spark, Grok 4.5)","modelMaker":null,"humanCollaborators":[],"aiRole":"Per the submitter's disclosure, the proof and submission preparation used Claude Opus 5, with the other models on bounded mechanical subtasks. A second, independent Lean proof of the same conjecture (ChatGPT 5.6 Sol and Codex) was submitted days earlier.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 proof; the axiom check includes native_decide (Lean.ofReduceBool / trustCompiler) for the exhaustive finite-graph certificates, which the submitter flags as the main trust assumption. Acceptance into the formal-conjectures repository is still pending, hence candidate status.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Written on the Wall II); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/google-deepmind/formal-conjectures/pull/4668","sourceName":"formal-conjectures PR #4668 - Mark WOWII Graph Conjecture 217 solved","collection":null,"collectionVersion":null,"links":[{"label":"Independent second Lean proof (PR #4656)","url":"https://github.com/google-deepmind/formal-conjectures/pull/4656","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fq-non-covering-congruence-systems","name":"Non-Covering Congruence Systems over Fq[x]","shortName":"Fq[x] covering systems","problemNumber":null,"field":"Function-field arithmetic","fieldGroup":"Number theory","statement":"Let $D_q(n)$ be the largest possible least degree of a polynomial omitted by a non-covering family of $n$ distinct-modulus congruence classes in $\\mathbb{F}_q[x]$. What is its asymptotic size? The answer is $D_q(n) = \\frac{n}{q-1} + O_q(1)$.","posedBy":"Rongyin Wang","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The leading asymptotic was determined up to a bounded q-dependent term. The author had conjectured whether $D_q(n) = |\\frac{n}{s}|$; this was proven to be false.","claimIssueNote":null,"solveDate":"2026-07-30","model":"ChatGPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Rongyin Wang"],"aiRole":"The model contributed the nested-modulus lower-bound construction and the idea of a truncated Chinese-remainder-theorem sieve for the upper bound; the author verified the arguments, added details and filled gaps.","verification":"unreviewed","verificationNote":"Author-verified arXiv preprint with theorem-specific AI attribution; relies on the known theorem that a non-covering family of n classes omits a polynomial of degree below n. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A function-field analogue question with a small literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27538","sourceName":"arXiv:2607.27538 - An asymptotic bound for non-covering congruence systems over Fq[x]","collection":null,"collectionVersion":null,"links":[{"label":"Wang 2026 - Conjecture statement","url":"https://www.sciencedirect.com/science/article/abs/pii/S1071579725001509?via%3Dihub","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":2},{"slug":"maxwell-three-charge-bound","name":"Maxwell's Three-Charge Equilibrium Bound","shortName":"Three-charge bound","problemNumber":null,"field":"Classical electrostatics","fieldGroup":"Mathematical physics","statement":"How many nondegenerate equilibrium points can the potential of three positive point charges have? Gabrielov, Novikov and Shapiro had proved at most $12$, and observed that their method would give $6$ if an auxiliary polynomial system had at least four solutions with multiplicity in each open quadrant. That four-solution statement holds, so the bound is $6$, and six is attained for special charge values.","posedBy":"Andrei Gabrielov, Dmitry Novikov, Boris Shapiro","yearPosed":null,"ageNote":"Maxwell's question about the number of equilibria of point charges dates to his 1873 Treatise; the sharp three-charge count is the modern form of it.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"the sharp bound for three charges; the general Maxwell bound was separately disproved in July 2026","claimIssueNote":null,"solveDate":"2026-07-30","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Andrei Gabrielov","Dmitry Novikov","T. Novikov","Boris Shapiro"],"aiRole":"The acknowledgement states that parts of the work, including the saddle-separation argument and the symbolic and numerical verification, were developed with the assistance of Claude, and that all results were independently verified by the authors. The abstract calls that separation argument at the unique saddle point of a separated-variable first integral the main new ingredient, so the model's contribution reaches the load-bearing step.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The three-charge case of Maxwell's equilibrium-counting problem, which the authors note has drawn renewed attention since the general conjecture fell.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.28785","sourceName":"arXiv:2607.28785 - From 12 to 6: Sharpening the Three-Charge Bound in Maxwell's Problem","collection":null,"collectionVersion":null,"links":[{"label":"The general Maxwell conjecture, disproved separately in July 2026, arXiv:2607.27197","url":"https://arxiv.org/abs/2607.27197","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"local-cohomology-nonclosed-support","name":"Local Cohomology Modules With Nonclosed Support","shortName":"Nonclosed local cohomology","problemNumber":null,"field":"Commutative Algebra, Local Cohomology","fieldGroup":"Algebra","statement":"Can a noetherian ring have a local cohomology module whose support is not closed - equivalently, one with infinitely many minimal primes? Huneke and Lyubeznik asked; the paper constructs such rings, so the answer is yes.","posedBy":"Craig Huneke and Gennady Lyubeznik","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-30","model":"ChatGPT 5.5 / 5.6","modelMaker":"OpenAI","humanCollaborators":["Bhargav Bhatt","Linquan Ma"],"aiRole":"Per the paper's AI disclosure, the examples arose through interactions with ChatGPT 5.5 and 5.6. The model's first attempt at Question 1.1 was flawed but carried the idea of using torsion points on elliptic curves; pushed further, it produced a counterexample, which the authors modified and simplified into the normal example of Theorem 2.4. In a separate interaction the model found the example behind Theorem 1.3. The authors wrote the paper with Codex assistance and take responsibility for correctness.","verification":"unreviewed","verificationNote":"No independent check. The authors state plainly that they are responsible for correctness, and the disclosure records that the model's first proposed solution was wrong before a usable idea emerged. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A named question of Huneke and Lyubeznik in commutative algebra, answered by construction; the finiteness behaviour of local cohomology is a long-running programme in the field.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.28096","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"graffiti-conjecture-6","name":"Graffiti Conjecture 6","shortName":"Graffiti 6","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Every finite connected simple graph G satisfies\n\n$$\\alpha(G)\\ge r(G)+\\ln(\\rho(G)),$$\n\nwhere $\\alpha(G)$ is the independence number, $r(G)$ is the radius, and $\\rho(G)$ is the minimum number of pairwise vertex-disjoint paths whose vertices cover $V(G)$.","posedBy":"Graffiti, reported by Ermelinda DeLaViña, Siemion Fajtlowicz, and Bill Waller","yearPosed":2002,"ageNote":"First documented in March 2002 and revised in May 2003; disproved approximately 24 years later.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Infinite family of counterexamples; mathematical argument internally checked, with external verification and novelty review pending.","claimIssueNote":null,"solveDate":"2026-07-30","model":"GPT-5.6 Thinking","modelMaker":"OpenAI","humanCollaborators":["Jackson (prompter)"],"aiRole":"GPT-5.6 Thinking produced and checked an infinite family of counterexamples. For each integer s >= 0, it considered a tree T_s formed from the path v_0v_1...v_{4s+7} by attaching leaves at v_{2s+2} and v_{2s+5}. It proved that\n\n$$\\alpha(T_s)=2s+5,\\qquad r(T_s)=2s+4,\\qquad \\rho(T_s)=3.$$\n\nSince $\\ln 3>1$, it follows that\n\n$$\\alpha(T_s)=2s+5<2s+4+\\ln 3=r(T_s)+\\ln\\rho(T_s).$$\n\nThus every T_s is a counterexample, disproving the conjecture and providing infinitely many counterexamples. The AI also audited the final proof line by line.","verification":"unreviewed","verificationNote":"The proof was checked line by line by GPT-5.6 Thinking. The radius, independence number, perfect matching, and path-covering number arguments were separately recomputed, including the smallest case s=0. The proof appears mathematically valid, but as of 2026-07-30 it has not been independently verified by an external graph theorist, a formal proof assistant, or peer review.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":"On Some Conjectures of Griggs and Graffiti — Ermelinda DeLaViña, Siemion Fajtlowicz, and Bill Waller","citationsSource":"https://www.uhd.edu/documents/academics/sciences/griggsngraffiti.pdf","citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Graffiti); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Lamptypeshi/graffiti-conjecture-6-counterexample/blob/main/graffiticonjecture6FALSE.pdf","sourceName":"Graffiti Conjecture 6 Counterexample","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"Lamp","upvotes":3,"downvotes":0,"commentCount":1},{"slug":"sombor-energy-conjecture","name":"Sombor-Energy Conjecture","shortName":"Sombor energy","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"Does every nontrivial finite simple graph have noninteger Sombor energy? If $\\rho_1,\\ldots,\\rho_n$ are the eigenvalues of the Sombor matrix of a graph $G$, its Sombor energy is\n\n$$E_{\\mathrm{SO}}(G)=\\sum_{i=1}^{n}|\\rho_i|.$$\n\nThe conjecture asserted that $E_{\\mathrm{SO}}(G)\\notin\\mathbb Z$ for every nontrivial graph. A connected graph on nine vertices is exhibited with $E_{\\mathrm{SO}}(G)=64$, disproving the conjecture.","posedBy":"Nima Ghanbari","yearPosed":2021,"ageNote":"First posted on arXiv in 2021 and published in 2022.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-30","model":"GPT-5.6 Thinking","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The AI constructed a connected bipartite graph on nine vertices and calculated its Sombor spectrum exactly. Writing its Sombor matrix in the block form\n\n$$S(G)=\\begin{pmatrix}0&B\\\\B^{T}&0\\end{pmatrix},$$\n\nthe singular values of $B$ were found to be\n\n$$5,\\quad 5,\\quad 11+\\sqrt{31},\\quad 11-\\sqrt{31}.$$\n\nTherefore,\n\n$$E_{\\mathrm{SO}}(G)\n=2\\left(5+5+(11+\\sqrt{31})+(11-\\sqrt{31})\\right)\n=64.$$\n\nThe AI also audited the edge list, degrees, connectivity, bipartition, matrix multiplication, characteristic polynomial, singular values and final energy calculation, and produced a self-contained proof.","verification":"unreviewed","verificationNote":"The proof has been internally audited using exact calculations. The graph has nine vertices, fifteen distinct edges and degree sequence $(4,4,4,3,3,3,3,3,3)$. Its connectivity, bipartition, Sombor matrix, product $B^{T}B$, singular values, complete spectrum and energy $64$ were independently recomputed within the AI conversation.\n\nThe result has not yet been checked by an independent graph-theory expert, peer reviewer or formal proof assistant. A literature search located the original conjecture, the 2023 partial-results paper and its 2024 corrigendum, but did not locate an equivalent connected counterexample. This search does not establish absolute priority, and no claim is made that this is the first or a new counterexample.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":6,"significanceNote":"A 2021 conjecture in chemical graph theory with a small literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Lamptypeshi/Sombor-Energy-Conjecture-Counterexample/blob/main/somborconnectedintegerenergyFALSE.pdf","sourceName":"Sombor Energy Conjecture Counterexample","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"Lamp","upvotes":3,"downvotes":0,"commentCount":0},{"slug":"fuglede-square-free-cyclic-rapid-primes","name":"Fuglede's Conjecture on Square-Free Cyclic Groups With Rapidly Growing Primes","shortName":"Fuglede, square-free cyclic","problemNumber":null,"field":"Harmonic Analysis, Spectral Sets","fieldGroup":"Analysis","statement":"Fuglede's conjecture asks whether a set tiles exactly when it is spectral. The paper proves it for an infinite sequence of square-free order cyclic groups: the tile-to-spectral direction for all square-free cyclic groups, and the spectral-to-tiling direction for those whose prime factors grow rapidly. Until now no cyclic group with an arbitrary number of distinct divisors was known to satisfy it.","posedBy":"Bent Fuglede","yearPosed":1974,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"A partial result. Fuglede's conjecture remains open for finite cyclic groups generally; this settles an infinite family and, for the spectral-to-tiling direction, only under rapid growth of the prime factors.","claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT + Gemini","modelMaker":null,"humanCollaborators":["Gábor Somlai"],"aiRole":"The author writes that the work at this pace would not have been possible without ChatGPT and Gemini, which assisted in verifying the proof, collecting the relevant literature and earlier results, and formatting the text. Verification and search are the bottom tier by our own definition; the mathematics is the author's.","verification":"unreviewed","verificationNote":"No independent check. The models are credited with verifying the author's proof rather than producing it. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Fuglede's conjecture is a well-known problem in harmonic analysis; this is the first time it is known for cyclic groups with an arbitrary number of distinct divisors.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26534","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tournaments-three-five-voters","name":"Tournaments Determined by Three and Five Voters","shortName":"Tournaments 3/5 voters","problemNumber":null,"field":"Social Choice, Tournaments","fieldGroup":"Combinatorics","statement":"Around the Kemeny median problem, which stays open for $m=3$ and $m=5$ voters, the paper refutes three conjectures on tournament inducibility: both conjectures of Milosz, Hamel and Pierrot (the 3-cycle extension for odd $m\\ge5$, and $\\mathrm{FAS}=\\mathrm{HS}_3$ at $n=11$), and the Shepardson-Tovey threshold conjecture.","posedBy":"Milosz, Hamel and Pierrot; Shepardson, Tovey","yearPosed":2018,"ageNote":"Two main conjectures posed by Milosz, Hamel, Pierot 2019; third conjecture only implicitly posed by Shepardson-Tovey 2009","solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Leonid Chindelevitch","Ararat Harutyunyan"],"aiRole":"The authors report using Claude for exploratory reasoning, implementation assistance, drafting and editing. Exploration and implementation are mathematical work; drafting and editing are not, and the disclosure does not separate them, so the lowest tier applies.","verification":"unreviewed","verificationNote":"No independent check, and the disclosure bundles mathematical exploration together with drafting, leaving the model's actual share unclear. Preprint, not refereed. All theorems in the paper LP or ILP-verified.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"Refutes three stated conjectures at once in the inducibility theory around the Kemeny median problem, a specialist but active corner of social choice.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26690","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Milosz, Hamel, Pierot 2019 - Problem statement","url":"https://www-labs.iro.umontreal.ca/~miloszro/iwoca/presentation_iwoca2018.pdf","kind":"problem-record"},{"label":"Github repo of the computations used","url":"https://github.com/Leonardini/Tournaments","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"maxwell-conjecture","name":"Maxwell's Conjecture on Point-Charge Equilibria","shortName":"Maxwell conjecture","problemNumber":null,"field":"Classical electrostatics","fieldGroup":"Mathematical physics","statement":"Do $n$ point charges whose electrostatic potential has only non-degenerate critical points always have at most $(n-1)^2$ of them? A configuration of five charges - three at the vertices of an equilateral triangle plus two small central charges pulled apart into a shallow bipyramid - has at least $24 > 16$ non-degenerate critical points, so the conjecture is false.","posedBy":"James Clerk Maxwell","yearPosed":2004,"ageNote":"The bound traces to Maxwell's 1873 Treatise; the precise non-degenerate form was formulated by Gabrielov, Novikov and Shapiro in 2007.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Philip Arathoon","Gavin Ball","Matthew D. Kvalheim"],"aiRole":"The idea behind the counterexample construction was suggested by the model; the authors verified all mathematical details and wrote the note.","verification":"unreviewed","verificationNote":"Short arXiv note with an explicit five-charge configuration and non-degeneracy verification; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Traces to Maxwell's 1873 Treatise; the modern form drove work in potential theory and real algebraic geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27197","sourceName":"arXiv:2607.27197 - The Maxwell Conjecture is False","collection":null,"collectionVersion":null,"links":[{"label":"Original conjecture formalization","url":"https://ui.adsabs.harvard.edu/abs/2004math.ph...9009G/abstract","kind":"lean-statement"}],"relations":[],"submittedBy":null,"upvotes":3,"downvotes":0,"commentCount":0},{"slug":"dm-stack-positivity-without-torsion-free","name":"Positivity on Deligne–Mumford Stacks Without the Torsion-Free Hypothesis","shortName":"DM positivity","problemNumber":null,"field":"Algebraic Geometry, Stacks","fieldGroup":"Geometry & topology","statement":"Casalaina-Martin and Zhjeqi proved that the first Chern class of every torsion-free coherent quotient of a tensor power of the logarithmic cotangent sheaf is pseudo-effective, noting in Remark 4.5 that torsion-freeness was imposed only for technical reasons. Can it be dropped? Yes.","posedBy":"Casalaina-Martin and Zhjeqi","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT 5.5 Pro + Danus","modelMaker":null,"humanCollaborators":["Shengyu Hou","Jihao Liu"],"aiRole":"The main result was obtained with ChatGPT 5.5 Pro and the Danus system, an agent built on Rethlas; human verification and polishing came afterwards. The authors caution that AI limitations mean related literature may have been missed.","verification":"unreviewed","verificationNote":"No independent check. The authors verified and polished the AI-produced argument themselves and explicitly flag that relevant references may have been missed. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Removes a hypothesis its own authors described as technical. A real but modest closure of a gap noted in a remark.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26989","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"optimal-exponent-relating-sumsets-and-difference-sets","name":"Optimal Exponent Relating Sumsets and Difference Sets","shortName":"Sum–difference exponent","problemNumber":null,"field":"Additive combinatorics","fieldGroup":"Number theory","statement":"For every finite set $A\\subset\\mathbb Z$ with $|A|\\ge 2$, define\n\n$$C(A)=\\frac{\\log\\left(|A+A|/|A|\\right)}\n{\\log\\left(|A-A|/|A|\\right)}.$$\n\nDetermine the largest possible value of $C(A)$, equivalently the least universal exponent $c$ such that\n\n$$\\frac{|A+A|}{|A|}\n\\le\n\\left(\\frac{|A-A|}{|A|}\\right)^c$$\n\nfor every such set $A$. The result proves that the supremum is exactly $2$, although no individual admissible set attains it.","posedBy":null,"yearPosed":null,"ageNote":"The result has been described publicly as solving a “50-year-old problem,” but the paper does not identify a precise original posing date. The optimality of the lower exponent was explicitly recorded as open by Merlijn Staps in 2014–2015. The 1969 reference concerns an early sum-dominant-set construction, not a clearly documented posing of this exact exponent question.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"New arXiv preprint with an author-provided Lean formalization; not yet peer-reviewed.","claimIssueNote":null,"solveDate":"2026-07-29","model":"Hy3","modelMaker":"Tencent Hunyuan","humanCollaborators":["Haowei Lin","Shanda Li"],"aiRole":"Tencent Hunyuan’s Hyra research agent, powered by the Hy3 model, was used to explore and optimize finite-set constructions. During an approximately 24-hour run, Hyra produced the construction underlying the paper after moving from finite numerical searches toward natural-language proposals of general constructions and supporting arguments.\n\nThe human authors independently checked the construction, corrected and rewrote the exposition, and prepared the final mathematical proof manually. GPT-5.6 Sol was used as an exploration judge and later helped translate the natural-language argument into a Lean 4 formalization. The language-model judgments were not used as proof certificates.","verification":"lean-verified","verificationNote":"This is a newly released arXiv v1 preprint and has not yet been peer-reviewed. It contains an explicit, self-contained mathematical construction and proof.\n\nThe authors also provide a Lean 4/mathlib formalization. The repository reports that `lake build` completes successfully with no `sorry` declarations or warnings. The principal asymptotic and supremum results use three `native_decide` certificates for elementary finite computations concerning a 12-element base-39 digit block. Consequently, those parts additionally trust Lean’s compiler and native execution, rather than relying exclusively on kernel reduction.\n\nThe formalization is strong supporting evidence, but it is author-provided, and no independent expert review was located as of 2026-07-30.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A classic Ruzsa-lineage question of additive combinatorics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27199","sourceName":"Lin and Li, “Settling the Optimal Exponent Relating Sumsets and Difference Sets,” arXiv:2607.27199","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"matthew","upvotes":4,"downvotes":0,"commentCount":2},{"slug":"son-pham","name":"Huneke-Wiegand Conjecture","shortName":"Huneke-Wiegand Conjecture","problemNumber":null,"field":"Commutative Algebra","fieldGroup":"Algebra","statement":"The Huneke–Wiegand Conjecture: Let $R$ be a one-dimensional Gorenstein local domain, and let $M$ be a finitely generated, non-zero, torsion-free $R$-module. If the tensor product $M \\otimes_R M^*$ is torsion-free, then $M$ is a projective (hence free) $R$-module.","posedBy":"Craig Huneke & Roger Wiegand","yearPosed":1994,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Verified by author of the conjecture","claimIssueNote":null,"solveDate":"2026-07-29","model":"GPT-5.6-Pro","modelMaker":null,"humanCollaborators":["Son Pham & Craig Huneke"],"aiRole":"Came up with the counterexample on shot. GPT-share chat for proof:\n\nhttps://chatgpt.com/c/6a6529a6-fb04-83ea-a397-a64ffed0b3d6","verification":"expert-verified","verificationNote":"The proposed data are:\n\nΓ = ⟨56,57,58,63,64,70,71,72,73,74,75,76,77,78,79,80,81,82,83, 87,89,90,93,95,96,97⟩,\n\nR = ℚ[t^Γ]_𝔪\n\nI = (t^56,t^70)R\n\nFull link: https://github.com/sonpham-org/huneke-wiegand-candidate-verification\nContains my own counter example proof and an independent verification by the conjecture author","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The Huneke-Wiegand conjecture, a known 1994 target in commutative algebra.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/sonpham-org/huneke-wiegand-candidate-verification","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"HiddenLemur412","upvotes":4,"downvotes":0,"commentCount":0},{"slug":"syk-ground-state-energy-sharp-bounds","name":"Sharp Bounds on the Ground State Energy of the SYK Model","shortName":"SYK ground state","problemNumber":null,"field":"Mathematical Physics, Random Matrices","fieldGroup":"Mathematical physics","statement":"For the Sachdev-Ye-Kitaev Hamiltonian on $n$ Majorana modes with $k$-body interactions, the paper proves $\\mathbb{E}\\|H\\|_{op} = (1-o(1))\\sqrt{2n}/k$ for super-constant $k \\le o(\\sqrt{n})$, confirming predictions of Garcia-Garcia, Jia and Verbaarschot and answering a question of Feng, Tian and Wei.","posedBy":"Feng, Tian and Wei; predicted by Garcia-Garcia, Jia and Verbaarschot","yearPosed":2019,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"GPT-5.5 Pro / GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Arpon Basu","Pravesh K. Kothari","Siddhant Midha"],"aiRole":"The authors analysed a preliminary version of their explicit deterministic operator with GPT-5.5 Pro and GPT-5.6 Pro, and credit GPT-5.6 Pro with suggesting Lemma 4.6, Proposition 4.14 and Lemma 4.15. They also used it to search related literature and to assist with verifying technical proofs.","verification":"unreviewed","verificationNote":"No independent check. The disclosure is unusually specific, naming the three results the model contributed, which is what makes the co-developed tier defensible here. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Settles the operator-norm asymptotics of a heavily studied model in quantum many-body physics, confirming a 2018 prediction and answering a question posed in 2019.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27185","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bosonic-quantum-capacity-non-gaussian","name":"Bosonic Quantum Communication Beyond the Thermal Threshold","shortName":"Non-Gaussian bosonic capacity","problemNumber":null,"field":"Quantum Information, Bosonic Channels","fieldGroup":"Quantum information & computing","statement":"Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a non-Gaussian state that beats it, giving positive quantum capacity in a region where every single-mode Gaussian input yields none.","posedBy":"Alexander Holevo and Reinhard Werner","yearPosed":1999,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Francesco Anna Mele","Giuseppe Catalano","Marco Fanizza","Vittorio Giovannetti","Ludovico Lami"],"aiRole":"From the paper's own AI-assisted research statement: some of the authors had tried hard to find such a non-Gaussian counterexample several years ago and failed; they tried again with ChatGPT 5.5 and failed again; ChatGPT 5.6 Sol then gave them the families of non-Gaussian counterexamples. They call the AI crucial to the work while taking full responsibility for its content.","verification":"unreviewed","verificationNote":"No independent review, but stronger than a bare claim: Appendix A turns the numerical evaluation into a rigorous bound by enclosing every quantity in certified real intervals, and a second certified witness is given in Appendix B. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Breaks a lower bound standing since Holevo and Werner in 1999 and shows a complete theory of noisy bosonic communication cannot stay within single-mode Gaussian states.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27449","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lami-regula-entanglement-irreversibility","name":"Lami-Regula Conjecture on Entanglement Irreversibility","shortName":"Entanglement irreversibility","problemNumber":null,"field":"Quantum information","fieldGroup":"Quantum information & computing","statement":"Is the irreversibility of entanglement manipulation robust in the strong-converse sense - a strict separation between the exponential strong-converse distillable entanglement and the entanglement cost, as conjectured by Lami and Regula? Yes: there are states for which any attempt to restore reversibility incurs an error growing exponentially in the number of copies, and the irreversibility persists even at polynomially growing error.","posedBy":"Ludovico Lami, Bartosz Regula","yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT (GPT-5.6 Sol)","modelMaker":"OpenAI","humanCollaborators":["Tulja Varun Kondra","Raphael Brinster","Hermann Kampermann","Dagmar Bruß","Nikolai Wyderka"],"aiRole":"Technical details of the proofs were developed with the help of ChatGPT (GPT-5.6 Sol, accessed July 2026); the authors thoroughly checked all output.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A 2023 Nature Physics conjecture at the center of entanglement reversibility.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27195","sourceName":"arXiv:2607.27195 - Very Strong Irreversibility of Quantum Entanglement","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-106","name":"Erdős Problem #106","shortName":"Erdős #106","problemNumber":106,"field":"Discrete Geometry, Packing","fieldGroup":"Geometry & topology","statement":"If $f(n)$ is the maximum total side length of $n$ interior-disjoint squares packed in the unit square, is $f(k^2 + 1) = k$? An exact rational configuration packs $17$ squares with total side length greater than $4$, refuting the identity at $k = 4$.","posedBy":"Paul Erdős","yearPosed":1932,"ageNote":"Dated to 1932 by Soifer, who heard it from Erdős directly (\"In 1932, the 19-year old Paul Erdős poses the following problem\"). Erdős independently dated it to \"more than 60 years ago\" writing in 1994.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"Claude Opus 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"The public Lean source credits Codex as formal author; The packing was discovered by Claude Opus 5 (Anthropic) against search infrastructure built and run by the submitter, and verified independently in exact rational arithmetic via separating-axis certificates.","verification":"lean-verified","verificationNote":"A 613-line, placeholder-free Lean proof of the counterexample; the erdosproblems.com page has now the result and is showing the conjecture as disproved.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":11,"significanceNote":"A numbered Erdos problem that an unusually dense reference trail on erdosproblems.com (9 sources), setting it above the typical entry in the catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/106","sourceName":"erdosproblems.com/106","collection":null,"collectionVersion":null,"links":[{"label":"Construction and verification code (GitHub)","url":"https://github.com/Sprite143/erdos-106-counterexample","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":4,"downvotes":0,"commentCount":0},{"slug":"optimal-partial-plank-coverings","name":"Optimal Partial Plank Coverings","shortName":"Partial plank coverings","problemNumber":null,"field":"Discrete Geometry, Coverings","fieldGroup":"Geometry & topology","statement":"Given planks of fixed total width, how should they be placed to cover as much of a convex body's volume as possible? Karoly Bezdek asked whether, for a Euclidean ball, the optimum is a single plank centred at the origin. It is, and the paper also settles every planar convex body.","posedBy":"Karoly Bezdek","yearPosed":2009,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Egor Bakaev","Alexander Polyanskii"],"aiRole":"The paper credits iterative discussions between the authors and ChatGPT with the proof of concavity of Minkowski subtraction - explicitly within the authors' own existing Minkowski-subtraction approach - and with the idea for the planar case. The overall approach is the authors'.","verification":"unreviewed","verificationNote":"No independent check, and the model's contribution is bounded by the authors to two steps inside their own framework. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A question of Bezdek in the lineage of Tarski's plank problem and Bang's theorem, answered for the ball and for all planar convex bodies.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27483","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Bezdek 2009, Section 5 - Conjecture statement","url":"https://arxiv.org/abs/0903.4637","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pencil-cubics-twelve-common-flex-lines","name":"Twelve Common Flex Lines in a General Pencil of Cubics","shortName":"12 flex lines","problemNumber":null,"field":"Algebraic Geometry, Plane Curves","fieldGroup":"Geometry & topology","statement":"Does a general pencil of plane cubics over $\\mathbb{C}$ have exactly $12$ common flex lines? Ciliberto, Miranda and Roé asked this in Remark 5.3 of their paper; the answer is yes.","posedBy":"Ciro Ciliberto, Rick Miranda, Joaquim Roé","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT 5.5 Pro + Danus","modelMaker":null,"humanCollaborators":["Jihao Liu","Yanze Wang"],"aiRole":"The main result was obtained with ChatGPT 5.5 Pro and the Danus system, an agent built on Rethlas; human verification and polishing came afterwards. The authors caution that AI limitations mean related literature may have been missed.","verification":"unreviewed","verificationNote":"No independent check. The authors verified and polished the AI-produced argument themselves and explicitly flag that relevant references may have been missed. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"A specialist question in classical projective geometry, posed in a 2026 remark and answered within months.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26396","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"regts-sevenster-mixed-partition-functions","name":"Mixed Partition Functions and Exponentially Bounded Edge-Connection Rank","shortName":"Regts–Sevenster conjecture","problemNumber":null,"field":"Graph Parameters, Tensor Categories","fieldGroup":"Combinatorics","statement":"Regts and Sevenster conjectured that a complex-valued graph parameter $f$ with $f(\\varnothing)=1$ has exponentially bounded edge-connection rank precisely when it is a mixed partition function. The paper proves it, with the numbers of even and odd colours bounded in terms of the rank bound.","posedBy":"Guus Regts and Bart Sevenster","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"Claude Fable 5 + GPT-5.6 Sol Pro","modelMaker":"Anthropic, OpenAI","humanCollaborators":["William Whistler"],"aiRole":"The acknowledgements say only that Claude Fable 5 and GPT-5.6 Sol Pro \"were used extensively in the development and preparation of this work\". That does not separate mathematical contribution from writing, so the lowest tier applies; read the disclosure rather than the tier.","verification":"lean-checked","verificationNote":"No independent check, and the AI disclosure is the vaguest in this batch - a single acknowledgements line covering development and preparation together. Preprint, not refereed. Lean proof available but not yet audited externally.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture of Regts and Sevenster characterising which graph parameters are partition functions, a central question in the area.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27198","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Regts & Sevenster 2021, footnotes - Conjecture statement","url":"https://dx.doi.org/10.4171/aihpd/100","kind":"problem-record"},{"label":"GitHub repo of the proof","url":"https://github.com/WillWhistler/Regts-Sevenster","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"wow-284-refuted","name":"Written on the Wall II, Conjecture 284","shortName":"WOW-284","problemNumber":null,"field":"Graph Theory (automated conjecture)","fieldGroup":"Combinatorics","statement":"WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five is at most the negative of its least distance eigenvalue. The paper refutes it with exact counterexamples of orders 38, 39, 40, 42 and 50, and develops a structural theory of the failure.","posedBy":"Written on the Wall II (automated conjecturing)","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":["Samuil Petkov"],"aiRole":"The author credits ChatGPT-5.6 Sol Pro with adversarial proof checking, proof exploration and Lean formalization, and assumes full responsibility for the mathematics, attribution and conclusions.","verification":"unreviewed","verificationNote":"No independent review, but the counterexamples are explicit and self-checking, and the source, exact certificates and build instructions are published at release v2.2.8 of the author's repository. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A machine-generated conjecture from the Written on the Wall II program, which by our rule sits at the bottom of the ladder regardless of how clean the refutation is.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27452","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Counterexample certificates and Lean sources","url":"https://github.com/SamPetkov/wow284","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lukic-conjecture","name":"The Lukic Conjecture","shortName":"Lukic conjecture","problemNumber":null,"field":"Spectral theory","fieldGroup":"Analysis","statement":"Let $\\mu$ be a probability measure on the unit circle with Verblunsky coefficients $\\alpha$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\\alpha$ into components localized at those points. A counterexample with two critical points of multiplicity three refutes it: the sequence satisfies the decomposition conditions while the corresponding weighted entropy is $-\\infty$.","posedBy":"Milivoje Lukić","yearPosed":2012,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Jun Yan"],"aiRole":"The counterexample - two phase modes with common power-decay exponent $3/20$ - was found by GPT-5.6, as the abstract states directly; the author developed and verified the construction.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint with the counterexample stated explicitly; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture in OPUC spectral theory (higher-order Szegő theorems).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26419","sourceName":"arXiv:2607.26419 - A Mixed-Resonance Counterexample to the Lukic Conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Lukic 2012 - Conjecture statement","url":"https://arxiv.org/abs/1210.6953","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"distinct-multiples-in-intervals","name":"Improved Bounds for Distinct Multiples in Intervals","shortName":"Distinct multiples","problemNumber":null,"field":"Number Theory, Erdős–Pomerance Functions","fieldGroup":"Number theory","statement":"For the Erdős–Pomerance functions $F(n)$ and $h_{\\mathbb{P}}(n)$ counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to $n$, the paper proves $F(n) \\ge h_{\\mathbb{P}}(n) \\ge n\\exp\\left(\\left(\\frac{\\log 2}{2} - o(1)\\right)\\frac{\\log n}{\\log\\log n}\\right)$, disproving Kominers' conjecture that $F(n) \\ll n\\log n$. The paper also significantly improves known upper bounds (which were on the order of $n^{3/2}$) to $F(n) \\le n^{4/3 + o(1)}$ and $h_{\\mathbb{P}}(n) \\le n^{4/3 - o(1)}$.","posedBy":"Scott Duke Kominers; functions introduced by Erdős and Pomerance","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT 5.x","modelMaker":"OpenAI","humanCollaborators":["Kaizhe Chen","Samuel Korsky"],"aiRole":"The note carries a dedicated Statement on AI saying that the main proofs in it were developed with the assistance of ChatGPT 5.x. That is a claim about the mathematics rather than the exposition, which is why this sits a tier above the rest of its batch.","verification":"unreviewed","verificationNote":"No independent check. The disclosure credits the model with the main proofs, so the result rests entirely on the author's own verification. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"Disproves a stated conjecture of Kominers and improves bounds of Ruzsa, van Doorn and Kominers on functions introduced by Erdős and Pomerance.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26450","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Kominers 2026 - Conjecture statement","url":"https://arxiv.org/abs/2607.10431","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"atom-centered-descriptor-completeness","name":"Completeness of Fixed-Order Atom-Centered Descriptors","shortName":"Fixed-order descriptor completeness","problemNumber":null,"field":"Invariants of point configurations","fieldGroup":"Geometry & topology","statement":"Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since was dissolved by going to a higher correlation order, leaving open whether the trispectrum (5-body correlations), or any fixed finite order, separates all noncongruent environments. It does not. There are noncongruent three-dimensional environments agreeing on every cluster of up to seven neighbours, and for each finite correlation order and angular cutoff there are continuous families of noncongruent environments with identical retained features.","posedBy":"Sergey N. Pozdnyakov, Michael J. Willatt, Albert P. Bartók, Christoph Ortner, Gábor Csányi, Michele Ceriotti","yearPosed":2020,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The key ingredients were already in the literature, decades old and in distant fields; what was missing was anyone connecting them to this question.","claimIssueNote":null,"solveDate":"2026-07-29","model":"Claude Opus 4.8 and Fable, Codex 5.5 and Sol 5.6","modelMaker":null,"humanCollaborators":["Michelangelo Domina","Michele Ceriotti"],"aiRole":"The paper is as much about the usage pattern as the result, and says so in its title. The authors had been stuck on these questions for years and could not get past the known Pozdnyakov examples. Coding agents built on Claude (Opus 4.8 and Fable) and Codex (5.5 and Sol 5.6) were given a summary of the field's literature and, across unstructured iteration, produced the degenerate configurations by locating results in unrelated communities and recognising what they implied here, among them the homometric structures studied in musical rhythm theory. The authors verified the constructions manually and with model-generated code, and lifted the cyclic degeneracies to three dimensions themselves. They also ran a reproducibility experiment, eight runs per model under each of two prompt conditions, to test how reliably a query of this kind lands.","verification":"unreviewed","verificationNote":"arXiv preprint, not peer-reviewed. The validation scripts and input structures are promised only on publication, so the constructions cannot yet be re-run from the public record; the authors state they checked them by hand and with model-generated code. The counterexamples are explicit point configurations, so they are checkable once released.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A six-year-old completeness question from the atomistic machine-learning literature, settled negatively at every fixed order. The authors themselves note the degenerate structures are far from plausible chemistry and largely reinforce a view the field had already reached.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26984","sourceName":"arXiv:2607.26984 - Using large language models to probe the limits of atom-centered structural descriptors","collection":null,"collectionVersion":null,"links":[{"label":"Pozdnyakov et al., the 2020 paper that posed it","url":"https://arxiv.org/abs/2001.11696","kind":"paper"}],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"modified-lyons-sidorova-conjecture","name":"The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths","shortName":"Lyons–Sidorova","problemNumber":null,"field":"Rough Paths, Path Signatures","fieldGroup":"Analysis","statement":"For a continuous bounded-variation path with signature $g$, logarithmic signature $l$ and increment $v$, the modified Lyons–Sidorova conjecture predicts the structure of $g$ when $R(l)=\\infty$. The paper proves it: $g=1$ when $v=0$, and otherwise a prefix $\\alpha$ of the centred path gives $S(\\gamma) = S(\\alpha)e^{v}S(\\alpha)^{-1}$.","posedBy":"Horatio Boedihardjo; formulated systematically by Boedihardjo, Geng and Wang, after the original conjecture of Lyons and Sidorova","yearPosed":2020,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.","claimIssueNote":null,"solveDate":"2026-07-29","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Elena Boguslavskaya"],"aiRole":"The author reports using ChatGPT for mathematical exploration, critical examination of arguments, testing of intermediate proof strategies, and drafting and editing, and states she reviewed and revised all AI-assisted material and takes full responsibility. The model is not credited with producing the proof.","verification":"unreviewed","verificationNote":"No independent check, and the disclosure places the model in a supporting role rather than crediting it with the argument. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture of Lyons and Sidorova about when a path signature determines the path up to conjugacy, in the setting of bounded variation.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26377","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"k-distinct-4k-barrier","name":"The 4^k Barrier for the k-Distinct Language","shortName":"k-distinct barrier","problemNumber":null,"field":"Parameterized automata","fieldGroup":"Theoretical computer science","statement":"Can the $k$-distinct language - words over $[n]$ of length at most $k$ with no repeated symbol - be recognized by an acyclic NFA of size $c^k n^{O(1)}$ for some $c < 4$? A construction of size $2^{1.96992k} n^{O(1)} < 3.918^k n^{O(1)}$ answers yes.","posedBy":"Ran Ben-Basat, Ariel Gabizon & Meirav Zehavi","yearPosed":2016,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-28","model":"ChatGPT / Codex 5.4-5.6 Pro, Gemini 3.1 Pro","modelMaker":"OpenAI / Google DeepMind","humanCollaborators":[],"aiRole":"The gadget-amplification framework - hashing symbols into many copies of a small local NFA gadget - was developed across ChatGPT/Codex 5.4-5.6 Pro and Gemini 3.1 Pro sessions.","verification":"unreviewed","verificationNote":"arXiv preprint with interval-arithmetic verification of the exponent and pinned verification code. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A concrete parameterized-complexity barrier question.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.25381","sourceName":"arXiv:2607.25381 - Breaking the 4^k barrier for the k-distinct language","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bipartite-bound-information","name":"Existence of Bipartite Bound Information","shortName":"Bound information","problemNumber":null,"field":"Classical & quantum information theory","fieldGroup":"Quantum information & computing","statement":"Does bipartite bound information exist: classical correlations between two parties and an eavesdropper that cost secret bits to create, yet from which no secret key can ever be distilled?","posedBy":"Nicolas Gisin & Stefan Wolf","yearPosed":2000,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-28","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The explicit example - a distribution on two bits and a trit with zero distillable key but positive secrecy cost - was found with GPT-5.6 Sol; the authors reconstructed the proof line by line.","verification":"unreviewed","verificationNote":"Authors reconstructed the proof line by line with exact ancillary checks; public arXiv preprint, not yet peer-reviewed. The paper also shows the distributions that originally motivated the conjecture are not themselves examples.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Gisin-Wolf 2000; a named open question of quantum key distillation for 25 years.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.25838","sourceName":"arXiv:2607.25838 - Bipartite bound information exists","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"boucksom-local-analytic-bertini","name":"Boucksom's Local Analytic Bertini Conjecture","shortName":"Local analytic Bertini","problemNumber":null,"field":"Pluripotential theory","fieldGroup":"Algebra","statement":"Does the analytic Bertini restriction theorem for multiplier ideals hold locally, outside a pluripolar exceptional set of fibers? Proved in full generality.","posedBy":"Sébastien Boucksom","yearPosed":2018,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-28","model":"Rethlas (GPT-5.6 Sol)","modelMaker":"OpenAI","humanCollaborators":["Mingchen Xia"],"aiRole":"The author had the proof idea before the AI era; Rethlas running GPT-5.6 Sol first carried out the details, and Xia then simplified and largely rewrote the proof.","verification":"unreviewed","verificationNote":"Author-rewritten arXiv preprint. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named Boucksom conjecture in complex/non-archimedean geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.25230","sourceName":"arXiv:2607.25230 - Analytic Bertini theorem II: the local case","collection":null,"collectionVersion":null,"links":[{"label":"Fujina 2018, Question 1.2 - Problem statement","url":"https://arxiv.org/html/1709.01406v3","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"probabilistic-automatic-complexity-at-most-three","name":"Probabilistic Automatic Complexity Is At Most Three","shortName":"Prob. automatic complexity","problemNumber":null,"field":"Automata Theory, Descriptional Complexity","fieldGroup":"Theoretical computer science","statement":"Gill introduced the probabilistic automatic complexity $A_P(w)$ of a string: the least number of states of a probabilistic finite automaton for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, no string with $A_P>3$ being known. The paper proves $A_P(w)\\le 3$ for every string over every finite alphabet, with an explicit three-state witness.","posedBy":"Christopher Gill","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-28","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":["Bjørn Kjos-Hanssen"],"aiRole":"The author states the construction was found in conversation with, and verified with the assistance of, Claude Fable 5, and that the proof was formalized in Lean with assistance from Harmonic's Aristotle.","verification":"unreviewed","verificationNote":"No independent review. A Lean formalization is reported but no artifact was located to check, so this does not carry the Lean-verified tier. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"Settles a question posed in the paper that introduced the measure, closing it completely rather than improving a bound.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26275","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"wow-conjecture-109","name":"Written on the Wall II, Graph Conjecture 109","shortName":"WoW 109","problemNumber":null,"field":"Graph invariants","fieldGroup":"Combinatorics","statement":"Must every connected graph satisfy the proposed upper bound on its independence number in terms of residue and largest induced-bipartite-subgraph order? The family $\\overline{K}_{2r+1} \\vee (K_r \\sqcup K_r)$ violates it for every $r \\ge 3$.","posedBy":"Graffiti (Written on the Wall II)","yearPosed":1996,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-28","model":"GPT-5.6 Sol Max (Codex)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The infinite counterexample family was found with GPT-5.6 Sol Max running in Codex; verified in Lean plus independent Python and C++ enumeration.","verification":"lean-verified","verificationNote":"Lean-checked disproof in the google-deepmind/formal-conjectures repository, with independent computational enumeration.