{"title":"VibeMathed - math problems solved by AI","url":"https://vibemathed.com","license":"CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)","methodology":"https://vibemathed.com/methodology","generated":"2026-08-02T19:46:13.943Z","count":252,"problems":[{"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"},{"label":"Written on the Wall II conjecture list","url":"http://cms.dt.uh.edu/faculty/delavinae/research/wowII/"}],"submittedBy":null,"upvotes":1,"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"},{"label":"Lean certificate (QuantumParallelRepetition.lean)","url":"https://github.com/openai/ten-proofs/blob/main/QuantumParallelRepetition.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"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":20,"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"},{"label":"Lean certificate (MulticolorTriangleRamsey.lean)","url":"https://github.com/openai/ten-proofs/blob/main/MulticolorTriangleRamsey.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"},{"label":"erdosproblems.com/183","url":"https://www.erdosproblems.com/183"}],"submittedBy":null,"upvotes":2,"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"},{"label":"Lean certificate (GapCVP.lean)","url":"https://github.com/openai/ten-proofs/blob/main/GapCVP.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"submittedBy":null,"upvotes":1,"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"},{"label":"Lean certificate (ConnesRigidity.lean)","url":"https://github.com/openai/ten-proofs/blob/main/ConnesRigidity.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"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"},{"label":"Lean certificate (EhrhartVolumeInequality.lean)","url":"https://github.com/openai/ten-proofs/blob/main/EhrhartVolumeInequality.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":2},{"slug":"non-sofic-groups-exist","name":"Existence of Non-Sofic Groups","shortName":"Non-sofic groups","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":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":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"},{"label":"Lean certificate (NonSoficGroup.lean)","url":"https://github.com/openai/ten-proofs/blob/main/NonSoficGroup.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"submittedBy":null,"upvotes":3,"downvotes":0,"commentCount":1},{"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":40,"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"},{"label":"Lean certificate (Permanent.lean)","url":"https://github.com/openai/ten-proofs/blob/main/Permanent.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"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"}],"submittedBy":null,"upvotes":1,"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"},{"label":"Lean certificate (CompactnessAndDegeneracy.lean)","url":"https://github.com/openai/ten-proofs/blob/main/CompactnessAndDegeneracy.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"},{"label":"erdosproblems.com/146","url":"https://www.erdosproblems.com/146"}],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"sphere-packing-upper-bounds-cohn-elkies","name":"Upper Bounds for High-Dimensional Sphere Packing","shortName":"Sphere packing bounds","problemNumber":null,"field":"Discrete geometry","fieldGroup":"Geometry & topology","statement":"How dense can a sphere packing in $\\mathbb{R}^n$ be as $n \\to \\infty$? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.","posedBy":null,"yearPosed":1978,"ageNote":"Dated to the Kabatiansky-Levenshtein bound of 1978, the asymptotic barrier this result improves on; asking how dense sphere packings can be is of course far older.","solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open","claimIssueNote":null,"solveDate":"2026-08-01","model":"Astra (internal preview)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.","verification":"lean-verified","verificationNote":"Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":50,"significanceNote":"The asymptotic sphere-packing problem is classical and central across geometry, coding and number theory; the Kabatiansky-Levenshtein bound stood since 1978.","solveCostUsd":182,"solveCostNote":"Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rates across all ten results together; this is that aggregate split evenly over the 11 entries, so the column sums to the published figure rather than claiming per-result precision.","sourceUrl":"https://openai.com/index/ten-advances-in-mathematics/","sourceName":"OpenAI: Ten advances in mathematics and theoretical computer science","links":[{"label":"Manuscripts (ten-proofs paper, PDF)","url":"https://cdn.openai.com/pdf/ten-proofs-oai.pdf"},{"label":"Lean certificate (SpherePacking.lean)","url":"https://github.com/openai/ten-proofs/blob/main/SpherePacking.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"submittedBy":null,"upvotes":2,"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"},{"label":"Lean certificate (CompactnessAndDegeneracy.lean)","url":"https://github.com/openai/ten-proofs/blob/main/CompactnessAndDegeneracy.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"},{"label":"erdosproblems.com/180","url":"https://www.erdosproblems.com/180"}],"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":40,"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"},{"label":"Lean certificate (MetricCodes.lean)","url":"https://github.com/openai/ten-proofs/blob/main/MetricCodes.lean"},{"label":"Reasoning walkthroughs (PDF)","url":"https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf"}],"submittedBy":null,"upvotes":2,"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: we present a seventeen-terminal common-point interval instance that certifies\n$$ C\\ \\ge\\ \\frac{1282494797984843521}{10^{18}}=1.28249\\ldots $$\n\nUpper bounds: we prove the first unconditional upper bound below 2 for a nontrivial family of extremal cells, representing the class as a weighted two-permutation prefix system. We also prove additional structural results on the limits of techniques used for the lower bound. The ceilings apply to the common-point / two-order class from which the lower bounds are drawn, not to the universal constant $C$ itself.","posedBy":"Dinitz, Garg, Goemans","yearPosed":1999,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"Record lower bound plus class-restricted ceilings; the existence and exact value of a finite universal constant remain open","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 — no independent expert review, 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"},{"label":"Paper I's Zenodo record","url":"https://zenodo.org/records/21701162"}],"submittedBy":"BraveDingo215","upvotes":2,"downvotes":0,"commentCount":3},{"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"}],"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":5,"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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"graffiti-conjecture-6","name":"Graffiti Conjecture 6","shortName":"Graffiti 6","problemNumber":null,"field":"Graph theory","fieldGroup":"Combinatorics","statement":"Every finite connected simple graph G satisfies\n\n$$\\alpha(G)\\ge r(G)+\\ln(\\rho(G)),$$\n\nwhere $\\alpha(G)$ is the independence number, $r(G)$ is the radius, and $\\rho(G)$ is the minimum number of pairwise vertex-disjoint paths whose vertices cover $V(G)$.","posedBy":"Graffiti, reported by Ermelinda DeLaViña, Siemion Fajtlowicz, and Bill Waller","yearPosed":2002,"ageNote":"First documented in March 2002 and revised in May 2003; disproved approximately 24 years later.","solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"Infinite family of counterexamples; mathematical argument internally checked, with external verification and novelty review pending.","claimIssueNote":null,"solveDate":"2026-07-30","model":"GPT-5.6 Thinking","modelMaker":"OpenAI","humanCollaborators":["Jackson (prompter)"],"aiRole":"GPT-5.6 Thinking produced and checked an infinite family of counterexamples. For each integer s >= 0, it considered a tree T_s formed from the path v_0v_1...v_{4s+7} by attaching leaves at v_{2s+2} and v_{2s+5}. It proved that\n\n$$\\alpha(T_s)=2s+5,\\qquad r(T_s)=2s+4,\\qquad \\rho(T_s)=3.$$\n\nSince $\\ln 3>1$, it follows that\n\n$$\\alpha(T_s)=2s+5<2s+4+\\ln 3=r(T_s)+\\ln\\rho(T_s).$$\n\nThus every T_s is a counterexample, disproving the conjecture and providing infinitely many counterexamples. The AI also audited the final proof line by line.","verification":"unreviewed","verificationNote":"The proof was checked line by line by GPT-5.6 Thinking. The radius, independence number, perfect matching, and path-covering number arguments were separately recomputed, including the smallest case s=0. The proof appears mathematically valid, but as of 2026-07-30 it has not been independently verified by an external graph theorist, a formal proof assistant, or peer review.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":"On Some Conjectures of Griggs and Graffiti — Ermelinda DeLaViña, Siemion Fajtlowicz, and Bill Waller","citationsSource":"https://www.uhd.edu/documents/academics/sciences/griggsngraffiti.pdf","citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (Graffiti); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/Lamptypeshi/graffiti-conjecture-6-counterexample/blob/main/graffiticonjecture6FALSE.pdf","sourceName":"Graffiti Conjecture 6 Counterexample","links":[],"submittedBy":"Lamp","upvotes":2,"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":[],"submittedBy":null,"upvotes":2,"downvotes":0,"commentCount":2},{"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":5,"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":[],"submittedBy":"Lamp","upvotes":2,"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. New independent sets in strong graph powers give $\\Theta(C_7) > 3.258020$, $\\Theta(C_{11}) > 5.289773$, $\\Theta(C_{13}) > 6.300109$ and $\\Theta(C_{15}) > 7.301399$.","posedBy":"Claude Shannon","yearPosed":1956,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"four record lower bounds; the exact capacities remain open for every odd cycle beyond C5","claimIssueNote":null,"solveDate":"2026-07-30","model":"ChatGPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The model generated and executed search programs across repeated prompts and returned the explicit independent-set constructions; the four authors checked that every reported set is independent.","verification":"unreviewed","verificationNote":"Author-checked constructions with public data, prompts and checking code; arXiv preprint (the C15 record was added in the 30 July revision). Not yet 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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lukic-conjecture","name":"The Lukic Conjecture","shortName":"Lukic conjecture","problemNumber":null,"field":"Spectral theory","fieldGroup":"Analysis","statement":"Let $\\mu$ be a probability measure on the unit circle with Verblunsky coefficients $\\alpha$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\\alpha$ into components localized at those points. A counterexample with two critical points of multiplicity three refutes it: the sequence satisfies the decomposition conditions while the corresponding weighted entropy is $-\\infty$.","posedBy":"Milivoje Lukić","yearPosed":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":[],"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":"candidate","aiContribution":"ai-assisted","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-29","model":"OpenAI Codex","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The public Lean source credits Codex as formal author; the available record does not establish that Codex discovered the packing, so this entry tracks the AI formalization rather than assigning mathematical priority.","verification":"lean-verified","verificationNote":"A 613-line, placeholder-free Lean proof of the counterexample; the erdosproblems.com page had not yet incorporated the result at audit time.","publication":"announcement","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","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"}],"submittedBy":null,"upvotes":2,"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":[],"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":[],"submittedBy":"matthew","upvotes":3,"downvotes":0,"commentCount":2},{"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"}],"submittedBy":null,"upvotes":2,"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":["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":[],"submittedBy":"HiddenLemur412","upvotes":3,"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":[],"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":[],"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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"exact-two-scenario-ssuf-bound-on-the-four-terminal-planar-gadget","name":"Exact two-scenario SSUF bound on the four-terminal planar gadget","shortName":"Four-terminal two-scenario SSUF","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 two positive cost scenarios. What is the worst-case normalized additive upper arc deviation? Theorem RB-003 proves $$\\beta_G^{(2\\mathrm{sc})}=\\frac{17}{8}=2.125,$$ as a non-attained supremum.","posedBy":"Matthew Protti","yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"variant","aiContribution":"ai-co-developed","resultNote":"Fixed graph; exactly two positive E-minus-C scenarios; cost non-increase; normalized additive upper arc deviation. Not an unrestricted planar, one-scenario, or many-scenario constant.","claimIssueNote":null,"solveDate":"2026-07-28","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Matthew Protti"],"aiRole":"GPT-5.6 Pro developed substantial portions of the construction and parameter search, symbolic derivations, analytic proof, exact verifiers, adversarial analyses, and manuscript drafts. Matthew Protti selected and framed the target, required exact enumeration, mutation tests and hostile review, forced correction of material errors and overclaims, chose the final theorem scope, and authorized the release. Codex and other model sessions reconstructed checks and release controls. Role separation reduced some shared-context risk but was not independent human review.","verification":"unreviewed","verificationNote":"The immutable v0.2.1 release contains a self-contained human-readable proof, exact rational construction, and deterministic verification artifacts. For the finite epsilon=1/1000 certificate, the graph-native checker enumerates all 16 routings and 13 arcs and obtains 1061/500; scaling demands by 4000 gives unavoidable upper deviation 8488. Exact algebraic checks, mutation tests, and two role-separated AI-assisted hostile-review rounds are included. The human-readable proof is authoritative. No external human mathematical or journal peer review is documented; v0.2.1 changes provenance and release hygiene only and leaves the v0.2.0 mathematics unchanged.","publication":"preprint","resolutionMethod":"computation","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A sharp constant inside one gadget construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/releases/download/v0.2.1/rb003_two_scenario_note_v2.pdf","sourceName":"RB-003 paper, revision 2 (v0.2.1)","links":[{"label":"Immutable v0.2.1 release","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/releases/tag/v0.2.1"},{"label":"Exact verification suite","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/blob/v0.2.1/scripts/verify_all.py"},{"label":"AI use and provenance","url":"https://github.com/matthewprotti/planar-ssuf-four-terminal-bound/blob/v0.2.1/AI_USE_AND_PROVENANCE.md"},{"label":"Related VibeMathed entry: four-terminal planar DGG counterexample","url":"https://vibemathed.com/problem/planar-four-terminal-dgg"}],"submittedBy":"LuckyHawk816","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":[],"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":[],"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":[],"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":[],"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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"feiges-conjecture","name":"Feige's Conjecture","shortName":"Feige's conjecture","problemNumber":null,"field":"Probability","fieldGroup":"Probability & statistics","statement":"Let $X_1,\\ldots,X_n$ be independent nonnegative random variables with $\\mathbb{E}X_i \\le 1$, and let $S$ be their sum. Is $\\mathbb{P}(S < \\mathbb{E}S + 1) \\ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\\delta \\ge 1$.","posedBy":"Uriel Feige","yearPosed":2004,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-27","model":"ChatGPT 5.6 Pro, GPT-5.6 Sol, Codex","modelMaker":"OpenAI","humanCollaborators":["Weibo Fu","Yanjun Han","Guanyang Wang","Jun Yan","Peng Zhang","Zhengqing Zhou","Zipei Nie","Jiaye Wei","Mark Stander"],"aiRole":"The primary paper states plainly that the proof was found by ChatGPT 5.6 Pro, combining the Vlassis-Thomas Dirichlet calibration theorem with Grünbaum-type convex geometry; the authors checked, revised and rewrote the argument, and the accompanying Lean formalization was developed with Codex. The independent second proof by Nie and Wei was obtained with the assistance of GPT-5.6 Sol. A further independent proof was found by Stander.","verification":"lean-verified","verificationNote":"An end-to-end Lean formalization of the $e^{-1}$ conjecture accompanies the primary paper, formalizing the Vlassis-Thomas theorem, Grünbaum's centroid theorem and the combining argument. Three independent AI-assisted proofs appeared within days; neither preprint is peer-reviewed yet.