{"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":"2026-09-16T22:08:56.082Z","count":719,"problems":[{"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)","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","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":"nontriviality-of-number-restricted-arithmetic-over-subdmq","name":"Nontriviality of number-restricted arithmetic over subDMQ","shortName":"subDMQ arithmetic is trivial","problemNumber":null,"field":"Paraconsistent logic and foundations","fieldGroup":"Logic & foundations","statement":"Weber's programme of paraconsistent mathematics keeps unrestricted comprehension and revises the logic of inference so that contradictions do not make every statement provable. Ripley and Weber's 2026 proposal restricts induction to numbers as part of a strategy for blocking paradox, and Ripley's presentation of it records that the resulting theory is not known to be trivial, the programme working \"without a net: no nontriviality proofs\". Is number-restricted arithmetic over subDMQ nontrivial: does it, together with naive comprehension and induction over the combined language, avoid proving everything?","posedBy":"Ellie Ripley and Zach Weber","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Number-restricted induction over subDMQ derives A ⇒ A ⊗ A for every formula A to which induction applies, without using quantifier splitting. Restricted quantifier splitting follows as a corollary. A second argument obtains contraction from unrestricted quantifier splitting and a separated domain partition. With suitable Curry fixed points, contraction yields triviality.","claimIssueNote":null,"solveDate":"2026-09-10","model":"GPT-6 Astra","modelMaker":"OpenAI","humanCollaborators":["Ryan Simonelli"],"aiRole":"The paper's author line is \"GPT-6 Astra (context 1e44c278f25b)\", dated 10 September 2026, with a footnote: \"Ryan Simonelli initiated and guided the research conversation that produced this note. The argument emerged from repeated unsuccessful attempts, at his prompting, to find an elegant sequent calculus with syntactic cut elimination for Ripley's reconstruction of Weber's mathematics.\" The submitter adds that after those attempts failed, the model instead established that the intended theory is trivial, and produced the proof constructions, the manuscript, the proof certificates and the checking code; Simonelli directed the investigation and assessed the outputs. Discovered rather than co-developed on that record: the model is credited as the author, and the human role was direction and assessment.","verification":"unreviewed","verificationNote":"Filed as Unreviewed rather than the submitted Expert-verified, for the same reason as this submitter's earlier subDL entry, and the trace is stronger this time. The paper's footnote 3 reads: \"I thank Ellie Ripley for confirming that the argument poses a problem for the intended programme, and for feedback on an earlier draft that helped to clarify and shorten this note.\" Ripley is exactly the right person - the proposer of subDMQ and of the restricted-induction strategy, confirming a result against their own proposal - and that footnote records agreement, not merely a correction. But it is the author's report of a private exchange, not the expert's own words a reader can follow to the source, and the Expert-verified rung's worked example is a published statement by the experts themselves. A public statement from Ripley would lift it. The paper supplies finite Hilbert-style proof certificates replayed by a custom Python checker and Isabelle replay scripts that the submitter reports have not been executed; neither was run here. This site checked the surrounding facts, not the derivations: Ripley's slides state the question as open, and the paper's source comparison pins Ripley's formalisation to the commit it examined.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"One step above this submitter's subDL entry at 7. That one settled a relevance property of a logic; this shows a proposed foundation proves everything once number-restricted induction and naive comprehension are combined, which forces a revision of the programme it was meant to support. The question was stated as open by the proposers. The community is the few dozen people working in paraconsistent mathematics, which is what keeps it at the level of a typical numbered Erdos problem rather than above it.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.ryansimonelli.com/uploads/1/3/3/4/133499356/contraction_from_number-restricted_induction_in_subdqm.pdf","sourceName":"Contraction from Number-Restricted Induction in subDMQ","links":[{"label":"Presentation of the Ripley–Weber restricted-quantification proposal","url":"https://negation.rocks/docs/rqsa-slides.pdf","kind":"other"},{"label":"Ripley’s formalization of subDMQ and arithmetic (source version examined)","url":"https://github.com/ellie-ripley/paraconsistent-maths/tree/5d717c32d5b293384655efb811fc837a6a8ae882","kind":"code"}],"relations":[],"submittedBy":"BraveEgret318","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 $j0$ 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)","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":"an-exponential-mortality-bound-for-finite-real-matrix-monoids","name":"Minimum-Rank and Mortality Bounds for Finite Real Matrix Monoids","shortName":"Finite-monoid minimum-rank and mortality bound","problemNumber":null,"field":"Linear algebra","fieldGroup":"Algebra","statement":"Let $n>0$, 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","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":"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","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":"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","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"reciprocal-triple-free-sets-the-finite-plateau-at-732","name":"Reciprocal-triple-free sets: the finite plateau at 732","shortName":"A390395: plateau at 732","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":"Proved f(732)=f(731) unconditionally as a structural finite theorem. The isolated component G={122,183,244,366,732} has exactly the edges {122,183,366}, {183,244,732}, {244,366,732}. Every admissible selection in G has size at most 3; replacement by {122,183,244} preserves admissibility and cardinality or increases it while removing 732. Combining this with the published f(731)=606 gives the candidate new table term f(732)=606. The historical asymptotic problem remains open. A general large-prime recurrence is proved separately in prose, not in Lean; it 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 finite target, computed a witness, identified an isolated five-vertex component, developed a replacement proof, wrote and compiled the Lean formalization, and checked prior work. Separate assistant instances reviewed the arithmetic and formal statement. The human project owner initiated and directed the research. No independent human expert review has yet occurred.","verification":"lean-checked","verificationNote":"Lean 4.33.1 compiles the complete plateau theorem f(732)=f(731), including support, duplicate-free cardinality, and the exact rational-equation bridge. The isolated-gadget lemma uses bv_decide with its native LRAT checker axiom, in addition to propext, Classical.choice and Quot.sound; this is not kernel-only verification. An independent exact Python check validates a 606-element witness and all 738 forbidden triples. The numerical upper baseline f(731)=606 remains external published OEIS data, not a formal theorem here. This is a partial finite result in a public repository, not peer-reviewed. Novelty remains provisional, and an independent human statement/priority audit is requested.","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/0d2885f58c5fe6eee8703cdee83a843a1594a7ea","sourceName":"GitHub repository with Lean proof and exact verifier","links":[{"label":"Passing Lean build, leanchecker, and exact witness check","url":"https://github.com/Sodelin/oldest-conjecture-/actions/runs/34127973633","kind":"lean-proof"}],"relations":[],"submittedBy":"ZestyDingo473","upvotes":0,"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","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":"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","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","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":"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","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":"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","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":"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":"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","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":18,"downvotes":1,"commentCount":4},{"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)","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":"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","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":"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)","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":"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","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-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)","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":"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","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":"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)","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":"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","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":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.","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":"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. No mathematician has read the argument. 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)","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"}],"relations":[],"submittedBy":"BoldPanther302","upvotes":5,"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":"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","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":"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":"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","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":"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","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":"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","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":1,"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","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","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":"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","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","links":[],"relations":[],"submittedBy":"VibeGene","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","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"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","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":"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","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","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":"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","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":"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","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","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":"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","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":"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","links":[],"relations":[],"submittedBy":"VibeGene","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","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":"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)","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":"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","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":1,"downvotes":0,"commentCount":1},{"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","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":"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)","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","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","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","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","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","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":"erdos-problem-74-locally-almost-bipartite-graphs","name":"Erdős Problem #74: locally almost bipartite graphs of infinite chromatic number","shortName":"Erdős #74","problemNumber":74,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Erdős, Hajnal and Szemerédi asked whether, for every function $f(n)\\to\\infty$, however slowly, there exists a graph of infinite chromatic number such that every finite $n$-vertex subgraph can be made bipartite by deleting at most $f(n)$ edges.\n\nThe answer is no: there exists a function $f(n)\\to\\infty$ such that every graph for which every finite $n$-vertex subgraph is within $f(n)$ edge deletions of bipartite has finite chromatic number.","posedBy":"Paul Erdős, András Hajnal, Endre Szemerédi","yearPosed":1982,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Astra proves that there exists a divergent function\n$$\nf:\\mathbb N\\to\\mathbb N,\\qquad f(n)\\to\\infty,\n$$\nsuch that no graph $G$ of infinite chromatic number can satisfy\n$$\nd_{\\mathrm{bip}}(H)\\le f(n)\n$$\nfor every finite $n$-vertex subgraph $H\\subseteq G$, where $d_{\\mathrm{bip}}(H)$ is the minimum number of edges that must be deleted to make $H$ bipartite.