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Written on the Wall II); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/google-deepmind/formal-conjectures","sourceName":"google-deepmind/formal-conjectures (WrittenOnTheWallII)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kemeny-three-voters","name":"Kemeny Rank Aggregation for Three Voters","shortName":"Kemeny, 3 voters","problemNumber":null,"field":"Computational social choice","fieldGroup":"Theoretical computer science","statement":"Is computing a Kemeny-optimal aggregate ranking NP-hard when the input consists of exactly three complete rankings? Hardness was known for every even $n \\ge 4$; three voters was the minimal open case, and $n = 2$ is polynomial-time solvable.","posedBy":"Cynthia Dwork, Ravi Kumar, Moni Naor & D. Sivakumar","yearPosed":2001,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-28","model":"GPT-5.6 Sol Ultra, Claude Fable 5","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Dominik Peters"],"aiRole":"GPT-5.6 Sol Ultra found the reduction from MAX CUT; Claude Fable 5 helped simplify parts of it. Together with earlier results, every fixed number of voters $n \\ge 3$ is now hard.","verification":"lean-verified","verificationNote":"The reduction is Lean-checked, alongside an author-written arXiv preprint.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The three-voter case left open by Dwork-Kumar-Naor-Sivakumar's foundational paper.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.25540","sourceName":"arXiv:2607.25540 - Kemeny rank aggregation is NP-hard for three voters","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hadwiger-debrunner-line-arrangements","name":"General Position for Planar Line Arrangements and $HD_2(p,3)$","shortName":"Hadwiger-Debrunner bound","problemNumber":null,"field":"Discrete geometry","fieldGroup":"Geometry & topology","statement":"For every $\\delta > 0$ and infinitely many $n$ there is a set of $n$ lines in the plane with no intersecting quadruple such that every subset of size at least $n^{4/5+\\delta}$ contains three concurrent lines. This improves the bound for a dual form of a theorem of Balogh and Solymosi, and yields an improved lower bound for the Hadwiger-Debrunner number $HD_2(p,3)$.","posedBy":"Hugo Hadwiger, Hans Debrunner","yearPosed":1957,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"improved bounds; the exact Hadwiger-Debrunner numbers remain open","claimIssueNote":null,"solveDate":"2026-07-28","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Oliver Roche-Newton"],"aiRole":"The AI disclaimer says the work was carried out in collaboration with ChatGPT, while the paper is human-written and the author takes full responsibility for its contents. No individual step is attributed, so the lowest tier applies.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Hadwiger-Debrunner numbers sit in the classical (p,q)-problem tradition solved by Alon and Kleitman, and their growth rates are actively tracked in discrete geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.25742","sourceName":"arXiv:2607.25742 - A general-position problem for planar line arrangements","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stanley-problem-4-differential-posets","name":"Stanley's Problem 4 on Differential Posets","shortName":"Stanley Problem 4","problemNumber":null,"field":"Differential posets","fieldGroup":"Combinatorics","statement":"For a differential poset $P$, must the weighted $2$-multichain series $M_{P,2}(q)$ be a rational multiple of $F_P(q)^2$, the square of its rank generating series?","posedBy":"Richard Stanley","yearPosed":1988,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-28","model":"TARS agent system","modelMaker":null,"humanCollaborators":[],"aiRole":"The construction was found by the TARS agent system (foundation model not disclosed); the proof was reconstructed and manually verified by the human authors.","verification":"unreviewed","verificationNote":"A locally finite $1$-differential poset with nonrational quotient series over every characteristic-zero field; the construction yields continuum many such series. Author-verified arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"From Stanley's 1988 differential-posets paper, a recognized source of problems.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24541","sourceName":"arXiv:2607.24541 - A negative answer to Stanley's Problem 4 on differential posets","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bondal-polishchuk-smooth-projective-counterexample","name":"Bondal-Polishchuk Conjecture for a Smooth Projective Variety","shortName":"Bondal-Polishchuk","problemNumber":null,"field":"Algebraic geometry","fieldGroup":"Algebra","statement":"Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no counterexample of the form $D^b(X)$ for a smooth projective variety was known. A particular weak Fano threefold provides one.","posedBy":"Alexei Bondal, Alexander Polishchuk","yearPosed":1993,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"first counterexample of the form D^b(X) for X smooth projective","claimIssueNote":null,"solveDate":"2026-07-28","model":"ChatGPT 5.6","modelMaker":"OpenAI","humanCollaborators":["Anya Nordskova"],"aiRole":"The AI disclosure is carefully bounded and worth quoting in full: the model most significantly found numerous mistakes in earlier attempted constructions, leading to the successful modifications, and it located the reference containing the threefold used. The author states that it did not produce the actual counterexample and was not used to generate any of the text.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A 1993 conjecture about exceptional collections that is standard background in derived categories of coherent sheaves and had resisted a smooth projective counterexample.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.25391","sourceName":"arXiv:2607.25391 - A smooth projective counterexample to Bondal-Polishchuk's conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"curvature-smoothing-flame-speed","name":"Monotone Slowdown of Turbulent Flame Speed under Curvature","shortName":"Curvature flame speed","problemNumber":null,"field":"Partial differential equations","fieldGroup":"Differential equations","statement":"Curvature is expected to smooth flame-front wrinkles and so reduce turbulent flame speed, and in two-dimensional shear flows this was proved. In three dimensions it fails: there is a smooth periodic shear flow for which introducing Markstein curvature diffusivity increases the effective flame speed predicted by the level-set G-equation.","posedBy":"combustion-theory expectation, proved in 2D by Liu, Xin and Yu","yearPosed":2018,"ageNote":"Not a formally posed conjecture but the prevailing expectation in the combustion literature, made rigorous in the two-dimensional case in 2018.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"within the curvature G-equation model, in three dimensions","claimIssueNote":null,"solveDate":"2026-07-28","model":"ChatGPT 5.5 Plus","modelMaker":"OpenAI","humanCollaborators":["Hung V. Tran","Jack Xin","Yifeng Yu"],"aiRole":"The paper devotes a section titled Exploratory Journey Leading to the Proof to this, and says the model played a significant role in developing the proof. Asked directly about monotonicity, several models including this one took the natural route of differentiating the cell problem and applying a maximum principle, which did not work. Once the authors switched to searching for an example with positive derivative, ChatGPT 5.5 Plus suggested the main formal steps leading to the construction: it helped identify promising ansatzes, organize the linearized calculation, and formulate the orbit-average mechanism showing how a positive value could arise. The authors verified, refined and made the ideas rigorous.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A widely held expectation in combustion theory with a rigorous two-dimensional theorem behind it, familiar to the front-propagation and homogenization community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.25185","sourceName":"arXiv:2607.25185 - Turbulent Flame Speed Can Increase under Curvature Smoothing","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stanley-differential-poset-rank-bound","name":"Stanley's Rankwise Lower-Bound Conjecture for Differential Posets","shortName":"Differential poset ranks","problemNumber":null,"field":"Differential posets","fieldGroup":"Combinatorics","statement":"Must every $r$-differential poset have at least as many elements in each rank as $Y^r$, the $r$-th Cartesian power of Young's lattice? For $r = 3$ the new construction has fourth-rank size $50$ against $51$ for $Y^3$.","posedBy":"Richard Stanley","yearPosed":1988,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-28","model":"TARS agent system","modelMaker":null,"humanCollaborators":[],"aiRole":"The counterexample was generated by the TARS agent system (underlying foundation model not disclosed) through autonomous mathematical search, then examined and independently verified by the human authors.","verification":"unreviewed","verificationNote":"Explicit construction verified by the human authors and published as an arXiv preprint. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"From Stanley's 1988 differential-posets paper, a recognized source of problems.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.22988","sourceName":"arXiv:2607.22988 - An explicit counterexample to Stanley's rankwise lower-bound conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"martinsson-steiner-fractional-chromatic","name":"Martinsson-Steiner Conjecture on Fractional Chromatic Number","shortName":"Fractional chromatic bound","problemNumber":null,"field":"Graph coloring","fieldGroup":"Combinatorics","statement":"Is the fractional chromatic number of every $d$-degenerate triangle-free graph at most $(1+o(1))\\frac{d}{\\log d}$, with a matching lower bound, as conjectured by Martinsson and Steiner? The upper bound is confirmed constructively for graphs of girth at least $5$, and the conjectured lower bound is established in a stronger form for every fixed girth; the original triangle-free case remains open.","posedBy":"Anders Martinsson, Raphael Steiner","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"girth >= 5 case; the triangle-free case remains open","claimIssueNote":null,"solveDate":"2026-07-28","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Peter Allen","Abhishek Dhawan","Jonathan A. Noel"],"aiRole":"The model solved an optimization problem the authors formulated to determine the correct shape of the fractional clique function, checked and simplified probabilistic and algebraic estimates, helped draft some calculations, and pointed the authors to a key reference. The construction and overall strategy are the authors', who take full responsibility.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A sharp recent conjecture in graph coloring.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.26271","sourceName":"arXiv:2607.26271 - Sharp bounds for the fractional chromatic number of high-girth d-degenerate graphs","collection":null,"collectionVersion":null,"links":[{"label":"Martinsson & Steiner 2025, Conjecture 5.2","url":"https://dx.doi.org/10.1017/fms.2025.10112","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kazhdan-lusztig-matroids-unimodality","name":"Unimodality of Kazhdan-Lusztig Polynomials of Matroids","shortName":"KL matroid unimodality","problemNumber":null,"field":"Algebraic combinatorics","fieldGroup":"Combinatorics","statement":"Are the Kazhdan-Lusztig polynomials of matroids always unimodal - in particular log-concave, or even real-rooted, as conjectured? No: representable matroids obtained by deleting points from finite projective geometries have non-unimodal Kazhdan-Lusztig polynomials over every finite field, so the log-concavity and real-rootedness conjectures are both false.","posedBy":"Katie Gedeon, Nicholas Proudfoot, Benjamin Young","yearPosed":2017,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-27","model":"Rethlas agent (GPT-5.6 Sol)","modelMaker":null,"humanCollaborators":["Ronnie Cheng","Shurui Liu"],"aiRole":"The result came from running the Rethlas research agent with base model GPT-5.6 Sol at maximum reasoning effort - the same agent behind the Analytic Bertini entry; the paper documents the run and the authors verified the constructions.","verification":"unreviewed","verificationNote":"arXiv preprint (v2) with explicit constructions from projective geometries; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Gedeon-Proudfoot-Young conjectures in the June Huh school's active program.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24186","sourceName":"arXiv:2607.24186 - Kazhdan-Lusztig polynomials of matroids need not be unimodal","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hall-ho-heat-flow-conjecture","name":"The Hall-Ho Heat Flow Conjecture for Random Matrices","shortName":"Heat flow conjecture","problemNumber":null,"field":"Random matrix theory","fieldGroup":"Probability & statistics","statement":"Hall and Ho conjectured how the zeros of the heat-flow-evolved characteristic polynomial of a random matrix behave in the large-$n$ limit. General cases are proved; in particular, for a complex Ginibre matrix the empirical measure of those zeros converges almost surely to the semicircle law.","posedBy":"Brian Hall, Ching-Wei Ho","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Proves general cases of the conjecture rather than every case.","claimIssueNote":null,"solveDate":"2026-07-27","model":"ChatGPT Pro 5.5, ChatGPT 5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Theodoros Assiotis"],"aiRole":"An unusually direct disclosure: \"In the course of the research presented here I have been using AI tools extensively, most significantly ChatGPT Pro 5.5 and ChatGPT 5.6 Sol Ultra for ideation, technical help, editing and checking the proofs and general editing and proofreading of the manuscript, at the level of a co-author. All mistakes are my own responsibility.\" The phrase \"at the level of a co-author\" is the author's own.","verification":"unreviewed","verificationNote":"A preprint, with no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture in the active Hall-Ho line on heat flow and random matrix zeros, specialist but with a clear attribution and a real following.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05185","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Hall & Ho 2025 - Conjecture 2.3","url":"https://link.springer.com/article/10.1007/s11005-025-01946-9","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"crouzeix-s-conjecture","name":"Crouzeix's Conjecture","shortName":"Crouzeix's conjecture","problemNumber":null,"field":"Matrix analysis","fieldGroup":"Analysis","statement":"Crouzeix conjectured in 2004 that for every square complex matrix $A$ and every polynomial $p$, $\\lVert p(A)\\rVert \\leq 2 \\max_{z \\in W(A)} |p(z)|$, where $W(A)$ is the numerical range of $A$ - that is, the numerical range is a 2-spectral set. Crouzeix proved a constant of 11.08 in 2007 and Crouzeix and Palencia lowered it to $1+\\sqrt{2}$ in 2017; the conjectured constant 2 is attained by $2\\times 2$ matrices. Jin proves the sharp bound by a function-theoretic route whose key theorem reduces the problem, via a sampling strategy, to a positivity condition; Lorist and Schwenninger independently prove it days later by combining double-layer potential machinery with a perturbation lemma for 2-dilations.","posedBy":"Michel Crouzeix","yearPosed":2004,"ageNote":"Posed by Crouzeix in 2004, with a documented ladder of constants since: 11.08 (Crouzeix, 2007), 1 + √2 (Crouzeix and Palencia, 2017), plus many special cases.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma for 2-dilations - produced with ChatGPT 5.6 Pro exploring proof strategies. The entry's headline axes record Jin's proof; the earlier version of this entry recorded Lorist-Schwenninger's as primary while Jin's AI provenance was still unknown. The complete (matrix-coefficient) form, which contains this scalar theorem as the case m = 1, is claimed in the OpenAI math release (September 2026); see complete-crouzeix-conjecture.","claimIssueNote":null,"solveDate":"2026-07-27","model":"GPT-5.6 Sol; ChatGPT 5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Shanmu Jin","Emiel Lorist","Felix L. Schwenninger"],"aiRole":"For the first proof: Jin, a neurosurgery resident with no specialized mathematical training, reports that the key result (Theorem 2) emerged during an approximately sixteen-hour autonomous run of GPT-5.6 Sol in ChatGPT Work mode - a public prompt adapted from the Cycle Double Cover run, web access denied, a branching portfolio of subagent strategies under adversarial audit, and no human intervention once started. Jin then simplified and verified the output; the repository publishes the prompt, successive manuscripts, a Lean formalization and an axiom audit. For the independent second proof, Lorist and Schwenninger disclose that ChatGPT 5.6 Pro was used to review previous approaches to the weaker spectral constant $1 + \\sqrt{2}$ and to identify a possible source of improvement in estimates involving iterates $f^n$ of extremal or approximately extremal functions.","verification":"expert-verified","verificationNote":"Independently expert-verified, publicly on record: Townsend and Greenbaum's essay of 14 August 2026 states that both authors and Michel Crouzeix himself \"have checked the proof thoroughly and believe that Dr. Jin's manuscript is correct\" - the conjecture's own poser among the verifiers, and Greenbaum co-organized the 2017 AIM workshop on the problem. This site read that essay in full and audited Jin's repository (commit 9df0783): 82 Lean files with zero sorry, zero axiom declarations and zero native_decide with comments stripped, on toolchain v4.28.0, alongside an Annals-formatted manuscript and the complete autonomous-run prompt - though the Lean was not compiled here and its statement-to-conjecture correspondence not audited, so the tier rests on the expert endorsement, not the formalization. The independent second proof by Lorist and Schwenninger (arXiv:2608.03841) has no comparable public endorsement yet and the essay stops short of vouching for it. Neither manuscript is refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":2,"renownNote":"English and Polish Wikipedia articles.","significance":35,"significanceNote":"A named 2004 conjecture at the centre of matrix analysis and operator theory: two decades of partial results, its own AIM workshop (2017), its own survey, and Wikipedia articles in two languages. Field-famous rather than household - level with Feige and Krauth-Mezard at 35, above the strong specialist band at 30 where it previously sat; the AIM workshop and the constant-lowering literature are the concrete differentiators.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.preprints.org/manuscript/202607.1919","sourceName":"Jin, The Numerical Range Is a 2-Spectral Set","collection":null,"collectionVersion":null,"links":[{"label":"Townsend and Greenbaum, The Neurosurgery Resident Who Proved Crouzeix's Conjecture (14 Aug 2026)","url":"https://alextownsend.net/essays/SIAMNews_CrouzeixConjecture.pdf","kind":"independent"},{"label":"Jin's repository: prompt, successive manuscripts, Lean formalization, axiom audit","url":"https://github.com/jinshanmu/CrouzeixConjecture","kind":"code"},{"label":"Lorist and Schwenninger, A solution to Crouzeix's conjecture (arXiv, 4 Aug 2026) - the independent second proof","url":"https://arxiv.org/abs/2608.03841","kind":"independent"},{"label":"AIM workshop on Crouzeix's conjecture (2017)","url":"https://aimath.org/pastworkshops/crouzeix.html","kind":"problem-record"},{"label":"Wikipedia: Crouzeix's conjecture","url":"https://en.wikipedia.org/wiki/Crouzeix%27s_conjecture","kind":"wikipedia"},{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/complete-crouzeix-conjecture","kind":"other"}],"relations":[],"submittedBy":"NimbleRaven553","upvotes":3,"downvotes":0,"commentCount":1},{"slug":"auslander-reiten-smalo-perfect-fields","name":"Infinitely Many Components in Auslander–Reiten Quivers over Perfect Fields","shortName":"Auslander–Reiten–Smalø","problemNumber":null,"field":"Representation Theory of Algebras","fieldGroup":"Algebra","statement":"For a perfect field $k$ and a representation-infinite finite-dimensional $k$-algebra $A$, the Auslander–Reiten quiver of $A$ has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for finite-dimensional algebras over perfect fields.","posedBy":"Auslander, Reiten and Smalø","yearPosed":1995,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-27","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Wen Chang","Quanyu Tang"],"aiRole":"The authors credit ChatGPT with the technical construction and verification of the semilinear twist argument in Lemma 4, and say it was used more substantially in extending the result from algebraically closed fields to arbitrary perfect fields - the step that gives the paper its stated generality.","verification":"unreviewed","verificationNote":"No independent review. The disclosure names the specific lemma and the extension the model contributed. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture of Auslander, Reiten and Smalø, established over perfect fields rather than only in special cases.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24466","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Auslander et al 1995, Conj. 3, p. 409 - Conjecture statement","url":"https://www.cambridge.org/core/books/abs/representation-theory-of-artin-algebras/conjectures/B0333BD84A593BE4AFF106E9362098B2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"feiges-conjecture","name":"Feige's Conjecture","shortName":"Feige's conjecture","problemNumber":null,"field":"Probability","fieldGroup":"Probability & statistics","statement":"Let $X_1,\\ldots,X_n$ be independent nonnegative random variables with $\\mathbb{E}X_i \\le 1$, and let $S$ be their sum. Is $\\mathbb{P}(S < \\mathbb{E}S + 1) \\ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\\delta \\ge 1$.","posedBy":"Uriel Feige","yearPosed":2004,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-27","model":"ChatGPT 5.6 Pro, GPT-5.6 Sol, Codex","modelMaker":"OpenAI","humanCollaborators":["Weibo Fu","Yanjun Han","Guanyang Wang","Jun Yan","Peng Zhang","Zhengqing Zhou","Zipei Nie","Jiaye Wei","Mark Stander"],"aiRole":"The primary paper states plainly that the proof was found by ChatGPT 5.6 Pro, combining the Vlassis-Thomas Dirichlet calibration theorem with Grünbaum-type convex geometry; the authors checked, revised and rewrote the argument, and the accompanying Lean formalization was developed with Codex. The independent second proof by Nie and Wei was obtained with the assistance of GPT-5.6 Sol. A further independent proof was found by Stander.","verification":"lean-verified","verificationNote":"An end-to-end Lean formalization of the $e^{-1}$ conjecture accompanies the primary paper, formalizing the Vlassis-Thomas theorem, Grünbaum's centroid theorem and the combining argument. Three independent AI-assisted proofs appeared within days; neither preprint is peer-reviewed yet.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Feige's 2004 bound, known across probability and TCS with two decades of partial results.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23980","sourceName":"arXiv:2607.23980 - Sharp small-deviation inequalities for sums of independent nonnegative random variables","collection":null,"collectionVersion":null,"links":[{"label":"Independent second proof (Nie and Wei), arXiv:2607.24528","url":"https://arxiv.org/abs/2607.24528","kind":"independent"},{"label":"Lean formalization of the e^-1 conjecture","url":"https://github.com/pengzhang91/Feige","kind":"lean-proof"},{"label":"Stander's proof","url":"https://doi.org/10.5281/zenodo.21622950","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"chafai-dadoun-youssef-log-energy-monotonicity","name":"Chafai-Dadoun-Youssef Questions on Logarithmic Energy Monotonicity","shortName":"Log-energy monotonicity","problemNumber":null,"field":"Random matrix theory","fieldGroup":"Probability & statistics","statement":"Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a finite-energy Wigner counterexample and a one-parameter family of Gaussian-regularised Bernoulli entry laws answer both questions negatively.","posedBy":"Djalil Chafai, Benjamin Dadoun, Pierre Youssef","yearPosed":2024,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-27","model":"ChatGPT Pro 5.5, ChatGPT 5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Theodoros Assiotis"],"aiRole":"The disclosure says the tools were used for ideation, technical help, editing and checking the proofs, and puts the contribution at the level of a co-author, while the author retains responsibility for mistakes. It does not separate which of the two counterexamples came from where.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"Two numbered questions from a recent random-matrix paper, documented but read within one community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24170","sourceName":"arXiv:2607.24170 - On non-monotonicity of logarithmic energy for random matrices","collection":null,"collectionVersion":null,"links":[{"label":"Chafai et al 2024 - Conjecture statement","url":"https://doi.org/10.1142/S2010326324500084","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kls-quadratic-forms","name":"KLS Conjecture for Quadratic Forms","shortName":"KLS for quadratic forms","problemNumber":null,"field":"Asymptotic convex geometry","fieldGroup":"Geometry & topology","statement":"Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is $\\operatorname{Var}\\langle MX, X\\rangle \\le C\\, \\mathbb{E}|\\nabla\\langle MX, X\\rangle|^2$ for every symmetric $M$?","posedBy":"Ravi Kannan, László Lovász & Miklós Simonovits","yearPosed":1995,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"with constant 2; also improves the global KLS bound to $O(\\log^{1/4} n)$","claimIssueNote":null,"solveDate":"2026-07-27","model":"ChatGPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Brayden Letwin"],"aiRole":"The key argument was developed in collaboration with ChatGPT-5.6 Pro and checked by the author.","verification":"unreviewed","verificationNote":"Author-checked arXiv preprint proving the quadratic-form case with constant 2 and deriving the global estimate $\\psi_n \\le C \\log^{1/4} n$. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The heart case of the KLS conjecture, the organizing problem of asymptotic convex geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24164","sourceName":"arXiv:2607.24164 - The KLS constant is O(log^(1/4) n)","collection":null,"collectionVersion":null,"links":[{"label":"The full KLS conjecture, claimed October 2026","url":"https://vibemathed.com/problem/kannan-lovasz-simonovits-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"bellman-lost-in-forest-golden-gnomon","name":"Bellman's Lost-in-a-Forest Problem for the Golden Gnomon","shortName":"Lost in a forest (gnomon)","problemNumber":null,"field":"Convex geometry","fieldGroup":"Geometry & topology","statement":"What is the shortest curve guaranteed to reach the boundary of the golden gnomon - the isosceles triangle with equal sides $1$ and apex angle $108^\\circ$ - from an unknown starting position and heading? The optimum is a symmetric seven-piece path of segments, circular shoulders and tangents, of exactly determined transcendental length $C = 1.282676\\ldots$ - the first proved exact optimum for an isosceles triangle with base angle below $45^\\circ$.","posedBy":"Richard E. Bellman","yearPosed":1956,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Bellman's problem for general regions remains open","claimIssueNote":null,"solveDate":"2026-07-27","model":"Claude Fable 5, GPT-5.6 Sol, Claude Opus 5","modelMaker":"Anthropic / OpenAI","humanCollaborators":["Alexander Temerev","Alessio Doria"],"aiRole":"The models were used throughout: to search out the extremal curve, to draft the arguments, and to write the accompanying Lean 4 development. The paper states precisely which steps are machine-checked, and notes those checks hold regardless of how the statements were found.","verification":"unreviewed","verificationNote":"Lean 4 verifies the two finite algebraic certificate families and the discrete ledger identities, but not the full argument end to end; the paper's appendix states exactly which steps are machine-checked. arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":1,"renownNote":null,"significance":30,"significanceNote":"Bellman's 1956 lost-in-a-forest problem, a fixture of unsolved-problem collections.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24483","sourceName":"arXiv:2607.24483 - The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon","collection":null,"collectionVersion":null,"links":[{"label":"Lean verification development","url":"https://github.com/atemerev/gnomon","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":8,"downvotes":0,"commentCount":0},{"slug":"gaussian-kinetic-trace-question","name":"Kinetic Trace Estimates in the Gaussian Model","shortName":"Kinetic trace estimates","problemNumber":null,"field":"Kinetic theory","fieldGroup":"Differential equations","statement":"Does the natural trace estimate hold for kinetic energy spaces in the unrestricted Gaussian velocity model on bounded domains (Question 1.8 of Albritton, Armstrong, Mourrat and Novack)? No: for each $1 \\le p < 2$ there are counterexamples on every bounded $\\mathrm{C}^{1,1}$ domain in dimension $d \\ge 2$. The paper also identifies the sharp boundary-regularity threshold $\\mathrm{C}^{1,1/2}$ for the natural trace weight.","posedBy":"Dallas Albritton, Scott Armstrong, Jean-Christophe Mourrat, Matthew Novack","yearPosed":2024,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"the named open question is answered negatively; the paper's positive theory goes further","claimIssueNote":null,"solveDate":"2026-07-27","model":"GPT-5.5 Pro, GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Lukas Niebel","Lisa Valentini"],"aiRole":"GPT-5.5 Pro provided preliminary counterexamples and GPT-5.6 Sol an initial proof of the natural half-space trace estimate; generative AI was also used in developing some of the subsequent arguments. The authors developed and verified the final theory.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A named open question from a 2024 kinetic-theory paper by leading authors.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24708","sourceName":"arXiv:2607.24708 - Sharp kinetic trace theory","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kuperberg-six-cylinder","name":"Kuperberg's Six-Cylinder Conjecture","shortName":"Kuperberg cylinders","problemNumber":null,"field":"Discrete geometry","fieldGroup":"Geometry & topology","statement":"How many pairwise non-overlapping infinite circular cylinders of unit radius can simultaneously touch a unit ball? Kuperberg conjectured in 1990 that the maximum is six.","posedBy":"Włodzimierz Kuperberg","yearPosed":1990,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-27","model":"Claude (version not disclosed)","modelMaker":"Anthropic","humanCollaborators":["Ivan Matić","Radoš Radoičić"],"aiRole":"The proof reduces the upper bound to 2,954,984 exact rational-polynomial cases, each an elementary arithmetic check by a deterministic verifier; the reduction and certificates were developed with Claude.","verification":"unreviewed","verificationNote":"Computer-assisted proof with a fully reproducible exact certificate, published as an arXiv preprint. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A known 1990 packing conjecture with decades of attention.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24691","sourceName":"arXiv:2607.24691 - A computer-assisted proof of Kuperberg's six-cylinder conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"quantum-conditional-entropy-continuity","name":"Sharp Continuity Bound for Quantum Conditional Entropy","shortName":"Entropy continuity bound","problemNumber":null,"field":"Quantum information theory","fieldGroup":"Quantum information & computing","statement":"What is the optimal uniform continuity bound for quantum conditional entropy in trace distance, depending only on the dimension of the conditioned system? The sharp bound $h_2(\\delta) + \\delta \\log(d^2 - 1)$ up to $\\delta = 1 - d^{-2}$, conjectured by Wilde, is proved.","posedBy":"Mark M. Wilde","yearPosed":2019,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-27","model":"ChatGPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The key proof idea, adapting the tight classical argument of Alhejji and Smith to the fully quantum setting, was developed with ChatGPT-5.6 Sol.","verification":"unreviewed","verificationNote":"Five-author arXiv preprint. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A sharpness question posed in the quantum Shannon theory literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24687","sourceName":"arXiv:2607.24687 - Sharp continuity of quantum conditional entropy","collection":null,"collectionVersion":null,"links":[{"label":"Wilde 2019 - Conjecture statement","url":"https://arxiv.org/abs/1909.01755","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"vandermonde-powers-non-snp","name":"Powers of the Vandermonde Determinant Are Eventually Non-SNP","shortName":"Vandermonde non-SNP","problemNumber":null,"field":"Algebraic combinatorics","fieldGroup":"Combinatorics","statement":"Monical, Tokcan and Yong conjectured that every fixed positive power of the Vandermonde determinant fails to have saturated Newton polytope in sufficiently many variables. For every even power $k \\ge 4$ there is an explicit lattice point of the Newton polytope of $a_{\\delta_k}^k$ with vanishing coefficient, obtained from a Dyson constant-term identity; the odd case follows by alternation, proving the conjecture.","posedBy":"Cara Monical, Neriman Tokcan, Alexander Yong","yearPosed":2017,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-26","model":"Codex (GPT-5.6 Sol Extra High)","modelMaker":"OpenAI","humanCollaborators":["Thien Le","Melanie Weber"],"aiRole":"The key even-power construction and the proof strategy arose from prompting OpenAI Codex; the complete transcript appears in the paper's appendix. The authors subsequently checked and organized the argument.","verification":"unreviewed","verificationNote":"arXiv preprint with the prompting transcript in an appendix and an accompanying Lean formalization (coverage per its repository); not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A Monical-Tokcan-Yong conjecture in the SNP program.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23828","sourceName":"arXiv:2607.23828 - Powers of the Vandermonde determinant are eventually non-SNP","collection":null,"collectionVersion":null,"links":[{"label":"Lean formalization","url":"https://github.com/steven-le-thien/vandermonde-snp","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"covering-number-c-12-6-4","name":"The Covering Number $C(12,6,4)$","shortName":"Covering number C(12,6,4)","problemNumber":null,"field":"Design theory","fieldGroup":"Combinatorics","statement":"A $t$-$(v,k,\\lambda)$ covering is a family of $k$-subsets of a $v$-set meeting every $t$-subset at least $\\lambda$ times, and $C(v,k,t)$ is the least number of blocks. The recorded bounds for $C(12,6,4)$ were $40 \\le C(12,6,4) \\le 41$. No $4$-$(12,6,1)$ covering with $40$ blocks exists, so $C(12,6,4) = 41$.","posedBy":"covering design tables","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"closes a one-block gap in the covering tables; the analogous next case is not reachable by this method","claimIssueNote":null,"solveDate":"2026-07-26","model":"GPT-5.6 Sol, GPT-5.6 Terra, Claude Fable 5","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Charlie Krug"],"aiRole":"The disclosure lists exploratory analysis, computational search, supporting code, manuscript drafting and revision, and proofreading, without separating which step came from where, so the lowest tier applies.","verification":"unreviewed","verificationNote":"The non-existence argument is a forced-structure search: a counting argument pins every point to degree exactly 20 and forces each point link to be an optimal 3-(11,5,1) covering, collapsing the search space. Single-author arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"An open value in the covering design tables, the kind of documented specific gap that design theorists track but that has no audience beyond them.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23766","sourceName":"arXiv:2607.23766 - The covering number C(12, 6, 4) is 41","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"djament-problem-grothendieck-category","name":"Djament's Problem on Locally Noetherian Grothendieck Categories","shortName":"Djament's problem","problemNumber":null,"field":"Category theory","fieldGroup":"Algebra","statement":"Djament asked whether a Grothendieck category satisfying suitable finiteness and exactness conditions must be equivalent to a module category. In the locally noetherian case the answer is no: there is a Grothendieck category with a noetherian generator satisfying AB4* that is not equivalent to a module category, built as a Gabriel quotient of a module category over an endomorphism ring of Herbera, Prihoda and Wiegand.","posedBy":"Aurelien Djament","yearPosed":2025,"ageNote":null,"solveType":"disproved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"the locally noetherian case; whether such a category can fail to admit a projective generator is left open as Problem 1.3","claimIssueNote":null,"solveDate":"2026-07-26","model":"ChatGPT (GPT-5.5, GPT-5.6)","modelMaker":"OpenAI","humanCollaborators":["Ryo Kanda"],"aiRole":"The disclosure attributes two specific things. The tool suggested examining the example in Section 8 of Herbera-Prihoda-Wiegand as a possible source of a negative answer, which is the example the paper is built on, and it suggested a proof strategy. The author independently checked that strategy and supplied the proof.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named problem about when Grothendieck categories are module categories, restated as Problem 5.13 by Martini, Parra, Saorin and Virili.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23520","sourceName":"arXiv:2607.23520 - A Grothendieck category with a noetherian generator and exact products","collection":null,"collectionVersion":null,"links":[{"label":"Martini et al 2025, Problem 5.13 - Problem statement","url":"https://arxiv.org/abs/2508.00670","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"zero-forcing-versus-independence","name":"Zero Forcing versus Independence in Subcubic Graphs","shortName":"Zero forcing vs independence","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Is the zero forcing number of every connected graph with maximum degree $3$ at most its independence number plus one? A connected 24-vertex subcubic graph with independence number $9$ and zero forcing number $11$ refutes this 2017 TxGraffiti conjecture, and a 36-vertex cubic variant refutes the cubic form: $Z = \\alpha + 2$ is attained.","posedBy":"TxGraffiti (automated conjecturing program)","yearPosed":2017,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-26","model":"Claude Opus 5","modelMaker":"Anthropic","humanCollaborators":["Mikko Fischer"],"aiRole":"The counterexamples were found with the assistance of Claude Opus 5, directed by the author, who independently verified them - a conjecture generated by one automated system falling to a search assisted by another.","verification":"unreviewed","verificationNote":"Explicit finite counterexamples, independently checked by the author; arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (TxGraffiti); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23664","sourceName":"arXiv:2607.23664 - A counterexample to the zero forcing versus independence conjecture for cubic and subcubic graphs","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"brezis-open-problem-5-6-fourier-summation","name":"Brezis's Open Problem 5.6 on Universal Fourier Summation","shortName":"Brezis OP 5.6","problemNumber":null,"field":"Harmonic analysis","fieldGroup":"Analysis","statement":"Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does $\\sum_n \\sigma_{n,\\varepsilon} n |\\hat f(n)|^2 \\to \\deg f$ hold for Holder maps below the threshold? No. For every $0 < \\alpha < 1/3$ there is an $f \\in C^{0,\\alpha}(S^1;S^1)$ for which the sum fails to converge to $\\deg f$, answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all $p > 3$. The endpoint $C^{0,1/3}$ is left unresolved.","posedBy":"Haim Brezis","yearPosed":2023,"ageNote":null,"solveType":"disproved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"negative below the 1/3 threshold; the endpoint case is still open","claimIssueNote":null,"solveDate":"2026-07-26","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Michal Cieszynski"],"aiRole":"The disclosure states that ChatGPT generated preliminary drafts of the proofs in the manuscript. The author then verified each argument in detail, revised the proofs where necessary, checked the cited sources, determined the final formulation of all results, and takes sole responsibility for the content.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; the construction combines degree-zero quotients of Blaschke factors with a Baire category argument. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A numbered entry on Brezis's published list of favourite open problems, a recognized problem list in analysis, with the degree-and-Fourier-moduli circle of questions attached to it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23598","sourceName":"arXiv:2607.23598 - Nonexistence of universal Fourier summation formulas for the degree below the Holder threshold","collection":null,"collectionVersion":null,"links":[{"label":"Brezis 2023, Problems 5.6-5.8 - Problem statement","url":"https://doi.org/10.4171/RLM/1008","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"carlson-depth-conjecture","name":"Carlson's Associated-Prime Depth Conjecture","shortName":"Carlson depth","problemNumber":null,"field":"Group cohomology","fieldGroup":"Algebra","statement":"Is the depth of the mod-$p$ cohomology ring of every finite group realized as the dimension of one of its associated primes? For $G = \\operatorname{SmallGroup}(128, 859)$ over $\\overline{\\mathbb{F}}_2$ the ring has depth $2$ while every associated-prime quotient has dimension at least $3$.","posedBy":"Jon F. Carlson","yearPosed":1995,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-26","model":"TARS agent system","modelMaker":null,"humanCollaborators":["Xinan Dai","Wenhao Deng","Yingdong Shi","Tailin Wu","Yuchen Yang"],"aiRole":"The candidate group was found by the TARS agent system (foundation model not disclosed); the counterexample is certified by exact GAP/Singular computations audited by the human authors.","verification":"unreviewed","verificationNote":"Exact computational certificate (verifier and certificates ship with the paper, checkable with the Python standard library alone) plus a human proof audit. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Carlson's 1995 conjecture in modular representation theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23732","sourceName":"arXiv:2607.23732 - An exact counterexample to Carlson's associated-prime depth conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"two-adic-absolute-galois-presentation","name":"Explicit Presentation of the 2-adic Absolute Galois Group","shortName":"Galois group of Q2","problemNumber":null,"field":"Algebraic number theory","fieldGroup":"Algebra","statement":"Give an explicit profinite presentation of $\\operatorname{Gal}(\\overline{\\mathbb{Q}}_2 / \\mathbb{Q}_2)$. The tame local cases were settled by the early 1980s; the dyadic case was the last one missing. The new presentation has four generators, two word relations and a pro-$2$ condition on the wild generators.","posedBy":null,"yearPosed":1982,"ageNote":"Posed year approximate: the explicit local theory away from p = 2 was complete by the early 1980s, leaving the dyadic case.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-26","model":"ChatGPT-5.5 Pro (GPT-5.6 A/B), Claude Fable 5, Claude Opus 4.8","modelMaker":"OpenAI / Anthropic","humanCollaborators":["David Roe","David Turturean"],"aiRole":"A ChatGPT Pro conversation produced the candidate presentation with an informal proof; it passed Roe's finite-quotient verifier on all 5,402 test groups, and the proof was then formalized twice in Lean 4 with coding agents.","verification":"lean-verified","verificationNote":"Two separately initiated Lean 4 formalizations, checked modulo 7 and 9 named interfaces to the classical literature respectively. Manuscript public with an interactive web edition; not yet externally peer-reviewed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The p=2 gap left by Jannsen-Wingberg (1982), a known hole in Galois theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://roed314.github.io/gq2/","sourceName":"A presentation of the absolute Galois group of Q2 (project site)","collection":null,"collectionVersion":null,"links":[{"label":"Finite-quotient verifier (5,402 test groups)","url":"https://roed314.github.io/gq2/verifier/","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"carrasco-conjecture-ozf","name":"Carrasco's Conjecture on the O'Shea-Zames-Falb Test","shortName":"Carrasco/OZF conjecture","problemNumber":null,"field":"Control theory","fieldGroup":"Differential equations","statement":"Is the O'Shea-Zames-Falb multiplier test necessary for robust stability of Lur'e systems with slope-restricted nonlinearities, as conjectured by Carrasco? No: there is a stable Lur'e interconnection, certified by a full-block multiplier, that admits no OZF multiplier.","posedBy":"Joaquin Carrasco","yearPosed":2023,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-26","model":"ChatGPT 5.5","modelMaker":"OpenAI","humanCollaborators":["Andrey Kharitenko"],"aiRole":"A finite-horizon counterexample provided by ChatGPT 5.5 motivated the construction; the author built and certified the full interconnection.","verification":"unreviewed","verificationNote":"Single-author arXiv note; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A known conjecture on the standard multiplier test in absolute stability theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23599","sourceName":"arXiv:2607.23599 - Existence of stable Lur'e systems for which the O'Shea-Zames-Falb stability test fails","collection":null,"collectionVersion":null,"links":[{"label":"Su et al 2023 - Conjecture statement","url":"https://dx.doi.org/10.1016/j.automatica.2023.110872","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cerny-one-cluster","name":"Černý Conjecture for One-Cluster Automata","shortName":"Černý, one-cluster","problemNumber":null,"field":"Automata theory","fieldGroup":"Theoretical computer science","statement":"Does every synchronizing one-cluster automaton on $n$ states admit a reset word of length at most $(n-1)^2$? The new bound $(m-1)(n-1) + m\\ell \\le (n-1)^2$ settles the one-cluster case of the Černý conjecture.","posedBy":null,"yearPosed":2016,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-25","model":"OpenAI Codex (GPT-5.6 Sol Ultra)","modelMaker":"OpenAI","humanCollaborators":["Yinfeng Zhu"],"aiRole":"The annular spectral descent argument was obtained in interaction with OpenAI Codex running GPT-5.6 Sol Ultra and verified by the author; the paper also proves the positive-level relative-extending-word conjecture of Kisielewicz, Kowalski and Szykuła.","verification":"unreviewed","verificationNote":"Author-verified arXiv preprint. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A recognized major case of the Černý conjecture, automata theory's oldest open problem.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.19675","sourceName":"arXiv:2607.19675 - The Černý conjecture for one-cluster automata via annular spectral descent","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-768","name":"Erdős Problem #768","shortName":"Erdős #768","problemNumber":768,"field":"Number Theory, Multiplicative","fieldGroup":"Number theory","statement":"If $A(x)$ counts integers satisfying the Sylow divisor condition, determine the constant $c$ in $A(x)/x = \\exp(-(c + o(1)) \\sqrt{\\log x} \\log\\log x)$. The claimed exact value is $c = 1/(2\\sqrt{\\log 2})$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-25","model":"ChatGPT + Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Eric Li"],"aiRole":null,"verification":"lean-verified","verificationNote":"Three clean Lean checks; the community tracker update is pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.24872","sourceName":"arXiv - A Resolution of Erdős Problem 768: the Sylow Divisor Condition","collection":null,"collectionVersion":null,"links":[{"label":"Erdős Problem 768","url":"https://www.erdosproblems.com/768","kind":"problem-record"},{"label":"GitHub repo of the proof","url":"https://github.com/ericlisg/erdos768-lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-684","name":"Erdős Problem #684","shortName":"Erdős #684","problemNumber":684,"field":"Number Theory, Binomial Coefficients","fieldGroup":"Number theory","statement":"For the least $k$ at which the small-prime part of $\\binom{n}{k}$ exceeds $n^2$, how large can $f(n)$ be?","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"retracted","aiContribution":null,"resultNote":null,"claimIssueNote":"A preprint claimed $\\limsup f(n)/\\log n = \\infty$, but a deterministic audit later found a counterexample to its key Lemma 18. The stated conclusion is not established and the problem remains open.","solveDate":"2026-07-25","model":"Model not publicly disclosed","modelMaker":null,"humanCollaborators":[],"aiRole":null,"verification":"contested","verificationNote":"Key lemma refuted by a checked counterexample; see the claim issue.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/684","sourceName":"erdosproblems.com/684","collection":null,"collectionVersion":null,"links":[{"label":"Follow-up: unbounded logarithmic limsup in problem 684","url":"https://arxiv.org/abs/2604.23784","kind":"paper"},{"label":"Short proofs in combinatorics and number theory (Section 2): bound improved to 6.20219(log n)^2","url":"https://arxiv.org/abs/2603.29961","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"rota-matroid-flat-unimodality","name":"Rota's Unimodality Conjecture for Matroid Flats","shortName":"Rota flat unimodality","problemNumber":null,"field":"Matroid theory","fieldGroup":"Combinatorics","statement":"Is the sequence $W_0, W_1, \\dots, W_n$ counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.","posedBy":"Gian-Carlo Rota","yearPosed":1970,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-24","model":"ChatGPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Alexander Divoux","Chayim Lowen","Shouda Wang"],"aiRole":"A counterexample to the convexity of the sequence $W_i^{-1}$ was found with ChatGPT-5.6 Pro following a suggestion of the authors. This matroid was constructed using Whittle's $q$-lift mechanism, which the authors then adapted into a counterexample for unimodality.","verification":"unreviewed","verificationNote":"Public arXiv preprint with explicit counterexamples, checked by the human authors. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":36,"significanceNote":"Rota's 1970 unimodality circle, the family of conjectures behind a Fields-medal program.