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":35,"significanceNote":"Feige's 2004 bound, known across probability and TCS with two decades of partial results.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.23980","sourceName":"arXiv:2607.23980 - Sharp small-deviation inequalities for sums of independent nonnegative random variables","links":[{"label":"Independent second proof (Nie and Wei), arXiv:2607.24528","url":"https://arxiv.org/abs/2607.24528"},{"label":"Lean formalization of the e^-1 conjecture","url":"https://github.com/pengzhang91/Feige"},{"label":"Stander's proof","url":"https://doi.org/10.5281/zenodo.21622950"}],"submittedBy":null,"upvotes":2,"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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kls-quadratic-forms","name":"KLS Conjecture for Quadratic Forms","shortName":"KLS for quadratic forms","problemNumber":null,"field":"Asymptotic convex geometry","fieldGroup":"Geometry & topology","statement":"Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is $\\operatorname{Var}\\langle MX, X\\rangle \\le C\\, \\mathbb{E}|\\nabla\\langle MX, X\\rangle|^2$ for every symmetric $M$?","posedBy":"Ravi Kannan, László Lovász & Miklós Simonovits","yearPosed":1995,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"with constant 2; also improves the global KLS bound to $O(\\log^{1/4} n)$","claimIssueNote":null,"solveDate":"2026-07-27","model":"ChatGPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Brayden Letwin"],"aiRole":"The key argument was developed in collaboration with ChatGPT-5.6 Pro and checked by the author.","verification":"unreviewed","verificationNote":"Author-checked arXiv preprint proving the quadratic-form case with constant 2 and deriving the global estimate $\\psi_n \\le C \\log^{1/4} n$. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":45,"significanceNote":"The heart case of the KLS conjecture, the organizing problem of asymptotic convex geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.24164","sourceName":"arXiv:2607.24164 - The KLS constant is O(log^(1/4) n)","links":[],"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":[],"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"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"quantum-conditional-entropy-continuity","name":"Sharp Continuity Bound for Quantum Conditional Entropy","shortName":"Entropy continuity bound","problemNumber":null,"field":"Quantum information theory","fieldGroup":"Quantum information & computing","statement":"What is the optimal uniform continuity bound for quantum conditional entropy in trace distance, depending only on the dimension of the conditioned system? The sharp bound $h_2(\\delta) + \\delta \\log(d^2 - 1)$ up to $\\delta = 1 - d^{-2}$, conjectured by Wilde, is proved.","posedBy":"Mark M. Wilde","yearPosed":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":[],"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":[],"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":[],"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"}],"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":[],"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/"}],"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":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/768","sourceName":"erdosproblems.com/768","links":[],"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":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/684","sourceName":"erdosproblems.com/684","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"rota-matroid-flat-unimodality","name":"Rota's Unimodality Conjecture for Matroid Flats","shortName":"Rota flat unimodality","problemNumber":null,"field":"Matroid theory","fieldGroup":"Combinatorics","statement":"Is the sequence $W_0, W_1, \\dots, W_n$ counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.","posedBy":"Gian-Carlo Rota","yearPosed":1970,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-24","model":"ChatGPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Alexander Divoux","Chayim Lowen","Shouda Wang"],"aiRole":"The counterexample construction - a $q$-lift mechanism that turns failures of log-concavity for flat counts into failures of unimodality - was found with ChatGPT-5.6 Pro; the authors verified it and wrote the paper.","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":35,"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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"s-decoding-polynomial-sparsity","name":"Minimum Sparsity of S-Decoding Polynomials","shortName":"S-decoding sparsity","problemNumber":null,"field":"Private information retrieval","fieldGroup":"Theoretical computer science","statement":"Can an $S$-decoding polynomial modulo a suitable product of $k$ primes attain the lower-bound minimum of $k + 1$ nonzero coefficients? A construction matches the bound for special products of $k$ primes, yielding exponentially fewer-server PIR.","posedBy":"Fatemeh Ghasemi & Swastik Kopparty","yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"conditional on a plausible number-theoretic conjecture; unconditional through s = 15","claimIssueNote":null,"solveDate":"2026-07-24","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The sparse-polynomial framework was developed with GPT-5.5 Pro and validated empirically by the authors.","verification":"unreviewed","verificationNote":"Author-checked ePrint with empirical validation of the construction. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A recent coding-theory question from a single paper.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://eprint.iacr.org/2026/1515","sourceName":"ePrint 2026/1515 - Exponentially fewer-server PIR from sparser S-decoding polynomials","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","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"},{"label":"erdosproblems.com/131","url":"https://www.erdosproblems.com/131"}],"submittedBy":"Theofil Xeff","upvotes":2,"downvotes":0,"commentCount":1},{"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"}],"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"},{"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"},{"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"},{"label":"The general conjecture, disproved a day earlier","url":"https://vibemathed.com/problem/dinitz-garg-goemans-unsplittable-flow"}],"submittedBy":"LuckyHawk816","upvotes":2,"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1177","sourceName":"erdosproblems.com/1177","links":[],"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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-593","name":"Erdős Problem #593","shortName":"Erdős #593","problemNumber":593,"field":"Infinite Hypergraph Theory","fieldGroup":"Combinatorics","statement":"Which finite triple systems occur in every triple system of uncountable chromatic number? The claimed characterization: exactly those that, after removing isolated vertices, are linear, have every hyperedge-node of their Levi graph meeting a bridge, and have every Berge cycle even.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT + Aristotle","modelMaker":"OpenAI / Harmonic","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked end to end; not yet conventionally refereed or incorporated by the community tracker.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/593","sourceName":"erdosproblems.com/593","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"stanley-claw-free-schur-positivity","name":"Stanley's Claw-Free Schur-Positivity Conjecture","shortName":"Claw-free Schur positivity","problemNumber":null,"field":"Algebraic combinatorics","fieldGroup":"Combinatorics","statement":"Is the chromatic symmetric function $X_G$ Schur positive for every claw-free graph $G$? Two explicit $12$-vertex line graphs have Schur coefficients $-64$ and $-40$ at $s_{(3,3,3,3)}$.","posedBy":"Richard Stanley","yearPosed":1995,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-23","model":"ChatGPT-5.6 Sol Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Both counterexample graphs were found with ChatGPT-5.6 Sol Pro; the negative coefficients were confirmed by independent computations.","verification":"unreviewed","verificationNote":"Author preprint with independent computational checks of the negative Schur coefficients. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Stanley's claw-free positivity question, a standing target in symmetric function theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.21508","sourceName":"arXiv:2607.21508 - Chromatic symmetric functions of claw-free graphs are not Schur positive","links":[],"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 iff already one-copy distillable","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":"Independent proof: Bharti, Gajjala & Haug (arXiv:2607.24479)","url":"https://arxiv.org/abs/2607.24479"},{"label":"Independent proof via a partial-trace inequality (arXiv:2607.24309)","url":"https://arxiv.org/abs/2607.24309"},{"label":"Third independent proof via a rank-two partial-trace inequality (Song and Chen), arXiv:2607.23416","url":"https://arxiv.org/abs/2607.23416"}],"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":[],"submittedBy":"Rasmus Lindahl","upvotes":1,"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":[],"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":[],"submittedBy":null,"upvotes":0,"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":[],"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"}],"submittedBy":null,"upvotes":1,"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":[],"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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-469","name":"Erdős Problem #469","shortName":"Erdős #469","problemNumber":469,"field":"Number Theory, Divisors","fieldGroup":"Number theory","statement":"Does the sum of the reciprocals of all primitive pseudoperfect numbers converge?","posedBy":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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/469","sourceName":"erdosproblems.com/469","links":[],"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":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kourovka-21-8-class-transpositions","name":"Kourovka Problem 21.8 - Horizontal Class Transpositions","shortName":"Kourovka 21.8","problemNumber":null,"field":"Group theory","fieldGroup":"Algebra","statement":"If $\\operatorname{CT}_{(k)}$ is generated by all horizontal class transpositions with modulus at most $k$, is $\\operatorname{CT}_{(k)} \\cong S_{\\operatorname{lcm}(2,\\dots,k)}$ for every $k \\ge 4$?","posedBy":null,"yearPosed":2026,"ageNote":"New in the 21st edition of the Kourovka Notebook (2026).","solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-20","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"The solution was discovered autonomously by Aristotle and formalized in Lean; the human authors curated the exposition.","verification":"lean-verified","verificationNote":"Autonomously discovered and formally verified in Lean by Aristotle; author-curated arXiv preprint covering eight Kourovka Notebook problems.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":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":[],"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":[],"submittedBy":null,"upvotes":3,"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":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":[],"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":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":[],"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":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":[],"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","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":[],"submittedBy":"JollyJackal127","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":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":[],"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":[],"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":[],"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":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":[],"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"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"online-spencer-vector-balancing","name":"Online Spencer Vector-Balancing Question","shortName":"Online Spencer","problemNumber":null,"field":"Discrepancy theory","fieldGroup":"Theoretical computer science","statement":"Can online vector balancing in the Spencer setting achieve the optimal order of prefix discrepancy with an efficient algorithm?","posedBy":null,"yearPosed":2023,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-18","model":"ChatGPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The compactly supported Metropolis fixed-point walk at the heart of the algorithm was proposed by ChatGPT-5.6 Pro; the authors manually checked and rewrote the proof. The result also extends the offline Beck-Fiala bound to sparsity $d \\ge \\log(T)^{1+o(1)}$.","verification":"unreviewed","verificationNote":"Author-rewritten and checked 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 prominent question of the online discrepancy wave.