{"schema":"postcutoff/openai-math-families@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/science/openai-math/","md":"https://postcutoff.com/science/openai-math/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"read":"2026-10-29","source":"https://github.com/openai/math (CONTENTS.md, lean/formalization.yaml, history.md)","families":[{"family":"001","title":"Milne’s rationality conjecture and algebraic specialization","summary":"Proves Milne's rationality conjecture for abelian varieties over \\overline{\\mathbb Q} with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.","papers":["https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-hodge-conjecture-cm-abelian-varieties-openai/"],"name_hits":[],"tier":"B","note":"Milne's rationality conjecture for abelian varieties with good reduction. Milne: 'it seems, the statements in the article are now proven'; Madapusi wrote an improved exposition.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/AGENT_NOTES.md","https://www.jmilne.org/math/articles/abstracts.html"]},{"family":"002","title":"The full BSD formula from low Selmer corank","summary":"Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.","papers":["https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"A","note":"Full BSD formula when some p-power Selmer corank is 0 or 1. Calegari, Colmez: technical, no progress on general BSD; Burungale: correctness still being analysed.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/AGENT_NOTES.md","https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","https://galoisrepresentations.org/2026/10/15/bsd-and-the-p-converse-guest-post-ashay-burungale/","https://x.com/ColmezPierre/status/2107898549476180125"]},{"family":"003","title":"The quasi-Riemann hypothesis","summary":"Proves that every Dirichlet L-function, including \\zeta(s), is zero-free in \\Re s\\gt 7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over \\mathbb Q(\\sqrt{-3}). A companion gives a different proof of the zero-free half-plane \\Re s\\gt 11/12. ([Lean](lean/docs/003.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","https://postcutoff.com/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/","https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/","https://postcutoff.com/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/"],"name_hits":[],"tier":"A","note":"Zero-free half-plane Re s > 7/8 for ζ and Dirichlet L-functions, Lean (two kernels). Siegel-zero companion reproved by Kerr and Green; Tao sketch; Quanta feature.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/003.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/003.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.21099","https://terrytao.wordpress.com/2026/10/19/a-sketch-of-the-proof-of-non-existence-of-siegel-zeroes/","https://www.quantamagazine.org/ai-proved-the-quasi-riemann-hypothesis-what-happens-now-20261019/"]},{"family":"004","title":"Hilbert’s tenth problem over ℚ","summary":"Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.","papers":["https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-hilbert-tenth-problem-rationals-claim/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-08-hilbert-number-lower-bound-gpt-5-6-sol-lean","2026-10-16-ai-assisted-results-october-15-listing","2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition"],"tier":"A","note":"Rational solvability of integer polynomials is undecidable. Unformalized; uses families 006 and 010. Calegari: 'threw me for a loop'; Pasten: 'original'; Sun: 86 unknowns.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/AGENT_NOTES.md","https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/","https://arxiv.org/abs/2610.20691","https://blog.computationalcomplexity.org/2026/10/conjecture-that-i-thought-would-not-be.html"]},{"family":"005","title":"Irrationality of Catalan’s constant","summary":"Proves that Catalan's constant G=\\sum_{j\\ge0}(-1)^j/(2j+1)^2 is irrational. ([Lean](lean/docs/005.md))","papers":["https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-catalan-constant-irrational/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"A","note":"Catalan's constant is irrational; one-line Lean statement over Mathlib, comparator accepted. Calegari sees new ideas or optimizations; follow-ups derive μ(G) ≤ 740.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/005.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/005.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/AGENT_NOTES.md","https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","https://hexagonmath.org/abs/2610.00187v5","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"006","title":"Goldfeld’s conjecture: densities and mean analytic rank","summary":"Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.","papers":["https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"A","note":"Goldfeld's conjecture: quadratic twists split 50/50 between analytic ranks 0 and 1, mean 1/2. A GitHub check found a weak proof step, not a disproof.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/AGENT_NOTES.md","https://github.com/holimart/math/issues/2","https://github.com/borism/openai-math-slop/issues/1","https://galoisrepresentations.org/2026/10/15/bsd-and-the-p-converse-guest-post-ashay-burungale/"]},{"family":"007","title":"Ordinary two-point correlations and the corrected Elliott conjecture","summary":"Proves the ordinary two-point Chowla conjecture, with a bound O(X/(\\log X)^c) for Liouville correlation sums along fixed nonproportional affine forms, where c &gt; 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times n^{it} for |t|\\le X. ([Lean](lean/docs/007.md))","papers":["https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Ordinary (non-logarithmic) two-point Chowla and binary corrected Elliott conjectures; Lean main theorem accepted. Wang–Yi build on it; no expert check yet. Lead filed.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/007.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf","https://github.com/openai/math/blob/main/lean/docs/007.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.16394","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"008","title":"The Deligne–Drinfeld conjecture","summary":"Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture. ([Lean](lean/docs/008.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-deligne-drinfeld-conjecture-proof/"],"name_hits":[],"tier":"A","note":"Deligne–Drinfeld proved; Willwacher: 'remaining part … solved by OpenAI', simplified proof; Hirose independent proof; Lean main statement accepted; HMI workshop. Own entry (confirmed).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/008.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/008.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.15592","https://arxiv.org/abs/2610.23719","https://www.tcd.ie/news_events/articles/2026/delignedrinfeld-conjecture-workshop/","https://sites.google.com/view/hmi-workshop-2026/schedule","https://x.com/hirose_minoru/status/2112725404645826716"]},{"family":"009","title":"Function-field reconstruction from Milnor K-theory and Galois data","summary":"Reconstructs function fields of transcendence degree at least two over algebraically closed constants from K^{\\mathrm M}_1/\\ell, K^{\\mathrm M}_2/\\ell, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic. ([Lean](lean/docs/009.md))","papers":["https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf","https://github.com/openai/math/blob/main/preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Function fields recovered from mod-ℓ Milnor K-theory; Bogomolov–Pop reconstruction. Lean covers uniqueness only. AHGT seminar talks announced on these manuscripts.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/009.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf","https://github.com/openai/math/blob/main/lean/docs/009.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/AGENT_NOTES.md","https://ahgt.math.cnrs.fr/activities/workshops/Seminar-Taming_the_flow_2026/"]},{"family":"010","title":"Unrestricted pro-modularity at the prime two","summary":"Every continuous odd absolutely irreducible two-dimensional 2-adic representation of G_{\\mathbb Q} unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.","papers":["https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf","https://github.com/openai/math/blob/main/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf","https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Fontaine–Mazur for odd 2-dimensional 2-adic representations without residual hypotheses. Calegari: main difficulties 'technical'; it is an input to the Hilbert-tenth proof.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/AGENT_NOTES.md","https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/"]},{"family":"011","title":"Prime-factor statistics of $`p-1`$","summary":"Proves that the normalized ordered logarithms of the prime factors of p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law \\mathrm{PD}(1) as p ranges uniformly over primes up to x and x\\to\\infty. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than n^{1-\\varepsilon} totient preimages, for every ε &gt; 0.","papers":["https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu/","https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/"],"name_hits":[],"tier":"B","note":"Poisson–Dirichlet law for prime factors of p−1 (Ford–Konyagin–Luca) and totients with n^(1−ε) preimages (Erdős #821, per Bloom). Lü–Wang–Wu build on it.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"012","title":"Independent largest prime factors of consecutive integers","summary":"Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and n+1 are asymptotically independent in ordinary natural density. In particular, the integers satisfying P^+(n)\\lt P^+(n+1) have density 1/2. ([Lean](lean/docs/012.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-07-09-gpt-5-6-sol-terra-luna","2026-07-17-cycle-double-cover-conjecture-proved","2026-07-21-openai-agents-hugging-face-intrusion","2026-07-23-frontier-act-trahan-obernolte","2026-07-23-massachusetts-senate-frontier-ai-independent-reviews","2026-07-27-crouzeix-conjecture-proved"],"tier":"B","note":"Erdős–Pomerance: largest prime factors of n and n+1 asymptotically independent; P⁺(n) < P⁺(n+1) has density 1/2 (Erdős #371). Lean main accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/012.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/012.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"013","title":"Ostmann’s inverse Goldbach conjecture","summary":"Proves that no finite modification of the primes can be written as A+B with A,B\\subseteq\\mathbb Z_{\\ge0} each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability. ([Lean](lean/docs/013.md))","papers":["https://github.com/openai/math/blob/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Ostmann's inverse Goldbach: no finite modification of the primes is a sumset A+B (Erdős #431). Lean main. Leonetti and Elsholtz–Stadlmann follow up.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/013.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/013.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.15758","https://arxiv.org/abs/2610.20554","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"014","title":"Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan","summary":"Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over \\overline{\\mathbb F}_q under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p &gt; 0, the same conclusion holds assuming additionally that p is very good for the group and p\\nmid |W_G|. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.","papers":["https://github.com/openai/math/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf","https://github.com/openai/math/blob/main/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf","https://github.com/openai/math/blob/main/preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026/Tame-Hecke-Eigensheaves-with-Several-Marked-Points.pdf","https://github.com/openai/math/blob/main/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Restricted geometric Langlands in characteristic p and Ramanujan for generic cuspidal exceptional-group representations over function fields. Eight manuscripts; no expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/AGENT_NOTES.md"]},{"family":"015","title":"Torus-packet equidistribution in prime, quartic, and sextic degrees","summary":"Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity. ([Lean](lean/docs/015.md))","papers":["https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf","https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Haar equidistribution of torus packets in prime degree ≥ 5 and primitive quartic and sextic fields. Lean covers the prime-degree theorem.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/015.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/015.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/AGENT_NOTES.md"]},{"family":"016","title":"Zilber–Pink in abelian varieties and the Siegel threefold","summary":"Proves the abelian Zilber–Pink conjecture over \\overline{\\mathbb Q}: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold \\mathcal A_2 for Hodge-generic curves defined over \\overline{\\mathbb Q}, without boundary or reduction assumptions.","papers":["https://github.com/openai/math/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-abelian-zilber-pink-claim/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"A","note":"Abelian Zilber–Pink over Q̄ (hence over any char-0 field via Barroero–Dill) and curves in A2. Unformalized; Barroero–Dill–Kühne, Dogra–Saha cite it as announced.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.10212","https://arxiv.org/abs/2610.17325"]},{"family":"017","title":"The irrationality exponent of π is 2","summary":"Proves that the irrationality exponent of π is exactly 2: for every ε &gt; 0 and all sufficiently large denominators q, every rational p/q satisfies |\\pi-p/q|\\ge q^{-2-\\varepsilon}. This also proves convergence of the Flint–Hills series \\sum_{n\\ge1}1/(n^3\\sin^2 n), with angles in radians. ([Lean](lean/docs/017.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-irrationality-exponent-pi-openai/","https://postcutoff.com/e/2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt/"],"name_hits":[],"tier":"A","note":"Irrationality exponent of π is 2 (Lean), implying Flint–Hills convergence. Calegari: 'amazing result'; Zudilin notes closeness to Roth's argument (via Persiflage).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/017.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/017.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/AGENT_NOTES.md","https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/"]},{"family":"018","title":"The Margulis–Platonov conjecture over global fields","summary":"Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of G(k) is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.","papers":["https://github.com/openai/math/blob/main/preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/margulis-platonov-global-function-fields.pdf","https://github.com/openai/math/blob/main/preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Margulis–Platonov conjecture on noncentral normal subgroups of G(k) over every global field, including characteristic 2. No expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/margulis-platonov-global-function-fields.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/AGENT_NOTES.md"]},{"family":"019","title":"The local p-adic section conjecture and global consequences","summary":"Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚ<sub>p</sub> correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves X_0(N) and X_1(N) of genus at least two.","papers":["https://github.com/openai/math/blob/main/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"A","note":"Local p-adic section conjecture; Grothendieck's section conjecture over ℚ for X0(N), X1(N). Unformalized. Carlson–Kim–Srinivasan–Stix online seminar since 20 Oct. Lead filed.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/AGENT_NOTES.md","https://www.minhyongkim.net/home/p-adic-section-conjecture"]},{"family":"020","title":"Squarefree quartics and power-free polynomial values","summary":"Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the (d-2)-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4. ([Lean](lean/docs/020.md))","papers":["https://github.com/openai/math/blob/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Squarefree values of irreducible quartics with predicted density; (d−2)-free values in degrees 4–8, covering all d ≥ 4 with Browning. Lean main accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/020.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf","https://github.com/openai/math/blob/main/lean/docs/020.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/AGENT_NOTES.md"]},{"family":"021","title":"A quadratic bound for Jacobsthal’s function","summary":"Answers Jacobsthal's quadratic-bound question: every interval of Ck^2 consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss. ([Lean](lean/docs/021.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Jacobsthal's function: every Ck² consecutive integers contain one coprime to any number with k prime factors. Lean, with a log-log refinement.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/021.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/021.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/AGENT_NOTES.md"]},{"family":"022","title":"The weak inhomogeneous Duffin–Schaeffer conjecture","summary":"Proves that for every real shift γ and finite-valued \\psi:\\mathbb N\\to[0,\\infty), divergence of \\sum_q\\phi(q)\\psi(q)/q implies \\|qx-\\gamma\\|\\lt \\psi(q) for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.","papers":["https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026"],"tier":"B","note":"Weak inhomogeneous Duffin–Schaeffer conjecture for every real shift. Manuscript only; no expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/AGENT_NOTES.md"]},{"family":"023","title":"Patterson's first moment for cubic Gauss sums","summary":"Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order X^{5/6}/\\log X. Every fixed nonzero prime-angle Fourier mode has smaller order. ([Lean](lean/docs/023.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/"],"name_hits":[],"tier":"B","note":"Unconditional Patterson first moment for cubic Gauss sums over primary Eisenstein primes. Lean main; a 19 Oct arXiv paper uses its character moments.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/023.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/023.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/AGENT_NOTES.md"]},{"family":"024","title":"An asymptotic formula for the number of totients","summary":"Gives an asymptotic equivalent for the number V(x) of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, V(cx)/V(x)\\to c for every fixed c &gt; 0, answering Erdős and Hall’s scaling question. ([Lean](lean/docs/024.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Asymptotic formula for the number of totients up to x; V(cx)/V(x) → c answers Erdős–Hall (Erdős #416, per Bloom). Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/024.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/024.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"025","title":"Short Egyptian fractions","summary":"Every rational a/b with 1\\le a\\lt b is a sum of O(\\log\\log b) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions. ([Lean](lean/docs/025.md))","papers":["https://github.com/openai/math/blob/main/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Erdős: every a/b is a sum of O(log log b) distinct unit fractions (#304). Lean. Bloom–van Doorn human exposition and simplification; erdosproblems credits it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/025.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/025.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Short-Egyptian-fractions-September-25-2026/AGENT_NOTES.md","https://hexagonmath.org/abs/2610.00407v1","https://www.erdosproblems.com/304"]},{"family":"026","title":"Positive lower density of large prime gaps","summary":"For every fixed C &gt; 0, a positive proportion of consecutive prime gaps exceed C\\log p_n, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where p_n/n increases have positive lower density, answering Erdős and Prachar. ([Lean](lean/docs/026.md))","papers":["https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-06-collatz-positive-density-log-time-mazur-codex","2026-09-22-claude-opus-5-5","2026-10-07-lebrun-salamon-conjecture-proof","2026-10-22-metr-redwood-red-team-openai-agent-monitoring","2026-10-26-ai-assisted-results-october-26-listing"],"tier":"B","note":"A positive proportion of prime gaps exceed C·log p for each C; answers Erdős–Prachar (#968). Lean covers only the p_n/n corollary.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/026.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/026.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"027","title":"Potential integral density on curve character varieties","summary":"Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SL<sub>r</sub> character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes. ([Lean](lean/docs/027.md))","papers":["https://github.com/openai/math/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-28-anthropic-ipo-prospectus-leak"],"tier":"C","note":"Determinant-one curve case of Litt's integral-density question for SL_r character varieties. Lean covers a supporting construction only.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/027.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/027.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/AGENT_NOTES.md"]},{"family":"028","title":"Uniformly bounded components of Gaussian-prime graphs","summary":"Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes. ([Lean](lean/docs/028.md))","papers":["https://github.com/openai/math/blob/main/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Gaussian moat conjecture: no bounded-step walk to infinity through Gaussian primes; components uniformly bounded. Lean main accepted; nonexplicit bound. Lead filed.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/028.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/028.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"029","title":"Primitive roots for every admissible integer base","summary":"Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least c_a x/(\\log x)^2 primes in every sufficiently large interval (x,2x) have primitive root a, with c_a\\gt 0.","papers":["https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf","https://github.com/openai/math/blob/main/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-artin-primitive-root-conjecture-infinitude/"],"name_hits":[],"tier":"A","note":"Infinitude in Artin's primitive root conjecture for every admissible base, ≫ x/(log x)² primes. Calegari: 'not at all surprising' given QRH; Bhargava adds positive density.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/AGENT_NOTES.md","https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","https://hexagonmath.org/abs/2610.00228v1"]},{"family":"030","title":"Modularity of elliptic curves over imaginary quadratic fields","summary":"Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.","papers":["https://github.com/openai/math/blob/main/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-modularity-elliptic-curves-imaginary-quadratic/"],"name_hits":[],"tier":"A","note":"Every elliptic curve over every imaginary quadratic field is modular. Calegari: 'a really nice trick'; Bao Le Hung extends it to all totally real fields.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/AGENT_NOTES.md","https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","https://hexagonmath.org/abs/2610.00156v1"]},{"family":"031","title":"Uchida’s conjecture for open homomorphisms of Galois groups","summary":"Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.","papers":["https://github.com/openai/math/blob/main/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Uchida's conjecture: open homomorphisms between solvably closed Galois groups of number fields come from field embeddings. No expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/AGENT_NOTES.md"]},{"family":"032","title":"The rational Hodge conjecture for CM abelian varieties","summary":"Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic. ([Lean](lean/docs/032.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-hodge-conjecture-cm-abelian-varieties-openai/"],"name_hits":[],"tier":"A","note":"Rational Hodge for CM abelian varieties; three K3 companions withdrawn 7 Oct, Lean added 26 Oct. Van Hoften–Sempliner reprove it; Milne: 'in fact been proved'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/032.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/032.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/AGENT_NOTES.md","https://hexagonmath.org/abs/2610.00259v2","https://www.jmilne.org/math/articles/abstracts.html","https://github.com/openai/math/blob/main/history.md"]},{"family":"033","title":"Iitaka subadditivity, variation, and logarithmic additivity","summary":"Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-\\mathcal C manifolds with rational simple-normal-crossing boundaries. For projective fibrations f:U\\to V of smooth complex quasi-projective varieties with connected fibers, general fiber F, and \\bar\\kappa(V)\\ge0, proves Popa's inequality \\bar\\kappa(U)\\ge\\kappa(F)+\\max\\{\\bar\\kappa(V),\\mathop{\\mathrm{Var}}\\nolimits (f)\\}, where variation measures the whole geometric generic fiber. ([Lean](lean/docs/033.md))","papers":["https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/"],"name_hits":[],"tier":"A","note":"Campana's orbifold Iitaka subadditivity and Popa's inequality; companion claims ordinary Iitaka C_{n,m}. Lean covers one branch only. Covered in the abundance entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/033.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/033.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/AGENT_NOTES.md"]},{"family":"034","title":"Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity","summary":"Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.","papers":["https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-indices-for-semi-log-canonical-log-Calabi-Yau-pairs-October-5-2026/uniform-slc-index.pdf","https://github.com/openai/math/blob/main/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026/uniform-log-iitaka.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/"],"name_hits":[],"tier":"A","note":"Log abundance in characteristic zero and for compact Kähler spaces, effective Iitaka fibration; Kähler papers revised 7 Oct. Jihao Liu raises priority; no expert assessment.","lean_scope":"","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/AGENT_NOTES.md","https://github.com/openai/math/blob/main/history.md","https://arxiv.org/abs/2610.35913"]},{"family":"035","title":"Log-canonical threefold abundance in numerical dimension one","summary":"Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p &gt; 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.","papers":["https://github.com/openai/math/blob/main/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"C","note":"Log abundance for lc threefolds in characteristic p > 3 in numerical dimension one. Manuscript only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/AGENT_NOTES.md"]},{"family":"036","title":"Numerical semiampleness and generalized minimal models","summary":"Proves numerical semiampleness for nef adjoints K_X+B+M with K_X+B pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.","papers":["https://github.com/openai/math/blob/main/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/numerical-generalized-abundance.pdf","https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Minimal-models-in-numerical-dimension-one-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/"],"name_hits":[],"tier":"B","note":"Numerical semiampleness of nef generalized adjoints; minimal models for generalized lc pairs. Kähler paper revised 7 Oct; listed in the abundance entry.","lean_scope":"","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/numerical-generalized-abundance.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-6-2026/AGENT_NOTES.md","https://github.com/openai/math/blob/main/history.md"]},{"family":"037","title":"The ordinary-double-point volume gap","summary":"Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most 2(n-1)^n. Equality holds precisely for an analytic ordinary double point.","papers":["https://github.com/openai/math/blob/main/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Ordinary-double-point volume gap: singular klt germs have normalized volume ≤ 2(n−1)ⁿ, equality only for ordinary double points. No expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/AGENT_NOTES.md"]},{"family":"038","title":"Fujita’s freeness conjecture","summary":"Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint K_X+mL is globally generated for every integer m\\ge n+1.","papers":["https://github.com/openai/math/blob/main/preprints/Fujitas-freeness-conjecture-September-23-2026/Fujitas-freeness-conjecture-September-23-2026.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"A","note":"Fujita's freeness conjecture at the sharp bound m ≥ n+1 in every dimension. 23 Sep manuscript, unformalized; no expert assessment found. Lead filed.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Fujitas-freeness-conjecture-September-23-2026/Fujitas-freeness-conjecture-September-23-2026.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Fujitas-freeness-conjecture-September-23-2026/AGENT_NOTES.md"]},{"family":"039","title":"Nagata’s conjecture and maximal Seshadri constants","summary":"Proves Nagata's strict inequality \\sum_i m_i\\lt d\\sqrt r for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities m<sub>i</sub>. It also proves maximal multipoint Seshadri constants (L^n/r)^{1/n} for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic. ([Lean](lean/docs/039.md))","papers":["https://github.com/openai/math/blob/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026/seshadri-positive-characteristic.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/"],"name_hits":[],"tier":"A","note":"Nagata's conjecture for r ≥ 10 very general points and maximal Seshadri constants. Lean main. Tevelev's exposition calls the proof remarkably simple.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/039.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/039.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/AGENT_NOTES.md"]},{"family":"040","title":"Bloch’s conjecture for complex surfaces","summary":"Proves Bloch's conjecture: for every smooth connected projective complex surface S with p_g(S)=0, the Albanese map \\mathrm{CH}_0(S)^0\\to\\mathrm{Alb}(S)(\\mathbb C) on integral degree-zero zero-cycles is an isomorphism. This combines the new p_g=q=0 theorem with the classical theorem of Bloch, Kas, and Lieberman.","papers":["https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-bloch-conjecture-surfaces-claim/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-12-bloch-constant-conjecture-disproved"],"tier":"A","note":"Bloch's conjecture for every surface with p_g = 0, via Quot-scheme virtual localization. Unformalized; Litt flags a miscitation he thinks harmless.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/AGENT_NOTES.md","https://x.com/littmath/status/2107616410897678765"]},{"family":"041","title":"Hyperkähler SYZ and projective-space bases","summary":"Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.","papers":["https://github.com/openai/math/blob/main/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026/main.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-08-hyperkahler-syz-conjecture-engel-mauri/"],"name_hits":[],"tier":"B","note":"Strong hyperkähler SYZ (nef isotropic classes semiample) and projective-space bases of Lagrangian fibrations. Engel–Mauri posted an independent human proof.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.12277"]},{"family":"042","title":"Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII","summary":"Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex surface of class VII is Oka exactly when it is a Hopf or Enoki surface.","papers":["https://github.com/openai/math/blob/main/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-06-openai-log-abundance-nonvanishing-claim"],"tier":"B","note":"Every complex K3 surface is Oka; Oka classification of minimal class VII surfaces. An et al. (arXiv 2610.10475) credit its local-to-global argument.