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.22515","sourceName":"arXiv:2607.22515 - Matroid flat counts are not unimodal","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"slug":"graveyard-problem-spectral-invisibility","name":"The Graveyard Problem for Dissipative Barrier Truncations","shortName":"Graveyard problem","problemNumber":null,"field":"Computational spectral theory","fieldGroup":"Analysis","statement":"The dissipative barrier method suppresses spectral pollution when a differential operator is truncated, but can it hide genuine spectral points? Known as the graveyard problem, the question stayed open in dimension two and above for more than a decade. It cannot: for Schrodinger operators in dimensions $d \\ge 2$ no spectral point becomes invisible, which together with the known one-dimensional theorem settles no-invisibility in every dimension.","posedBy":"Marco Marletta, Sergey Naboko","yearPosed":2014,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-24","model":"ChatGPT-based agent system","modelMaker":"OpenAI","humanCollaborators":["Matthew J. Colbrook","Marco Marletta"],"aiRole":"An unusually candid account, and the reason this entry sits at the lowest tier despite the models mattering. The first author built a system of ChatGPT-based agents to propose, test and attack proofs of the no-invisibility theorem. After false starts, including arguments with real gaps and arguments leaning on results that do not exist, the agents converged on a strategy Marletta and Naboko had already dismissed for want of a reversed Hansmann estimate. The final two lemmas the agents produced were not trusted. Suspecting the idea was a distorted echo of human mathematics in the training data, the authors searched for the uncited human source and instead found two stronger 2024 papers of Gil that supply exactly the reversed estimate. So the agents supplied direction, not the proof.","verification":"unreviewed","verificationNote":"arXiv preprint that discloses all AI use in line with the Leiden Declaration; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A decade-old named obstruction in computational spectral theory, the remaining gap in whether dissipative barriers are a trustworthy numerical method.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.22120","sourceName":"arXiv:2607.22120 - No Spectral Invisibility for Dissipative Barrier Truncations in Any Dimension","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hessian-conjecture-five-variable-counterexample","name":"A Five-Variable Counterexample to the Hessian Conjecture","shortName":"Hessian conjecture, $n=5$","problemNumber":null,"field":"Affine Algebraic Geometry, Polynomial Maps","fieldGroup":"Algebra","statement":"The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian conjecture $\\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.","posedBy":"Hessian conjecture literature; after de Bondt and Alpöge","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-24","model":"ChatGPT Pro + Claude","modelMaker":null,"humanCollaborators":["Guowu Meng","Liang Yang"],"aiRole":"The disclosure credits the second author with finding the Schur descent of Section 6 with the assistance of ChatGPT Pro, and states that the exact-arithmetic verifications throughout were carried out with assistance from Claude. The descent is the step that produces the counterexample; the surrounding framework is the authors'.","verification":"unreviewed","verificationNote":"No independent review, but the result is an explicit polynomial with a stated Hessian determinant, so it is checkable by direct computation. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Settles the Hessian conjecture in dimension five, in the same circle of problems as the Jacobian conjecture, and narrows the remaining low-dimensional cases.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.22198","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"depth-of-the-in-tree-of-s-in-the-graph-of-q-qsq-1-in-the-symmetric-group","name":"Depth of the in-tree of $s$ under $q \\mapsto q s q^{-1}$ on $n$-cycles","shortName":"In-tree depth under cyclic conjugation","problemNumber":null,"field":"Permutation combinatorics / functional graphs","fieldGroup":"Combinatorics","statement":"Fix an $n$-cycle $s$ and map every $n$-cycle $q$ to its conjugate $D(q) = q s q^{-1}$, which is the same as reading the one-line word $(q(0), \\ldots, q(n-1))$ back as a cycle. Iterating $D$ turns the $(n-1)!$ $n$-cycles into a functional graph. Its only fixed point is $s$, and the cycles that eventually reach $s$ form a tree feeding into it. How deep is that tree?\n\nExactly $\\varphi(n)$ cycles map directly onto $s$, and the tree stays shallow - depth 1 - unless $8 \\mid n$ or $p^2 \\mid n$ for an odd prime $p$, which is the Hull-Dobell threshold for the existence of a full-period non-translation affine map on $\\mathbb{Z}/n$. Past it the depth is $p^{e-1}$ for $n = p^e$ with $p$ odd, $2^{e-1} - 1$ for $n = 2^e$, and for general $n$ the largest of these over the prime powers dividing $n$.","posedBy":"Frédéric Lefebvre-Naré","yearPosed":2006,"ageNote":"The author first asked the question as a school pupil around 1980 and posted his hand and Excel explorations to a dedicated blog, archived by the Wayback Machine with entries stamped September 2006 - the earliest citable reference, which is what the posed year records. The precise question answered here, the exact depth of the in-tree, was only formulated in the June 2026 dialogue with the model.","solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Opus 4.8 constructed a branch of the stated depth, giving a lower bound, and believed it had a matching upper bound; that proof was wrong and the statement stayed a conjecture. FABLE 5 later proved it. In the author's summary of the method: \"The proof turns conjugation, near $s$, into base-$p$ arithmetic.\" A cycle near the fixed point splits into a coarse base permutation and a vector of carries in $\\mathbb{Z}/p$, $D$ acts on the carries by the carrying of ordinary base-$p$ addition, and the depth comes out as the nilpotency length of a shift difference - exactly that for odd $p$, one less for $p = 2$. The single missing carry that odd primes absorb and $2$ cannot is what produces the two-branch answer.\n\nA companion survey paper covers the rest of the graph: the other periodic orbits, congruences on basin sizes, and a cyclic-sieving count. The depth theorem is the substantive part.","claimIssueNote":null,"solveDate":"2026-07-24","model":"Claude FABLE 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"Claude Opus 4.8 in a first phase, then Claude FABLE 5, which obtained the result, did 99.9% of the research, in manual mode (that is, with more than 50 human prompts by an amateur, and step-by-step approvals). OpenAI's GPT-5.5 and others were used to proofread; all corrections after human and AI proofreading were made by Claude. Diagrams were made by Claude under human instructions.","verification":"unreviewed","verificationNote":"Not refereed, not formalised, and the paper says so itself on page one: \"An end-to-end verification of the assembled whole, and human peer review, remain to be done; the argument should be examined critically before being relied upon.\" The author, an amateur, states plainly that he cannot check the proof. So this stays Unreviewed and Candidate.\n\nThe claim is unusually checkable, though, and this site checked it independently, from the statement rather than from the author's code. Enumerating every $n$-cycle and building the whole functional graph for $n \\le 10$: $|D^{-1}(s)| = \\varphi(n)$ at every $n$, and the depth matches the formula at every $n$, including both nontrivial cases in range, depth 3 at $n = 8$ and at $n = 9$. Walking the tree backwards, which costs $n$ checks per node instead of $(n-1)!$, reaches depth 7 at $n = 16$, 5 at $n = 25$, 3 at $n = 18$, and 9 at $n = 27$ across 472,392 nodes - every one the predicted value. The Hull-Dobell mechanism was checked directly: preimages of a translation are exactly the affine maps with that multiplier, Hull-Dobell decides which are $n$-cycles, and the threshold predicts a nonempty second level exactly, no exceptions at $n = 6, 8, 9, 12, 16, 18, 25, 27$. The survey's basin-count sequence recounts to 1, 2, 2, 6, 7, 18, 17, 29 for $n = 3 \\ldots 10$, matching, and OEIS returns nothing for it.\n\nWhat none of that touches is the proof, which is the novelty: a formula confirmed at every $n$ reachable is not a theorem for all $n$.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":3,"significanceNote":"Below the anchor at 5, which covers machine-generated conjectures and recent one-paper questions. This one is a real, precisely posed question about a natural object, but it was asked by a high-school pupil in about 1980, put on a personal blog in 2006, and engaged with by nobody in the twenty years since - no citations, no literature, no community. A Graffiti conjecture at least lands on a list a research community reads. Above the floor because the question is genuine rather than generated, and the submitter says the same thing: this is not about an important issue in mathematics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.datastrategies.fr/fr/blog/vibemathing-45-ans-plus-tard","sourceName":"Data Stratégies (the author's blog)","collection":null,"collectionVersion":null,"links":[{"label":"The depth theorem, with the full proof (companion paper, 24 July 2026)","url":"https://www.datastrategies.fr/sites/default/files/documents/2026-07/depth_companion_1.pdf","kind":"paper"},{"label":"The functional graph of the whole map (survey paper, 25 July 2026)","url":"https://www.datastrategies.fr/sites/default/files/documents/2026-07/derivation_2.pdf","kind":"paper"},{"label":"The author's 2006 blog on the same question, via the Wayback Machine","url":"https://web.archive.org/web/2012/http://permutationc.free.fr/B1703659376/","kind":"problem-record"}],"relations":[],"submittedBy":"StormyRaven240","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"s-decoding-polynomial-sparsity","name":"Minimum Sparsity of S-Decoding Polynomials","shortName":"S-decoding sparsity","problemNumber":null,"field":"Private information retrieval","fieldGroup":"Theoretical computer science","statement":"Can an $S$-decoding polynomial modulo a suitable product of $k$ primes attain the lower-bound minimum of $k + 1$ nonzero coefficients? A construction matches the bound for special products of $k$ primes, yielding exponentially fewer-server PIR.","posedBy":"Fatemeh Ghasemi & Swastik Kopparty","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"conditional on a plausible number-theoretic conjecture; unconditional through s = 15","claimIssueNote":null,"solveDate":"2026-07-24","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The sparse-polynomial framework was developed with GPT-5.5 Pro and validated empirically by the authors.","verification":"unreviewed","verificationNote":"Author-checked ePrint with empirical validation of the construction. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A recent coding-theory question from a single paper.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://eprint.iacr.org/2026/1515","sourceName":"ePrint 2026/1515 - Exponentially fewer-server PIR from sparser S-decoding polynomials","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strichartz-tse-radon-nikodym-integrability","name":"Strichartz-Tse $L^p$-Integrability on the Sierpinski Gasket","shortName":"Sierpinski energy densities","problemNumber":null,"field":"Fractal analysis","fieldGroup":"Analysis","statement":"Energy measures of any two nonconstant harmonic functions on the standard Sierpinski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence that the Radon-Nikodym densities are $L^p$-integrable for $1 < p < \\log 15 / \\log 9$. That range is confirmed: the associated quantities are uniformly bounded for arbitrary ordered pairs of nonconstant harmonic functions.","posedBy":"Robert S. Strichartz, Samuel Tse","yearPosed":2010,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-24","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Konstantinos Tsougkas"],"aiRole":"The acknowledgement says the work was developed with assistance from the model, used to explore ideas and support aspects of the technical development and the writing, with all results and arguments independently verified by the author.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numerically observed threshold reported by Strichartz and Tse, within the analysis-on-fractals community that Strichartz largely built.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.22784","sourceName":"arXiv:2607.22784 - Lp-Integrability of Radon-Nikodym Densities Between Harmonic Energy Measures","collection":null,"collectionVersion":null,"links":[{"label":"Strichartz & Tse 2010 - Conjecture statement","url":"https://doi.org/10.1524/anly.2010.1038","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ghasemi-kopparty-sparse-decoding-polynomials","name":"Ghasemi-Kopparty Problem on Sparse $S$-Decoding Polynomials","shortName":"Sparse S-decoding polynomials","problemNumber":null,"field":"Computational complexity","fieldGroup":"Theoretical computer science","statement":"Can $S$-decoding polynomials modulo a product of $k$ primes be built with only $k+1$ nonzero coefficients, the minimum their own lower bound allows? Yes, via a general framework for special prime products. The consequence is that for any constant $s$ there is an $s$-server private information retrieval protocol with communication $\\exp(O((\\log n)^{1/s}(\\log\\log n)^{1-1/s}))$ on an $n$-bit database, where previous constructions at that communication needed $2^{O(s)}$ servers.","posedBy":"Mahdi Ghasemi, Swastik Kopparty","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"the PIR consequence is conditional on a number-theoretic conjecture implied by either the generalized repunit conjecture or Schinzel's hypothesis H, and is unconditional for s <= 15","claimIssueNote":null,"solveDate":"2026-07-24","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Aparna Gupte","Seyoon Ragavan"],"aiRole":"Stated in the abstract itself rather than buried in an acknowledgement: the main result for constant $s$ and its proof were discovered in a GPT-5.5 Pro conversation prompted by the authors.","verification":"unreviewed","verificationNote":"arXiv preprint; the authors also empirically validate the construction and make the result unconditional for all $s \\le 15$. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"An open problem posed at ITCS 2026 with a matching lower bound already in hand, inside the matching-vector PIR line running from Efremenko to Ghasemi-Kopparty-Sudan.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.22033","sourceName":"arXiv:2607.22033 - Exponentially Fewer-Server PIR from Sparser S-Decoding Polynomials","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-131-non-dividing-sets","name":"Erdős Problem #131","shortName":"Erdős #131","problemNumber":131,"field":"Number Theory, Additive Combinatorics","fieldGroup":"Number theory","statement":"Let $F(N)$ be the maximal size of $A\\subseteq\\{1,\\ldots,N\\}$ such that no $a\\in A$ divides the sum of any nonempty subset of $A\\setminus\\{a\\}$. Estimate $F(N)$. The lower bound $F(N)\\gg N^{1/5}$ is classical, from constructions of Erdős and Csaba, and every non-dividing set is non-averaging, which gave $F(N)\\leq N^{1/4+o(1)}$. The claimed new result is the matching upper bound $F(N)\\leq N^{1/5+o(1)}$, obtained by running the Pham-Zakharov density-increment argument one dimension lower through a projective normalization, hence $F(N)=N^{1/5+o(1)}$.","posedBy":"Paul Erdős","yearPosed":1975,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The new content is the upper bound; the matching N^(1/5) construction is prior work of Erdős and Csaba. erdosproblems.com has not accepted the claim","claimIssueNote":null,"solveDate":"2026-07-24","model":"GPT-5.6 Sol, Claude","modelMaker":"OpenAI, Anthropic","humanCollaborators":[],"aiRole":"The paper states that the novel idea - the projective normalization that survives the divisibility constraints and drops the associated convex geometry by one dimension, moving the exponent from 1/4 to 1/5 - was found by GPT-5.6 Sol. The Lean formalization was then completed by a Claude agent loop working autonomously against a human-written route document until the development compiled with no sorry and a clean axiom audit.","verification":"lean-verified","verificationNote":"Built and audited by the site on 2026-08-02. A clean clone of the author's Lean 4 development (50 files, 17,408 lines) compiles against the pinned mathlib revision on Lean 4.32.0 with no sorry, admit or native_decide. `#print axioms Nondividing.main_log_limit` returns exactly the eleven whitelisted axioms - propext, Classical.choice, Quot.sound and the eight declared external interfaces - and notably no sorryAx, so no placeholder is load-bearing. Statement fidelity checked against the trusted Challenge.lean: the definitions of non-dividing and F, and the theorem type log F(N)/log N -> 1/5, match. NOT verified: the eight external axioms are assumed rather than proved. Each cites a published result (Schneider, Rogers-Shephard, Betke-Henk-Wills, Pham-Zakharov Lemmas 1, 7 and 13, Conlon-Fox-Pham) but none was checked line by line against its source, and the density-increment exponent in convex_density_set is where the 1/4 to 1/5 improvement lives. erdosproblems.com still lists the problem open with no comments.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://theofilxeff.github.io/Erdos_131.pdf","sourceName":"A projective approach to non-dividing sets","collection":null,"collectionVersion":null,"links":[{"label":"Lean 4 formalization","url":"https://github.com/theofilxeff/erdos_131","kind":"lean-proof"},{"label":"erdosproblems.com/131","url":"https://www.erdosproblems.com/131","kind":"problem-record"}],"relations":[],"submittedBy":"Theofil Xeff","upvotes":2,"downvotes":0,"commentCount":1},{"slug":"werner-two-copy-distillability","name":"Two-Copy Distillability of Werner States","shortName":"Werner 2-copy","problemNumber":null,"field":"Entanglement theory","fieldGroup":"Quantum information & computing","statement":"Is a Werner state that is not one-copy distillable ever two-copy distillable? The first open rung of the NPT bound-entanglement ladder, open since 2000.","posedBy":null,"yearPosed":2000,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"Two-copy distillable if and only if already one-copy distillable.\n\nFour independent papers settled this within five days of each other, and no single one of them is the account of record. Fu, Gao and Park posted first on 23 July (arXiv:2607.21367), followed by Song and Chen on 26 July (arXiv:2607.23416), then on 27 July both Fraser, Huber, Pozsgay and Vona (arXiv:2607.24309) and Bharti, Gajjala and Haug (arXiv:2607.24479). The headline axes here follow the first posting, which is a filing convention and not a claim about who solved it.\n\nPozsgay has stated publicly that his group had their AI-assisted proof before the first paper appeared. This site cannot verify a private completion date, so that is recorded as his account rather than as a finding. All four are linked below.","claimIssueNote":null,"solveDate":"2026-07-23","model":"GPT-5.5, GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The proof by Fu, Gao and Park was AI-assisted and independently verified by the authors.","verification":"unreviewed","verificationNote":"Author-verified arXiv preprint. Within days, two further groups posted independent proofs of the same theorem (arXiv:2607.24309, arXiv:2607.24479), which strengthens confidence but none is yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Gateway case of the NPT bound-entanglement problem, on the PRX Quantum open-problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21367","sourceName":"arXiv:2607.21367 - A solution to 2-copy distillability of Werner states","collection":null,"collectionVersion":null,"links":[{"label":"Song and Chen - A partial-trace matrix inequality and Werner-state distillability (26 Jul)","url":"https://arxiv.org/abs/2607.23416","kind":"independent"},{"label":"Fraser, Huber, Pozsgay and Vona - two-copy distillability and a new partial trace bound (27 Jul)","url":"https://arxiv.org/abs/2607.24309","kind":"independent"},{"label":"Bharti, Gajjala and Haug - sharp partial-trace inequalities (27 Jul)","url":"https://arxiv.org/abs/2607.24479","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"slug":"reiner-conjecture-corank-three","name":"Reiner's Conjecture on Higher Bruhat Orders in Corank 3","shortName":"Reiner's conjecture","problemNumber":null,"field":"Algebraic combinatorics","fieldGroup":"Combinatorics","statement":"Reiner conjectured a description of the homotopy types of intervals in higher Bruhat orders. In corank $3$ it holds: the facial intervals of $B(n,n-3)$ are exactly the spherical intervals, and every other interval is contractible.","posedBy":"Victor Reiner","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"corank 3; the general conjecture remains open, corank 2 being McConville's case","claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT Pro (5.5, 5.6 Sol)","modelMaker":"OpenAI","humanCollaborators":["Daria Poliakova"],"aiRole":"The AI use declaration lists the model as search engine, proof assistant, vector graphics artist and editor, and states that all outputs of the proof assistant were thoroughly digested by a human and that the exposition is by and for humans.","verification":"unreviewed","verificationNote":"Single-author arXiv note; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A named conjecture on higher Bruhat orders in the Manin-Schechtman and Ziegler tradition, with a small specialist readership in poset topology.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21420","sourceName":"arXiv:2607.21420 - Homotopy types of intervals in corank-three higher Bruhat orders","collection":null,"collectionVersion":null,"links":[{"label":"Reiner 1999, Conjecture 6.9 - Conjecture statement","url":"https://doi.org/10.1017/9781009701587.009","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"target-free-clique-conjecture","name":"The Target-Free Clique Conjecture for Threshold-Linear Networks","shortName":"Target-free clique","problemNumber":null,"field":"Mathematical neuroscience","fieldGroup":"Combinatorics","statement":"The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network are exactly its target-free cliques, the bidirected cliques no outside vertex receives an edge from every member of. An explicit six-neuron counterexample refutes it.","posedBy":"Carina Curto, Jesse Geneson, Katherine Morrison","yearPosed":2019,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"Codex (GPT-5.6), Claude Code (Opus 4.8, Fable 5)","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Jesse Geneson"],"aiRole":"The acknowledgement says the systems were used for proof exploration, proof criticism, exposition and revision, with no specific step attributed, so the lowest tier applies.","verification":"site-confirmed","verificationNote":"The counterexample is a single fixed six-vertex graph with an explicit parameter regime, so the refutation reduces to a finite determinant and clique computation set out in the paper. Single-author arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A named conjecture in the combinatorial threshold-linear network literature, real and cited within mathematical neuroscience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21396","sourceName":"arXiv:2607.21396 - A six-neuron counterexample to the target-free clique conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"planar-four-terminal-dgg","name":"Four-Terminal Planar Case of the Dinitz-Garg-Goemans Cost Conjecture","shortName":"Four-terminal planar DGG","problemNumber":null,"field":"Combinatorial optimization","fieldGroup":"Algorithms & optimization","statement":"Does the Dinitz-Garg-Goemans cost-preserving unsplittable-flow rounding conjecture survive on acyclic planar instances with only four terminals? An explicit instance answers no: every cost-nonincreasing unsplittable routing has upper overload at least $335$ while the maximum demand is $294$.","posedBy":"Yefim Dinitz, Naveen Garg & Michel Goemans","yearPosed":1999,"ageNote":"The posed year is that of the parent DGG conjecture; the restricted planar four-terminal case was not separately dated.","solveType":"disproved","resolution":"variant","aiContribution":"ai-co-developed","resultNote":"restricted planar four-terminal case","claimIssueNote":null,"solveDate":"2026-07-23","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Matthew Protti"],"aiRole":"GPT-5.6 Pro carried out much of the construction search, symbolic derivation, proof development, exact-verifier development, adversarial critique and manuscript preparation. The human author selected and framed the problem, directed the investigation, caught a cost-normalization error, required exact and adversarial checks, set the claim scope and approved the release. A later Codex session independently re-encoded the key graph, finite and symbolic checks and ran deterministic stress and release checks.","verification":"unreviewed","verificationNote":"The immutable v0.1.0 public disclosure ships a manuscript, exact data, an exhaustive verifier over all 16 routings and all 13 arcs, mutation tests, deterministic hashes and a separate AI-assisted computational cross-check. The attained certificate is $335/294$; the package also proves the limiting lower bound $(299 - 41sqrt{41})/32$, with sharpness only in the stated fixed-topology, equal-full-cost, two-cheap-choice model. Released 2026-07-23, one day after the first public disproof of the general conjecture, and developed independently of it.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A concrete special case of the Dinitz-Garg-Goemans conjecture.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/releases/tag/v0.1.0","sourceName":"Public research disclosure v0.1.0 (GitHub release)","collection":null,"collectionVersion":null,"links":[{"label":"Manuscript (PDF)","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/releases/download/v0.1.0/ssuf_four_terminal_note_v5.pdf","kind":"paper"},{"label":"Exact verification suite (verify_all.py)","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/blob/v0.1.0/scripts/verify_all.py","kind":"code"},{"label":"AI-use and provenance note","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/blob/v0.1.0/AI_USE_AND_PROVENANCE.md","kind":"transcript"},{"label":"The general conjecture, disproved a day earlier","url":"https://vibemathed.com/problem/dinitz-garg-goemans-unsplittable-flow","kind":"other"}],"relations":[],"submittedBy":"Matthew Protti","upvotes":3,"downvotes":0,"commentCount":0},{"slug":"unconditional-unclonable-encryption","name":"Unconditional One-Bit Unclonable Encryption","shortName":"Unclonable encryption","problemNumber":null,"field":"Quantum cryptography","fieldGroup":"Quantum information & computing","statement":"Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?","posedBy":"Anne Broadbent & Sébastien Lord","yearPosed":2020,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":null,"resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"GPT-5.6 Sol Ultra, Codex","modelMaker":"OpenAI","humanCollaborators":["Prabhanjan Ananth","Amit Sahai"],"aiRole":"Two simultaneous papers achieve the goal independently; one construction, using random Pauli eigenstates, attains the optimal exponent up to constants (Ragavan, ePrint 2026/1509).","verification":"unreviewed","verificationNote":"Two independent author-verified proofs posted the same week (Ananth-Sahai arXiv:2607.21551 and Ragavan ePrint 2026/1509); one development is Lean-checked. Neither peer-reviewed yet.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Broadbent-Lord 2020; the central question of unclonable cryptography with heavy partial work.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21551","sourceName":"arXiv:2607.21551 - Unconditional unclonable encryption","collection":null,"collectionVersion":null,"links":[{"label":"Simultaneous construction with the optimal exponent: Ragavan (ePrint 2026/1509)","url":"https://eprint.iacr.org/2026/1509","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stanley-claw-free-schur-positivity","name":"Stanley's Claw-Free Schur-Positivity Conjecture","shortName":"Claw-free Schur positivity","problemNumber":null,"field":"Algebraic combinatorics","fieldGroup":"Combinatorics","statement":"Is the chromatic symmetric function $X_G$ Schur positive for every claw-free graph $G$? Two explicit $12$-vertex line graphs have Schur coefficients $-64$ and $-40$ at $s_{(3,3,3,3)}$.","posedBy":"Richard Stanley","yearPosed":1995,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Both counterexample graphs were found with ChatGPT-5.6 Sol Pro; the negative coefficients were confirmed by independent computations.","verification":"unreviewed","verificationNote":"Author preprint with independent computational checks of the negative Schur coefficients. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Stanley's claw-free positivity question, a standing target in symmetric function theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21508","sourceName":"arXiv:2607.21508 - Chromatic symmetric functions of claw-free graphs are not Schur positive","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"petersen-coloring-conjecture","name":"Petersen Coloring Conjecture","shortName":"Petersen Coloring","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Jaeger conjectured that every bridgeless cubic graph $G$ admits a Petersen coloring: a map $\\varphi\\colon E(G)\\to E(P)$ into the edges of the Petersen graph $P$ such that, for every vertex $v$ of $G$, the three edges at $v$ are sent to three edges meeting at a common vertex of $P$. Equivalently, by Jaeger's theorem, every bridgeless cubic graph has a normal 5-edge-coloring. The conjecture implies both the Berge-Fulkerson conjecture and the 5-cycle-double-cover conjecture. False: there is an explicit simple connected bridgeless cubic graph on $112$ vertices, of girth five and edge- and vertex-connectivity three, with no Petersen coloring.","posedBy":"François Jaeger","yearPosed":1985,"ageNote":"Posed by Jaeger in 1985, in 'On five-edge-colorings of cubic graphs and nowhere-zero flow problems'; often cited as 1988 via his nowhere-zero flow survey. Open for 41 years, and verified by computer for all bridgeless cubic graphs up to 34 vertices, so a 112-vertex counterexample is well clear of the searched range.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"This preprint was not the first disproof. A 68-vertex counterexample was posted to X on 23 July 2026 by @NeuralReformist, credited to GPT-5.6 Sol Ultra, sixteen days earlier. This site decoded that sparse6 string and checked it independently: 68 vertices, 102 edges, simple, cubic, connected, bridgeless, girth five, and no Petersen coloring under the same encoder used for the 112-vertex graph. Whether the two are independent is unknown - the preprint does not cite the post. The headline axes still record the preprint, the only complete writeup with certificates.\n\nThe implication runs one way: the Petersen coloring conjecture implies Berge-Fulkerson and the 5-cycle-double-cover conjecture, so refuting it leaves both of those open. The paper does not claim 112 is minimum, and it supplies a second, nonisomorphic $D_3$-symmetric 112-vertex counterexample. Combined with a theorem of Ma, Mattiolo, Steffen and Wolf, one counterexample yields infinitely many.","claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Bryce Putman"],"aiRole":"The paper's \"Computational provenance and responsibility\" section states in full: \"OpenAI language-model systems were used extensively in the discovery, computational search, verification, and preparation of this work. The author reviewed the final claims and artifacts and accepts responsibility for the contents.\" No product name, model version or division of labour is given, so which of discovery, search, verification and write-up the model actually carried is not recoverable from the paper. The catalog records the model as ChatGPT because that is this catalog's convention for an unnamed OpenAI system; the paper itself names none.","verification":"site-confirmed","verificationNote":"Reproduced here on 12 August 2026, independently of the paper's certificates. The 112-vertex graph was rebuilt from the appendix edge table, and the SHA-256 of its normalized sorted edge list reproduces the digest in Theorem 1.1 exactly, pinning the object under review to the one claimed. Every property in that theorem was rederived: 112 vertices, 168 edges, simple, cubic, connected, bridgeless, girth five, connectivity three. Non-existence of a Petersen coloring was then re-proved with a CNF encoding written here from the definition - each edge carries one of the 15 edges of $KG(5,2)$, each vertex selects one of the 10 target stars, the three edges at a vertex land in that star and are pairwise distinct - and solved with CaDiCaL via PySAT. UNSAT. That re-derives the unsatisfiability rather than replaying the shipped DRAT certificates, and the encoder was written without reference to the paper's: the same 3640 variables, forced by the problem shape, but 31,360 clauses against their 68,324. It ran twice in separate processes with identical results. Six controls - $K_4$, $K_{3,3}$, the 3-cube, the prism, Desargues and the Petersen graph itself - all came back satisfiable through the same encoder. Petersen is the important one, being a snark: a coloring for it rules out the encoder having quietly tested 3-edge-colorability. Not checked: the second $D_3$-symmetric counterexample, the normal-5-edge-coloring formulation, and the DRAT proofs. Four-day-old arXiv preprint, unrefereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":"No Wikipedia article of its own, but it is a listed Open Problem Garden problem, standard background in the snark and cubic-graph literature, and the subject of a continuing stream of papers on normal edge-colorings.","significance":40,"significanceNote":"Jaeger's conjecture is one of the central conjectures on cubic graphs: the Open Problem Garden entry calls it an extraordinary conjecture, and it implies both the Berge-Fulkerson conjecture and the 5-cycle-double-cover conjecture, with a substantial literature on normal edge-colorings and sublinear approximations built around it. Placed above a well-tracked specialist conjecture and below the cycle double cover conjecture itself (55), which is more widely known outside the area.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.10012","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"First known counterexample: 68 vertices, posted to X, 23 July 2026","url":"https://x.com/NeuralReformist/status/2080153035045839069","kind":"announcement"},{"label":"Open Problem Garden: Petersen coloring conjecture","url":"http://www.openproblemgarden.org/op/petersen_coloring_conjecture","kind":"problem-record"},{"label":"Zenodo artifact record v1.1.0 (CNFs, DRAT certificates, hashes)","url":"https://doi.org/10.5281/zenodo.21845291","kind":"code"},{"label":"Ma, Mattiolo, Steffen, Wolf - Sets of r-graphs that color all r-graphs","url":"https://doi.org/10.1007/s00493-025-00144-4","kind":"paper"}],"relations":[],"submittedBy":"VibeGene","upvotes":5,"downvotes":0,"commentCount":4},{"slug":"one-relator-groups-commutator-relators","name":"Commutator Relators Do Not Force Hopficity, Residual Finiteness or Automaticity","shortName":"One-relator groups","problemNumber":null,"field":"Combinatorial Group Theory","fieldGroup":"Algebra","statement":"Baumslag asked whether a one-relator group $G=F/\\langle\\langle r\\rangle\\rangle$ with $r$ a commutator is Hopfian, residually finite or automatic. The paper constructs a family $G_m=\\langle a,t \\mid [t,a[a,t]^{-m}]\\rangle$ answering all three negatively.","posedBy":"Gilbert Baumslag","yearPosed":1998,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Ke Wang","Qiang Zhang"],"aiRole":"The paper's disclosure is a single sentence: ChatGPT assisted in constructing the examples. The examples are the result, but 'assisted' does not say how much, so the lowest tier applies.","verification":"unreviewed","verificationNote":"No independent check, and the one-line disclosure does not separate the model's contribution from the authors'. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Answers a question of Baumslag, a central figure in one-relator group theory, negatively on all three counts.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21493","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Baumslag et al 1998, Questions OR7, OR8: Problem statement","url":"https://web.stevens.edu/algebraic/alexeim/Publications/All_files_new/Openproblem_final_40.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"monical-snp-conjecture","name":"Monical's Saturated Newton Polytope Conjecture","shortName":"Monical SNP","problemNumber":null,"field":"Newton polytopes","fieldGroup":"Combinatorics","statement":"If a chromatic symmetric function is Schur positive, must every finite-variable specialization $X_G(x_1, \\dots, x_k)$ have a saturated Newton polytope? A $12$-vertex bipartite graph realizes weights $(6,6,0)$ and $(8,2,2)$ but omits their midpoint $(7,4,1)$.","posedBy":"Cara Monical, Neriman Tokcan & Alexander Yong","yearPosed":2017,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The finite witness was found with ChatGPT-5.6 Sol Pro and verified by direct computation.","verification":"unreviewed","verificationNote":"Author-checked finite witness with a combinatorial proof, in the same preprint that resolves the claw-free Schur-positivity conjecture. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"From the well-cited saturated-Newton-polytope program.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21508","sourceName":"arXiv:2607.21508 - Chromatic symmetric functions of claw-free graphs are not Schur positive","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"araujo-piga-schacht-tight-hamilton","name":"Araujo-Piga-Schacht Question on Tight Hamilton Cycles","shortName":"Tight Hamilton cycles","problemNumber":null,"field":"Hypergraph theory","fieldGroup":"Combinatorics","statement":"Araujo, Piga and Schacht asked whether density and codegree both above $1/4$ force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is $p_0 = \\max_{0 \\le x \\le 1}\\min\\{x^3, 1-x\\} \\approx 0.3177$, and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every $p > 1/3$ the asymptotically sharp minimum-codegree threshold is determined.","posedBy":"Araujo, Piga, Schacht","yearPosed":2021,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"the question is answered negatively and the correct threshold is determined","claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Xichao Shu"],"aiRole":"The author acknowledges using ChatGPT in the early stage of the project and during manuscript preparation, and states specifically that an initial idea leading to the first construction in the paper arose during an interaction with it.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named question in quasirandom hypergraph Hamiltonicity, a well-worked corner of extremal combinatorics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21568","sourceName":"arXiv:2607.21568 - Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs","collection":null,"collectionVersion":null,"links":[{"label":"Han et al 2021 - Problem statement","url":"https://doi.org/10.1016/j.jctb.2025.10.006","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"faber-krahn-position-convex-bodies","name":"Uniqueness of the Faber–Krahn Position of Convex Bodies","shortName":"Faber–Krahn position","problemNumber":null,"field":"Spectral Geometry, Convex Geometry","fieldGroup":"Analysis","statement":"A convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. The paper proves this position is unique up to orthogonal transformations, answering a question of Schmuckenschläger from 2011, via a new log-convexity property of the first eigenvalue under positive definite linear deformations.","posedBy":"Michael Schmuckenschläger","yearPosed":2011,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Dmitry Faifman","Iosif Polterovich"],"aiRole":"The AI usage disclosure says only that the authors made use of ChatGPT for mathematical discussions and editorial assistance. It does not separate the two, so the lowest tier applies.","verification":"unreviewed","verificationNote":"No independent check, and the disclosure is general rather than pointing at any specific result. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Answers a 2011 question and yields a new proof of the Pólya–Szegő theorem for triangles as a consequence.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21539","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"liu-morin-extension-conjecture","name":"The Liu-Morin Extension Conjecture","shortName":"Liu-Morin extension","problemNumber":null,"field":"Homological algebra","fieldGroup":"Algebra","statement":"Han's conjecture predicts that a finite-dimensional algebra with eventually vanishing Hochschild homology has finite global dimension. Within the tau-Hochschild framework it splits into persistence and survival, and the Liu-Morin extension conjecture concerns when the property passes to extensions. A protected corner theorem bounding Ext at a surviving vertex settles it beyond the monomial and special biserial cases, and yields a trichotomy for failures on three strongly connected vertices.","posedBy":"Liu, Morin","yearPosed":2004,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Han's conjecture itself remains open; this settles the Liu-Morin extension case","claimIssueNote":null,"solveDate":"2026-07-23","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":["Marco Armenta"],"aiRole":"The disclosure states that the author used the model for deep research, formulation of theorems, and drafts of their proofs, reviewing and editing the output as needed.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"An extension conjecture attached to Han's conjecture, itself a standard open problem about Hochschild homology and global dimension.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.20849","sourceName":"arXiv:2607.20849 - Protected corners and a trichotomy for Han's conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Liu & Morin 2004: Conjecture statement","url":"https://www.ams.org/journals/proc/2004-132-12/S0002-9939-04-07512-4/S0002-9939-04-07512-4.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-593","name":"Erdős Problem #593","shortName":"Erdős #593","problemNumber":593,"field":"Infinite Hypergraph Theory","fieldGroup":"Combinatorics","statement":"Which finite triple systems occur in every triple system of uncountable chromatic number? The claimed characterization: exactly those that, after removing isolated vertices, are linear, have every hyperedge-node of their Levi graph meeting a bridge, and have every Berge cycle even.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT + Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked end to end; not yet conventionally refereed or incorporated by the community tracker.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"A numbered Erdos problem that carries a $500 Erdos prize, setting it above the typical entry in the catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/593","sourceName":"erdosproblems.com/593","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"slug":"wowii-conjecture-72-two-induced-trees-pin-down-tree-g","name":"WOWII Conjecture 72: Two induced trees pin down tree($ G $)","shortName":"WOWII 72 (tree($ G $) bound)","problemNumber":null,"field":"Graph theory (induced trees / graph invariants)","fieldGroup":"Combinatorics","statement":"For a connected graph $ G $, let $ t= $ tree($ G $) (order of a largest induced tree), $ A= $ average eccentricity, and $ L= $ maximum independence number of a neighbourhood. Then\n$ \\lceil (A+L)/3 \\rceil \\le t. $\n(The evenly-divided reading of the conjecture holds; a stronger reading that divides only $ L $ by three is false.)","posedBy":"Ermelinda DeLaViña (Graffiti.pc / Written on the Wall II)","yearPosed":2001,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The evenly-divided reading of WOWII Conjecture 72 holds:\n$ \\lceil(A + L)/3\\rceil \\le t, $\nwhere $ t = $ tree($ G $) (order of a largest induced tree), $ A = $ average eccentricity and $ L = $ maximum neighbourhood independence number.\nA stronger reading that divides only $ L $ by three is false. The argument rests on two elementary observations (a diametral path is chordless and therefore induces a tree on $ D+1 $ vertices; a maximum independent set in a neighbourhood induces a star on $ L+1 $ vertices). The original conjecture’s precise wording is not yet pinned in a public formal repository, so statement fidelity remains to be audited.","claimIssueNote":null,"solveDate":"2026-07-23","model":"GPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The model was given a short prompt (copied from a successful earlier run on a different WOWII conjecture) and asked to find a solution or counterexample to an open conjecture of its choosing. It selected WOWII Conjecture 72, produced the two-lemma argument (diametral path induces a tree on $ D+1 $ vertices; maximum independent neighbourhood induces a star on $ L+1 $ vertices), derived the bound $ \\lceil(A+L)/3\\rceil\\le t $, and explicitly distinguished the evenly-divided reading (true) from the stronger reading that divides only $ L $ by three (false). The human then posted the diagram and commentary.","verification":"site-confirmed","verificationNote":"Re-derived in full by this site on 17 August 2026 - the argument is elementary and correct, and short enough to state: a shortest path between two vertices at maximum distance D is induced, so it induces a path (a tree) on D+1 vertices, giving tree(G) >= D+1 >= A+1 since average eccentricity is at most D; a maximum independent set in a neighbourhood plus its centre induces a star on L+1 vertices, giving tree(G) >= L+1; hence A + L <= 2*tree(G) - 2, and ceil((A+L)/3) <= tree(G) follows by integrality. The stronger reading (dividing only L by three) fails on the claimed counterexample family. What keeps this Candidate is not the mathematics but the statement: Conjecture 72's canonical wording is not publicly pinned (no formal statement exists in the Formal Conjectures repository), so which reading DeLaViña intended is unconfirmed, and the X post plus transcript is the only artifact. Site-confirmed records this site's independent re-derivation of the proved reading.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A machine-generated conjecture from DeLaViña's Written on the Wall II list - exactly the catalog's Graffiti anchor at 5. Open since around 2001 and part of the current wave of WOWII resolutions.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://x.com/4thRT/status/2080412225232400539","sourceName":"X post by @4thRT (with attached diagram and ChatGPT conversation link)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"QuietFerret326","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"dual-sequential-fat-shattering-bound","name":"Dual Sequential Fat-Shattering and Tight Threshold Extraction","shortName":"Sequential fat-shattering","problemNumber":null,"field":"Model theory and learning theory","fieldGroup":"Theoretical computer science","statement":"Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension, resolving the first. A new proof of Daskalakis-Golowich tight-threshold extraction, avoiding multicolored Ramsey numbers, resolves the second, which concerned repairing the bound in a result claimed by Jung, Kim and Tewari.","posedBy":"Constantinos Daskalakis, Noah Golowich; Jung, Kim and Tewari","yearPosed":2022,"ageNote":"Jung, Kim & Tewari 2020 posed the general problem first, but the error-free case was posed by Daskalakis & Golowich 2022","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Gabriel Conant","Caroline Terry"],"aiRole":"The acknowledgements separate human and AI credit explicitly and attribute specific steps. Proposition 3.4 was provided by ChatGPT on direct request. The authors' original Theorem 1.11 gave only a uniform analogue with a weaker bound; asked whether the bound could be improved, the model proposed the two-parameter inductive scheme that gives the current one, and it identified that the uniformity hypotheses were an artifact of an imprecise formulation, letting the authors drop them. For Theorem 1.14 the authors handed over their draft with a description of the obstacles and asked for a proof; the model generated an argument following their induction scheme and two-case structure. The authors checked and thoroughly rewrote everything.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"Open questions from recent papers at the junction of model theory and statistical learning theory, documented but read by a small community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21761","sourceName":"arXiv:2607.21761 - Encoding orders and trees in real-valued functions","collection":null,"collectionVersion":null,"links":[{"label":"Daskalakis & Golowich 2022 - Problem statement","url":"https://arxiv.org/abs/2111.08911","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"separable-jacobian-conjecture-characteristic-two","name":"The Separable Jacobian Conjecture in Characteristic 2","shortName":"Separable Jacobian (char 2)","problemNumber":null,"field":"Affine algebraic geometry","fieldGroup":"Algebra","statement":"Adjamagbo's positive-characteristic refinement of the Jacobian conjecture asks that a polynomial endomorphism with unit Jacobian determinant whose induced function-field extension has degree prime to the characteristic be an automorphism. It is false: an explicit $F: \\mathbb{A}_k^3 \\to \\mathbb{A}_k^3$ over any field of characteristic $2$ has Jacobian determinant identically $1$ and function-field degree $3$, yet is not injective.","posedBy":"Kossivi Adjamagbo (specific case)","yearPosed":1995,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"The counterexample to the original Jacobian conjecture does not specialize to characteristic 2; this is a modification that does.","claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Irit Huq-Kuruvilla"],"aiRole":"The AI usage section states that the text is the result of a discussion with ChatGPT 5.6 Sol, with conversation transcripts available on request, and that the proof itself was verified as correct by the named author.","verification":"site-confirmed","verificationNote":"The counterexample is a three-line explicit polynomial map; the non-injectivity is witnessed by three distinct source points and the Jacobian determinant is a direct computation, both checkable by hand. Single-author arXiv note, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The separable formulation is the version of the Jacobian conjecture designed to survive in positive characteristic, and the author notes no other counterexample to it is known.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.20968","sourceName":"arXiv:2607.20968 - An Explicit Characteristic-2 Counterexample to the Separable Jacobian Conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Dimension-two counterexample, Lean kernel-checked (Mondello)","url":"https://arxiv.org/abs/2608.02634","kind":"independent"},{"label":"Adjamagbo 1995 - Conjecture statement","url":"https://link.springer.com/chapter/10.1007/978-94-015-8555-2_5","kind":"problem-record"}],"relations":[{"to":"jacobian-conjecture","kind":"related","note":"General case in dimensions n >= 3"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1177","name":"Erdős Problem #1177","shortName":"Erdős #1177","problemNumber":1177,"field":"Infinite Hypergraph Theory","fieldGroup":"Combinatorics","statement":"For a finite forbidden triple system $G$, what exact uncountable chromatic cardinalities occur among $G$-free triple systems, and how do those spectra interact? The revised manuscript answers the three exact-cardinal questions and claims a complete spectrum dichotomy.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT + Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked end to end; not yet conventionally refereed or incorporated by the community tracker.