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.14238","sourceName":"arXiv:2607.14238 - Online Beck-Fiala down to logarithmic sparsity","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"feige-hypergraph-moore-bound","name":"Feige's Hypergraph Moore-Bound Conjecture","shortName":"Hypergraph Moore bound","problemNumber":null,"field":"Extremal combinatorics","fieldGroup":"Combinatorics","statement":"At the conjectured density, must every $k$-uniform hypergraph contain a short nontrivial even cover - a set of hyperedges covering each vertex an even number of times - with no superfluous polylogarithmic factors? Known up to polylog factors since 2022; now proved exactly for every $k \\ge 3$.","posedBy":"Uriel Feige","yearPosed":2008,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-17","model":"GPT-5.6 Sol, GPT-5.5 Pro, Claude Opus 4.8, Claude Fable 5","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"The colored-walk polynomial-interpolation argument over Kikuchi graphs was developed across several frontier models and written up by a five-author team, with an independent spectral proof alongside.","verification":"unreviewed","verificationNote":"Five-author arXiv preprint plus an independent spectral proof of the same bound. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":25,"significanceNote":"Feige's 2008 conjecture, well known in TCS with a substantial partial-results literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2607.14068","sourceName":"arXiv:2607.14068 - The hypergraph Moore bound","links":[],"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":[],"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":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/254","sourceName":"erdosproblems.com/254","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/130","sourceName":"erdosproblems.com/130","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1189","sourceName":"erdosproblems.com/1189","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/267","sourceName":"erdosproblems.com/267","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/709","sourceName":"erdosproblems.com/709","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/450","sourceName":"erdosproblems.com/450","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1188","sourceName":"erdosproblems.com/1188","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/584","sourceName":"erdosproblems.com/584","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/489","sourceName":"erdosproblems.com/489","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/769","sourceName":"erdosproblems.com/769","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/394","sourceName":"erdosproblems.com/394","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/796","sourceName":"erdosproblems.com/796","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1186","sourceName":"erdosproblems.com/1186","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"benjamini-hochberg-correlated-gaussian","name":"Benjamini-Hochberg FDR Under Correlated Gaussian Tests","shortName":"BH under correlation","problemNumber":null,"field":"Statistics","fieldGroup":"Probability & statistics","statement":"Does the Benjamini-Hochberg procedure always control the false-discovery rate at its nominal level for correlated two-sided Gaussian p-values? A factor model gives $\\mathrm{FDR} > 0.0104$ at nominal level $\\alpha = 0.01$.","posedBy":null,"yearPosed":2006,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-07-13","model":"GPT-5.6 Pro","modelMaker":"OpenAI","humanCollaborators":["Edgar Dobriban"],"aiRole":"The counterexample was obtained by GPT-5.6 Pro and carefully checked by the author, with a rigorous interval-arithmetic certificate valid for all sufficiently large numbers of hypotheses.","verification":"unreviewed","verificationNote":"Author-checked arXiv preprint with an interval-arithmetic certificate. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"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":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/538","sourceName":"erdosproblems.com/538","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/959","sourceName":"erdosproblems.com/959","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":1},{"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/336","sourceName":"erdosproblems.com/336","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/662","sourceName":"erdosproblems.com/662","links":[],"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":[],"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":[],"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":[],"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":"unreviewed","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.","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":[],"submittedBy":null,"upvotes":1,"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":5,"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":[],"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":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/866","sourceName":"erdosproblems.com/866","links":[],"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":10,"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":[],"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":[],"submittedBy":null,"upvotes":0,"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1038","sourceName":"erdosproblems.com/1038","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/320","sourceName":"erdosproblems.com","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/321","sourceName":"erdosproblems.com","links":[],"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":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/123","sourceName":"erdosproblems.com","links":[],"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":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/119","sourceName":"erdosproblems.com","links":[],"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":"site-confirmed","verificationNote":"Marked proved on erdosproblems.com via a proof claim by GPT-5.6 Sol; 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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/793","sourceName":"erdosproblems.com","links":[],"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":[],"submittedBy":null,"upvotes":1,"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"}],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1061","sourceName":"erdosproblems.com/1061","links":[],"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":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","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"}],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/865","sourceName":"erdosproblems.com","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","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"}],"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":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/176","sourceName":"erdosproblems.com/176","links":[],"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"}],"submittedBy":"QuietLemur253","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 Erdős catalog: real and documented, with a specialist audience.","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"},{"label":"Price's disproof of the literal reading","url":"https://www.overleaf.com/read/tgrrgqpbjpht#c5a085"}],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1197","sourceName":"erdosproblems.com","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/948","sourceName":"erdosproblems.com","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/306","sourceName":"erdosproblems.com/306","links":[],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/451","sourceName":"erdosproblems.com/451","links":[],"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":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/986","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"erdos-942","name":"Erdős Problem #942","shortName":"Erdős #942","problemNumber":942,"field":"Number Theory, Powerful Numbers","fieldGroup":"Number theory","statement":"Let $h(n)$ count powerful integers in $[n^2, (n+1)^2)$. What is the extremal order of $h(n)$?","posedBy":null,"yearPosed":1976,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"lower bound improved to ≫ log n/(log log n · log log log n) infinitely often; the extremal order remains open","claimIssueNote":null,"solveDate":"2026-06-14","model":"Claude, Codex, Aristotle","modelMaker":"Anthropic / OpenAI / Harmonic","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/942","sourceName":"erdosproblems.com/942","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-326","name":"Erdős Problem #326","shortName":"Erdős #326","problemNumber":326,"field":"Number Theory, Additive Bases","fieldGroup":"Number theory","statement":"Does there exist $A=\\{a_1<a_2<\\cdots\\}\\subset \\mathbb{N}$ which is a minimal basis of order $2$ (every large integer is the sum of $2$ elements from $A$, and no proper subset of $A$ has this property) such that $\\lim_{k\\to \\infty}a_k/k^2=c$ for some $c\\neq 0$? A claimed construction gives a minimal basis with $A(x)=C\\sqrt{x}+O(1)$, answering the question affirmatively; Erdős and Graham had conjectured a negative answer.","posedBy":"Paul Erdős, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"Affirmative answer claimed, contrary to the negative answer Erdős and Graham conjectured; erdosproblems.com still lists the problem open","claimIssueNote":null,"solveDate":"2026-06-14","model":"GPT-5.5, Aristotle, Codex","modelMaker":"OpenAI, Harmonic","humanCollaborators":["Aron Bhalla"],"aiRole":"Per the author's disclosure, most of the mathematics is his own, with GPT-5.5 used to stress-test ideas, suggest revisions, identify gaps and write up some proofs; the solution was then formalized over several weeks with Aristotle, Codex and GPT-5.5 into a roughly 15,000-line Lean proof confirming all claims in the manuscript.","verification":"lean-verified","verificationNote":"The author reports a ~15,000-line Lean formalization, type-checkable online, confirming all claims of the manuscript. It has not been independently audited for statement fidelity, and erdosproblems.com has not accepted the claim: the site's owner found the AI-written exposition hard to digest while stressing that this was not a correctness objection.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/326","sourceName":"erdosproblems.com/326","links":[{"label":"Manuscript (latest version)","url":"https://drive.google.com/file/d/1VKaFmiMWWMW7NME-L47HVGSWOoBt9sug/view?usp=sharing"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"graffiti-conjecture-143","name":"Graffiti Conjecture 143","shortName":"Graffiti 143","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"For every connected graph, is the variance of its positive adjacency eigenvalues at most its order divided by its average distance? Exact dumbbell-graph certificates refute the bound under both conventions for average distance.","posedBy":"Graffiti (Siemion Fajtlowicz's program)","yearPosed":1990,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-12","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":"unreviewed","verificationNote":"Exact certificate verified by two independently written checkers; public artifacts repository. Not externally refereed.","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://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pandey-parity-petersen","name":"Pandey Parity Conjecture for Generalized Petersen Graphs","shortName":"Pandey parity","problemNumber":null,"field":"Independence polynomials","fieldGroup":"Combinatorics","statement":"For every $n \\ge 2k + 1$, is the independence polynomial of $GP(n, k)$ real-rooted if and only if $k$ is even? Exact Sturm counts refute both directions.","posedBy":null,"yearPosed":2026,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-12","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":"unreviewed","verificationNote":"Exact certificate verified by two independently written checkers; public artifacts repository. Not externally refereed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A recent conjecture on one graph family.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"sun-trigonometric-permanents","name":"Sun's Conjecture 4.6(ii) on Trigonometric Permanents","shortName":"Sun permanents","problemNumber":null,"field":"Experimental number theory","fieldGroup":"Number theory","statement":"For an odd prime $p$, do Sun's normalized trigonometric permanents satisfy $s_p < 0 \\iff p \\equiv 5 \\pmod{12}$ and $s'_p < 0 \\iff p \\equiv 7 \\pmod 8$? Exact computation at $p = 29$ refutes both sign laws.","posedBy":"Zhi-Wei Sun","yearPosed":2021,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-12","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":"unreviewed","verificationNote":"Both sign clauses refuted by exact calculations in multiple independent implementations; public artifacts. Not externally refereed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"One of Zhi-Wei Sun's many posted conjectures; one-paper audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"koch-narayan-conjecture-1","name":"Koch-Narayan Conjecture 1","shortName":"Koch-Narayan 1","problemNumber":null,"field":"Extremal graph theory","fieldGroup":"Combinatorics","statement":"For a bipartite graph without isolated vertices and with a unique minimum dominating set, does the proposed function $m(n, \\gamma)$ bound the number of edges whenever $\\gamma \\ge 2$ and $n \\ge 3\\gamma$? A $13$-vertex bipartite graph with $22$ edges exceeds the conjectured maximum of $21$.","posedBy":"Koch & Narayan","yearPosed":2025,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-12","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":"unreviewed","verificationNote":"Exact certificate verified by two independently written checkers; public artifacts repository. Not externally refereed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A recent conjecture from a single paper.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"txgraffiti-davila-conjecture-9","name":"TxGraffiti-Davila Conjecture 9","shortName":"TxGraffiti-Davila 9","problemNumber":null,"field":"Graph domination & zero forcing","fieldGroup":"Combinatorics","statement":"If $G$ is connected, cubic and diamond-free, must the zero-forcing number satisfy $Z(G) \\le \\gamma(G) + 2$? A connected cubic triangle-free $14$-vertex graph has $Z = 7$ and $\\gamma = 4$.","posedBy":"Randy Davila (TxGraffiti)","yearPosed":2024,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-12","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":"unreviewed","verificationNote":"Exact certificate verified by two independently written checkers; public artifacts repository. Not externally refereed.","publication":"announcement","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://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"iris-conjecture-6-1","name":"IRIS Conjecture 6.1 on Simple 3-Polytopes","shortName":"IRIS 6.1","problemNumber":null,"field":"Polyhedral combinatorics","fieldGroup":"Geometry & topology","statement":"For a simple $3$-polytope with at least three faces of size at least $7$, must $p_6 \\ge \\frac{39}{20} + \\frac{p_3}{2} - \\frac{p_5}{4} - \\sum_{k \\ge 7} p_k$? Five minimal ten-face counterexamples refute the printed inequality.