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.10475"]},{"family":"043","title":"P = W for fixed-determinant SL<sub>n</sub> moduli spaces","summary":"Proves P_k=W_{2k}=W_{2k+1} on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.","papers":["https://github.com/openai/math/blob/main/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Fixed-determinant P = W in composite rank, completing all coprime ranks with the known prime-rank cases. No comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/AGENT_NOTES.md"]},{"family":"044","title":"The equivariant cohomological Hikita conjecture","summary":"Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable point is stable and the gauge action is free, the equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic cocharacter-fixed locus of its flavor-deformed Coulomb branch.","papers":["https://github.com/openai/math/blob/main/preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Equivariant cohomological Hikita conjecture for all quivers. Ben Webster's GitHub repo Hikita-AI hosts a reader's guide and rewrite, with text contributed by Hiraku Nakajima.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026/AGENT_NOTES.md","https://github.com/bwebste/Hikita-AI"]},{"family":"046","title":"Shafarevich counterexamples in dimension two and with large fundamental group","summary":"Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.","papers":["https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf","https://github.com/openai/math/blob/main/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-05-20-ai-disproves-erdos-unit-distance-conjecture","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Counterexamples to Shafarevich's holomorphic convexity conjecture: a projective surface and a fourfold with large fundamental group. No expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/AGENT_NOTES.md"]},{"family":"047","title":"Zariski cancellation and affine fibrations over the complex numbers","summary":"Constructs an integral complex affine fourfold X\\not\\cong\\mathbb A^4 with X\\times\\mathbb A^1\\cong\\mathbb A^5, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections X\\to\\mathbb A^1 and \\mathbb A^5\\to\\mathbb A^2 have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial. ([Lean](lean/docs/047.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Zariski cancellation fails over ℂ in dimension 4 (explicit fourfold); Dolgachev–Weisfeiler disproved. Lean accepted; Huang's entry cites it only. Lead filed.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/047.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/047.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.12581","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"048","title":"A characteristic-zero counterexample to Lipman–Zariski","summary":"Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.","papers":["https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Characteristic-zero counterexample to the Zariski–Lipman conjecture: a singular normal surface with free tangent sheaf. Manuscript only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2604.24508"]},{"family":"049","title":"A stable-coordinate counterexample in four variables","summary":"Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces. ([Lean](lean/docs/049.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf","https://github.com/openai/math/blob/main/preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"A stable coordinate that is not a coordinate in four variables; Abhyankar–Sathaye counterexample. Lean main accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/049.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf","https://github.com/openai/math/blob/main/lean/docs/049.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/AGENT_NOTES.md"]},{"family":"050","title":"A counterexample to Griffiths’ positivity conjecture","summary":"Constructs ample rank-two bundles on \\mathbb P^1\\times\\mathbb P^1 with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface. ([Lean](lean/docs/050.md))","papers":["https://github.com/openai/math/blob/main/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Ample rank-two bundles on P¹×P¹ without Griffiths-positive metrics, disproving Griffiths' positivity conjecture. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/050.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/050.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026/AGENT_NOTES.md"]},{"family":"051","title":"Kobayashi’s canonical-ampleness conjecture","summary":"Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.","papers":["https://github.com/openai/math/blob/main/preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026/canonical-ampleness.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Kobayashi: compact Kähler manifolds without entire curves have ample canonical bundle. Qin–Tian cite it in arXiv 2609.25762 v5.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026/canonical-ampleness.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2609.25762"]},{"family":"052","title":"Tangent splittings and product decompositions","summary":"A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition. ([Lean](lean/docs/052.md))","papers":["https://github.com/openai/math/blob/main/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Beauville splitting for two integrable summands and Höring's integrability conjecture on rationally connected manifolds. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/052.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/052.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/AGENT_NOTES.md"]},{"family":"053","title":"A counterexample to Pixton completeness in Chow","summary":"Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.","papers":["https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"A tautological relation vanishing in Chow but outside Pixton's span disproves Pixton's completeness conjecture. Johannes Schmitt signed up to digest it.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/AGENT_NOTES.md"]},{"family":"054","title":"Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations","summary":"For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.","papers":["https://github.com/openai/math/blob/main/preprints/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"A","note":"Very general cubic fourfolds of large admissible discriminant are irrational despite K3 associations, disproving Kuznetsov's conjecture. No expert comment found. Lead filed.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026/AGENT_NOTES.md"]},{"family":"055","title":"Gepner symmetry and large-volume stability on threefolds","summary":"Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift 2/5. Also constructs numerical Bridgeland stability conditions at every sufficiently large volume on all smooth projective complex threefolds with trivial canonical bundle, with the exact ordinary and square-root-Todd central charges.","papers":["https://github.com/openai/math/blob/main/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Toda's Gepner stability condition on every quintic threefold; large-volume Bridgeland stability on all Calabi–Yau threefolds. No expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/AGENT_NOTES.md"]},{"family":"056","title":"Termination of projective and Kähler fourfold minimal model programs","summary":"Proves termination of every existing generalized log canonical flip sequence on globally Weil ℚ-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small flip diagrams with the prescribed ample signs. Also proves termination of arbitrary permitted minimal model programs for projective log canonical fourfolds with rational boundary in characteristic zero.","papers":["https://github.com/openai/math/blob/main/preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-7-2026/termination-generalized-lc-kahler-fourfolds.pdf","https://github.com/openai/math/blob/main/preprints/Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-6-2026/termination-generalized-terminal-flips-compact-kahler-fourfolds.pdf","https://github.com/openai/math/blob/main/preprints/Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/"],"name_hits":[],"tier":"B","note":"Termination of MMPs for projective lc fourfolds and Kähler fourfolds; three manuscripts revised 7 Oct. Listed in the abundance entry.","lean_scope":"","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-7-2026/termination-generalized-lc-kahler-fourfolds.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-7-2026/AGENT_NOTES.md","https://github.com/openai/math/blob/main/history.md"]},{"family":"057","title":"Fundamental groups of special complex varieties and root orbifolds","summary":"Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective complex fourfolds along one nonempty smooth connected divisor, when special in the stated differential-line sense. For smooth special complex quasi-projective varieties, proves that every finite-dimensional complex linear representation of the fundamental group has virtually nilpotent image of class at most two.","papers":["https://github.com/openai/math/blob/main/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor-October-5-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. No expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/AGENT_NOTES.md"]},{"family":"058","title":"Semialgebraic universal covers and bounded domains","summary":"Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products D\\times\\mathbb C^m\\times F, with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂ<sup>n</sup> has a finite étale cover by an abelian variety. ([Lean](lean/docs/058.md))","papers":["https://github.com/openai/math/blob/main/preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Kollár–Pardon conjecture on semialgebraic universal covers. Lean covers the bounded-symmetric-domain companion only.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/058.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/058.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/AGENT_NOTES.md"]},{"family":"059","title":"Counterexamples to Zariski’s multiplicity conjecture","summary":"Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ<sup>4</sup> with isolated critical points and multiplicities four and five.","papers":["https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Counterexamples to Zariski's 1971 multiplicity conjecture: ambiently homeomorphic hypersurface germs with different multiplicities. Manuscript only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/AGENT_NOTES.md"]},{"family":"060","title":"The Global Spherical Shell conjecture","summary":"Every connected minimal compact complex surface of class VII with b_2\\gt 0 contains a global spherical shell, proving the positive-b<sub>2</sub> Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in \\mathbb C^2\\setminus\\{0\\} whose complement is connected.","papers":["https://github.com/openai/math/blob/main/preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Global spherical shell conjecture for class VII surfaces with b₂ > 0, completing their classification modulo known cases. No expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/AGENT_NOTES.md"]},{"family":"062","title":"Projective contact classification and the LeBrun–Salamon conjecture","summary":"Proves the LeBrun–Salamon conjecture: every closed connected positive quaternionic-Kähler manifold of real dimension at least eight is homothetic to a compact symmetric Wolf space. It also proves contact-Fano homogeneity and classifies smooth connected complex projective contact manifolds of complex dimension at least three: those with b_2=1 are adjoint varieties with their canonical contact structures, while those with b_2\\ge2 have underlying manifold \\mathbb P(T^*Z) for a smooth projective variety Z.","papers":["https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/"],"name_hits":[],"tier":"B","note":"LeBrun–Salamon conjecture and projective contact classification. Hu–Liu–Wan posted an independent proof; covered by our two-proofs entry.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.10410"]},{"family":"063","title":"The generalized Mukai conjecture","summary":"Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies \\rho(\\iota-1)\\le n, with equality exactly for (\\mathbb P^{\\iota-1})^\\rho. Here the pseudoindex is the least anticanonical degree of a rational curve.","papers":["https://github.com/openai/math/blob/main/preprints/The-Generalized-Mukai-Conjecture-September-24-2026/article.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Generalized Mukai conjecture ρ(ι−1) ≤ n for Fano manifolds. Pearson's arXiv v2 (2604.25023) notes the claim.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Generalized-Mukai-Conjecture-September-24-2026/article.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Generalized-Mukai-Conjecture-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2604.25023"]},{"family":"064","title":"Topological triviality of μ-constant surface singularities","summary":"Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ<sup>3</sup> with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.","papers":["https://github.com/openai/math/blob/main/preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/main.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026"],"tier":"B","note":"Surface case of the μ-constant (Lê–Ramanujam) problem: μ-constant families in ℂ³ are topologically trivial. Manuscript only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/AGENT_NOTES.md"]},{"family":"065","title":"Virasoro constraints for complete intersections and projective-bundle towers","summary":"Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.","papers":["https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf","https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026/article.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Virasoro constraints for smooth complete intersections and projective-bundle towers in all genera. No expert comment found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/AGENT_NOTES.md"]},{"family":"066","title":"Bounded klt complements for Fano contractions","summary":"Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ. ([Lean](lean/docs/066.md))","papers":["https://github.com/openai/math/blob/main/preprints/Bounded-klt-complements-for-Fano-contractions-September-25-2026/Bounded-klt-complements-for-Fano-contractions-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Shokurov's bounded klt complements for ε-lc Fano contractions. Lean covers a local compactness lemma only.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/066.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Bounded-klt-complements-for-Fano-contractions-September-25-2026/Bounded-klt-complements-for-Fano-contractions-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/066.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Bounded-klt-complements-for-Fano-contractions-September-25-2026/AGENT_NOTES.md"]},{"family":"067","title":"The Campana–Peternell conjecture in dimension six","summary":"Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.","papers":["https://github.com/openai/math/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Campana–Peternell conjecture in dimension six: Fano sixfolds with nef tangent bundle are rational homogeneous. Manuscript only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/AGENT_NOTES.md"]},{"family":"068","title":"Anticanonical nonvanishing in every dimension","summary":"If X is a smooth connected complex projective variety and -K_X admits a smooth Hermitian metric with nonnegative curvature, then H^0(X,-mK_X)\\ne0 for some m &gt; 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.","papers":["https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"C","note":"Smooth semipositive anticanonical bundle forces a section of −mK_X. Seven manuscripts; only AI-written digests found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/AGENT_NOTES.md"]},{"family":"069","title":"Global quantum geometric Langlands at irrational level","summary":"Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level c\\in\\mathbb C\\setminus\\mathbb Q. It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.","papers":["https://github.com/openai/math/blob/main/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Quantum geometric Langlands equivalence at irrational level for simple groups. An automated referee report (OSVerify) flags issues.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/AGENT_NOTES.md","https://osverify.github.io/cases/quantum-langlands.html"]},{"family":"071","title":"Koebe’s circle-domain conjecture","summary":"Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture. ([Lean](lean/docs/071.md))","papers":["https://github.com/openai/math/blob/main/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Koebe's 1908 circle-domain conjecture (existence) and the removability-to-rigidity half of He–Schramm. Lean covers both. No expert comment found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/071.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/071.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/AGENT_NOTES.md","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"072","title":"Brennan's conjecture and the integral-means spectrum","summary":"Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |\\phi'|^s is area-integrable for 4/3\\lt s\\lt 4. The sharp universal integral-means identity is B_{\\mathcal S}(t)=|t|-1 for t ≤ −2. A strict bound B_b(-1)\\lt 1/4 for bounded univalent functions disproves Kraetzer's prediction at that parameter. ([Lean](lean/docs/072.md))","papers":["https://github.com/openai/math/blob/main/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov/"],"name_hits":[],"tier":"A","note":"Brennan's conjecture and the integral-means spectrum, Lean main. Binder, Ivrii, Prause and Smirnov checked it and gave a shorter proof.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/072.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/072.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.23853"]},{"family":"073","title":"The Falconer distance conjecture","summary":"Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set E\\subset\\mathbb R^d with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure. ([Lean](lean/docs/073.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-falconer-distance-conjecture-lean/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-06-openai-release-3d-restriction-bochner-riesz-claims","2026-10-16-ai-assisted-results-october-15-listing"],"tier":"A","note":"Falconer's distance conjecture in every dimension; Lean statement uses Mathlib's dimH and Lebesgue measure, accepted. Cheng et al. build a quantitative planar version.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/073.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/073.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.22222","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"074","title":"Kakeya in three and four dimensions","summary":"Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps L^3(\\mathbb R^3) to L^3(S^2) with norm O_\\varepsilon(\\delta^{-\\varepsilon}) for every ε &gt; 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.","papers":["https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/"],"name_hits":[],"tier":"A","note":"Kakeya maximal conjecture in ℝ³ and full dimension of 4D Kakeya sets (manuscripts only); superseded by family 079's all-dimension Lean claim.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/AGENT_NOTES.md"]},{"family":"075","title":"The $`L\\log L`$ Fourier-convergence conjecture","summary":"Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in L\\log L(\\mathbb T) converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the L\\log L scale. ([Lean](lean/docs/075.md))","papers":["https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Fourier partial sums of every L log L function converge almost everywhere, the classical endpoint question beyond Carleson–Hunt. Lean main. No expert comment found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/075.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/075.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/AGENT_NOTES.md"]},{"family":"076","title":"Real ultraflat Littlewood polynomials and unbounded binary merit factors","summary":"Constructs polynomials with N consecutive coefficients in \\{-1,1\\} whose modulus is (1+o(1))\\sqrt N uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture. ([Lean](lean/docs/076.md))","papers":["https://github.com/openai/math/blob/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf","https://github.com/openai/math/blob/main/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf","https://github.com/openai/math/blob/main/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Ultraflat ±1 polynomials at every large length; disproves Turyn's bounded merit factor conjecture (Erdős #1150). Lean; el Abdalaoui disputes one scope.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/076.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf","https://github.com/openai/math/blob/main/lean/docs/076.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10","https://arxiv.org/abs/2610.30797"]},{"family":"077","title":"Fourier restriction for positively curved surfaces","summary":"Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface \\Sigma\\subset\\mathbb R^3, including surfaces with boundary, the extension operator is bounded from L^p(\\Sigma) to L^p(\\mathbb R^3) for every p &gt; 3.","papers":["https://github.com/openai/math/blob/main/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/"],"name_hits":[],"tier":"A","note":"Stein's Fourier restriction conjecture in ℝ³ (sphere, positively curved surfaces). Manuscripts only; Korobenko–Sawyer cite the diagonal theorem.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.22634"]},{"family":"078","title":"The three-dimensional Bochner–Riesz conjecture","summary":"Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on L^p(\\mathbb R^3) for every 1\\le p\\le\\infty whenever \\delta\\gt \\max\\{3|1/p-1/2|-1/2,0\\}.","papers":["https://github.com/openai/math/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/"],"name_hits":[],"tier":"A","note":"Three-dimensional Bochner–Riesz conjecture in its strict range. Manuscript only; also a corollary of family 079's all-dimension claim.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/AGENT_NOTES.md"]},{"family":"079","title":"Local smoothing for the Euclidean wave equation","summary":"Proves Sogge’s local smoothing conjecture in every spatial dimension n\\ge2, with arbitrarily small loss above the sharp Sobolev threshold. The wave estimate yields the Bochner–Riesz conjecture and, for 2\\le p\\lt\\infty, maximal bounds for these means and sharp bounds for Stein’s square function. It also gives Fourier restriction throughout the interior of the conjectured range for the sphere, compact paraboloid, and truncated cone, and local smoothing for the Schrödinger equation. The Kakeya and Nikodym maximal conjectures follow. In particular, every Euclidean Kakeya set has full Hausdorff dimension. ([Lean](lean/docs/079.md))","papers":["https://github.com/openai/math/blob/main/preprints/Local-smoothing-for-the-Euclidean-wave-equation-October-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/"],"name_hits":[],"tier":"A","note":"Sogge's local smoothing conjecture, 3D (Sep) and every dimension (manuscript and Lean added 26 Oct), with Kakeya and Bochner–Riesz corollaries. Leng: 'absolutely shocking'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/079.md","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Local-smoothing-for-the-Euclidean-wave-equation-October-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/079.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Local-smoothing-for-the-Euclidean-wave-equation-October-25-2026/AGENT_NOTES.md","https://github.com/openai/math/blob/main/history.md","https://x.com/JamesLeng15/status/2114875830228906072"]},{"family":"080","title":"The exact Sobolev endpoint for Schrödinger convergence","summary":"Proves almost-everywhere convergence e^{it\\Delta}f\\to f as t\\downarrow0 for every f\\in H^{n/(2(n+1))}(\\mathbb R^n) and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H<sup>1/3</sup>.","papers":["https://github.com/openai/math/blob/main/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/"],"name_hits":[],"tier":"B","note":"Sharp Sobolev endpoint H^{n/(2(n+1))} of Carleson's Schrödinger convergence problem in every dimension. Manuscripts only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/AGENT_NOTES.md"]},{"family":"081","title":"Riesz transforms and rectifiability in higher codimension","summary":"Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2\\le n\\le d-2, an n-Ahlfors–David regular Radon measure on ℝ<sup>d</sup> is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L<sup>2</sup>-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds. ([Lean](lean/docs/081.md))","papers":["https://github.com/openai/math/blob/main/preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-release-3d-restriction-bochner-riesz-claims","2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt","2026-10-17-ai-assisted-results-october-16-listing","2026-10-19-ai-assisted-results-october-19-listing","2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim"],"tier":"B","note":"Bounded Riesz transforms imply uniform rectifiability in the remaining higher codimensions (David–Semmes problem). Lean main. Hoffman–Jaye cite it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/081.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/081.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2510.26934"]},{"family":"082","title":"Annular variation and dyadic absolute bounds for the triangular Hilbert transform","summary":"Proves maximal and annular r-variation bounds, for every r &gt; 2, from complex L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2) to L^{3/2}(\\mathbb R^2). The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric L^3\\times L^3\\times L^3 point. ([Lean](lean/docs/082.md))","papers":["https://github.com/openai/math/blob/main/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf","https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-dyadic-triangular-Hilbert-form-October-5-2026/dyadic-triangular-hilbert.pdf","https://github.com/openai/math/blob/main/preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Maximal and variational L³×L³→L^{3/2} bounds for the triangular Hilbert transform at the symmetric point. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/082.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf","https://github.com/openai/math/blob/main/lean/docs/082.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/AGENT_NOTES.md"]},{"family":"083","title":"Hilbert transforms along Lipschitz directions","summary":"Proves a uniform strong L<sup>2</sup> bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L<sup>2</sup>-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale. ([Lean](lean/docs/083.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-13-ai-assisted-results-october-2026","2026-10-16-ai-assisted-results-october-15-listing"],"tier":"B","note":"Short-scale L² bound for Hilbert transforms along Lipschitz vector fields, a form of Stein's conjecture. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/083.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/083.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026/AGENT_NOTES.md"]},{"family":"084","title":"The geometric case of the Erdős similarity conjecture","summary":"For every fixed q\\in(0,1), constructs compact subsets of [0,1] with measure arbitrarily close to one containing no translated and nontrivially dilated copy of \\{q^n:n\\ge1\\}, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio. ([Lean](lean/docs/084.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf","https://github.com/openai/math/blob/main/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Erdős similarity conjecture for geometric sequences of every ratio (#120). Lean covers the dyadic case; Jiang–Xi extend the construction.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/084.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf","https://github.com/openai/math/blob/main/lean/docs/084.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.11449","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"085","title":"Endpoint Sobolev regularity of centered disk averages","summary":"Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real f\\in W^{1,1}(\\mathbb R^2), the centered disk maximal function satisfies \\|\\nabla Mf\\|_1\\le C\\|\\nabla f\\|_1 with an absolute constant. It belongs locally to W^{1,1} and has a globally integrable weak gradient. ([Lean](lean/docs/085.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026/article.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-release-3d-restriction-bochner-riesz-claims"],"tier":"B","note":"Hajłasz–Onninen problem for the planar centered disk maximal function: ‖∇Mf‖₁ ≤ C‖∇f‖₁. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/085.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/085.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026/AGENT_NOTES.md"]},{"family":"086","title":"An L<sup>3</sup> bound for the trilinear Hilbert transform","summary":"Proves that the principal-value trilinear Hilbert transform with shifts x-t, x-2t, x-3t is bounded from L^3(\\mathbb R)^3 to L^1(\\mathbb R). This resolves the L<sup>3</sup> exponent case of the standard conjecture for slopes 1, 2, 3.","papers":["https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-07-25-sharp-thin-shell-inequality-gpt-5-6-pro","2026-08-13-banach-isometric-conjecture-solved","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-06-openai-math-release-722-manuscripts","2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma","2026-10-08-alphaproof-nexus-science-paper"],"tier":"B","note":"L³×L³×L³ → L¹ bound for the trilinear Hilbert transform with slopes 1, 2, 3. OSVerify's automated report lists issues.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/AGENT_NOTES.md","https://osverify.github.io/cases/trilinear-hilbert.html"]},{"family":"087","title":"The Mahler conjectures, functional inequalities and polar-product symplectic width","summary":"Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product K\\times K^\\circ in dimension 2n has Gromov width 4. ([Lean](lean/docs/087.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-mahler-conjecture-lean/"],"name_hits":["2026-10-06-irrationality-exponent-pi-openai","2026-10-06-openai-math-release-722-manuscripts","2026-10-23-ai-assisted-results-october-23-listing","2026-10-24-oai-math-revised-community-hub"],"tier":"A","note":"Symmetric and general Mahler conjectures in every dimension with equality cases; Lean for both, accepted. Kalai highlights it; Winter: Mahler implies Kalai's flag conjecture.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/087.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/087.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/AGENT_NOTES.md","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","https://arxiv.org/abs/2610.09990","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"088","title":"Sharp projection-body inequalities and a counterexample to simplex maximization","summary":"Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension. ([Lean](lean/docs/088.md))","papers":["https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Petty's projection-volume conjecture for n ≥ 4, Lutwak–Petty inequalities, and a counterexample to Brannen's simplex conjecture. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/088.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/088.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/AGENT_NOTES.md"]},{"family":"089","title":"Bounded-distortion L<sub>1</sub> embeddings of planar and bounded-treewidth graphs","summary":"Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L<sub>1</sub> with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded. ([Lean](lean/docs/089.md))","papers":["https://github.com/openai/math/blob/main/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair L1-embedding conjecture. Lean main. Hung Le is reading the planar proof.