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1177","sourceName":"erdosproblems.com/1177","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"minimal-distance-problem-sharp-exponent","name":"The Minimal Distance Problem","shortName":"Minimal distance problem","problemNumber":null,"field":"Discrete geometry","fieldGroup":"Geometry & topology","statement":"How well separated can a family of point-line pairs in the unit square be? For every $\\varepsilon > 0$ there are arbitrarily large families $(x_1,\\ell_1),\\ldots,(x_n,\\ell_n)$ in $[0,1]^2$ with $x_i \\in \\ell_i$ and $\\mathrm{dist}(x_i,\\ell_j) \\ge n^{-2/3-\\varepsilon}$ for all $i \\ne j$. Combined with earlier work of Cohen, Pohoata and Zakharov, this settles the problem at the sharp exponent $2/3$. The same construction disproves a conjecture of Hunter, Pohoata, Verstraete and Zhang about induced point-line matchings over finite fields.","posedBy":"Cohen, Pohoata, Zakharov","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"also disproves a separate finite-field conjecture of Hunter, Pohoata, Verstraete and Zhang","claimIssueNote":null,"solveDate":"2026-07-22","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Cosmin Pohoata"],"aiRole":"The acknowledgement draws the line precisely. The author's own plan was to use a high-degree number field analogue of the Hunter-Pohoata-Verstraete-Zhang construction to reach the Ruzsa endpoint. In his words, the decisive new idea of using the codimension-one, square-difference-free, trace-zero lattice in place of a Ruzsa-like set, which is what upgrades the exponent to the sharp one, is entirely due to GPT-5.6 Pro.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named problem in incidence geometry with a documented ladder of exponents and connections to square-difference-free sets, Nikodym sets and minimal blocking sets.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.20422","sourceName":"arXiv:2607.20422 - The sharp exponent for the minimal distance problem","collection":null,"collectionVersion":null,"links":[{"label":"Cohen et al 2025: Problem statement","url":"https://doi.org/10.1007/s00222-025-01331-2","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dinitz-garg-goemans-unsplittable-flow","name":"Dinitz-Garg-Goemans Conjecture","shortName":"Dinitz-Garg-Goemans","problemNumber":null,"field":"Combinatorial Optimization","fieldGroup":"Algorithms & optimization","statement":"For single-source unsplittable flow, every fractional flow can be rounded to an unsplittable flow whose cost is no higher than the fractional cost, while each arc's load is exceeded by at most the maximum demand. (The cost version of Goemans' unsplittable-flow conjecture.)","posedBy":"Yefim Dinitz, Naveen Garg, Michel Goemans","yearPosed":1999,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-22","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Dmitry Rybin"],"aiRole":"Rybin used GPT-5.6 Pro to search for and construct an explicit counterexample: a graph whose fractional flow cost is 58, while every unsplittable flow with capacity violation at most 15 costs at least 60 - so no cost-preserving rounding exists.","verification":"unreviewed","verificationNote":"Announced on X by Dmitry Rybin (2026-07-22) with a shared GPT-5.6 Pro chat. The counterexample is a concrete finite graph checkable by direct computation (fractional cost 58 vs. minimum unsplittable cost 60 under capacity violation $\\le 15$), but it is not yet peer-reviewed or formally verified. Not to be confused with the separate 'Dinitz conjecture' on Latin-square colourings.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":"A Wikipedia article now exists, but it was created **in response** to the problem being solved, so it does not count as a valid dedicated article.","significance":20,"significanceNote":"A well-known 1999 conjecture in flow approximation algorithms.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://x.com/DmitryRybin1/status/2079904005652893709","sourceName":"Dmitry Rybin (X)","collection":null,"collectionVersion":null,"links":[{"label":"Independent restricted counterexample: planar, four terminals (Protti)","url":"https://vibemathed.com/problem/planar-four-terminal-dgg","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":3,"downvotes":0,"commentCount":0},{"slug":"an-explicit-counterexample-to-the-rank-two-poisson-conjecture","name":"An Explicit Counterexample to the Rank-Two Poisson Conjecture","shortName":"Rank-Two Poisson","problemNumber":null,"field":"Poisson algebra","fieldGroup":"Algebra","statement":"Let $\\mathcal{P}_{2}=\\mathbb{C}[x,q,p,z]$ carry the canonical Poisson bracket determined by $\\{p,x\\}=\\{z,q\\}=1$ and by the vanishing of the other brackets between distinct generators. Here and throughout, “rank two” means two canonical pairs in the standard indexing of the canonical Poisson algebras; thus there are four polynomial generators and the Poisson tensor has geometric rank four. We give explicit polynomials $R,T,D,S\\in\\mathbb{Q}[x,q,p,z]$ satisfying $\\{D,R\\}=1,\\qquad\\{S,T\\}=1,\\qquad\\{R,S\\}=\\{R,T\\}=\\{D,S\\}=\\{D,T\\}=0,$ while $R=x(2-3xq)$. Consequently, the assignment $(x,q,p,z)\\mapsto(R,T,D,S)$ defines a Poisson endomorphism of $\\mathcal{P}_{2}$ that is not an automorphism. This disproves the Poisson Conjecture for two canonical pairs, and hence for every number of canonical pairs at least two.","posedBy":null,"yearPosed":null,"ageNote":"No posed year, deliberately. PC(n) is the Poisson-algebra formulation used by Adjamagbo and van den Essen in their equivalence theorem rather than a conjecture either of them posed, and its content reaches back to Keller's 1939 Jacobian conjecture through that equivalence. Dating it to any one of those would be a guess, so the entry stays off the age chart.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The paper constructs explicit $R,T,D,S\\in\\mathbb Q[x,q,p,z]$ defining a Poisson endomorphism of the canonical rank-two Poisson algebra $\\mathcal P_2$ that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant $1$, and has an explicit fiber of exactly three points. Thus $\\mathrm{PC}(2)$ is false, and stabilization gives failure of $\\mathrm{PC}(n)$ for every $n\\ge2$. An appendix further constructs an explicit nonautomorphic endomorphism of the fourth Weyl algebra, proving $\\mathrm{DC}(4)$ false.","claimIssueNote":null,"solveDate":"2026-07-22","model":"ChatGPT 5.6 Sol; Claude Fable 5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Christopher D. Long"],"aiRole":"The four-variable rank-two Poisson construction, including its Hamiltonian correction, was produced during an interactive research session with ChatGPT 5.6 Sol. The model also assisted with organizing the differential-form proof, exact symbolic verification, literature checking, and manuscript drafting. Claude Fable 5 subsequently performed independent algebraic audits and supplied editorial comments. Christopher Long checked the mathematics and assumes responsibility for the final paper.","verification":"unreviewed","verificationNote":"Unreviewed. A preprint with no peer review and no proof-assistant verification. What it does have is unusual for the tier: every Poisson identity is verified directly in the paper, there is an exact symbolic audit of the polynomial identities, and the noninjectivity is exhibited as a fiber of exactly three explicit points. All of that is four polynomials in four variables, so any reader with a computer algebra system can check the whole claim in minutes. Claude Fable 5 supplied an independent algebraic audit, which is not independent human expert review. Nobody has done that on the record.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":4,"renownNote":"The Dixmier conjecture has Wikipedia articles in English, German, French and Swedish. The Poisson formulation has none of its own; it is known through that equivalence.","significance":35,"significanceNote":"The Poisson, Jacobian and Dixmier conjectures are equivalent, so once the Jacobian conjecture fell in July 2026 the others followed in principle. What is new is explicit and minimal: the counterexample descends to two canonical pairs, the smallest rank where the statement can fail, and the appendix kills the fourth Dixmier conjecture outright. Well below the Jacobian conjecture at 65, the result this is a consequence of and on Smale's list. Above the five-variable Hessian counterexample at 30: Dixmier is the more famous name, and this covers every rank at least two rather than one dimension.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.23777","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Dixmier conjecture","url":"https://en.wikipedia.org/wiki/Dixmier_conjecture","kind":"wikipedia"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"wow-conjecture-103","name":"Written on the Wall II, Graph Conjecture 103","shortName":"WoW 103","problemNumber":null,"field":"Graph invariants","fieldGroup":"Combinatorics","statement":"For every connected graph $G$, is $\\alpha(G) \\le \\lfloor b(G) - \\log(\\operatorname{ecc}_{avg}(G)) \\rfloor$, where $b(G)$ is the largest induced-bipartite-subgraph order? An $11$-vertex counterexample - a triangle with four leaves on each of two vertices - has $\\alpha = 9$ against bound $8$.","posedBy":"Graffiti (Written on the Wall II)","yearPosed":1996,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-22","model":"ChatGPT + Codex","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The counterexample was found with ChatGPT and Codex and verified in Lean, alongside exhaustive subset enumeration.","verification":"lean-verified","verificationNote":"Lean-checked counterexample merged into the google-deepmind/formal-conjectures repository.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Written on the Wall II); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/google-deepmind/formal-conjectures/pull/4482","sourceName":"formal-conjectures PR #4482 - Disprove WOWII Conjecture 103","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"symmetrically-colored-k-ap-bound","name":"Improved Bound for Colorings Without Symmetrically Colored k-APs","shortName":"Symmetric $k$-AP colorings","problemNumber":null,"field":"Additive Combinatorics, Ramsey Theory","fieldGroup":"Combinatorics","statement":"Deng, Tidor and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no symmetrically coloured 4-term arithmetic progression, giving an $O(N^{\\log_{22}3})$ coloring. The paper gives an $O_k(N^{4/k^2})$ coloring of $[N]$ avoiding symmetrically coloured $k$-APs for every even $k\\ge4$, improving the exponent.","posedBy":"Deng, Tidor and Zhao","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Improves the exponent rather than answering the asked question: whether an $N^{o(1)}$ colouring exists remains open.","claimIssueNote":null,"solveDate":"2026-07-22","model":"GPT-5","modelMaker":"OpenAI","humanCollaborators":["Ruizhe Shi","Yiqi Dong"],"aiRole":"The acknowledgements state the key construction idea was suggested during an interaction with GPT-5.","verification":"unreviewed","verificationNote":"No independent review. The model is credited with the key construction idea rather than the surrounding analysis. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"Progress on a question of Deng, Tidor and Zhao, improving the known exponent for every even k at once.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.20752","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Deng et al 2025, Conjecture 1.6: Conjecture statement","url":"https://dx.doi.org/10.1017/S0305004125000106","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"e-log-concavity-chromatic-quasisymmetric","name":"$e$-Log-Concavity of Chromatic Quasisymmetric Functions","shortName":"e-log-concavity","problemNumber":null,"field":"Algebraic combinatorics","fieldGroup":"Combinatorics","statement":"Chromatic quasisymmetric functions of natural unit interval graphs were conjectured to have log-concave coefficients in the elementary basis. A connected $13$-vertex example refutes it: for the Hessenberg function $h=(2,4,4,6,7,10,10,10,10,12,12,13,13)$ and $\\lambda=(6,5,1,1)$ the coefficients of $q^5,q^6,q^7$ are $1,6,38$, and $6^2 < 1 \\cdot 38$. The coefficient is still positive, palindromic and unimodal, so log-concavity is separated from the weaker shape properties that motivated the conjecture.","posedBy":"Bruce Sagan, Foster Tom","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-22","model":"ChatGPT with Codex","modelMaker":"OpenAI","humanCollaborators":["Boris Kafidov"],"aiRole":"The disclosure describes a computer-assisted investigation in which the model was used extensively as an interactive research assistant: it generated and revised the exploratory and verification code, ran independent computational cross-checks, searched the literature, and helped organize and draft the manuscript. The author states that no assertion is accepted on the authority of model output.","verification":"site-confirmed","verificationNote":"The counterexample is a single explicit graph and partition, and the paper ships an exact standard-library Python verifier with a recorded SHA-256 digest. arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A conjecture in the Shareshian-Wachs chromatic quasisymmetric function programme, verified computationally by Abreu and Nigro up to the order just below this witness.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.20595","sourceName":"arXiv:2607.20595 - A 13-vertex counterexample to e-log-concavity for chromatic quasisymmetric functions","collection":null,"collectionVersion":null,"links":[{"label":"Sagan & Tom 2026, Conjecture 4.2: Conjecture statement","url":"https://dx.doi.org/10.5802/alco.468","kind":"problem-record"},{"label":"Python code of the counterexample","url":"https://arxiv.org/src/2607.20595v2/anc/verify_counterexample.py","kind":"code"},{"label":"Python code of the rank-three verifier","url":"https://arxiv.org/src/2607.20595v2/anc/verify_rank_three.py","kind":"code"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"nikolov-ullman-pure-dp-square-root-rate","name":"Nikolov-Ullman Pure-DP Query Release Conjecture","shortName":"Pure-DP square-root rate","problemNumber":null,"field":"Differential privacy","fieldGroup":"Theoretical computer science","statement":"Nikolov and Ullman asked, as Open Problem 1 on DifferentialPrivacy.org, whether $k$ statistical queries over a universe of size $T$ can be released under pure differential privacy at the square-root error rate that the known lower bounds suggest, rather than the cube-root rate of the classical small-database method. They can: for every $n$ and $\\varepsilon > 0$ there is an $\\varepsilon$-differentially private mechanism with expected error $O(\\min\\{1, \\sqrt{\\log(2T)\\log(2k)/(\\varepsilon n)}\\})$.","posedBy":"Aleksandar Nikolov, Jonathan Ullman","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"information-theoretic; a polynomial-time implementation remains open","claimIssueNote":null,"solveDate":"2026-07-22","model":"Codex, Harmonic Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Jack Fitzsimons"],"aiRole":"The generative-AI disclosure states that Codex and Aristotle were used in connection with Lean formalization and proof search, and that Codex also gave editorial feedback on clarity and organization. Proof search is a mathematical contribution, but the disclosure does not say which steps came from where.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint with a companion Lean 4 development that the paper says machine-checks the finite construction, pure privacy after deterministic decoding, and the all-regimes upper bound, with an axiom audit and a paper-to-Lean crosswalk in the artifact. We have not compiled it. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Listed as Open Problem 1 on DifferentialPrivacy.org, the differential-privacy community's recognized problem list, and the last gap between pure and approximate DP for query release.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.20418","sourceName":"arXiv:2607.20418 - Pure-DP Statistical Query Release at the Conjectured Square-Root Rate","collection":null,"collectionVersion":null,"links":[{"label":"Nikolov & Ullman 2021, Open Problem 1: Problem statement","url":"https://differentialprivacy.org/open-problem-optimal-query-release/","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"graffiti-conjecture-284","name":"Graffiti Conjecture 284","shortName":"Graffiti 284","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"If a finite graph has girth at least five, must its minimum dual degree satisfy $\\delta^*(G) \\le -\\partial_n(G)$, where $\\partial_n(G)$ is the smallest eigenvalue of its distance matrix? The Hoffman-Singleton graph violates it: dual degree $7$ against eigenvalue bound $4$.","posedBy":"Graffiti (Siemion Fajtlowicz's program)","yearPosed":1996,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-22","model":"Grok 4.5 Medium (Capy build)","modelMaker":"xAI","humanCollaborators":[],"aiRole":"The Capy agent running Grok 4.5 Medium identified the Hoffman-Singleton graph as a counterexample; the certificate was reproduced independently under adversarial review.","verification":"unreviewed","verificationNote":"Publicly posted exact certificate on a classical, independently checkable graph (Hoffman-Singleton); no formal writeup yet.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Graffiti); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://x.com/justinsunyt/status/2080116559352316409","sourceName":"Public certificate thread (X)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"slug":"polynomial-time-low-degree-conjecture","name":"The Polynomial-Time Low-Degree Conjecture","shortName":"Low-degree conjecture","problemNumber":null,"field":"Average-case complexity","fieldGroup":"Theoretical computer science","statement":"The low-degree conjecture predicts that when the low-degree advantage between a planted distribution and a uniform null distribution stays bounded, no polynomial-time algorithm can distinguish them. It is false. There is a planted distribution that agrees with the null through the relevant degree, is invariant under vertex relabeling, and is nevertheless distinguished in polynomial time by a rank argument.","posedBy":"Samuel B. Hopkins","yearPosed":2018,"ageNote":"The conjecture crystallized in Hopkins's 2018 thesis and became the standard evidence format for average-case hardness in the years after.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"the construction is probabilistic; an explicit uniformly samplable example remains open","claimIssueNote":null,"solveDate":"2026-07-22","model":"ChatGPT 5.4, 5.5, 5.6","modelMaker":"OpenAI","humanCollaborators":["Songtao Mao"],"aiRole":"The statement on AI use is specific in both directions. The author first put forward constructions from two-dimensional Reed-Muller-style codes and asked whether they could be made invariant under all vertex relabelings while keeping efficient decoding; the model's responses established that they could not, closing off that route. The rank argument that carries the paper was later developed with ChatGPT 5.6 after the author fed it ideas in the spirit of his Remark 2.5. The author independently checked, simplified and organized every proof.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The low-degree method is the standard tool for arguing average-case hardness across high-dimensional statistics, inference and random optimization, and this conjecture is what licensed reading those bounds as evidence about polynomial time.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.20318","sourceName":"arXiv:2607.20318 - The Polynomial-Time Low-Degree Conjecture is False","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"lamplighter-stability-radius","name":"Stability Radius of the Lamplighter Group","shortName":"Lamplighter stability","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"Dogon, Levit and Vigdorovich asked for an explicit upper bound on the stability radius of an infinitely presented group. The lamplighter group provides the first: explicit polynomial bounds on both its Hilbert-Schmidt stability rate and its stability radius, obtained through approximately invariant measures and an effective marker construction.","posedBy":"Alon Dogon, Arie Levit, Itamar Vigdorovich","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-22","model":"ChatGPT 5.5, Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Alon Dogon","Thomas Vidick"],"aiRole":"The disclosure separates the mathematics from the formalization. The authors had an exponential bound with a greedy marker construction; on being given the marker lemma, ChatGPT 5.5 produced the polynomial improvement, which is the paper's headline. The authors then recognized that the polynomial marker lemma follows from known descriptive-combinatorics techniques and holds for general group actions. The model also supplied the statement and proof of the Appendix A lower bound. Separately, after the paper was complete, Aristotle auto-formalized the main theorem in Lean over 64 prompts and roughly 14 partial days.","verification":"lean-checked","verificationNote":"An Aristotle-produced Lean formalization of the main statement accompanies the paper, including background material not already in Mathlib, with a comparator file supplied so the formalized statement can be checked against the paper. We have not compiled it yet. arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A question from a recent paper on quantitative group stability, with a real but small readership in approximate representation theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.20135","sourceName":"arXiv:2607.20135 - Polynomial Hilbert-Schmidt stability of the lamplighter group","collection":null,"collectionVersion":null,"links":[{"label":"Dogon et al 2024, Question 12.10: Problem statement","url":"https://arxiv.org/abs/2407.11608","kind":"problem-record"},{"label":"GitHub repo of the statement in Lean","url":"https://github.com/vidick/lamplighter-comparator/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Github repo of the proof","url":"https://github.com/vidick/lamplighter-lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"levit-mandrescu-unimodality-conjecture","name":"Levit–Mandrescu Unimodality Conjecture","shortName":"Levit–Mandrescu Unimodality","problemNumber":null,"field":"Graph Theory, Independence Polynomials","fieldGroup":"Combinatorics","statement":"A graph on $n$ vertices is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu conjectured that the independence polynomial $i(G,x)$ of every very well-covered graph is unimodal, i.e. its coefficient sequence is nondecreasing and then nonincreasing.","posedBy":"Vadim E. Levit, Eugen Mandrescu","yearPosed":2006,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-22","model":"GPT-5.6 Sol, Claude Fable 5","modelMaker":null,"humanCollaborators":["Lucas B."],"aiRole":"The models produced an explicit counterexample: the whiskering of $E_{588} \\vee (39K_{11} \\sqcup 85K_{12})$, a very well-covered graph on 4,074 vertices, whose independence polynomial $(1+x)^{1449}(1+2x)^{588} + (1+x)^{1913}(1+12x)^{39}(1+13x)^{85} - (1+x)^{2037}$ has a strict local valley at $a_{1095}$. Per the announcement, the search took a few hours once the question was posed.","verification":"unreviewed","verificationNote":"Announced on LinkedIn by Lucas B. (Head of AI Research, Jump Trading); no preprint yet, and the reviewers credited are internal to the team rather than independent. The stated polynomial was recomputed with exact integer arithmetic: $a_{1094} > a_{1095} < a_{1096}$ holds, so the polynomial given is genuinely not unimodal, and its low-order coefficients ($a_0 = 1$, $a_1 = 4074$) are consistent with a graph on 4,074 vertices. What remains unchecked is that the whiskered graph's independence polynomial equals the polynomial stated.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A known conjecture on independence polynomials.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.linkedin.com/posts/tesuji_levit-mandrescu-counterexample-activity-7487528446403768320-b1pc","sourceName":"Lucas B. (Jump Trading), LinkedIn announcement","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"Rasmus Lindahl","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"quartic-syk-spectral-edge","name":"Spectral Edge of the Quartic SYK Model","shortName":"Quartic SYK edge","problemNumber":null,"field":"Random matrices & quantum many-body theory","fieldGroup":"Mathematical physics","statement":"Determine the leading asymptotic of the largest eigenvalue of the $N$-Majorana quartic SYK Hamiltonian as $N \\to \\infty$. The preprint proves $\\lambda_1/\\sqrt{N} \\to 4\\int_0^\\infty g_0(t)^4\\,dt \\approx 0.32504$ almost surely, via the limiting free energy at every fixed positive temperature.","posedBy":null,"yearPosed":2016,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-22","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Yukun He"],"aiRole":"Developed with GPT-5.6; the finite-bath interpolation reduces the quartic SYK pressure to a local cavity-kernel identity.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; independent review pending, so this is recorded as a candidate rather than an accepted resolution.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Spectral edge of the SYK model, a heavily studied physics ensemble.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.18998","sourceName":"arXiv:2607.18998 - The spectral edge of the quartic SYK model","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"xz-conjecture-mathieu-su2-counterexamples","name":"Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)","shortName":"xz and Mathieu, SU(2)","problemNumber":null,"field":"Mathieu–Zhao Spaces, Integral Conjectures","fieldGroup":"Algebra","statement":"For the integral $\\mathcal{I}(h)$ over the unit interval and the torus, the paper gives the three-term Laurent polynomial $f(x,z)=(1-z^{-1})((1-x)+xz)$ with $\\mathcal{I}(f^n)=0$ but $\\mathcal{I}(z^{-1}f^n)=(-1)^{n-1}/(n+1)\\neq0$. This disproves the $xz$-conjecture with one interval and one torus variable, shows $\\ker\\mathcal{I}$ is not a Mathieu–Zhao subspace, and by padding yields counterexamples for SU(2).","posedBy":"Olivier Mathieu; the xz-conjecture in the Mathieu–Zhao literature","yearPosed":1995,"ageNote":"The year posed is for the Mathieu conjecture, the date of the xz conjecture is not known","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-21","model":"ChatGPT 5","modelMaker":"OpenAI","humanCollaborators":["Christopher D. Long"],"aiRole":"Under a heading on AI provenance and author responsibility, the paper states the counterexample and its lift were discovered by ChatGPT 5.","verification":"unreviewed","verificationNote":"No independent review, but the counterexample is a three-term Laurent polynomial with closed-form moment identities, checkable by hand. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The Mathieu conjecture is a well-known problem connected to the Jacobian conjecture; this refutes the xz form and the SU(2) case with an explicit witness.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.19012","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Mathieu 1995: Mathieu conjecture statement","url":"https://smf.emath.fr/publications/some-conjectures-about-invariant-theory-and-their-applications","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"boots-royle-cao-vince-conjecture","name":"Boots-Royle/Cao-Vince Conjecture on Planar Spectral Radius","shortName":"Boots-Royle/Cao-Vince","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge with a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for every $n \\ge 9$. Tait and Tobin proved it for sufficiently large $n$ in 2017; the conjecture now holds for all $n \\ge 9$.","posedBy":"Barry Boots, Gordon Royle; Dasong Cao, Andrew Vince","yearPosed":1991,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-21","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Lele Liu","Bo Ning","Yi Wang"],"aiRole":"The declaration credits the model with generating the code that searched for extremal planar graphs, and with assisting in several computations and symbolic derivations, alongside language polishing.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A conjecture posed independently by two groups around 1991 and a standard reference point in spectral extremal graph theory, with Tait and Tobin's asymptotic solution in JCTB.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.19268","sourceName":"arXiv:2607.19268 - A complete solution to the Boots-Royle/Cao-Vince conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pak-slonim-stretched-schubert-polynomiality","name":"Pak-Slonim Conjecture on Stretched Schubert Structure Constants","shortName":"Stretched Schubert","problemNumber":null,"field":"Algebraic combinatorics","fieldGroup":"Combinatorics","statement":"Pak and Slonim conjectured that stretched Schubert structure constants are eventually polynomial. They are. Monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial, and the Schubert duality of Watanabe carries this to the structure constants. The same result settles the polynomiality half of a conjecture of Alexandersson and Alhajjar for key polynomials.","posedBy":"Igor Pak, Zachary Slonim","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-21","model":"Codex","modelMaker":"OpenAI","humanCollaborators":["Per Alexandersson"],"aiRole":"The acknowledgement says only that Codex was used during preparation of the paper for proof exploration and editorial assistance. The two are not separated and no step is attributed, so the lowest tier applies.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture in Schubert calculus from Pak and Slonim, with a companion conjecture of Alexandersson and Alhajjar attached, familiar within algebraic combinatorics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.18746","sourceName":"arXiv:2607.18746 - Polynomiality of Stretched Schubert Structure Constants and Key Coefficients","collection":null,"collectionVersion":null,"links":[{"label":"Pak & Slonim 2026: Problem statement","url":"https://arxiv.org/abs/2604.27107","kind":"announcement"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"johnson-freyd-ostrik-yu-cocycle-question","name":"Johnson-Freyd-Ostrik-Yu Question on Categorical Cocycles","shortName":"Categorical 3-cocycles","problemNumber":null,"field":"Tensor categories","fieldGroup":"Algebra","statement":"Twisted Deligne products categorify the tensor product of two Grothendieck rings. Classifying them leads to categorical $n$-cocycles, and Johnson-Freyd, Ostrik and Yu asked whether these are always pullbacks of ordinary group cocycles on the universal grading group of the underlying based ring. For $3$-cocycles they are.","posedBy":"Theo Johnson-Freyd, Victor Ostrik, Matthew Yu","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"answered for 3-cocycles, inside a broader partial classification of twisted Deligne products.","claimIssueNote":null,"solveDate":"2026-07-21","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Pavel Etingof","Dmitri Nikshych","Victor Ostrik"],"aiRole":"The acknowledgement says only that the authors collaborated with the model in this paper, especially in the Appendix, which is where the cocycle question is answered. No step is attributed, so the lowest tier applies.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A question from the fusion-category literature, posed by leading names in the area but specialized.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.19560","sourceName":"arXiv:2607.19560 - Twisted Deligne products of semisimple tensor categories","collection":null,"collectionVersion":null,"links":[{"label":"Johnson-Freyd et al 2026: Conjecture statement","url":"https://arxiv.org/abs/2601.09060","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"norine-antipodal-coloring-conjecture","name":"Norine's Antipodal-Colouring Conjecture","shortName":"Norine's conjecture","problemNumber":null,"field":"Graph Theory, Topological Combinatorics","fieldGroup":"Combinatorics","statement":"Norine conjectured every red-blue edge-colouring of the $n$-dimensional hypercube $Q_n$ in which antipodal edges get opposite colours contains a monochromatic path from some vertex to its antipode. The paper proves it, via a chain-level Borsuk–Ulam obstruction.","posedBy":"Serguei Norine","yearPosed":2008,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-21","model":"GPT-5","modelMaker":"OpenAI","humanCollaborators":["Hehui Wu","Ningyuan Yang"],"aiRole":"The authors state the central proof idea was first generated with the assistance of GPT-5, and that the manuscript was drafted with Codex.","verification":"unreviewed","verificationNote":"No independent check. The model is credited with the central idea rather than the whole argument. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture on hypercube colourings, proved outright rather than in special cases.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.19276","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Norine 2008: Conjecture statement","url":"https://www.openproblemgarden.org/op/edge_antipodal_colorings_of_cubes","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dittert-conjecture-dimension-16","name":"Dittert's Conjecture in Dimension 16","shortName":"Dittert n=16","problemNumber":null,"field":"Linear Algebra, Permanents","fieldGroup":"Algebra","statement":"Dittert's conjecture asserts that among nonnegative $n\\times n$ matrices whose entries sum to $n$, the functional $\\varphi(A)=\\prod_i r_i+\\prod_j c_j-\\operatorname{per}(A)$ is uniquely maximized by $J_n/n$. The paper proves the case $n=16$ which, with Pang's result for $n\\ge17$, establishes the conjecture for every $n\\ge16$.","posedBy":"Eberhard Dittert","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Partial: this settles $n=16$ only. Combined with Pang's $n≥17$ the conjecture holds for all $n≥16$, leaving the small cases open.","claimIssueNote":null,"solveDate":"2026-07-21","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Boris Kafidov"],"aiRole":"The disclosure states the proof strategy, the joint-deficit scaling lemma, and most of the original proof text were produced by GPT-5.6 Sol through ChatGPT in response to the author's prompts; ChatGPT also revised the exposition and prepared the manuscript.","verification":"unreviewed","verificationNote":"No independent check, and the model is credited with the strategy and the central lemma rather than with support. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture on permanents open since the 1980s, advanced to the point where only small dimensions remain.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.19439","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[{"to":"dittert-s-conjecture-in-dimension-five","kind":"related","note":"Last open case of the conjecture, proved sometime after this entry"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mod-4-kawauchi-conjecture","name":"The mod 4 Kawauchi Conjecture","shortName":"mod 4 Kawauchi","problemNumber":null,"field":"Knot theory","fieldGroup":"Geometry & topology","statement":"Kawauchi conjectured that the Conway polynomial of an amphicheiral knot factors as $\\nabla_K(z) = f(z)f(-z)$ for an integer polynomial $f$. Hartley proved it for negative amphicheiral knots and Ermotti, Hongler and Weber published the first counterexample to the general case. The mod 4 form of the conjecture, equivalent to a statement the author conjectured independently in 2006, is true.","posedBy":"Akio Kawauchi","yearPosed":1979,"ageNote":"Kawauchi's factorization question dates to the late 1970s; the mod 4 form is equivalent to a statement Conant conjectured independently in 2006.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"the mod 4 form; the general conjecture is false by Ermotti, Hongler and Weber","claimIssueNote":null,"solveDate":"2026-07-21","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":["Jim Conant"],"aiRole":"The paper states the proof was produced with the help of Claude Fable 5 and that the draft was prepared in the course of an extended research conversation. The acknowledgements attribute three specific things to the model: the quotient-tower strategy, the level-wise torsion identities, and an initial draft of the main argument.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture on Conway polynomials of amphicheiral knots with a four-decade literature running through Hartley and Ermotti-Hongler-Weber.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.18655","sourceName":"arXiv:2607.18655 - A proof of the mod 4 Kawauchi Conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"wow-conjecture-143","name":"Written on the Wall II, Graph Conjecture 143","shortName":"WoW 143","problemNumber":null,"field":"Graph invariants","fieldGroup":"Combinatorics","statement":"For every finite connected graph, is $\\operatorname{girth}(G) + 1$ at most the product of its largest induced-tree order and its second-smallest degree?","posedBy":"Graffiti (Written on the Wall II)","yearPosed":1996,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-21","model":"GPT-5.6 Thinking","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Proved with GPT-5.6 Thinking and formalized in Lean.","verification":"lean-verified","verificationNote":"Lean-checked in the google-deepmind/formal-conjectures repository; maintainer review completed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Written on the Wall II); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/google-deepmind/formal-conjectures","sourceName":"google-deepmind/formal-conjectures (WrittenOnTheWallII)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"batyrev-stringy-hodge-numbers","name":"Batyrev's Stringy Hodge Number Conjecture","shortName":"Batyrev's Conjecture","problemNumber":null,"field":"Algebraic Geometry","fieldGroup":"Algebra","statement":"For a projective variety $X$ with at worst Gorenstein canonical singularities whose stringy $E$-function $E_{\\mathrm{st}}(X; u, v)$ is a polynomial, all stringy Hodge numbers $h^{p,q}_{\\mathrm{st}}(X)$ are non-negative. (Batyrev 1998, Conjecture 3.10.)","posedBy":"Victor Batyrev","yearPosed":1998,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-21","model":"GPT","modelMaker":"OpenAI","humanCollaborators":["Matthew Satriano","Jeremy Usatine"],"aiRole":"Satriano and Usatine found the counterexample with the assistance of GPT: $X = M_0 \\times \\mathbb{P}^1$, where $M_0$ is the coarse moduli space of rank-2 semistable bundles with trivial determinant over a genus-3 curve. $X$ is a 7-dimensional projective variety with Gorenstein terminal singularities whose stringy $E$-function is a polynomial, yet its stringy Hodge number $h^{2,5}_{\\mathrm{st}}(X) = -1$ is negative.","verification":"unreviewed","verificationNote":"arXiv preprint 2607.19184 (21 Jul 2026) by Matthew Satriano and Jeremy Usatine. The proof is short and fully explicit: the stringy $E$-function is written out and its $u^2 v^5$ coefficient gives $h^{2,5}_{\\mathrm{st}} = -1$, so it is hand-verifiable. A domain-expert preprint, not yet peer-reviewed. Distinct from the unrelated Batyrev-Manin conjecture on rational points.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Batyrev's stringy invariants are foundational in birational geometry and mirror symmetry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.19184","sourceName":"arXiv:2607.19184","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-469","name":"Erdős Problem #469","shortName":"Erdős #469","problemNumber":469,"field":"Number Theory, Divisors","fieldGroup":"Number theory","statement":"Does the sum of the reciprocals of all primitive pseudoperfect numbers converge?","posedBy":"SJ Benkoski, Paul Erdos","yearPosed":1974,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-21","model":"GPT-5.6 Sol Ultra, Claude Fable 5","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Zachary J. Lewis"],"aiRole":"The public proof, which also yields density results for pseudoperfect numbers, was developed with GPT-5.6 Sol Ultra and reviewed with Claude Fable 5.","verification":"lean-verified","verificationNote":"Lean-checked in two independent public developments.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/469","sourceName":"erdosproblems.com/469","collection":null,"collectionVersion":null,"links":[{"label":"Benkoski & Erdos 1974 - Problem statement","url":"http://www.math-inst.hu/~p_erdos/1974-24.pdf","kind":"problem-record"},{"label":"Lewis' preprint on GitHub","url":"https://github.com/ZachL111/erdos469-preprint/blob/main/paper/output/pdf/erdos469_v1_2_1_preprint.pdf","kind":"transcript"},{"label":"GitHub repo of the proof","url":"https://github.com/ZachL111/erdos469-preprint/tree/main/formal","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"chernoff-density-strong-log-concavity","name":"Strong Log-Concavity of Chernoff's Density","shortName":"Chernoff log-concavity","problemNumber":null,"field":"Probability & statistics","fieldGroup":"Probability & statistics","statement":"Is the density of Chernoff's distribution - the law of $\\operatorname{argmax}_t \\{W(t) - t^2\\}$ for two-sided Brownian motion $W$ - strongly log-concave, as conjectured by Balabdaoui and Wellner in 2014?","posedBy":"Fadoua Balabdaoui & Jon A. Wellner","yearPosed":2014,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-21","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Xianyang Zhang","Quan Zhou"],"aiRole":"The proof was generated in its entirety by GPT-5.6 Sol; the authors checked it and prepared the manuscript.","verification":"unreviewed","verificationNote":"Author-checked public arXiv preprint. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A specialist question in shape-constrained statistics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.18619","sourceName":"arXiv:2607.18619 - Chernoff's density is strongly log-concave","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"full-rsb-in-the-sherrington-kirkpatrick-spin-glass","name":"Full-RSB in the Sherrington–Kirkpatrick spin glass","shortName":"FRSB in the SK spin glass","problemNumber":null,"field":null,"fieldGroup":"Mathematical physics","statement":"This work proves full replica symmetry breaking for the zero-field Sherrington–Kirkpatrick model at zero temperature $\\beta=\\infty$: the Parisi minimizer is absolutely continuous, has a smooth density, and has support $[0,1)$, thereby confirming the prediction by Parisi.","posedBy":"Giorgio Parisi","yearPosed":1979,"ageNote":"Dated from Parisi's 1979 Physical Review Letters paper \"Infinite number of order parameters for spin-glasses\", which introduced the hierarchical order parameter whose full-RSB structure is confirmed here. The paper cites it as the origin of the prediction rather than as a formally posed conjecture.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"For the Sherrington-Kirkpatrick model with no external field, at zero temperature $\\beta=\\infty$, the paper proves that the zero-temperature Parisi minimizer is absolutely continuous with a smooth density and has support $[0,1)$ - full replica symmetry breaking, confirming the Parisi picture at the ground state. It also shows $q_\\beta\\to1$ as $\\beta\\to\\infty$.\n\nThe positive-temperature input is Lopatto's: for every $\\beta>1$ the Parisi measure is supported on the closed interval $[0,q_\\beta]$, with a smooth density on $[0,q_\\beta)$ and a single atom at the right endpoint $q_\\beta$. That endpoint atom is what the paper's own quantitative estimates target, so it is not incidental. This entry is cited as Theorem 1.1 rather than reproved.\n\nRemark 1.4 is worth reading beside the support claim: the half-open interval is essential, because the zero-temperature functional cannot see an endpoint atom at all, so there is no canonical mass there to converge to.","claimIssueNote":null,"solveDate":"2026-07-20","model":"ChatGPT 5.6","modelMaker":null,"humanCollaborators":["Hong-Bin Chen"],"aiRole":"The most explicit authorship disclosure in this catalog. The manuscript's opening note says the proof arguments and prose \"were generated by the same model from prompts supplied by Hong-Bin Chen\", and then states on its own line:\n\n\"ChatGPT 5.6 is the author of the manuscript.\"\n\nIt continues: \"Since arXiv's policy on generative AI language tools does not permit such a tool to be listed as an author, Hong-Bin Chen is only formally listed as the author for submission purposes and assumes full responsibility for the submitted text. This formal attribution reflects arXiv's policy rather than the division of labor in producing the manuscript: Hong-Bin Chen's role was limited to prompting, editing, proofreading, and verifying the arguments; in particular, he did not construct the proof arguments. He has read and verified the proofs, although errors or oversights may remain.\"\n\nThe Lean development was also written by the same model. AI-discovered is unambiguous here: a human posed the problem and checked the output, and states outright that he did not construct the argument.","verification":"unreviewed","verificationNote":"Unreviewed, and the tier was lowered from Lean-checked on inspection of the repository - a labelling correction, not a doubt about the mathematics.\n\nWhat holds: across 75 files and 5,544 lines of Lean 4 there is no $\\texttt{sorry}$, no $\\texttt{admit}$ and no $\\texttt{native\\_decide}$, confirmed on 28 August 2026. What does not: the Lean-checked rung requires no stray axioms, and this project declares seven mathematical axioms in $\\texttt{ExternalInputs.lean}$. Three cite prior work (Lopatto, Auffinger-Chen, Chen-Handschy-Lerman), but four stand in for the paper's own unformalized analysis - the minimizer itself, and the analytic data behind Propositions 4.1-4.2, 4.3 and 4.4. The repository's own ledger also marks several items \"Open analytic\", and notes that four terminal approximation modules are \"excluded from the root target and omitted from this GitHub bundle\".\n\nWhat the formalization does establish is real and is credited: given that analytic data, the gap exclusions and the smooth-density conclusion are Lean deductions rather than assumptions, as AXIOMS.md is careful to state. The repository is unusually candid - it ships an axiom ledger and a dependency table, and the paper's own footnote says \"This is not an assumption-free verification of the entire paper\". The project was not built here, for want of a toolchain. The paper is an unrefereed preprint and no human has independently reviewed the analysis.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Full replica symmetry breaking in the SK model is a central prediction of Parisi's theory, and establishing it rigorously at zero temperature - absolute continuity, smooth density, support all of $[0,1)$ - is a substantial result rather than a technical refinement. It sits above the SK-adjacent entries already here, Talagrand's critical overlap conjecture at 30 and the Gardner transition at 28. Held below the top band because it is unrefereed, no human has reviewed the analysis, and it takes Lopatto's positive-temperature structure theorem as an input.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.48550/arXiv.2607.18032","sourceName":"FRSB IN THE SK SPIN GLASS: CONVERGENCE TO FULL-INTERVAL SUPPORT AT ZERO TEMPERATURE","collection":null,"collectionVersion":null,"links":[{"label":"Full support of the zero-temperature SK order parameter (OpenAI math release)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf","kind":"other"}],"relations":[],"submittedBy":"SpryRaven345","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"kourovka-18-50-permuted-products","name":"Kourovka Problem 18.50 - Prescribed Permuted-Product Cardinality","shortName":"Kourovka 18.50","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"Given $n$ and $1 \\le c \\le n!$, can $n$ distinct group elements be chosen so that their $n!$ ordered products take exactly $c$ distinct values? Constructions realize every $c$.","posedBy":null,"yearPosed":2014,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"From the Kourovka Notebook, group theory's recognized standing problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17477","sourceName":"arXiv:2607.17477 - On some problems from the Kourovka Notebook","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gaussian-moments-conjecture","name":"Gaussian Moments Conjecture","shortName":"Gaussian Moments Conj.","problemNumber":null,"field":"Probability, Commutative Algebra","fieldGroup":"Probability & statistics","statement":"The Gaussian Moments Conjecture asks whether, for complex polynomials $P,Q$ in $n$ independent standard real Gaussian variables, $\\mathbb{E}(P^m)=0$ for all $m\\geq 1$ forces $\\mathbb{E}(QP^m)=0$ for all large $m$. Explicit counterexamples with $\\mathbb{E}(P^m)=0$ and $\\mathbb{E}(QP^m)=m!\\neq 0$ exist in three variables (a five-term quartic $P$) and four variables, so the conjecture is false in every dimension $n\\geq 3$.","posedBy":"Harm Derksen, Arno van den Essen, Wenhua Zhao","yearPosed":2017,"ageNote":"Conjectured by Derksen, van den Essen and Zhao (Israel J. Math 219, 2017) as part of the Jacobian-conjecture circle; the search here was prompted by Alpöge's announced counterexample to the Jacobian conjecture.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively","claimIssueNote":null,"solveDate":"2026-07-20","model":"GPT-5.6 Sol Pro, Claude Fable 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Christopher D. Long"],"aiRole":"Per the paper's AI-provenance section, the four-variable construction was produced by ChatGPT 5.6 Sol Pro without human intervention after the initial prompt, which told it the Jacobian conjecture had been disproved and asked whether a small Gaussian-moments counterexample might follow; shown that example, Claude Fable 5 found the three-variable construction and supplied independent algebraic checks. The author bears responsibility for the mathematics and exposition.","verification":"site-confirmed","verificationNote":"Re-derived by the site on 2026-08-02: both counterexamples were rebuilt from the paper's stated polynomials and evaluated in exact rational arithmetic against the standard Gaussian moment rules (E(W^a Z^b) = a! when a = b, else 0; E(T^c) the double factorial), independently of the paper's own algebra. For m = 1 through 10 both give E(P^m) = 0 and E(QP^m) = m! exactly, and the term counts and degrees match the paper (five terms of degree four in three variables, six of degree three in four). The surrounding exposition has had no independent review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A named conjecture in the Jacobian-conjecture circle, equivalent in part to the Image Conjecture: a real if specialist target, and its failure follows the Jacobian disproof it was prompted by.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.18186","sourceName":"arXiv:2607.18186","collection":null,"collectionVersion":null,"links":[{"label":"Wilson, proof of the two-variable case (no AI involved)","url":"https://arxiv.org/abs/2607.23887","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kourovka-21-8-class-transpositions","name":"Kourovka Problem 21.8 - Horizontal Class Transpositions","shortName":"Kourovka 21.8","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"If $\\operatorname{CT}_{(k)}$ is generated by all horizontal class transpositions with modulus at most $k$, is $\\operatorname{CT}_{(k)} \\cong S_{\\operatorname{lcm}(2,\\dots,k)}$ for every $k \\ge 4$?","posedBy":null,"yearPosed":2026,"ageNote":"New in the 21st edition of the Kourovka Notebook (2026).","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"From the Kourovka Notebook, group theory's recognized standing problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17477","sourceName":"arXiv:2607.17477 - On some problems from the Kourovka Notebook","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gaussian-product-inequality-conjecture","name":"Gaussian product inequality conjecture","shortName":"GPI","problemNumber":null,"field":null,"fieldGroup":"Probability & statistics","statement":"Let $\\boldsymbol{X} = (X_1,\\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. Then, for every $\\alpha_1,\\ldots,\\alpha_n > 0$,\n$$\\mathsf{E}\\left[\\prod_{i=1}^n |X_i|^{\\alpha_i}\\right] \\geq \\prod_{i=1}^n \\mathsf{E}\\left[|X_i|^{\\alpha_i}\\right].