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-11","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":"Exact certificate verified by two independently written checkers; public artifacts repository. Not externally refereed.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"Machine-generated (IRIS); real but unfamous by construction.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"graffiti-conjecture-154","name":"Graffiti Conjecture 154 (Standard-Deviation Reading)","shortName":"Graffiti 154","problemNumber":null,"field":"Spectral graph theory","fieldGroup":"Combinatorics","statement":"For every connected graph, is the deviation of its adjacency eigenvalues at most its order divided by its average distance? Exact lollipop-graph certificates refute the inequality when deviation means population standard deviation, under both common average-distance conventions.","posedBy":"Graffiti (Siemion Fajtlowicz's program)","yearPosed":1990,"ageNote":null,"solveType":"disproved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"refuted under the standard-deviation reading; the statement is reading-sensitive and other readings remain open","claimIssueNote":null,"solveDate":"2026-06-11","model":"Demonstrandum multi-agent pipeline","modelMaker":null,"humanCollaborators":[],"aiRole":"Found by the Demonstrandum multi-agent pipeline; the certificate ships with mutation-tested checkers and an independent clean-room recomputation.","verification":"unreviewed","verificationNote":"Exact certificate verified by two independently written checkers; public artifacts repository. Not externally refereed.","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://github.com/demonstrandum-research/artifacts","sourceName":"Demonstrandum artifacts repository (RESULTS.md)","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-539","name":"Erdős Problem #539","shortName":"Erdős #539","problemNumber":539,"field":"Number Theory, Multiplicative Combinatorics","fieldGroup":"Number theory","statement":"For $|A| = n$, how small can the cofactor set $Q(A) = \\{a / \\gcd(a,b) : a, b \\in A\\}$ be? The answer is $h(n) = n^{1/2 + o(1)}$: a new upper bound $h(n) \\le n^{1/2} \\exp(O(\\sqrt{\\log n}))$ matches the classical lower bound.","posedBy":null,"yearPosed":1973,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"main exponent determined; sharper subpolynomial factors remain open","claimIssueNote":null,"solveDate":"2026-06-10","model":"ProofCouncil (GPT-5.5 Pro)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The upper-bound construction was found by the ProofCouncil harness running GPT-5.5 Pro.","verification":"lean-verified","verificationNote":"Lean record alongside the official Erdős problems update marking the exponent determined.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/539","sourceName":"erdosproblems.com/539","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-619","name":"Erdős Problem #619","shortName":"Erdős #619","problemNumber":619,"field":"Graph Theory","fieldGroup":"Combinatorics","statement":null,"posedBy":"Paul Erdős, András Gyárfás, Miklós Ruszinkó","yearPosed":1998,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-09","model":"Claude Fable 5, Codex, GPT-5.5","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":25,"citationsPaper":"Paul Erdős, András Gyárfás, Miklós Ruszinkó (1998), \"How to decrease the diameter of triangle-free graphs\", Combinatorica","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W2063190536","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/619","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-696","name":"Erdős Problem #696","shortName":"Erdős #696","problemNumber":696,"field":"Number Theory, Divisors","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-05","model":"Aristotle, Claude Code, Claude Opus 4.7, GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Jake Mallen","David Turturean"],"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":40,"citationsPaper":"Paul Erdős (1979), \"Some unconventional problems in number theory\", Astérisque","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W3173535233","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/696","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-623","name":"Erdős Problem #623","shortName":"Erdős #623","problemNumber":623,"field":"Set Theory, Infinite Combinatorics","fieldGroup":"Combinatorics","statement":"Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ a function from the finite subsets of $X$ to $X$ such that $f(A)\\not\\in A$ for all $A$. Must there exist an infinite independent $Y\\subseteq X$, i.e. with $f(B)\\not\\in Y$ for all finite $B\\subset Y$? Claimed resolution: the positive assertion is equivalent to Koepke's free-subset property, hence independent of ZFC, with consistency strength exactly a measurable cardinal.","posedBy":"Paul Erdős, András Hajnal","yearPosed":1958,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"Resolved (if correct) by an independence result rather than a proof or disproof in ZFC: consistency of the positive answer is equivalent to a measurable cardinal, of the negative to ZFC alone","claimIssueNote":null,"solveDate":"2026-06-04","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Sungchul Lee"],"aiRole":"The equivalence to Koepke's free-subset property and the resulting consistency analysis were obtained with the assistance of GPT-5.5 Pro; the author checked the mathematical details.","verification":"unreviewed","verificationNote":"An AI screening on the forum found no issues and forum readers concur it would fully resolve the problem in the set-theoretic sense, but there is no independent expert review and 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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/623","sourceName":"erdosproblems.com/623","links":[{"label":"Paper and LaTeX source","url":"https://github.com/lsngchl/Erdos623"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fullrsb-jamming-identity","name":"FullRSB Jamming Identity a + b = 1","shortName":"Jamming exponents","problemNumber":null,"field":"Statistical physics","fieldGroup":"Mathematical physics","statement":"Can the critical-exponent relation $a + b = 1$ at the jamming transition, observed numerically to high precision in the full replica-symmetry-breaking solution of hard spheres, be derived analytically from the scaling equations?","posedBy":null,"yearPosed":2014,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-02","model":"Claude Sonnet 4.6, Claude Opus 4.7","modelMaker":"Anthropic","humanCollaborators":["Giorgio Parisi","Francesco Zamponi"],"aiRole":"Parisi and Zamponi asked Claude for help; it quickly proposed the essentially correct idea - integration-by-parts identities combined with a maximum principle - whose first formal write-up contained errors the authors then fixed and verified.","verification":"expert-verified","verificationNote":"Peer-reviewed and published in the Journal of Statistical Mechanics (2026); also public as an arXiv preprint.","publication":"peer-reviewed","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"A central identity of replica theory and jamming physics (Parisi school).","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.03300","sourceName":"arXiv:2606.03300 - A proof of an identity for the critical exponents of jamming","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"generalized-totient-divisibility","name":"Divisibility Set of the Generalized Euler Totient","shortName":"Totient divisibility","problemNumber":null,"field":"Multiplicative number theory","fieldGroup":"Number theory","statement":"Define $\\varphi_k(n) = \\sum_{1 \\le a \\le n, (a,n)=1} a^k$ and $\\mathcal{D}_s = \\{k \\ge s : \\varphi_s(n) \\mid \\varphi_k(n) \\text{ for every } n\\}$. Is $\\mathcal{D}_1 = \\{1, 3, 15\\}$, as conjectured by Büyükaşik and collaborators?","posedBy":"Engin Büyükaşik et al.","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06-01","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["John M. Campbell"],"aiRole":"The exact classification was proved via an argument based on interactions with GPT-5.5 Pro.","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":5,"significanceNote":"A recent question on a generalized totient with a one-paper audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2606.01633","sourceName":"arXiv:2606.01633 - On a problem on a generalization of Euler's totient function","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-477","name":"Erdős Problem #477","shortName":"Erdős #477","problemNumber":477,"field":"Additive Number Theory","fieldGroup":"Number theory","statement":"Does there exist an integer polynomial $f$ of degree at least two and a set $A \\subseteq \\mathbb{Z}$ such that every integer has a unique representation $n = a + f(k)$? A manuscript claims the thirteenth powers admit a tiling complement.","posedBy":null,"yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-06","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/477","sourceName":"erdosproblems.com/477","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-borwein-binary-blocks","name":"Binary Digits of the Erdős-Borwein Constant","shortName":"Erdős-Borwein digits","problemNumber":null,"field":"Digital number theory","fieldGroup":"Number theory","statement":"Does the block $11$ occur infinitely often in the base-$2$ expansion of the Erdős-Borwein constant $E = \\sum_{n \\ge 1} \\frac{1}{2^n - 1}$? Posed by Crandall in 2012.","posedBy":"Richard Crandall","yearPosed":2012,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-22","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["John M. Campbell"],"aiRole":"The proof - a congruence construction in the spirit of Erdős combined with the Alford-Granville-Pomerance estimate for primes in arithmetic progressions - was developed through extensive interactions with GPT-5.5 Pro.","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":10,"significanceNote":"A concrete question on a named constant, posed by Crandall.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.24160","sourceName":"arXiv:2605.24160 - On the binary digits of the Erdős-Borwein constant","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-138","name":"Erdős Problem #138","shortName":"Erdős #138","problemNumber":138,"field":"Ramsey Theory","fieldGroup":"Combinatorics","statement":"If $W(k)$ is the least $N$ such that every two-colouring of $\\{1, \\dots, N\\}$ contains a monochromatic $k$-term arithmetic progression, must $W(k+1) - W(k) \\to \\infty$?","posedBy":null,"yearPosed":1981,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"the stronger question $W(k)^{1/k} \\to \\infty$ remains open","claimIssueNote":null,"solveDate":"2026-05-21","model":"AlphaProof Nexus","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"Proved by AlphaProof Nexus with a Lean-checked argument.","verification":"lean-verified","verificationNote":"Lean-checked; official Erdős problems record updated.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/138","sourceName":"erdosproblems.com/138","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"wow-graph-conjecture-2","name":"Written on the Wall II, Graph Conjecture 2","shortName":"WoW Conjecture 2","problemNumber":null,"field":"Extremal graph theory","fieldGroup":"Combinatorics","statement":"For a finite connected graph $G$, let $L_s(G)$ be the maximum number of leaves in a spanning tree and $\\ell(G)$ the average local independence number. Must $L_s(G) \\ge 2(\\ell(G) - 1)$?","posedBy":"Graffiti (Written on the Wall II)","yearPosed":1996,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-21","model":"AlphaProof Nexus","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"Solved autonomously by AlphaProof Nexus, with the proof formally verified in Lean.","verification":"lean-verified","verificationNote":"Lean-checked; formal proofs published with DeepMind's AlphaProof Nexus report (arXiv:2605.22763) and its accompanying repository.","publication":"preprint","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://arxiv.org/abs/2605.22763","sourceName":"arXiv:2605.22763 - AlphaProof Nexus report","links":[{"label":"Independent second Lean proof by Kenta Kitamura (formal-conjectures PR #4654)","url":"https://github.com/google-deepmind/formal-conjectures/pull/4654"},{"label":"Kitamura Lean proof repository","url":"https://github.com/KitaKen1/wowii-graph-conjecture-2-lean"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"green-open-problem-57","name":"Ben Green's Open Problem 57","shortName":"Green Problem 57","problemNumber":null,"field":"Higher-order Fourier analysis","fieldGroup":"Analysis","statement":"For a finite abelian group $G$, let $\\Phi(G)$ be the absolutely convex hull of the specified trilinear kernels and $\\Phi'(G)$ its restriction where the third factor depends only on $x_1 + x_2$. Is $\\Phi(G) = \\Phi'(G)$? A counterexample over $\\mathbb{Z}/3\\mathbb{Z}$ separates the hulls.","posedBy":"Ben Green","yearPosed":2024,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"intended complex form disproved, with a certified strict support-function gap","claimIssueNote":null,"solveDate":"2026-05-21","model":"AlphaProof Nexus","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"Solved autonomously by AlphaProof Nexus, with the proof formally verified in Lean.","verification":"lean-verified","verificationNote":"Lean-checked; formal proofs published with DeepMind's AlphaProof Nexus report (arXiv:2605.22763) and its accompanying repository.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"From Ben Green's public open-problem list.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.22763","sourceName":"arXiv:2605.22763 - AlphaProof Nexus report","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"pure-o-sequences-log-concavity","name":"Log-Concavity of Codimension-Three Pure O-Sequences","shortName":"Pure O-sequences","problemNumber":null,"field":"Commutative algebra","fieldGroup":"Algebra","statement":"For a pure O-sequence $h = (h_0, \\dots, h_e)$ of codimension three and type two, is $h_i^2 \\ge h_{i-1} h_{i+1}$ for every interior index $i$? The stated monomial case is proved; the broader level-Hilbert-function case remains open.","posedBy":null,"yearPosed":2022,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-21","model":"AlphaProof Nexus","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"Solved autonomously by AlphaProof Nexus, with the proof formally verified in Lean.","verification":"lean-verified","verificationNote":"Lean-checked; formal proofs published with DeepMind's AlphaProof Nexus report (arXiv:2605.22763) and its accompanying repository.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"An explicit conjecture inside the pure O-sequence program.