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/089.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/089.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/AGENT_NOTES.md","https://minorfree.github.io/OpenAIDump/"]},{"family":"090","title":"Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy","summary":"Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for 0\\lt s\\lt 2 and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy. ([Lean](lean/docs/090.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Universal optimality of the triangular lattice, Sandier–Serfaty, Brauchart–Hardin–Saff and planar Cohn–Elkies sharpness. Lean for main parts. No expert comment found. Lead filed.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/090.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/090.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/AGENT_NOTES.md","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"091","title":"Logarithmic and L<sub>p</sub> Brunn–Minkowski inequalities and the B-conjecture","summary":"Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive L<sub>p</sub> Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout 0\\lt p\\lt 1. ([Lean](lean/docs/091.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Logarithmic Brunn–Minkowski conjecture for symmetric bodies and the B-conjecture for even log-concave measures. Lean main; Livshyts explores it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/091.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/091.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/AGENT_NOTES.md","https://glivshyts6.math.gatech.edu/"]},{"family":"092","title":"The optimal order of convex-body covering density","summary":"Determines the optimal worst-case covering density as \\Theta(n\\log n), for both lattice and unrestricted translative coverings. Every convex body in ℝ<sup>n</sup>, n ≥ 2, admits a lattice covering of density at most Cn\\log n; centrally symmetric examples in every sufficiently large dimension require at least cn\\log n even without the lattice restriction, for absolute c,C\\gt 0. ([Lean](lean/docs/092.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Worst-case covering density of convex bodies is Θ(n log n), lattice and translative. Lean main; Kalai lists it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/092.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/092.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/AGENT_NOTES.md","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/"]},{"family":"093","title":"Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures","summary":"Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.","papers":["https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"C","note":"Dimension-free log-Sobolev inequality for centered log-concave densities with subgaussian marginals. Manuscript only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/AGENT_NOTES.md"]},{"family":"094","title":"Subpolynomial dimension reduction in L<sub>p</sub>","summary":"For every fixed 1\\lt p\\lt \\infty and distortion D &gt; 1, every n-point subset of real L<sub>p</sub> embeds into \\ell_p^d with distortion at most D and dimension d=n^{o(1)}, answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension \\Theta(n^2) when p ≠ 2. ([Lean](lean/docs/094.md))","papers":["https://github.com/openai/math/blob/main/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Subpolynomial dimension reduction in L_p for p ≠ 2 at any distortion D > 1, answering Naor's question. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/094.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/094.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/AGENT_NOTES.md"]},{"family":"095","title":"Hyperbolicity cones without semidefinite lifts","summary":"Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral. ([Lean](lean/docs/095.md))","papers":["https://github.com/openai/math/blob/main/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf","https://github.com/openai/math/blob/main/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf","https://github.com/openai/math/blob/main/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"A hyperbolicity cone that is not a spectrahedral shadow disproves the generalized and projected Lax conjectures. Lean main; Kevin Shu's exposition.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/095.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf","https://github.com/openai/math/blob/main/lean/docs/095.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/AGENT_NOTES.md","https://www.quasicoherent.blog/posts/generalized_lax/"]},{"family":"096","title":"The Gaussian propeller conjecture in every dimension","summary":"Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8\\pi). In dimension at least two, three planar sectors of angle 2\\pi/3, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8\\pi/9)(1-1/k) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs. ([Lean](lean/docs/096.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-23-propeller-conjecture-openai-proof-heilman-simplification/"],"name_hits":[],"tier":"B","note":"Khot–Naor Gaussian propeller conjecture in every dimension, Lean. Heilman simplified the proof and treats it as proved.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/096.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/096.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.32158"]},{"family":"097","title":"The Euclidean Steinitz–Bergström bound","summary":"Proves that any finite sequence in the Euclidean unit ball of ℝ<sup>d</sup> admits signs keeping every partial sum within C\\sqrt d, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order S_2(d)=\\Theta(\\sqrt d). ([Lean](lean/docs/097.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Euclidean Steinitz–Bergström constant is Θ(√d). Lean main; Russo–Wei cite it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/097.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/097.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.16715"]},{"family":"098","title":"Compact counterexamples to bi-Lipschitz dimension reduction","summary":"Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space. ([Lean](lean/docs/098.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Compact doubling subsets of every infinite-dimensional Banach space with no bi-Lipschitz embedding into finite dimensions (Lang–Plaut). Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/098.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/098.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/AGENT_NOTES.md"]},{"family":"099","title":"The sharp exponential scale of edit-distance distortion","summary":"Determines the least distortion of embedding edit distance on words of length at most d into real ℓ<sub>1</sub>: it is \\exp(\\Theta(\\sqrt{\\log d\\,\\log\\log d})). Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d; binary words already force the lower bound. ([Lean](lean/docs/099.md))","papers":["https://github.com/openai/math/blob/main/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"ℓ1 distortion of edit distance is exp(Θ(√(log d log log d))). Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/099.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/099.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/AGENT_NOTES.md"]},{"family":"100","title":"Cylinder coverings below the half-area bound","summary":"Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three. ([Lean](lean/docs/100.md))","papers":["https://github.com/openai/math/blob/main/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Finite cylinder covers of the regular tetrahedron below the half-area bound disprove Bang's conjecture. Lean; Pinho gives an explicit 4,096-cylinder cover.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/100.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/100.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/AGENT_NOTES.md","https://hexagonmath.org/abs/2610.00304v2"]},{"family":"101","title":"The sharp simplex conjecture for isotropic constants","summary":"Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.","papers":["https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Simplices uniquely maximize the isotropic constant (strong slicing conjecture); sharp entropy bound for log-concave laws. Manuscript only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/AGENT_NOTES.md"]},{"family":"102","title":"The Unique Games Conjecture and optimal approximation thresholds","summary":"Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results. ([Lean](lean/docs/102.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Constant-factor-hardness-of-Min-UnCut-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Constant-factor-hardness-of-directed-feedback-vertex-set-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-unique-games-conjecture-openai/"],"name_hits":[],"tier":"A","note":"Khot's Unique Games Conjecture plus direct Max-Cut and Vertex Cover hardness, Lean. Moshkovitz (via Aaronson): 'The Unique Games Conjecture is true'; writeup 'poorly written'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/102.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/102.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Unique-Games-Theorem-September-23-2026/AGENT_NOTES.md","https://scottaaronson.blog/?p=10169","https://zenodo.org/records/23265572"]},{"family":"103","title":"Exact derandomization of logarithmic space: $`\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}`$","summary":"Proves \\mathsf L=\\mathsf{RL}=\\mathsf{BPL}, resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds. ([Lean](lean/docs/103.md))","papers":["https://github.com/openai/math/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim/"],"name_hits":[],"tier":"A","note":"L = RL = BPL, Lean added 7 Oct; configuration fixed 26 Oct; a community re-check found statement definitions identical to the proof's.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/103.md","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/103.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/AGENT_NOTES.md","https://github.com/openai/math/blob/main/history.md"]},{"family":"104","title":"Quasipolynomial algorithms for mean-payoff, stochastic and parity games","summary":"Gives deterministic algorithms using 2^{O((\\log(L+2))^2)} bit operations, for complete binary input length L, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal positional strategies in ordinary games, the nonnegative expectation-of-liminf value set in turn-based stochastic games, and the winning set for nonnegative liminf mean payoff conjoined with parity. Signed rewards, rational chance probabilities, and parity priorities are unrestricted and binary-encoded. ([Lean](lean/docs/104.md))","papers":["https://github.com/openai/math/blob/main/preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/stochastic-mean-payoff-games.pdf","https://github.com/openai/math/blob/main/preprints/Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026/mean-payoff-parity.pdf","https://github.com/openai/math/blob/main/preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Deterministic quasipolynomial algorithms for mean-payoff, stochastic mean-payoff and mean-payoff parity games. Lean covers the randomized version only.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/104.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/stochastic-mean-payoff-games.pdf","https://github.com/openai/math/blob/main/lean/docs/104.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.25188"]},{"family":"105","title":"Perfect completeness for 2-to-1 games","summary":"Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational \\delta\\in(0,1), it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map. ([Lean](lean/docs/105.md))","papers":["https://github.com/openai/math/blob/main/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-unique-games-conjecture-openai/"],"name_hits":[],"tier":"B","note":"Khot's 2-to-1 Games Conjecture with perfect completeness, Lean. Fei–Minzer–Wang posted a 4-to-1 version on 14 Sep (in the UGC entry).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/105.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/105.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/AGENT_NOTES.md","https://eccc.weizmann.ac.il/report/2026/179/"]},{"family":"106","title":"Hardness of coloring three-colorable graphs","summary":"It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed 0\\lt \\delta\\lt 1/3, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size \\delta n otherwise, where n is the number of vertices. ([Lean](lean/docs/106.md))","papers":["https://github.com/openai/math/blob/main/preprints/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-unique-games-conjecture-openai/"],"name_hits":["2026-10-06-unique-games-conjecture-openai"],"tier":"A","note":"NP-hard to colour 3-colourable graphs with any constant number of colours, Lean. Fei–Minzer–Wang derive the same from 4-to-1 games (UGC entry).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/106.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/106.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026/AGENT_NOTES.md","https://eccc.weizmann.ac.il/report/2026/179/"]},{"family":"107","title":"Matrix multiplication with exponent at most 9/4","summary":"Proves \\omega\\le9/4 over ℂ, giving O_\\varepsilon(n^{9/4+\\varepsilon}) arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension n<sup>a</sup> with a &gt; 0.465 permits n^{2+o(1)} rectangular multiplication. Further square bounds give ω &lt; 2.258 outside finitely many positive characteristics and ω &lt; 2.371054886006746 over every fixed field. ([Lean](lean/docs/107.md))","papers":["https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-matrix-multiplication-omega-9-4/"],"name_hits":["2026-08-28-matrix-spencer-conjecture-proved","2026-10-06-openai-math-release-722-manuscripts"],"tier":"A","note":"ω ≤ 9/4 for matrix multiplication, Lean. Re-proved over every field by Amanov–van den Berg–Zuiddam and constructively by Pass; algorithms galactic.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/107.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/107.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.30054","https://arxiv.org/abs/2610.28627"]},{"family":"108","title":"A cubic permanent–determinant lower bound","summary":"Proves an \\Omega(n^3) lower bound for the border determinantal complexity of the n\\times n permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least cn^3, for an absolute c &gt; 0 and all sufficiently large n; the same bound therefore holds for exact representations. ([Lean](lean/docs/108.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Ω(n³) lower bound for the border determinantal complexity of the permanent (best was quadratic). Lean main; Guo reuses its Lean declarations.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/108.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/108.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2609.29568"]},{"family":"109","title":"Integer multiplication below $`n\\log n`$","summary":"Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(\\log n)^{1-\\kappa}), with \\kappa=2^{-182}, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen n\\log n optimality conjecture in the ordinary multitape bit model.","papers":["https://github.com/openai/math/blob/main/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/"],"name_hits":[],"tier":"A","note":"Integer multiplication in O(n (log n)^{1−κ}), κ = 2^−182. David Harvey: 'the claimed o(n log n) bound is plausible', details unchecked; community raised κ.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Integer-multiplication-below-n-log-n-September-23-2026/AGENT_NOTES.md","https://web.maths.unsw.edu.au/~davidharvey/research/subnlogn.pdf","https://github.com/CrocSwap/integer-mult-bounds"]},{"family":"110","title":"Optimal-order randomized k-server on arbitrary metrics","summary":"Establishes a randomized competitive ratio O(\\log^2(k+1)) for k-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and \\log(t+1) at request t, with a finite instance-dependent additive movement constant. ([Lean](lean/docs/110.md))","papers":["https://github.com/openai/math/blob/main/preprints/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Randomized k-server O(log²k) on every metric, matching the known lower-bound order. Lean main; configuration changed 26 Oct.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/110.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/110.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026/AGENT_NOTES.md","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"111","title":"One-sample matroid prophet inequalities against an almighty adversary","summary":"For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule's entire random seed; no polynomial-time implementation is asserted. ([Lean](lean/docs/111.md))","papers":["https://github.com/openai/math/blob/main/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"One-sample matroid prophet inequality with constant ratio against an almighty adversary. Lean (2^−310); bottlebrushes PR improves to 2^−38.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/111.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf","https://github.com/openai/math/blob/main/lean/docs/111.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/AGENT_NOTES.md","https://github.com/bottlebrushes/math/pull/1"]},{"family":"112","title":"Beyond the square-root exponent for depth-three circuits","summary":"Constructs a single language in deterministic polynomial time whose n-bit membership function requires 2^{\\omega(\\sqrt n)} total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds. ([Lean](lean/docs/112.md))","papers":["https://github.com/openai/math/blob/main/preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-21-courtade-kumar-conjecture-proved","2026-09-26-axios-tens-of-thousands-frontier-model-incidents","2026-10-07-multiple-unicast-conjecture-false-gpt-6","2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma"],"tier":"B","note":"Explicit language requiring 2^{ω(√n)} depth-3 circuits, past the square-root barrier. Lean main. Chukhin (ECCC) and Li–Zhong extend it; Goldreich comments.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/112.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/112.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/AGENT_NOTES.md","https://eccc.weizmann.ac.il/report/2026/248/","https://arxiv.org/abs/2610.27127"]},{"family":"113","title":"Approximate counting and entropy of perfect matchings","summary":"Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points. ([Lean](lean/docs/113.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-19-perfect-matchings-fpras-general-graphs-openai-liu-meng/"],"name_hits":["2026-09-30-barvinok-log-concavity-totally-unimodular-astra"],"tier":"A","note":"FPRAS for counting perfect matchings in general graphs; AOV entropy conjecture. Lean. Liu–Meng give a simplified self-contained proof. Own entry (30 Oct).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/113.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/113.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.26803","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"114","title":"Approximate counting of common integer polymatroid bases","summary":"Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ. ([Lean](lean/docs/114.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/polymatroid-fpras.pdf","https://github.com/openai/math/blob/main/preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-30-barvinok-log-concavity-totally-unimodular-astra"],"tier":"B","note":"FPRAS for common bases of two matroids and integer polymatroids, Lean. Chen–Liu (arXiv 2610.06724, 5 Oct) independently claim the matroid case.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/114.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/polymatroid-fpras.pdf","https://github.com/openai/math/blob/main/lean/docs/114.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.06724"]},{"family":"115","title":"Sampling and counting contingency tables with arbitrary margins","summary":"For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution. ([Lean](lean/docs/115.md))","papers":["https://github.com/openai/math/blob/main/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Exact uniform sampling and FPRAS counting of contingency tables with arbitrary margins and cell bounds. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/115.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/115.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/AGENT_NOTES.md"]},{"family":"116","title":"Uniform black-box noncommutative identity testing across characteristics","summary":"For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation. ([Lean](lean/docs/116.md))","papers":["https://github.com/openai/math/blob/main/preprints/Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026/uniform-matrix-hitting-points-positive-characteristic.pdf","https://github.com/openai/math/blob/main/preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-spin-one-haldane-gap-proof","2026-10-06-quasi-riemann-hypothesis-openai","2026-10-11-landau-siegel-zeros-banks-explicit-account","2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean","2026-10-27-ramsey-r3333-upper-bound-61-ai-lean-sat"],"tier":"B","note":"Deterministic black-box identity testing for noncommutative formulas in every characteristic. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/116.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026/uniform-matrix-hitting-points-positive-characteristic.pdf","https://github.com/openai/math/blob/main/lean/docs/116.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026/AGENT_NOTES.md"]},{"family":"117","title":"Uniform sparsest cut: hardness and semidefinite gaps","summary":"Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least \\sqrt{\\log n}/(\\log\\log n)^3, approaching the square-root-logarithmic upper bound. ([Lean](lean/docs/117.md))","papers":["https://github.com/openai/math/blob/main/preprints/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-spin-one-haldane-gap-proof","2026-10-06-quasi-riemann-hypothesis-openai","2026-10-11-landau-siegel-zeros-banks-explicit-account","2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean","2026-10-27-ramsey-r3333-upper-bound-61-ai-lean-sat"],"tier":"B","note":"Constant-factor NP-hardness of uniform sparsest cut and near √log n SDP gaps. Lean covers the integrality gaps only.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/117.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/117.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026/AGENT_NOTES.md"]},{"family":"118","title":"Bin packing and unbounded configuration-LP gaps","summary":"Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items. ([Lean](lean/docs/118.md))","papers":["https://github.com/openai/math/blob/main/preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Bin packing: configuration-LP additive gap unbounded (MIRUP false); fixed additive approximation NP-hard. Lean main.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/118.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/118.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026/AGENT_NOTES.md"]},{"family":"119","title":"The Courtade–Kumar and Hellinger conjectures","summary":"Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation. ([Lean](lean/docs/119.md))","papers":["https://github.com/openai/math/blob/main/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/main.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-09-21-courtade-kumar-conjecture-proved/"],"name_hits":[],"tier":"B","note":"Courtade–Kumar and Hellinger conjectures, Lean. Courtade–Kumar had been proved three times in September; Ky proves the antipodal Hellinger case.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/119.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/119.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.10618"]},{"family":"120","title":"Almost-linear-time exact matching and prescribed-degree factors in general graphs","summary":"Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in (n+m)^{1+o(1)} word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.","papers":["https://github.com/openai/math/blob/main/preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Randomized almost-linear-time maximum matching and prescribed-degree factors in general graphs. Manuscript only.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/AGENT_NOTES.md"]},{"family":"121","title":"Almost-linear approximation of edit distance","summary":"For every fixed rational \\varepsilon\\in(0,1), gives a randomized (1+\\varepsilon) approximation to unit-cost edit distance in worst-case expected time N^{1+o(1)}, with success probability at least 2/3. The strings have total length N and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy. ([Lean](lean/docs/121.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"(1+ε)-approximation of edit distance in N^{1+o(1)} time. Lean main; Hu–Kociumaka–Nogler–Saha footnote it as claimed.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/121.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/121.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.22250"]},{"family":"122","title":"Quantitative trace-reconstruction bounds with a uniform decoder","summary":"At every fixed deletion probability in (0,1), reconstructing an arbitrary length-n binary string requires n^{\\Omega(\\log\\log n)} independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most n^{-\\varepsilon} for fixed ε &gt; 0, both bounds become polynomial in the input and parameter encoding. ([Lean](lean/docs/122.md))","papers":["https://github.com/openai/math/blob/main/preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026/uniform-trace-reconstruction.pdf","https://github.com/openai/math/blob/main/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/quantitative-lower-bounds-for-trace-reconstruction-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Worst-case trace reconstruction needs n^{Ω(log log n)} traces; quasipolynomial uniform decoder. Lean covers the lower bound.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/122.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026/uniform-trace-reconstruction.pdf","https://github.com/openai/math/blob/main/lean/docs/122.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026/AGENT_NOTES.md"]},{"family":"124","title":"Polynomial-time scheduling on three identical machines","summary":"Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule. ([Lean](lean/docs/124.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Polynomial-time minimum-makespan scheduling of unit jobs with precedences on three machines, a Garey–Johnson open problem since 1979. Lean main accepted. Lead filed.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/124.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/124.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/AGENT_NOTES.md","https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md"]},{"family":"125","title":"The metric k-median approximation threshold and recovery","summary":"Gives a deterministic polynomial-time (1+2/e+\\varepsilon)-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε &gt; 0. Assuming P\\ne NP, the optimal infimum approximation factor is 1+2/e. ([Lean](lean/docs/125.md))","papers":["https://github.com/openai/math/blob/main/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e/","https://postcutoff.com/e/2026-10-13-k-median-better-than-2-approximation-epfl/"],"name_hits":[],"tier":"B","note":"Tight (1+2/e+ε) approximation for metric k-median, Lean. Svensson: 'completely solved a problem' his students worked on; EPFL group posts other algorithms.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/125.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/125.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/AGENT_NOTES.md","https://x.com/AlgoSvensson/status/2107736070192287922"]},{"family":"126","title":"Exponential semidefinite complexity of perfect matching","summary":"Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift 0\\lt \\rho\\lt 1, allowing arbitrary real positive semidefinite factors. ([Lean](lean/docs/126.md))","papers":["https://github.com/openai/math/blob/main/preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026/shifted-matching-psd.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-08-dualization-eth-lower-bound-gpt-6-astra","2026-10-17-ai-assisted-results-october-16-listing","2026-10-26-openai-eth-false-subexponential-sat-claim","2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim","2026-10-26-openai-yang-mills-qcd-existence-claim"],"tier":"B","note":"Claims every exact semidefinite lift of the perfect-matching polytope has exponential size (Rothvoss's question); Lean proves only superpolynomial bounds.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/126.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026/shifted-matching-psd.pdf","https://github.com/openai/math/blob/main/lean/docs/126.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026/AGENT_NOTES.md"]},{"family":"127","title":"Average sensitivity of polynomial threshold functions","summary":"Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8d\\sqrt n, uniformly for 1\\le d\\le n. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with \\mathop{\\mathrm{sign}}\\nolimits (0)=1. ([Lean](lean/docs/127.md))","papers":["https://github.com/openai/math/blob/main/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-08-derek-thompson-26-facts-ai-economy"],"tier":"B","note":"Claims average sensitivity of degree-d polynomial threshold functions is at most 8d√n, the asymptotic Gotsman–Linial conjecture; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/127.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/127.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/AGENT_NOTES.md"]},{"family":"128","title":"A factor-two approximation for shortest common superstring","summary":"Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels. ([Lean](lean/docs/128.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims a polynomial-time factor-2 approximation for shortest common superstring; Lean-formalized; several community implementations benchmark it against greedy.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/128.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/128.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/AGENT_NOTES.md"]},{"family":"129","title":"Exponential state costs for two-way automata","summary":"Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size. ([Lean](lean/docs/129.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-08-dualization-eth-lower-bound-gpt-6-astra","2026-10-17-ai-assisted-results-october-16-listing","2026-10-26-openai-eth-false-subexponential-sat-claim","2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim","2026-10-26-openai-yang-mills-qcd-existence-claim"],"tier":"B","note":"Claims exponential lower bounds for 2NFA complementation and for simulating 1NFAs by 2DFAs over growing alphabets (Sakoda–Sipser); both Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/129.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/129.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026/AGENT_NOTES.md"]},{"family":"130","title":"Exact Fourier transforms below $`n\\log n`$","summary":"Gives a deterministic length-n discrete Fourier transform algorithm using O(n(\\log n)^{1-\\delta}) operations for every n, with explicit \\delta=10^{-13}. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing. ([Lean](lean/docs/130.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-17-ai-assisted-results-october-16-listing"],"tier":"A","note":"Claims exact DFT in O(n(log n)^(1−10⁻¹³)) operations, below n log n; Lean; Recht calls it galactic; Aaronson: barrier 'since the 1960s'; community raised the saving.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/130.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/130.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/AGENT_NOTES.md","https://www.argmin.net/p/faster-fourier-transforms","https://scottaaronson.blog/?p=10169","https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/","https://www.johndcook.com/blog/2026/10/07/faster-fourier-transform/"]},{"family":"131","title":"Rapid mixing of graph switches for every degree sequence","summary":"Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in O(n^8) time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time. ([Lean](lean/docs/131.md))","papers":["https://github.com/openai/math/blob/main/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-24-openai-agent-medicare-breach-australia","2026-10-03-jay-clayton-ai-czar-super-intelligence-force","2026-10-05-vanity-fair-altman-new-establishment-interview"],"tier":"B","note":"Claims the switch chain mixes in O(n⁸) for every graphical degree sequence (Kannan–Tetali–Vempala conjecture); Lean-formalized mixing bound.