\n\nIn fact, every included resolution proves the stronger statement that graphs satisfying the constructed local bound are 3-colorable. The formal challenge advertises only the weaker conclusion that their chromatic number must be finite.","claimIssueNote":null,"solveDate":"2026-08-28","model":"GPT-6 Astra (pre-release)","modelMaker":"OpenAI","humanCollaborators":["Tom Adamczewski"],"aiRole":"GPT-6 Astra autonomously found the disproof and wrote the Lean proofs in the FrontierMath Erdős benchmark, with no human seeing or steering the proof search. Six successful resolutions are included. They construct a sufficiently slowly diverging local bipartization budget and show that any graph satisfying it must actually be 3-colorable, contradicting the required infinite chromatic number. The proofs use several apparently distinct mechanisms, including finite profile exclusion, local defect witnesses, odd-cycle elimination with controlled gluing, and compactness.","verification":"lean-verified","verificationNote":"Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos74 at a626ecc: 11,481 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/74.lean at commit 488aade2 and the theorem is its negation. Six independent resolutions are included; all prove the stronger 3-colourability, while the compared theorem asserts only finite chromatic number. The rate f(n) ~ log n / log log n is not part of the certified theorem. 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":38,"significanceNote":"Erdős, Hajnal and Szemerédi (1982), $500 prize: can a graph of infinite chromatic number have every n-vertex subgraph within f(n) edges of bipartite, for f growing arbitrarily slowly? Rödl had the positive result for f(n) = epsilon n. The answer is no, with an explicit rate log n / log log n reported informally. A well-known question from the infinite-combinatorics side of Erdős's work.","solveCostUsd":218,"solveCostNote":null,"sourceUrl":"https://github.com/tadamcz/erdos74","sourceName":"Github","links":[{"label":"erdosproblems.com/74: status and Thomas Bloom's proof exposition","url":"https://www.erdosproblems.com/74","kind":"problem-record"},{"label":"Challenge.lean: the compared statement","url":"https://github.com/tadamcz/erdos74/blob/main/Challenge.lean","kind":"lean-statement"},{"label":"Solution.lean and the Lean development","url":"https://github.com/tadamcz/erdos74/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":"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","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)","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":"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","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","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"graffiti-conjecture-806","name":"Graffiti Conjecture 806","shortName":"Graffiti 806","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"Let $S$ be the set of square-free integers in $[2, n]$ and $G = PR[S]$ the graph on $S$ in which two integers are adjacent when they are not coprime. From the cases $n \\le 100$ and about twenty further values $n \\le 200$, Graffiti conjectured that the largest adjacency eigenvalue $\\lambda_1(G)$ is at most the number of distinct vertex degrees.\n\nFalse. At $n = 51$ the graph has $31$ vertices, $11$ distinct degrees and $\\lambda_1 > 11.846$; 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)","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":"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","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":1,"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","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.","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","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":"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 $\\mathrm{wt}(a)+\\mathrm{wt}(b)+\\mathrm{wt}(c)1$, 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","links":[{"label":"Problem source","url":"https://arxiv.org/abs/1701.01907","kind":"paper"}],"relations":[],"submittedBy":"RustyKestrel290","upvotes":0,"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","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":2,"downvotes":0,"commentCount":0},{"slug":"nevanlinna-s-half-plane-omitted-values-problem","name":"Nevanlinna’s half-plane omitted-values problem","shortName":"Nevanlinna half-plane problem","problemNumber":null,"field":"Complex analysis","fieldGroup":"Analysis","statement":"We construct a real meromorphic function $F$ on $\\mathbb{C}$ such that $F^{-1}(\\{0,1,\\infty\\})\\subset\\mathbb{R}$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\\in\\widehat{\\mathbb{C}}\\setminus\\{0,1,\\infty\\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction provides an independent negative answer to a question going back to Nevanlinna’s 1925 work that had remained open for over a century.\n\nPostcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.","posedBy":"Rolf Nevanlinna","yearPosed":1925,"ageNote":"The question goes back to Nevanlinna's 1925 work on value distribution in angular domains, which is the date recorded here. Its precise modern form is not his wording, though: the paper notes that Eremenko, Kulikov and Sodin \"recently returned to this question and explicitly recorded it\" as Question 2 of arXiv:2604.06136, and that is the statement answered in the negative.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The paper constructs a real meromorphic function $F$ on $\\mathbb{C}$ satisfying $F^{-1}(\\{0,1,\\infty\\})\\subset\\mathbb{R}$, with each of the three fibers $F^{-1}(0)$, $F^{-1}(1)$, and $F^{-1}(\\infty)$ infinite, such that for every $a\\in\\widehat{\\mathbb{C}}\\setminus\\{0,1,\\infty\\}$, the $a$-point divisor in each of the upper and lower half-planes fails the Blaschke condition. Consequently, $F$ is not of bounded type in either half-plane.\n\nThis gives a negative answer to Nevanlinna’s century-old question asking whether an entire-plane meromorphic function that omits three distinct values in a half-plane must be of bounded type there. By postcomposition with Möbius transformations, the three exceptional values $\\{0,1,\\infty\\}$ may be replaced by any prescribed triple of distinct values in $\\widehat{\\mathbb{C}}$. By affine change of variables, the construction applies to any Euclidean half-plane.","claimIssueNote":null,"solveDate":"2026-08-24","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Quanyu Tang","Bokai Cui","Wei He","Tao Hu","Yanyang Li","Ke Wang","Zijun Yu"],"aiRole":"Disclosed twice, in the abstract and in a dedicated section 1.2 \"Declaration of AI usage\": \"The core mathematical construction and proof underlying this paper were generated during an autonomous run of GPT-5.6 Sol using OpenAI's ultra setting. The run lasted 7 hours, 14 minutes, and 14 seconds. The resulting candidate proof was subsequently subjected to human checking, mathematical auditing, and revision. The initial draft of the manuscript was also generated by AI and was subsequently reviewed and lightly revised by the authors.\" For a disproof the construction is the whole result, and an autonomous run produced it, so this is AI-discovered on the authors' own account.\n\nWorth noting alongside: the independent concurrent counterexample of He and Zhang is also AI-involved, its acknowledgements stating that \"This counterexample was obtained through AI-assisted exploration under the authors' mathematical supervision and guidance\" - though it names no model. Both routes to this century-old problem, within two days of each other, had a model in the loop.","verification":"unreviewed","verificationNote":"An arXiv preprint (v1, 26 August 2026, math.CV), unrefereed, with no formalization and no computational certificate, so there was nothing mechanical to re-run here and no mathematics was checked. What was verified on 27 August 2026: the paper exists at arXiv:2608.26062 with the title and all seven authors this entry lists; the AI declaration is quoted above verbatim from section 1.2; the problem it answers is real and traceable to Eremenko, Kulikov and Sodin's Question 2 (arXiv:2604.06136, confirmed by title and authors); the concurrent work of He and Zhang exists at arXiv:2608.24829 dated 25 August and does contain its own AI-assistance statement; and the timestamped GitHub release of 24 August that the priority claim rests on resolves and is live. So the chronology the paper asserts is independently checkable, which is more than most priority claims offer.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"A question traceable to Nevanlinna's 1925 work, open for a century, and live enough that Eremenko, Kulikov and Sodin restated it explicitly in 2026 - a named problem in classical value-distribution theory with real pedigree rather than a corner case. Two groups answered it independently within two days, itself a mark of the attention it was getting. Placed with the KLS quadratic-forms band at 45: above the specialist complex-geometry entries in the twenties and thirties, below the Carathéodory and Yau-Tian-Donaldson tier at 55-60.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.26062","sourceName":"arXiv","links":[{"label":"ChatGPT prompt","url":"https://chatgpt.com/s/cx_6a8c620c99308191972b12d53e3d314f","kind":"transcript"},{"label":"Independent contemporary result by He and Zhang","url":"https://arxiv.org/abs/2608.24829","kind":"paper"},{"label":"Eremenko, Kulikov, Sodin - where the problem is explicitly recorded (Question 2)","url":"https://arxiv.org/abs/2604.06136","kind":"problem-record"},{"label":"Timestamped candidate-proof archive, 24 Aug 2026 (the priority record)","url":"https://github.com/QuanyuTang/three-omitted-values-bounded-type-candidate-proof/releases/tag/private-archive-2026-08-24-v1","kind":"other"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"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","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","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","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":"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)","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":"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":[],"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","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":"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","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":"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","links":[],"relations":[],"submittedBy":"PluckyWombat559","upvotes":0,"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","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":"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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":0},{"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","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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","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":"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","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","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":"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","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":"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","links":[],"relations":[],"submittedBy":"SilentIbis759","upvotes":0,"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","links":[],"relations":[],"submittedBy":null,"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","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":"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)","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":"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","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":"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","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":"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.","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","links":[{"label":"Balog's announcement thread","url":"https://x.com/matejbalog/status/2089597390369984794","kind":"announcement"}],"relations":[],"submittedBy":null,"upvotes":2,"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","links":[],"relations":[],"submittedBy":"BoldJackal580","upvotes":0,"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","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]","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":"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”","links":[],"relations":[],"submittedBy":"shemshallah","upvotes":0,"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)","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":"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":null,"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","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","links":[],"relations":[],"submittedBy":null,"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","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":"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":null,"ageNote":null,"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","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","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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","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\"","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":"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","links":[],"relations":[],"submittedBy":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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)","links":[],"relations":[],"submittedBy":"PluckyCobra527","upvotes":1,"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","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":"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)","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":"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","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"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.","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","links":[],"relations":[],"submittedBy":"VibeGene","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","links":[],"relations":[],"submittedBy":"LucidManta102","upvotes":2,"downvotes":0,"commentCount":0},{"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","links":[],"relations":[],"submittedBy":null,"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)","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":"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","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":4,"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","links":[],"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","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","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":"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","links":[],"relations":[],"submittedBy":"FrostyVulture999","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","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":"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","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.","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","links":[],"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","links":[],"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)","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":"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","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":"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":null,"ageNote":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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":"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","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","links":[],"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":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":"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","links":[],"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":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":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","links":[],"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","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","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":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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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":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":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","links":[],"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","links":[],"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","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":"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","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","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":"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","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","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","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":1,"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","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[],"relations":[],"submittedBy":null,"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","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","links":[{"label":"X.com","url":"https://x.com/DimitrisPapail/status/2086158118354887060","kind":"discussion"}],"relations":[],"submittedBy":"VibeGene","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","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":"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","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","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":"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","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","links":[],"relations":[],"submittedBy":null,"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","links":[],"relations":[],"submittedBy":null,"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","links":[],"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","links":[],"relations":[],"submittedBy":null,"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","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":"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","links":[],"relations":[],"submittedBy":"VibeGene","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)","links":[],"relations":[],"submittedBy":"LuckyMongoose479","upvotes":1,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[],"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","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":"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","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","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":"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":null,"ageNote":null,"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","links":[{"label":"AxiomProver Lean certificate","url":"https://github.com/AxiomMath/RR_a3","kind":"lean-proof"}],"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","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":"schiffer-conjecture","name":"Schiffer's Conjecture and the Pompeiu Problem","shortName":"Schiffer's Conjecture","problemNumber":null,"field":"Spectral geometry","fieldGroup":"Analysis","statement":"If a smooth bounded domain in $\\mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian that is constant on the boundary, must the domain be a ball? Pompeiu posed an equivalent integral-equation form in 1929; Schiffer's 1957 reformulation via Neumann eigenfunctions is the version on Yau's 1982 list (Problem 80), and Williams proved the two formulations logically equivalent for simply connected domains in 1976. Cao-Labora and de Dios Pont construct infinitely many planar domains with large $N$-fold symmetry that are not balls and admit such an eigenfunction, disproving Schiffer's conjecture; applying Williams' classical reduction to the same domains (their Corollary 1.2) disproves Pompeiu's problem as well.","posedBy":"D. Pompeiu (1929); reformulated via Neumann eigenfunctions by M. M. Schiffer (1957)","yearPosed":1929,"ageNote":"Pompeiu posed the original integral-equation form in 1929; Schiffer's 1957 reformulation via Neumann eigenfunctions is the version that appears as Problem 80 on Yau's 1982 list. Williams proved the two formulations logically equivalent for simply connected domains in 1976.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"Also refutes the 1929 Pompeiu problem: Corollary 1.2 applies Williams' classical 1976 equivalence to the same constructed domains, so this is one construction settling both, not two separate results.","claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.6, Claude Opus 4.8, Claude Fable 5","modelMaker":"OpenAI, Anthropic","humanCollaborators":["Gonzalo Cao-Labora","Jaume de Dios Pont"],"aiRole":"Models with coding harnesses were used in multiple parts of the research: numerically verifying the asymptotic estimates, producing first drafts of the proofs of the Bessel function estimates, and helping with exposition. \n\nThe Lean4 verification of the proof was written by GPT 5.6 from an early draft of the paper. The novel construction strategy is the authors' own.","verification":"lean-verified","verificationNote":"A day-old preprint. The paper states that a Lean4 verification of the proof was written by GPT 5.6, available at https://github.com/jaumededios/Schiffer. It solves the Pompeiu Problem challenge provided by https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/PompeiuProblem.lean","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":53,"significanceNote":"A flagship problem of spectral geometry for seven decades: Problem 80 on Yau's list, equivalent to the 1929 Pompeiu problem, with a partial-results literature running since the 1970s.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05114","sourceName":"arXiv","links":[{"label":"Lean formalization of Theorem 1.1 and Corollary 1.2","url":"https://github.com/jaumededios/Schiffer","kind":"lean-proof"},{"label":"Pompeiu problem Wikipedia article","url":"https://en.wikipedia.org/wiki/Pompeiu_problem","kind":"wikipedia"}],"relations":[],"submittedBy":null,"upvotes":8,"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":null,"yearPosed":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generalized-vanishing-conjecture-five-variables","name":"The Generalized Vanishing Conjecture","shortName":"Generalized Vanishing Conjecture","problemNumber":null,"field":"Commutative algebra; differential operators","fieldGroup":"Algebra","statement":"Zhao's Generalized Vanishing Conjecture asks whether, for a differential operator with constant coefficients, $\\Lambda^m(P^m) = 0$ for all large $m$ forces $\\Lambda^m(P^m Q) = 0$ for all large $m$.","posedBy":"Wenhua Zhao","yearPosed":2007,"ageNote":"Posed by Zhao in 2007, in the circle of ideas around the Mathieu and Jacobian conjectures. The construction here is motivated by the recent counterexample to the Mathieu conjecture for SU(2).","solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":"For n <= 2, we proof that the Generalized Vanishing Conjecture holds, and we have a failure at dimension 3 (including a counter-example).\n\nThis improves the original failure at dimension 5.","claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Alexander Dvorsky","Roy van Rijn"],"aiRole":"The initial paper (Alexander Dvorsky) used \"as an interactive research assistant\": the paper credits it with testing the homogenized five-variable construction, generating computational verification code and locating relevant literature, with all computations and references independently verified afterwards.\n\nA second paper (Roy van Rijn) used OpenAI's Codex to close the conjecture completely. It proofs that the conjecture holds for n <= 2 and fails at dimension 3 (counter-example provided in the paper). Codex monitored/ran several experiments and the human steered the experiments.","verification":"lean-verified","verificationNote":"The preprints have (as of yet) no independent review. The counterexample can be checked directly, the proof has additional Python calculations and Lean verification.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A named 2007 conjecture in the Mathieu and Jacobian conjecture circle, where a positive answer had been pursued as a route to the Jacobian conjecture itself.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://zenodo.org/records/21803325","sourceName":"The Generalized Vanishing Conjecture: The Two-Variable Theorem and the First Failing Dimension","links":[{"label":"An Explicit Five-Variable Counterexample to the Generalized Vanishing Conjecture","url":"https://arxiv.org/abs/2608.07338","kind":"paper"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":1},{"slug":"mais-o60-fourier-alignment","name":"Problem MAIS-O60: Single-Neuron Fourier Alignment","shortName":"Fourier alignment (MAIS-O60)","problemNumber":null,"field":"Neural network training dynamics","fieldGroup":"Algorithms & optimization","statement":"Does a single ReLU neuron trained on modular addition align to one Fourier frequency? Open Problem MAIS-O60, itself posed by Claude Fable 5 under the direction of Lionel Levine, is answered negatively: an explicit construction reaches a frozen state whose Fourier energy is spread equally across all nonzero frequency classes, on an open set of initial conditions.","posedBy":"Claude Fable 5 (directed by Lionel Levine)","yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Gautam Neelakantan Memana"],"aiRole":"The main counterexample is the author's, with GPT-5.6 Sol assisting the literature review and problem clarification. The strengthened results in Appendix A were developed through conversations with GPT-5.6 Sol, which also produced the appendix's initial draft. A rare full loop: the problem was posed by one model and its strengthening drafted by another.","verification":"unreviewed","verificationNote":null,"publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A machine-posed 2026 problem from a curated open-problems list; real and documented, with no accumulated literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.04451","sourceName":"arXiv","links":[],"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","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","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","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":"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= 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","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":"multivariate-independence-polynomial-bounds","name":"Lower Bounds for Multivariate Independence Polynomials","shortName":"Multivariate independence polynomials","problemNumber":null,"field":"Extremal combinatorics; statistical physics","fieldGroup":"Combinatorics","statement":"The multivariate independence polynomial is the partition function of the hard-core model with per-vertex fugacities. The paper proves a lower bound extending to the multivariate setting a result Tao proved in the univariate case, and settles a conjectured generalization for a multiaffine version of the semiproper colouring partition function with two proper colours.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-05","model":"Aletheia (Gemini Deep Think), Harmonic Aristotle","modelMaker":"Google DeepMind, Harmonic","humanCollaborators":["Joonkyung Lee","Jaehyeon Seo"],"aiRole":"\"The key steps in both proofs were obtained, at least in part, by using Aletheia, a mathematics research agent built upon Gemini Deep Think at Google DeepMind.\" The paper marks where the agent contributed, publishes raw prompts and outputs in a repository, and calls the work a benchmark demonstrating that current models can in part assist with mathematical research. Theorem 1.4 was separately formalized in Lean 4 with Harmonic Aristotle.","verification":"lean-checked","verificationNote":"Theorem 1.4 is formalized in Lean 4 using Harmonic Aristotle, with the files public; the rest of the paper is not formalized, and nobody independent has audited the informal-to-formal correspondence.