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.22763","sourceName":"arXiv:2605.22763 - AlphaProof Nexus report","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-12","name":"Erdős Problem #12","shortName":"Erdős #12","problemNumber":12,"field":"Extremal Number Theory","fieldGroup":"Number theory","statement":"Let $A \\subset \\mathbb{N}$ be infinite with no distinct $a, b, c \\in A$ such that $a \\mid (b + c)$ with $b, c > a$. Can $|A \\cap [1, N]|/\\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?","posedBy":null,"yearPosed":1970,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"parts (i) and (ii) resolved - a near-linear-density construction exists, refuting the N^{1-c} decay; the reciprocal-sum part remains open","claimIssueNote":null,"solveDate":"2026-05-21","model":"AlphaProof Nexus","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked; formal proofs published with the AlphaProof Nexus report (arXiv:2605.22763).","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/12","sourceName":"erdosproblems.com/12","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"monochromatic-quantum-graphs-four-particles","name":"Four-Particle Monochromatic Quantum Graphs","shortName":"Quantum graphs, N=4","problemNumber":null,"field":"Quantum optics & graph amplitudes","fieldGroup":"Quantum information & computing","statement":"For four particles and local dimension $D \\ge 4$, can a complete edge-coloured, complex-weighted graph have unit perfect-matching amplitude for every monochromatic inherited colouring and zero for every nonmonochromatic one? Ruled out for the whole family, including the real-, integer- and trinary-weight variants.","posedBy":null,"yearPosed":2017,"ageNote":null,"solveType":"disproved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"the N = 4, D ≥ 4 family is fully ruled out; the general two-parameter problem remains open","claimIssueNote":null,"solveDate":"2026-05-21","model":"AlphaProof Nexus","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"Solved autonomously by AlphaProof Nexus, with the proofs formally verified in Lean.","verification":"lean-verified","verificationNote":"Lean-checked; formal proofs published with DeepMind's AlphaProof Nexus report (arXiv:2605.22763).","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"From the quantum-graph existence program of Krenn's catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.22763","sourceName":"arXiv:2605.22763 - AlphaProof Nexus report","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"monochromatic-quantum-graphs-diagonal","name":"Monochromatic Quantum Graphs in the Diagonal Family","shortName":"Quantum graphs, N=D","problemNumber":null,"field":"Quantum optics & graph amplitudes","fieldGroup":"Quantum information & computing","statement":"Can a complete edge-coloured, complex-weighted graph realize perfect-matching amplitudes of one on every monochromatic inherited vertex colouring and zero otherwise? Nonexistence is proved in the diagonal family $N = D$ for every even $N \\ge 4$, alongside further finite cases.","posedBy":null,"yearPosed":2017,"ageNote":null,"solveType":"disproved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"the diagonal family is ruled out; the broader two-parameter problem remains open","claimIssueNote":null,"solveDate":"2026-05-21","model":"AlphaProof Nexus","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"Solved autonomously by AlphaProof Nexus, with the proofs formally verified in Lean.","verification":"lean-verified","verificationNote":"Lean-checked; formal proofs published with DeepMind's AlphaProof Nexus report (arXiv:2605.22763).","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"From the quantum-graph existence program of Krenn's catalog.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.22763","sourceName":"arXiv:2605.22763 - AlphaProof Nexus report","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"fisher-information-log-convexity","name":"Log-Convexity of Fisher Information Along Heat Flow","shortName":"Fisher log-convexity","problemNumber":null,"field":"Information theory","fieldGroup":"Probability & statistics","statement":"For every smooth positive density $f$ on $\\mathbb{R}^d$, must the Fisher information $t \\mapsto I(f * \\gamma_t)$ be log-convex along the heat flow?","posedBy":null,"yearPosed":2015,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-18","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The hexagonal counterexample - a smooth positive Gaussian-decaying density on the plane - was found with GPT-5.5 Pro; tensorization extends the disproof to every dimension at least two.","verification":"unreviewed","verificationNote":"Public arXiv preprint with an explicit construction and numerics. 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 question in the information-theoretic entropy-power circle.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.18081","sourceName":"arXiv:2605.18081 - A hexagonal counterexample to log-convexity of Fisher information","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1039","name":"Erdős Problem #1039","shortName":"Erdős #1039","problemNumber":1039,"field":"Complex Analysis","fieldGroup":"Analysis","statement":"For $f(z) = \\prod_{i=1}^n (z - z_i)$ with all $|z_i| \\le 1$, let $\\rho(f)$ be the radius of the largest disc contained in $\\{z : |f(z)| < 1\\}$. Is $\\rho(f) \\gg 1/n$? The worst case is now known to be $\\Theta(1/n)$, with the explicit bound $\\rho(f) \\ge (\\log 2)/n$.","posedBy":null,"yearPosed":1958,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":"order of magnitude determined; the exact asymptotic constant remains open","claimIssueNote":null,"solveDate":"2026-05-17","model":"GPT-5.5 Pro, Codex 5.5","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The bounds were developed with GPT-5.5 Pro and Codex 5.5.","verification":"lean-verified","verificationNote":"Lean-checked and expert-vouched; official record updated.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1039","sourceName":"erdosproblems.com/1039","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"integral-invariant-cycles-degree-one","name":"Integral Local Invariant Cycles in Degree One","shortName":"Invariant cycles","problemNumber":null,"field":"Algebraic geometry","fieldGroup":"Algebra","statement":"For a semistable one-parameter family of complex projective varieties with smooth nearby fiber $X_t$ and monodromy $T$, is the map $H^1(X, \\mathbb{Z}) \\to H^1(X_t, \\mathbb{Z})^T$ surjective? True in degree one, although the integral statement fails in higher degree.","posedBy":null,"yearPosed":2026,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-17","model":"QED (GPT-5.5)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"QED found an independent proof of the degree-one theorem.","verification":"expert-verified","verificationNote":"Verified by the contributing domain expert; documented in the QED system paper.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":5,"significanceNote":"A recent technical statement with a one-paper audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2604.24021","sourceName":"arXiv:2604.24021 - QED: an open-source multi-agent system for mathematical proofs","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lamplighter-total-variation","name":"Total Variation for the Lamplighter Walk on Z","shortName":"Lamplighter TV","problemNumber":null,"field":"Probability on groups","fieldGroup":"Probability & statistics","statement":"For the switch-walk-switch walk on $\\mathbb{Z}_2 \\wr \\mathbb{Z}$ started at $(0,0)$ and $(0,2)$, prove $\\|P_t^x - P_t^y\\|_{TV} \\asymp t^{-1/2}$.","posedBy":null,"yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-15","model":"QED (GPT-5.5)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"Proved by the QED multi-agent system in decomposition mode; the expert who posed the problem provided no mathematical input beyond the statement.","verification":"expert-verified","verificationNote":"Verified by the contributing domain expert; documented in the QED system paper.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"An expert-posed question in random walk theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2604.24021","sourceName":"arXiv:2604.24021 - QED: an open-source multi-agent system for mathematical proofs","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"lamplighter-return-probability","name":"Return Probability for the Lamplighter Walk on a Tree","shortName":"Lamplighter return","problemNumber":null,"field":"Probability on groups","fieldGroup":"Probability & statistics","statement":"For the switch-walk-switch lamplighter walk on $\\mathbb{Z}_2 \\wr T_d$, prove the sharp asymptotic $p_{2n}(e,e) = \\rho_d^{2n} \\exp[-(\\pi^2 (\\log(d-1))^2 + o(1)) \\frac{n}{\\log^2 n}]$ with $\\rho_d = \\frac{2\\sqrt{d-1}}{d}$.","posedBy":null,"yearPosed":2025,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-15","model":"QED (GPT-5.5 Pro)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The QED multi-agent system produced the proof from the problem statement alone, through multiple rounds of decomposition and refinement.","verification":"expert-verified","verificationNote":"Verified by the contributing domain expert who posed the problem; public preprint.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"An expert-posed question in random walk theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.21744","sourceName":"arXiv:2605.21744 - Return probability for the switch-walk-switch lamplighter walk","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"gaussian-completely-monotone-conjecture","name":"Gaussian Completely Monotone Conjecture","shortName":"Gaussian CMC","problemNumber":null,"field":"Entropy & heat flow","fieldGroup":"Probability & statistics","statement":"Along the heat flow, do the successive time derivatives of the entropy of $X + \\sqrt{t}\\,Z$ alternate in sign, as conjectured by Cheng and Geng? An explicit measure on $\\mathbb{R}$ has a fifth derivative with the forbidden sign.","posedBy":"Fan Cheng & Yanlin Geng","yearPosed":2015,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"also refutes the McKean and Toscani conjectures","claimIssueNote":null,"solveDate":"2026-05-12","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Yuzhou Gu","Mark Sellke"],"aiRole":"The explicit counterexample measure was found by GPT-5.5 Pro.","verification":"unreviewed","verificationNote":"Author-checked arXiv preprint by Gu and Sellke. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"Cheng-Geng 2015, a named conjecture in network information theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.11656","sourceName":"arXiv:2605.11656 - A counterexample to the Gaussian completely monotone conjecture","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-690","name":"Erdős Problem #690","shortName":"Erdős #690","problemNumber":690,"field":"Number Theory","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-08","model":"Multiscalar Fields System","modelMaker":null,"humanCollaborators":["Davide Crapis","Shouqiao Wang"],"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":40,"citationsPaper":"Paul Erdős (1979), \"Some unconventional problems in number theory\", Astérisque","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W3173535233","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/690","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-7","name":"Erdős Problem #7","shortName":"Erdős #7","problemNumber":7,"field":"Number Theory, Covering Systems","fieldGroup":"Number theory","statement":"Can there be a finite covering system of the integers with distinct moduli, all of which are odd and greater than $1$?","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"proved","resolution":"retracted","aiContribution":null,"resultNote":null,"claimIssueNote":"The claimed Lean proof that no such covering system exists was withdrawn after audit: its central axiom asserted that a product of factors greater than one is less than one, and a statement-fidelity audit confirmed the gap. The problem remains open.","solveDate":"2026-05-07","model":"Aristotle","modelMaker":"Harmonic","humanCollaborators":[],"aiRole":"Both the failed formalization and the audit that exposed its false axiom were AI-assisted.","verification":"contested","verificationNote":"Claim withdrawn; see the claim issue. Recorded because failed formalizations are part of the honest history of AI mathematics.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/7","sourceName":"erdosproblems.com/7","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1032","name":"Erdős Problem #1032","shortName":"Erdős #1032","problemNumber":1032,"field":"Critical Graph Theory","fieldGroup":"Combinatorics","statement":"Do arbitrarily large 4-chromatic edge-critical graphs exist with minimum degree bounded below by a positive constant times the number of vertices?","posedBy":null,"yearPosed":1973,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"a new density-degree inequality gives δ(G) ≤ (3/10 + o(1))|V(G)|, improving 0.328; existence of a linear construction remains open","claimIssueNote":null,"solveDate":"2026-05-07","model":"GPT-5.5 Pro, Codex","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"lean-verified","verificationNote":"Lean-checked with expert screening.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1032","sourceName":"erdosproblems.com/1032","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-750","name":"Erdős Problem #750","shortName":"Erdős #750","problemNumber":750,"field":"Graph Theory, Chromatic Number","fieldGroup":"Combinatorics","statement":null,"posedBy":"Paul Erdős","yearPosed":1994,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-03","model":"GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Przemek Chojecki"],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/750","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-351","name":"Erdős Problem #351","shortName":"Erdős #351","problemNumber":351,"field":"Number Theory, Complete Sequences","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-03","model":"GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Kevin Barreto","Liam Price"],"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":371,"citationsPaper":"P. Erdős, R. Graham (1980), \"Old and new problems and results in combinatorial number theory\", Monographies de L'Enseignement Mathematique","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W1489006728","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/351","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-283","name":"Erdős Problem #283","shortName":"Erdős #283","problemNumber":283,"field":"Number Theory, Unit Fractions","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-03","model":"GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Kevin Barreto","Liam Price"],"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":371,"citationsPaper":"P. Erdős, R. Graham (1980), \"Old and new problems and results in combinatorial number theory\", Monographies de L'Enseignement Mathematique","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W1489006728","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/283","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-870","name":"Erdős Problem #870","shortName":"Erdős #870","problemNumber":870,"field":"Number Theory, Additive Bases","fieldGroup":"Number theory","statement":"Let $k\\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\\geq c\\log n$ for all large $n$ (where $r(n)$ counts representations of $n$ as a sum of at most $k$ elements of $A$) then $A$ must contain a minimal basis of order $k$? The claimed answer is no, for every $k\\geq 3$.","posedBy":"Paul Erdős, Melvyn Nathanson","yearPosed":1979,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":"A total refutation is claimed for all k>=3, building on the Larsen-Larsen resolution of problem #868; erdosproblems.com still lists the problem open","claimIssueNote":null,"solveDate":"2026-05-02","model":"GPT-5.4 Pro, GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["David Turturean"],"aiRole":"The proof was developed via an automated multi-turn scaffold that iteratively queried GPT-5.4 Pro and GPT-5.5 Pro over roughly forty turns, with constructions inspired by the Larsen-Larsen order-2 basis; the author later reworked the k=3 case after community concerns and verified the write-up himself and with GPT-5.5 Pro.","verification":"unreviewed","verificationNote":"Verification so far is by the author and by GPT-5.5 model runs he links; a Lean formalization attempt is blocked because the underlying Larsen-Larsen probabilistic construction resists autoformalization. No independent human review.