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/131.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/131.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/AGENT_NOTES.md"]},{"family":"132","title":"A superquadratic separation of sensitivity and block sensitivity","summary":"Constructs total Boolean functions with block sensitivity \\mathop{\\mathrm{bs}}\\nolimits (f)\\ge s(f)^\\alpha for a fixed α &gt; 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here s(f) counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together. ([Lean](lean/docs/132.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Disproves the quadratic sensitivity vs block-sensitivity bound (bs ≥ s^α, α > 2); Lean; Kothari and others raised α past 2.13 with Lean.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/132.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/132.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/AGENT_NOTES.md","https://x.com/RobinKothari/status/2110829273108127765","https://prove2.me/campaigns/block-sensitivity-vs-sensitivity"]},{"family":"133","title":"The computational complexity of Weisfeiler–Leman refinement","summary":"Proves unconditional n^{\\Omega(k)} deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete. ([Lean](lean/docs/133.md))","papers":["https://github.com/openai/math/blob/main/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Variable-dimension-Weisfeiler-Leman-equivalence-on-general-and-subcubic-graphs-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Claims n^Ω(k) time lower bounds and EXPTIME-completeness for Weisfeiler–Leman equivalence; Lean in part; concurrent AI-assisted work by Curticapean, Liu, Neuen, Seppelt.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/133.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/133.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/AGENT_NOTES.md","https://zenodo.org/records/23213174"]},{"family":"134","title":"Generalized star height at most three","summary":"Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem. ([Lean](lean/docs/134.md))","papers":["https://github.com/openai/math/blob/main/preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/Generalized-Star-Height-at-Most-Four-September-25-2026/Generalized-Star-Height-at-Most-Four-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/Generalized-Star-Height-at-Most-Three-September-25-2026/article.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-22-odlyzko-poonen-conjecture-proved","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover","2026-10-23-ai-assisted-results-october-23-listing"],"tier":"B","note":"Claims every regular language has generalized star height at most 3 (uniform bound in the generalized star-height problem); Lean-formalized; height 1 vs 2 untouched.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/134.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/Generalized-Star-Height-at-Most-Four-September-25-2026/Generalized-Star-Height-at-Most-Four-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/134.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026/AGENT_NOTES.md"]},{"family":"135","title":"Homogeneous depth-five lower bounds for iterated matrix multiplication","summary":"Over every characteristic-zero field, the (1,1) entry of a product of n independent n\\times n variable matrices requires n^{\\Theta(\\sqrt n)} gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables. ([Lean](lean/docs/135.md))","papers":["https://github.com/openai/math/blob/main/preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Claims sharp n^Θ(√n) homogeneous depth-five circuit bounds for iterated matrix multiplication over characteristic zero; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/135.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/135.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026/AGENT_NOTES.md"]},{"family":"136","title":"A quasilinear PCP theorem for PPAD","summary":"Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length N(\\log N)^{O(1)} such that any polynomially encoded rational assignment satisfying all but a fixed fraction to fixed accuracy yields an endpoint solution. Such assignments always exist, giving robust local verification with only quasilinear size overhead.","papers":["https://github.com/openai/math/blob/main/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Claims the quasilinear PCP-for-PPAD conjecture; only partial Lean; Daskalakis et al. (arXiv 2610.28141) cite the claim but do not rely on it.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.28141"]},{"family":"137","title":"One-tape time simulation in two-fifths-power space","summary":"Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using O(T^{2/5}\\log^C(T+2)) space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted. ([Lean](lean/docs/137.md))","papers":["https://github.com/openai/math/blob/main/preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026/article.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Claims one-tape time-T machines can be simulated in O(T^(2/5) polylog T) space, improving the square-root exponent; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/137.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/137.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026/AGENT_NOTES.md"]},{"family":"138","title":"Subset Sum in $`O(2^{0.49n})`$ time","summary":"Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary O(2^{0.49n}) word-RAM time on polynomial-bit inputs, where n counts the integers. The time bound holds on every execution and success probability is at least 2/3 on every input. Inputs may repeat positive integers; words have O(n+b) bits for maximum input bit length b.","papers":["https://github.com/openai/math/blob/main/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/subset-sum.pdf","https://github.com/openai/math/blob/main/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-09-arxiv-ai-disclosed-results-early-october-2026","2026-10-09-subset-sum-near-linear-time-gpt-astra-ultra","2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra","2026-10-26-openai-eth-false-subexponential-sat-claim"],"tier":"B","note":"Claims worst-case Subset Sum in randomized O(2^0.49n) time, beating meet-in-the-middle; no Lean; Kapila claims 0.4877 building on it (Hexagon).","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/subset-sum.pdf","https://github.com/openai/math/blob/main/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/AGENT_NOTES.md","https://hexagonmath.org/2610.00261v1"]},{"family":"139","title":"Subpolynomial query complexity for log-concave sampling","summary":"For C<sup>2</sup> potentials with a supplied minimizer and I\\preceq\\nabla^2V\\preceq2I, proves that sampling within total variation 1/10 requires only C_\\varepsilon d^\\varepsilon exact value-and-gradient queries for every fixed ε &gt; 0. The bound holds on every run, with unrestricted computation between queries. A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero. ([Lean](lean/docs/139.md))","papers":["https://github.com/openai/math/blob/main/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026/article.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Claims well-conditioned log-concave sampling needs only C_ε·d^ε value-and-gradient queries, with a log d lower bound; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/139.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/139.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026/AGENT_NOTES.md"]},{"family":"140","title":"Memory–sample lower bounds for noiseless Gaussian regression","summary":"For fixed A &gt; 0, a one-pass learner with Ad^2 persistent bits needs \\Omega_A(d\\log(1/\\epsilon)) noiseless Gaussian samples to recover a unit vector to angular error 0\\lt \\epsilon\\le1/10 with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness. ([Lean](lean/docs/140.md))","papers":["https://github.com/openai/math/blob/main/preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Claims one-pass learners with O(d²) memory bits need Ω(d log(1/ε)) noiseless Gaussian samples; six manuscripts; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/140.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/140.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026/AGENT_NOTES.md"]},{"family":"141","title":"Existential–universal real sentences in the counting hierarchy","summary":"Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.","papers":["https://github.com/openai/math/blob/main/preprints/Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026/etr-counting-hierarchy.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims the existential theory of the reals lies in the counting hierarchy; no Lean; Bürgisser notes 'several ChatGPT generated papers' claiming this.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026/etr-counting-hierarchy.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026/AGENT_NOTES.md","https://arxiv.org/abs/2606.25121"]},{"family":"142","title":"Deterministic polynomial factorization over prime fields","summary":"Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽<sub>p</sub>, including multiplicities, in bit complexity polynomial in (n+1)\\log p. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.","papers":["https://github.com/openai/math/blob/main/preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026/Deterministic-Polynomial-Factorization-over-Prime-Fields.pdf"],"lean":false,"entries":[],"name_hits":["2026-07-14-gpt-5-6-zeroth-order-convex-lower-bound","2026-09-04-reed-solomon-list-decoding-capacity-ai","2026-09-14-k-server-conjecture-proved","2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding","2026-10-25-ai-assisted-results-october-25-listing","2026-10-26-ai-assisted-results-october-26-listing"],"tier":"A","note":"Claims deterministic polynomial-time factoring over F_p without GRH, a classic derandomization problem; no Lean; relies on a companion Hecke zero-free claim; Voloch posted a reformulation.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026/Deterministic-Polynomial-Factorization-over-Prime-Fields.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026/AGENT_NOTES.md","https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","https://github.com/mathcommunity/oai-math-revised/issues/26"]},{"family":"143","title":"Hilbert's sixteenth problem: uniform bounds for limit cycles","summary":"Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles. ([Lean](lean/docs/143.md))","papers":["https://github.com/openai/math/blob/main/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-hilbert-sixteenth-uniform-limit-cycle-bound-claim/"],"name_hits":[],"tier":"A","note":"Claims a uniform bound on limit cycles of planar polynomial fields (Hilbert's 16th, second part); Lean covers only quintic Liénard; no expert check.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/143.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/143.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/AGENT_NOTES.md","https://carlostoledo.co/reports/openai-math.html","https://arxiv.org/abs/2610.12218","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"144","title":"Banach’s simple Lebesgue-spectrum problem","summary":"Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L<sup>2</sup> space: the bilateral iterates of one real observable form an orthonormal basis of that space. ([Lean](lean/docs/144.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims a smooth volume-preserving T³ diffeomorphism with simple Lebesgue spectrum, answering Banach's problem in smooth dynamics; Lean; no expert reaction found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/144.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/144.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/AGENT_NOTES.md"]},{"family":"145","title":"Rokhlin’s multiple-mixing problem","summary":"Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge. ([Lean](lean/docs/145.md))","papers":["https://github.com/openai/math/blob/main/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Claims mixing implies mixing of all orders (Rokhlin, 1949); Lean accepted; Bryna Kra singled it out but said the writing blocked her understanding.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/145.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/145.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/AGENT_NOTES.md","https://www.europesays.com/ai/200309/","https://carlostoledo.co/reports/openai-math.html"]},{"family":"146","title":"Positive metric entropy for the standard map","summary":"Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail. ([Lean](lean/docs/146.md))","papers":["https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-06-collatz-positive-density-log-time-mazur-codex","2026-09-22-claude-opus-5-5","2026-10-07-lebrun-salamon-conjecture-proof","2026-10-22-metr-redwood-red-team-openai-agent-monitoring","2026-10-26-ai-assisted-results-october-26-listing"],"tier":"B","note":"Claims positive metric entropy of the Chirikov standard map for all large parameters (Sinai's conjecture); Lean; Oliver Knill notes small parameters remain open.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/146.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/146.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/AGENT_NOTES.md","https://www.quantumcalculus.org/positive-entropy-for-standard-map-by-open-ai/"]},{"family":"147","title":"The near-boundary Birkhoff conjecture","summary":"Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.","papers":["https://github.com/openai/math/blob/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-09-pierce-birkhoff-conjecture-disproved"],"tier":"B","note":"Claims the near-boundary Birkhoff conjecture: a foliated grazing collar of caustics forces an ellipse; no Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/AGENT_NOTES.md"]},{"family":"148","title":"The entropy-rate dimension formula for self-similar measures","summary":"For every self-similar measure on the line generated by finitely many contracting similarities, proves \\dim_{\\mathrm H}\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}, where h_{\\mathrm{RW}} is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. ([Lean](lean/docs/148.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/","https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/"],"name_hits":[],"tier":"B","note":"Claims the exact-overlaps dimension formula for self-similar measures on the line; Lean; Kittle–Kogler's independent AI-found, human-rewritten proof has its own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/148.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/148.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.10511"]},{"family":"149","title":"Classwise permanence for weakly reversible mass-action systems","summary":"Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds. ([Lean](lean/docs/149.md))","papers":["https://github.com/openai/math/blob/main/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf","https://github.com/openai/math/blob/main/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-23-ai-assisted-results-october-23-listing/"],"name_hits":[],"tier":"B","note":"Claims the permanence conjecture for weakly reversible mass-action systems; Lean; Mislan and Lv et al. reuse its Lean development (mention in 23 Oct listing).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/149.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf","https://github.com/openai/math/blob/main/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/149.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/AGENT_NOTES.md","https://hexagonmath.org/abs/2610.00321v1"]},{"family":"150","title":"Weak mixing of triangular billiards with an irrational angle","summary":"Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions. ([Lean](lean/docs/150.md))","papers":["https://github.com/openai/math/blob/main/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf","https://github.com/openai/math/blob/main/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims every triangular billiard with an irrational angle is weakly mixing; ergodicity is Lean-formalized; previously known only for generic triangles.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/150.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf","https://github.com/openai/math/blob/main/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/150.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/AGENT_NOTES.md"]},{"family":"151","title":"A C<sup>1</sup> counterexample to the entropy conjecture","summary":"Constructs a noninvertible C<sup>1</sup> self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C<sup>1</sup> self-maps: homological growth need not force positive orbit complexity. ([Lean](lean/docs/151.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/"],"name_hits":[],"tier":"B","note":"Disproves Shub's entropy conjecture for C¹ self-maps; Lean; Andrey Gogolev wrote an elementary account extending the construction to every finite C^r.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/151.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/151.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/AGENT_NOTES.md","https://people.math.osu.edu/gogolyev.1/index_files/EC_example7.pdf"]},{"family":"152","title":"Zero entropy does not guarantee a smooth positive-volume model","summary":"Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C<sup>∞</sup> diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension. ([Lean](lean/docs/152.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2017-12-05-alphazero","2021-01-05-dall-e-clip","2025-01-20-deepseek-r1","2026-07-30-gpt-5-6-price-cut","2026-08-10-claude-riemann-zeta-zeros-two-thirds","2026-09-22-claude-opus-5-5"],"tier":"B","note":"Claims a zero-entropy system with no smooth positive-volume model (smooth realization problem); Lean proves only a finite-entropy version.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/152.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/152.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/AGENT_NOTES.md"]},{"family":"153","title":"Arithmetic classification and non-Pisot singularity for Bernoulli convolutions","summary":"Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every \\lambda\\in(0,1) by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in (1,2), giving examples beyond reciprocal Pisot parameters.","papers":["https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims an arithmetic classification of singular Bernoulli convolutions and singularity at quartic Salem reciprocals; no Lean; Kogler plans an exposition; follow-ups cite it.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.09902","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"154","title":"Pointwise multiple ergodic averages for mixing transformations","summary":"Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.","papers":["https://github.com/openai/math/blob/main/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf","https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf","https://github.com/openai/math/blob/main/preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf","https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims a.e. convergence of multiple ergodic averages for mixing transformations (Furstenberg's question, mixing case); four manuscripts; no Lean; no reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf","https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/AGENT_NOTES.md"]},{"family":"155","title":"A counterexample to periodic tiling in dimension three","summary":"Constructs a finite translational tile in ℤ<sup>3</sup> that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ<sup>3</sup>, even with arbitrary real translation vectors. ([Lean](lean/docs/155.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/"],"name_hits":[],"tier":"A","note":"Finite ℤ³ tile with no fully periodic tiling (lowest possible dim), ℝ³ thickening too; Lean incl. minimality accepted. Demaine–Langerman build on it (undecidability). No expert check yet. Own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/155.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/155.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.12392","https://www.quantamagazine.org/update/168049/","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","https://github.com/edemaine/translation-tiling-paper","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv"]},{"family":"156","title":"Borsuk's conjecture fails in dimension nine","summary":"Constructs a compact subset of ℝ<sup>9</sup> that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ<sup>4</sup>, with the Frobenius metric. ([Lean](lean/docs/156.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-borsuk-conjecture-dimension-nine-counterexample/"],"name_hits":["2026-05-01-borsuk-conjecture-dimension-63","2026-10-20-double-cap-conjecture-witsenhausen-lean-ai"],"tier":"A","note":"Compact (infinite) projector set of ℝP³ in ℝ⁹ needs >10 parts; record was 63 (finite). Lean accepted; Kalai: 'major progress', 'very beautiful'. Finite subsets 8-colourable (Damásdi). Own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/156.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/156.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/AGENT_NOTES.md","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","https://en.wikipedia.org/wiki/Borsuk%27s_conjecture","https://gilkalai.wordpress.com/2026/10/21/progress-on-borsuks-problem-and-answer-to-tyi-60/","https://erdosproblemaday.com/day/505-openai-borsuk-fails-in-dimension-nine-refereed","https://github.com/google-deepmind/formal-conjectures/pull/6898","https://teorth.github.io/optimizationproblems/constants/28a.html","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv"]},{"family":"157","title":"Graph coloring, clique minors, and Colin de Verdière invariants","summary":"Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy \\chi_f(G)\\gt h(G), where h(G) is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound \\chi_f(G)\\le\\mu(G)+1. In the positive direction, every finite nonempty graph satisfies \\chi_{\\mathrm{list}}(G)\\le C h(G) for a universal constant C. ([Lean](lean/docs/157.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-hadwiger-conjecture-counterexample-claim/"],"name_hits":[],"tier":"A","note":"Claims counterexamples to Hadwiger's conjecture, even fractional, with independence number 2; Lean proves the ordinary form (added 7 Oct); no expert check.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/157.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/157.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/AGENT_NOTES.md","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","https://github.com/EdouardBonnet/hadwiger-counterexample","https://carlostoledo.co/reports/openai-math.html"]},{"family":"158","title":"The Euclidean plane cannot be colored with five colors","summary":"Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane. ([Lean](lean/docs/158.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-claim/"],"name_hits":[],"tier":"A","note":"Claims the plane is not 5-colourable, so χ(R²) is 6 or 7; Lean (definition caveat closed 26 Oct); Koshelev–Kupavskii–Voronov build on it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/158.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/158.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/AGENT_NOTES.md","https://x.com/DmitryRybin1/status/2107852898617229724","https://arxiv.org/abs/2610.18907","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"159","title":"Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds","summary":"Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of \\{1,\\ldots,N\\} with no nonconstant k-term progression has size at most C_kN\\exp[-c_k(\\log N)^{\\varepsilon_k}], with positive constants depending only on k. ([Lean](lean/docs/159.md))","papers":["https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-erdos-arithmetic-progressions-conjecture-claim/"],"name_hits":["2026-05-03-erdos-1196-primitive-sets","2026-05-27-sum-product-conjecture-false-over-reals","2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1","2026-10-06-openai-math-release-722-manuscripts","2026-10-14-ai-assisted-results-mid-october-2026","2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro"],"tier":"A","note":"Claims Erdős's $5000 arithmetic-progression conjecture via quasipolynomial Szemerédi bounds; Lean covers the qualitative statement; Bloom: synthesis of existing techniques.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/159.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/159.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10","https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/3.lean","https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/"]},{"family":"160","title":"Superexponential van der Waerden numbers","summary":"Resolves Erdős's superexponential-growth question for van der Waerden numbers. If W_r(k) is the least interval length forcing a monochromatic k-term progression in every r-coloring, then W_r(k)\\gt k^{ck\\lfloor\\log_2 r\\rfloor} for an absolute c &gt; 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, W_r(k)^{1/k}\\to\\infty for each fixed r. ([Lean](lean/docs/160.md))","papers":["https://github.com/openai/math/blob/main/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims W_r(k) > k^(ck log r), so W_2(k)^(1/k) → ∞ (Erdős #138); Lean; Bloom notes Fox–Hunter had r ≥ 3 by similar methods.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/160.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/160.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"161","title":"Counterexamples to Sidorenko’s conjecture and the forcing conjecture","summary":"Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness. ([Lean](lean/docs/161.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-sidorenko-conjecture-counterexample-claim/"],"name_hits":[],"tier":"A","note":"Disproves Sidorenko's conjecture with a 35-vertex, 66-edge bipartite graph, and the forcing conjecture; Lean proves a host exists, none published.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/161.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/161.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/AGENT_NOTES.md","https://x.com/HamedHa26534517/status/2107621391616876611","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","https://carlostoledo.co/reports/openai-math.html"]},{"family":"162","title":"Counterexamples to Ryser’s covering conjecture","summary":"Disproves Ryser's covering conjecture by constructing intersecting (q+1)-partite, (q+1)-uniform hypergraphs with covering number q+1, rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture. ([Lean](lean/docs/162.md))","papers":["https://github.com/openai/math/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims intersecting (q+1)-partite hypergraphs needing q+1 cover vertices, disproving Ryser's conjecture and Gyárfás's tree-cover conjecture; Lean; no explicit instance.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/162.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/162.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/AGENT_NOTES.md","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","https://carlostoledo.co/reports/openai-math.html"]},{"family":"164","title":"Hindman’s finite sums and products conjecture","summary":"Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.","papers":["https://github.com/openai/math/blob/main/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims Hindman's finite sums-and-products conjecture over the integers; no OpenAI Lean, but Yao Xu's AI-assisted Lean proof is linked by formal-conjectures (Erdős #172).","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10","https://github.com/cadamcat/hindman-sums-products-lean4"]},{"family":"165","title":"The Harary–Hill and Zarankiewicz crossing-number formulas","summary":"Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs. ([Lean](lean/docs/165.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-harary-hill-zarankiewicz-crossing-numbers-openai/"],"name_hits":[],"tier":"A","note":"Claims exact crossing numbers of K_n and K_m,n (Harary–Hill, Zarankiewicz); Lean; Balko published a simplified human proof; own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/165.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/165.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.34444"]},{"family":"166","title":"The higher-dimensional Erdős distinct-distances conjecture","summary":"For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝ<sup>d</sup> determine at least c_dn^{2/d} distinct distances, with c_d\\gt 0 depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.","papers":["https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims ≫ n^(2/d) distinct distances in every dimension d ≥ 3 (Erdős #1083); no OpenAI Lean; Bloom notes it builds on Tidor–Yu–Zakharov.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"167","title":"Planar distinct distances and unit-distance bounds","summary":"Proves the weak pinned Erdős distance conjecture: for every fixed ε &gt; 0, all but o(n) points of any n-point planar set determine at least n^{1-\\varepsilon} distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by O(n^{4/3-\\delta}) for an absolute δ &gt; 0. ([Lean](lean/docs/167.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-unit-distance-upper-bound-power-saving-claim/"],"name_hits":["2026-09-22-mumford-shah-conjecture-astra-claim","2026-10-19-ai-assisted-results-zenodo-october-17-19","2026-10-24-oai-math-revised-community-hub"],"tier":"A","note":"Claims u(n) = O(n^(4/3−δ)), first gain over Spencer–Szemerédi–Trotter since 1984, plus weak pinned distances; Lean; WillSawin sketched an effective version.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/167.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/167.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10","https://github.com/teorth/optimizationproblems/pull/204"]},{"family":"168","title":"Combinatorial invariance of Kazhdan–Lusztig polynomials","summary":"Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems. ([Lean](lean/docs/168.md))","papers":["https://github.com/openai/math/blob/main/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Claims the combinatorial invariance conjecture for Kazhdan–Lusztig polynomials in all Coxeter systems (studied by DeepMind in 2021); Lean; cited as claimed by Caselli–Marietti.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/168.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/168.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.21647","https://arxiv.org/abs/2610.14275","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/"]},{"family":"169","title":"Shareshian–Wachs elementary positivity","summary":"Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in \\mathbb N[q]. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity. ([Lean](lean/docs/169.md))","papers":["https://github.com/openai/math/blob/main/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims Shareshian–Wachs e-positivity of chromatic quasisymmetric functions; Lean (config fixed 26 Oct); Swanson calls it opaque and independent of Hikita's proof.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/169.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/169.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/AGENT_NOTES.md","https://mathoverflow.net/questions/481860/status-of-the-stanley-stembridge-conjecture/482378"]},{"family":"170","title":"Sharp logarithmic exponents for off-diagonal Ramsey numbers","summary":"For every fixed integer s ≥ 5, proves r(s,t)=t^{s-1}/(\\log t)^{s-2+o(1)} as t\\to\\infty, determining the logarithmic exponent and matching the classical upper bound at that scale. Here r(s,t) is the least number of vertices forcing an s-clique or a t-vertex independent set. ([Lean](lean/docs/170.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-08-01-openai-astra-ten-advances","2026-09-26-mubayi-zarankiewicz-astra","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind","2026-10-18-ai-assisted-results-october-17-listing","2026-10-20-ai-assisted-results-october-20-listing"],"tier":"B","note":"Claims r(s,t) = t^(s−1)/(log t)^(s−2+o(1)) for s ≥ 5, sharpening Bradač's 2026 exponent; Lean; Lin–Zhang use it for r(4,t).