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"Extends a result of Tao to the multivariate hard-core model, a documented question at the combinatorics and statistical physics boundary, specialist in reach.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2602.02450","sourceName":"arXiv","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"hrt-conjecture","name":"HRT Conjecture","shortName":"HRT Conjecture","problemNumber":null,"field":"Harmonic Analysis","fieldGroup":"Analysis","statement":"Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent time-frequency shifts.","posedBy":"Christopher Heil, Jayakumar Ramanathan, Pankaj Topiwala","yearPosed":1996,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-05","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Markus Faulhuber","Philipp Petersen","Jordy Timo van Velthoven","Felix Voigtlaender"],"aiRole":"ChatGPT (GPT-5.6 Pro) developed the counterexample and its proof strategy in dialogue with the authors. They initiated the search, told it to enlarge the phase-space configuration when it stalled, steered it from abstract methods toward elementary ones, and had it turn an existential construction into a fixed finite one. The authors then replaced several of the model's arguments with elementary proofs and closed gaps they call occasionally substantial: the whole of Section 6.2 and Appendix A was originally two sentences.","verification":"unreviewed","verificationNote":"A preprint one day old with no external confirmation. The authors certify their numerical estimates with interval arithmetic, but that is their own rigor rather than third-party review.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":33,"significanceNote":"The central open problem of time-frequency and Gabor analysis for thirty years, with a continuous partial-results literature, but little known outside harmonic analysis.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.05044","sourceName":"arXiv","links":[{"label":"Lean-certified four-point partial results (April 2026, pre-disproof)","url":"https://arxiv.org/abs/2604.21228","kind":"paper"}],"relations":[],"submittedBy":"QuietMarten105","upvotes":3,"downvotes":0,"commentCount":0},{"slug":"asymptotically-attaining-the-moore-bound","name":"Asymptotically attaining the Moore bound","shortName":"Degree-Diameter Problem","problemNumber":null,"field":null,"fieldGroup":"Combinatorics","statement":"For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. It is shown that\n$$\\lim_{d \\to \\infty}\\frac{n_k(d)}{d^k} = 1$$\nfor every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter. \n\nAlso proved a similar lower bound on the edge-variant of the problem, and a tight asymptotic for the bipartite variant of the edge problem.","posedBy":"Béla Bollobás","yearPosed":1978,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.","claimIssueNote":null,"solveDate":"2026-08-04","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Wouter Cames van Batenburg","Samuel Korsky"],"aiRole":"The paper's tool disclosure states that GPT-5.6 Pro, used in exploratory brainstorming directed by the authors, suggested splitting complete flags into their odd- and even-rank subflags. That suggestion arose in connection with the edge problem but became the halved-flag construction carrying Theorem 1.1 itself, the graph being named for it. The authors developed it, formulated and verified every argument, and take full responsibility. Generative AI also assisted the Lean 4 formalization.","verification":"lean-verified","verificationNote":"Lean 4 formalization at github.com/woutercvb/wewantmoore, checked by the site on 2026-08-06 at commit 32beb227. `DegreeDiameter.theorem_1_1` states Theorem 1.1 itself, as a limit of nKD k d / d^k, and `corollary_1_2` states Corollary 1.2; neither is a weakened lemma, and a second independent route is proved alongside each under `_via_big_cell`. No sorry or admit appears in the sources, and the committed axiom audit shows both final theorems resting only on propext, Classical.choice and Quot.sound. That audit is not taken on trust: the repository's CI builds the project from the pinned toolchain and manifest, regenerates the axiom and dependency reports, and fails if they differ from the committed ones. It passes on this commit. The formalization was itself AI-assisted, per the paper's disclosure, and the authors note it is not a line-by-line transcription: k = 1 is handled by the same construction rather than by complete graphs, and the order and cap asymptotics go through leading terms rather than the displayed O_k(q^-1) estimates.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":6,"renownNote":null,"significance":38,"significanceNote":"The degree-diameter problem carries its own Electronic Journal of Combinatorics dynamic survey (DS14, Miller and Siran), and the asymptotic form is Bollobas's own conjecture, recorded in Extremal Graph Theory and Random Graphs. It stood 48 years with the asymptotic known only for k in {2, 3, 5}, through generalized polygons, and the best uniform coefficient for large k was 0.629.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.03965","sourceName":"Asymptotically attaining the Moore bound","links":[{"label":"Asymptotically attaining the Moore bound — Lean 4 formalization","url":"https://github.com/woutercvb/wewantmoore","kind":"code"},{"label":"Degree diameter problem","url":"https://en.wikipedia.org/wiki/Degree_diameter_problem","kind":"wikipedia"}],"relations":[],"submittedBy":"GoldenMongoose827","upvotes":2,"downvotes":0,"commentCount":1},{"slug":"period-index-conjecture","name":"The Period-Index Conjecture","shortName":"Period-index","problemNumber":null,"field":"Algebraic geometry","fieldGroup":"Algebra","statement":"For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field $k$ of characteristic $0$ and any $d \\geq 3$ there is a $d$-dimensional variety over $k$ carrying a Brauer class that violates it, for Hodge-theoretic reasons. For $d = 3$ the construction needs no uncountability, so the conjecture fails already over $\\overline{\\mathbf{Q}}$.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-04","model":"ChatGPT","modelMaker":"OpenAI","humanCollaborators":["Alexander Perry"],"aiRole":"The paper's AI disclosure is unusually specific about a partial success. Prompted to find a counterexample by the Hodge-theoretic strategy, ChatGPT produced an example that was flawed, but whose shape survived into the final solution: a quotient of the same form, with an abelian surface, a genus two curve and the group Z/4. The author refined that into the working construction.","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":30,"significanceNote":"A named conjecture in the arithmetic of Brauer groups, standing in the literature on period-index bounds over function fields and over algebraically closed fields, and a reference point for Hodge-theoretic obstructions.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.03684","sourceName":"arXiv:2608.03684 - The period-index conjecture is false","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"finite-generation-for-klt-generalized-pairs","name":"Finite Generation for klt Generalized Pairs","shortName":"klt generalized pairs","problemNumber":null,"field":"Birational geometry","fieldGroup":"Algebra","statement":"The BCHM theorem makes the log canonical ring of a projective klt pair finitely generated. Generalized pairs, introduced by Birkar and Zhang, add an auxiliary nef part and have become a central tool in birational geometry, so it is natural to ask whether finite generation survives. It does not: Liu and Wang construct a smooth projective klt generalized pair whose generalized log canonical ring is not a finitely generated algebra.","posedBy":"Yoshinori Gongyo","yearPosed":2023,"ageNote":"Traced to a research summary in Gongyo's 2023 annual report. Gongyo and Takayama had since proved finite generation for klt generalized pairs in many special cases, which sharpened the general question rather than settling it.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"The paper records how stuck this was: the first author had discussed the question with Caucher Birkar, Osamu Fujino, Christopher D. Hacon, Junpeng Jiao, Vladimir Lazic and Lingyao Xie, and writes that despite a general feeling that a negative answer was likely, no precise counterexample could be found. The authors also note that, given the limitations of generative AI, they may have missed related literature and welcome corrections.","claimIssueNote":null,"solveDate":"2026-08-04","model":"GPT-5.6-sol-ultra, Fable 5, Danus","modelMaker":"OpenAI, Anthropic, Rethlas","humanCollaborators":["Jihao Liu","Yanze Wang"],"aiRole":"Stated in the abstract, which is unusual: \"The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol-ultra, Fable 5, and the Danus system.\" Remark 1.5 adds that Danus is a specialised agent built on the Rethlas system, and that human verification and polishing were done afterwards.","verification":"unreviewed","verificationNote":"A preprint days old, with no independent review. The claim is an explicit construction rather than an existence argument, so it is checkable by anyone who works in the area.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":22,"significanceNote":"Asks whether the BCHM finite generation theorem, one of the landmark results of modern birational geometry, extends to generalized pairs. Only three years old, which caps it, but leading figures in the field had tried it without success.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.03258","sourceName":"arXiv","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"perfectly-complete-quantum-key-agreement-owf","name":"Perfectly Complete Quantum Key Agreement from One-Way Functions","shortName":"Perfect QKA from OWFs","problemNumber":null,"field":"Quantum cryptography","fieldGroup":"Quantum information & computing","statement":"Whether perfectly complete quantum key agreement can be built from quantumly secure one-way functions in a black-box way. It cannot: for any protocol where Alice and Bob exchange only classical messages, make at most $q_A$ and $q_B$ quantum queries to a Boolean random oracle and agree on a key with certainty, an eavesdropper given the classical messages recovers the key with certainty in $O((q_A + q_B)^5)$ classical oracle queries, independent of the number of rounds and the transcript length.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-04","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Longcheng Li","Qian Li","Xingjian Li","Qipeng Liu"],"aiRole":"The acknowledgment states that GPT-5.6 Sol Ultra discovered the proof in a one-shot conversation and produced a preliminary draft, with the authors independently verifying every statement and proof, then simplifying and refining the argument.","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 black-box separation in the quantum random oracle model, in the line of Impagliazzo-Rudich style impossibility results for key agreement from one-way functions.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.03824","sourceName":"arXiv:2608.03824 - Impossibility of Perfectly Complete Many-Round Key Agreement in the QROM","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"averkov-hofscheier-nill-h-star-real-rootedness","name":"Real-Rootedness of Ehrhart h*-Polynomials at Large Width","shortName":"h* real-rootedness","problemNumber":null,"field":"Polyhedral combinatorics","fieldGroup":"Combinatorics","statement":"A question of Averkov, Hofscheier and Nill on whether the Ehrhart $h^*$-polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and unimodality of the $h^*$-vector, with the analogous statement for the local $h^*$-polynomial of a lattice simplex.","posedBy":"Gennadiy Averkov, Johannes Hofscheier, Benjamin Nill","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-04","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Benjamin Nill"],"aiRole":"The acknowledgments state that the proofs were found using ChatGPT 5.6 Sol, which also produced a first draft of the paper, with the author solely responsible for the final version.","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 stated question in Ehrhart theory, where unimodality of the h*-vector has been a recurring target; the answer follows from a result of Basu and Oertel once the right reduction is seen.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.03635","sourceName":"arXiv:2608.03635 - Lattice polytopes of large width have real-rooted Ehrhart h*-polynomials","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"signed-depth-relevance-of-subdl","name":"Signed Depth Relevance of subDL","shortName":"Signed Depth Relevance of subDL","problemNumber":null,"field":"Paraconsistent logic and foundations","fieldGroup":"Logic & foundations","statement":"subDL is a logic developed for paraconsistent mathematics by Zach Weber (2021), combining elements of relevant logic and affine logic. Tore Øgaard (2026) shows that subDL satisfies two important relevance properties: the signed variable-sharing property, which requires premises and conclusions of a valid inference to share a propositional variable with the appropriate polarity, and the depth-relevance property, which requires such a shared variable to occur at matching implicational depths. He leaves open whether subDL satisfies the stronger signed depth-relevance property, which combines these two constraints by requiring a variable to occur with both the appropriate sign and the appropriate implicational depth.\n\nThe result proved here answers Øgaard’s question affirmatively: subDL satisfies the signed depth-relevance property. The proof proceeds by constructing, from any counterexample to signed depth relevance, an interpretation for subDL under which the premises receive designated values while the conclusion does not, contradicting validity. The construction can also be viewed as a simplification of Brady’s ω-rule technique for establishing relevance properties.","posedBy":"Tore Øgaard","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"subDL satisfies the signed depth relevance property, answering an open question posed by Øgaard (2026). More precisely, every valid inference in subDL contains a propositional variable that occurs in both the premises and conclusion with matching sign and at matching implicational depth.","claimIssueNote":null,"solveDate":"2026-08-04","model":"ChatGPT 5.6-Sol","modelMaker":"OpenAI","humanCollaborators":["Ryan Simonelli"],"aiRole":"The strongest AI attribution in this catalog, and it is not a disclosure statement but a byline: the paper's author line reads \"GPT-5.6 Sol\", dated 22 August 2026, with a single footnote - \"Initially prompted by Ryan Simonelli.