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/870","sourceName":"erdosproblems.com/870","links":[{"label":"Write-up","url":"https://www.overleaf.com/read/gknkvvxrymfv#956531"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-694","name":"Erdős Problem #694","shortName":"Erdős #694","problemNumber":694,"field":"Number Theory","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1979,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05-01","model":"GPT-5.5 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":"argument","citations":40,"citationsPaper":"Paul Erdős (1979), \"Some unconventional problems in number theory\", Astérisque","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W3173535233","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/694","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":1,"downvotes":0,"commentCount":0},{"slug":"erdos-planar-unit-distance","name":"Erdős's Planar Unit Distance Conjecture","shortName":"Unit Distance Conj.","problemNumber":90,"field":"Combinatorial Geometry","fieldGroup":"Geometry & topology","statement":"Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.","posedBy":"Paul Erdős","yearPosed":1946,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-05","model":"OpenAI frontier model (specific version not disclosed)","modelMaker":"OpenAI","humanCollaborators":["Noga Alon","Thomas Bloom","Timothy Gowers","Daniel Litt","Will Sawin","Jacob Tsimerman","Melanie Matchett Wood"],"aiRole":"Model-assisted construction of a point configuration with more than $n^{1.014}$ unit-distance pairs, beating the conjectured bound.","verification":"expert-verified","verificationNote":"Independently checked by nine mathematicians and reported as meeting the bar for publication in a top journal.","publication":"announcement","resolutionMethod":"construction","citations":201,"citationsPaper":"Erdős (1946), \"On Sets of Distances of n Points\", Amer. Math. Monthly","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W4211205179","renownLangs":10,"renownNote":null,"significance":40,"significanceNote":"Erdős's 1946 unit-distance problem, a founding question of combinatorial geometry.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://asiliconvalleyinsider.com/2026/05/27/how-openai-models-helped-solve-erdos-problem-1196-and-disprove-erdoss-planar-unit-distance-conjecture/","sourceName":"A Silicon Valley Insider","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"carbery-almost-orthogonality","name":"Carbery's Almost-Orthogonality Inequality in Lp","shortName":"Carbery inequality","problemNumber":null,"field":"Functional analysis","fieldGroup":"Analysis","statement":"For $p \\ge 2$, does Carbery's proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power $2$ - and if not, what is the largest possible exponent?","posedBy":"Anthony Carbery","yearPosed":2009,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"exponent 2 fails for every p > 2; the sharp exponent p' form is proved for integer p ≥ 2","claimIssueNote":null,"solveDate":"2026-05","model":"Grok Heavy, Grok 4.20 Heavy","modelMaker":"xAI","humanCollaborators":["Ziang Chen","Jaume de Dios Pont","Paata Ivanisvili","Jose Madrid","Haozhu Wang"],"aiRole":"The authors knew a counterexample should exist from unstructured brute-force search; Grok produced a construction with a clear structural pattern, which revealed the optimal exponent p'.","verification":"unreviewed","verificationNote":"Author-checked public 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 Carbery question in harmonic analysis.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.05192","sourceName":"arXiv:2605.05192 - Almost-orthogonality in Lp spaces: a case study with Grok","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1151","name":"Erdős Problem #1151","shortName":"Erdős #1151","problemNumber":1151,"field":"Analysis, Polynomials","fieldGroup":"Analysis","statement":"Let $\\mathcal{L}^nf$ be the Lagrange interpolation polynomials of a continuous $f$ on the Chebyshev nodes. Prove that, for any closed $A\\subseteq [-1,1]$, there exists a continuous function $f$ such that $A$ is the set of limit points of $\\mathcal{L}^nf(x)$.","posedBy":"Paul Erdős","yearPosed":1999,"ageNote":"As recorded in the 1999 problem collection [Va99]; the question itself is older.","solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"An elementary solution via a primitive-row decomposition of the Chebyshev-node measures; the main theorem is formalized in Lean, but erdosproblems.com still lists the problem open","claimIssueNote":null,"solveDate":"2026-04-30","model":"GPT-5.5 Pro, Codex","modelMaker":"OpenAI","humanCollaborators":["Przemysław Chojecki","Allen Hart"],"aiRole":"The solution was obtained with GPT-5.5 Pro using an explicit primitive-row decomposition of the Chebyshev-node measures; Theorem 1.1(a), the main contribution, was subsequently formalized largely autonomously by ChatGPT and Codex.","verification":"lean-verified","verificationNote":"Theorem 1.1(a), the main part of the contribution, is formalized in Lean and the formalization was confirmed correct on the forum; part (b) is unformalized because it depends on an Erdős result absent from mathlib. erdosproblems.com still lists the problem 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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1151","sourceName":"erdosproblems.com/1151","links":[{"label":"Write-up","url":"https://www.ulam.ai/research/erdos1151.pdf"},{"label":"Lean formalization of Theorem 1.1(a)","url":"https://github.com/AllenGrahamHart/FormalConjectures-Bench/tree/main/formalizations/erdos1151"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1201","name":"Erdős Problem #1201","shortName":"Erdős #1201","problemNumber":1201,"field":"Number Theory, Primes","fieldGroup":"Number theory","statement":"Is it true that for every $\\epsilon,\\eta>0$ there exists a $k$ such that the density of $n$ for which $P(n(n+1)\\cdots(n+k))>n^{1-\\epsilon}$ is at least $1-\\eta$, where $P(m)$ is the greatest prime divisor of $m$? A short argument via the Matomäki-Radziwiłł theorem establishes the lower-density version.","posedBy":"Paul Erdős","yearPosed":1976,"ageNote":"Erdős proved the $n^{1/2-\\epsilon}$ version in a 1976 paper and conjectured the strengthening there; catalogued from his 1980 problem list.","solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":"As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open","claimIssueNote":null,"solveDate":"2026-04-30","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Przemysław Chojecki"],"aiRole":"The deduction from the Matomäki-Radziwiłł theorem on multiplicative functions was written by GPT-5.5 Pro; Tao and Sawin's forum discussion pinned down exactly what the known results do and do not give for this problem.","verification":"unreviewed","verificationNote":"Discussed on the problem's forum, including by Tao, who delineated the remaining natural-density gap; no independent review of the note itself.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1201","sourceName":"erdosproblems.com/1201","links":[{"label":"GPT-5.5 Pro note","url":"https://www.ulam.ai/research/erdos1201.pdf"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1133","name":"Erdős Problem #1133","shortName":"Erdős #1133","problemNumber":1133,"field":"Approximation Theory","fieldGroup":"Analysis","statement":"Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below $(1+\\varepsilon)n$ to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.","posedBy":null,"yearPosed":1967,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-29","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"unreviewed","verificationNote":"Public manuscript with a community-standard check only; the official record is still 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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1133","sourceName":"erdosproblems.com/1133","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"kirby-5-16-dga-undecidability","name":"Kirby Problem 5.16 for Noncommutative Semifree DGAs","shortName":"Kirby 5.16 (DGAs)","problemNumber":null,"field":"Decision problems in topology","fieldGroup":"Geometry & topology","statement":"For semifree noncommutative differential graded algebras over a nontrivial computable unital commutative ring, are stable tame isomorphism, quasi-isomorphism, or derived Morita equivalence algorithmically decidable? All three are undecidable.","posedBy":null,"yearPosed":null,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"resolves the noncommutative half of the Kirby-list problem","claimIssueNote":null,"solveDate":"2026-04-28","model":"Aletheia (Gemini Deep Think)","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"Two essentially autonomous solutions were produced by the Aletheia research agent iterating generate-verify-revise on Gemini Deep Think.","verification":"unreviewed","verificationNote":"Human-checked public proofs with released transcripts. Not yet peer-reviewed.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":20,"significanceNote":"From the Kirby problem list, low-dimensional topology's recognized ledger.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2605.08122","sourceName":"arXiv:2605.08122 - Undecidability problems for semifree DG algebras","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1092","name":"Erdős Problem #1092","shortName":"Erdős #1092","problemNumber":1092,"field":"Graph Theory, Chromatic Number","fieldGroup":"Combinatorics","statement":null,"posedBy":"Paul Erdős","yearPosed":1976,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-28","model":"GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Przemek Chojecki"],"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":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1092","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-43","name":"Erdős Problem #43","shortName":"Erdős #43","problemNumber":43,"field":"Number Theory, Sidon Sets","fieldGroup":"Number theory","statement":"If Sidon sets $A, B \\subseteq \\{1, \\dots, N\\}$ satisfy $(A-A) \\cap (B-B) = \\{0\\}$, must $\\binom{|A|}{2} + \\binom{|B|}{2} \\le \\binom{f(N)}{2} + O(1)$, where $f(N)$ is the largest Sidon-set size in $[N]$ - and can the bound be improved by a fixed proportion when $|A| = |B|$?","posedBy":null,"yearPosed":1982,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":"both proposed bounds fail","claimIssueNote":null,"solveDate":"2026-04-27","model":"GPT-5.5 Pro, Aristotle, Claude","modelMaker":"OpenAI / Harmonic / Anthropic","humanCollaborators":[],"aiRole":"The equal-size bound is disproved by an explicit construction; the unrestricted bound fails as a consequence of the resolution of Erdős Problem #42.","verification":"site-confirmed","verificationNote":"The official Erdős problems record marks both questions answered negatively, with component Lean proofs.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/43","sourceName":"erdosproblems.com/43","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-953","name":"Erdős Problem #953","shortName":"Erdős #953","problemNumber":953,"field":"Geometric Measure Theory","fieldGroup":"Geometry & topology","statement":"What is the largest possible measure of a subset of a radius-$R$ disk in $\\mathbb{R}^2$ containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives $M(R) \\ll R^{1/2}$; with Sárközy's lower construction, $M(R) = R^{1/2 + o(1)}$.","posedBy":null,"yearPosed":1977,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-27","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"unreviewed","verificationNote":"Public proof; expert digestion ongoing.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/953","sourceName":"erdosproblems.com/953","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1101","name":"Erdős Problem #1101","shortName":"Erdős #1101","problemNumber":1101,"field":"Sieve Theory","fieldGroup":"Number theory","statement":"Does there exist a good pairwise-coprime sequence $u_n$ with $\\sum 1/u_n < \\infty$ and polynomial growth? What if one only requires $u_n \\le e^{o(n)}$?","posedBy":null,"yearPosed":1981,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"a subexponential good sequence is constructed; the polynomial-growth question remains open","claimIssueNote":null,"solveDate":"2026-04-27","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"unreviewed","verificationNote":"Community note reporting the construction.