$$\nMoreover, if $\\mathsf{Var}(X_i) > 0$ for every $i$, then equality holds if and only if $X_1,\\ldots,X_n$ are independent.","posedBy":"Péter E. Frenkel","yearPosed":2007,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"ChatGPT 5.6 Sol (Work Max)","modelMaker":null,"humanCollaborators":["Frédéric Ouimet","Dylan Greaves"],"aiRole":"The AI provided a complete and correct solution without the characterization of equality in terms of independence (but only because the equality case was not in the original prompt by Dylan Greaves).","verification":"lean-verified","verificationNote":"The prompt and output are available at https://chatgpt.com/share/6a5ea69b-1648-83e8-80b1-014ae0b1003c. This early version of the proof was formalized in Lean using Codex; see https://github.com/dylgre/gaussian-product-inequality. The proof has also been checked by ChatGPT 5.6 Sol (Pro), Gemini 3.1 Pro (Extended Thinking), Grok 4.5 (Expert), and Frédéric Ouimet.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":"Research Gate","citationsUrl":"https://doi.org/10.13140/RG.2.2.17569.77923/1","renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The GPI conjecture has a real two-decade literature across probability.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://doi.org/10.13140/RG.2.2.17569.77923/1","sourceName":"A proof of the strong Gaussian product inequality conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":"JollyJackal127","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"kourovka-21-147-relatively-convex","name":"Kourovka Problem 21.147 - Relatively Convex Subgroups","shortName":"Kourovka 21.147","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"Must the right-relatively convex subgroups of a right-orderable nonabelian group form a sublattice of its subgroup lattice? A construction shows they need not.","posedBy":null,"yearPosed":2026,"ageNote":"New in the 21st edition of the Kourovka Notebook (2026).","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"From the Kourovka Notebook, group theory's recognized standing problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17477","sourceName":"arXiv:2607.17477 - On some problems from the Kourovka Notebook","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"semi-streaming-matching-greedy-optimal","name":"Optimality of Greedy for Single-Pass Semi-Streaming Matching","shortName":"Semi-streaming matching","problemNumber":null,"field":"Streaming algorithms","fieldGroup":"Theoretical computer science","statement":"Can any single-pass semi-streaming algorithm beat the naive greedy $1/2$-approximation for maximum matching? No. No single-pass semi-streaming algorithm, deterministic or randomized, achieves a better-than-half approximation, so greedy is optimal. The same construction settles the optimal competitive ratio of online matching with preemption at $1/2$.","posedBy":"Feigenbaum, Kannan, McGregor, Suri, Zhang","yearPosed":2005,"ageNote":"Open since the graph streaming model was introduced in the mid-2000s; the paper describes it as outstanding for over two decades.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Claude Fable 5, GPT-5.6 Sol, Claude Opus 5, Gemini","modelMaker":"Anthropic / OpenAI / Google","humanCollaborators":["Sepehr Assadi","Max Jiang","Mars Xiang"],"aiRole":"The AI acknowledgement is unusually precise. The authors supplied an optimal solution to a relaxation of blueprints, and a combination of Claude Fable and GPT-5.6 Sol gave the idea of using random walks to lift that solution from the relaxation to blueprints. The authors write that these random walks formed the crux of their blueprints. GPT-5.6 Sol also solved the optimization problem behind Lemma 3.1 after the authors directed it to formulate the problem as a linear program, though the proof in the paper is the authors' own. The authors state that no AI-generated text appears in the paper and that they wrote all statements and proofs themselves.","verification":"unreviewed","verificationNote":"arXiv preprint by established authors in the area, building on their own earlier blueprint framework; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The central open question of the graph streaming model, cited across theoretical computer science for two decades, and the reason the model's flagship problem stayed unresolved.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.14656","sourceName":"arXiv:2607.14656 - Semi-Streaming Matching in a Single Pass II: Greedy is Optimal","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kourovka-20-125-rota-baxter","name":"Kourovka Problem 20.125 - Noninjective Rota-Baxter Operator","shortName":"Kourovka 20.125","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"Can a nonabelian group admit a Rota-Baxter operator that is surjective but not injective? A construction shows yes.","posedBy":null,"yearPosed":2022,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"From the Kourovka Notebook, group theory's recognized standing problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17477","sourceName":"arXiv:2607.17477 - On some problems from the Kourovka Notebook","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kourovka-21-150-rank-inequality","name":"Kourovka Problem 21.150 - Rank Inequality for p-Group Extensions","shortName":"Kourovka 21.150","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"For an extension $G = A \\rtimes B$ of elementary abelian $p$-groups with $a \\in A$ satisfying $C_B(a) = 1$, must $H = \\langle a, B\\rangle$ satisfy $\\operatorname{rank}(Z(H) \\cap H') \\le \\operatorname{rank}(B)$? An explicit extension violates the bound.","posedBy":null,"yearPosed":2026,"ageNote":"New in the 21st edition of the Kourovka Notebook (2026).","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"From the Kourovka Notebook, group theory's recognized standing problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17477","sourceName":"arXiv:2607.17477 - On some problems from the Kourovka Notebook","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"canonical-closure-completeness-conp","name":"Completeness of Canonical Closure Representations Is coNP-Complete","shortName":"Closure completeness","problemNumber":null,"field":"Computational Complexity, Closure Systems","fieldGroup":"Theoretical computer science","statement":"A finite closure system can be given by implications or by a list of subsets closed under intersection. Deciding whether one specification of each kind defines the same family had remained open in several settings; the paper proves the problem coNP-complete.","posedBy":"Open in the closure-systems literature","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Mikhail Babin"],"aiRole":"The disclosure states the main proof was obtained by GPT-5.6 Pro through ChatGPT and checked by the author.","verification":"unreviewed","verificationNote":"No independent review; the author reports checking the model's proof himself. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Settles the complexity of a decision problem left open across several formulations in the closure-systems literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.18534","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bh-fdr-universal-multiplicative-bound","name":"Universal Multiplicative FDR Bound for Benjamini-Hochberg","shortName":"BH multiplicative bound","problemNumber":null,"field":"Statistics","fieldGroup":"Probability & statistics","statement":"The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \\downarrow 0$, with an explicit two-sided lower bound $q\\sqrt{\\log(1/q)}/(2\\sqrt{\\pi}) + 0.6493 q + o(q)$.","posedBy":"folklore in the FDR literature","yearPosed":null,"ageNote":"Not attributable to a single paper; the paper cites Reiner-Benaim 2007, Benjamini 2010, Roux 2018, Sarkar 2023 and Sarkar-Zhang 2025 as the record of the belief.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Lihua Lei"],"aiRole":"The disclosure says only that the author used the model for assistance with proof exploration, exposition and editing. It does not attribute any specific step, so the lowest tier applies.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"Benjamini-Hochberg is one of the most used procedures in applied statistics, and how badly dependence can break it is a question the multiple-testing community has tracked for two decades.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.14812","sourceName":"arXiv:2607.14812 - How Much Can Gaussian Dependence Inflate the Benjamini-Hochberg Procedure's FDR?","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kourovka-3-46-locally-soluble","name":"Kourovka Problem 3.46 - Maximal Locally Soluble Normal Subgroups","shortName":"Kourovka 3.46","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"Does there exist a group with more than one but only finitely many maximal locally soluble normal subgroups? An explicit group with exactly two settles it.","posedBy":null,"yearPosed":1969,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":16,"significanceNote":"From the Kourovka Notebook, group theory's recognized standing problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17477","sourceName":"arXiv:2607.17477 - On some problems from the Kourovka Notebook","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kourovka-21-24-power-graph-cograph","name":"Kourovka Problem 21.24 - Cograph Power Graphs Are Chordal","shortName":"Kourovka 21.24","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"If the power graph of a finite group contains no induced path on four vertices, must it also contain no induced cycle of length at least four - that is, is every cograph power graph chordal?","posedBy":null,"yearPosed":2026,"ageNote":"New in the 21st edition of the Kourovka Notebook (2026).","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"From the Kourovka Notebook, group theory's recognized standing problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17477","sourceName":"arXiv:2607.17477 - On some problems from the Kourovka Notebook","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kourovka-19-25-totient-sums","name":"Kourovka Problem 19.25 - Totient Sums and Simplicity","shortName":"Kourovka 19.25","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"Do a finite group's order together with $\\sum_{g \\in G} \\varphi(|g|)$ determine whether the group is simple? A simple and a non-simple group of order $6048$ share the statistic $23984$.","posedBy":null,"yearPosed":2018,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"From the Kourovka Notebook, group theory's recognized standing problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17477","sourceName":"arXiv:2607.17477 - On some problems from the Kourovka Notebook","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-424","name":"Erdős Problem #424","shortName":"Erdős #424","problemNumber":424,"field":"Number Theory, Integer Sequences","fieldGroup":"Number theory","statement":"Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \\dots, a_n$ all possible values of $a_ia_j - 1$ with $i \\ne j$. Is it true that the set of integers which eventually appear has positive density?","posedBy":"Douglas Hofstadter","yearPosed":1977,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"Proves positive lower density. The Formal Conjectures encoding asks for Set.HasPosDensity, a density that exists and is positive; erdosproblems.com says Erdos most likely meant lower density.","claimIssueNote":null,"solveDate":"2026-07-20","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Samuel Korsky"],"aiRole":"GPT-5.6 Pro developed the argument together with Samuel Korsky, in particular searching for the transition matrices the interval-partition argument needs. The Lean formalization was produced separately, with Codex, by Boris Alexeev.","verification":"lean-verified","verificationNote":"Lean 4.32.0 and Mathlib v4.32.0 formalization by Boris Alexeev, produced with Codex, from the informal argument of Samuel Korsky and GPT-5.6 Pro. Rebuilt independently on 2026-08-02 against Lean 4.32.0 and Mathlib v4.32.0 (the file as published, sha256 ca4a2371918b1a7c66dccfe324305298): all 6,394 lines compile in 1,215 s with no sorry and no admit, and #print axioms reports the top theorem depending on exactly [propext, Classical.choice, Quot.sound], the three standard Lean axioms, with no sorryAx and nothing assumed. The formalized conclusion is positive lower density, stated against the same nextGeneration, sequenceSet and generatedSet definitions the Formal Conjectures statement of #424 uses. Status is candidate rather than resolved because erdosproblems.com has not accepted the claim: its proof-claims page states plainly that appearing there is no guarantee of correctness and does not mean anyone associated with the site examined any part of the proof.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"Promoted from the numbered-Erdos baseline of 10 on a reference trail denser than most: Problem 63 of Ben Green's 100 Open Problems, section E31 of Guy's Unsolved Problems in Number Theory, OEIS A005244, a Formal Conjectures entry, two Erdos source citations, and Steinerberger's observation that the literal 1980 phrasing is trivially false. Green's list is a curated signal rather than a compendium - he says he avoided both notorious problems and ones that look hopeless - so inclusion means a live question. Level with Erdos #390 at 13, below #1196 at 15, which has a literature around it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/forum/thread/424","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[{"label":"Lean proof (Erdos424.lean)","url":"https://github.com/plby/lean-proofs/blob/main/src/latest/ErdosProblems/Erdos424.lean","kind":"lean-proof"},{"label":"Proof claim on erdosproblems.com","url":"https://www.erdosproblems.com/forum/thread/424/proof-claims#proof-claim-91","kind":"discussion"},{"label":"Manuscript (PDF)","url":"https://drive.google.com/file/d/1SGSUQhNB8KL75VPqdfAq9yzFNEYL6Mwn/view","kind":"paper"},{"label":"Formal Conjectures statement of #424","url":"https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/424.lean","kind":"lean-statement"},{"label":"Problem 63 of Ben Green's 100 Open Problems","url":"https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf","kind":"problem-record"}],"relations":[],"submittedBy":"GoldenMongoose827","upvotes":0,"downvotes":0,"commentCount":1},{"slug":"jacobian-conjecture","name":"Jacobian Conjecture","shortName":"Jacobian Conjecture","problemNumber":null,"field":"Algebraic Geometry","fieldGroup":"Algebra","statement":"Every polynomial map $\\mathbb{C}^n \\to \\mathbb{C}^n$ with constant nonzero Jacobian determinant is invertible, with a polynomial inverse.","posedBy":"Ott-Heinrich Keller","yearPosed":1939,"ageNote":"Only n ≥ 3 is disproved; the two-variable (plane) case of Keller's 1939 conjecture remains open. Counterexample found by Levent Alpöge with Claude Fable 5.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"n ≥ 3; plane case open","claimIssueNote":null,"solveDate":"2026-07-20","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":["Levent Alpöge"],"aiRole":"Alpöge used Fable 5 as a research collaborator to hunt down an explicit counterexample map in three variables, rather than running a generic formal-proof search. The two-variable (plane) case of the conjecture remains open.","verification":"expert-verified","verificationNote":"The counterexample is hand-checkable by direct substitution and was independently confirmed by outside mathematicians within hours of posting. Briefly caught in a Wikipedia edit war over whether to record it. No formal peer-reviewed publication yet.","publication":"announcement","resolutionMethod":"construction","citations":759,"citationsPaper":"Bass, Connell & Wright (1982), \"The Jacobian conjecture: Reduction of degree and formal expansion of the inverse\", Bull. AMS","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W2081461845","renownLangs":13,"renownNote":null,"significance":65,"significanceNote":"On Smale's problem list and notorious across algebraic geometry since 1939.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://officechai.com/ai/an-anthropic-researcher-says-fable-just-helped-him-disprove-the-85-year-old-jacobian-conjecture/","sourceName":"OfficeChai","collection":null,"collectionVersion":null,"links":[{"label":"Gao's self-contained account, generalizing the tangent-sweep mechanism","url":"https://arxiv.org/abs/2608.00222","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":13,"downvotes":0,"commentCount":0},{"slug":"virtual-surjection-conjecture","name":"The Virtual Surjection Conjecture for Discrete Groups","shortName":"Virtual surjection","problemNumber":null,"field":"Geometric group theory","fieldGroup":"Algebra","statement":"If a subgroup of a product of groups of type $F_k$ virtually surjects onto every $k$-tuple of factors, must it be of type $F_k$ itself? Yes, for discrete groups, and likewise for $FP_k$. The homological $n$-$(n+1)$-$(n+2)$ Conjecture follows for discrete groups when the common quotient is finitely presented, and that hypothesis cannot be dropped.","posedBy":"Martin Bridson, James Howie, Charles Miller III, Hamish Short","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"GPT","modelMaker":"OpenAI","humanCollaborators":["Tal Cohen","Mark Shusterman"],"aiRole":"The authors write that they developed the main arguments in the body of the paper in the course of interactions with GPT. The disclosure does not break the contribution down further.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture on finiteness properties of subgroups of direct products, standard in geometric group theory alongside the n-(n+1)-(n+2) conjecture it implies.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.18079","sourceName":"arXiv:2607.18079 - Virtual Surjection and the n-(n+1)-(n+2) Theorem for Discrete Groups","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"unital-banach-algebra-not-a-calkin-algebra","name":"A Unital Banach Algebra That Is Not a Calkin Algebra","shortName":"Non-Calkin algebra","problemNumber":null,"field":"Banach Algebras, Operator Theory","fieldGroup":"Analysis","statement":"The realisation problem asks which unital Banach algebras arise as the Calkin algebra $\\mathcal{B}(X)/\\mathcal{K}(X)$ of some Banach space. Recorded in Tarbard's thesis and studied by Horváth and Kania. The paper exhibits a unital Banach algebra that cannot be one.","posedBy":"Tomasz Kania, M. Tarbard","yearPosed":2013,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-19","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Antonio Acuaviva","Pablo Acuaviva"],"aiRole":"ChatGPT 5.5 Pro, driven directly and through Codex agents over a filesystem, generated the key ideas and the proof; the authors verified and refined it. The paper reports that the model sometimes misattributed a theorem, cited a result imprecisely, or presented steps as immediate when they still needed checking, so the human verification was load-bearing. The authors state none of these results came from the newer generation of models.","verification":"unreviewed","verificationNote":"No independent check. The proof stands on verification by the paper's authors, who describe correcting misattributions and filling in steps the model presented as immediate. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Answers a recorded realisation question for Calkin algebras by explicit construction; the surrounding programme includes the Argyros–Haydon theorem.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17388","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Tarbard 2013 thesis - Problem statement","url":"https://doi.org/10.5287/ora-8n7rzq1ny","kind":"problem-record"}],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dfjp-interpolation-schauder-basis","name":"Weakly Compact Factorization Through a Space With a Basis","shortName":"DFJP basis factorization","problemNumber":null,"field":"Banach Space Factorization","fieldGroup":"Analysis","statement":"Davis, Figiel, Johnson and Pełczyński showed their interpolation space admits a Schauder basis when the range space has a shrinking one. Can the DFJP space always be chosen with a basis whenever the range space has a basis? The paper proves it can.","posedBy":"Davis, Figiel, Johnson and Pełczyński; raised again by Kevin Beanland","yearPosed":1974,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-19","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Antonio Acuaviva","Pablo Acuaviva"],"aiRole":"ChatGPT 5.5 Pro, driven directly and through Codex agents over a filesystem, generated the key ideas and the proof; the authors verified and refined it. The paper reports that the model sometimes misattributed a theorem, cited a result imprecisely, or presented steps as immediate when they still needed checking, so the human verification was load-bearing. The authors state none of these results came from the newer generation of models.","verification":"unreviewed","verificationNote":"No independent check. The proof stands on verification by the paper's authors, who describe correcting misattributions and filling in steps the model presented as immediate. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The oldest of the batch: a question about the DFJP interpolation construction that the paper states remained open until this proof.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17388","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"MathOverflow 240472 (the question)","url":"https://mathoverflow.net/questions/240472","kind":"discussion"}],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sentos","name":"Erdős Problem #390: the second-order constant for $f(n)-2n$","shortName":"Erdős #390","problemNumber":390,"field":"factorials","fieldGroup":"Number theory","statement":"Let $f(n)$ be the least $m$ for which $n!$ can be written as $a_1\\cdots a_k$ with $n < a_1 < \\cdots < a_k = m$ - the smallest possible largest factor in a factorization of $n!$ into distinct integers all exceeding $n$. Erdős, Guy and Selfridge proved $f(n) - 2n \\asymp n/\\log n$. Erdős asked whether there is a constant $c$ with\n$$f(n) - 2n \\sim c\\,\\frac{n}{\\log n},$$\nand what it is.\n\nThis preprint answers yes and names the constant:\n$$\\lim_{n\\to\\infty}\\frac{(f(n)-2n)\\log n}{n} = \\frac{4029639598}{25970038185} \\approx 0.15516.$$","posedBy":"Paul Erdos","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"The headline is the constant, and its two halves have different histories. The lower bound, $\\liminf (f(n)-2n)/(n/\\log n) \\ge 4029639598/25970038185$, is not new here: it is Mausberg's thirteen-layer valuation cut, posted to the erdosproblems.com forum in May 2026 and credited as such in the paper. Its author wrote there that it \"does not prove an upper bound, nor does it prove that an asymptotic constant exists.\"\n\nThe novelty is the matching upper bound, so the claim is that the thirteen-layer bound is exactly tight. It is assembled from an exact cofactor-allocation certificate, central-binomial anchors, a guarded rough-signature selector, a friable-number covariance bridge, a finite-band tangent correction, and column-sparse rounding. That construction is what a reader should scrutinize; everything else is inherited or machine-checked.\n\nerdosproblems.com still lists #390 as open, and the paper calls itself a proposed solution.","claimIssueNote":null,"solveDate":"2026-07-19","model":"ChatGPT 5.6","modelMaker":"OpenAI","humanCollaborators":["Shouqiao Wang"],"aiRole":"AI wrote 117 pages proof and wrote 382k lines of Lean 4 code to verify this proof using custom prompt by the author which is also available, prompt does not help AI with any math hints or anything.","verification":"lean-verified","verificationNote":"Both halves of the top tier are present, and both were checked here rather than taken on trust.\n\nStatement fidelity: the repository ships a bridge module holding a namespaced copy of the Formal Conjectures extremal function. Diffed against `FormalConjectures/ErdosProblems/390.lean` upstream, the two definitions are identical up to a bound variable's name, and the bridge's terminal `formalF_rhs` is character-for-character the right-hand side of FC's `erdos_390`, which upstream is still marked open. So the formal statement is the problem as posed, pinned in a community-reviewed repository.\n\nKernel check: the project's CI run of 28 July 2026 is green at the commit that is still the state of `390/lean` today - the only later commit adds a LICENSE. Reading the log rather than the badge: 8,635 targets, zero `declaration uses 'sorry'` warnings against `warningAsError = true`, a source grep that fails the build on sorry, admit, axiom, opaque, sorryAx or unsafe, and an audit step printing `formalF_rhs depends on axioms: [propext, Classical.choice, Quot.sound]` with the workflow asserting that exact string. The work-in-progress dependency PrimeNumberTheoremAnd reports `MediumPNT` axiom-clean in the same log, and the tree declares no axioms of its own.\n\nWhat this site did not do is re-run the build: the kernel evidence above is the author's public CI, read at the exact commit. And no human has reviewed the mathematics of the upper bound.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"Above the anchor at 10 for a typical numbered Erdos problem, on the reference trail rather than on judgment. #390 has a published theorem behind it (Erdos, Guy and Selfridge 1982, the paper whose title asks about 239), an OEIS sequence, a formalized statement in Formal Conjectures, an active forum thread, a substantive comment from Terence Tao setting out the obstruction argument, and a 2026 seven-author paper on its sibling #391. That is a denser trail than the large tie at 10. Below Erdos #1196 at 15, which sits in a better-populated corner of the subject.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/ShouqiaoW/erdos/blob/main/390/paper.pdf","sourceName":"A Proposed Solution to Erdős Problem 390","collection":null,"collectionVersion":null,"links":[{"label":"The Lean 4 development, with the Formal Conjectures bridge","url":"https://github.com/ShouqiaoW/erdos/tree/main/390/lean","kind":"lean-proof"},{"label":"Formal Conjectures: the upstream statement of Erdős 390, still marked open","url":"https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/390.lean","kind":"lean-statement"},{"label":"erdosproblems.com #390 - still listed open","url":"https://www.erdosproblems.com/390","kind":"problem-record"},{"label":"Forum thread: Tao on the obstruction, and Mausberg's thirteen-layer lower bound","url":"https://www.erdosproblems.com/forum/thread/390","kind":"discussion"},{"label":"The green CI run whose log this review read (28 July 2026)","url":"https://github.com/ShouqiaoW/erdos/actions/runs/30332889070","kind":"code"}],"relations":[],"submittedBy":"QuietNarwhal605","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"toroidal-elton-odell","name":"The Toroidal Elton–Odell Theorem","shortName":"Toroidal Elton–Odell","problemNumber":null,"field":"Banach Space Geometry","fieldGroup":"Analysis","statement":"The Elton–Odell theorem gives, in every infinite-dimensional normed space, a unit-sphere sequence with mutual distances at least $1+\\varepsilon$. Over $\\mathbb{C}$, identifying vectors differing by a unimodular scalar gives a toroidal distance. Does every infinite-dimensional complex normed space admit such a uniformly separated sequence for that distance? Yes.","posedBy":"Tomasz Kania","yearPosed":2018,"ageNote":"Stated in Kania et al 2018 according to the paper's authors","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-19","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Antonio Acuaviva","Pablo Acuaviva"],"aiRole":"ChatGPT 5.5 Pro, driven directly and through Codex agents over a filesystem, generated the key ideas and the proof; the authors verified and refined it. The paper reports that the model sometimes misattributed a theorem, cited a result imprecisely, or presented steps as immediate when they still needed checking, so the human verification was load-bearing. The authors state none of these results came from the newer generation of models.","verification":"unreviewed","verificationNote":"No independent check. The proof stands on verification by the paper's authors, who describe correcting misattributions and filling in steps the model presented as immediate. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The complex projective analogue of a named classical theorem, raised explicitly in the literature and solved in full.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17388","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Kania et al 2018, 5.3.1 - Conjecture statement","url":"https://doi.org/10.1016/j.jfa.2018.01.008","kind":"problem-record"}],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strict-cosingularity-adjoints-separable-range","name":"Strict Cosingularity and Adjoints for Separable Range","shortName":"Strict cosingularity","problemNumber":null,"field":"Banach Space Operator Ideals","fieldGroup":"Analysis","statement":"Pełczyński's duality between strictly singular and strictly cosingular operators fails without weak compactness. Beanland asked, in work with Androulakis and later on MathOverflow, for the separable-range case: the paper answers it affirmatively and shows that for separable $X$, $T$ is strictly cosingular exactly when $T^{*}$ is strictly singular.","posedBy":"Kevin Beanland","yearPosed":2008,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-19","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Antonio Acuaviva","Pablo Acuaviva"],"aiRole":"ChatGPT 5.5 Pro, driven directly and through Codex agents over a filesystem, generated the key ideas and the proof; the authors verified and refined it. The paper reports that the model sometimes misattributed a theorem, cited a result imprecisely, or presented steps as immediate when they still needed checking, so the human verification was load-bearing. The authors state none of these results came from the newer generation of models.","verification":"unreviewed","verificationNote":"No independent check. The proof stands on verification by the paper's authors, who describe correcting misattributions and filling in steps the model presented as immediate. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A duality question left open in a 2008 paper and later posted publicly on MathOverflow, now settled in the form asked.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.17388","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"MathOverflow 98449 (the question)","url":"https://mathoverflow.net/questions/98449","kind":"discussion"}],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"primariness-lp-l1","name":"Primariness of the Mixed-Norm Space $L_p(L_1)$","shortName":"Primariness of $L_p(L_1)$","problemNumber":null,"field":"Banach Space Decomposition Theory","fieldGroup":"Analysis","statement":"A Banach space is primary if in every decomposition into two complemented subspaces one summand is isomorphic to the whole. Lechner, Motakis, Müller and Schlumprecht identified the primariness of $L_p(L_1)$ as a prominent remaining open case; the paper proves it is primary for $10$ such that for all large $N$ some $A \\subseteq \\{1,\\ldots,N\\}$ of size $> cN$ has that property, which also gives the best known lower bounds for related unit-fraction avoidance problems.","posedBy":"Paul Erdos, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-16","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Will Sawin"],"aiRole":"The author gives a narrative rather than a blanket acknowledgement. Starting from a computation of Stijn Cambie, he asked ChatGPT to look for patterns in Cambie's extremal set that might suggest a generalization; it observed that in a pair with a given ratio the larger element is usually absent unless the smaller is absent for other reasons, and described a change of variables. Dropping the hedges in that observation gives the set the paper analyzes, which turns out to be essentially where Hooley's function takes its minimum value. The model was also used for reference search and proofreading.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A question from the Erdos-Graham problem collection on unit fractions, the kind of documented named problem that has a real if specialized following.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.15419","sourceName":"arXiv:2607.15419 - Sets of unit fractions without two members whose average is a unit fraction","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gao-huo-ma-consecutive-cycle-lengths","name":"Gao-Huo-Ma Question on Cycle Lengths in Critical Graphs","shortName":"Consecutive cycle lengths","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Gao, Huo and Ma asked whether for every fixed $k \\ge 3$ there is a function $f_k(n) \\to \\infty$ such that every $n$-vertex $(k+1)$-critical graph contains $f_k(n)$ consecutive cycle lengths. The paper settles this and two related problems on cycle lengths and cycles with chords under chromatic and degree constraints.","posedBy":"Gao, Huo, Ma","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-16","model":"ChatGPT 5.5","modelMaker":"OpenAI","humanCollaborators":["Xiaozheng Chen","Bo Ning"],"aiRole":"The declaration says the model was used in the search for the construction in Theorem 1.2, starting from a requirement based on the Hajos join that the second author supplied, and that the present construction emerged after several rounds of refinement, discussion and checking. It also helped with proofreading.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered question from a 2021 paper on cycle lengths in critical graphs, with a documented but specialized following.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.15501","sourceName":"arXiv:2607.15501 - Cycle lengths and chords under chromatic and degree constraints","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"jungic-rainbow-ap-order","name":"The Order of Long Rainbow Arithmetic Progressions","shortName":"Rainbow APs","problemNumber":null,"field":"Ramsey theory","fieldGroup":"Combinatorics","statement":"Let $T_k$ be the least $t$ such that every equinumerous $t$-coloring of $[tn]$ contains a rainbow $k$-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured $T_k = \\Theta(k^2)$; Conlon, Fox and Sudakov proved $T_k = O(k^2 \\log k)$. The matching lower bound $T_k = \\Omega(k^2 \\log k)$ holds, so $T_k = \\Theta(k^2 \\log k)$ and the conjectured order is wrong.","posedBy":"Veselin Jungic, Jacob Licht, Mohammad Mahdian, Jaroslav Nesetril, Rados Radoicic","yearPosed":2003,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"the true order is determined, and it is not the conjectured one","claimIssueNote":null,"solveDate":"2026-07-16","model":"Codex (GPT-5.6), Claude Code (Fable 5)","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Jesse Geneson"],"aiRole":"The acknowledgement says the two systems were used for proof exploration, proof criticism, exposition and revision. Proof exploration and criticism are mathematical work rather than prose work, but no specific step is attributed, so the lowest tier applies.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A 2003 Combinatorics, Probability and Computing conjecture in anti-Ramsey theory that Conlon, Fox and Sudakov had already worked on, well known within extremal combinatorics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.15116","sourceName":"arXiv:2607.15116 - The order of long rainbow arithmetic progressions","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"schur-number-recurrence","name":"The Abbott-Hanson Recurrence for Schur Numbers","shortName":"Schur number recurrence","problemNumber":null,"field":"Ramsey theory","fieldGroup":"Combinatorics","statement":"The classical Abbott-Hanson recurrence gives $S(k+2) \\ge 9S(k)+4$ for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted $S$-templates, a more flexible form of Rowley's template construction, yield $S(k+2) \\ge 10S(k)+2$ and hence improved lower bounds.","posedBy":"Harvey Abbott, Denis Hanson","yearPosed":1972,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown","claimIssueNote":null,"solveDate":"2026-07-16","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Nils Bengone","Amine Brouk","Max Grinsztajn","Terence Helbert","Bao Lugherini","Arpad Rimmel","Joanna Tomasik"],"aiRole":"The abstract says the shifted $S$-template construction was discovered during a conversation with ChatGPT 5.5 Pro and then refined, verified and extended to multiple templates by the authors, and the conclusion describes the paper as building on an original idea of the model.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Schur numbers are a classical Ramsey-theoretic quantity where progress is rare and the Abbott-Hanson recurrence had been the standard tool for over fifty years.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.15034","sourceName":"arXiv:2607.15034 - Shifted S-templates and improved lower bounds for Schur numbers","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"belinskaya-theorem-measure-preserving-flows","name":"Belinskaya's Theorem for Measure-Preserving Flows","shortName":"Belinskaya for flows","problemNumber":null,"field":"Ergodic Theory, Orbit Equivalence","fieldGroup":"Analysis","statement":"Two free ergodic measure-preserving flows whose $\\mathrm{L}^1$ full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This proves the flow analogue of Belinskaya's theorem, answering a question posed by François Le Maître and the author.","posedBy":"François Le Maître and Konstantin Slutsky","yearPosed":2021,"ageNote":"arXiv preprint posing the problem first published in 2021, later revised in 2025 before publication in a journal (EMS)","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-16","model":"GPT-5 + Claude Opus 4","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Konstantin Slutsky"],"aiRole":"The paper states the criterion the proof turns on and its application were discovered autonomously by a two-agent AI system running GPT-5 and Claude Opus 4 in a research loop; Codex was used separately for proofreading and stylistic editing.","verification":"unreviewed","verificationNote":"No independent review. The paper attributes the key criterion to an autonomous agent loop rather than to prompted assistance. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Answers a question the author had posed with Le Maître, transferring a classical rigidity theorem from transformations to flows.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.14444","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Le Maitre & Slutsky 2021, Questions 10.10 & 10.17: Problem statement","url":"https://arxiv.org/abs/2108.09009","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"weak-simplex-conjecture","name":"The Weak Simplex Conjecture","shortName":"Weak Simplex Conjecture","problemNumber":null,"field":"Information theory","fieldGroup":"Probability & statistics","statement":"Among $d+1$ equiprobable equal-energy signals in Gaussian noise, is the regular simplex optimal for average error probability? Yes. The underlying comparison is that for any $m \\times m$ correlation matrix $R$ with $R - \\mathbf{1}\\mathbf{1}^{\\mathsf T}/m \\succeq 0$ and $X \\sim \\mathcal{N}(0,R)$, the maximum of the $X_i$ is stochastically dominated by the maximum of $m$ independent standard Gaussians.","posedBy":"Claude Shannon; Balakrishnan; Weber","yearPosed":1965,"ageNote":"The simplex conjectures go back to the 1960s coding-theory literature; the strong form was disproved by Steiner in 1994.","solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-15","model":"GPT-5.6 Pro, GPT-5.6 Sol Max","modelMaker":"OpenAI","humanCollaborators":["Abhijeet Mulgund"],"aiRole":"The disclosure says the models were used during the development of the work to assist with mathematical exploration, proof development and checking, literature discovery and organization, and drafting. It attributes no specific step, so the lowest tier applies.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named conjecture of long standing in information theory about optimal signal sets under Gaussian noise, with the strong form already known false.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.14087","sourceName":"arXiv:2607.14087 - Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"quantum-memory-advantage-process-tomography","name":"Quantum Memory Advantage for Process Tomography","shortName":"Quantum memory advantage","problemNumber":null,"field":"Quantum information","fieldGroup":"Quantum information & computing","statement":"Does quantum memory give a query-complexity advantage for learning an unknown quantum channel, when protocols without it must measure after each channel use and keep only a classical transcript? It does, and the paper also determines how little coherent memory suffices for the advantage to appear.","posedBy":"open question in quantum learning theory","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-15","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Carlos Bravo-Prieto","Weiyuan Gong","Antonio Anna Mele"],"aiRole":"The authors acknowledge the model for assistance in discussing and refining proof ideas, as well as improving the presentation. No specific step is attributed, so the lowest tier applies.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named open question in quantum process tomography, part of the memory-versus-no-memory separation programme that has been active across quantum learning theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.13476","sourceName":"arXiv:2607.13476 - Quantum memory advantage for quantum process tomography","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kac-walk-cutoff-sphere","name":"Cutoff for Kac's Walk on the Sphere","shortName":"Kac walk cutoff","problemNumber":null,"field":"Markov chains","fieldGroup":"Probability & statistics","statement":"The discrete-time Kac walk on $S^{n-1}$ started from a coordinate vector exhibits total variation cutoff at $C_{\\mathrm{BRW}} n \\log n$, where $C_{\\mathrm{BRW}} \\approx 3.8916$ is set by the speed of the leftmost particle in a branching random walk. The cutoff is therefore not at the conjectured $2n\\log n$.","posedBy":"Mark Kac","yearPosed":1956,"ageNote":"Kac introduced the walk in 1956; the 2n log n cutoff location was the standing conjecture that this work overturns.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"the conjectured cutoff location of 2n log n is wrong","claimIssueNote":null,"solveDate":"2026-07-15","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Vishesh Jain","Clayton Mizgerd"],"aiRole":"The acknowledgement says the authors used ChatGPT extensively, at the level of a co-author, for brainstorming, help with proofs, literature review, checking for mistakes, writing code, and preparing the manuscript. It does not separate which parts came from where.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Kac's walk is a standard model in the mixing-time literature and its cutoff location had a specific conjectured value that this determines and refutes.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.13401","sourceName":"arXiv:2607.13401 - Total variation cutoff for Kac's walk on the sphere","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"davenport-constant-heisenberg-125","name":"The Small Davenport Constant of the Heisenberg Group of Order 125","shortName":"Davenport, $H_{125}$","problemNumber":null,"field":"Additive Combinatorics, Zero-Sum Theory","fieldGroup":"Combinatorics","statement":"Godara and Sarkar proved $\\mathsf{d}(H_{27})=6$ for the exponent-$p$ Heisenberg group and posed $\\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$, leaving $p\\ge5$ open. The paper settles the first open case, $\\mathsf{d}(H_{125})=12$, the upper bound reducing to a finite spread bound verified by exhaustive search and independently reproduced.","posedBy":"Naveen K. Godara, Siddhartha Sarkar","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Settles the case p=5. The posed formula for every odd prime remains open.","claimIssueNote":null,"solveDate":"2026-07-15","model":"Claude + GPT-5","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Patrick White"],"aiRole":"An attribution note states the results were obtained by an AI research collaboration in which an AI system occupied the principal research seat under human direction. Claude produced the search implementations, the independent reproduction of the load-bearing computation and the write-up; a tool-free GPT-5 produced the reduction to the lemma chain and the corrected hypotheses of several lemmas.","verification":"unreviewed","verificationNote":"No independent review, though the load-bearing computation was reproduced by a second search with a different pruning strategy. Preprint, not refereed.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"First open case of a posed formula in zero-sum theory over non-abelian groups.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.14379","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Godara & Sarkar 2025: Conjecture statement","url":"https://doi.org/10.1142/S0219498825502688","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"witten-zeta-leading-residue","name":"A Universal Leading-Residue Formula for Witten Zeta Functions","shortName":"Witten zeta residue","problemNumber":null,"field":"Zeta Functions, Root Systems","fieldGroup":"Number theory","statement":"For an irreducible crystallographic root system of rank $r$ with Coxeter number $h$, the paper proves that Au's normalized Witten zeta function has a simple pole at $2/h$ and evaluates its residue in closed form in terms of the Cartan determinant, the Weyl group order and the invariant degrees.","posedBy":"Arising from Au's work on Witten zeta functions","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-14","model":"GPT-5.6 Sol via Codex","modelMaker":"OpenAI","humanCollaborators":["Jonas Matuzas"],"aiRole":"The most complete authorship claim in this batch: the paper states that all mathematical derivations, proofs, exposition, computational code and publication materials were generated and written entirely in Codex using GPT-5.6 Sol, with Codex Ultra mode for multi-agent work, and that GPT-5.6 Pro was used separately in the ChatGPT web app to review the derivations.","verification":"unreviewed","verificationNote":"No independent review, and the author describes the entire mathematical content as machine-generated, reviewed by a second model rather than by a person. Treat accordingly. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"A specialist closed-form evaluation rather than a named open problem, listed for the unusually complete AI authorship.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.12728","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"derivative-free-convex-oracle-gap","name":"Oracle-Complexity Gap in Derivative-Free Convex Optimization","shortName":"Zeroth-order oracle gap","problemNumber":null,"field":"Optimization (Oracle Complexity)","fieldGroup":"Algorithms & optimization","statement":"For deterministically minimizing a convex 1-Lipschitz function on the $d$-dimensional ball using only exact function values, the query complexity sat between $\\Omega(d)$ and $O(d^2 \\log^2 d)$ since 1996. The paper proves a near-quadratic lower bound $\\Omega(d^2 / \\log(d+1))$, closing the gap: $Q(d, \\sim d^{-1/2}) = \\Theta(d^2)$, a polynomial separation from full first-order information.","posedBy":"Vladimir Protasov (gap since 1996)","yearPosed":1996,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-14","model":"GPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":["Phillip Kerger"],"aiRole":"Kerger reports that GPT-5.6 Sol Pro solved the problem rather than the author, following a workflow like OpenAI's Cycle Double Cover effort. It first proved a $\\tilde{\\Omega}(d^2)$ lower bound at accuracy of order $d^{-3}$ (after ~148 minutes), which was then refined to the order-$d^{-1/2}$ result via a further ~230-minute run. The author verified the arguments by hand and takes full responsibility.","verification":"unreviewed","verificationNote":"arXiv preprint 2607.13335 (14 Jul 2026) by Phillip Kerger (UC Berkeley), not yet peer-reviewed. The weaker-accuracy $\\tilde{\\Omega}(d^2)$-at-$d^{-3}$ lower bound was formally verified in Lean (github.com/PhillipKerger/zero-order-bounds-lean-verification); the headline improvement to accuracy $d^{-1/2}$ is not yet Lean-formalized (it needs convex-geometry results like Urysohn's inequality absent from current Lean libraries) and rests on the author's hand verification.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A 30-year oracle-complexity gap in zeroth-order optimization.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.13335","sourceName":"arXiv:2607.13335","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"brualdi-interchange-graph-hamiltonicity","name":"Brualdi's Question on Hamiltonicity of Interchange Graphs","shortName":"Interchange graphs","problemNumber":null,"field":"Combinatorial matrix theory","fieldGroup":"Combinatorics","statement":"The interchange graph $G(R,S)$ has the $(0,1)$-matrices with row sums $R$ and column sums $S$ as vertices, adjacent when they differ by a single $2\\times 2$ interchange. Brualdi asked whether $G(R,S)$ is always Hamiltonian. It satisfies more: it is maximally Hamiltonian, Hamilton-laceable when bipartite and Hamilton-connected when not.","posedBy":"Richard A. Brualdi","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-14","model":"Claude, GPT/Codex","modelMaker":"Anthropic / OpenAI","humanCollaborators":["Jeffrey S. Baggett","Huiya Yan"],"aiRole":"The declaration says the computational search and verification programs and the Lean 4 formalization were developed with AI-assisted tools under author direction, and that no AI system is an author. The structural induction carrying the proof is the authors'.","verification":"unreviewed","verificationNote":"The paper reports a Lean 4 formalization alongside computational search and verification programs. We have not compiled it. arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A long-standing question of Brualdi in combinatorial matrix theory, standard background for anyone working with interchange classes of 0-1 matrices.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.13165","sourceName":"arXiv:2607.13165 - Interchange graphs of (0,1)-matrices are maximally Hamiltonian","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"thakur-carlitz-wieferich-conjecture","name":"Thakur's Conjecture on Carlitz-Wieferich Primes","shortName":"Carlitz-Wieferich primes","problemNumber":null,"field":"Function field arithmetic","fieldGroup":"Number theory","statement":"A monic prime $P$ of $\\mathbb{F}_q[T]$ is a $c$-Wieferich prime if $\\rho_P(1) \\equiv 1 \\bmod P^2$ for the Carlitz module $\\rho$. On limited data and proofs in degrees $2$ and $3$, Thakur suggested in 2015 that in odd characteristic every $c$-Wieferich prime has degree divisible by $p$. It is false: an explicit irreducible $c$-Wieferich prime has degree not divisible by $p$, and the resulting common factor has a closed form.","posedBy":"Dinesh Thakur","yearPosed":2015,"ageNote":"Restated as open in 2024; Bamunoba and Bergstrom, after extensive computation, expressed the belief that it holds in odd characteristic.