\" The model is credited as the author of the paper, not thanked in an acknowledgment. The submitter, who is Simonelli, describes his own role as having prompted it and summarises the model's as having \"entirely constructed the proof\", which the byline corroborates rather than merely asserts. Øgaard is thanked separately for correcting a notational error in an earlier draft and for observing the connection to Brady's $\\omega$-rule.","verification":"unreviewed","verificationNote":"Filed as Unreviewed rather than the submitted Expert-verified, and the distinction is narrow enough to spell out. The paper's acknowledgments thank Tore Fjetland Øgaard - who posed the question, and so is both a named domain expert and a person with no stake in this proof - \"for identifying the notational error concerning $\\Rightarrow_m$ and $\\to$ in an earlier draft and for pointing out the relation between the terminal clause and Brady's $\\omega$-rule\". That is genuine engagement by the right person: he read it closely enough to catch an error. It is not an endorsement of the final proof, and it is the only publicly checkable trace of his involvement. The submission states that he has verified the proof correct; that may well be so, but it rests on a private communication a reader cannot follow, and the Expert-verified rung on this site requires a checkable endorsement (its worked example is a published essay stating outright that named experts checked a proof and believe it correct). This site checked the surrounding facts, not the mathematics: Øgaard's paper exists as cited and leaves exactly this question open, and the proof itself - five pages over the Anderson-Belnap matrix $M_0$ - was read but not audited.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":7,"significanceNote":"A question left open months earlier in a single 2026 paper (Øgaard, JoLLI, March 2026), inside a small but real research programme - Weber's paraconsistent mathematics, with a lineage running back through Brady, Logan and Anderson-Belnap. Sits with the 2026-posed cluster in this catalog: above the 4-5 band of one-off questions, level with Nathanson's product intersection problems and the dynamical-sampling conjecture at 8, below the Kourovka list problems at 13, which carry a named collection's standing that this does not.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.ryansimonelli.com/uploads/1/3/3/4/133499356/signed_depth_relevance_for_subdl.pdf","sourceName":"Signed Depth Relevance for subDL","links":[{"label":"Øgaard, subDL is Relevant (JoLLI, 2026) - where the question is posed","url":"https://doi.org/10.1007/s10849-026-09462-2","kind":"problem-record"}],"relations":[],"submittedBy":"BraveEgret318","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"exact-two-scenario-ssuf-bound-on-the-four-terminal-planar-gadget","name":"Exact SSUF scenario-count ladder on the four-terminal planar gadget","shortName":"Four-terminal SSUF scenario ladder","problemNumber":null,"field":"Combinatorial optimization; unsplittable flow","fieldGroup":"Algorithms & optimization","statement":"For the released four-terminal planar acyclic single-source unsplittable-flow gadget, require one unsplittable routing to be no more expensive than a prescribed fractional routing under each of $m$ strictly positive full-demand route-cost-difference scenarios. What is the worst-case normalized additive upper-arc deviation? The v0.3.0 fixed-gadget scenario-count ladder proves\n\n$$\\beta_G^{(m,+)}=\n\\begin{cases}\n\\dfrac{299-41\\sqrt{41}}{32},&m=1,\\\\\n\\dfrac{17}{8},&m=2,\\\\\n3,&m=3,\\\\\n4,&m\\ge4.\n\\end{cases}$$\n\nThe $m\\ge2$ values are non-attained suprema; no attainment or non-attainment assertion is made for $m=1$. The one-scenario value also holds for legally realizable signed and zero route-cost differences; no such multi-scenario extension is claimed.","posedBy":"Matthew Protti","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"variant","aiContribution":"ai-co-developed","resultNote":"Fixed four-terminal planar DAG; positive route-cost differences for m≥2, with non-attained suprema; signed/zero only for m=1, whose attainment is unstated. No topology-wide/unrestricted-planar claim.","claimIssueNote":null,"solveDate":"2026-08-04","model":"GPT-5.6 Pro; Codex","modelMaker":"OpenAI","humanCollaborators":["Matthew Protti"],"aiRole":"GPT-5.6 Pro generated and developed substantial portions of the construction and parameter searches, symbolic derivations, analytic proofs, exact verifiers, adversarial analyses, and manuscript drafts. Codex sessions implemented or reconstructed code paths, performed repository-level checks, and attacked proposed claims. Matthew Protti selected and framed the targets, required exact arithmetic, exhaustive finite checks, mutation tests and hostile review, identified and forced correction of material errors and overclaims, chose the final theorem scope and public wording, reviewed the work, and authorized the release. Role-separated model review reduced some shared-context risk but was not independent human verification.","verification":"unreviewed","verificationNote":"The unrefereed v0.3.0 release contains complete analytic proof sources, exact certificates, deterministic replay, claim ledgers, and review evidence for the fixed-gadget ladder. The $m=2$ rung was first released on 28 July 2026 in immutable v0.2.1; Matthew Protti's no-error review covered that corpus only. v0.3.0 adds the exact $m=1$, $m=3$, and $m\\ge4$ branches while preserving the earlier source. Exact-arithmetic checks, exhaustive finite enumeration, mutation tests, claim-scoped role-separated AI reviews, a blind independent reconstruction of the fixed finite atlas, and one scope-limited human-directed AI-assisted adversarial report are included. The latter did not inspect every theorem component. No Lean proof, full independent expert verification, or journal peer review is documented, so the catalog status remains Unreviewed. The analytic proofs are authoritative; computation is corroborative.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":4,"significanceNote":"A sharp constant inside one gadget construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/releases/tag/v0.3.0","sourceName":"v0.3.0 fixed-gadget scenario-cover release","links":[{"label":"Fixed-gadget scenario-cover synopsis (v0.3.0)","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/releases/download/v0.3.0/ssuf_fixed_gadget_scenario_cover_synopsis.pdf","kind":"paper"},{"label":"Exact verification suite (v0.3.0)","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/blob/v0.3.0/scripts/verify_all.py","kind":"code"},{"label":"AI use and research provenance (v0.3.0)","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/blob/v0.3.0/AI_USE_AND_PROVENANCE.md","kind":"transcript"},{"label":"Related VibeMathed entry: four-terminal planar DGG counterexample","url":"https://vibemathed.com/problem/planar-four-terminal-dgg","kind":"problem-record"},{"label":"Original two-scenario release (28 July 2026)","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/releases/tag/v0.2.1","kind":"other"},{"label":"Complete proof corpus and claim ledger (v0.3.0)","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/tree/v0.3.0/research/fixed_gadget_scenario_cover","kind":"code"}],"relations":[{"to":"dinitz-garg-goemans-unsplittable-flow","kind":"related","note":"A scenario-count ladder on one fixed four-terminal planar gadget, in the same cost-preserving unsplittable-flow setting the disproof opened up."}],"submittedBy":"Matthew Protti","upvotes":1,"downvotes":0,"commentCount":0},{"slug":"chip-firing-middle-stair","name":"The Middle Stair of Parallel Chip-Firing","shortName":"Chip-firing middle stair","problemNumber":null,"field":"Combinatorial dynamics","fieldGroup":"Combinatorics","statement":"Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between $2|E|-|V|$ and $2|E|$ has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.","posedBy":"David Ji, Michael Li, Daniel Wang","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-04","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Daniel Wang","Nathan Lannan"],"aiRole":"The paper presents the proof plainly as found by the model: \"The proof presented in Section 3 was found by the large language model GPT-5.6-Sol. The authors verified the resulting argument and take full responsibility.\"","verification":"expert-verified","verificationNote":"Beyond the authors' own verification, the acknowledgments thank David Ji and Michael Li, two of the conjecture's posers, for assisting with reviewing the proof.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":12,"significanceNote":"A documented 2024 conjecture unifying thirty years of partial results on the devil's staircase, known within the chip-firing community.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.04153","sourceName":"arXiv","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"uniform-szekelyhidi-conjectures-for-complex-hessian-equations-on-projective-mani","name":"Uniform Székelyhidi conjectures for complex Hessian equations on projective manifolds","shortName":"Uniform Székelyhidi conjectures","problemNumber":null,"field":"Kähler geometry","fieldGroup":"Geometry & topology","statement":"We prove a Nakai-Moishezon-type criterion for complex Hessian-type equations on projective manifolds whose associated degree-n polynomials are strongly strictly right-Noetherian. For strictly right-Noetherian polynomials of arbitrary degree, we prove a uniform Nakai-Moishezon-type criterion. This class includes the complex Hessian and Hessian quotient equations.","posedBy":"Gábor Székelyhidi; Ryosuke Murakami","yearPosed":2018,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Chen, Nie and Xu prove a Nakai–Moishezon-type numerical criterion for a broad class of complex Hessian-type equations on compact projective manifolds. In particular, Corollary 1.3 gives a uniform version of Székelyhidi’s conjecture for complex Hessian quotient equations, while Corollary 1.5 proves the uniform version, formulated by Murakami, for complex k-Hessian equations. However, the results assume projectivity and a uniform numerical condition, whereas Székelyhidi’s original conjecture is formulated for arbitrary compact Kähler manifolds. Thus this should not be recorded as a full solution of the unrestricted original conjecture.","claimIssueNote":null,"solveDate":"2026-08-04","model":"ChatGPT 5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Gao Chen","Sijie Nie","Yulun Xu"],"aiRole":"The authors state that the proofs of the cone-inclusion lemmas in Section 3 are revised versions of arguments generated by ChatGPT 5.6 Sol. The authors subsequently checked and edited those arguments for mathematical clarity. ChatGPT 5.6 Sol also identified gaps in an earlier version of the manuscript and assisted with grammar correction. The Section 3 cone-inclusion lemmas are then used in Section 4 to prove the uniform Székelyhidi-type results.","verification":"unreviewed","verificationNote":"Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.03815, Chen-Nie-Xu). The declaration is verbatim as this entry describes: the Section 3 cone-inclusion lemma proofs are revised versions of arguments generated by ChatGPT 5.6 Sol, checked and edited by the authors, and the model was also used to find gaps in an earlier version - the v2 comment records that a gap in v1 (omega not a priori Kahler) was fixed, which is worth knowing when weighing a fresh preprint. The mathematics was not checked here. No independent review.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"Szekelyhidi's Nakai-Moishezon-type criteria for complex Hessian equations are a standing target in Kahler geometry with a real partial-results literature. This settles a broad class (strictly right-Noetherian polynomials) on projective manifolds - substantial, specialist, partial.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.03815","sourceName":"arXiv","links":[],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"bandelt-dress-quartet-distance","name":"The Bandelt-Dress Quartet Distance Conjecture","shortName":"Quartet distance","problemNumber":null,"field":"Combinatorics","fieldGroup":"Combinatorics","statement":"The quartet distance counts the four-leaf subsets on which two binary phylogenetic trees display different topologies. Bandelt and Dress conjectured the maximum over trees on $n$ leaves. Proved: it is $(2/3 + o(1))\\binom{n}{4}$, by reducing arbitrary pairs of trees to caterpillars through a common-root planarization and an identity on five-leaf trees.","posedBy":"Hans-Jurgen Bandelt, Andreas Dress","yearPosed":1986,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-04","model":"GPT-5.6","modelMaker":"OpenAI","humanCollaborators":["Lior Pachter"],"aiRole":"The paper states plainly that the author used GPT-5.6 to prove the theorem and to draft an initial version of the manuscript.","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 1986 conjecture calibrating the scale of the quartet distance, which is a standard metric for comparing phylogenetic trees in computational biology.