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1101","sourceName":"erdosproblems.com/1101","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-42","name":"Erdős Problem #42","shortName":"Erdős #42","problemNumber":42,"field":"Number Theory, Sidon Sets, Additive Combinatorics","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1995,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-27","model":"GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Harjas Sandhu"],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/42","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-896","name":"Erdős Problem #896","shortName":"Erdős #896","problemNumber":896,"field":"Number Theory","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1972,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-26","model":"GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Przemek Chojecki"],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/896","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1138","name":"Erdős Problem #1138","shortName":"Erdős #1138","problemNumber":1138,"field":"Number Theory, Primes","fieldGroup":"Number theory","statement":null,"posedBy":"Various","yearPosed":1999,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-25","model":"GPT-5.5 Pro, GPT-5.5 Thinking","modelMaker":null,"humanCollaborators":["Kireet Cheri","Sourish Kumrawat","Hrishi Sunder"],"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":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1138","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-888","name":"Erdős Problem #888","shortName":"Erdős #888","problemNumber":888,"field":"Number Theory, Squares","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1998,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-25","model":"Aristotle, GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["Przemek Chojecki"],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/888","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-906","name":"Erdős Problem #906","shortName":"Erdős #906","problemNumber":906,"field":"Analysis, Entire Functions","fieldGroup":"Analysis","statement":"Is there an entire non-zero function $f:\\mathbb{C}\\to \\mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\\cdots$, the set $\\{ z: f^{(n_k)}(z)=0 \\textrm{ for some }k\\geq 1\\}$ is everywhere dense? The literal question is trivial for polynomials, so the claims address the transcendental entire case, in the affirmative.","posedBy":"Paul Erdős","yearPosed":1956,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"Two independent affirmative claims (Adriano's, posted first, and a GPT-5.5 Pro note); Erdős himself wrote in 1982 that the problem had been solved affirmatively long before, without a locatable reference","claimIssueNote":null,"solveDate":"2026-04-25","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":["Przemysław Chojecki"],"aiRole":"The probabilistic argument via the cofinite reformulation, using Sodin's Edelman-Kostlan and Offord-type estimates for Gaussian analytic functions, was developed with GPT-5.5 Pro; an independent solution by another contributor was posted first the same day.","verification":"unreviewed","verificationNote":"AI screenings reported one minor issue on each claim; no formalization (the required tools are not in mathlib) and no independent expert review; 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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/906","sourceName":"erdosproblems.com/906","links":[{"label":"GPT-5.5 Pro note","url":"https://www.ulam.ai/research/erdos906.pdf"},{"label":"Independent proposed solution","url":"https://github.com/Drill23/erdos-problem-906-proposed-solution"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-38","name":"Erdős Problem #38","shortName":"Erdős #38","problemNumber":38,"field":"Number Theory","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1956,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-25","model":"GPT-5.5 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":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/38","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"optimal-strategies-in-the-all-heads-coin-game","name":"Optimal Strategies in the All-Heads Coin Game","shortName":"All-heads coin game","problemNumber":null,"field":"Markov decision processes","fieldGroup":"Probability & statistics","statement":"In the all-heads coin game a player starts with $n$ coins, each showing heads with\nprobability $p$; each round all remaining coins are flipped, the player must set aside at\nleast one head (losing if none shows), and wins once all coins are set aside. Determine\noptimal strategies and the winning probability $w_{n,p}$. Resolved: for $p=\\tfrac12$ every\nstrategy achieves $w_{n,1/2}=\\tfrac12$; for $p>\\tfrac12$ the single-head strategy One is\noptimal, $n\\mapsto w_{n,p}$ is strictly increasing, and $W(p)=\\lim_n w_{n,p}$ has an explicit\nseries representation. In the regime $p<\\tfrac12$, explicitly left open by van Doorn, a\nfirst-order perturbation in $\\delta=\\tfrac12-p$ gives a closed-form description: the deficit\nsatisfies $\\tfrac12-w_{n,1/2-\\delta}\\approx\\delta c_n$, where $c_n$ obeys a linear recursion\nfor $n\\ge7$ with limit $L\\approx1.7035$, and to first order the optimal-value sequence has a\nstrict local minimum at $n=5$ and no local maximum.","posedBy":"W. van Doorn (small-$p$ regime left open; game builds on a question of J. Breitner)","yearPosed":2024,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-24","model":"Claude Opus 4.6 / 4.7 / 4.8","modelMaker":"Anthropic","humanCollaborators":["Peter Pfaffelhuber"],"aiRole":"Per the paper's authorship disclosure: Claude (Anthropic; versions Opus 4.6, 4.7, 4.8), used\ninteractively, produced the mathematical text, the numerical code, and the complete Lean\n4/Mathlib formalization. The underlying ideas, choice of research question, the structuring\nof the joint induction, and the decision to formally verify are the author's; Claude's role\nwas execution — drafting exposition, proposing and debugging Lean proof tactics, selecting\nMathlib lemmas, producing numerical scripts — with every edit reviewed by the author.","verification":"lean-verified","verificationNote":"Every numbered result, including the perturbation analysis, is formally verified in Lean 4 with Mathlib: no `sorry`, no custom axioms (only propext, Classical.choice, Quot.sound), no `native_decide`/`unsafe`. Trust surface is two files (`CoinsLean/Challenge.lean`, `CoinsLean/CoinsLean/Defs.lean`), independently checkable via the Lean comparator on the public repository; manuscript↔Lean map in Appendix A. arXiv preprint (v2, June 2026), not peer-reviewed.\n\nStatus set to partially resolved (2026-08-02): p = 1/2 and p > 1/2 are fully resolved, but for p < 1/2 the paper gives only a first-order expansion in δ = 1/2 − p near 1/2, leaving the range of validity δ₀(n) open and the numerically observed local maxima outside its reach. Verification tier unchanged: the Lean checks what the paper claims, and the paper does not claim the full small-p regime.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":null,"significanceNote":null,"solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2604.22991","sourceName":"arXiv:2604.22991 [math.PR]","links":[{"label":"Lean formalization + full transcript:","url":"https://github.com/pfaffelh/coins"}],"submittedBy":"PluckyGecko226","upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-330","name":"Erdős Problem #330","shortName":"Erdős #330","problemNumber":330,"field":"Number Theory, Additive Basis","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-24","model":"GPT-5.5 Pro","modelMaker":null,"humanCollaborators":["David Turturean"],"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":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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/330","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-202","name":"Erdős Problem #202","shortName":"Erdős #202","problemNumber":202,"field":"Covering Systems","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1961,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-23","model":"GPT-5.4 Pro","modelMaker":null,"humanCollaborators":["Boon Suan Ho"],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/202","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1014","name":"Erdős Problem #1014","shortName":"Erdős #1014","problemNumber":1014,"field":"Graph Theory, Ramsey Theory","fieldGroup":"Combinatorics","statement":null,"posedBy":"Paul Erdős","yearPosed":1971,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-23","model":"OpenAI internal model","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":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1014","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1190","name":"Erdős Problem #1190","shortName":"Erdős #1190","problemNumber":1190,"field":"Number Theory, Covering Systems","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-23","model":"GPT-5.4 Pro","modelMaker":null,"humanCollaborators":["Boon Suan Ho"],"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":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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1190","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-863","name":"Erdős Problem #863","shortName":"Erdős #863","problemNumber":863,"field":"Number Theory, Sidon Sets, Additive Combinatorics","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1992,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-22","model":"GPT-5.4 Pro","modelMaker":null,"humanCollaborators":["Boon Suan Ho"],"aiRole":"Ho and GPT-5.4 Pro observed that a positive answer follows by connecting existing results: a routine adaptation of the Erdős-Turán bound for Sidon sets, together with a construction of Cilleruelo, Ruzsa, and Trujillo, gives $c'_r < c_r$ for all $r\\ge 2$.","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":3,"citationsPaper":"P. Erdős (1992), \"Some of my forgotten problems in number theory\", Hardy-Ramanujan J.","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W2611456884","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/863","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-603","name":"Erdős Problem #603","shortName":"Erdős #603","problemNumber":603,"field":"Combinatorics, Set Theory","fieldGroup":"Combinatorics","statement":null,"posedBy":"Paul Erdős","yearPosed":1987,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-21","model":"GPT-5.4 Pro","modelMaker":null,"humanCollaborators":["Przemek Chojecki"],"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":9,"citationsPaper":"P. Erdős (1985), \"Some problems on finite and infinite graphs\", Logic and combinatorics (Arcata, Calif.,","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W4249085394","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/603","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-610","name":"Erdős Problem #610","shortName":"Erdős #610","problemNumber":610,"field":"Graph Theory","fieldGroup":"Combinatorics","statement":null,"posedBy":"Paul Erdős, Tibor Gallai, Zsolt Tuza","yearPosed":1992,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-21","model":"Aristotle, GPT-5.4 Pro","modelMaker":null,"humanCollaborators":["Przemek Chojecki"],"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":67,"citationsPaper":"Paul Erdős, Tibor Gallai, Zsolt Tuza (1992), \"Covering the cliques of a graph with vertices\", Discrete Math.","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W2086357278","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/610","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-996","name":"Erdős Problem #996","shortName":"Erdős #996","problemNumber":996,"field":"Analysis, Fourier Series","fieldGroup":"Analysis","statement":"Let $n_1<n_2<\\cdots$ be a lacunary sequence of integers and $f\\in L^2([0,1])$ with $n$th Fourier partial sum $f_n$. Is there an absolute constant $C>0$ such that if $\\| f-f_n\\|_2 \\ll (\\log\\log\\log n)^{-C}$ then $\\frac{1}{N}\\sum_{k\\leq N}f(\\{\\alpha n_k\\})\\to\\int_0^1 f$ for almost every $\\alpha$? A preprint answers this negatively via a dyadic spike-block counterexample.","posedBy":"Paul Erdős","yearPosed":1964,"ageNote":null,"solveType":"disproved","resolution":"candidate","aiContribution":"ai-assisted","resultNote":"Answered negatively in a preprint that also settles the p=2 case of problem #995; erdosproblems.com still lists the problem open","claimIssueNote":null,"solveDate":"2026-04-21","model":"GPT-5.4 Pro","modelMaker":"OpenAI","humanCollaborators":["Boon Suan Ho"],"aiRole":"Per the paper's acknowledgements, GPT-5.4 Pro was used during development to explore proof strategies, test intermediate formulations and assist with exposition; all arguments were independently verified by the author, who takes full responsibility.","verification":"unreviewed","verificationNote":"An arXiv preprint with no independent review yet; erdosproblems.com still lists the problem open.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/996","sourceName":"erdosproblems.com/996","links":[{"label":"arXiv preprint","url":"https://arxiv.org/abs/2604.18535"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-522","name":"Erdős Problem #522","shortName":"Erdős #522","problemNumber":522,"field":"Random Polynomials","fieldGroup":"Probability & statistics","statement":"For $P_n(z) = \\sum_{k=0}^n \\varepsilon_k z^k$ with independent uniform signs, does the number $R_n$ of roots in $|z| \\le 1$ satisfy $R_n/(n/2) \\to 1$ almost surely? The manuscript proves the strong law with $R_n = n/2 + O_\\omega(n^{149/150})$.","posedBy":null,"yearPosed":1961,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-20","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"unreviewed","verificationNote":"Public manuscript; a full expert review has not been located.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/522","sourceName":"erdosproblems.com/522","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1195","name":"Erdős Problem #1195","shortName":"Erdős #1195","problemNumber":1195,"field":"Analysis, Number Theory","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-19","model":"GPT-5.4 Pro","modelMaker":null,"humanCollaborators":["Boon Suan Ho"],"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":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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1195","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1217","name":"Erdős Problem #1217","shortName":"Erdős #1217","problemNumber":1217,"field":"Number Theory, Divisors, Primitive Sets","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős, András Sárközy, Endre Szemerédi","yearPosed":1966,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-16","model":"GPT-5.4 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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1217","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"avidor-zwick-max-cut-sdp","name":"Avidor-Zwick Question on Low-Dimensional Max-Cut SDP","shortName":"Avidor-Zwick Max-Cut","problemNumber":null,"field":"Approximation algorithms","fieldGroup":"Theoretical computer science","statement":"For fixed $d$, can every $d$-dimensional feasible solution of the triangle-strengthened Max-Cut SDP be rounded in polynomial time with ratio strictly larger than $\\alpha_{GW}$? A rounding achieving $\\alpha_{GW} + 2^{-O(d)}$ answers yes.","posedBy":"Adi Avidor & Uri Zwick","yearPosed":2005,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-16","model":"Gemini (internal), ChatGPT-5.2 Extended Pro, Gemini 3.0 Pro DeepThink","modelMaker":"Google DeepMind / OpenAI","humanCollaborators":[],"aiRole":"The key anti-concentration lemma for signs of low-dimensional Gaussian projections was first proved by Google's internal Gemini model with a weaker bound; the optimal $2^{-\\Theta(d)}$ form was then obtained with ChatGPT-5.2 Extended Pro and Gemini 3.0 Pro DeepThink, with proofs edited by the authors.","verification":"unreviewed","verificationNote":"Author-edited and checked 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 from the Max-Cut SDP literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2604.13971","sourceName":"arXiv:2604.13971 - Max Cut with small-dimensional SDP solutions","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-741","name":"Erdős Problem #741","shortName":"Erdős #741","problemNumber":741,"field":"Additive Combinatorics","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1994,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-16","model":"DeepMind prover agent","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":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/741","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-858","name":"Erdős Problem #858","shortName":"Erdős #858","problemNumber":858,"field":"Number Theory, Primitive Sets","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1970,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-15","model":"GPT-5.4 Pro","modelMaker":null,"humanCollaborators":["Przemek Chojecki"],"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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/858","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-856","name":"Erdős Problem #856","shortName":"Erdős #856","problemNumber":856,"field":"Number Theory","fieldGroup":"Number theory","statement":"Let $k\\geq 3$ and $f_k(N)$ be the maximum of $\\sum_{n\\in A}\\frac{1}{n}$ over all $A\\subseteq\\{1,\\ldots,N\\}$ containing no $k$ subsets with the same pairwise least common multiple. Estimate $f_k(N)$. The claimed answer: $f_k(N)=(\\log N)^{\\gamma_k+o(1)}$, where $\\gamma_k$ is a weighted generalization of the Tang-Zhang sunflower capacity.","posedBy":"Paul Erdős","yearPosed":1970,"ageNote":null,"solveType":"proved","resolution":"candidate","aiContribution":"ai-co-developed","resultNote":"Identifies the exponent as a variational sunflower-capacity constant, sharpening the Tang-Zhang bounds; the value of that constant itself remains open, as does site acceptance","claimIssueNote":null,"solveDate":"2026-04-15","model":"GPT-5.4 Pro","modelMaker":"OpenAI","humanCollaborators":["Przemysław Chojecki"],"aiRole":"A weighted version of the Tang-Zhang sunflower-capacity argument giving the exact logarithmic exponent was developed with GPT-5.4 Pro, using a mass-transport idea from the forum's discussion of problem #1196.","verification":"unreviewed","verificationNote":"An AI screening found no issues and no prior literature with the result; the site's owner unpacked and restated the main claim without checking details, a Lean formalization attempt hit missing mathlib prerequisites, and the problem is still listed 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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/856","sourceName":"erdosproblems.com/856","links":[{"label":"Write-up","url":"https://www.ulam.ai/research/erdos856-final.pdf"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-258","name":"Erdős Problem #258","shortName":"Erdős #258","problemNumber":258,"field":"Irrationality","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-14","model":"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":"argument","citations":371,"citationsPaper":"P. Erdős, R. Graham (1980), \"Old and new problems and results in combinatorial number theory\", Monographies de L'Enseignement Mathematique","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W1489006728","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/258","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-960","name":"Erdős Problem #960","shortName":"Erdős #960","problemNumber":960,"field":"Geometry","fieldGroup":"Geometry & topology","statement":null,"posedBy":"Paul Erdős","yearPosed":1984,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-09","model":"OpenAI internal model","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":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/960","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-987","name":"Erdős Problem #987","shortName":"Erdős #987","problemNumber":987,"field":"Analysis, Discrepancy","fieldGroup":"Analysis","statement":null,"posedBy":"Paul Erdős","yearPosed":1964,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-09","model":"OpenAI internal model","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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/987","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-990","name":"Erdős Problem #990","shortName":"Erdős #990","problemNumber":990,"field":"Analysis","fieldGroup":"Analysis","statement":null,"posedBy":"Paul Erdős","yearPosed":1964,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-09","model":"OpenAI internal model","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":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/990","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1091","name":"Erdős Problem #1091","shortName":"Erdős #1091","problemNumber":1091,"field":"Graph Theory, Chromatic Number","fieldGroup":"Combinatorics","statement":null,"posedBy":"Paul Erdős","yearPosed":1976,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-09","model":"OpenAI internal model","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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1091","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1141","name":"Erdős Problem #1141","shortName":"Erdős #1141","problemNumber":1141,"field":"Number Theory, Primes","fieldGroup":"Number theory","statement":null,"posedBy":"Various","yearPosed":1999,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-09","model":"OpenAI internal model","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":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1141","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"adaboost-cycling","name":"Exhaustive AdaBoost Cycling Question","shortName":"AdaBoost cycling","problemNumber":null,"field":"Learning theory","fieldGroup":"Theoretical computer science","statement":"Does exhaustive AdaBoost always converge to a finite cycle of weak classifiers and weight vectors on every finite training set? A finite instance whose orbit never becomes periodic answers no.","posedBy":"Cynthia Rudin, Robert Schapire & Ingrid Daubechies","yearPosed":2012,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-co-developed","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-08","model":"GPT-5.4 Pro, Claude Opus 4.6","modelMaker":"OpenAI / Anthropic","humanCollaborators":[],"aiRole":"The block-product gadget - two factors sharing an exact period-2 orbit whose linearized return maps have dominant eigenvalues with an irrational logarithmic ratio - was developed with GPT-5.4 Pro and Claude Opus 4.6.","verification":"unreviewed","verificationNote":"All assertions certified by exact rational arithmetic; computer-assisted 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":"Posed by Rudin, Schapire and Daubechies; cited across boosting theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2604.07055","sourceName":"arXiv:2604.07055 - AdaBoost does not always cycle","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-26","name":"Erdős Problem #26","shortName":"Erdős #26","problemNumber":26,"field":"Number Theory, Divisors","fieldGroup":"Number theory","statement":"Let $A\\subset\\mathbb{N}$ be infinite. Must there exist some $k\\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\\in A$? The question as posed follows negatively from Davenport–Erdős (1951). The AI result settles Tenenbaum's harder variant, also negatively: there is an infinite $A$ such that for every $k\\geq 1$ the set of multiples of $A+k$ has upper density below $0.34$.","posedBy":"Paul Erdős, Gérald Tenenbaum","yearPosed":1995,"ageNote":null,"solveType":"disproved","resolution":"variant","aiContribution":"ai-discovered","resultNote":"The question as posed was implicit in Davenport–Erdős (1951); the AI result settles Tenenbaum's open variant negatively","claimIssueNote":null,"solveDate":"2026-04-06","model":"DeepMind prover agent","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"A DeepMind prover agent constructed an infinite set $A$ such that for every $k\\geq 1$ the set of multiples of $A+k$ has upper density less than $0.34$, resolving Tenenbaum's variant of the problem in the negative.","verification":"site-confirmed","verificationNote":"erdosproblems.com marks the problem DISPROVED and documents the DeepMind construction in the page remarks; the variant result is recorded there without a separate formal artifact.","publication":"announcement","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/26","sourceName":"erdosproblems.com/26","links":[{"label":"Formalised statement (formal-conjectures)","url":"https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/26.lean"}],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"anderson-quasi-completeness","name":"Anderson's Quasi-Completeness Question","shortName":"Anderson quasi-complete","problemNumber":null,"field":"Commutative algebra","fieldGroup":"Algebra","statement":"Is every weakly quasi-complete Noetherian local ring quasi-complete? Asked by D. D. Anderson in 2014. The ring $A = k^p[[X, Y]][k]$ with $k = \\mathbb{F}_p(u_1, u_2, \\dots)$ is weakly quasi-complete but not quasi-complete.","posedBy":"D. D. Anderson","yearPosed":2014,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-04","model":"Rethlas + Archon (GPT-5.4 Pro)","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":"The dual-agent framework (Rethlas for informal reasoning, Archon for formal verification) ran roughly 80 hours; the decisive example is a classical ring going back to Nagata, which the system recognized as answering Anderson's question.","verification":"lean-verified","verificationNote":"Lean-checked with a statement comparator guarding against misformalization; arXiv preprint documents the pipeline.","publication":"preprint","resolutionMethod":"construction","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A named but specialist question in commutative ring theory.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2604.03789","sourceName":"arXiv:2604.03789 - Automated conjecture resolution with formal verification","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"anchored-gda-last-iterate","name":"Last-Iterate Rate for Anchored Gradient Descent-Ascent","shortName":"Anchored GDA rate","problemNumber":null,"field":"Convex optimization","fieldGroup":"Algorithms & optimization","statement":"For smooth convex-concave min-max problems, can anchored gradient descent-ascent be scheduled so that its exact last-iterate squared-gradient residual is $O(1/t)$, closing the gap left by the 2019 analysis?","posedBy":null,"yearPosed":2019,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-04","model":"AlphaProof Nexus","modelMaker":"Google DeepMind","humanCollaborators":[],"aiRole":"The agent searched for the anchoring schedule and its proof simultaneously, discovering a parameter choice yielding the stronger guarantee via a discrete-time recurrence argument rather than the usual continuous-time ODE analysis.","verification":"lean-verified","verificationNote":"Lean-checked; accompanying arXiv preprint by the DeepMind team.","publication":"preprint","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A specialist rate question in min-max optimization.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://arxiv.org/abs/2604.03782","sourceName":"arXiv:2604.03782 - An improved last-iterate convergence rate for anchored gradient descent ascent","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-152","name":"Erdős Problem #152","shortName":"Erdős #152","problemNumber":152,"field":"Sidon Sets","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős, András Sárközy, Vera T. Sós","yearPosed":1994,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04-03","model":"DeepMind prover agent","modelMaker":null,"humanCollaborators":[],"aiRole":null,"verification":"site-confirmed","verificationNote":"Marked solved by erdosproblems.com's official status. 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Solve credited via Terence Tao's AI-contributions wiki.","publication":"announcement","resolutionMethod":"argument","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 Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1202","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-1196-primitive-sets","name":"Erdős Problem #1196 — Primitive Sets","shortName":"Primitive Sets (#1196)","problemNumber":1196,"field":"Number Theory","fieldGroup":"Number theory","statement":"Bounds the weighted sum $\\sum 1/(a \\log a)$ taken over primitive sets of integers (sets where no element divides another).","posedBy":"Paul Erdős, András Sárközy, Endre Szemerédi","yearPosed":1966,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-04","model":"GPT-5.4 Pro","modelMaker":"OpenAI","humanCollaborators":["Boris Alexeev","Kevin Barreto","Yanyang Li","Jared Duker Lichtman","Liam Price","Jibran Iqbal Shah","Quanyu Tang","Terence Tao"],"aiRole":"Price, an amateur with no advanced math training, fed GPT-5.4 Pro the bare problem statement with no historical context. It found the key move: reweighting the random walk via the von Mangoldt function.","verification":"lean-verified","verificationNote":"The raw output needed cleanup, but the underlying idea held up. Examined, corrected, generalized and written up in a joint account by Alexeev, Barreto, Li, Lichtman, Price, Shah, Tang and Tao. Formalised in Lean.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":15,"significanceNote":"The Erdős-Sárközy-Szemerédi primitive-sets line, with a real literature.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/1196","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-514","name":"Erdős Problem #514","shortName":"Erdős #514","problemNumber":514,"field":"Entire Functions","fieldGroup":"Analysis","statement":"For a transcendental entire function, how fast can $|f(z)|$ be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus $M(r, f)$?","posedBy":null,"yearPosed":1961,"ageNote":null,"solveType":"proved","resolution":"partial","aiContribution":"ai-discovered","resultNote":"an escape path dominating every power of |z| with bounded initial length is constructed, and universal positive-power lower bounds are ruled out; the broader variant remains open","claimIssueNote":null,"solveDate":"2026-04","model":"GPT-5.5 Pro","modelMaker":"OpenAI","humanCollaborators":[],"aiRole":null,"verification":"unreviewed","verificationNote":"Public manuscript with community discussion.","publication":"announcement","resolutionMethod":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/514","sourceName":"erdosproblems.com/514","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-997","name":"Erdős Problem #997","shortName":"Erdős #997","problemNumber":997,"field":"Analysis, Discrepancy, Primes","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős","yearPosed":1964,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-03-31","model":"OpenAI internal model","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":"argument","citations":null,"citationsPaper":null,"citationsSource":null,"citationsUrl":null,"renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/997","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-380","name":"Erdős Problem #380","shortName":"Erdős #380","problemNumber":380,"field":"Number Theory","fieldGroup":"Number theory","statement":null,"posedBy":"Paul Erdős, Ronald Graham","yearPosed":1980,"ageNote":null,"solveType":"proved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-03-31","model":"GPT-5.4 Pro","modelMaker":null,"humanCollaborators":["Terence Tao"],"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":371,"citationsPaper":"P. Erdős, R. Graham (1980), \"Old and new problems and results in combinatorial number theory\", Monographies de L'Enseignement Mathematique","citationsSource":"OpenAlex","citationsUrl":"https://openalex.org/W1489006728","renownLangs":0,"renownNote":null,"significance":10,"significanceNote":"A numbered problem from the Erdős catalog: real and documented, with a specialist audience.","solveCostUsd":null,"solveCostNote":null,"sourceUrl":"https://www.erdosproblems.com/380","sourceName":"erdosproblems.com","links":[],"submittedBy":null,"upvotes":0,"downvotes":0,"commentCount":0},{"slug":"erdos-125","name":"Erdős Problem #125","shortName":"Erdős #125","problemNumber":125,"field":"Number Theory, Base Representations","fieldGroup":"Number theory","statement":null,"posedBy":"Stefan Burr, Paul Erdős, Ronald Graham, Wen-Ching Winnie Li","yearPosed":1996,"ageNote":null,"solveType":"disproved","resolution":"resolved","aiContribution":"ai-discovered","resultNote":null,"claimIssueNote":null,"solveDate":"2026-03-30","model":"DeepMind prover agent","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":10,"citationsPaper":"S. A. Burr, P. Erdős, R. L. Graham, W. 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