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/170.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/170.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10","https://arxiv.org/abs/2610.17891"]},{"family":"171","title":"The hypercube Ramsey conjecture","summary":"Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is \\Theta(2^n). Thus every red-blue coloring of a complete graph on a universal constant times the cube's number of vertices contains a monochromatic copy of the cube.","papers":["https://github.com/openai/math/blob/main/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-08-01-openai-astra-ten-advances","2026-09-26-mubayi-zarankiewicz-astra","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind","2026-10-18-ai-assisted-results-october-17-listing","2026-10-20-ai-assisted-results-october-20-listing"],"tier":"B","note":"Claims the hypercube Ramsey number is O(2^n) (Burr–Erdős); no OpenAI Lean, but Yao Xu's AI-assisted Lean proof is linked by formal-conjectures (Erdős #181).","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10","https://github.com/google-deepmind/formal-conjectures/pull/7128"]},{"family":"172","title":"Classification of finite Euclidean Ramsey configurations","summary":"Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set. ([Lean](lean/docs/172.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims a tensor-criterion classification of finite Euclidean Ramsey configurations, disproving Leader–Russell–Walters; Lean; several Lean follow-ups build on it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/172.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/172.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"173","title":"Seymour’s second-neighborhood conjecture","summary":"Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs. ([Lean](lean/docs/173.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-24-ai-assisted-results-october-24-listing"],"tier":"B","note":"Claims Seymour's second-neighbourhood conjecture for all oriented graphs; Lean-formalized; no expert assessment found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/173.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/173.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/AGENT_NOTES.md"]},{"family":"174","title":"Deterministic construction of strong thin spanning trees","summary":"Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal C/k fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities. ([Lean](lean/docs/174.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-07-14-gpt-5-6-zeroth-order-convex-lower-bound","2026-09-04-reed-solomon-list-decoding-capacity-ai","2026-09-14-k-server-conjecture-proved","2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding","2026-10-25-ai-assisted-results-october-25-listing","2026-10-26-ai-assisted-results-october-26-listing"],"tier":"B","note":"Claims Goddyn's strong thin-tree conjecture with a deterministic polynomial-time construction; Lean-formalized; no reaction found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/174.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/174.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-strong-thin-tree-conjecture-September-23-2026/AGENT_NOTES.md"]},{"family":"175","title":"Talagrand’s expectation thresholds, discrete convexity, and graph decompositions","summary":"Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices. ([Lean](lean/docs/175.md))","papers":["https://github.com/openai/math/blob/main/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf","https://github.com/openai/math/blob/main/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026"],"tier":"B","note":"Claims Talagrand's expectation-threshold equivalence and discrete-convexity conjectures; Lean; Fang–Wang's v2 says OpenAI's result supersedes their log-star bound.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/175.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf","https://github.com/openai/math/blob/main/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/175.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/AGENT_NOTES.md","https://arxiv.org/abs/2609.30680","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/"]},{"family":"176","title":"The second Kahn–Kalai conjecture with an edge-count bound","summary":"Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in G(n,p) is at most C p_{\\mathrm E}(n,H)(1+\\log_2 h), with universal C. Here p_{\\mathrm E} is the least density at which every subgraph of H has expected copy count at least 1/2. ([Lean](lean/docs/176.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-14-friedgut-coalition-conjecture-resolved","2026-09-16-chvatal-conjecture-proved","2026-10-06-openai-math-release-722-manuscripts"],"tier":"B","note":"Claims the second Kahn–Kalai conjecture with a 1 + log₂h factor; Lean-formalized; Kalai listed it among related results.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/176.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/176.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/AGENT_NOTES.md","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/"]},{"family":"177","title":"Bounded-degree coboundary expanders","summary":"Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽<sub>2</sub> coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension. ([Lean](lean/docs/177.md))","papers":["https://github.com/openai/math/blob/main/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims bounded-degree F₂ coboundary expanders in every dimension ≥ 3; Lean; Michael Chapman counts it among problems he 'grew up on'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/177.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/177.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/AGENT_NOTES.md","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/"]},{"family":"178","title":"Deterministic nonbipartite Ramanujan graphs in every fixed degree","summary":"For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between -2\\sqrt{d-1} and 2\\sqrt{d-1}. A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.","papers":["https://github.com/openai/math/blob/main/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims deterministic nonbipartite Ramanujan graphs for every degree ≥ 3; no Lean; a public gist alleges a gap in Theorem 8.3.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/AGENT_NOTES.md","https://gist.github.com/Malt7635/4569b09c5ec295f0c1c4b294ee58fa0c","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/"]},{"family":"179","title":"The circulant Hadamard and Barker-sequence conjectures","summary":"Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n &gt; 1 exactly when n\\in\\{2,3,4,5,7,11,13\\}. ([Lean](lean/docs/179.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims Ryser's circulant Hadamard conjecture (orders 1 and 4 only) and the Barker-sequence classification; Lean; formal-conjectures marks it solved.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/179.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/179.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-circulant-Hadamard-conjecture-September-23-2026/AGENT_NOTES.md","https://github.com/google-deepmind/formal-conjectures/pull/6898"]},{"family":"180","title":"Barnette’s Hamiltonian-cycle conjecture","summary":"Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle. ([Lean](lean/docs/180.md))","papers":["https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-21-barnette-conjecture-openai-proof-zhentao-li-exposition/"],"name_hits":[],"tier":"A","note":"Claims Barnette's Hamiltonicity conjecture; Lean; Zhentao Li's hand-written, partly re-proved version (arXiv 2610.30490); own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/180.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/180.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.30490"]},{"family":"181","title":"The Erdős–Gallai cycle-decomposition conjecture","summary":"Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most Cn simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs. ([Lean](lean/docs/181.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/"],"name_hits":[],"tier":"B","note":"Claims O(n) cycle-and-edge decompositions (Erdős–Gallai); Lean; Jaehoon Kim proved it independently with AI help (arXiv 2610.07840).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/181.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/181.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.07840","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"182","title":"Power savings for intersective polynomial differences and prime arguments","summary":"For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of \\{1,\\ldots,N\\} avoiding nonzero values h(1),h(2),\\ldots as differences has size O_h(N^{1-c_k}), with c_k\\gt 0 depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h. ([Lean](lean/docs/182.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf","https://github.com/openai/math/blob/main/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf","https://github.com/openai/math/blob/main/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-06-02-microsoft-mai-models-build-2026","2026-07-01-grothendieck-group-scheme-counterexample","2026-07-23-ai-kill-switch-act","2026-09-03-gpt-6-astra","2026-09-08-jacob-coxon-resigns-anthropic","2026-09-08-meta-muse-personal-agent"],"tier":"B","note":"Claims power-saving Sárközy-type bounds for intersective polynomial differences; Lean for squares; Fan–Lott call the draft 'largely unreadable'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/182.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf","https://github.com/openai/math/blob/main/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf","https://github.com/openai/math/blob/main/lean/docs/182.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.07765"]},{"family":"183","title":"Power savings for planar halving lines and k-sets","summary":"Improves the planar halving-line bound to O(n^{4/3-\\varepsilon}) for sets with no three collinear and an absolute ε &gt; 0. More generally, an n-point set with no three collinear has O(n(k+1)^{1/3-\\varepsilon_0}) strictly separable k-subsets for 1\\le k\\le n/2, with an absolute \\varepsilon_0\\gt 0. The constants and positive exponents are nonquantitative. ([Lean](lean/docs/183.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-06-02-microsoft-mai-models-build-2026","2026-07-01-grothendieck-group-scheme-counterexample","2026-07-23-ai-kill-switch-act","2026-09-03-gpt-6-astra","2026-09-08-jacob-coxon-resigns-anthropic","2026-09-08-meta-muse-personal-agent"],"tier":"B","note":"Claims O(n^(4/3−ε)) halving lines for points with no three collinear, improving Dey's bound; Lean-formalized; constants non-explicit.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/183.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/183.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/AGENT_NOTES.md"]},{"family":"184","title":"Correspondence coloring with a fixed forbidden subgraph","summary":"Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need O_F(\\Delta/\\log\\Delta) colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex K<sub>r</sub>-free graph of average degree d ≥ 2 has an independent set of size \\Omega_r(n\\log d/d). ([Lean](lean/docs/184.md))","papers":["https://github.com/openai/math/blob/main/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf","https://github.com/openai/math/blob/main/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-hodge-conjecture-cm-abelian-varieties-openai","2026-10-06-openai-math-release-722-manuscripts"],"tier":"B","note":"Claims the Ajtai–Erdős–Komlós–Szemerédi independence conjecture and Alon–Krivelevich–Sudakov colouring; Lean for independence; Bloom notes it beats Shearer.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/184.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf","https://github.com/openai/math/blob/main/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/184.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"185","title":"Counterexamples to infinite matroid intersection and packing/covering","summary":"Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result. ([Lean](lean/docs/185.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-07-01-grothendieck-group-scheme-counterexample","2026-07-24-rota-flats-unimodality-disproved","2026-09-09-pierce-birkhoff-conjecture-disproved","2026-09-21-grad-conjecture-counterexamples","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-09-arxiv-ai-disclosed-results-early-october-2026"],"tier":"B","note":"Disproves unrestricted infinite matroid intersection and packing/covering conjectures in ZFC; Lean; Nash-Williams's finitary version untouched.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/185.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/185.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/AGENT_NOTES.md"]},{"family":"186","title":"Uniform influence and sharp thresholds for graph and hypergraph properties","summary":"Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed 0\\lt \\varepsilon\\lt 1/2, every nontrivial increasing relabeling-invariant property crosses from probability ε to 1-\\varepsilon within width O((\\log n)^{-2}) for graphs and O_r((\\log n)^{-r/(r-1)}) for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties. ([Lean](lean/docs/186.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf","https://github.com/openai/math/blob/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-spin-one-haldane-gap-proof","2026-10-06-quasi-riemann-hypothesis-openai","2026-10-11-landau-siegel-zeros-banks-explicit-account","2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean","2026-10-27-ramsey-r3333-upper-bound-61-ai-lean-sat"],"tier":"B","note":"Claims Friedgut–Kalai threshold-width bounds for symmetric graph and hypergraph properties; Lean for graphs; Kalai highlighted it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/186.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf","https://github.com/openai/math/blob/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/186.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/AGENT_NOTES.md","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/"]},{"family":"187","title":"Snaky in 21 Maker moves","summary":"Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins. ([Lean](lean/docs/187.md))","papers":["https://github.com/openai/math/blob/main/preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Settles Harary's Snaky game: Maker wins within 21 moves; Lean; Nándor Sieben built proof sequences from it (20 moves); later 19.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/187.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/187.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Snaky-in-21-Maker-moves-September-25-2026/AGENT_NOTES.md","https://github.com/nandorsieben/Snaky","https://x.com/carl_feynman/status/2107973332037415229"]},{"family":"188","title":"The sharp terminal leave in random triangle removal","summary":"Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to n^{3/2}/(2\\sqrt2), with mean-square convergence after normalization by n<sup>3/2</sup>. This proves the triangle case of the Joos–Kühn sharp-constant conjecture. ([Lean](lean/docs/188.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims random triangle removal leaves ~n^(3/2)/(2√2) edges (Joos–Kühn constant); Lean; Keevash proved it independently with essentially the same method.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/188.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/188.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.20414"]},{"family":"189","title":"Cycle–clique Ramsey numbers","summary":"Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(C_m,K_n)=(m-1)(n-1)+1 for every m\\ge n\\ge3, except R(C_3,K_3)=6. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph. ([Lean](lean/docs/189.md))","papers":["https://github.com/openai/math/blob/main/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims the Erdős–Faudree–Rousseau–Schelp formula R(C_m,K_n) = (m−1)(n−1)+1 for all m ≥ n ≥ 3; Lean-formalized (Erdős #551).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/189.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/189.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/AGENT_NOTES.md","https://www.erdosproblems.com/forum/thread/blog:10"]},{"family":"190","title":"Polynomial removal fails for ordered binary matrices","summary":"Disproves polynomial ordered binary matrix removal with one fixed 66\\times66 zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones. ([Lean](lean/docs/190.md))","papers":["https://github.com/openai/math/blob/main/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/"],"name_hits":["2026-09-22-odlyzko-poonen-conjecture-proved","2026-09-27-erdos-993-forest-independence-polynomials-unimodal","2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe","2026-10-06-polynomial-littlewood-offord-gpt-6-pro","2026-10-13-ai-assisted-results-october-2026","2026-10-14-ai-assisted-results-mid-october-2026"],"tier":"C","note":"Disproves polynomial removal for ordered binary matrices with a 66×66 pattern; Lean; Zinan Wang found a 4×4 obstruction (arXiv 2610.15370).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/190.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/190.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.15370"]},{"family":"191","title":"A power improvement in the Heilbronn triangle lower bound","summary":"For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least n^{-2+c} for one absolute c &gt; 0. This disproves the conjectured almost-n<sup>−2</sup> upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be. ([Lean](lean/docs/191.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims n points with every triangle of area ≥ n^(−2+c), refuting the conjectured n^(−2+o(1)) Heilbronn upper bound; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/191.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/191.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/AGENT_NOTES.md"]},{"family":"192","title":"Boolean functions violate the square-root degree bound by arbitrary factors","summary":"Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C &gt; 0, there is a sign-valued Boolean function f with \\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. ([Lean](lean/docs/192.md))","papers":["https://github.com/openai/math/blob/main/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-14-friedgut-coalition-conjecture-resolved","2026-09-15-typesafe-jev-system-one-model","2026-09-16-chvatal-conjecture-proved","2026-09-21-courtade-kumar-conjecture-proved","2026-10-06-polynomial-littlewood-offord-gpt-6-pro","2026-10-07-multiple-unicast-conjecture-false-gpt-6"],"tier":"C","note":"Disproves the Gopalan–Servedio square-root degree bound on linear Fourier coefficients by arbitrary factors; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/192.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/192.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/AGENT_NOTES.md"]},{"family":"193","title":"Serre’s intersection-multiplicity conjecture","summary":"Proves strict positivity of Serre's intersection multiplicity \\chi^R(M,N) for nonzero finitely generated modules over any regular local ring, provided M\\otimes_R N has finite length and \\dim M+\\dim N=\\dim R. This resolves the positivity conjecture, including ramified mixed characteristic.","papers":["https://github.com/openai/math/blob/main/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey"],"tier":"A","note":"Claims Serre's positivity of intersection multiplicities for all regular local rings, incl. ramified mixed characteristic; no Lean; formal-conjectures keeps it open.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/AGENT_NOTES.md","https://github.com/google-deepmind/formal-conjectures/pull/7215"]},{"family":"194","title":"Lech’s multiplicity conjecture","summary":"Proves e(R)\\le e(S) for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic. ([Lean](lean/docs/194.md))","papers":["https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims Lech's multiplicity conjecture in all dimensions and characteristics; Lean covers a supporting lemma; Iyengar–Ma–Walker had an AI-found equal-characteristic proof.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/194.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/194.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/AGENT_NOTES.md","https://mark-walker-22.github.io/Lech.html"]},{"family":"195","title":"A counterexample to the small Cohen–Macaulay module conjecture","summary":"Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture. ([Lean](lean/docs/195.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey/"],"name_hits":[],"tier":"B","note":"Counterexample to Hochster's small Cohen–Macaulay module conjecture (domain form); Lean covers only the ring; Anghel's survey; own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/195.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/195.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/AGENT_NOTES.md"]},{"family":"196","title":"A counterexample to Kaplansky’s zero-divisor conjecture","summary":"Constructs a finitely presented torsion-free group G whose group algebra \\mathbb F_2[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space. ([Lean](lean/docs/196.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim/"],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-06-openai-math-release-722-manuscripts","2026-10-13-ai-assisted-results-october-2026","2026-10-19-quasitrace-monotone-completeness-counterexamples-openai"],"tier":"A","note":"Claims a finitely presented torsion-free group with zero divisors in F₂[G], refuting Kaplansky's zero-divisor conjecture; Lean-formalized; no explicit elements published.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/196.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/196.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/AGENT_NOTES.md","https://carlostoledo.co/reports/openai-math.html"]},{"family":"197","title":"A torsion-free group algebra that is not directly finite","summary":"Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽<sub>2</sub> is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture. ([Lean](lean/docs/197.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf","https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim/"],"name_hits":[],"tier":"A","note":"Claims Kaplansky direct-finiteness counterexamples, refuting Gottschalk's surjunctivity and the Determinant Conjecture; Lean only for torsion examples; torsion-free case unformalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/197.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf","https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/197.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/AGENT_NOTES.md","https://carlostoledo.co/reports/openai-math.html","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","https://github.com/google-deepmind/formal-conjectures/pull/7440"]},{"family":"198","title":"A counterexample to finitistic-dimension finiteness","summary":"Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture. ([Lean](lean/docs/198.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims a finite-dimensional algebra of infinite little finitistic dimension, refuting the finitistic dimension conjecture; Lean; Tang extended it to all fields.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/198.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/198.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/AGENT_NOTES.md","https://hexagonmath.org/abs/2610.00140v5"]},{"family":"199","title":"Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures","summary":"Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field. ([Lean](lean/docs/199.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims counterexamples to Auslander–Reiten, Tachikawa and Nakayama-type conjectures; Lean; Marczinzik urges careful checking; Tang and Kitamura extend them.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/199.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/199.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/AGENT_NOTES.md","https://sites.google.com/view/rene-marczinzik/homological-conjectures-and-problems"]},{"family":"200","title":"Eisenbud–Green–Harris and lex-plus-powers","summary":"Proves the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a lex-plus-powers ideal with the same Hilbert function and no smaller graded Betti numbers.","papers":["https://github.com/openai/math/blob/main/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims the Eisenbud–Green–Harris and lex-plus-powers conjectures in characteristic zero; no Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/AGENT_NOTES.md"]},{"family":"201","title":"A counterexample to Kurosh’s division-ring problem","summary":"Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.","papers":["https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims an algebraic two-generated division ring of infinite dimension, answering Kurosh's problem negatively; no catalogued Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/AGENT_NOTES.md"]},{"family":"202","title":"The blockwise Alperin weight conjecture","summary":"Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights.","papers":["https://github.com/openai/math/blob/main/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-21-ai-assisted-results-october-21-listing"],"tier":"A","note":"Claims the blockwise Alperin weight conjecture for every finite group and prime; no Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/AGENT_NOTES.md"]},{"family":"203","title":"Donovan's conjecture over fields and complete mixed-characteristic DVRs","summary":"Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.","papers":["https://github.com/openai/math/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims Donovan's conjecture over algebraically closed fields and its integral form over complete DVRs; no Lean; no reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/AGENT_NOTES.md"]},{"family":"204","title":"Tensor saturation for even spin groups","summary":"Proves saturation factor one for \\mathop{\\mathrm{Spin}}\\nolimits (2n), n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-D part of the simply-laced saturation conjecture.","papers":["https://github.com/openai/math/blob/main/preprints/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-07-colkitt-codex-integer-multiplication-exponent"],"tier":"C","note":"Claims saturation factor one for Spin(2n), the type-D case of the simply-laced saturation conjecture; no Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026/AGENT_NOTES.md"]},{"family":"205","title":"Saxl’s conjecture and universal tensor squares","summary":"Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every S<sub>n</sub> with n\\notin\\{2,4,9\\} has an irreducible representation whose tensor square contains all irreducibles. ([Lean](lean/docs/205.md))","papers":["https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims Saxl's conjecture and universal tensor squares for S_n (n ∉ {2,4,9}); Lean-formalized; volunteers re-derived parts.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/205.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/205.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/AGENT_NOTES.md"]},{"family":"206","title":"Finite lattice representation and undecidability","summary":"Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group. ([Lean](lean/docs/206.md))","papers":["https://github.com/openai/math/blob/main/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2025-09-17-deepmind-unstable-singularities-fluids","2026-05-27-sum-product-conjecture-false-over-reals","2026-09-07-anandkumar-euler-singularity-r3","2026-10-02-meta-muse-spark-six-math-papers","2026-10-06-finite-index-over-treeable-jkl-problem","2026-10-07-colkitt-codex-integer-multiplication-exponent"],"tier":"B","note":"Claims some finite lattices are not congruence lattices of finite algebras, plus undecidability; Lean; DeMeo's AI-assisted reading found the negative answer sound.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/206.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/206.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/AGENT_NOTES.md","https://williamdemeo.org/blog/openai-flrp-claim/"]},{"family":"207","title":"The ℓ¹-Bass conjecture for all discrete groups","summary":"Proves the ℓ<sup>1</sup>-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over \\ell^1(G) are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain. ([Lean](lean/docs/207.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf","https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim/"],"name_hits":["2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Claims the ℓ¹-Bass conjecture and Kaplansky's idempotent conjecture in characteristic zero; Lean; Chatterji doubts it, Sauer calls it 'deeply creative'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/207.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf","https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/207.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/AGENT_NOTES.md","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"208","title":"Finite symmetric tensor categories and the Verlinde tower","summary":"Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p &gt; 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category \\mathrm{Ver}_{p^n}. The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.","papers":["https://github.com/openai/math/blob/main/preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/"],"name_hits":[],"tier":"B","note":"Claims the finite Benson–Etingof–Ostrik conjecture; Coulembier and Flake proved it first with GPT-6 Astra help (arXiv 2610.04096); no Lean.","lean_scope":"","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.04096"]},{"family":"209","title":"Integral counterexamples to Gersten’s conjecture","summary":"Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic (0,5) have nonzero integral K-theory classes that vanish over their fraction fields.","papers":["https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims integral counterexamples to Gersten's conjecture for ramified mixed-characteristic (0,5) regular local rings; no Lean; no reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/AGENT_NOTES.md"]},{"family":"210","title":"Foulkes' conjecture for sixth powers and quadratic stabilization","summary":"Proves the sixth case of Foulkes’ conjecture: \\mathop{\\mathrm{Sym}}\\nolimits ^6(\\mathop{\\mathrm{Sym}}\\nolimits ^bV) embeds equivariantly in \\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^6V) for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map \\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^aV)\\to\\mathop{\\mathrm{Sym}}\\nolimits ^a(\\mathop{\\mathrm{Sym}}\\nolimits ^bV) is surjective for a ≥ 2 and b\\ge a(a-1), giving dimension-independent quadratic stabilization. ([Lean](lean/docs/210.md))","papers":["https://github.com/openai/math/blob/main/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims Foulkes' conjecture for sixth powers and quadratic stabilization of the Foulkes–Howe map; Lean; a volunteer re-checked 6 ≤ b ≤ 25 by computer.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/210.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/210.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/AGENT_NOTES.md","https://realopjt.github.io/2026/10/07/checking-an-ais-maths-part-2.html"]},{"family":"211","title":"The geometric phase diagram, diffusion, and spectra of random planar maps","summary":"Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for 0\\lt q\\le4 and to the Brownian continuum random tree for q &gt; 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs. ([Lean](lean/docs/211.md))","papers":["https://github.com/openai/math/blob/main/preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026/fk-ising-walk-limit.pdf","https://github.com/openai/math/blob/main/preprints/Spectral-convergence-for-critical-FK-Ising-planar-maps-October-5-2026/spectral-convergence-critical-fk-ising-planar-maps.pdf","https://github.com/openai/math/blob/main/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf","https://github.com/openai/math/blob/main/preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims FK planar maps converge to LQG (q≤4) or the Brownian tree (q>4); Lean for q>4; Holden's independent work complements it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/211.