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-14","model":"Claude Opus 4.8","modelMaker":"Anthropic","humanCollaborators":["David Niedbala Giraudin"],"aiRole":"The methodology section is the most explicit division of labour in this batch. The author is an independent researcher with no formal mathematical training. He set the research direction and the criteria for which problems to pursue and contributed a structural, visual reading of the objects; the model proposed problems meeting those criteria and supplied the mathematical domain knowledge, the formalization, the drafting, and the design and execution of all computations, under his direction. The strategy emerged from the dialogue. Lacking the training to verify the mathematics directly, the author relied on exact computational checks reproduced across independent systems.","verification":"site-confirmed","verificationNote":"Independently reproduced. We recomputed the claim from the definitions in arXiv:2607.15305, from scratch and with no computer-algebra dependency, so the check shares no code with the author's appendix. Confirmed: $x^3-8x^2-4x-11$ is irreducible over $\\mathbb{F}_{19}$, so $\\mathbb{F}_{19^3}$ is a field; $P$ is monic of degree $5$, irreducible over $\\mathbb{F}_{19^3}$, and genuinely uses the cubic extension; and $\\rho_P(1) \\equiv 1 \\bmod P^2$, which is the definition of a $c$-Wieferich prime, computed through the Carlitz recursion inside $\\mathbb{F}_q[T]/(P^2)$. The Bamunoba-Bergstrom criterion the paper cites, $M_5(\\theta) = 0$, was computed by a separate route and agrees. We also confirmed that $\\mu(X)$ divides $X + X^q + \\cdots + X^{q^4}$, which is what makes $G = \\mu(T^q - T)$ divide $[5]$. Since $19 \\nmid 5$, the counterexample stands. This matters more than usual here because the author states he cannot verify the mathematics directly. arXiv preprint (v2), not peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A documented suggestion of Thakur restated as an open question in 2024 and believed true after extensive computation, but confined to the function-field arithmetic community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.15305","sourceName":"arXiv:2607.15305 - A counterexample to a conjecture of Thakur on Carlitz-Wieferich primes","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"record-compositions-alternating-permutations","name":"Record Compositions of Alternating Permutations","shortName":"Record compositions","problemNumber":null,"field":"Enumerative Combinatorics, Symmetric Functions","fieldGroup":"Combinatorics","statement":"Amdeberhan, Shareshian and Stanley showed a function from the theory of partition Eisenstein series counts alternating permutations with a given record partition, and asked whether a similar theory exists for record compositions, suggesting a role for noncommutative symmetric functions. The paper solves that open problem with a product formula.","posedBy":"Amdeberhan, Shareshian and Stanley","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-14","model":"AxiomProver","modelMaker":"Axiom Math","humanCollaborators":["Evan Chen","Ken Ono","Michal Mogielnicki"],"aiRole":"The paper states that AxiomProver autonomously produced and verified the results in Lean, and devotes a section to the protocol, the formal files and the verification environment.","verification":"lean-checked","verificationNote":"The paper reports an accompanying Lean/Mathlib formalization produced autonomously by AxiomProver, described in its own section. Not rebuilt here, and not independently reviewed. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Solves a problem posed by Stanley and coauthors in the theory of partition Eisenstein series, and does so with a machine-checked proof.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.12873","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Amdeberhan et al 2026 - Problem statement","url":"https://arxiv.org/abs/2605.27828","kind":"problem-record"},{"label":"GitHub link to the problem statement in Lean","url":"https://github.com/AxiomMath/record-compositions/blob/main/RecordCompositions/problem.lean","kind":"lean-statement"},{"label":"Github link to the proof in Lean","url":"https://github.com/AxiomMath/record-compositions/blob/main/RecordCompositions/solution.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sabidussi-compatibility","name":"Sabidussi's Compatibility Conjecture","shortName":"Sabidussi compatibility","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Can the edges of a finite connected multigraph, given a closed eulerian trail, be partitioned into circuits so that no circuit contains two edges used consecutively in the trail? The proof in fact four-colours the edges to satisfy the constraints.","posedBy":"Gert Sabidussi","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-14","model":"GPT-5.6 Pro, GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Nikolay Ulyanov"],"aiRole":"Developed with GPT-5.6 Pro and GPT-5.6 Sol; the author reviewed the proof.","verification":"lean-verified","verificationNote":"Lean 4 formalization available in the author's repository, alongside the arXiv preprint.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"An old named conjecture in structural graph theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.13225","sourceName":"arXiv:2607.13225 - A proof of Sabidussi's compatibility conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Fleischner 1980: Conjecture statement","url":"https://www.sciencedirect.com/science/article/pii/0095895680900775","kind":"problem-record"},{"label":"GitHub link to the conjecture statement (Challenge.lean)","url":"https://github.com/gexahedron/sabidussi-lean/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"GitHub repo of the proof","url":"https://github.com/gexahedron/sabidussi-lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"benjamini-hochberg-correlated-gaussian","name":"Benjamini-Hochberg FDR Under Correlated Gaussian Tests","shortName":"BH under correlation","problemNumber":null,"field":"Statistics","fieldGroup":"Probability & statistics","statement":"Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives $\\mathrm{FDR} > 0.0104$ at nominal level $\\alpha = 0.01$.","posedBy":null,"yearPosed":2006,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Edgar Dobriban"],"aiRole":"The counterexample was obtained by GPT-5.6 Pro and carefully checked by the author, with a rigorous interval-arithmetic certificate valid for all sufficiently large numbers of hypotheses.","verification":"unreviewed","verificationNote":"Author-checked arXiv preprint with an interval-arithmetic certificate. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":14,"significanceNote":"FDR under correlation is a widely felt applied-statistics question, but diffuse as a single problem.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.12208","sourceName":"arXiv:2607.12208 - The Benjamini-Hochberg procedure can fail to control the FDR","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-336","name":"Erdős Problem #336","shortName":"Erdős #336","problemNumber":336,"field":"Number Theory, Additive Basis","fieldGroup":"Number theory","statement":"If $h(r)$ is the maximal finite exact order attainable by an additive basis of order at most $r$, what is $\\lim_{r \\to \\infty} h(r)/r^2$? The candidate proof identifies the sharp limit $1/3$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/336","sourceName":"erdosproblems.com/336","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"classical-smith-ward-problem","name":"The Classical Smith-Ward Problem","shortName":"Smith-Ward problem","problemNumber":null,"field":"Operator algebras","fieldGroup":"Analysis","statement":"The Smith-Ward theorem realizes the first $k$ essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of $k$, equivalently whether the identity map on a three-dimensional operator system $\\mathrm{span}\\{1,q(D),q(K)\\}$ in the Calkin algebra always lifts. It need not: an explicit three-dimensional hyperrigid operator system has no unital completely positive lift, and its dual is the first three-dimensional operator system that fails to be exact.","posedBy":"R. R. Smith, J. D. Ward","yearPosed":1980,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"Harris had settled the generalized problem in dimension four; this reaches dimension three","claimIssueNote":null,"solveDate":"2026-07-13","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Marcel Scherer"],"aiRole":"The one-line disclosure says the model was used to perform literature search and to accelerate the search for the operator system. Since that operator system is the counterexample, the contribution touches the mathematics, but the wording does not say the model found it.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint (v2), isolating and strengthening an argument of Harris; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A named lifting problem in operator algebras with a documented line of attack through Paulsen and more recently Harris.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.04274","sourceName":"arXiv:2607.04274 - A Three-Dimensional Operator System without the Smith-Ward Property","collection":null,"collectionVersion":null,"links":[{"label":"Smith & Ward 1980: Problem statement","url":"https://www.jstor.org/stable/2374180","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-538","name":"Erdős Problem #538","shortName":"Erdős #538","problemNumber":538,"field":"Number Theory, Multiplicative Combinatorics","fieldGroup":"Number theory","statement":"If each integer has at most $r$ representations $m = pa$ with $p$ prime and $a \\in A \\subseteq [1, N]$, what is the best upper bound for $\\sum_{a \\in A} 1/a$? The candidate proof gives the matching order $\\Theta_r(\\log N / \\log\\log N)$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/538","sourceName":"erdosproblems.com/538","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"two-counterexamples-geometry-of-numbers","name":"Two Counterexamples in the Geometry of Numbers","shortName":"Geometry of numbers","problemNumber":null,"field":"Geometry of Numbers, Lattices","fieldGroup":"Number theory","statement":"The paper gives counterexamples in dimensions eight and nine to two problems: the Cartesian-product problem posed by Cassels for critical determinants and formulated by Zong for lattice packings, and a question raised by Sarnak, formulated as a conjecture by Chiu, on whether height among unit-volume flat tori is minimized by a lattice maximizing its shortest nonzero vector.","posedBy":"J. W. S. Cassels and Chuanming Zong; Peter Sarnak, formulated by Chiu","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"ChatGPT 5 + Codex","modelMaker":"OpenAI","humanCollaborators":["Nihar Gargava"],"aiRole":"The acknowledgements state the counterexamples were found using ChatGPT, and that the final paper is partly the author's own writing and partly written with the help of Codex.","verification":"unreviewed","verificationNote":"No independent check, but the results are explicit counterexamples in dimensions eight and nine and are therefore checkable by inspection. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Refutes two separate stated problems at once, one of them a question of Sarnak.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.11695","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"type-d-asep-tracy-widom","name":"Type-D ASEP Tracy-Widom Marginals","shortName":"Type-D ASEP","problemNumber":null,"field":"Interacting particle systems","fieldGroup":"Probability & statistics","statement":"Do the one-species current marginals of type-D ASEP have the predicted Tracy-Widom long-time asymptotics despite the model's two-species interactions?","posedBy":"Jeffrey Kuan","yearPosed":2022,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"Claude Opus 4.8, Claude Fable 5, Aristotle","modelMaker":"Anthropic / Harmonic","humanCollaborators":["Jeffrey Kuan"],"aiRole":"The exact current-decoupling identity behind the proof was developed with Claude models; Aristotle checked foundational tiers in Lean.","verification":"unreviewed","verificationNote":"Author-reviewed arXiv preprint with foundational Lean tiers machine-checked. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A concrete KPZ-universality question for a named process.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.11376","sourceName":"arXiv:2607.11376 - Long-time asymptotics of type D ASEP","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ising-perceptron-storage-capacity","name":"Krauth-Mezard Storage Capacity of the Ising Perceptron","shortName":"Ising perceptron capacity","problemNumber":null,"field":"Probability","fieldGroup":"Probability & statistics","statement":"Krauth and Mezard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant $\\alpha_\\star \\approx 0.8330786$. Ding and Sun proved the matching lower bound and Huang the upper bound, but each was conditional on a global sign condition nobody had verified. Both conditions now hold rigorously, so $M_N/N \\to \\alpha_\\star$ in probability with $\\alpha_\\star \\in [0.833078599, 0.833078600]$.","posedBy":"Werner Krauth, Marc Mezard","yearPosed":1989,"ageNote":"The constant was predicted by the replica method in 1989; Ding-Sun and Huang reduced the problem to two global sign conditions that stayed unverified until now.","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"the last conditional step in the Ding-Sun and Huang bounds is discharged","claimIssueNote":null,"solveDate":"2026-07-13","model":"Claude Fable 5, Codex","modelMaker":"Anthropic / OpenAI","humanCollaborators":["Yitzchak Shmalo"],"aiRole":"The declaration is among the most explicit in the catalog: most of the work reported was done by two AI systems. Fable 5 developed the main mathematical architecture, named as the moment-coordinate reduction, the ray majorants and their pinned identities, the bound, the sublevel-set localization and the corrected near-one chain, and wrote the bulk of the proof code and the initial manuscript. Codex ran a separate internal re-audit of the final source and proof boundary and validated the repaired verifier blocks.","verification":"unreviewed","verificationNote":"A computer-assisted proof using Arb interval arithmetic. The paper is unusually candid about the failure modes: two formula-level errors, a too-tight ball rule and a sign error in a tilted constraint term, survived until a 30-digit quadrature and a finite-difference cross-check caught them, and the author notes that Arb guarantees the arithmetic and not the formulas. All reported certificates postdate both corrections. arXiv preprint, not peer-reviewed, and we have not re-run the certificates.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"The capacity of the Ising perceptron is a landmark of rigorous statistical physics, and the Ding-Sun proof was celebrated precisely because it came so close; what remained was an unverified numerical condition.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.11010","sourceName":"arXiv:2607.11010 - The storage capacity of the Ising perceptron: verification of the outstanding numerical conditions","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-267","name":"Erdős Problem #267","shortName":"Erdős #267","problemNumber":267,"field":"Number Theory, Irrationality","fieldGroup":"Number theory","statement":"If $n_1 < n_2 < \\cdots$ with $n_{k+1}/n_k \\ge c > 1$, must $\\sum_k 1/F_{n_k}$ be irrational? The proposed proof closes the range $1 < c < 2$ left open by earlier criteria.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/267","sourceName":"erdosproblems.com/267","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1188","name":"Erdős Problem #1188","shortName":"Erdős #1188","problemNumber":1188,"field":"Number Theory, Covering Systems","fieldGroup":"Number theory","statement":"Estimate the number $F(x)$ of minimal distinct covering systems whose moduli all lie in $[1, x]$. The candidate proof gives $\\log\\log F(x)/\\log x \\to 1$, i.e. $F(x) = \\exp(x^{1+o(1)})$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1188","sourceName":"erdosproblems.com/1188","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-769","name":"Erdős Problem #769","shortName":"Erdős #769","problemNumber":769,"field":"Discrete Geometry","fieldGroup":"Geometry & topology","statement":"For the least cutoff $c(n)$ after which every $k$ occurs as the number of homothetic cubes in a decomposition of the unit $n$-cube, is $c(n) \\gg n^n$? The Lean proof shows $c(n) = o(n^n)$ along odd dimensions.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"the conjectured lower bound is disproved; good bounds for c(n) remain open","claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; community status pending.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/769","sourceName":"erdosproblems.com/769","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-662","name":"Erdős Problem #662","shortName":"Erdős #662","problemNumber":662,"field":"Discrete Geometry","fieldGroup":"Geometry & topology","statement":"Among sufficiently large one-separated planar point sets, does the triangular lattice maximize the number of distances below each threshold? Explicit rational oblique lattices beat the triangular lattice under several closed- and strict-shell readings.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"natural readings of the ambiguous historical statement are disproved","claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Finite certificates and variant statements Lean-checked; community status pending.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/662","sourceName":"erdosproblems.com/662","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1189","name":"Erdős Problem #1189","shortName":"Erdős #1189","problemNumber":1189,"field":"Number Theory, Covering Systems","fieldGroup":"Number theory","statement":"For irreducible covering sets of size $k$, determine their count, the possible largest modulus, the maximal reciprocal sum, and whether divisor-set examples occur infinitely often.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"exact largest modulus 3·2^{k-3} for k ≥ 5, near-linear least maximum, reciprocal mass Θ(log k), and an infinite divisor family; the counting asymptotic rests on the cited BBMST theorem","claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 Star Fleet (Claude Fable 5 referee)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked with one named literature input made explicit; community status pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1189","sourceName":"erdosproblems.com/1189","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-254","name":"Erdős Problem #254","shortName":"Erdős #254","problemNumber":254,"field":"Number Theory, Complete Sequences","fieldGroup":"Number theory","statement":"If $A \\subseteq \\mathbb{N}$ has unbounded dyadic-shell counts and $\\sum_{n \\in A} \\|\\theta n\\| = \\infty$ for every $0 < \\theta < 1$, must $A$ be complete - is every sufficiently large integer a sum of distinct elements of $A$?","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/254","sourceName":"erdosproblems.com/254","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-394","name":"Erdős Problem #394","shortName":"Erdős #394","problemNumber":394,"field":"Number Theory, Multiplicative","fieldGroup":"Number theory","statement":"For the least $t_k(n)$ with $n \\mid t_k(n)(t_k(n)+1)\\cdots(t_k(n)+k-1)$, do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with $c = 1/2048$ admissible in the $t_2$ bound.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/394","sourceName":"erdosproblems.com/394","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-489","name":"Erdős Problem #489","shortName":"Erdős #489","problemNumber":489,"field":"Number Theory, Sieve Theory","fieldGroup":"Number theory","statement":"If $A$ is a forbidden-divisor set with $|A \\cap [1,x]| = o(\\sqrt{x})$ and $B = \\{b_1 < b_2 < \\cdots\\}$ the sifted set, must $x^{-1} \\sum_{b_i < x} (b_{i+1} - b_i)^2$ converge to a finite limit?","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/489","sourceName":"erdosproblems.com/489","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-796","name":"Erdős Problem #796","shortName":"Erdős #796","problemNumber":796,"field":"Number Theory, Multiplicative Combinatorics","fieldGroup":"Number theory","statement":"If $g_3(n)$ is the largest size of $A \\subseteq [1,n]$ with fewer than three representations of every product $a_1 a_2$, does its conjectured second-order normalized term converge? The candidate proof gives an explicit limit constant.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/796","sourceName":"erdosproblems.com/796","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-709","name":"Erdős Problem #709","shortName":"Erdős #709","problemNumber":709,"field":"Extremal Divisibility","fieldGroup":"Number theory","statement":"How long must an interval be to contain distinct representatives $x_i$, with $a_i \\mid x_i$, for every $n$-element set of moduli $A = \\{a_1, \\dots, a_n\\}$?","posedBy":null,"yearPosed":1959,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"upper bound improved to f(n) ≤ 14n^{3/7} with an explicit logarithmic lower bound; matching bounds remain open","claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked construction; community status pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/709","sourceName":"erdosproblems.com/709","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-450","name":"Erdős Problem #450","shortName":"Erdős #450","problemNumber":450,"field":"Number Theory, Divisors","fieldGroup":"Number theory","statement":"How large must $y(\\varepsilon, n)$ be so that every interval $(x, x+y)$ contains at most $\\varepsilon y$ integers having a divisor in $(n, 2n)$? The candidate proof gives the sharp fixed-$\\varepsilon$ order $y = \\Theta_\\varepsilon(n)$, uniformly in the translate.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked; the erdosproblems.com community status is still pending, so this is a candidate rather than an accepted resolution.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/450","sourceName":"erdosproblems.com/450","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1186","name":"Erdős Problem #1186","shortName":"Erdős #1186","problemNumber":1186,"field":"Arithmetic Ramsey Theory","fieldGroup":"Combinatorics","statement":"What is the minimum asymptotic density $\\delta_k$ of monochromatic $k$-term arithmetic progressions in every two-colouring of $\\{1, \\dots, n\\}$? The exact certificate gives $\\delta_3 = 117/2192$, matching the known 548-bead colouring.","posedBy":"Paul Erdos (general case); Pablo Parrilo, Aaron Robertson and Dan Saracino (specific case)","yearPosed":2008,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open.","claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 Star Fleet (Claude Fable 5 referee)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":null,"verification":"lean-checked","verificationNote":"Dual exact checkers plus a partial Lean formalization; community status pending. Not yet mentioned on Thomas Bloom's database.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1186","sourceName":"erdosproblems.com/1186","collection":null,"collectionVersion":null,"links":[{"label":"Parrilo et al 2008 - Problem statement","url":"https://www.sciencedirect.com/science/article/pii/S009731650700043X/pdf?md5=71549e9db4846e8ec3abc201f620bdf9&pid=1-s2.0-S009731650700043X-main.pdf","kind":"problem-record"},{"label":"Star Fleet's repo of the proof","url":"https://www.starfleetmath.com/downloads/verify/erdos-1186/erdos-1186-solution.zip","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cohn-elkies-bound-dimension-36","name":"Tightness of the Cohn-Elkies Bound in Dimension 36","shortName":"Sphere packing in dim 36","problemNumber":null,"field":"Sphere packing","fieldGroup":"Geometry & topology","statement":"Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension $36$ as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight-$18$ modular forms for $\\Gamma_0(24)$, shows the two-point linear programming bound in dimension $36$ exceeds the density of the Kschischang-Pasupathy packing by a factor of at least $32.91$.","posedBy":"Henry Cohn, Noam Elkies","yearPosed":2003,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown","claimIssueNote":null,"solveDate":"2026-07-13","model":"Claude Fable 5, Claude Opus 4.8, Codex (GPT-5.6)","modelMaker":"Anthropic / OpenAI","humanCollaborators":["Rifat Jumagulov"],"aiRole":"The disclosure reports substantial assistance: the Claude models were used for the construction of the certificate itself, for the verification tooling and for drafting, and Codex was used as an independent cross-check of the certificate computations.","verification":"unreviewed","verificationNote":"The result is an explicit dual-feasible certificate, so it is checkable in principle by evaluating the constructed function; the paper reports an independent cross-check of the computations by a second model. We have not reproduced it. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The Cohn-Elkies linear program is the method behind the dimension 8 and 24 solutions, so where it provably cannot work is a live question in the sphere packing programme.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.11319","sourceName":"arXiv:2607.11319 - A dual linear programming bound for sphere packing in dimension 36","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-959","name":"Erdős Problem #959","shortName":"Erdős #959","problemNumber":959,"field":"Distinct Distances","fieldGroup":"Geometry & topology","statement":"How large can the difference between the largest and second-largest distance multiplicities be among $n$ planar points?","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"superlinear lower bound M(n) ≥ n^{1 + 1/(50000 log log n)}, improving Ω(n log n); the exact order remains open","claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 starships (Claude Fable 5 reviewer)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"Produced by the GPT-5.6 starships pipeline with Claude Fable 5 as reviewer.","verification":"lean-verified","verificationNote":"Lean-checked construction; community status pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/959","sourceName":"erdosproblems.com/959","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"slug":"tsp-optimal-chain-density","name":"Optimal Chain Density and Space-Time Tradeoffs for the TSP","shortName":"Chain density, TSP","problemNumber":null,"field":"Extremal Set Theory, Algorithms","fieldGroup":"Algorithms & optimization","statement":"The paper nearly settles the tradeoff between the size of a set system over $[n]$ and its number of full chains, an extremal question raised by Johnson, Leader and Russell as a counterpart to Sperner-type results, and linked by recent work to the space and time complexity of Bellman–Held–Karp dynamic programming for permutation problems.","posedBy":"Johnson, Leader and Russell","yearPosed":2015,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"The authors describe the tradeoff as nearly settled rather than settled.","claimIssueNote":null,"solveDate":"2026-07-13","model":"ChatGPT 5","modelMaker":"OpenAI","humanCollaborators":["Alexandr Andoni","Justin Dallant","László Kozma","Hantao Yu"],"aiRole":"The disclosure is one sentence: the results of the paper were obtained with the help of ChatGPT 5. It says the results rather than the exposition, but does not say how much, so the lowest tier applies.","verification":"unreviewed","verificationNote":"No independent review, and the single-sentence disclosure does not separate the model's contribution from the authors'. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A 2015 extremal question with a direct bearing on the complexity of classical dynamic programming for permutation problems.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.11311","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-584","name":"Erdős Problem #584","shortName":"Erdős #584","problemNumber":584,"field":"Extremal Graph Theory","fieldGroup":"Combinatorics","statement":"Must every graph with $n$ vertices and $\\delta n^2$ edges contain large subgraphs in which every two edges lie on specified short cycles? A dense high-girth construction refutes the statement when $\\delta$ may shrink with $n$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"the literal wording is refuted; the intended variant remains open","claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 Star Fleet (Claude Fable 5 referee)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Sorry-free Lean construction for the literal statement.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/584","sourceName":"erdosproblems.com/584","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ross-nondeficient-conjectures","name":"Ross's Two Conjectures on Nondeficient Numbers","shortName":"Ross's conjectures","problemNumber":null,"field":"Number theory","fieldGroup":"Number theory","statement":"Ross introduced $\\mathcal{S}$-perfect numbers, integers expressible as $1 + \\sum \\lambda_j d_j$ over their proper divisors with coefficients in $\\mathcal{S}$, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd nondeficient numbers to $\\mathcal{S}$-perfection. Both are false.","posedBy":"Ross","yearPosed":2024,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["John M. Campbell"],"aiRole":"The paper says in its introduction that the disproofs are built on extensive interactions with GPT-5.5 Pro, and the acknowledgements place that interaction in the exploratory and proof-development stages. All AI suggestions were substantially revised, corrected and independently verified by the author.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"Two conjectures from a 2024 paper introducing the notion; recent and specialized, in the classical perfect-numbers tradition.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.11043","sourceName":"arXiv:2607.11043 - Disproofs of two conjectures concerning nondeficient numbers","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-130","name":"Erdős Problem #130","shortName":"Erdős #130","problemNumber":130,"field":"Discrete Geometry, Chromatic Number","fieldGroup":"Geometry & topology","statement":"For an infinite planar set in strong general position, how large can the chromatic and clique numbers of its positive-integer-distance graph be - in particular, can the chromatic number be infinite? Yes: there is such a set, no three collinear and no four concyclic, with infinite chromatic number.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"the infinite-chromatic subquestion is proved; the rest of the problem remains open","claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 Star Fleet (Claude Fable 5 referee)","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked end to end; community review pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/130","sourceName":"erdosproblems.com/130","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"coxeter-code-minimum-distance","name":"The Coxeter Code Minimum Distance Conjecture","shortName":"Coxeter codes","problemNumber":null,"field":"Coding theory","fieldGroup":"Combinatorics","statement":"Coble and Barg introduced binary Coxeter codes, the span of indicators of standard cosets of fixed rank in a finite Coxeter system, generalizing Reed-Muller codes, and proposed a conjectural value for the minimum distance of a general Coxeter code. The conjecture is true, and it yields a decoding consequence.","posedBy":"Nolan Coble, Alexander Barg","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-12","model":"ChatGPT-5.4, Claude Sonnet 4.6","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Alexander Barg","Qendrim R. Gashi","Tianyuan Xu"],"aiRole":"The authors say the models were used as exploratory tools in connection with the problem, and bound that use explicitly: brainstorming, discussion of possible approaches, and preliminary checking of ideas.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"A conjectural minimum distance proposed in the paper that introduced the codes; recent and specialized, though Reed-Muller codes give it wider context.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.10774","sourceName":"arXiv:2607.10774 - Minimum distance and decoding of Coxeter codes","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bounded-circumference-cycle-counting","name":"Counting Fixed Cycles in Graphs with Bounded Circumference","shortName":"Bounded circumference cycles","problemNumber":null,"field":"Extremal graph theory","fieldGroup":"Combinatorics","statement":"Zhu, Gyori, He, Lv, Salia and Xiao conjectured the maximum number of copies of a fixed cycle in an $n$-vertex graph of bounded circumference, attained by the join of a clique with an independent set. For every fixed $s \\ge 3$ and $L \\ge 2s+2$ and all large $n$, $\\mathrm{ex}(n, C_{2s+1}, \\mathcal{C}_{\\ge L+1}) = N(C_{2s+1}, H(n,L))$. Together with the companion even-cycle result this settles the conjecture.","posedBy":"Zhu, Gyori, He, Lv, Salia, Xiao","yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"the odd-cycle half; the even-cycle half is a companion paper by the same authors","claimIssueNote":null,"solveDate":"2026-07-12","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Xiamiao Zhao","Yuanpei Wang"],"aiRole":"The declaration credits the model with solving one case of Theorem 1.2, in particular the calculations in that proof, and with rewriting the Section 2.4 argument in the language of directed graphs; the rest is readability and exposition. The authors reviewed and verified the proofs and take sole responsibility.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":9,"significanceNote":"A 2023 Bulletin of the LMS conjecture in generalized Turan theory, real and cited but recent and specialized.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.10779","sourceName":"arXiv:2607.10779 - Counting Odd Cycles in Graphs with Bounded Circumference","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"elizalde-luo-pattern-avoidance","name":"Elizalde-Luo Pattern-Avoidance Conjecture","shortName":"Elizalde-Luo","problemNumber":null,"field":"Enumerative combinatorics","fieldGroup":"Combinatorics","statement":"Is the number of nonnesting permutations of $\\{1,1,\\dots,n,n\\}$ avoiding both $1132$ and $3312$ equal to $3^n - 3 \\cdot 2^{n-1} + 1$ for every $n \\ge 1$?","posedBy":"Sergi Elizalde & Luo","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-12","model":"Demonstrandum multi-agent pipeline","modelMaker":null,"humanCollaborators":[],"aiRole":"Found by the Demonstrandum multi-agent pipeline; every refutation ships a finite certificate, a mutation-tested checker, and an independent clean-room recomputation.","verification":"lean-verified","verificationNote":"Proved and Lean-checked end to end; not externally refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A recent conjecture from a single combinatorics paper.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"grothendieck-finite-flat-group-schemes","name":"Grothendieck's Finite Flat Group Scheme Order Question","shortName":"Grothendieck group schemes","problemNumber":null,"field":"Algebraic Geometry","fieldGroup":"Algebra","statement":"Grothendieck asked whether every finite locally free group scheme of order $n$ is killed by $n$ (its $n$-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne settled the commutative case, it is necessarily non-commutative over a non-reduced base.","posedBy":"Alexander Grothendieck","yearPosed":1966,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-11","model":"GPT-5.6 Sol, Claude Fable 5","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Akhil Mathew","Kevin Buzzard"],"aiRole":"OpenAI's Sol found an explicit counterexample, a rank-4 Hopf algebra over $\\mathbb{Z}[a,b]/(a^3, b^3, a^2 b + 2)$ whose order-4 group scheme is not killed by 4, and Claude Fable 5 autoformalized the full argument in Lean within hours. Akhil Mathew directed the work and submitted it to Mathlib; Kevin Buzzard independently compiled and checked the 1076-line proof.","verification":"lean-verified","verificationNote":"Machine-checked in Lean and submitted to Mathlib (PR #41748, opened 2026-07-14, disclosed as built with OpenAI's Codex and Anthropic's Claude under the author's direction). Kevin Buzzard independently compiled the 1076-line proof and confirmed it uses only standard mathlib definitions. Under active expert review (Wieser, Brasca) and not yet merged; no journal publication yet, but the counterexample is explicit and kernel-checked.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A Grothendieck question open for sixty years, known through arithmetic geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/leanprover-community/mathlib4/pull/41748","sourceName":"Mathlib PR #41748","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"kotzig-perfect-1-factorisation-asymptotic","name":"Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically","shortName":"Perfect 1-factorisation","problemNumber":null,"field":"Design theory","fieldGroup":"Combinatorics","statement":"Kotzig conjectured that for every even $n \\ge 4$ the complete graph $K_n$ decomposes into $n-1$ perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: $K_n$ decomposes into $n-1$ perfect matchings of which $(1-o(1))n$ have the property that any pair forms a Hamilton cycle.","posedBy":"Anton Kotzig","yearPosed":1964,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"asymptotic form only; Kotzig's conjecture itself remains far from solved","claimIssueNote":null,"solveDate":"2026-07-10","model":"ChatGPT 5.4","modelMaker":"OpenAI","humanCollaborators":["Yangyang Cheng","Amedeo Sgueglia"],"aiRole":"The authors state that the construction in Theorem 1.3, a generalisation of the one in their introduction, was generated by ChatGPT 5.4, which also supplied a correct but long proof of it. The proof they present is a cleaner and substantially different one of their own, and the proof of the main theorem, Theorem 1.2, was obtained entirely by the authors. The model's construction is nonetheless load-bearing: the main result is built by deleting a random subset of the matchings it produces.","verification":"unreviewed","verificationNote":"arXiv preprint that routes through a robust-expander result of Kuhn and Osthus; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"A famous 1960s conjecture in design theory and graph decomposition, attacked for six decades and known throughout combinatorics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.09459","sourceName":"arXiv:2607.09459 - The perfect 1-factorisation conjecture holds asymptotically","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"shellsort-tokuda-lower-bound","name":"Worst-Case Complexity of Shellsort with Tokuda's Gap Sequence","shortName":"Shellsort lower bound","problemNumber":null,"field":"Analysis of algorithms","fieldGroup":"Algorithms & optimization","statement":"Shellsort's worst-case running time is unknown for the gap sequences actually used in practice. Encoding a permutation as the polynomial $\\sigma(1)z + \\cdots + \\sigma(n)z^n$ gives a framework for lower bounds, and yields $\\Omega(N^{1.26})$ for Tokuda's 1992 sequence, extending to any strictly decreasing sequence staying within a fixed distance of a rational geometric one.","posedBy":"Donald Shell; Naoyuki Tokuda","yearPosed":1992,"ageNote":"Shellsort dates to 1959 and its worst-case complexity has been open in general ever since; Tokuda proposed the gap sequence at issue in 1992.","solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"a lower bound for Tokuda's sequence; the general Shellsort complexity question stays open","claimIssueNote":null,"solveDate":"2026-07-10","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Zhenghan Zang"],"aiRole":"The acknowledgement credits the model with providing the initial framework of the proof of Lemma 2, which the author then verified, refined and wrote up.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Shellsort's complexity is a classic unresolved question in the analysis of algorithms, and Tokuda's is one of the gap sequences that made it into practice.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.08997","sourceName":"arXiv:2607.08997 - Improved lower bounds of the time complexity of shellsort","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"double-covers-discrete-box","name":"New Bounds for Double Covers of the Discrete Box","shortName":"Double covers of ${0,1,2}^d$","problemNumber":null,"field":"Extremal Combinatorics, Covering Problems","fieldGroup":"Combinatorics","statement":"For $A=\\{0,1,2\\}^d$, write $f(d)$ for the fewest proper sub-boxes covering every point exactly twice. Leader, Miličević and Tan asked whether $f(d)\\ge 2^d$ for all $d$, as Question 4.1 of the PatternBoost paper. The paper gives new bounds on $f(d)$.","posedBy":"Leader, Miličević and Tan","yearPosed":2017,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Improved bounds rather than a settled question: the asked-for inequality is not established in general.","claimIssueNote":null,"solveDate":"2026-07-10","model":"Claude","modelMaker":"Anthropic","humanCollaborators":["Patrick White"],"aiRole":"The author credits large language models used as reasoning engines with producing several of the key ideas, naming the modular refinement of Section 3, the crossing lemma of Section 5 and the constructions of Section 7.","verification":"lean-checked","verificationNote":"Lean formalization available. No independent check. The disclosure is unusually specific about which sections the model produced. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"Progress on a question posed in the PatternBoost paper, itself a study of machine-assisted mathematics.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.09014","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Leader et al 2017, Question 4: Problem statement","url":"https://www.sciencedirect.com/science/article/pii/S0097316517301024","kind":"problem-record"},{"label":"Github repo of the proof","url":"https://github.com/pw/box-double-covers","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"nivat-conjecture-non-convex-window-counterexample","name":"A Counterexample to Nivat's Conjecture for a Non-Convex Window","shortName":"Nivat, non-convex window","problemNumber":null,"field":"Symbolic Dynamics, Tilings","fieldGroup":"Combinatorics","statement":"The paper constructs an exact cluster $F\\subseteq\\mathbb{Z}^2$ of cardinality 8 with full affine span and an $F$-tiling whose orbit closure contains no 1-periodic $F$-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows. This answers negatively a question of Kari and Moutot from 2023.","posedBy":"Maurice Nivat; the specific question by Jarkko Kari and Etienne Moutot","yearPosed":2023,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-10","model":"GPT-5 + Claude Opus 4","modelMaker":null,"humanCollaborators":["Abhishek Khetan"],"aiRole":"The paper's note on the use of AI credits the construction of Section 2 - the cluster and tiling that constitute the counterexample - to GPT-5, alongside use of Claude Opus 4.","verification":"unreviewed","verificationNote":"No independent review. The object is an explicit finite cluster with an explicit tiling, so the construction can be checked directly. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Nivat's conjecture is a well-known problem on low-complexity two-dimensional configurations; this closes the non-convex window case in the strong form Kari and Moutot asked about.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.09830","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"cycle-double-cover-conjecture","name":"Cycle Double Cover Conjecture","shortName":"Cycle Double Cover","problemNumber":null,"field":"Graph Theory","fieldGroup":"Combinatorics","statement":"Conjectures that every bridgeless graph has a collection of cycles covering each edge exactly twice.","posedBy":"George Szekeres, Paul Seymour","yearPosed":1973,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-10","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Running in Ultra mode with 64 parallel subagents, GPT-5.6 Sol produced a claimed proof of the full Cycle Double Cover Conjecture in under an hour. OpenAI released both the proof manuscript and the task prompt; a public Lean formalization was added afterwards.","verification":"lean-checked","verificationNote":"Announced by OpenAI researcher Ethan Knight on 10 July 2026, timed to the GPT-5.6 Sol Ultra release. Not peer-reviewed; the Cycle Double Cover Conjecture has a history of claimed proofs later found to have gaps, so mathematicians are treating it cautiously pending independent review. Lean released here: https://github.com/openai/cdc-lean","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":2,"renownNote":null,"significance":55,"significanceNote":"Szekeres-Seymour; one of the most famous open problems in graph theory, in every textbook.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.scientificamerican.com/article/chatgpt-just-proved-another-50-year-old-math-conjecture/","sourceName":"Scientific American","collection":null,"collectionVersion":null,"links":[{"label":"Independent Codex-assisted proof claim the same week (Kintali), unreviewed, arXiv:2607.14140","url":"https://arxiv.org/abs/2607.14140","kind":"independent"},{"label":"Lean-checked proof","url":"https://github.com/openai/cdc-lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":5,"downvotes":0,"commentCount":0},{"slug":"klopp-zadik-node-private-community","name":"Klopp-Zadik Question on Polynomial-Time Node-Private Recovery","shortName":"Node-private recovery","problemNumber":null,"field":"Differential privacy","fieldGroup":"Theoretical computer science","statement":"Klopp and Zadik gave an exponential-time node-private algorithm for exact community recovery in stochastic block models and asked whether a polynomial-time algorithm could match it. One can: a Lipschitz surrogate for the penalized likelihood plus an accept-reject sampler gives a high-probability polynomial-time node-private algorithm that nearly matches the exponential-time guarantee.","posedBy":"Olga Klopp, Ilias Zadik","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-10","model":"ChatGPT 5.5 Plus","modelMaker":"OpenAI","humanCollaborators":["Laurentiu Marchis","Olga Klopp","Po-Ling Loh","Ilias Zadik"],"aiRole":"The AI declaration says the paper was written with the model's help and that it played a critical role in brainstorming the initial idea for a polynomial-time algorithm and in providing proof outlines for the main results. Two of the authors are the pair who posed the question.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"An open question from a 2026 paper on private community detection, at the junction of differential privacy and the well-studied stochastic block model.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.09441","sourceName":"arXiv:2607.09441 - Near-optimal node-private community estimation in polynomial-time","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"minimum-edge-outerplanarity","name":"Minimum Edge-Outerplanar Embedding","shortName":"Edge-outerplanarity","problemNumber":null,"field":"Graph algorithms","fieldGroup":"Theoretical computer science","statement":"Can the minimum edge-outerplanarity of a finite loopless planar graph, minimized over all planar embeddings, be computed in polynomial time? Asked by Bentz in 2009.","posedBy":"Cédric Bentz","yearPosed":2009,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-09","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Hantao Yu"],"aiRole":"The reduction to computing an embedding of minimum face-depth was initially produced by GPT-5.5 Pro, then verified and polished manually by the author.","verification":"unreviewed","verificationNote":"Author-checked and polished arXiv preprint. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"A specialist complexity question with a small audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.08110","sourceName":"arXiv:2607.08110 - Minimum edge-outerplanar embeddings are polynomial-time computable","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"purdy-inequality-hyperplane-arrangements","name":"Purdy's Inequality for Hyperplane Arrangements","shortName":"Purdy's inequality","problemNumber":null,"field":"Discrete geometry","fieldGroup":"Geometry & topology","statement":"For an arrangement of $n$ hyperplanes in $\\mathbb{P}^3_{\\mathbb{C}}$ with $\\ell$ intersection lines and $p$ intersection points where at least three hyperplanes meet, the refined form of Purdy's inequality expects $p - \\ell + n + 2 \\ge 0$. An explicit arrangement built from roots of unity violates it.","posedBy":"George Purdy","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"the refined form for essential arrangements in projective three-space","claimIssueNote":null,"solveDate":"2026-07-09","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Mateusz Michalek","Piotr Pokora"],"aiRole":"The acknowledgement credits the model with generating the computer programs used to carry out the symbolic computations over configurations of points and planes, which is how the violating arrangement was checked.","verification":"site-confirmed","verificationNote":"The counterexample is an explicit root-of-unity arrangement whose point, line and plane counts the paper works out in closed form, so the violation is a finite check. arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named inequality in the incidence-geometry tradition of Purdy, with a documented expected refinement for hyperplane arrangements.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.08463","sourceName":"arXiv:2607.08463 - A counterexample to Purdy's inequality for hyperplane arrangements","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"strichartz-cantor-fourier-frame","name":"Strichartz's Question on Fourier Frames for the Cantor Measure","shortName":"Cantor Fourier frames","problemNumber":null,"field":"Harmonic analysis","fieldGroup":"Analysis","statement":"Does the middle-third Cantor measure admit a Fourier frame, that is, a countable set of exponentials giving two-sided frame bounds on its $L^2$ space? No. The Cantor measure with base $b$ admits no Fourier frame for any odd integer $b > 1$, which answers Strichartz's question for the middle-third case.","posedBy":"Robert S. Strichartz","yearPosed":2000,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-09","model":"GPT-5.5, GPT-5.5 in Codex","modelMaker":"OpenAI","humanCollaborators":["Jaume de Dios Pont","Lukas Liehr","Mitchell A. Taylor"],"aiRole":"The paper devotes a section to it. The authors were trying to build a frame, not to rule one out. With GPT-5.5 they analyzed why their translated ternary digit set candidates fail to give scale-uniform frame bounds, and it is that failed construction which suggested the obstruction the final proof turns on. The model also simplified the key normalized polynomial into a more concise equivalent form. GPT-5.5 in Codex then wrote the Lean formalization, and the authors state that the proof files were generated by language models while they curated and checked the statement.","verification":"lean-verified","verificationNote":"Lean 4 formalization of the main theorem at the linked repository. Showcase.lean carries a self-contained statement the authors curated and reviewed for human readability; the proof files themselves were LLM-generated, and the trust rests on Mathlib's definitions. We have not recompiled it. arXiv preprint, not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A well-known question of Strichartz sitting in the Fuglede and frame-spectrality circle, with two decades of partial nonexistence results around it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.08656","sourceName":"arXiv:2607.08656 - Cantor measures with odd base do not admit Fourier frames","collection":null,"collectionVersion":null,"links":[{"label":"Lean 4 formalization","url":"https://github.com/jaumededios/Cantor_Measure_Frames","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-866","name":"Erdős Problem #866","shortName":"Erdős #866","problemNumber":866,"field":"Additive Combinatorics","fieldGroup":"Number theory","statement":"Estimate the least excess $g_k(N)$ forcing $k$ integers whose pairwise sums all lie in a dense subset of $\\{1, \\dots, 2N\\}$; in particular, determine the positive variant $h_4(n)$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"h₄(n) = 4 for every n ≥ 331,777, with improved global bounds; the broader problem remains open","claimIssueNote":null,"solveDate":"2026-07-08","model":"Demonstrandum multi-agent pipeline","modelMaker":null,"humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Headline theorems Lean-checked; 298 exact finite cells independently certified.