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.03542","sourceName":"arXiv:2608.03542 - The maximum quartet distance between phylogenetic trees","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"wowii-graph-conjecture-144","name":"Written on the Wall II, Graph Conjecture 144","shortName":"WOWII 144","problemNumber":null,"field":"Graph Theory (automated conjecture)","fieldGroup":"Combinatorics","statement":"For every finite connected simple graph $G$, is the order of the largest induced tree at least $\\mathrm{girth}(G) - 1 + \\mathrm{ecc}(G, \\mathrm{center}(G))$, where the last term is the eccentricity of the centre set? Answered affirmatively, with a Lean proof.","posedBy":"Written on the Wall II (automated conjecturing)","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"The Formal Conjectures pull request flipping this from open to solved is still open rather than merged, so the canonical repository has not yet accepted it.","claimIssueNote":null,"solveDate":"2026-08-03","model":"ChatGPT + Codex","modelMaker":"OpenAI","humanCollaborators":["Chris Maki"],"aiRole":"The author states that ChatGPT and Codex assisted with computational exploration, proof analysis, Lean API discovery and proof engineering, and that he reviewed the work thoroughly and takes full responsibility for it.","verification":"lean-verified","verificationNote":"Checked here on 2026-08-03, statically rather than by rebuilding. The theorem statement was diffed against the upstream Formal Conjectures statement and is identical apart from a hypothesis binder name, which is the fidelity check that matters. All 16 Lean files at the pinned commit (5,873 lines) contain no sorry, no admit, no axiom declarations and no native_decide. The author reports lake build --wfail and axiom checks passing; that build was NOT reproduced here.","publication":"announcement","resolutionMethod":"argument","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 however clean the proof.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/google-deepmind/formal-conjectures/pull/4696","sourceName":"Formal Conjectures PR #4696","links":[{"label":"Lean proof at the pinned commit","url":"https://github.com/beowulf127/wowii144-lean/blob/046429d509b28c90ee2ec38ae27c1ad377c6a5fc/WOWII144/Main.lean","kind":"lean-proof"},{"label":"Upstream statement (Formal Conjectures)","url":"https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/WrittenOnTheWallII/GraphConjecture144.lean","kind":"lean-statement"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"axiotis-sviridenko-condition-number-barrier","name":"The Axiotis-Sviridenko Condition-Number Conjecture","shortName":"Condition-number barrier","problemNumber":null,"field":"Approximation algorithms","fieldGroup":"Algorithms & optimization","statement":"Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. Their conjectured lower bound is established for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer and Tulsiani.","posedBy":"Kyriakos Axiotis, Maxim Sviridenko","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.","claimIssueNote":null,"solveDate":"2026-08-03","model":"Gemini-based agentic system (internal)","modelMaker":"Google","humanCollaborators":["Honghao Lin","Vahab Mirrokni","David P. Woodruff"],"aiRole":"The acknowledgements state that the proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google, with the authors verifying it and editing for presentation.","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 2021 conjecture on the price of sparsity in least squares, sitting in the hardness-of-approximation literature that grew from the Small-Set Expansion Hypothesis.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.02588","sourceName":"arXiv:2608.02588 - The Condition-Number Barrier in Sparse Least Squares","links":[],"relations":[],"submittedBy":null,"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","links":[],"relations":[],"submittedBy":"October","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"denjoy-theorem-sharpness","name":"Sharpness of Denjoy's Theorem","shortName":"Denjoy sharpness","problemNumber":null,"field":"Dynamical systems","fieldGroup":"Analysis","statement":"Denjoy's 1932 theorem says a $C^{1+\\mathrm{bv}}$ circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity $\\omega$ weaker than Lipschitz, there is a $C^{1+\\omega}$ circle diffeomorphism with irrational rotation number and a wandering interval. The case $\\omega(t) = t\\log(1/t)$ settles an open problem going back to Herman's 1979 work, which had constructions only for $\\omega(t) = t\\log(1/t)^{1+\\varepsilon}$.","posedBy":"Michael Herman","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-03","model":"GPT-5.6 Sol Ultra; Claude Fable 5","modelMaker":null,"humanCollaborators":["Rohil Prasad"],"aiRole":"The AI use section says the author prompted GPT-5.6 Sol Ultra to construct a Denjoy example for the modulus $t\\log(1/t)$, which is the corollary settling Herman's case, and used Claude Fable 5 to search for errors.","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":"Closes the regularity gap left by Herman's 1979 constructions, which had stood as the boundary of Denjoy's theorem for over forty years.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.02380","sourceName":"arXiv:2608.02380 - On the sharpness of Denjoy's theorem","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"mihail-vazirani-conjecture","name":"The Mihail-Vazirani Conjecture","shortName":"Mihail-Vazirani","problemNumber":null,"field":"Polyhedral combinatorics","fieldGroup":"Combinatorics","statement":"Mihail and Vazirani conjectured that the graph of every $0/1$-polytope has edge expansion at least one. Disproved by a family of $0/1$-polytopes whose edge expansion decreases exponentially in the dimension.","posedBy":"Milena Mihail, Umesh Vazirani","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-03","model":"GPT-5.6 Sol","modelMaker":"OpenAI","humanCollaborators":["Xiongxin Yang"],"aiRole":"The AI use section states that the construction and its analysis were generated by GPT-5.6 Sol in a one-shot manner, from a prompt adapted from OpenAI's cycle double cover prompt by replacing the problem statement. The author verified the result independently.","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":30,"significanceNote":"A named conjecture underpinning rapid-mixing arguments for random walks on 0/1-polytopes, cited across approximate counting and sampling for three decades.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.01870","sourceName":"arXiv:2608.01870 - 0/1-Polytopes with Exponentially Small Edge Expansion","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"simonovits-product-conjecture","name":"The Simonovits Product Conjecture","shortName":"Product conjecture","problemNumber":null,"field":"Extremal graph theory","fieldGroup":"Combinatorics","statement":"Simonovits conjectured that if a forbidden family $\\mathcal{F}$ with $p(\\mathcal{F}) > 1$ 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 Furedi and Simonovits.","posedBy":"Miklos Simonovits","yearPosed":null,"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 no date is claimed here.","solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"one construction disproves both the product conjecture and its weak form","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.","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":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","links":[],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"connes-rigidity-icc-property-t","name":"Connes' Rigidity Conjecture for ICC Property (T) Groups","shortName":"Connes rigidity (ICC, T)","problemNumber":null,"field":"Operator algebras","fieldGroup":"Algebra","statement":"Connes' rigidity conjecture asks whether an ICC group with Kazhdan's property (T) is determined by its group von Neumann algebra. Disproved for this class: two explicit countable discrete groups $\\Gamma_1$ and $\\Gamma_2$, both ICC and property (T), are non-isomorphic as groups while $L(\\Gamma_1) \\cong L(\\Gamma_2)$.","posedBy":"Alain Connes","yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"Settles the ICC property (T) case. The paper records that the result was obtained independently of and concurrently with work by OpenAI.","claimIssueNote":null,"solveDate":"2026-08-03","model":"GPT-5.6 Sol; Codex; Danus","modelMaker":"OpenAI","humanCollaborators":["Shuoxing Zhou"],"aiRole":"The AI use statement says the construction underlying the main result was found mainly by GPT-5.6 Sol, Codex and the Danus multi-agent research system, under the author's mathematical guidance, with Lean 4.32.1 used alongside.","verification":"unreviewed","verificationNote":"arXiv preprint, not yet peer-reviewed. The paper reports Lean 4.32.1 used during the work; the site has not checked that development.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Connes' rigidity conjecture is among the best-known open problems in von Neumann algebras, and W*-superrigidity for property (T) groups has organised a substantial part of that literature since the 1980s.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.02327","sourceName":"arXiv:2608.02327 - ICC property(T) groups without W*-superrigidity","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"two-terminal-reliability-fpras","name":"Approximating Two-Terminal Network Reliability","shortName":"Two-terminal reliability","problemNumber":null,"field":"Randomized algorithms","fieldGroup":"Algorithms & optimization","statement":"Does two-terminal reliability, the probability that $s$ still reaches $t$ when edges fail independently, admit a fully polynomial-time randomised approximation scheme? Asked explicitly in Kannan's 1994 survey and left open while the all-terminal cases were settled by Karger and by Guo and Jerrum. Answered positively for general graphs, both directed and undirected. The complementary unreliability question is shown to be BIS-hard, so it is unlikely to admit one.","posedBy":"Sampath Kannan","yearPosed":1994,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-03","model":"GPT-5.6 Sol Ultra","modelMaker":"OpenAI","humanCollaborators":["Weiming Feng","Yucheng Fu","Heng Guo"],"aiRole":"The abstract credits GPT-5.6 Sol Ultra with the key idea of the algorithm. The three authors develop the analysis, the BIS-hardness result and the write-up.","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":"The approximability of two-terminal reliability was asked explicitly in Kannan's 1994 survey and stood while the neighbouring all-terminal problems fell to Karger and to Guo and Jerrum. The paper settles it in both the directed and undirected cases and pairs it with a hardness result for the complementary quantity.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.02523","sourceName":"arXiv:2608.02523 - Approximating two-terminal network reliability","links":[],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"graffiti-residue-common-divisor","name":"Graffiti's Residue Problem for Common-Divisor Graphs","shortName":"Graffiti residue problem","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"A problem from Fajtlowicz's Graffiti program, studied by Erdős and Staton, on the Havel-Hakimi residue of common-divisor graphs. The paper resolves the problem and extends it, determining the residue's first-order scale and its nontrivial constant from the degree sequence.","posedBy":"Graffiti (S. Fajtlowicz's program); studied by Erdős and Staton","yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-03","model":"Theo-Conjecture, OpenAI Codex","modelMaker":null,"humanCollaborators":["Randy Davila"],"aiRole":"The study deliberately retained Graffiti's conjecture-example-proof workflow through the TxGraffiti / Theo-Conjecture line: the systems supported registry construction, conjecture generation, exact stress testing, proof search and proof auditing, with the surviving conjecture selected by discrepancy tracking against a quarter-million-graph registry.","verification":"unreviewed","verificationNote":null,"publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A machine-generated Graffiti conjecture, the band the scoring ladder pins at 5, though this one carries Erdős and Staton's names in its history.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.04040","sourceName":"arXiv","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kalton-peck-space-hyperplanes","name":"Kalton-Peck Space and its Hyperplanes","shortName":"Kalton-Peck hyperplanes","problemNumber":null,"field":"Banach space theory","fieldGroup":"Analysis","statement":"Whether the real Kalton-Peck space $Z_2$ is isomorphic to its hyperplanes. It is not: no hyperplane of $Z_2$ is isomorphic to $Z_2$, proved through a rank parity theorem for symplectic spaces applied to the Kalton-Swanson symplectic structure.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-08-03","model":"GPT-5.5 Pro-Extended","modelMaker":"OpenAI","humanCollaborators":["A. Das","V. Ferenczi","Ch. Rosendal"],"aiRole":"The AI use and Lean formalization section states that the principal result was first obtained through an extended interaction with ChatGPT using GPT-5.5 with Pro-Extended reasoning, which supplied an ingredient of the argument.","verification":"unreviewed","verificationNote":"arXiv preprint, not yet peer-reviewed. The paper reports a Lean formalization alongside; the site has not checked it.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"The Kalton-Peck space is a standard counterexample-generating object in Banach space theory, and the hyperplane question sits in the tradition of Banach's hyperplane problem settled by Gowers.