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026/fk-ising-walk-limit.pdf","https://github.com/openai/math/blob/main/preprints/Spectral-convergence-for-critical-FK-Ising-planar-maps-October-5-2026/spectral-convergence-critical-fk-ising-planar-maps.pdf","https://github.com/openai/math/blob/main/lean/docs/211.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.10501"]},{"family":"212","title":"Planar first-passage geometry and the absence of bigeodesics","summary":"Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C<sup>1</sup> boundary. Differentiability also holds for every Gamma law with positive shape and rate. ([Lean](lean/docs/212.md))","papers":["https://github.com/openai/math/blob/main/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-22-mumford-shah-conjecture-astra-claim","2026-10-19-ai-assisted-results-zenodo-october-17-19","2026-10-24-oai-math-revised-community-hub"],"tier":"B","note":"Claims no bigeodesics in planar first-passage percolation and strict convexity of the exponential limit shape; Lean covers differentiability.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/212.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/212.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/AGENT_NOTES.md"]},{"family":"213","title":"Critical percolation on every quasi-transitive graph","summary":"Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with p_c\\lt 1: at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ<sup>3</sup>. ([Lean](lean/docs/213.md))","papers":["https://github.com/openai/math/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-09-03-dying-percolation-theta-pc-zero/"],"name_hits":["2026-06-09-gemini-3-5-live-translate","2026-08-18-openai-pauses-rl-training","2026-09-01-openai-astra-critical-cyber-threshold","2026-09-03-dying-percolation-theta-pc-zero","2026-09-03-gpt-6-astra","2026-09-03-openai-daybreak-frontline-defenders"],"tier":"A","note":"Claims θ(p_c) = 0 on Z³ and on all quasi-transitive graphs (Benjamini–Schramm); Z³ covered in our Anthropic entry; quasi-transitive part not.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/213.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/213.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/AGENT_NOTES.md","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","https://arxiv.org/abs/2610.30063"]},{"family":"214","title":"The Benjamini–Schramm nonuniqueness conjecture","summary":"Proves p_c\\lt p_u for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition. ([Lean](lean/docs/214.md))","papers":["https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims p_c < p_u on every nonamenable quasi-transitive graph (Benjamini–Schramm 1996); Lean; Tasmin Chu says it builds on Hutchcroft's approach.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/214.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/214.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/AGENT_NOTES.md","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/"]},{"family":"215","title":"Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics","summary":"Constructs the canonical continuum limit of the two-dimensional nearest-neighbor O(3) model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice O(4) model, determines the exact leading asymptotic of the full transfer gap, m_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor O(n) models with n ≥ 3 at every positive temperature. ([Lean](lean/docs/215.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf","https://github.com/openai/math/blob/main/preprints/An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026/o3-particle-pole.pdf","https://github.com/openai/math/blob/main/preprints/Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026/exact-mass-o4.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-polyakov-o3-sigma-model-mass-gap-claim/"],"name_hits":[],"tier":"A","note":"Claims Polyakov's 2D O(n) mass gap (n ≥ 3), an O(3) continuum theory with a gap, and O(4) asymptotics; Lean for decay only.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/215.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf","https://github.com/openai/math/blob/main/preprints/An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026/o3-particle-pole.pdf","https://github.com/openai/math/blob/main/lean/docs/215.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/AGENT_NOTES.md","https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/","https://x.com/ZoharKo/status/2107976691855626565"]},{"family":"216","title":"Critical and near-critical XY scaling and BKT universality","summary":"For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations C_{\\beta_c}(r)\\sim Ar^{-1/4}(\\log r)^{1/8} and the Berezinskii–Kosterlitz–Thouless essential singularity \\sqrt{\\beta_c-\\beta}\\log\\xi(\\beta)\\to B, with A,B\\gt 0 after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.","papers":["https://github.com/openai/math/blob/main/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/BKT-universality-for-height-and-planar-spin-fields-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-06-09-gemini-3-5-live-translate","2026-08-18-openai-pauses-rl-training","2026-09-01-openai-astra-critical-cyber-threshold","2026-09-03-dying-percolation-theta-pc-zero","2026-09-03-gpt-6-astra","2026-09-03-openai-daybreak-frontline-defenders"],"tier":"B","note":"Claims the critical XY correlation r^(−1/4)(log r)^(1/8) and the BKT essential singularity; six manuscripts; no Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/AGENT_NOTES.md"]},{"family":"217","title":"The low-temperature Sherrington–Kirkpatrick fluctuation law","summary":"For every fixed inverse temperature β &gt; 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to c_\\beta n^{1/3}, with c_\\beta\\gt 0, confirming the predicted n<sup>1/6</sup> standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.","papers":["https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Claims n^(1/6) fluctuations and a limiting law for the low-temperature SK free energy; conditional Lean; Aronow–Lopatto reprove one step.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/AGENT_NOTES.md","https://hexagonmath.org/abs/2610.00402v1"]},{"family":"218","title":"Conformal universality for weakly interacting and random-bond Ising models","summary":"Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE<sub>3</sub> interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments. ([Lean](lean/docs/218.md))","papers":["https://github.com/openai/math/blob/main/preprints/Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026/general-weak-random-bond-ising.pdf","https://github.com/openai/math/blob/main/preprints/Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026/quenched-sle3-weak-random-bond-ising.pdf","https://github.com/openai/math/blob/main/preprints/Logarithmic-Relative-Fluctuations-in-the-Weakly-Disordered-Planar-Ising-Model-October-5-2026/critical-relative-second-moment-weak-random-bond-ising.pdf","https://github.com/openai/math/blob/main/preprints/Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims conformal (SLE₃) universality for weakly interacting and weak random-bond Ising models; Lean covers only a comparison lemma.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/218.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026/general-weak-random-bond-ising.pdf","https://github.com/openai/math/blob/main/preprints/Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026/quenched-sle3-weak-random-bond-ising.pdf","https://github.com/openai/math/blob/main/lean/docs/218.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026/AGENT_NOTES.md"]},{"family":"219","title":"GOE bulk universality for regular graphs with weak Anderson disorder","summary":"For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.","papers":["https://github.com/openai/math/blob/main/preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026/fixed-energy-universality-weak-anderson-disorder-random-regular-graphs.pdf","https://github.com/openai/math/blob/main/preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026","2026-10-18-ai-assisted-results-october-17-listing","2026-10-19-army-echotribble-autonomous-cyber-agents","2026-10-21-ai-assisted-results-october-21-listing"],"tier":"B","note":"Claims GOE bulk universality for fixed-degree random regular graphs, even cubic; He, Huang and Wang proved the clean case independently (arXiv 2610.08127).","lean_scope":"","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026/fixed-energy-universality-weak-anderson-disorder-random-regular-graphs.pdf","https://github.com/openai/math/blob/main/preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.08127"]},{"family":"220","title":"Directional zero–one laws beyond iid environments and iid ballisticity","summary":"On ℤ<sup>d</sup>, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture. ([Lean](lean/docs/220.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf","https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Directional-transience-implies-ballisticity-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims directional zero–one laws (d ≥ 3) and ballisticity from directional transience for random walks in random environment; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/220.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf","https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/220.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/AGENT_NOTES.md"]},{"family":"221","title":"The Mézard–Parisi formula for diluted spin glasses","summary":"Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability. ([Lean](lean/docs/221.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"B","note":"Claims the Mézard–Parisi cavity formula for diluted even-arity spin glasses (Viana–Bray); Lean; OpenAI published a reasoning summary.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/221.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/221.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/AGENT_NOTES.md","https://github.com/openai/math/blob/main/reasoning_traces/mezard-parisi-formula.pdf"]},{"family":"222","title":"Perceptron free energies and microscopic jamming exponents","summary":"Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order. ([Lean](lean/docs/222.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims perceptron free energies (Ising and spherical) and jamming exponents; Lean only for finiteness; Fu–Sohn's independent draft overlaps.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/222.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/222.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/AGENT_NOTES.md","https://hexagonmath.org/abs/2610.00154v1"]},{"family":"223","title":"Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling","summary":"Proves chordal SLE<sub>κ</sub> limits for critical square-lattice random-cluster interfaces for 0\\lt q\\le4, with \\kappa=4\\pi/\\arccos(-\\sqrt q/2): bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q &lt; 1, under the stated marked-boundary approximations. For 1\\le q\\le4, complete nested plane loops converge to CLE<sub>κ</sub>.","papers":["https://github.com/openai/math/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf","https://github.com/openai/math/blob/main/preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf","https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"A","note":"Claims SLE limits of critical square-lattice FK interfaces for 0<q≤4, including bond percolation (q=1); no Lean; an independent audit alleges a gap.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/AGENT_NOTES.md","https://github.com/JJardine919/openai-math-audit"]},{"family":"224","title":"Critical and quenched near-critical universality for Poisson–Voronoi percolation","summary":"Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment probability to the triangular-lattice reference law. This establishes quenched near-critical universality for crossing thresholds.","papers":["https://github.com/openai/math/blob/main/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf","https://github.com/openai/math/blob/main/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf","https://github.com/openai/math/blob/main/preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims Cardy's formula for critical Poisson–Voronoi percolation and quenched near-critical universality; no Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf","https://github.com/openai/math/blob/main/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/AGENT_NOTES.md"]},{"family":"225","title":"Gaussian free field limits throughout the balanced six-vertex regime","summary":"The balanced square-lattice six-vertex height field with a=b=1 and 0\\lt c\\le2 converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit height increments and Green kernel -(2\\pi)^{-1}\\log|x-y|, the exact variance multiplier is 1/\\arcsin(c/2).","papers":["https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf"],"lean":false,"entries":[],"name_hits":["2026-05-12-gaussian-completely-monotone-disproved","2026-08-28-matrix-spencer-conjecture-proved","2026-10-02-meta-muse-spark-six-math-papers","2026-10-06-openai-math-release-722-manuscripts","2026-10-07-colkitt-codex-integer-multiplication-exponent","2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Claims Gaussian free field limits for the balanced six-vertex model for all 0 < c ≤ 2, variance multiplier 1/arcsin(c/2); no Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/AGENT_NOTES.md"]},{"family":"226","title":"The double-dimer loop ensemble converges to CLE<sub>4</sub>","summary":"Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested CLE<sub>4</sub>. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.","papers":["https://github.com/openai/math/blob/main/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims half-plane double-dimer loops converge to nested CLE₄ as curves (Temperleyan case of Kenyon's conjecture); no Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/AGENT_NOTES.md"]},{"family":"227","title":"Critical SK autocorrelation processes and dynamics across the temperature transition","summary":"For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the \\log n scale for fixed 0\\le\\beta\\lt 1, mixing time n^{2/3+o(1)} at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β &gt; 1, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit. ([Lean](lean/docs/227.md))","papers":["https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf","https://github.com/openai/math/blob/main/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/"],"name_hits":["2026-06-09-gemini-3-5-live-translate","2026-08-18-openai-pauses-rl-training","2026-09-01-openai-astra-critical-cyber-threshold","2026-09-03-dying-percolation-theta-pc-zero","2026-09-03-gpt-6-astra","2026-09-03-openai-daybreak-frontline-defenders"],"tier":"B","note":"Claims SK Glauber mixing across the transition (cutoff, n^(2/3) at criticality); Lean for several; Du–Huang–Mohanty–Rajaraman and Hou–Liu–Yu treat it as concurrent work.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/227.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf","https://github.com/openai/math/blob/main/lean/docs/227.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.17717","https://arxiv.org/abs/2610.09882"]},{"family":"228","title":"Continuum phase transitions for radial pair potentials","summary":"Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem. ([Lean](lean/docs/228.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims stable radial pair potentials with a first-order continuum phase transition, the type sought in Barry Simon's problem; Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/228.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/228.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/AGENT_NOTES.md"]},{"family":"229","title":"Exact three- and four-state reconstruction thresholds and four-state tree capacity","summary":"Proves the exact reconstruction threshold d\\lambda^2\\gt 1, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d &gt; 1 and d &gt; 0, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of λ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model. ([Lean](lean/docs/229.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/four-state-potts.pdf","https://github.com/openai/math/blob/main/preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026/four-state-capacity.pdf","https://github.com/openai/math/blob/main/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-02-05-gpt-5-3-codex","2026-07-14-gpt-5-6-zeroth-order-convex-lower-bound","2026-09-25-conjectures-io-erdos-green-bounty-solves","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e","2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim"],"tier":"B","note":"Claims exact Kesten–Stigum reconstruction thresholds for 3- and 4-state tree channels and the 3-community block-model threshold; partial Lean.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/229.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/four-state-potts.pdf","https://github.com/openai/math/blob/main/preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026/four-state-capacity.pdf","https://github.com/openai/math/blob/main/lean/docs/229.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/AGENT_NOTES.md"]},{"family":"230","title":"Exact Hausdorff gauges for SLE","summary":"Resolves Schramm’s Hausdorff-measure question for chordal SLE<sub>κ</sub>, 0\\lt \\kappa\\lt 8. The explicit gauge r^d(\\log\\log(1/r))^{(2-d)/2}, d=1+\\kappa/8, gives almost surely positive finite measure to every trace segment \\gamma([s,t]) with 0\\lt s\\lt t\\lt \\infty, and finite expected measure to the trace in every bounded disk. ([Lean](lean/docs/230.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/"],"name_hits":[],"tier":"B","note":"Claims exact Hausdorff gauges for SLE_κ (Schramm's question); partial Lean; Dapeng Zhan used the companion to identify natural length.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/230.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/230.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.15066"]},{"family":"231","title":"The free uniform spanning forest is a factor of IID","summary":"On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID. ([Lean](lean/docs/231.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims the free uniform spanning forest is a factor of IID on every graph (Lyons's question); Lean-formalized; no reaction found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/231.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/231.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/AGENT_NOTES.md"]},{"family":"232","title":"Gaussian fields and interfaces for triangular-lattice Lipschitz heights","summary":"Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments 0,\\pm2 and two-arc boundary values \\pm1, and zero-boundary integer Lipschitz heights weighted by fixed x\\in[1/\\sqrt2,1]. Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE<sub>4</sub>, establishing Schramm’s real-field/interface predictions.","papers":["https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Claims GFF limits for triangular-lattice Lipschitz heights and an SLE₄ interface; three manuscripts revised 7 Oct after proof repairs; no Lean.","lean_scope":"","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026/paper.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/AGENT_NOTES.md","https://github.com/openai/math/blob/main/history.md"]},{"family":"233","title":"The joint critical Ashkin–Teller current limit","summary":"Identifies the joint scaling limit of Ashkin–Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible lattice approximations and wired primal/free dual boundaries, the height converges to the predicted Gaussian free field, and the clusters to canonical recursive sets of that same field, retaining every nesting depth.","papers":["https://github.com/openai/math/blob/main/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/main.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"C","note":"Claims the joint critical Ashkin–Teller height and current scaling limit; revised 7 Oct (transmission, boundary-collar repairs); no Lean.","lean_scope":"","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/main.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/AGENT_NOTES.md","https://github.com/openai/math/blob/main/history.md"]},{"family":"234","title":"All-temperature pressure of orthogonally invariant Ising spin glasses","summary":"Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero. ([Lean](lean/docs/234.md))","papers":["https://github.com/openai/math/blob/main/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Claims all-temperature variational pressure formulas for orthogonally invariant Ising spin glasses; Lean-formalized; specialist result.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/234.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/234.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/AGENT_NOTES.md"]},{"family":"235","title":"Limiting random SAT thresholds, sharp variance and computability","summary":"For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance \\Theta_k(n) for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent [ECCC TR26-229](https://eccc.weizmann.ac.il/report/2026/229/), made public October 5, 2026; this family supplies another proof and the sharper variance and computability results. ([Lean](lean/docs/235.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf","https://github.com/openai/math/blob/main/preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf","https://github.com/openai/math/blob/main/preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026/article.pdf","https://github.com/openai/math/blob/main/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai/"],"name_hits":[],"tier":"A","note":"Threshold for random k-SAT, every k ≥ 3, plus Θ(n) variance and computability of α₃; all 4 Lean challenges accepted (26 Oct). OpenAI credits Carenini's human ECCC proof (5 Oct) with priority. Own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/235.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf","https://github.com/openai/math/blob/main/preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf","https://github.com/openai/math/blob/main/lean/docs/235.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/AGENT_NOTES.md","https://eccc.weizmann.ac.il/report/2026/229/","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv","https://arxiv.org/abs/2609.26222"]},{"family":"236","title":"The exact factor-of-IID threshold for free Ising spins on trees","summary":"Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when \\tanh\\beta\\le(d-1)^{-1/2}, including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question. ([Lean](lean/docs/236.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims the free Ising state on d-regular trees is a factor of IID exactly when tanh β ≤ 1/√(d−1) (Lyons); Lean-formalized.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/236.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/236.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/AGENT_NOTES.md"]},{"family":"237","title":"The three-quarter exponent for honeycomb self-avoiding walk","summary":"Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter n^{3/4+o(1)}. Its local mass and covering numbers have exponent 4/3. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability. ([Lean](lean/docs/237.md))","papers":["https://github.com/openai/math/blob/main/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Claims honeycomb self-avoiding walks have diameter n^(3/4+o(1)), Nienhuis's prediction, via 13 manuscripts; Lean covers only supporting finiteness lemmas.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/237.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/237.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/AGENT_NOTES.md"]},{"family":"238","title":"Optimal logarithmic mixing of the Thorp shuffle","summary":"Proves that the Thorp shuffle randomizes N=2^d labeled cards in \\Theta(\\log N) physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions. ([Lean](lean/docs/238.md))","papers":["https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims Θ(log N) mixing of the Thorp shuffle for N = 2^d cards, settling its order; 12 manuscripts with Lean; cutoff open.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/238.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/238.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/AGENT_NOTES.md"]},{"family":"239","title":"Sharp singularity rates for symmetric random sign matrices","summary":"Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give \\Pr(\\det A_n=0)=(1/2+o(1))^n; for fixed bias p\\in(0,1)\\setminus\\{1/2\\}, the rate is (p^2+(1-p)^2+o(1))^n. In the biased case, agreeing rows attain this rate.","papers":["https://github.com/openai/math/blob/main/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf","https://github.com/openai/math/blob/main/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Claims P(det A_n = 0) = (1/2+o(1))^n for symmetric random sign matrices, and the biased-sign rate; no Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf","https://github.com/openai/math/blob/main/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/AGENT_NOTES.md"]},{"family":"240","title":"Shelah's eventual categoricity and the prescribed-threshold obstruction","summary":"Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice. ([Lean](lean/docs/240.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims Shelah's eventual categoricity conjecture in ZFC; Lean covers only the CH obstruction; Espíndola disputes Proposition 7.2.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/240.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/240.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/AGENT_NOTES.md","https://x.com/christianrubene/status/2111425004667572321"]},{"family":"241","title":"Rigidity of the Turing degrees","summary":"Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability. ([Lean](lean/docs/241.md))","papers":["https://github.com/openai/math/blob/main/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/"],"name_hits":[],"tier":"B","note":"Claims the Turing degrees are rigid; Lean; Andrew Marks re-proved the key lemma; Kjos-Hanssen adapted it to truth-table degrees.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/241.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/241.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/AGENT_NOTES.md","https://math.berkeley.edu/~marks/notes/aut.pdf","https://blog.computationalcomplexity.org/2026/10/conjecture-that-i-thought-would-not-be.html"]},{"family":"242","title":"Single-fold Diophantine representations and undecidability under an at-most-one-solution promise","summary":"Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise. ([Lean](lean/docs/242.md))","papers":["https://github.com/openai/math/blob/main/preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims every r.e. set has a single-fold Diophantine representation (Davis–Matiyasevich–Robinson question); Lean-formalized; no reaction found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/242.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/242.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Single-fold-Diophantine-representations-September-24-2026/AGENT_NOTES.md"]},{"family":"243","title":"Separating choiceless counting from polynomial time and witnessed choice","summary":"Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽<sub>3</sub> defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank. ([Lean](lean/docs/243.md))","papers":["https://github.com/openai/math/blob/main/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims choiceless polynomial time with counting does not capture PTIME (Blass–Gurevich–Shelah); Lean-formalized; no reaction found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/243.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/243.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/AGENT_NOTES.md"]},{"family":"244","title":"The Partition Principle does not imply Choice","summary":"Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements. ([Lean](lean/docs/244.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-08-karagila-openai-partition-principle/","https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/"],"name_hits":[],"tier":"A","note":"Claims Con(ZF) gives a model of the Partition Principle without Choice; Lean; Karagila criticised the paper and declined to review; own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/244.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf","https://github.com/openai/math/blob/main/lean/docs/244.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/AGENT_NOTES.md","https://karagila.org/2026/openai-pp/"]},{"family":"245","title":"Weak normalization implies strong normalization in pure type systems","summary":"Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. ([Lean](lean/docs/245.md))","papers":["https://github.com/openai/math/blob/main/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims weak normalization implies strong normalization for all pure type systems (Barendregt–Geuvers–Klop); Lean-formalized; no reaction found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/245.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/245.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/AGENT_NOTES.md"]},{"family":"246","title":"Cannon's conjecture","summary":"Every word-hyperbolic group with boundary homeomorphic to S<sup>2</sup> admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold. ([Lean](lean/docs/246.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-cannon-conjecture-proof-claim/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"A","note":"Claims Cannon's conjecture: 2-sphere-boundary hyperbolic groups act geometrically on H³; Lean; Mario Bonk presented 'the AI proof' at Quasiworld (22 Oct).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/246.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/246.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/AGENT_NOTES.md","https://www.youtube.com/watch?v=L2DU0VupdlU","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"247","title":"An infinite finitely presented residually finite 2-group and a finitely presented nil algebra","summary":"Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽<sub>2</sub>-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions. ([Lean](lean/docs/247.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf","https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-finitely-presented-burnside-periodic-group-claim/"],"name_hits":[],"tier":"A","note":"Claims an infinite finitely presented periodic group (Steinberg group with property (T)) and a residually finite 2-group; Lean for the first.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/247.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf","https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/247.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/AGENT_NOTES.md","https://github.com/KitaKen1/all-prime-residually-finite-p-groups"]},{"family":"248","title":"Thompson's group F is nonamenable","summary":"Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem. ([Lean](lean/docs/248.md))","papers":["https://github.com/openai/math/blob/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition/"],"name_hits":[],"tier":"A","note":"Claims Thompson's group F is nonamenable; Lean; Kim–Lodha's human exposition and follow-ups; own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/248.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/248.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/AGENT_NOTES.md","https://arxiv.org/abs/2610.18360"]},{"family":"249","title":"A finitely generated Eilenberg–Ganea counterexample","summary":"Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells. ([Lean](lean/docs/249.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims a finitely generated group with cohomological dimension 2 and geometric dimension 3, refuting Eilenberg–Ganea; Lean; Avramidi ties it to Bestvina–Brady.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/249.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/249.