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/866","sourceName":"erdosproblems.com/866","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"quantum-monge-kantorovich-distance-disproof","name":"The Quantum Wasserstein Semidistance Is Not a Distance","shortName":"Quantum Monge–Kantorovich","problemNumber":null,"field":"Quantum Information, Optimal Transport","fieldGroup":"Quantum information & computing","statement":"Friedland and coauthors proposed a quantum analogue of the $p$-Wasserstein distance and conjectured that, though only a semidistance in general, it is a true distance for a particular quantum cost matrix and for cost matrices near it. The paper disproves both conjectures with an explicit family of triples of states violating the triangle inequality.","posedBy":"Friedland, Eckstein, Cole and Życzkowski","yearPosed":2022,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-08","model":"ChatGPT 5","modelMaker":"OpenAI","humanCollaborators":["Tomasz Miller"],"aiRole":"The acknowledgements state the counterexamples were found with the assistance of ChatGPT 5.","verification":"unreviewed","verificationNote":"No independent review, but the refutation is an explicit family of state triples and the triangle inequality can be checked directly. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Refutes a conjecture from a Physical Review Letters paper about the metric structure of quantum optimal transport.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.07764","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"involution-equivariant-topological-recursion","name":"Remodeling for the Affine Binary Dihedral Calabi–Yau Threefold","shortName":"Equivariant remodeling","problemNumber":null,"field":"Mirror Symmetry, Topological Recursion","fieldGroup":"Mathematical physics","statement":"The paper proves a closed-string remodeling statement for the affine binary dihedral Calabi–Yau orbifold threefold, a target outside the toric setting of the Bouchard–Klemm–Mariño–Pasquetti remodeling conjecture, replacing the toric mirror curve by a type-$D_l$ logarithmic Toda curve and the topological recursion by its $\\mathbb{Z}_2$-equivariant form.","posedBy":"Bouchard, Klemm, Mariño and Pasquetti (remodeling conjecture)","yearPosed":2009,"ageNote":null,"solveType":"proved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"A variant rather than the BKMP conjecture itself: the target lies outside the toric setting the conjecture addresses.","claimIssueNote":null,"solveDate":"2026-07-08","model":"Rethlas-based system","modelMaker":null,"humanCollaborators":["Bohan Fang","Zhuoming Lan","Jingxiang Ma"],"aiRole":"The paper states plainly that the mathematics of the paper was generated by a Rethlas-based system under a human-provided strategy and initial input, using an orchestrator the authors built to spawn agents.","verification":"unreviewed","verificationNote":"No independent check, and the authors describe the mathematical content as machine-generated under their strategy. Preprint, not refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Extends the remodeling correspondence beyond the toric case the original conjecture covers.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.07355","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Bouchard et al 2009: Conjecture statement","url":"https://dx.doi.org/10.1007/s00220-008-0620-4","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kurkov-fubini-sum","name":"Kurkov's Fubini-Number Sum Conjecture","shortName":"Kurkov Fubini sum","problemNumber":null,"field":"Enumerative combinatorics","fieldGroup":"Combinatorics","statement":"For the Fubini numbers $a(n)$, is $a(n) = \\sum_{k=0}^{2^{n-1}-1} A284005(k)$ for every $n > 0$, as conjectured on the OEIS in 2018?","posedBy":null,"yearPosed":2018,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-08","model":"Demonstrandum multi-agent pipeline","modelMaker":null,"humanCollaborators":[],"aiRole":"Found by the Demonstrandum multi-agent pipeline; every refutation ships a finite certificate, a mutation-tested checker, and an independent clean-room recomputation.","verification":"unreviewed","verificationNote":"Refined ordered-set-partition proof, audited, with an exhaustive checker; not externally refereed.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A concrete sum identity with a one-paper audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stable-phase-retrieval-random-spans","name":"Stable Phase Retrieval for Spans of Independent Random Variables","shortName":"Stable phase retrieval","problemNumber":null,"field":"Functional Analysis, Phase Retrieval","fieldGroup":"Analysis","statement":"After $L^2$ normalization, stable phase retrieval holds over the $L^2$-spans of independent real-valued centered random variables exactly when all but possibly one coordinate satisfies a uniform two-sided $L^1$ bound. This confirms the characterization conjectured by Calderbank, Daubechies, Freeman and Freeman.","posedBy":"Calderbank, Daubechies, Freeman and Freeman","yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-07","model":"GPT-5 + Claude Opus 4","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Pedro Abdalla","Jaume de Dios Pont","João P. G. Ramos","Mitchell A. Taylor"],"aiRole":"Under a heading on the usage of large language models, the authors say the models played a significant role in the development of the work, describing arguments proposed by the authors and then found by GPT-5, with Claude Opus 4 also used.","verification":"lean-checked","verificationNote":"No independent review. The disclosure describes a back-and-forth in which the model supplied arguments the authors had proposed in outline. Preprint, not refereed. Lean formalization available, not audited by a third party yet.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Confirms a characterization conjectured by the authors of the foundational work on stable phase retrieval, and does so completely rather than in special cases.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.06693","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Calderbank et al 2023: Conjecture statement","url":"https://arxiv.org/abs/2203.03135","kind":"problem-record"},{"label":"GitHub repo of the proof","url":"https://github.com/jaumededios/RandomVarSPR","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"faber-harris-isolation-lemma-conjecture","name":"Faber-Harris Conjecture on the Isolation Lemma","shortName":"Isolation Lemma extremal","problemNumber":null,"field":"Extremal combinatorics","fieldGroup":"Combinatorics","statement":"For an inclusion-free hypergraph on $n$ vertices, a weight assignment $w:[n]\\to[d]$ is isolating when a unique edge attains minimum weight. Faber and Harris conjectured that the number of isolating assignments is at least $n\\sum_{j=0}^{d-1} j^{n-1}$, attained by the hypergraph of $n$ singleton edges. The bound holds, and extends to a more general class of objective functions.","posedBy":"Vance Faber, David G. Harris","yearPosed":2018,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-07","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Vance Faber","David G. Harris"],"aiRole":"The acknowledgement credits the model with one specific and load-bearing step: the proof benefited from its assistance in discovering the fractional charging argument. The authors are the pair who stated the conjecture in 2018.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A published 2018 conjecture pinning down the extremal case of the Isolation Lemma, a workhorse of randomized algorithms and derandomization.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.06171","sourceName":"arXiv:2607.06171 - The singleton hypergraph is extremal for the Isolation Lemma","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"amdeberhan-medina-moll-arctan-conjecture","name":"Amdeberhan-Medina-Moll Arctangent Sum Conjecture","shortName":"Arctangent sums","problemNumber":null,"field":"Number theory","fieldGroup":"Number theory","statement":"Let $x_n = \\tan\\left(\\sum_{k=1}^{n} \\arctan k\\right)$. Amdeberhan, Medina and Moll conjectured that $x_n \\notin \\mathbb{Z}$ for every $n \\ge 5$. Any integer value $x_n = m$ must satisfy $|m| \\ge e^{(1/2+o(1)) n \\log n}$, which forces $\\#\\{1 \\le n \\le N : x_n \\in \\mathbb{Z}\\} = O(\\log N)$. The conjecture therefore holds for a density-one set of $n$, improving on the previously known density of $120/817 \\approx 0.147$.","posedBy":"Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll","yearPosed":2008,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"density-one set of n; the conjecture itself remains open","claimIssueNote":null,"solveDate":"2026-07-07","model":"AxiomProver","modelMaker":"Axiom Math","humanCollaborators":["Ken Ono"],"aiRole":"The system was given only a natural-language statement of the definitions and the three target results, plus an instruction to formalize and prove them with no sorry. From that input AxiomProver autonomously produced both the Lean formalization of the problem and a complete Lean proof. The human author then wrote the paper's exposition using the formal development as his reference, which reverses the usual order: the Lean came first and the prose was derived from it.","verification":"unreviewed","verificationNote":"A public Lean 4.28.0 development accompanies the paper, containing a formalization of the problem and a proof the author states is sorry-free and adds no axioms. We attempted to compile it and did not complete the build, so the axiom claim here rests on the author's statement rather than on our own check.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture from a 2008 Journal of Number Theory paper with a documented line of partial results, familiar within the arctangent-sums and Gaussian-integer literature but not beyond it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.05739","sourceName":"arXiv:2607.05739 - Integer values of tan(arctan 1 + arctan 2 + ... + arctan n) are rare","collection":null,"collectionVersion":null,"links":[{"label":"AxiomProver Lean development","url":"https://github.com/AxiomMath/TanArctan","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"odd-dimensional-rank-bound-counterexample","name":"Counterexample to the Odd-Dimensional Rank Bound for Abelian p-Group Actions","shortName":"Odd-dim rank bound","problemNumber":null,"field":"Algebraic Geometry, Calabi–Yau Varieties","fieldGroup":"Geometry & topology","statement":"Moraga conjectured, and Kollár and Zhuang recorded, an odd-dimensional extension of the rank bound for faithful abelian $p$-group actions on smooth Calabi–Yau varieties. The paper disproves it.","posedBy":"Joaquín Moraga; recorded by Kollár and Zhuang","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-06","model":"ChatGPT 5.5 Pro + Danus","modelMaker":"OpenAI; Rethlas","humanCollaborators":["Jihao Liu","Yanze Wang"],"aiRole":"The abstract states the main result was obtained by ChatGPT 5.5 Pro and the Danus system, an agent built on Rethlas; the sketch of the proof was likewise produced by the model.","verification":"unreviewed","verificationNote":"No independent check. The Danus papers carry a standing caution from their authors that AI limitations mean related literature may have been missed. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Refutes a conjectured extension recorded in the literature by Kollár and Zhuang.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.04891","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Kollár & Zhang 2026, Remark 22: Conjecture statement","url":"https://arxiv.org/abs/2402.15362","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"online-discrepancy-linear-time","name":"Optimal Online Discrepancy in Linear Time","shortName":"Linear-time discrepancy","problemNumber":null,"field":"Discrepancy theory","fieldGroup":"Theoretical computer science","statement":"Given online vectors $v_t \\in \\mathbb{R}^d$ with $\\|v_t\\|_2 \\le 1$, can signs $\\varepsilon_t \\in \\{-1, 1\\}$ be chosen in $O(dT)$ total time so that every prefix has $\\ell_\\infty$ discrepancy $O(\\sqrt{\\log T})$ with high probability? The previous optimal algorithm ran in time exponential in $T$ and $d$.","posedBy":null,"yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-06","model":"GPT-5.5 Pro Extended","modelMaker":"OpenAI","humanCollaborators":["Ishaq Aden-Ali"],"aiRole":"The algorithm and main proof were discovered in a GPT-5.5 Pro Extended conversation prompted by the author; every prefix sum is written as a sum of three coupled Gaussian vectors.","verification":"unreviewed","verificationNote":"Author-checked arXiv preprint. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":9,"significanceNote":"An algorithmic follow-up in online discrepancy.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.04388","sourceName":"arXiv:2607.04388 - Optimal online discrepancy minimization in linear time","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"k-polystable-toric-fano-small-alpha","name":"K-Polystable Toric Fano Varieties With Small Alpha Invariants","shortName":"Small alpha invariants","problemNumber":null,"field":"Algebraic Geometry, K-Stability","fieldGroup":"Geometry & topology","statement":"For every $n\\ge2$ the paper exhibits an $n$-dimensional K-polystable toric $\\mathbb{Q}$-Fano variety whose alpha invariant is exactly $\\tfrac{2}{2n+1}$, answering a question of Liu and Zhuang on whether a K-semistable example exists with alpha invariant between $\\tfrac{1}{n+1}$ and $\\tfrac1n$.","posedBy":"Yuchen Liu and Ziquan Zhuang","yearPosed":2022,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-04","model":"ChatGPT 5.5 Pro + Danus","modelMaker":"OpenAI; Frenzy Math","humanCollaborators":["Jihao Liu","Ziwen Zhu"],"aiRole":"The abstract states the main result was obtained by ChatGPT 5.5 Pro and the Danus system, an agent built on Rethlas; the proof sketch was likewise produced by the model.","verification":"unreviewed","verificationNote":"No independent check, and the same standing caution about missed literature applies. Preprint, not refereed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Answers a stated question in K-stability by explicit construction, for every dimension at once.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.04005","sourceName":"arXiv","collection":null,"collectionVersion":null,"links":[{"label":"Liu & Zhuang 2022, Question 1.5 (1): Conjecture statement","url":"https://dx.doi.org/10.1017/nmj.2020.28","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"signed-bar-uniqueness-harrison-reiman","name":"The Signed BAR Uniqueness Problem","shortName":"Signed BAR conjecture","problemNumber":null,"field":"Stochastic processes","fieldGroup":"Probability & statistics","statement":"For a multidimensional reflected diffusion, does the basic adjoint relationship uniquely characterize the stationary distribution? The question had stood unresolved for more than thirty-five years since the BAR approach was introduced. For stable Harrison-Reiman data with a nonsingular $M$-matrix reflection matrix, the finite-signed uniqueness problem is settled, via pathwise differentiability of the reflected process; the nonsigned version is also shown unique within the Harrison-Reiman class.","posedBy":"J. Michael Harrison, Martin I. Reiman","yearPosed":1990,"ageNote":"The BAR approach dates to around 1990; this uniqueness question is described in the paper as unresolved for over thirty-five years.","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-03","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Yiping Lu","Youheng Zhu"],"aiRole":"The paper names the model contribution in its title and states that the proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors, with the chat logs published. It also records a negative result worth having: on a harder related task both ChatGPT 5.5 Pro extended and Claude Opus 4.8 max failed.","verification":"unreviewed","verificationNote":"arXiv preprint with the originating chat logs linked, so the provenance claim is checkable even though the mathematics is not yet refereed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A foundational uniqueness question underpinning the BAR method for reflected diffusions, open since the approach was introduced and central to the heavy-traffic queueing literature that uses it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.03639","sourceName":"arXiv:2607.03639 - An AI-Assisted Solution to the Signed BAR Conjecture: Uniqueness in the Harrison-Reiman Class","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"tree-product-conjecture","name":"The Tree Product Conjecture","shortName":"Tree product conjecture","problemNumber":null,"field":"Structural graph theory","fieldGroup":"Combinatorics","statement":"Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood conjectured that every graph of degree-$d$ polynomial growth embeds into the strong product of $d$ trees of linear growth and a bounded clique. False for $d = 4$: a counterexample built from the discrete Heisenberg group.","posedBy":"Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar, Wood","yearPosed":2022,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"disproved at d = 4; the conjecture for smaller d is untouched","claimIssueNote":null,"solveDate":"2026-07-03","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Freddie Illingworth","Sergey Norin","Raphael Steiner"],"aiRole":"The AI disclosure is narrow and specific: the model was used to help work out the details of the compactness argument. The Heisenberg group counterexample is the authors'.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A 2022 conjecture from the product structure theory programme, put forward by eight authors and cited as a target in that literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.03041","sourceName":"arXiv:2607.03041 - Disproof of the tree product conjecture via the Heisenberg group","collection":null,"collectionVersion":null,"links":[{"label":"Second counterexample construction via subdivided expanders (Aug 2026)","url":"https://arxiv.org/abs/2608.04659","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"signed-bar-conjecture","name":"Signed BAR Conjecture for Reflected Brownian Motion","shortName":"Signed BAR","problemNumber":null,"field":"Stochastic networks","fieldGroup":"Probability & statistics","statement":"Does the finite signed basic adjoint relation determine the invariant signed measure uniquely, and how far beyond the Harrison-Reiman class can uniqueness extend?","posedBy":null,"yearPosed":1990,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"uniqueness proved for stable Harrison-Reiman systems with a nonsingular M-matrix reflection; an infinite-dimensional obstruction is shown in the larger completely-S class","claimIssueNote":null,"solveDate":"2026-07-03","model":"ChatGPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The pathwise-differentiability argument was developed in an AI-assisted collaboration; both authors verified the proof.","verification":"unreviewed","verificationNote":"Two-author verification with a public arXiv proof. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A long-open technical conjecture in queueing/diffusion theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.03639","sourceName":"arXiv:2607.03639 - An AI-assisted solution to the signed BAR conjecture","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"white-s-conjecture-on-matroids","name":"White's Conjecture on Matroids","shortName":"White's conjecture","problemNumber":null,"field":"Matroid theory","fieldGroup":"Combinatorics","statement":"White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank $9$ binary matroid constitutes a counterexample.","posedBy":"Neil White","yearPosed":1980,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-02","model":"ChatGPT-5.5 Pro, Claude Opus 4.8","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Matt Larson"],"aiRole":"The author prompted ChatGPT-5.5 Pro to check White's conjecture and it claimed to prove the result for ranks $5$ and $6$. After further prompting to check for random matroids of ranks $7$, $8$, $9$, and $10$, it found a matroid isomorphic to the given counterexample.\n\nChatGPT-5.5 Pro and Claude Opus 4.8 were used for proofreading and generating the figures.","verification":"unreviewed","verificationNote":"arXiv preprint with explicit binary matroid construction; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":40,"significanceNote":"A named conjecture from 1980 on the toric ideal of a matroid, with a substantial literature of proved special cases behind it. Placed just above Rota's flat unimodality conjecture at 36: better known in the algebraic-combinatorics community, and forty-five years old when it fell.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.02208","sourceName":"Counterexamples to two conjectures about matroids","collection":null,"collectionVersion":null,"links":[{"label":"GPT-5.5 Chat Log","url":"https://chatgpt.com/share/6a27ed96-0374-83ea-8daf-8057d7d6ceb5","kind":"transcript"}],"relations":[],"submittedBy":"HiddenPanther560","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"north-east-lattice-paths-with-few-collinear-vertices","name":"North-East Lattice Paths with Few Collinear Vertices","shortName":"North-East Lattice Paths","problemNumber":null,"field":"Discrete geometry - lattice paths","fieldGroup":"Combinatorics","statement":"Let $A(k)$ be the largest possible number of moves in a north-east lattice path whose visited vertices contain no $k$ collinear points. Gerver (1979) and Gerver and Ramsey (1979) bounded $A(k)$ by\n$$\\exp\\left(\\Omega\\left(\\log(k)^2\\right)\\right) \\le A(k) \\le \\exp\\left(O\\left(k^4\\right)\\right),$$\nand determining the true growth rate has been open since. Both bounds are improved to\n$$\\exp\\left(\\Omega\\left(k^{1/3}\\right)\\right) \\le A(k) \\le \\exp\\left(O\\left(k^2\\right)\\right),$$\nwith the upper bound proved in the sharper form $\\exp\\left(\\left(\\tfrac{2}{e}+o(1)\\right)(k-1)^2\\right)$.","posedBy":"Joseph L. Gerver, L. Thomas Ramsey","yearPosed":1979,"ageNote":"Posed in a pair of 1979 Pacific J. Math. papers by Gerver and by Gerver and Ramsey, and catalogued in Brass, Moser and Pach's Research Problems in Discrete Geometry. Open 47 years, with the bounds improved by Lidbetter in 2024 and the small cases attacked by SAT solvers by Barnoff and Bright in 2026.","solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"Both bounds move, and the gap stays enormous: the lower bound rises from $\\exp(\\Omega(\\log^2 k))$ to $\\exp(\\Omega(k^{1/3}))$ and the upper falls from $\\exp(O(k^4))$ to $\\exp(O(k^2))$, so $A(k)$ is still undetermined between an exponent of $k^{1/3}$ and one of $k^2$. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness, a sharper line-counting step, or a different model entirely.","claimIssueNote":null,"solveDate":"2026-07-02","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Samuel Korsky"],"aiRole":"The acknowledgement in full: the author was assisted by GPT-5.5 Pro in preparing the paper, but \"the main construction ideas, including the dyadic-interval random variables in the lower bound and the density-increment framework in the upper bound, were due to the author\". AI tools checked computations, assisted with drafting, and improved the upper-bound constant by suggesting the use of the mediant of the relevant Farey fractions. That last contribution is traceable in the text: it lifts the density increment from $(1/8-o(1))(k-1)^{-2}$ to $(1/4-o(1))(k-1)^{-2}$, which is what produces the $2/e$ constant. So the model sharpened the constant inside the new upper bound rather than the exponent, which is the lower tier by this site's definition.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2607.02832, Korsky, 2 July 2026). The abstract matches this entry, and the acknowledgement is verbatim as the AI-role note now quotes it - including the sentence attributing the main construction ideas to the author, which the submission's quote had omitted. The model's named contribution was traced through the text to the density-increment step it actually improves. The prior bounds attribute correctly: Gerver, Pacific J. Math. 83 (1979) 349-355, and Gerver-Ramsey, same volume, 357-363. The proofs themselves - a dyadic slope-field random construction and a Farey-mediant density increment - were not checked here and need a discrete geometer. Unrefereed preprint, no independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named problem - the Gerver-Ramsey collinearity problem - from two 1979 Pacific J. Math. papers, catalogued in Brass-Moser-Pach's standard problem book and still drawing work in 2024 and 2026. Forty-seven years open with a genuine literature, but firmly inside discrete geometry: the named specialist band at 15.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.02832","sourceName":"North-East Lattice Paths with Few Collinear Vertices (arXiv)","collection":null,"collectionVersion":null,"links":[{"label":"Gerver, Long walks in the plane with few collinear points (Pacific J. Math. 83, 1979)","url":"https://doi.org/10.2140/pjm.1979.83.349","kind":"problem-record"},{"label":"Gerver and Ramsey, On certain sequences of lattice points (Pacific J. Math. 83, 1979)","url":"https://doi.org/10.2140/pjm.1979.83.357","kind":"problem-record"},{"label":"Lidbetter, Improved bound for the Gerver-Ramsey collinearity problem (Discrete Math., 2024)","url":"https://doi.org/10.1016/j.disc.2023.113718","kind":"paper"},{"label":"Barnoff and Bright, North-East lattice paths avoiding k collinear points via satisfiability (2026)","url":"https://doi.org/10.1016/j.aam.2026.103112","kind":"paper"}],"relations":[],"submittedBy":"GoldenMongoose827","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"log-concavity-of-flats-of-matroids","name":"Log-Concavity of Flats of Matroids","shortName":"Mason flat log-concavity","problemNumber":null,"field":"Matroid theory","fieldGroup":"Combinatorics","statement":"Mason conjectured the following: let $M$ be a matroid of rank $r$, and let $W_i$ denote the number of flats of $M$ of rank $i$. Is it true that for all $1 \\leq i \\leq r - 1$, we have $W_i^2 \\geq W_{i + 1}W_{i - 1}$? This is false; a counterexample is given by a graphic matroid whose graph is a generalized theta graph with $79$ edges.","posedBy":"J. H. Mason","yearPosed":1972,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.","claimIssueNote":null,"solveDate":"2026-07-02","model":"ChatGPT-5.5 Pro, Claude Opus 4.8","modelMaker":"OpenAI; Anthropic","humanCollaborators":["Matt Larson"],"aiRole":"The author prompted ChatGPT-5.5 Pro to search for counterexamples to Mason's conjecture. After finding none on at most $9$ elements, the author expanded his search to consider matroids on large ground sets realizable over $\\mathbb{F}_5$ and at failures of log-concavity at high indices. ChatGPT-5.5 Pro found a variant of the given counterexample.\n\nChatGPT-5.5 Pro and Claude Opus 4.8 were used for proofreading and generating the figures.","verification":"unreviewed","verificationNote":"arXiv preprint with explicit graphic matroid construction; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The flats version of Mason's 1972 conjecture, and the less cited of the two statements that carry his name - the famous one concerns independent sets. Placed below Rota's flat unimodality conjecture at 36, which asks the weaker question about the same sequence and was refuted three weeks later.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.02208","sourceName":"Counterexamples to two conjectures about matroids","collection":null,"collectionVersion":null,"links":[{"label":"GPT-5.5 Chat Log","url":"https://chatgpt.com/share/6a30bef8-af94-83ea-babc-574f4598d841","kind":"transcript"}],"relations":[],"submittedBy":"HiddenPanther560","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"erdos-1038","name":"Erdős Problem #1038","shortName":"Erdős #1038","problemNumber":1038,"field":"Extremal Polynomials","fieldGroup":"Analysis","statement":"Among all nonconstant monic polynomials $f$ whose roots lie in $[-1, 1]$, determine $\\inf_f |\\{x \\in \\mathbb{R} : |f(x)| < 1\\}|$.","posedBy":null,"yearPosed":1958,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"A July 2026 manuscript by Darvas, Peng and Tao, developed with GPT-5.5 Pro, claims the exact extremal value and measure.","verification":"unreviewed","verificationNote":"Author-checked manuscript; the official record is still open.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1038","sourceName":"erdosproblems.com/1038","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-321","name":"Erdős Problem #321","shortName":"Erdős #321","problemNumber":321,"field":"Number Theory, Unit Fractions","fieldGroup":"Number theory","statement":"What is the largest $A\\subseteq\\{1,\\dots,N\\}$ such that all subset sums $\\sum_{n\\in S}1/n$ (over $S\\subseteq A$) are distinct?","posedBy":"Paul Erdős, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Young","Zhu","Luo"],"aiRole":"GPT-5.6 Sol (prompted by Young, Zhu, and Luo) proved the matching upper bound $R(N)\\asymp \\frac{N}{\\log N}\\prod_{j\\ge 3}\\log_j N$ (companion to #320).","verification":"site-confirmed","verificationNote":"Marked solved on erdosproblems.com via a proof claim; follows from the resolution of #320.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/321","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-119","name":"Erdős Problem #119","shortName":"Erdős #119","problemNumber":119,"field":"Analysis, Polynomials","fieldGroup":"Analysis","statement":"For unit-modulus complex numbers $z_i$, let $p_n(z)=\\prod_{i\\le n}(z-z_i)$ and $M_n=\\max_{|z|=1}|p_n(z)|$. Erdős's prize question: is there $c>0$ with $\\sum_{k\\le n} M_k > n^{1+c}$?","posedBy":"Paul Erdős","yearPosed":1957,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Samuel Korsky"],"aiRole":"GPT-5.6, with Samuel Korsky, resolved Erdős's prize question, proving $\\sum_{k\\le n} M_k \\gg n^{5/4}/\\sqrt{\\log n}$ (hence $M_n > n^{1/4-o(1)}$ infinitely often).","verification":"site-confirmed","verificationNote":"Marked solved on erdosproblems.com; carried an Erdős prize of USD 100. Resolved via a proof claim by GPT-5.6 and Samuel Korsky; not formally Lean-verified.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"A numbered Erdos problem that carries a $100 Erdos prize and an unusually dense reference trail on erdosproblems.com (10 sources), setting it above the typical entry in the catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/119","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-123","name":"Erdős Problem #123","shortName":"Erdős #123","problemNumber":123,"field":"Number Theory","fieldGroup":"Number theory","statement":"Let $a,b,c>1$ be pairwise coprime integers. Is every large integer a sum of distinct numbers of the form $a^k b^l c^m$ ($k,l,m\\ge 0$), none dividing another?","posedBy":"Paul Erdős, Mordechai Lewin","yearPosed":1996,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Colin Snyder"],"aiRole":"GPT-5.6 (prompted by Colin Snyder) resolved the Erdős-Lewin conjecture in the affirmative.","verification":"lean-verified","verificationNote":"Marked proved (Lean) on erdosproblems.com; carried an Erdős prize of USD 250. Formally verified in Lean.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":11,"significanceNote":"A numbered Erdos problem that an unusually dense reference trail on erdosproblems.com (6 sources), setting it above the typical entry in the catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/123","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-320","name":"Erdős Problem #320","shortName":"Erdős #320","problemNumber":320,"field":"Number Theory, Unit Fractions","fieldGroup":"Number theory","statement":"Let $S(N)$ count the distinct values of $\\sum_{n\\in A} 1/n$ over $A\\subseteq\\{1,\\dots,N\\}$. Estimate $S(N)$.","posedBy":"Paul Erdős, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Young","Zhu","Luo"],"aiRole":"GPT-5.6 Sol (prompted by Young, Zhu, and Luo) proved a matching upper bound, pinning $\\log S(N)$ to order $\\frac{N}{\\log N}\\prod_{j\\ge 3}\\log_j N$.","verification":"site-confirmed","verificationNote":"Marked solved on erdosproblems.com via a proof claim; not formally Lean-verified.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/320","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-793","name":"Erdős Problem #793","shortName":"Erdős #793","problemNumber":793,"field":"Number Theory","fieldGroup":"Number theory","statement":"Let $F(n)$ be the largest $A\\subseteq\\{1,\\dots,n\\}$ with $a\\nmid bc$ for distinct $a,b,c\\in A$. Is $F(n)=\\pi(n)+(C+o(1))\\,n^{2/3}(\\log n)^{-2}$ for some constant $C$?","posedBy":"Paul Erdős","yearPosed":1969,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Przemek Chojecki"],"aiRole":"GPT-5.6 Sol (prompted by Przemek Chojecki) proved $F(n)=\\pi(n)+(\\tfrac{27}{2}+o(1))\\frac{n^{2/3}}{(\\log n)^2}$, a refined form of Erdős's 1938 argument.","verification":"lean-verified","verificationNote":"Marked proved (Lean) on erdosproblems.com via a proof claim by GPT-5.6 Sol.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/793","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bertoin-yor-moment-determinacy","name":"Bertoin-Yor Moment Determinacy Conjecture","shortName":"Bertoin-Yor determinacy","problemNumber":null,"field":"Probability","fieldGroup":"Probability & statistics","statement":"For an unkilled Levy process $\\xi$ drifting to $+\\infty$ with all positive exponential moments, let $I_\\xi = \\int_0^\\infty e^{-\\xi_t}\\,dt$ and $X_\\xi = 1/I_\\xi$. Bertoin and Yor proved $X_\\xi$ is moment-determinate when $\\xi$ has no positive jumps and conjectured that this condition is necessary. The conjecture is settled.","posedBy":"Jean Bertoin, Marc Yor","yearPosed":2002,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-30","model":"GPT-5.4 Thinking, GPT-5.5 Thinking and Pro","modelMaker":"OpenAI","humanCollaborators":["Martin Minchev"],"aiRole":"The paper has a dedicated Use of AI tools section stating the models were used during the exploratory and editorial stages of the work. Exploration is mathematical work rather than prose work, but no individual step is attributed, so the lowest tier applies.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named 2002 conjecture on exponential functionals of Levy processes, a well-worked corner of probability.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.00132","sourceName":"arXiv:2607.00132 - On a moment determinacy conjecture of Bertoin and Yor","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ziegler-cross-polytope-01","name":"Ziegler's Cross-Polytope Conjecture (simplicial 0/1-polytopes)","shortName":"Ziegler cross-polytope","problemNumber":null,"field":"Combinatorics, Discrete Geometry","fieldGroup":"Geometry & topology","statement":"Ziegler proved every simplicial $d$-dimensional 0/1-polytope has at most $2d$ vertices, and asked whether attaining $2d$ vertices forces central symmetry (i.e. a 0/1 cross-polytope). Known true for $d \\le 6$; open since ~2000.","posedBy":"Günter M. Ziegler","yearPosed":2000,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-30","model":"DeepSeek V4 Flash, GLM 5.2","modelMaker":"DeepSeek / Zhipu AI","humanCollaborators":["Volker Kaibel","Sebastian Pokutta"],"aiRole":"An agentic research framework (locally-deployed open-weights DeepSeek V4 Flash + GLM 5.2, augmented with reflection prompts) first produced a flawed proof that the conjecture holds, then attempted a Lean 4 formalization as verification. The formalization failed, and from that failure the agent extracted the combinatorial condition that yielded an explicit counterexample: 14 vertices in $\\{0,1\\}^7$ whose convex hull is simplicial but not centrally symmetric.","verification":"unreviewed","verificationNote":"arXiv preprint 2606.31640 (30 Jun 2026) by Volker Kaibel and Sebastian Pokutta. The counterexample is explicit and computer-checkable (exhaustive enumeration finds exactly five such non-centrally-symmetric polytopes in dimension 7, of two combinatorial types); a domain-expert preprint, not yet peer-reviewed. Notably found with locally-run open-weights models, not closed frontier LLMs.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"One of Ziegler's known 0/1-polytope questions.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.31640","sourceName":"arXiv:2606.31640","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"rectangles-triangles-lattice-sets","name":"Rectangles versus Isosceles Triangles in Lattice Sets","shortName":"Rectangles and triangles","problemNumber":null,"field":"Combinatorial geometry","fieldGroup":"Combinatorics","statement":"Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions in the Mizohata-Takeuchi circle of problems.","posedBy":"Jonathan Bennett and coauthors","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-29","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Jonathan Bennett","Vjekoslav Kovac","Shohei Nakamura","Itamar Oliveira"],"aiRole":"The paper says in its introduction, and again in a dedicated AI usage declaration, that the combinatorial construction of the finite lattice sets achieving the stated bounds was found by ChatGPT 5.5 Pro; the authors ran it repeatedly and developed the surrounding estimates. Gemini 3.1 Pro produced the TikZ for the figures.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A combinatorial geometry question whose answer feeds directly into Mizohata-Takeuchi type obstructions in harmonic analysis.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.30178","sourceName":"arXiv:2606.30178 - Rectangles, triangles and Schrodinger waves","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lower-bound-0-38055470-for-erdos-s-minimum-overlap-constant","name":"Lower bound $0.38055470$ for Erdős's minimum overlap constant","shortName":"Minimum overlap lower bound","problemNumber":36,"field":"Additive combinatorics","fieldGroup":"Combinatorics","statement":"Erdős's minimum overlap problem (1955): split $\\{1,\\dots,2N\\}$ into two sets $A, B$ of size $N$, and let $M(N)$ be the minimum, over all such splits, of the maximum over $k$ of the number of solutions of $a - b = k$ with $a \\in A$, $b \\in B$. Determine the constant $c = \\lim_{N\\to\\infty} M(N)/N$. Before this result the best bounds were $0.379005 < c$ (White 2022) and $c < 0.380868$ (SimpleTES 2026, literature).","posedBy":"Paul Erdős (1955); Erdős Problem #36","yearPosed":1955,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"A rigorous lower bound $c > 0.38055470$, improving White's $0.379005$ (2022). The problem stays open: the best public upper bound is $0.3808585749$ (Einstein Arena, August 2026; $0.380868$ in the literature, SimpleTES), so $c$ is now known to within about $3 \\times 10^{-4}$. The bound rests on a finite certificate checked in ball arithmetic plus a step-function transfer from the continuous to the finite problem. The later Station paper (arXiv 2608.23691) reaches $0.380552$, below this; a higher claim of $0.3805634$ posted to the tracker on 19 September is unchecked.","claimIssueNote":null,"solveDate":"2026-06-29","model":"GPT Pro (version not stated)","modelMaker":"OpenAI","humanCollaborators":["Liam Price"],"aiRole":"The author, Liam Price, states on the erdosproblems.com thread for problem 36 that GPT Pro produced the improvement (\"GPT Pro improves the lower bound to 0.38055470\", 29 June 2026), and the tracker's proof-claim record lists the claim as made \"using GPT Pro\". The repository itself carries no disclosure. The division of labour is not described beyond that sentence.","verification":"site-confirmed","verificationNote":"Re-run by this site on 4 October 2026: the repository's run_arb_verification_chunks.sh at commit 6bc610e, Python 3.14.0 and python-flint 0.9.0. All three chunks reported success, proved_all_bins true over 172 mean bins, worst margin about 1.9e-8, certified lower bound 0.3805547027625940..., matching the shipped report to about thirty digits. The check is rigorous: Arb ball arithmetic at 160 bits on decimal-string inputs, with floating point used only to choose interval splits. Not machine-checked: the proof note's step-function transfer from the continuous to the finite problem, which was not audited here. erdosproblems.com still lists White's 0.379005 as the lower bound.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Erdős problems score 10 by rule; this one is raised because the constant has a seventy-year bound history (Scherk, Swierczkowski, Moser, Haugland, White), is in Guy's Unsolved Problems (C17) and Tao's optimisation-constants list, and is under active attack from both sides. Still only a bound. Level with other well-known Erdős constants and below named conjectures at 20 and up.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Leeham06972452/erdos-36-lower-bound","sourceName":"GitHub","collection":null,"collectionVersion":null,"links":[{"label":"erdosproblems.com #36 thread (author's announcement)","url":"https://www.erdosproblems.com/forum/thread/36","kind":"discussion"},{"label":"Bounds table, teorth/optimizationproblems (constant 1b)","url":"https://github.com/teorth/optimizationproblems/blob/main/constants/1b.md","kind":"problem-record"},{"label":"erdosproblems.com problem 36","url":"https://www.erdosproblems.com/36","kind":"problem-record"},{"label":"Station paper, a later and lower AI-assisted bound (0.380552)","url":"https://arxiv.org/abs/2608.23691","kind":"paper"}],"relations":[],"submittedBy":"CobaltJackal234","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"seymour-second-neighborhood-outdegree-7","name":"Seymour's Second Neighborhood Conjecture","shortName":"Second neighborhood","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Seymour conjectured that every oriented graph has a vertex $x$ with $|N^{++}(x)| \\ge |N^{+}(x)|$. It holds for oriented graphs of minimum out-degree exactly $7$, the first improvement to the out-degree threshold since Kaneko and Locke settled degree $6$ in 2001.","posedBy":"Paul Seymour","yearPosed":1990,"ageNote":"Seymour posed it around 1990; the out-degree threshold had not moved since Kaneko and Locke reached 6 in 2001.","solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"minimum out-degree 7; the conjecture is open in general Superseded by a claimed proof of the full conjecture in the OpenAI math release (September 2026), see seymour-second-neighborhood-conjecture.","claimIssueNote":null,"solveDate":"2026-06-29","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Arpan Sadhukhan","R. B. Sandeep","Sagnik Sen"],"aiRole":"The CP-SAT models behind the computational part were developed with assistance from ChatGPT 5.5 Pro; the resulting OR-Tools encodings were then run and independently checked by the authors.","verification":"unreviewed","verificationNote":"The proof leans on a CP-SAT computation whose encodings the authors state they verified independently. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":32,"significanceNote":"One of the best known open problems on tournaments and oriented graphs, attacked steadily for three decades with the out-degree threshold as the standard measure of progress.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.30588","sourceName":"arXiv:2606.30588 - A proof of Seymour's second neighborhood conjecture for oriented graphs with minimum out-degree seven","collection":null,"collectionVersion":null,"links":[{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/seymour-second-neighborhood-conjecture","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"da-silva-machado-seade-conjecture","name":"The da Silva Machado-Seade Conjecture","shortName":"Seade conjecture","problemNumber":null,"field":"Singularity theory","fieldGroup":"Geometry & topology","statement":"da Silva Machado and Seade conjectured that weighted homogeneous isolated hypersurface singularities are exactly those admitting a logarithmic vector field transverse to the link. True: for a reduced isolated hypersurface germ in $\\mathbb{C}^{n+1}$ with $n \\ge 2$, or $n = 1$ and the germ irreducible, the criterion holds.","posedBy":"Diogo da Silva Machado, Jose Seade","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"An independent human proof of the same conjecture appeared the same week.","claimIssueNote":null,"solveDate":"2026-06-29","model":"Danus (built on Rethlas)","modelMaker":"Frenzy Math","humanCollaborators":["Jihao Liu","Xiping Zhang"],"aiRole":"The paper states its main result was obtained by the Danus system, a specialized agent built on Rethlas and described as substantially more capable for fundamental mathematical research. A separate paper published the same week gives a different, human proof of the same conjecture and cites this one as the agent's.","verification":"unreviewed","verificationNote":"No independent check. arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture characterizing weighted homogeneity of isolated singularities, in the Saito logarithmic vector field tradition.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.29891","sourceName":"arXiv:2606.29891 - Criteria of isolated weighted homogeneous hypersurface singularities using logarithmic vector fields","collection":null,"collectionVersion":null,"links":[{"label":"Independent human proof of the same conjecture, arXiv:2606.29886","url":"https://arxiv.org/abs/2606.29886","kind":"independent"},{"label":"da Silva Machado & Seade 2026: Conjecture statement","url":"https://arxiv.org/abs/2605.09210","kind":"announcement"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fgg-qaoa-ring-of-disagrees","name":"FGG Conjecture for QAOA on the Ring of Disagrees","shortName":"QAOA ring of disagrees","problemNumber":null,"field":"Quantum optimization","fieldGroup":"Quantum information & computing","statement":"For an even cycle of size $N$ and depth $p$ with $2p + 2 \\le N$, is the optimal QAOA approximation ratio for MaxCut exactly $\\frac{2p+1}{2p+2}$, as Farhi, Goldstone and Gutmann conjectured?","posedBy":"Edward Farhi, Jeffrey Goldstone & Sam Gutmann","yearPosed":2014,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-29","model":"Claude Fable 5","modelMaker":"Anthropic","humanCollaborators":[],"aiRole":"Claude Fable 5 found a dynamical-symmetry and quantum-signal-processing argument; the complete proof is checked by the Lean 4 kernel. An independent group proved the same result simultaneously via Laurent-polynomial optimization (arXiv:2606.29562).","verification":"lean-verified","verificationNote":"Machine-verified end to end in Lean 4, with an independent simultaneous human proof of the same theorem.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The original Farhi-Goldstone-Gutmann QAOA benchmark question.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.29687","sourceName":"arXiv:2606.29687 - A machine-verified proof of a quantum-optimization conjecture","collection":null,"collectionVersion":null,"links":[{"label":"Independent simultaneous proof via quantum signal processing (arXiv:2606.29562)","url":"https://arxiv.org/abs/2606.29562","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-sos-pairwise-sums-5-8","name":"The Erdos-Sos Pairwise-Sums Problem","shortName":"Erdos-Sos pairwise sums","problemNumber":null,"field":"Additive combinatorics","fieldGroup":"Combinatorics","statement":"Let $f_3(N)$ be the least size forcing a set $A \\subseteq \\{1,\\ldots,N\\}$ to contain distinct $a,b,c$ with $a+b$, $a+c$ and $b+c$ all in $A$. The upper bound $f_3(N) \\le 5N/8 + O(1)$ matches the standard construction $[N/8,N/4] \\cup [N/2,N]$, so $f_3(N) = 5N/8 + O(1)$.","posedBy":"Paul Erdos, Vera T. Sos (before them by Choi, Erdos & Szemerédi)","yearPosed":1975,"ageNote":"First posed as a problem by Choi, Erdos and Szemerédi 1975","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-28","model":"GPT-5.5 Pro, Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Ricky Cipollini"],"aiRole":"The paper states that the manuscript was written by GPT-5.5 Pro from a proof developed by the author together with GPT-5.5 Pro, and that the accompanying Lean formalization was carried out with Aristotle. Both the mathematics and the write-up are joint with the model rather than checked by it.","verification":"lean-verified","verificationNote":"The paper reports a Lean formalization against Mathlib with no sorries and no added axioms. We have not compiled it. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"An Erdos-Sos problem on pairwise sums with a standing construction and a gap that this closes exactly.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.29361","sourceName":"arXiv:2606.29361 - A sharp 5/8 bound for an Erdos-Sos pairwise-sums problem","collection":null,"collectionVersion":null,"links":[{"label":"Thomas Bloom's record of the problem (#865)","url":"https://www.erdosproblems.com/865","kind":"problem-record"},{"label":"Choi et al 1975 - Earliest known problem statement","url":"https://mathscinet.ams.org/mathscinet/relay-station?mr=369305","kind":"problem-record"},{"label":"GitHub repo of the proof","url":"https://github.com/mrricky22/erdos-865-lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mossel-peres-multivariable-bernoulli-factory","name":"Mossel-Peres Multivariable Bernoulli Factory Claim","shortName":"Bernoulli factories","problemNumber":null,"field":"Probability","fieldGroup":"Probability & statistics","statement":"Mossel and Peres showed that a single-variable function admits a finite-automata Bernoulli factory exactly when it is rational, and their Theorem 2.9 claimed the same extension to multivariable functions. The multivariable claim is false, refuted by an explicit counterexample.","posedBy":"Elchanan Mossel, Yuval Peres","yearPosed":2005,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"the published multivariable claim fails; the single-variable theorem stands","claimIssueNote":null,"solveDate":"2026-06-28","model":"Gemini 3.1 Pro (DeepThink)","modelMaker":"Google","humanCollaborators":["Renato Paes Leme","Jon Schneider"],"aiRole":"The tool use disclosure states plainly that the authors used the DeepThink mode of Gemini 3.1 Pro to derive the counterexample, and to refine the presentation.","verification":"unreviewed","verificationNote":"The refutation is an explicit counterexample, so it is a finite check. arXiv note, not peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A named theorem in the Bernoulli factory literature that had been cited as settled for twenty years.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.29595","sourceName":"arXiv:2606.29595 - Note on Finite-Automata Bernoulli Factories for Rational Functions","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-731-dyadic-regularity","name":"Erdos Problem #731","shortName":"Erdos 731","problemNumber":731,"field":"Number theory","fieldGroup":"Number theory","statement":"Let $A(n)$ be the least positive integer not dividing $\\binom{2n}{n}$. Erdos asked for the behaviour of $A(n)$ for reasonable $n$. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals against the scale $F_X = \\sqrt{2}(\\log 2)^{1/4} L^{1/4} \\exp\\sqrt{(\\log 2)L}$ with $L = \\log(2X)$.","posedBy":"Paul Erdos","yearPosed":1975,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Resolved under an explicit formalization of 'reasonable', not in full generality.","claimIssueNote":null,"solveDate":"2026-06-27","model":"ChatGPT, Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Eric Li"],"aiRole":"The declaration says large language models, primarily ChatGPT, were used extensively throughout the research while the author originated the ideas, and that the accompanying Lean formalization was developed by the author with Aristotle.","verification":"lean-checked","verificationNote":"A Lean formalization accompanies the paper; we have not compiled it. The result is conditional on the paper's own formalization of Erdos's informal word 'reasonable', which the entry's resolution status reflects. arXiv preprint, not peer-reviewed, and problem still marked open on the Erdos problems database.