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2608.02126","sourceName":"arXiv:2608.02126 - Kalton-Peck space is not isomorphic to its hyperplanes","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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":"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","links":[],"relations":[],"submittedBy":"Curator34","upvotes":1,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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":"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","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"}],"relations":[],"submittedBy":null,"upvotes":9,"downvotes":0,"commentCount":3},{"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","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":"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":"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","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"}],"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":"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)","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":"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","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":"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","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":"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","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":"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","links":[],"relations":[],"submittedBy":"Curator34","upvotes":0,"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","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":"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","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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":"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","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 $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf","kind":"paper"},{"label":"Lean certificate (CompactnessAndDegeneracy.lean)","url":"https://github.com/openai/ten-proofs/blob/main/CompactnessAndDegeneracy.lean","kind":"lean-proof"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf","kind":"transcript"},{"label":"erdosproblems.com/146","url":"https://www.erdosproblems.com/146","kind":"problem-record"}],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes","name":"An Erdős–Kac law for base-$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","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":"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","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":"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":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":3,"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)","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":"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","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":"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":null,"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","links":[],"relations":[],"submittedBy":null,"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","links":[],"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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","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":"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","links":[],"relations":[],"submittedBy":"Lamp","upvotes":3,"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","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","links":[],"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":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"leading asymptotic determined up to a bounded q-dependent term","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]","links":[],"relations":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":2},{"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","links":[],"relations":[],"submittedBy":"Lamp","upvotes":3,"downvotes":0,"commentCount":1},{"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","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":"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","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":"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","links":[],"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","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":"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":null,"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":null,"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":"unreviewed","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.","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[],"relations":[],"submittedBy":"Curator34","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","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":"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":null,"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","links":[],"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","links":[],"relations":[],"submittedBy":"Curator34","upvotes":0,"downvotes":0,"commentCount":0},{"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","links":[],"relations":[],"submittedBy":"HiddenLemur412","upvotes":4,"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 Shepard's threshold conjecture.","posedBy":"Milosz, Hamel and Pierrot; Shepard","yearPosed":null,"ageNote":null,"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.","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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":null,"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","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","links":[],"relations":[],"submittedBy":null,"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","links":[],"relations":[],"submittedBy":null,"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","links":[],"relations":[],"submittedBy":"matthew","upvotes":4,"downvotes":0,"commentCount":2},{"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","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":"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":null,"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","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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":null,"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","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","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","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","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":null,"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","links":[],"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","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)","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","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","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","links":[],"relations":[],"submittedBy":null,"upvotes":1,"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","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","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[],"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[{"label":"Lean verification development","url":"https://github.com/atemerev/gnomon","kind":"lean-proof"}],"relations":[],"submittedBy":null,"upvotes":8,"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":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":1,"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","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":"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.","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","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"}],"relations":[],"submittedBy":"NimbleRaven553","upvotes":3,"downvotes":0,"commentCount":1},{"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)","links":[],"relations":[],"submittedBy":null,"upvotes":2,"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":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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":null,"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","links":[],"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","links":[],"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","links":[],"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":null,"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","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)","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":"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","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":null,"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","links":[],"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","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":"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":null,"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","links":[],"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","links":[],"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","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":"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://www.erdosproblems.com/768","sourceName":"erdosproblems.com/768","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","links":[],"relations":[],"submittedBy":null,"upvotes":1,"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","links":[],"relations":[],"submittedBy":null,"upvotes":2,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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)","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":"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":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[],"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","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","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":"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":null,"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","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","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":"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","links":[],"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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":"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":"Pascal Adjamagbo","yearPosed":null,"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","links":[{"label":"Dimension-two counterexample, Lean kernel-checked (Mondello)","url":"https://arxiv.org/abs/2608.02634","kind":"independent"}],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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)","links":[],"relations":[],"submittedBy":"QuietFerret326","upvotes":1,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"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":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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":null,"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","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)","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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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":null,"ageNote":null,"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","links":[],"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","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":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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","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":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","links":[],"relations":[],"submittedBy":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[{"label":"Dixmier conjecture","url":"https://en.wikipedia.org/wiki/Dixmier_conjecture","kind":"wikipedia"}],"relations":[],"submittedBy":"VibeGene","upvotes":0,"downvotes":0,"commentCount":0},{"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","links":[],"relations":[],"submittedBy":null,"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","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":null,"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","links":[],"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)","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":"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":null,"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","links":[],"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)","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"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":null,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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":null,"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":"unreviewed","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. 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","links":[],"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":null,"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","links":[],"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","links":[],"relations":[],"submittedBy":"Rasmus Lindahl","upvotes":1,"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":null,"yearPosed":null,"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","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":null,"ageNote":null,"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","links":[],"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 that 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","links":[],"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 that 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":null,"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","links":[],"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","links":[],"relations":[],"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","links":[],"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","links":[],"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":null,"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","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","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":null,"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","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)","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","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","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":"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","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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","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","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?","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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","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":"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","links":[],"relations":[],"submittedBy":"SpryRaven345","upvotes":1,"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","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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","links":[],"relations":[],"submittedBy":"JollyJackal127","upvotes":2,"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","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","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-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","links":[],"relations":[],"submittedBy":null,"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":null,"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":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","links":[],"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","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","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","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","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":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-16","model":"GPT-5 + Claude Opus 4","modelMaker":null,"humanCollaborators":["Konstantin Slutsky"],"aiRole":"The paper states that 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","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","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","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":"Godara and Sarkar","yearPosed":null,"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":null,"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","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","links":[],"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","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","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":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-14","model":"AxiomProver","modelMaker":null,"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":"unreviewed","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","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","links":[],"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":null,"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","links":[],"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","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","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","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","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","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","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","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","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","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":null,"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","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","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","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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","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","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","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","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","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","links":[],"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","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"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","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","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","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","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":null,"yearPosed":null,"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":"unreviewed","verificationNote":"Dual exact checkers plus a partial Lean formalization; 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/1186","sourceName":"erdosproblems.com/1186","links":[],"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","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","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)","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","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","links":[],"relations":[],"submittedBy":null,"upvotes":1,"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","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":"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":null,"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":"unreviewed","verificationNote":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[],"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","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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","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","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","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","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":"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","links":[],"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)","links":[],"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","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":null,"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","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":null,"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":null,"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":"unreviewed","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.","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","links":[],"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","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","links":[{"label":"AxiomProver Lean development","url":"https://github.com/AxiomMath/TanArctan","kind":"lean-proof"}],"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","links":[],"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":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-06","model":"ChatGPT 5.5 Pro + Danus","modelMaker":null,"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","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":null,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-04","model":"ChatGPT 5.5 Pro + Danus","modelMaker":null,"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","links":[],"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","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","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-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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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)","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":"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","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":"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","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","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","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","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","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","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","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","links":[],"relations":[],"submittedBy":null,"upvotes":1,"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","links":[],"relations":[],"submittedBy":null,"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","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","links":[],"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":"da Silva Machado, Jose Seade","yearPosed":null,"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":null,"humanCollaborators":["Jihao Liu","Xiping Zhang"],"aiRole":"The paper states that 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","links":[{"label":"Independent human proof of the same conjecture, arXiv:2606.29886","url":"https://arxiv.org/abs/2606.29886","kind":"independent"}],"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","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":"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","links":[],"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","yearPosed":null,"ageNote":null,"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","links":[],"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","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":null,"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":"unreviewed","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.","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","links":[],"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","links":[],"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":null,"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","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":null,"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","links":[],"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","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":"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","links":[],"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","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":null,"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","links":[],"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":null,"ageNote":null,"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":"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":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","links":[],"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","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":null,"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","links":[],"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","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":null,"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":"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 question of Baker recorded by Poonen in the Bertini-over-finite-fields literature, with a stated prediction that this confirms.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.23513","sourceName":"arXiv:2606.23513 - Hyperplane anti-Bertini embeddings over finite fields","links":[],"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","links":[],"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","links":[],"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","links":[{"label":"arXiv preprint","url":"https://arxiv.org/abs/2606.23659","kind":"paper"}],"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","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-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","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","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-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","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","links":[{"label":"Lean repository","url":"https://github.com/guanyangwang/ktv-swap-lean","kind":"lean-proof"}],"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":null,"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","links":[],"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","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","links":[],"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","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":null,"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":"unreviewed","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.","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","links":[],"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","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","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":"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":null,"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":"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":"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"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","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"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":null,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"part (1) of the conjecture","claimIssueNote":null,"solveDate":"2026-06-16","model":"Rethlas","modelMaker":null,"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 that 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","links":[],"relations":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"yun-sra-jadbabaie-shuffling-inequalities","name":"The Yun-Sra-Jadbabaie SS-RS-GD Inequalities","shortName":"SS-RS-GD inequalities","problemNumber":null,"field":"Optimization","fieldGroup":"Algorithms & optimization","statement":"Yun, Sra and Jadbabaie posed as a COLT 2021 open question whether, for well-conditioned symmetric matrices, the operators encoding the expected iterate of single-shuffle SGD, random-reshuffle SGD and gradient descent on a quadratic finite sum satisfy $\\|W_{ss}\\| \\le \\|W_{rs}\\| \\le \\|W_{gd}\\|$. They do.","posedBy":"Chulhee Yun, Suvrit Sra, Ali Jadbabaie","yearPosed":2021,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-16","model":"GPT-5.5 Pro (extended)","modelMaker":"OpenAI","humanCollaborators":["Binghui Peng"],"aiRole":"The paper states the proof was found via GPT-5.5 Pro prompted by the author, and a footnote says the proof idea was completely generated by the model, linking the shared conversation. The write-up was assembled by the author.","verification":"unreviewed","verificationNote":"Single-author arXiv preprint with the originating conversation linked; not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"A COLT open question about why shuffling beats sampling in SGD, a question the optimization community has tracked since 2021.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.22620","sourceName":"arXiv:2607.22620 - A Resolution of the SS-RS-GD Inequalities","links":[],"relations":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-986","name":"Erdős Problem #986","shortName":"Erdős #986","problemNumber":986,"field":"Graph Theory, Ramsey Theory","fieldGroup":"Combinatorics","statement":null,"posedBy":"Paul Erdős","yearPosed":1990,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"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","links":[],"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