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/AGENT_NOTES.md","https://scottaaronson.blog/?p=10169","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"250","title":"Boone–Higman embeddings with higher finiteness","summary":"A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F<sub>∞</sub>: a classifying space with finitely many cells in each dimension. A single group of type F<sub>∞</sub> can also contain every finitely presented group. ([Lean](lean/docs/250.md))","papers":["https://github.com/openai/math/blob/main/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims the Boone–Higman conjecture with F∞ simple overgroups; Lean; Zaremsky says it shows his group's sufficient condition always works.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/250.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/250.md","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/AGENT_NOTES.md","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"251","title":"Amenability, unitarizability, and strong Ulam stability","summary":"Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space. ([Lean](lean/docs/251.md))","papers":["https://github.com/openai/math/blob/main/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Claims Dixmier's unitarizability problem (unitarizable groups are amenable) and strong Ulam stability. Lean covers Dixmier part. Kourovka 20.62: 'unverified AI solution'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/251.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf","https://github.com/openai/math/blob/main/lean/docs/251.md","https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21upd-3.pdf","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","https://github.com/certusoid/unitarisability-and-amenability"]},{"family":"252","title":"A torsion-free hyperbolic group that is neither residually finite nor linear over any field","summary":"Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field. ([Lean](lean/docs/252.md))","papers":["https://github.com/openai/math/blob/main/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-17-non-sofic-hyperbolic-group-sandhu-thom/"],"name_hits":[],"tier":"A","note":"Torsion-free hyperbolic group that is not residually finite, a long-standing question; Lean accepted. Basis of Sandhu–Thom's non-sofic group. Kourovka: unverified AI solution.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/252.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/252.md","https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21upd-3.pdf","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"253","title":"An infinite finitely presented simple amenable group","summary":"Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously. ([Lean](lean/docs/253.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Infinite finitely presented simple amenable group; Lean (existence) accepted. Zaremsky struck it from his list: 'I'll just cross out problems solved by AI.'","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/253.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/253.md","https://zaremsky.github.io/open_problems.pdf"]},{"family":"254","title":"Classifying spaces and geometric obstructions for Artin groups","summary":"The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin K(\\pi,1) conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT(0) space. ([Lean](lean/docs/254.md))","papers":["https://github.com/openai/math/blob/main/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"A","note":"Artin K(π,1) conjecture (finite rank), parabolic intersection conjecture, Artin group without geometric CAT(0) action; all Lean-accepted. Tiozzo: Astra shrank 116 generators to 12.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/254.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/254.md","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","https://github.com/frogscooper/artin-cat0-29"]},{"family":"255","title":"Quasi-isometric recognition of virtually polycyclic groups","summary":"Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one. ([Lean](lean/docs/255.md))","papers":["https://github.com/openai/math/blob/main/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Eskin–Fisher–Whyte conjecture: groups quasi-isometric to virtually polycyclic groups are virtually polycyclic. Lean scope note exists, Comparator accepted. No reactions found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/255.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/255.md","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv"]},{"family":"256","title":"Nonsingular systems of equations over arbitrary groups","summary":"Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group. ([Lean](lean/docs/256.md))","papers":["https://github.com/openai/math/blob/main/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf","https://github.com/openai/math/blob/main/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev/"],"name_hits":[],"tier":"B","note":"Kervaire's conjecture and its nonsingular-systems extension (Howie). Kourovka 17.94 and 18.87(a): 'AI solution confirmed by N. V. Bogachev'. Lean covers single-relator case. Own entry (30 Oct).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/256.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf","https://github.com/openai/math/blob/main/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/256.md","https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21upd-3.pdf","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"257","title":"A hyperbolic group without a geometric CAT(0) action","summary":"Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete \\mathop{\\mathrm{CAT}}\\nolimits (0) space, in any dimension. ([Lean](lean/docs/257.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Word-hyperbolic group with no geometric CAT(0) action, so not every hyperbolic group is CAT(0). Lean accepted; two-dimensionality not formalized. Lupini builds on it.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/257.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/257.md","https://zenodo.org/records/23424960"]},{"family":"258","title":"Gersten’s conjecture and virtual compact specialness of one-relator groups","summary":"Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup \\mathrm{BS}(m,n), with m,n\\ne0, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.","papers":["https://github.com/openai/math/blob/main/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Gersten's conjecture: Baumslag–Solitar-free one-relator groups are hyperbolic; hyperbolic ones virtually compact special. Kharlampovich–Miasnikov use it, explicitly conditionally.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf","https://arxiv.org/abs/2610.25305"]},{"family":"259","title":"A group without fixed price","summary":"Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.","papers":["https://github.com/openai/math/blob/main/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Negative answer to Gaboriau's fixed price problem: a finitely generated group with free actions of different costs. No Lean; no reactions found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf"]},{"family":"260","title":"Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions","summary":"Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem. ([Lean](lean/docs/260.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance/"],"name_hits":[],"tier":"A","note":"Spacetime Penrose inequality in all dimensions plus charged, rotating and AdS versions. Lean covers supporting results only. Bray–Dolmen–Khuri–Schmidt prove a maximally parameterized case.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/260.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Charged-Reduction-of-the-Spacetime-Penrose-Inequality-in-Spatial-Dimensions-at-Least-Four-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/260.md","https://arxiv.org/abs/2610.20559"]},{"family":"261","title":"Localization and delocalization in the Anderson model","summary":"Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight. ([Lean](lean/docs/261.md))","papers":["https://github.com/openai/math/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-anderson-localization-2d-delocalization-3d/"],"name_hits":[],"tier":"A","note":"2D Anderson localization at every disorder and 3D absolutely continuous spectrum at weak disorder (Simon's Problems 1–2). OpenAI Lean: spectrum only; third-party Lean of 3D.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/261.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/261.md","https://arxiv.org/abs/2610.11611","https://github.com/cadamcat/anderson-ac-lean4"]},{"family":"262","title":"Sharp finite-matrix Lieb–Thirring inequalities and all equality cases","summary":"Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2\\lt \\gamma\\lt 3/2 and arbitrary finite-matrix potentials W ≥ 0 with \\int\\mathop{\\mathrm{tr}}\\nolimits (W^{\\gamma+1/2})\\lt \\infty: the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech<sup>2</sup> solitons with independent scales and centers, and zero channels. ([Lean](lean/docs/262.md))","papers":["https://github.com/openai/math/blob/main/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Sharp one-dimensional Lieb–Thirring constant for 1/2<γ<3/2 (scalar conjecture), matrix version and equality cases. Lean (scalar constant) Comparator-accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/262.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/262.md","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv"]},{"family":"263","title":"The ionization and generalized ionization conjectures","summary":"For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most Z+CM electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as m\\to\\infty and Z/m\\to\\infty; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking Z\\to\\infty first. ([Lean](lean/docs/263.md))","papers":["https://github.com/openai/math/blob/main/preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Ionization conjecture forms: molecules bind at most Z+CM electrons; universal ionization-energy and radius bounds; generalized asymptotics (Lean). Chen–Wu prove a TFDW analogue.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/263.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/263.md","https://arxiv.org/abs/2610.09997","https://github.com/cadamcat/atomic-ionization-lean4"]},{"family":"264","title":"Strong cosmic censorship near two-ended Kerr data","summary":"Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense G<sub>δ</sub> subset of a weighted smooth neighborhood of smooth complete two-ended asymptotically flat vacuum data has full maximal globally hyperbolic developments with no future continuous nondegenerate extension whose weak connection is locally square-integrable. No symmetry is imposed; extensions need not satisfy the vacuum equations.","papers":["https://github.com/openai/math/blob/main/preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-strong-cosmic-censorship-near-kerr/"],"name_hits":["2025-11-20-openai-gpt-5-science-acceleration","2026-10-06-quasi-riemann-hypothesis-openai","2026-10-11-landau-siegel-zeros-banks-explicit-account","2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance","2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean","2026-10-16-ai-assisted-results-october-15-listing"],"tier":"A","note":"Local strong cosmic censorship near Kerr in Christodoulou's C⁰∩W^{1,2}_loc formulation, with C¹ and C² versions. No Lean. Sung-Jin Oh organised a reading group.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf","https://math.berkeley.edu/~sjoh/pdfs/understanding-essay.pdf"]},{"family":"265","title":"Area laws and tensor networks for two-dimensional gapped systems","summary":"Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open L\\times L squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L<sup>−1</sup>.","papers":["https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-16-openai-2d-area-law-lean-formalized-axiom/"],"name_hits":[],"tier":"A","note":"2D area law and polynomial PEPS for gapped ground states. Formalized independently (Axiom's AreaLaw2D, TNLean). Aaronson: 'One of the main open problems in Hamiltonian complexity.'","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf","https://scottaaronson.blog/?p=10169","https://github.com/AxiomMath/AreaLaw2D","https://arxiv.org/abs/2610.15199"]},{"family":"266","title":"Exactly three mutually unbiased bases in dimension six","summary":"Proves N(6)=3, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ<sup>6</sup>, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class. ([Lean](lean/docs/266.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-14-ai-assisted-results-mid-october-2026"],"tier":"A","note":"Zauner's conjecture: exactly three MUBs in ℂ⁶, computer-assisted. Code restored 26 Oct; Lean proves only ≤5. Jaswal reports an independent certified computation (Zenodo, 13 Oct).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/266.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/266.md","https://zenodo.org/doi/10.5281/zenodo.23255857","https://github.com/maximusrufus/mub6-certified-rerun"]},{"family":"267","title":"Positive-temperature Bose–Einstein condensation and exact quantum depletion","summary":"Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit. ([Lean](lean/docs/267.md))","papers":["https://github.com/openai/math/blob/main/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf","https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf","https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf","https://github.com/openai/math/blob/main/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-heisenberg-ferromagnet-bose-einstein-condensation/"],"name_hits":[],"tier":"A","note":"Bose–Einstein condensation in every ground state of the dilute hard-sphere gas (Lean accepted) and at fixed positive temperature; Bogoliubov depletion law. SciAm highlight.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/267.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf","https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf","https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf","https://github.com/openai/math/blob/main/lean/docs/267.md","https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv"]},{"family":"268","title":"The spin-one Haldane gap","summary":"Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field h=3/5 and gives boundary-selected infinite-volume states with topological index −1.","papers":["https://github.com/openai/math/blob/main/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-spin-one-haldane-gap-proof/"],"name_hits":[],"tier":"A","note":"Spin-one Haldane gap for even periodic chains (computer-assisted). Hal Tasaki studied the analytic part favourably; computer part not independently checked. Own entry.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf","https://haltasaki.github.io/writings/AI/OpenAI_Haldane.html"]},{"family":"269","title":"Uniform Laughlin gap and stability under bounded scalar disorder","summary":"Proves the fermionic Laughlin spectral-gap conjecture for the full V<sub>1</sub> interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles. ([Lean](lean/docs/269.md))","papers":["https://github.com/openai/math/blob/main/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf","https://github.com/openai/math/blob/main/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Laughlin spectral-gap conjecture at filling 1/3, stable under weak disorder. Lean covers unperturbed gap. Qiang Gao's SDP improves the bound from 1/25 to 1/5.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/269.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf","https://github.com/openai/math/blob/main/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/269.md","https://arxiv.org/abs/2610.12700"]},{"family":"270","title":"Threshold and positive-energy bound states of the BFSS matrix model","summary":"Proves that the undeformed relative \\mathrm{SU}(N) BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For \\mathrm{SU}(2), a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.","papers":["https://github.com/openai/math/blob/main/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"BFSS matrix model: unique SU(N) threshold bound state; infinitely many SU(2) positive-energy bound states, contradicting the original BFSS paper. No Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf"]},{"family":"271","title":"Bloch's law, its lattice correction, and the spherical magnetization law","summary":"Proves Bloch's T<sup>3/2</sup> law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ<sup>3</sup>. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings. ([Lean](lean/docs/271.md))","papers":["https://github.com/openai/math/blob/main/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf","https://github.com/openai/math/blob/main/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf","https://github.com/openai/math/blob/main/preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf","https://github.com/openai/math/blob/main/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-heisenberg-ferromagnet-bose-einstein-condensation/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-12-bloch-constant-conjecture-disproved"],"tier":"A","note":"Spontaneous magnetization of the quantum Heisenberg ferromagnet in d≥3 (Lean accepted) and Bloch's T^{3/2} law. OpenAI published a reasoning summary; SciAm highlight.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/271.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf","https://github.com/openai/math/blob/main/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf","https://github.com/openai/math/blob/main/preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf","https://github.com/openai/math/blob/main/lean/docs/271.md","https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/","https://github.com/openai/math/blob/main/reasoning_traces/spontaneous-magnetization-quantum-heisenberg-ferromagnet.pdf"]},{"family":"272","title":"Entanglement without distillable secret key","summary":"Constructs an entangled state on \\mathbb C^{10}\\otimes\\mathbb C^{10} with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on M_{21}(\\mathbb C) whose square is not entanglement breaking disproves Christandl's PPT-square conjecture. ([Lean](lean/docs/272.md))","papers":["https://github.com/openai/math/blob/main/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-08-ppt-entanglement-distillation-rains-bound-chatgpt-astra","2026-10-16-ai-assisted-results-october-15-listing"],"tier":"B","note":"Entangled state with zero distillable secret key; disproves Christandl's PPT-squared conjecture with a 21-dimensional channel. Lean covers the PPT-squared counterexamples only.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/272.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/272.md"]},{"family":"273","title":"The entropy photon-number inequality","summary":"Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ. ([Lean](lean/docs/273.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Entropy photon-number inequality (Guha–Shapiro conjecture) for beam splitters in any number of modes. Lean Comparator-accepted. No expert reaction found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/273.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/273.md","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv"]},{"family":"274","title":"Parity is not in QAC<sup>0</sup>","summary":"Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li's reductions give the same bounded-error obstruction for strict majority. ([Lean](lean/docs/274.md))","papers":["https://github.com/openai/math/blob/main/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu"],"tier":"B","note":"Parity is not in QAC⁰ (Moore's conjecture, measured-output model). Lean accepted. Aaronson: 'one of the great questions of quantum complexity theory since 1999'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/274.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/274.md","https://scottaaronson.blog/?p=10169"]},{"family":"275","title":"QMA-hardness of continuum Coulomb energy","summary":"Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one. ([Lean](lean/docs/275.md))","papers":["https://github.com/openai/math/blob/main/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"QMA-hardness of approximating the continuum Coulomb ground-state energy. Lean Comparator-accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/275.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026.pdf","https://github.com/openai/math/blob/main/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/275.md"]},{"family":"276","title":"Classical capacity of generalized amplitude damping","summary":"Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel. ([Lean](lean/docs/276.md))","papers":["https://github.com/openai/math/blob/main/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-03-dying-percolation-theta-pc-zero","2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra","2026-09-06-collatz-positive-density-log-time-mazur-codex","2026-09-27-poincare-conjecture-lean-formalization","2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e","2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim"],"tier":"C","note":"Exact classical capacity of every qubit generalized amplitude-damping channel, with additivity. Lean (capacity) accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/276.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/276.md"]},{"family":"277","title":"Threshold repetition for entangled games","summary":"Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v &lt; 1, the probability of winning at least a fraction v+\\delta of k independent repetitions decays exponentially in k, for 0\\lt \\delta\\lt 1-v. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed. ([Lean](lean/docs/277.md))","papers":["https://github.com/openai/math/blob/main/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-01-new-mexico-frontier-ai-safety-accountability-act","2026-10-02-meta-muse-spark-six-math-papers","2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e","2026-10-16-china-proposes-export-controls-ai-models-agents"],"tier":"C","note":"Exponential threshold parallel repetition for entangled two-player games. Lean accepted; Rahul Jain compares rates in arXiv 2610.21410.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/277.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/277.md","https://arxiv.org/abs/2610.21410"]},{"family":"278","title":"Failure of Kohn–Sham ensemble representation","summary":"Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in L^{3/2}(\\mathbb R^3)+L^\\infty(\\mathbb R^3). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.","papers":["https://github.com/openai/math/blob/main/preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"C","note":"Kohn–Sham ensemble representability fails for a three-electron molecule in L^{3/2}+L^∞. No Lean; Haycraft's note argues representability in a larger class.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf","https://github.com/Hadron90/kohn-sham-representability"]},{"family":"279","title":"Exact quantum factoring over a fixed finite gate set","summary":"Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices. ([Lean](lean/docs/279.md))","papers":["https://github.com/openai/math/blob/main/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-02-05-gpt-5-3-codex","2026-07-14-gpt-5-6-zeroth-order-convex-lower-bound","2026-09-25-conjectures-io-erdos-green-bounty-solves","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e","2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim"],"tier":"C","note":"Exact (probability-one) polynomial-time quantum factoring over one fixed finite gate set. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/279.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/279.md"]},{"family":"280","title":"Unitary vertex operator algebras and conformal nets","summary":"Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category. ([Lean](lean/docs/280.md))","papers":["https://github.com/openai/math/blob/main/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Strongly rational unitary vertex operator algebras give completely rational conformal nets (strong locality). Lean covers the first construction step only.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/280.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/280.md"]},{"family":"281","title":"QAOA attains the SK optimum in the thermodynamic-first limit","summary":"Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree. ([Lean](lean/docs/281.md))","papers":["https://github.com/openai/math/blob/main/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"QAOA reaches the Sherrington–Kirkpatrick ground-state energy when size grows before depth. Lean covers Parisi identities, not QAOA convergence.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/281.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/281.md"]},{"family":"282","title":"From scale symmetry to local conformal symmetry in four-dimensional QFT","summary":"Under the stated bounded-local-net and field-reconstruction hypotheses, proves that scale symmetry implies local conformal symmetry for four-dimensional unitary positive-energy theories with a discrete bounded-below scaling spectrum of finite multiplicity, finite scaling support and a physical local scale current. The stress tensor has a traceless improvement with unchanged spacetime charges. The conclusion concerns local Ward identities, not a global conformal action on the whole net.","papers":["https://github.com/openai/math/blob/main/preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Scale invariance implies local conformal invariance in 4D QFT under stated algebraic-QFT hypotheses (conditional). A 2013 claim by Farnsworth–Luty–Prelipina was withdrawn.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf"]},{"family":"283","title":"Polynomial-time unitary synthesis from a Boolean oracle","summary":"Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every n-qubit unitary channel within diamond-norm error 1/2, after a suitable Boolean oracle is chosen. Gates, qubits, oracle calls and query length are polynomially bounded. The target-dependent oracle may have an unrestricted truth table; its efficient classical construction is not asserted.","papers":["https://github.com/openai/math/blob/main/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Positive answer to the Aaronson–Kuperberg unitary synthesis problem. Aaronson: 'the opposite of what most of us expected'; SciAm highlight; Dakshita Khurana is working through it.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf","https://scottaaronson.blog/?p=10169","https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/","https://dakshitakhurana.substack.com/p/classical-at-heart"]},{"family":"284","title":"The optimal quartic separation between randomized and quantum queries","summary":"Shows that the universal bound R(f)=O((1+Q(f))^4) for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and quantum worst-case bit-query complexities with error at most 1/3; computation between queries is unrestricted.","papers":["https://github.com/openai/math/blob/main/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"R(f)=O(Q(f)^4) is tight for total Boolean functions, refuting the cubic conjecture. Aaronson: 'this finally closes that story'. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf","https://scottaaronson.blog/?p=10169"]},{"family":"285","title":"Counterexamples to Baum–Connes and Kadison–Kaplansky","summary":"Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C<sup>∗</sup>-algebra, disproving the Kadison–Kaplansky conjecture.","papers":["https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-baum-connes-kadison-kaplansky-counterexamples/"],"name_hits":[],"tier":"A","note":"Coefficient-free Baum–Connes fails (torsion-free injectivity; irrational-trace surjectivity); Kadison–Kaplansky fails. Guoliang Yu builds on the irrational-trace paper (arXiv 2610.35674). No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf","https://arxiv.org/abs/2610.35674"]},{"family":"286","title":"Rigidity and arithmetic of lattice von Neumann algebras","summary":"Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-(T) lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.","papers":["https://github.com/openai/math/blob/main/preprints/Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Lattice case of Connes' rigidity: classification of finite-index bimodules of property-(T) lattice factors. Cyril Houdayer 'genuinely thrilled' (Proofs and Prompts).","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026/paper.pdf","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"287","title":"Isomorphism of the free group factors","summary":"Resolves the free group factor isomorphism problem: L(\\mathbb F_2)\\cong L(\\mathbb F_3), and hence all interpolated free group factors, including L(\\mathbb F_\\infty), are isomorphic. Their common factor has fundamental group \\mathbb R_{\\gt 0}. ([Lean](lean/docs/287.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-free-group-factors-isomorphism-openai/"],"name_hits":[],"tier":"A","note":"Free group factors are all isomorphic, L(F₂)≅L(F₃). Lean accepted; Shlyakhtenko proved a C*-algebraic version (our entry marks it confirmed).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/287.md","status":"confirmed","sources":["https://github.com/openai/math/blob/main/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/287.md","https://arxiv.org/abs/2610.19590"]},{"family":"288","title":"Kadison's similarity conjecture","summary":"Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C<sup>∗</sup>-algebra to operators on a Hilbert space becomes a *-homomorphism after conjugation by a bounded invertible operator. ([Lean](lean/docs/288.md))","papers":["https://github.com/openai/math/blob/main/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-kadison-similarity-problem-lean/"],"name_hits":[],"tier":"A","note":"Kadison's 1955 similarity problem answered positively, plus one hyperreflexivity constant; Lean full statement accepted (oai-math-revised, MathVet). AIM marks solved; no expert reaction. Own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/288.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/288.md","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/AGENT_NOTES.md","https://math.vet/openai-math/","https://github.com/MColbrook/AIM/blob/main/problems/037-kadison-similarity.md","https://www.math.uwaterloo.ca/~j37peter/problems.html"]},{"family":"289","title":"Strong Kadison–Kastler stability and its spatial boundaries","summary":"Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C<sup>∗</sup>-algebras need not be ambiently unitarily conjugate. ([Lean](lean/docs/289.md))","papers":["https://github.com/openai/math/blob/main/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf","https://github.com/openai/math/blob/main/preprints/Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"B","note":"Strong Kadison–Kastler conjecture: nearby von Neumann algebras are conjugate by a near-identity unitary, plus boundary counterexamples. Lean (uniform statement) accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/289.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf","https://github.com/openai/math/blob/main/preprints/Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/289.md"]},{"family":"290","title":"Relative bicentralizers and modular spectral recovery","summary":"Proves Connes' bicentralizer conjecture for every type III<sub>1</sub> factor with separable predual and every faithful normal state. More generally, for every inclusion N\\subset M of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra P\\subset N with P'\\cap c(M)=N'\\cap c(M), resolving the relative bicentralizer conjecture. ([Lean](lean/docs/290.md))","papers":["https://github.com/openai/math/blob/main/preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026.pdf","https://github.com/openai/math/blob/main/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026/Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Relative Connes bicentralizer conjecture. Houdayer: it builds on Marrakchi's methods; Houdayer–Marrakchi solved the absolute case in September. Lean partial.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/290.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026.pdf","https://github.com/openai/math/blob/main/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026/Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/290.md","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"291","title":"Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability","summary":"Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C<sup>∗</sup>-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting. ([Lean](lean/docs/291.md))","papers":["https://github.com/openai/math/blob/main/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma"],"tier":"B","note":"Unital Toms–Winter conjecture and equivariant Jiang–Su stability (Szabó). Overlaps human papers (Evington et al., Thiel); our Toms–Winter entry covers the human proof.