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.29062","sourceName":"arXiv:2606.29062 - A Resolution of Erdos Problem 731 under Dyadic Regularity","collection":null,"collectionVersion":null,"links":[{"label":"Erdos et al 1975 - Problem statement","url":"https://mathscinet.ams.org/mathscinet/relay-station?mr=369288","kind":"problem-record"},{"label":"Github repo of the proof","url":"https://github.com/ericlisg/erdos731","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"unit-distance-independence-ratio","name":"Erdos's Question on the Independence Ratio of Unit-Distance Graphs","shortName":"Unit-distance independence","problemNumber":null,"field":"Combinatorial geometry","fieldGroup":"Geometry & topology","statement":"Erdos asked whether a finite unit-distance graph in the plane can have independence ratio below $1/4$. One exists, built on the geometric fractional chromatic number framework of Matolcsi, Ruzsa, Varga and Zsamboki plus a carefully chosen two-vertex augmentation.","posedBy":"Paul Erdos","yearPosed":1986,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-26","model":"ChatGPT, Codex (GPT-5.5)","modelMaker":"OpenAI","humanCollaborators":["Akos Ducz","Daniel Varga"],"aiRole":"The models were used for software development including code drafting and debugging, plus improvement and formatting suggestions. The search that produces the graph is computational, so the tooling is load-bearing even though no mathematical step is attributed.","verification":"unreviewed","verificationNote":"The result is a specific finite graph, so it is checkable by computation. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A question of Erdos in the Hadwiger-Nelson circle, where the independence ratio bound is a standard measure of progress on colouring the plane.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.28157","sourceName":"arXiv:2606.28157 - A unit-distance graph in the plane with independence ratio below 1/4","collection":null,"collectionVersion":null,"links":[{"label":"Erdos 1986 - Problem statement","url":"https://www.renyi.hu/~p_erdos/1986-09.pdf","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"k-antichains-unit-cube","name":"Conjecture on $k$-Antichains in the Unit Cube","shortName":"k-antichains","problemNumber":null,"field":"Combinatorics","fieldGroup":"Combinatorics","statement":"A subset $A$ of the pointwise-ordered cube $[0,1]^n$ is a $k$-antichain when it meets every chain in at most $k$ points. The conjecture concerns the largest possible $(n-1)$-dimensional Hausdorff measure of such a set; it is settled here, following work of Janzer.","posedBy":"conjecture in the antichain-measure literature","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-26","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["John M. Campbell"],"aiRole":"The paper states that the construction carrying the proof is largely based on the author's extensive interactions with GPT-5.5 Pro.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A specific conjecture on Hausdorff measure of antichains in the cube, with a small but real following after Janzer's work on it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.28606","sourceName":"arXiv:2606.28606 - On a conjecture on k-antichains in the unit n-cube","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lassak-reduced-body-area","name":"Lassak's Area Bound for Reduced Planar Bodies","shortName":"Lassak's conjecture","problemNumber":null,"field":"Convex geometry","fieldGroup":"Geometry & topology","statement":"Lassak conjectured that a reduced planar convex body of thickness $\\Delta$ has area at most $(\\pi/4)\\Delta^2$, the value for the disc. False: an explicit reduced body of thickness $1$ has area $0.786215\\ldots > \\pi/4 = 0.785398\\ldots$, given by a closed-form support function.","posedBy":"Marek Lassak","yearPosed":2005,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-26","model":"GPT-5.5 Pro, Claude Opus 4.8","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Scott Duke Kominers"],"aiRole":"The acknowledgement credits the models with assisting the computations and coding in preparing the article, without separating which step came from where, so the lowest tier applies.","verification":"site-confirmed","verificationNote":"The counterexample is given by an explicit support function with the area stated to six decimal places, so the refutation reduces to evaluating a closed-form integral. Single-author arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture on reduced bodies, a standard object in convex geometry where the disc was the natural extremal guess.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.28612","sourceName":"arXiv:2606.28612 - A reduced planar body with area greater than pi/4","collection":null,"collectionVersion":null,"links":[{"label":"Lassak 2005: Conjecture statement","url":"https://www.semanticscholar.org/paper/Area-of-reduced-polygons-Lassak/4f9efbf36b60b5606c5ef6b6b8277678e5e90040","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"regular-gabor-frame-existence","name":"The Existence Problem for Regular Gabor Frames","shortName":"Gabor frame existence","problemNumber":null,"field":"Harmonic analysis","fieldGroup":"Analysis","statement":"Does every lattice of density above one admit a Gabor frame with a nice window? No. For every dimension $d > 1$ there are explicit criteria on lattices $\\Lambda \\subset \\mathbb{R}^{2d}$ with $D(\\Lambda) > 1$ such that no function with continuous Zak transform generates a Gabor frame along $\\Lambda$, which answers the existence problem negatively for Schwartz-class windows.","posedBy":"time-frequency analysis literature","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-24","model":"GPT-5.4, Claude Opus 4.7","modelMaker":"OpenAI / Anthropic","humanCollaborators":["Jaume de Dios Pont","Lukas Liehr","Mitchell A. Taylor"],"aiRole":"The disclosure separates the two roles: GPT-5.4 was used mainly for mathematical exploration, including exploring whether homology-theoretic methods could give a common-zero criterion for two quasiperiodic functions, while Claude Opus 4.7 assisted with the Lean formalization. The authors verified everything independently. The same group's Cantor Fourier frame paper is also in this catalog.","verification":"unreviewed","verificationNote":"A Lean formalization accompanies the paper; we have not compiled it. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The existence question for Gabor frames on general lattices, a central problem in time-frequency analysis where the density condition was long hoped to suffice.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.26052","sourceName":"arXiv:2606.26052 - On the existence problem of regular Gabor frames","collection":null,"collectionVersion":null,"links":[{"label":"Same group's Cantor Fourier frame result, arXiv:2607.08656","url":"https://arxiv.org/abs/2607.08656","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1061","name":"Erdős Problem #1061","shortName":"Erdős #1061","problemNumber":1061,"field":"Analytic Number Theory","fieldGroup":"Number theory","statement":"For $S(x) = \\#\\{(a,b) : a + b \\le x,\\ \\sigma(a) + \\sigma(b) = \\sigma(a+b)\\}$, is $S(x) \\sim cx$? The preprint claims $S(x)$ grows faster than $x (\\log x)^R$ for every fixed $R$, ruling out the linear asymptotic.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-24","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"unreviewed","verificationNote":"Public self-contained preprint; independent expert review pending and the official record still open.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1061","sourceName":"erdosproblems.com/1061","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"furthest-pair-seth-hardness","name":"Quadratic-Time Hardness of Furthest Pair in Superconstant Dimension","shortName":"Furthest Pair under SETH","problemNumber":null,"field":"Fine-grained complexity","fieldGroup":"Theoretical computer science","statement":"Furthest Pair and its relatives admit $f(d)\\,n^{2-\\Theta(1/d)}$ algorithms, making them the standard examples of barely subquadratic computation, and whether that is optimal in superconstant dimension was open. Under SETH it is: Furthest Pair requires quadratic time once the dimension is superconstant.","posedBy":"fine-grained complexity literature","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-24","model":"ChatGPT 5.5 Pro (with Codex, Claude Opus, Gemini for feedback)","modelMaker":"OpenAI / Anthropic / Google","humanCollaborators":["Barna Saha","Yinzhan Xu","Christopher Ye"],"aiRole":"The paper states the proof was initially discovered by ChatGPT 5.5 Pro, that the initial prompt was essential, and that other systems were used to generate feedback on it. The authors validated and substantially edited the proof to improve it.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The barely-subquadratic class is a well-known frontier in fine-grained complexity, and this pins the dimension at which the speedup dies.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.25887","sourceName":"arXiv:2606.25887 - Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1061-sum-of-divisors","name":"Erdos Problem #1061","shortName":"Erdos 1061","problemNumber":1061,"field":"Number theory","fieldGroup":"Number theory","statement":"Let $S(x)$ count ordered pairs $(a,b)$ with $a+b \\le x$ and $\\sigma(a)+\\sigma(b) = \\sigma(a+b)$. Erdos asked whether $S(x) \\sim cx$. The opposite extreme holds: for every $R > 0$, $S(x)/(x(\\log x)^R) \\to \\infty$, so the count beats every fixed logarithmic scale.","posedBy":"Paul Erdos","yearPosed":2004,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-24","model":"ChatGPT, Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Eric Li"],"aiRole":"The declaration says large language models, primarily ChatGPT, were used extensively throughout the research, with the author originating the ideas, and that the accompanying Lean formalization was produced with Harmonic's Aristotle under the author's direction and audit.","verification":"unreviewed","verificationNote":"A Lean formalization produced with Aristotle accompanies the paper; we have not compiled it. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.25849","sourceName":"arXiv:2606.25849 - A resolution of Erdos Problem 1061 on the sum-of-divisors function","collection":null,"collectionVersion":null,"links":[{"label":"Guy 2004, B15 - Problem statement","url":"https://mathscinet.ams.org/mathscinet/relay-station?mr=2076335","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-593-1177-obligatory-triple-systems","name":"Erdos Problems #593 and #1177","shortName":"Erdos 593 and 1177","problemNumber":593,"field":"Infinite combinatorics","fieldGroup":"Combinatorics","statement":"Erdos Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system. The answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and one-point amalgamations. Problem #1177 on exact spectra is settled alongside it.","posedBy":"Paul Erdos","yearPosed":1975,"ageNote":"1975 is the year problem #593 was posed, problem #1177 was recorded in 1999 and posed much earlier, but an exact date is not available","solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Settles two numbered Erdos problems at once.","claimIssueNote":null,"solveDate":"2026-06-23","model":"ChatGPT, Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Eric Li"],"aiRole":"The declaration says large language models, primarily ChatGPT, were used extensively throughout the research, with the author originating the ideas, and that the accompanying Lean formalization was produced with Harmonic's Aristotle under the author's direction and audit.","verification":"lean-checked","verificationNote":"A Lean formalization produced with Aristotle accompanies the paper; we have not compiled it. arXiv preprint, not peer-reviewed. Both problems still listed as open on Thomas Bloom's database.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":13,"significanceNote":"A numbered Erdos problem that carries a $500 Erdos prize, setting it above the typical entry in the catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.24882","sourceName":"arXiv:2606.24882 - A Resolution of Erdos Problems 593 and 1177: Obligatory Triple Systems and Exact Spectra","collection":null,"collectionVersion":null,"links":[{"label":"Erdos et al 1975 - #593 Problem statement","url":"https://mathscinet.ams.org/mathscinet/relay-station?mr=398876","kind":"problem-record"},{"label":"Bollobas et al 1999, p. 14 - #1177 Problem statement","url":"https://web.math.pmf.unizg.hr/~vjekovac/EP/Some_of_Pauls_favorite_problems.pdf","kind":"problem-record"},{"label":"GitHub repo of the proof","url":"https://github.com/ericlisg/erdos-593-1177-lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-768-sylow-divisor-condition","name":"Erdos Problem #768","shortName":"Erdos 768","problemNumber":768,"field":"Number theory","fieldGroup":"Number theory","statement":"Let $A(x)$ count $n \\le x$ such that every prime $p \\mid n$ has a divisor $d > 1$ of $n$ with $d \\equiv 1 \\pmod p$. Erdos asked whether $A(x)/x = \\exp(-(c+o(1))\\sqrt{\\log x}\\log\\log x)$. It does, with $c = 1/(2\\sqrt{\\log 2})$.","posedBy":"Paul Erdos","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-23","model":"ChatGPT, Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Eric Li"],"aiRole":"The declaration says large language models, primarily ChatGPT, were used extensively throughout the research, with the author originating the ideas, and that the accompanying Lean formalization was produced with Harmonic's Aristotle under the author's direction and audit. The author also notes rejecting flawed model suggestions along the way.","verification":"unreviewed","verificationNote":"A Lean formalization produced with Aristotle accompanies the paper; we have not compiled it. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.24872","sourceName":"arXiv:2606.24872 - A Resolution of Erdos Problem 768: the Sylow Divisor Condition","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kontsevich-strata-asphericity","name":"Kontsevich's Asphericity Conjecture for Strata of Differentials","shortName":"Strata asphericity","problemNumber":null,"field":"Teichmuller theory","fieldGroup":"Geometry & topology","statement":"A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold $K(\\pi,1)$ spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an orbifold $K(\\pi,1)$, giving infinitely many counterexamples, along with counterexamples for associated stability spaces.","posedBy":"Maxim Kontsevich","yearPosed":1997,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-23","model":"ChatGPT 5.5 Plus","modelMaker":"OpenAI","humanCollaborators":["Dawei Chen","Jingyin Huang","Yu Qiu","Fei Yu"],"aiRole":"The authors state that the criterion in Appendix A arose first from an iterative, guided conversation with ChatGPT 5.5 Plus, and that they then verified, corrected and revised the proof.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"An asphericity conjecture attributed to Kontsevich about strata of differentials, standard background in Teichmuller dynamics and stability conditions.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.24135","sourceName":"arXiv:2606.24135 - Non-asphericity of strata of genus-one differentials and stability spaces","collection":null,"collectionVersion":null,"links":[{"label":"Kontsevich & Zorich 1997, Section 4: Conjecture statement","url":"https://arxiv.org/abs/hep-th/9701164","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-lovasz-cover-number","name":"The Erdos-Lovasz Cover Number Problem","shortName":"Erdos-Lovasz cover number","problemNumber":null,"field":"Extremal combinatorics","fieldGroup":"Combinatorics","statement":"Let $g(r)$ be the fewest edges in an $r$-uniform intersecting hypergraph with cover number $r$. Erdos and Lovasz proved $g(r) \\ge 8r/3 - 3$. An elementary argument gives $g(r) \\ge 3r - 4$, and building on it with Kahn's small-codegree edge-colouring theorem pushes the bound further.","posedBy":"Paul Erdos, Laszlo Lovasz","yearPosed":1975,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"an improved lower bound; the true order of g(r) remains open","claimIssueNote":null,"solveDate":"2026-06-23","model":"ChatGPT 5.5 Pro, Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Varun Sivashankar"],"aiRole":"The acknowledgement states the proof was discovered with the help of ChatGPT 5.5 Pro, and that Theorem 1 was then formalized in Lean with Harmonic's Aristotle.","verification":"unreviewed","verificationNote":"The Lean formalization is partial by the author's own account: part (i) of Theorem 1 is formalized in full and part (ii) only conditional on Kahn's theorem. We have not compiled it. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A classical Erdos-Lovasz quantity from the 1975 paper that founded the study of intersecting hypergraphs with large cover number, where the constant had barely moved.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.24878","sourceName":"arXiv:2606.24878 - An Improved Lower Bound for the Erdos-Lovasz Cover Number Problem","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"baker-anti-bertini-hyperplane","name":"Baker's Question on Smooth Hyperplane Sections over Finite Fields","shortName":"Anti-Bertini embeddings","problemNumber":null,"field":"Algebraic geometry","fieldGroup":"Algebra","statement":"Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must acquire a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted no for every positive-dimensional variety, and that prediction is correct.","posedBy":"Matthew Baker; prediction by Bjorn Poonen","yearPosed":2004,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-22","model":"GPT-based models, Rethlas","modelMaker":"OpenAI / Frenzy Math","humanCollaborators":["Yutong Zhang","Yaoran Yang"],"aiRole":"The acknowledgement says the author used GPT-based large language models to generate and compare possible proof strategies, some of which was incorporated into the final manuscript, and used the Rethlas agent as an auxiliary proof-checker.","verification":"lean-checked","verificationNote":"arXiv preprint; not yet peer-reviewed. Lean formalization available, but not yet audited by a third party.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A question of Baker recorded by Poonen in the Bertini-over-finite-fields literature, with a stated prediction this confirms.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.23513","sourceName":"arXiv:2606.23513 - Hyperplane anti-Bertini embeddings over finite fields","collection":null,"collectionVersion":null,"links":[{"label":"Poonen 2004, Question 4.1: Conjecture statement","url":"https://arxiv.org/abs/math/0204002","kind":"problem-record"},{"label":"Github repo of the proof","url":"https://github.com/anetigone/lean-research-formalizations","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"finite-field-restriction-paraboloid","name":"The Finite Field Restriction Problem for the Paraboloid","shortName":"Finite field restriction","problemNumber":null,"field":"Harmonic analysis","fieldGroup":"Analysis","statement":"For the three-dimensional paraboloid $P_3$ over a prime field in which $-1$ is not a square, the Fourier extension operator maps $L^2$ to $L^r$ for $r > 176/51 = 3.45098\\ldots$, improving the exponent by combining a bilinear approach with point-line incidence bounds.","posedBy":"Gerd Mockenhaupt, Terence Tao","yearPosed":2004,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"a record exponent; the conjectured range is not yet reached","claimIssueNote":null,"solveDate":"2026-06-22","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Mark Lewko"],"aiRole":"The author states he used ChatGPT to help organize the argument and to identify the cutoffs used to optimize the estimates. Choosing those cutoffs is what fixes the exponent, so the contribution touches the result rather than only the write-up.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"The finite field restriction problem, posed by Mockenhaupt and Tao as a model for Euclidean restriction, where the exponent is the standard measure of progress.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.22882","sourceName":"arXiv:2606.22882 - A bilinear approach to the finite field restriction problem, II","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-671","name":"Erdős Problem #671","shortName":"Erdős #671","problemNumber":671,"field":"Analysis, Interpolation","fieldGroup":"Analysis","statement":"For triangular arrays of nodes $a_i^n\\in[-1,1]$ let $\\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$, with fundamental polynomials $p_i^n$. Is there a choice of nodes such that for every continuous $f$ there is some $x$ where $\\limsup_n \\sum_i\\lvert p_{i}^n(x)\\rvert=\\infty$ and yet $\\mathcal{L}^nf(x) \\to f(x)$? Is there a choice with $\\limsup_n \\sum_i\\lvert p_{i}^n(x)\\rvert=\\infty$ for every $x$, yet for every continuous $f$ some $x$ has $\\mathcal{L}^nf(x)\\to f(x)$? Both questions are claimed resolved in the affirmative.","posedBy":"Paul Erdős","yearPosed":1982,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Both parts claimed answered affirmatively, with a Lean formalization; two proof claims are filed on erdosproblems.com but the problem is still listed open","claimIssueNote":null,"solveDate":"2026-06-22","model":"GPT-5.5 Pro, Codex","modelMaker":"OpenAI","humanCollaborators":["Liam Price"],"aiRole":"GPT Pro produced the affirmative resolutions of both questions and Codex the Lean formalization; the humans directed the models with a writing-style prompt and cleaned up terminology, a workflow the site's owner singled out as unusually readable for AI-assisted papers.","verification":"lean-verified","verificationNote":"The argument comes with a Codex-produced Lean formalization checkable online; no independent audit of statement fidelity, and erdosproblems.com still lists the problem open with the claims filed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A numbered Erdos problem that carries a $250 Erdos prize, setting it above the typical entry in the catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/671","sourceName":"erdosproblems.com/671","collection":null,"collectionVersion":null,"links":[{"label":"Write-up","url":"https://www.overleaf.com/read/gqmfrhsprtqm#59ac11","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-550","name":"Erdős Problem #550","shortName":"Erdős #550","problemNumber":550,"field":"Graph Theory, Ramsey Theory","fieldGroup":"Combinatorics","statement":"Let $m_1\\leq\\cdots\\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\\ldots,m_k$, prove that $R(T,G)\\leq (\\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1$.","posedBy":"Paul Erdős, Ralph Faudree, Cecil Rousseau, Richard Schelp","yearPosed":1985,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Claimed proved in a preprint of E. Li; erdosproblems.com still lists the problem open pending human review","claimIssueNote":null,"solveDate":"2026-06-22","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Eric Li"],"aiRole":"The paper's disclosure states ChatGPT was used for ideation, formulation, proof exploration and refinement, narrowing the search space, programming and orchestration, with the author taking responsibility for the final contents; forum readers describe it as an affirmative paper almost purely by AI.","verification":"unreviewed","verificationNote":"AI screenings on the forum initially flagged issues that turned out to be PDF-parsing artifacts; a re-run against the TeX source found no issues. No independent human review yet, and the site's owner is explicitly reserving judgement.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/550","sourceName":"erdosproblems.com/550","collection":null,"collectionVersion":null,"links":[{"label":"arXiv preprint","url":"https://arxiv.org/abs/2606.23659","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-865","name":"Erdős Problem #865","shortName":"Erdős #865","problemNumber":865,"field":"Number Theory, Additive Combinatorics","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1972,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-22","model":"GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Ricky Cipollini"],"aiRole":null,"verification":"lean-verified","verificationNote":"Listed as solved on erdosproblems.com and the proof is verified in Lean. Solve credited via Terence Tao's AI-contributions wiki.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/865","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kannan-tetali-vempala-conjecture","name":"Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case)","shortName":"KTV conjecture","problemNumber":null,"field":"Markov chain mixing time","fieldGroup":"Probability & statistics","statement":"The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have spectral gap at least $\\binom{m}{2}^{-1}\\binom{n}{2}^{-1}$ on $m \\times n$ matrices, which is worst-case tight and settles the bipartite case.","posedBy":"Ravindran Kannan, Prasad Tetali, Santosh Vempala","yearPosed":1997,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-21","model":"ChatGPT 5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Weibo Fu (Princeton)","Qian Qin (Minnesota)","Guanyang Wang (Rutgers)"],"aiRole":null,"verification":"lean-verified","verificationNote":null,"publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The standard rapid-mixing conjecture for fixed-margin sampling, cited across the MCMC community since 1997.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.22636","sourceName":"arxiv","collection":null,"collectionVersion":null,"links":[{"label":"Lean repository","url":"https://github.com/guanyangwang/ktv-swap-lean","kind":"lean-proof"},{"label":"Claimed undirected case (OpenAI math release)","url":"https://vibemathed.com/problem/kannan-tetali-vempala-undirected","kind":"other"}],"relations":[],"submittedBy":"QuietLemur253","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"zhi-wei-sun-legendre-determinant","name":"Zhi-Wei Sun's Conjecture 3.4 on a Truncated Legendre-Symbol Determinant","shortName":"Sun's determinant conjecture","problemNumber":null,"field":"Number theory","fieldGroup":"Number theory","statement":"Zhi-Wei Sun conjectured a closed evaluation of a truncated Legendre-symbol determinant. For every prime $p \\equiv 3 \\pmod 4$ it equals $\\lfloor (p-2)/3 \\rfloor^2 x$, proved by reducing to inverse data for Chapman's full Legendre-symbol matrix and evaluating that with Vsemirnov's factorization and a Schur-Pfaffian resolvent identity.","posedBy":"Zhi-Wei Sun","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-21","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Yaoran Yang","Gaishi Yang","Yutong Zhang"],"aiRole":"The abstract states flatly that OpenAI's ChatGPT produced the proof, which the authors independently checked and confirmed. The authors separately used a GPT-assisted framework to develop Lean 4 formalizations of selected components.","verification":"unreviewed","verificationNote":"The authors report Lean 4 formalizations of selected proof components rather than the whole argument. arXiv preprint, not peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"One of Zhi-Wei Sun's numbered conjectures on Legendre-symbol determinants, a well-defined target with a small dedicated following.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.22548","sourceName":"arXiv:2606.22548 - A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant","collection":null,"collectionVersion":null,"links":[{"label":"Sun 2024, Conjecture 3.4: Conjecture statement","url":"https://dx.doi.org/10.48550/arXiv.2405.03626","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-176","name":"Erdős Problem #176","shortName":"Erdős #176","problemNumber":176,"field":"Discrepancy Theory","fieldGroup":"Combinatorics","statement":"Let $N(k, \\ell)$ be the least $N$ such that every $f : [N] \\to \\{-1, 1\\}$ has a $k$-term arithmetic progression $P$ with $|\\sum_{n \\in P} f(n)| \\ge \\ell$. In particular, is $N(k, 2) \\le C^k$?","posedBy":null,"yearPosed":1965,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"a polynomial bound for N(k,2), stronger than the exponential bound asked for; the two-parameter problem remains open","claimIssueNote":null,"solveDate":"2026-06-21","model":"Codex 5.5, ChatGPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Public Lean proof of the N(k,2) clause.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":11,"significanceNote":"A numbered Erdos problem that an unusually dense reference trail on erdosproblems.com (7 sources), setting it above the typical entry in the catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/176","sourceName":"erdosproblems.com/176","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-948","name":"Erdős Problem #948","shortName":"Erdős #948","problemNumber":948,"field":"Number Theory, Ramsey Theory","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1977,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-21","model":"Aristotle, GPT-5.5 Pro","modelMaker":null,"humanCollaborators":[],"aiRole":null,"verification":"site-confirmed","verificationNote":"Marked solved by erdosproblems.com's official status. Solve credited via Terence Tao's AI-contributions wiki.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/948","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-346","name":"Erdős Problem #346","shortName":"Erdős #346","problemNumber":346,"field":"Number Theory, Complete Sequences","fieldGroup":"Number theory","statement":"Let $A=\\{1\\leq a_1< a_2<\\cdots\\}$ be a set of integers such that $A\\backslash B$ is complete for any finite subset $B$ and not complete for any infinite subset $B$. If $a_{n+1}/a_n \\geq 1+\\epsilon$ for all $n$, must $\\lim_n a_{n+1}/a_n=(1+\\sqrt{5})/2$? Under the reading where the ratio limit is assumed to exist, a Lean-verified argument forces the limit to be the golden ratio; a separate construction disproves the literal statement where convergence is not assumed.","posedBy":"Paul Erdős, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"The problem statement is ambiguous: the limit-exists reading is claimed proved (Lean), while the convergence-from-hypotheses reading was disproved by a Lean-checked construction of Price that the community classes as a variant","claimIssueNote":null,"solveDate":"2026-06-21","model":"ChatGPT, Codex","modelMaker":"OpenAI","humanCollaborators":["Kenta Kitamura"],"aiRole":"Kitamura's affirmative Lean 4 formalization of the limit-exists reading was produced with ChatGPT and Codex; days earlier, GPT Pro with Codex had produced a Lean-checked disproof of the literal reading (Liam Price), which the forum classes as solving a variant with precursors in Burr-Erdős 1981.","verification":"lean-verified","verificationNote":"A community screening found the Lean of the variant disproof correct and corresponding to its paper (one typo); the affirmative limit-exists formalization reports standard axioms only. erdosproblems.com still lists the problem open.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/346","sourceName":"erdosproblems.com/346","collection":null,"collectionVersion":null,"links":[{"label":"Kitamura's Lean formalization","url":"https://github.com/KitaKen1/erdos346-ratio-limit-lean","kind":"lean-proof"},{"label":"Price's disproof of the literal reading","url":"https://www.overleaf.com/read/tgrrgqpbjpht#c5a085","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1197","name":"Erdős Problem #1197","shortName":"Erdős #1197","problemNumber":1197,"field":"Analysis","fieldGroup":"Analysis","statement":null,"posedBy":"Paul Erdős","yearPosed":1980,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-21","model":"Aristotle, Claude Opus 4.7, GPT-5.4 Pro","modelMaker":null,"humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Listed as solved on erdosproblems.com and the proof is verified in Lean. Solve credited via Terence Tao's AI-contributions wiki.","publication":"announcement","resolutionMethod":"construction","citations":49,"citationsPaper":"Paul Erdős (1980), \"A survey of problems in combinatorial number theory\", Ann. Discrete Math.","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W1663749032","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1197","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-306","name":"Erdős Problem #306","shortName":"Erdős #306","problemNumber":306,"field":"Number Theory, Unit Fractions","fieldGroup":"Number theory","statement":"If $a/b \\in \\mathbb{Q}_{>0}$ and $b$ is squarefree, can $a/b$ always be written as a finite sum of reciprocals of distinct products of two distinct primes?","posedBy":null,"yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-19","model":"AI-assisted Lean development (models not itemized)","modelMaker":null,"humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked modulo two explicitly isolated Rosser-Schoenfeld analytic inputs; the official problem page still lists the problem as open.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/306","sourceName":"erdosproblems.com/306","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-consecutive-integers-prime-factors","name":"Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors","shortName":"Consecutive integers","problemNumber":null,"field":"Number theory","fieldGroup":"Number theory","statement":"Let $n_k$ be the least $n > 2k$ such that $(n-k)(n-k+1)\\cdots(n-1)$ has no prime factor in $(k, 2k)$. Erdos conjectured a superpolynomial lower bound; for all large $k$, $n_k > e^{\\log^2 k / (20 \\log\\log k)}$.","posedBy":"Paul Erdos","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-18","model":"ChatGPT 5.5 Pro, Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":["Wouter van Doorn","Quanyu Tang"],"aiRole":"The paper says the bound was conceived of by ChatGPT 5.5 Pro and that the original paper the model wrote remains publicly available, so the provenance is checkable rather than asserted. The supporting proofs were obtained by Harmonic's Aristotle and are stated to be fully self-contained.","verification":"lean-checked","verificationNote":"Proofs produced by an automated theorem prover and described as fully self-contained; we have not independently checked them. arXiv preprint, not peer-reviewed. Lean formalization available but not yet audited externally.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A conjecture of Erdos on prime factors of blocks of consecutive integers, in the Erdos-Selfridge tradition.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.19863","sourceName":"arXiv:2606.19863 - Consecutive integers free of certain prime factors","collection":null,"collectionVersion":null,"links":[{"label":"Erdos 1979 - Problem statement","url":"https://mathscinet.ams.org/mathscinet/relay-station?mr=515121","kind":"problem-record"},{"label":"Thomas Bloom's record of the problem (#451)","url":"https://www.erdosproblems.com/451","kind":"problem-record"},{"label":"Github repo of the proof","url":"https://github.com/Woett/Lean-files/blob/main/ErdosProblem451.lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"dynamical-sampling-normal-orbit-frame","name":"Conjecture 3 of the Dynamical Sampling Survey","shortName":"Normalized orbit frames","problemNumber":null,"field":"Frame theory","fieldGroup":"Analysis","statement":"Aldroubi, Cabrelli, Krishtal and Molter conjectured that for a bounded normal operator $T$ and any vector $g$, the normalized orbit $\\{T^k g / \\|T^k g\\| : k \\ge 0\\}$ is never a frame. It can be: an explicit construction produces a normalized orbit that is a frame.","posedBy":"Akram Aldroubi, Carlos Cabrelli, Ilya Krishtal, Ursula Molter","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-18","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Ilya A. Krishtal","Gotz E. Pfander"],"aiRole":"The paper states the constructions were achieved using ChatGPT, whose assistance was also used in preparing the manuscript. One of the authors is among those who posed the conjecture.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":8,"significanceNote":"A numbered conjecture from the 2026 dynamical sampling survey, recent and specialized but explicitly posed.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.20848","sourceName":"arXiv:2606.20848 - The normalized orbit of a bounded normal operator can be a frame","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sabok-s-prime-simplex","name":"Sabok's S-Prime Simplex Questions","shortName":"Sabok's simplex questions","problemNumber":null,"field":"Convex geometry","fieldGroup":"Geometry & topology","statement":"Sabok asked whether the compact convex set $S'(X)$ attached to a separable metric space of diameter at most one is always a simplex, and whether $S'(\\mathbb{U}_1)$ is the Poulsen simplex. Both answers are negative, with obstructions already visible for finite $X$ and for the Urysohn space.","posedBy":"Marcin Sabok","yearPosed":2013,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-18","model":"GPT-based models","modelMaker":"OpenAI","humanCollaborators":["Yutong Zhang","Yaoran Yang"],"aiRole":"The acknowledgement says the authors used GPT-based large language models to generate and compare possible proof strategies. Same authors as the Baker anti-Bertini entry in this catalog.","verification":"lean-checked","verificationNote":"arXiv preprint; not yet peer-reviewed. Lean formalization available, but not yet audited by a third party.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"Two explicitly posed questions of Sabok about a convex invariant of metric spaces, documented but specialized.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.27215","sourceName":"arXiv:2607.27215 - Finite and Urysohn obstructions to Sabok's S-prime simplex questions","collection":null,"collectionVersion":null,"links":[{"label":"Sabok 2013, Question 9.4: Problem statement","url":"https://arxiv.org/abs/1306.1049","kind":"announcement"},{"label":"GitHub repo of the proof","url":"https://github.com/anetigone/sabok-sprime-obstructions-lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-451","name":"Erdős Problem #451","shortName":"Erdős #451","problemNumber":451,"field":"Number Theory, Primes","fieldGroup":"Number theory","statement":"Let $n_k$ be the least integer greater than $2k$ for which $\\prod_{i=1}^k (n_k - i)$ has no prime factor in $(k, 2k)$. How rapidly must $n_k$ grow?","posedBy":null,"yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"the conjectured superpolynomial growth is established; the sharper order remains open","claimIssueNote":null,"solveDate":"2026-06-18","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"unreviewed","verificationNote":"Human-checked arXiv proof.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/451","sourceName":"erdosproblems.com/451","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"ramachandra-natarajan-correlation-gap","name":"The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture","shortName":"Correlation gap","problemNumber":null,"field":"Optimization","fieldGroup":"Algorithms & optimization","statement":"Ramachandra and Natarajan conjectured a bound on the pairwise independent correlation gap in their 2025 Operations Research Letters paper. An explicit counterexample refutes it.","posedBy":"Arjun Ramachandra, Karthik Natarajan","yearPosed":2025,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-18","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Arjun Ramachandra","Karthik Natarajan"],"aiRole":"The abstract credits the counterexample to GPT-5.5 Pro in its first sentence. The authors add that earlier attempts with free tiers of ChatGPT and Claude made no progress, which is a useful data point on where the capability threshold sat. They are refuting their own conjecture.","verification":"lean-checked","verificationNote":"The refutation is an explicit counterexample, so it is a finite check. Short arXiv note, not peer-reviewed.\n\nA Lean 4 / Mathlib formalization of the counterexample was contributed in August 2026 by its author, produced with Codex. Curator source audit: all 579 lines read, with no sorry, admit, native_decide, unsafe declaration, user-declared axiom, implemented_by or partial def anywhere; finite checks go through kernel decide and rational identities through norm_num, and Mathlib is pinned to an exact revision on toolchain v4.33.0-rc2. The curator has not compiled it, and it is the work of the same person who reported the result, so it is not third-party corroboration.\n\nWhat the formalization does and does not settle is worth stating exactly. Its final theorem is a seven-part conjunction certifying the witness and its bounds: the three-atom distribution attains the target marginals with expected coverage 4, no distribution exceeds 4, the product distribution is pairwise feasible, every pairwise-feasible distribution is bounded by $479/160$, and $4 \\div (479/160) = 640/479 > 4/3$. Both bounds are universally quantified rather than spot-checked. What the file never states is the Ramachandra-Natarajan conjecture itself, so the step from this instance to the conjecture being refuted stays informal and rests on the conjectured bound really being $4/3$. That is the difference between a kernel-checked artifact and an audited claim, and why this sits on the unaudited Lean rung.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"A conjecture the same authors published in 2025; recent and narrow, but explicitly stated in the literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.19663","sourceName":"arXiv:2606.19663 - Counterexample to a conjecture on the pairwise independent correlation gap using AI","collection":null,"collectionVersion":null,"links":[{"label":"Lean 4 / Mathlib formalization of the counterexample (Codex, author-contributed)","url":"https://github.com/Olangu/pairwise-correlation-gap-lean","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":1},{"slug":"vinzant-phase-retrieval-injectivity","name":"Vinzant's Conjecture on Phase Retrieval Injectivity","shortName":"Phase retrieval injectivity","problemNumber":null,"field":"Frame theory","fieldGroup":"Analysis","statement":"Vinzant conjectured, in a form later restated by Bandeira, that the $4M-4$ threshold for injective complex phase retrieval is sharp. Part (1) holds: for $A \\in \\mathbb{C}^{N \\times M}$ with $N = 4M-5$ and i.i.d. standard complex Gaussian entries, the phase retrieval map generated by $A$ fails to be injective with positive probability.","posedBy":"Cynthia Vinzant; restated by Afonso S. Bandeira","yearPosed":2015,"ageNote":"First posed by Vinzant in 2015, later restated by Bandeira in 2026, the earlier date is used","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"part (1) of the conjecture","claimIssueNote":null,"solveDate":"2026-06-16","model":"Rethlas","modelMaker":"Frenzy Math","humanCollaborators":["Zhangsong Li"],"aiRole":"The paper states that the main result was obtained using generative AI, in particular the Rethlas system.","verification":"unreviewed","verificationNote":"Short single-author arXiv note; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named conjecture on the sharp measurement count for complex phase retrieval, a central constant in frame theory Bandeira put back into circulation.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.17922","sourceName":"arXiv:2606.17922 - On Injectivity of Phase Retrieval","collection":null,"collectionVersion":null,"links":[{"label":"Bandeira 2026: Problem statement","url":"https://arxiv.org/abs/2603.29571","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"erdos-pairwise-coprime-reciprocal-sum","name":"Erdos's Question on Shifted Pairwise-Coprime Reciprocal Sums","shortName":"Pairwise-coprime sums","problemNumber":null,"field":"Number theory","fieldGroup":"Number theory","statement":"Let $\\mathcal{M}(n)$ be the supremum of $\\sum_{a \\in A} 1/(n-a)$ over pairwise coprime $A \\subset [1,n)$. Erdos asked whether $\\mathcal{M}(n) \\le \\sum_{p= 5, the OpenAI math release (September 2026) claims the sharper lower bound r(s,t) >= t^(s-1)/(log t)^(s-2+eps), matching the Ajtai-Komlos-Szemeredi upper bound in the logarithmic exponent up to (log t)^o(1); see off-diagonal-ramsey-log-exponent.","claimIssueNote":null,"solveDate":"2026-06-16","model":"Claude, OpenAI internal model","modelMaker":null,"humanCollaborators":["Domagoj Bradač"],"aiRole":null,"verification":"site-confirmed","verificationNote":"Marked solved by erdosproblems.com's official status. Solve credited via Terence Tao's AI-contributions wiki.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":11,"significanceNote":"A numbered Erdos problem that an unusually dense reference trail on erdosproblems.com (8 sources), setting it above the typical entry in the catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/986","sourceName":"erdosproblems.com","collection":null,"collectionVersion":null,"links":[{"label":"Stronger result in the OpenAI math release","url":"https://vibemathed.com/problem/off-diagonal-ramsey-log-exponent","kind":"other"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"erdos-hajnal-high-girth-subgraph","name":"The Erdos-Hajnal High-Girth Subgraph Conjecture","shortName":"High-girth subgraphs","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Erdos and Hajnal asked whether $h_r(G) = \\max\\{\\chi(H) : H \\subseteq G,\\ \\mathrm{girth}(H) \\ge r\\}$ tends to infinity as $\\chi(G)$ does, for every fixed $r \\ge 4$. It does in every fixed polynomial edge-density regime.","posedBy":"Paul Erdos, Andras Hajnal","yearPosed":1966,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"in polynomial edge-density regimes; the general question remains open","claimIssueNote":null,"solveDate":"2026-06-16","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Eric Li"],"aiRole":"The declaration says ChatGPT was used for ideation and formalization during preparation, with the author responsible for the mathematics. Part of the same series of Erdos-problem resolutions in this catalog.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"A long-standing Erdos-Hajnal question tying chromatic number to high-girth subgraphs, one of the classical hard questions about chromatic number.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.17901","sourceName":"arXiv:2606.17901 - The Erdos-Hajnal High-Girth Subgraph Conjecture Holds in the Polynomial Chromatic-Sparsity Regime","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"elekes-ronyai-expander-conjecture","name":"The Near-Quadratic Elekes-Ronyai Expander Conjecture","shortName":"Elekes-Ronyai expander","problemNumber":null,"field":"Additive combinatorics","fieldGroup":"Combinatorics","statement":"The near-quadratic Elekes-Ronyai expander conjecture over $\\mathbb{R}$ predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers, has image with a fixed power saving from quadratic size.","posedBy":"Gyorgy Elekes, Lajos Ronyai","yearPosed":2000,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-15","model":"ChatGPT 5.5 Pro, Rethlas","modelMaker":"OpenAI / Frenzy Math","humanCollaborators":["Jihao Liu"],"aiRole":"The abstract states that the main result was obtained by generative AI, particularly ChatGPT 5.5 Pro and the Rethlas system. The proof builds on a recent OpenAI construction of an infinite tower, so this is an AI result standing on another AI result.","verification":"unreviewed","verificationNote":"No independent check. Single-author arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The Elekes-Ronyai expansion phenomenon is a pillar of modern incidence geometry and sum-product theory, and the near-quadratic strengthening was the standing expectation.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.16738","sourceName":"arXiv:2606.16738 - A counterexample to the near-quadratic Elekes-Ronyai expander conjecture over R","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"matrix-spencer-algebraic-case","name":"The Matrix Spencer Conjecture for C*-Algebra Contractions","shortName":"Matrix Spencer (algebraic)","problemNumber":null,"field":"Discrepancy theory","fieldGroup":"Probability & statistics","statement":"A structured special case of the Matrix Spencer conjecture, reached through the representation theory of finite-dimensional C*-algebras: the conjectured discrepancy bound holds for every family of contractions contained in a suitable algebra.","posedBy":"Nikhil Bansal, Haotian Jiang, Raghu Meka","yearPosed":2022,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"a structured special case, proved the same week as the independent group version","claimIssueNote":null,"solveDate":"2026-06-14","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Emrullah Akbas","Suvrit Sra"],"aiRole":"The statement on LLM use says the model was used extensively throughout the project as an interactive proof assistant, including for expanding on arguments.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":30,"significanceNote":"The Matrix Spencer conjecture is the matrix analogue of Spencer's six standard deviations theorem and one of the best known open problems in discrepancy theory, with a decade of partial results behind it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.16005","sourceName":"arXiv:2606.16005 - An Algebraic Matrix Spencer Theorem","collection":null,"collectionVersion":null,"links":[{"label":"The independent group version, arXiv:2606.12181","url":"https://arxiv.org/abs/2606.12181","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-942","name":"Erdős Problem #942","shortName":"Erdős #942","problemNumber":942,"field":"Number Theory, Powerful Numbers","fieldGroup":"Number theory","statement":"Let $h(n)$ count powerful integers in $[n^2, (n+1)^2)$. What is the extremal order of $h(n)$?","posedBy":null,"yearPosed":1976,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"lower bound improved to ≫ log n/(log log n · log log log n) infinitely often; the extremal order remains open","claimIssueNote":null,"solveDate":"2026-06-14","model":"Claude, Codex, Aristotle","modelMaker":"Anthropic / OpenAI / Harmonic","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/942","sourceName":"erdosproblems.com/942","collection":null,"collectionVersion":null,"links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"brezis-mironescu-circle-minimizers","name":"Brezis-Mironescu Open Problems 23 and 24 on Minimizing Maps to the Circle","shortName":"Circle minimizers","problemNumber":null,"field":"Calculus of variations","fieldGroup":"Analysis","statement":"For $s \\in (1/4,1)$ and any degree, the only $W^{s,1/s}$-minimizers among maps $\\mathbb{S}^1 \\to \\mathbb{S}^1$ are Blaschke products. This resolves Open Problems 23 and 24 of Brezis and Mironescu's book on mappings to the circle, and Brezis's Favorite Open Problem 5.4 in the same range of $s$.","posedBy":"Haim Brezis, Petru Mironescu","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-assisted","resultNote":"in the range s in (1/4,1); Brezis's Problem 5.4 outside that range is untouched","claimIssueNote":null,"solveDate":"2026-06-14","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Dorian Martino","Katarzyna Mazowiecka","Armin Schikorra"],"aiRole":"The LLM usage note says the authors used ChatGPT to assist with conceptualization and computations, with all mathematical validation their own.","verification":"unreviewed","verificationNote":"arXiv preprint; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Numbered open problems from the Brezis-Mironescu monograph plus an entry on Brezis's own favourite-problems list, both recognized reference lists in analysis.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.15713","sourceName":"arXiv:2606.15713 - On minimizing W^{s,1/s}-maps between circles","collection":null,"collectionVersion":null,"links":[{"label":"Brezis's Open Problem 5.6, also settled with AI in July 2026, arXiv:2607.23598","url":"https://arxiv.org/abs/2607.23598","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-326","name":"Erdős Problem #326","shortName":"Erdős #326","problemNumber":326,"field":"Number Theory, Additive Bases","fieldGroup":"Number theory","statement":"Does there exist $A=\\{a_1