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/291.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/291.md","https://arxiv.org/abs/2610.09390"]},{"family":"292","title":"Kirchberg's $`\\mathcal O_2`$ norm-ultrapower embedding problem","summary":"Constructs an explicit separable unital full group C<sup>∗</sup>-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C<sup>∗</sup>-algebra, for any free ultrafilter on the natural numbers. Taking the target to be \\mathcal O_2 answers Kirchberg's norm-ultrapower embedding problem negatively. ([Lean](lean/docs/292.md))","papers":["https://github.com/openai/math/blob/main/preprints/An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Kirchberg's O₂ norm-ultrapower embedding problem answered negatively with an explicit full group C*-algebra. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/292.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/292.md"]},{"family":"293","title":"Invariant projections, hyperinvariant subspaces, and transitive algebras","summary":"Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II<sub>1</sub> factor. ([Lean](lean/docs/293.md))","papers":["https://github.com/openai/math/blob/main/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Operator on Hilbert space with no nontrivial hyperinvariant subspace; proper transitive commutant; II₁ operator with no invariant projection. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/293.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/293.md"]},{"family":"294","title":"Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness","summary":"Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C<sup>∗</sup>-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C<sup>∗</sup>-algebras can have a properly infinite minimal tensor product, with one factor C_r^*(\\mathbb F_2). ([Lean](lean/docs/294.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-19-quasitrace-monotone-completeness-counterexamples-openai/"],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-06-openai-math-release-722-manuscripts","2026-10-13-ai-assisted-results-october-2026","2026-10-19-quasitrace-monotone-completeness-counterexamples-openai"],"tier":"B","note":"Kaplansky's quasitrace conjecture disproved; Lean accepted. Arulseelan–Hanson fully reproduce it. Our entry wrongly called the manuscript unpublished; corrected 29 Oct.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/294.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/294.md","https://arxiv.org/abs/2610.27184"]},{"family":"295","title":"The Kadison–Ringrose cohomology conjecture","summary":"Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture. ([Lean](lean/docs/295.md))","papers":["https://github.com/openai/math/blob/main/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Kadison–Ringrose conjecture: bounded Hochschild cohomology of a von Neumann algebra in itself vanishes in degrees ≥2. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/295.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/295.md"]},{"family":"296","title":"The generator problem for finite factors","summary":"Proves that every type II<sub>1</sub> factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion P\\subset M of such factors, the unitaries u with M=W^*(P,u) form a dense G<sub>δ</sub> subset in the trace 2-norm topology. ([Lean](lean/docs/296.md))","papers":["https://github.com/openai/math/blob/main/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Generator problem: every II₁ factor with separable predual is singly generated. Lean accepted. Elliott–Li–Niu apply it (arXiv 2610.17411).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/296.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/296.md","https://arxiv.org/abs/2610.17411"]},{"family":"297","title":"A ZFC counterexample to Naimark's problem","summary":"Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C<sup>∗</sup>-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable. ([Lean](lean/docs/297.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Second ZFC counterexample to Naimark's problem, after Tanaka's. Nik Weaver (MathOverflow): against his expectation. Lean accepted with a definition-body caveat.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/297.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf","https://github.com/openai/math/blob/main/lean/docs/297.md","https://mathoverflow.net/a/515860"]},{"family":"298","title":"Two notions of free entropy differ even when both are finite","summary":"Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy -\\infty\\lt \\chi\\lt \\chi^*\\lt \\infty. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite. ([Lean](lean/docs/298.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Microstates and non-microstates free entropy differ while both finite (Voiculescu's question). Lean accepted; Jekel cites it as the first example.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/298.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/298.md","https://arxiv.org/abs/2610.27243"]},{"family":"299","title":"The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability","summary":"A nonzero unital separable complex C<sup>∗</sup>-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question. ([Lean](lean/docs/299.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Kirchberg–Rørdam character criterion for Jiang–Su stability; Dadarlat–Toms question. Lean accepted; Bruce–Geffen apply it to Thompson's group F.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/299.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/299.md","https://arxiv.org/abs/2610.28473"]},{"family":"300","title":"Approximation and quadratic strong-operator paving","summary":"Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with O(\\varepsilon^{-2}) blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.","papers":["https://github.com/openai/math/blob/main/preprints/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-06-free-group-factors-isomorphism-openai","2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e","2026-10-13-k-median-better-than-2-approximation-epfl"],"tier":"C","note":"Popa–Vaes quadratic strong-operator paving conjecture. No Lean; no reactions found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf","https://github.com/openai/math/blob/main/preprints/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf"]},{"family":"301","title":"Trace cones and Razak–Jacelon stabilization","summary":"Classifies separable nuclear complex C<sup>∗</sup>-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients. ([Lean](lean/docs/301.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Robert's trace-cone question: classification after Razak–Jacelon stabilization. Lean covers a supporting ideal-transport result.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/301.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/301.md"]},{"family":"302","title":"Radius of comparison equals half the mean dimension","summary":"For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of C(X)\\rtimes_h\\mathbb Z equals \\tfrac12\\mathrm{mdim}(X,h), including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.","papers":["https://github.com/openai/math/blob/main/preprints/Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026"],"tier":"B","note":"Radius of comparison equals half the mean dimension for minimal homeomorphisms (Phillips–Toms). Elliott–Niu build on it (arXiv 2610.11143). No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026/paper.pdf","https://arxiv.org/abs/2610.11143"]},{"family":"303","title":"Weak pure infiniteness and Cuntz-algebra absorption","summary":"Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C<sup>∗</sup>-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs \\mathcal O_\\infty, without unitality or simplicity assumptions. ([Lean](lean/docs/303.md))","papers":["https://github.com/openai/math/blob/main/preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Kirchberg–Rørdam question: proper infiniteness of positive elements implies strong pure infiniteness. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/303.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/303.md"]},{"family":"304","title":"The Hilbert–Smith conjecture in every dimension","summary":"Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.","papers":["https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-hilbert-smith-conjecture-claim/"],"name_hits":["2026-10-06-openai-math-release-722-manuscripts"],"tier":"A","note":"Hilbert–Smith conjecture in every dimension; no Lean. A separate AI-agent proof is Lean-checked on Palomar (adbrw). Formal Conjectures issue #6914 still open.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf","https://github.com/adbrw/HS_proof","https://palomar-registry.org/entry?id=PALOMAR-2026-10-16-000006&version=1"]},{"family":"305","title":"Four-dimensional disk embedding and Wall's conjecture","summary":"The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F<sub>2</sub> is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.","papers":["https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-borel-conjecture-dimension-four-free-group-not-good/"],"name_hits":[],"tier":"A","note":"Disk embedding conjecture fails, so F₂ is not good (Freedman–Quinn); a PD4 group without aspherical manifold model (Wall). No Lean; no expert check found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"306","title":"The purely cosmetic surgery conjecture","summary":"Distinct Dehn surgery slopes on a nontrivial smooth knot in S<sup>3</sup> never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.","papers":["https://github.com/openai/math/blob/main/preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Purely cosmetic surgery conjecture for knots in S³. No Lean. Ichihara–Kishimoto posted independent partial results (arXiv 2610.10019).","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026/paper.pdf","https://arxiv.org/abs/2610.10019"]},{"family":"307","title":"Failure of rational injectivity for maximal coarse assembly","summary":"Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture. ([Lean](lean/docs/307.md))","papers":["https://github.com/openai/math/blob/main/preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026","2026-10-14-first-proof-third-batch"],"tier":"B","note":"Rational coarse Novikov conjecture disproved: maximal and reduced coarse assembly fail rational injectivity. Lean (reduced case) accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/307.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/307.md"]},{"family":"308","title":"Finite Smith–Toda complexes at every height","summary":"For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology BP_*/(p,v_1,\\ldots,v_n) and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes V(4) at p = 1009.","papers":["https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-16-smith-toda-v4-angeltveit-improves-openai/"],"name_hits":[],"tier":"B","note":"Finite Smith–Toda complexes V(n) at every height for varying primes; V(4) at p=1009. Angeltveit extends V(4) to p≥11 (own entry).","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf","https://arxiv.org/abs/2610.23215"]},{"family":"309","title":"The Kervaire invariant problem at the prime three","summary":"Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.","papers":["https://github.com/openai/math/blob/main/preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Odd-primary Kervaire invariant problem at p=3: classes survive only in stems 10, 106 and 322. No Lean; no reactions found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/paper.pdf"]},{"family":"310","title":"Quillen's conjecture in rational homology","summary":"Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.","papers":["https://github.com/openai/math/blob/main/preprints/Rational-homology-and-Quillens-conjecture-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-09-arxiv-ai-disclosed-results-early-october-2026","2026-10-10-erdos-problem-1117-maximum-modulus-gpt-6-codex"],"tier":"B","note":"Quillen's conjecture on p-subgroup posets, in rational homology, for every finite group and prime. No Lean; no reactions found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Rational-homology-and-Quillens-conjecture-September-24-2026/paper.pdf"]},{"family":"311","title":"The Hovey–Strickland and Chai conjectures","summary":"Proves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height n. Through the implication of Barthel–Heard–Naumann, this proves the Hovey–Strickland conjecture: dualizable K(n)-local spectra have exactly n+2 thick tensor ideals, and their Balmer spectrum is a chain of n+1 points.","papers":["https://github.com/openai/math/blob/main/preprints/Stabilizer-Orbits-and-Thick-Tensor-Ideals-of-Dualizable-Kn-Local-Spectra-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-07-chenglong-ma-hovey-strickland-openai-chatgpt-upload-provenance/"],"name_hits":[],"tier":"B","note":"Chai and Hovey–Strickland conjectures. Chenglong Ma (arXiv 2610.10072): OpenAI's strategy and notation 'identical' to his draft uploaded to ChatGPT on 8 Sept; Ningyi Li proved it independently. Own entry.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Stabilizer-Orbits-and-Thick-Tensor-Ideals-of-Dualizable-Kn-Local-Spectra-September-24-2026/paper.pdf","https://arxiv.org/abs/2610.10072","https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Stabilizer-Orbits-and-Thick-Tensor-Ideals-of-Dualizable-Kn-Local-Spectra-September-24-2026/AGENT_NOTES.md","https://github.com/mahrud/openai-math-comments/pull/311","https://news.ycombinator.com/item?id=50010118"]},{"family":"312","title":"The Grothendieck homotopy hypothesis","summary":"Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces. ([Lean](lean/docs/312.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-07-01-grothendieck-group-scheme-counterexample","2026-07-25-sharp-thin-shell-inequality-gpt-5-6-pro","2026-09-10-con-leche-consistent-lean-checker-claude","2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe","2026-10-20-ai-assisted-results-october-20-listing","2026-10-27-ai-assisted-results-october-27-listing"],"tier":"B","note":"Grothendieck's homotopy hypothesis for Grothendieck ∞-groupoids (Ara–Henry convention). Lean covers the elementary-expansion step only; Yao Xu's project claims more.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/312.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/312.md","https://github.com/cadamcat/homotopy-hypothesis-lean4"]},{"family":"313","title":"Finite generation for the $`K(n)`$-local sphere","summary":"Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the K(n)-local sphere is a finitely generated ℤ<sub>p</sub>-module, for every prime, positive height and integer degree. The same conclusion holds after K(n)-localizing any finite p-local spectrum.","papers":["https://github.com/openai/math/blob/main/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf"],"lean":false,"entries":[],"name_hits":["2025-09-17-deepmind-unstable-singularities-fluids","2026-05-27-sum-product-conjecture-false-over-reals","2026-09-07-anandkumar-euler-singularity-r3","2026-10-02-meta-muse-spark-six-math-papers","2026-10-06-finite-index-over-treeable-jkl-problem","2026-10-07-colkitt-codex-integer-multiplication-exponent"],"tier":"B","note":"Every homotopy group of the K(n)-local sphere is finitely generated over ℤ_p (Hovey–Strickland question). No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf"]},{"family":"314","title":"Cyclic length and chromatic fixed-point loss","summary":"Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.","papers":["https://github.com/openai/math/blob/main/preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-07-cyclicity-conjecture-positive-operators-dobrick","2026-10-13-ai-assisted-results-october-2026"],"tier":"C","note":"Optimal chromatic fixed-point loss equals cyclic subnormal chain length for p-groups. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026.pdf"]},{"family":"315","title":"The four-dimensional Singer conjecture","summary":"Proves that the L<sup>2</sup>-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.","papers":["https://github.com/openai/math/blob/main/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-08-08-kimi-k3-cheats-aisi-inspect-cyber-eval","2026-09-29-nyt-anthropic-faith-leaders-claude-consciousness"],"tier":"B","note":"Four-dimensional Singer conjecture for aspherical 4-manifolds and PD4 complexes. No Lean; no reactions found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf"]},{"family":"316","title":"Curtis’s conjecture","summary":"Proves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes η, ν, σ and the Kervaire-invariant-one classes that exist.","papers":["https://github.com/openai/math/blob/main/preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Curtis's conjecture on the mod-2 stable Hurewicz image of the sphere. No Lean; no reactions found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026/paper.pdf"]},{"family":"317","title":"Thomason model structures in all strict higher dimensions","summary":"Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension. ([Lean](lean/docs/317.md))","papers":["https://github.com/openai/math/blob/main/preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Ara–Maltsiniotis conjecture: Thomason model structures on strict n-categories for all n. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/317.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/317.md"]},{"family":"318","title":"Chromatic splitting: filtrations and counterexamples","summary":"Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and p+1 (p ≥ 7). Nevertheless, for n ≥ 1 and p\\gt n+1, the overlap L_{n-1}L_{K(n)}S_p^\\wedge admits a 2^n-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of E(2)-local modules over the derived completed sphere.","papers":["https://github.com/openai/math/blob/main/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-02-tomon-triangle-free-box-graphs-astra"],"tier":"B","note":"Chromatic splitting conjecture counterexamples (strong at height 3; weak at heights p, p+1). Beaudry–Schlank, with Claude, show weak splitting fails at n=p.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf","https://hexagonmath.org/2610.00138v1"]},{"family":"319","title":"Counterexamples to finite generation at chromatic height two","summary":"Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed \\mathrm{BP}\\langle2\\rangle by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.","papers":["https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-07-01-grothendieck-group-scheme-counterexample","2026-07-24-rota-flats-unimodality-disproved","2026-09-09-pierce-birkhoff-conjecture-disproved","2026-09-21-grad-conjecture-counterexamples","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-09-arxiv-ai-disclosed-results-early-october-2026"],"tier":"C","note":"Hahn–Wilson finite-generation conjecture refuted at chromatic height two for large primes. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf"]},{"family":"320","title":"Nonhomeomorphic closed aspherical four-manifolds","summary":"Constructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.","papers":["https://github.com/openai/math/blob/main/preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-borel-conjecture-dimension-four-free-group-not-good/"],"name_hits":[],"tier":"A","note":"Homotopy-equivalent, non-homeomorphic closed aspherical 4-manifolds with hyperbolic fundamental group: topological Borel conjecture fails in dimension 4. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026/paper.pdf","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"321","title":"A counterexample to Wall's finite D(2) problem","summary":"Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.","papers":["https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Counterexample to Wall's D(2) problem. No OpenAI Lean; Pyronis WallD2 offers a conditional Lean formalization.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf","https://github.com/Pyronis-Inc/WallD2"]},{"family":"322","title":"Tingley’s sphere-isometry problem","summary":"Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry. ([Lean](lean/docs/322.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Tingley's problem solved for all real Banach spaces. Lean accepted. Kudaybergenov et al. independently treat a finite-dimensional class (arXiv 2610.19626).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/322.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/322.md","https://arxiv.org/abs/2610.19626"]},{"family":"323","title":"Independence of the separable quotient problem","summary":"Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields. ([Lean](lean/docs/323.md))","papers":["https://github.com/openai/math/blob/main/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-27-erdos-993-forest-independence-polynomials-unimodal"],"tier":"B","note":"Separable quotient problem is independent of ZFC (relative to a measurable cardinal). Lean covers the CH counterexample. Mitchell Taylor: 'truly spectacular'.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/323.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/323.md","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"324","title":"Lipschitz equivalent Banach spaces need not be linearly isomorphic","summary":"Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class. ([Lean](lean/docs/324.md))","papers":["https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026","2026-10-16-ai-assisted-results-october-15-listing","2026-10-17-ai-assisted-results-october-16-listing"],"tier":"B","note":"Lipschitz-equivalent separable Banach spaces need not be linearly isomorphic. Lean accepted. No reactions found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/324.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/324.md"]},{"family":"325","title":"The complete Crouzeix conjecture","summary":"Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has \\lVert P[A]\\rVert\\le2\\sup_{z\\in W(A)}\\lVert P(z)\\rVert, where W(A) is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions. ([Lean](lean/docs/325.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-07-27-crouzeix-conjecture-proved","2026-10-06-openai-math-release-722-manuscripts"],"tier":"B","note":"Complete (matrix-valued) Crouzeix conjecture with constant 2 on any Hilbert space. Lean accepted. The scalar case was proved in July (our 2026-07-27 entry).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/325.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/325.md"]},{"family":"326","title":"The cotype–cotype conjecture under the approximation property","summary":"Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type. ([Lean](lean/docs/326.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Cotype–cotype conjecture for real Banach spaces with the approximation property. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/326.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/326.md"]},{"family":"327","title":"Markov type characterizes superreflexivity","summary":"Proves that every real Banach space with Markov type p for some p &gt; 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type. ([Lean](lean/docs/327.md))","papers":["https://github.com/openai/math/blob/main/preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-05-03-erdos-1196-primitive-sets","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus","2026-10-17-ai-assisted-results-october-16-listing","2026-10-26-ai-assisted-results-october-26-listing"],"tier":"B","note":"Naor's renorming question: nontrivial Markov type characterizes superreflexivity. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/327.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/327.md"]},{"family":"328","title":"Nonexpansive fixed points in reflexive Banach spaces","summary":"Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity. ([Lean](lean/docs/328.md))","papers":["https://github.com/openai/math/blob/main/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Kirk's problem: every reflexive Banach space has the fixed-point property for nonexpansive maps. Lean accepted. No reactions found.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/328.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/328.md"]},{"family":"329","title":"A counterexample to metric-entropy duality","summary":"Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube. ([Lean](lean/docs/329.md))","papers":["https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Pietsch's dimension-free duality conjecture for metric entropy disproved. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/329.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/329.md"]},{"family":"330","title":"A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces","summary":"Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound. ([Lean](lean/docs/330.md))","papers":["https://github.com/openai/math/blob/main/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-16-ai-assisted-results-october-15-listing"],"tier":"C","note":"Kalton's question: uniformly discrete space whose Lipschitz-free space has AP but not BAP. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/330.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/330.md"]},{"family":"331","title":"Reflexive midpoint convexity and diamond distortion","summary":"Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least \\sqrt{1+k/12}, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity. ([Lean](lean/docs/331.md))","papers":["https://github.com/openai/math/blob/main/preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Reflexive space separating midpoint from asymptotic uniform convexity; diamond distortion bounds. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/331.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/lean/docs/331.md"]},{"family":"332","title":"Metric Markov cotype of ℓ<sub>1</sub> and Hilbert-space Lipschitz extension","summary":"Proves that real ℓ<sub>1</sub> has metric Markov cotype two, answering Mendel and Naor's question. Consequently, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ<sub>1</sub> extends to the whole space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.","papers":["https://github.com/openai/math/blob/main/preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Real ℓ₁ has metric Markov cotype 2 (Mendel–Naor question), giving Ball-type extension from Hilbert space into ℓ₁. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf"]},{"family":"333","title":"Smooth isometric immersions of surfaces into ℝ<sup>4</sup>","summary":"Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ<sup>4</sup>, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature. ([Lean](lean/docs/333.md))","papers":["https://github.com/openai/math/blob/main/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-08-19-caratheodory-loewner-smooth-counterexamples-alpoge-claude","2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-14-ai-assisted-results-mid-october-2026","2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey","2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim"],"tier":"B","note":"Every closed smooth surface isometrically immerses in ℝ⁴. Lean accepted. Kitamura claims a strengthening to embeddings on GitHub.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/333.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/333.md","https://github.com/KitaKen1/smooth-isometric-embeddings-r4"]},{"family":"334","title":"A smooth surface metric with no local isometric immersion in ℝ<sup>3</sup>","summary":"Constructs a smooth positive-definite metric on (-1,1)^2 for which no neighborhood of the origin admits a smooth isometric immersion into ℝ<sup>3</sup>. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood. ([Lean](lean/docs/334.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Smooth surface metric with no local isometric immersion in ℝ³, answering the classical local problem negatively. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/334.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/334.md"]},{"family":"335","title":"Gromov’s integral scalar-curvature bound for simplicial volume","summary":"Proves \\int_M(\\mathrm{Scal}_g^-)^{n/2}\\,dV_g\\ge a_n\\lVert M\\rVert for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with a_n\\gt 0 depending only on dimension. Here \\mathrm{Scal}_g^-=\\max\\{0,-\\mathrm{Scal}_g\\} and \\lVert M\\rVert is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.","papers":["https://github.com/openai/math/blob/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf","https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc/"],"name_hits":["2026-08-01-openai-astra-ten-advances","2026-08-13-banach-isometric-conjecture-solved"],"tier":"A","note":"Gromov–Lawson aspherical conjecture in all dimensions (positive scalar curvature forces rational inessentiality) and Gromov's integral simplicial-volume bound. No Lean. Sauer: 'I am stunned.'","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf","https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"336","title":"Spectral scalar curvature, Urysohn width, and macroscopic dimension","summary":"Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying -4\\Delta+\\mathrm{Scal}\\ge1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 whose entire fibers have diameter bounded only by n in the original metric. This strengthens Gromov's width conclusion to spectral scalar curvature. Universal covers of closed positive-scalar-curvature manifolds also have continuous macroscopic dimension at most n-2 for every n ≥ 2.","papers":["https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf","https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Gromov-type width and macroscopic-dimension bounds under spectral scalar curvature. No Lean. Yuchen Bi adapts its ideas (arXiv 2610.28390).","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf","https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf","https://arxiv.org/abs/2610.28390"]},{"family":"337","title":"Sharp Cartan–Hadamard isoperimetry and rigidity","summary":"Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT(0) spaces. ([Lean](lean/docs/337.md))","papers":["https://github.com/openai/math/blob/main/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-integral-fillings-in-CAT(0"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Cartan–Hadamard isoperimetric conjecture in every dimension with rigidity; sharp CAT(0) fillings. Lean covers only the CAT(0) integral-current form.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/337.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Sharp-integral-fillings-in-CAT%280%29-spaces-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/337.md"]},{"family":"338","title":"Yau's uniformization conjecture","summary":"Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂ<sup>n</sup>.","papers":["https://github.com/openai/math/blob/main/preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-11-yau-math-ai-lab-alliance","2026-10-13-ai-assisted-results-october-2026","2026-10-20-ai-assisted-results-october-20-listing"],"tier":"A","note":"Yau's uniformization conjecture: complete noncompact Kähler manifolds with positive bisectional curvature are biholomorphic to ℂⁿ. No Lean; no expert reaction found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf"]},{"family":"339","title":"Katok's entropy rigidity conjecture","summary":"Proves Katok's entropy rigidity conjecture for closed connected Riemannian manifolds of dimension at least three with strictly negative sectional curvature: normalized Liouville measure maximizes entropy for the geodesic flow if and only if the metric is locally symmetric.","papers":["https://github.com/openai/math/blob/main/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Katok's entropy rigidity conjecture in dimension ≥3. Tristan Humbert (Proofs and Prompts) called the paper 'mostly unreadable slop'. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/"]},{"family":"340","title":"A counterexample to the nearby Lagrangian conjecture","summary":"Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in T^*(S^9\\times S^{N-1}) that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.","papers":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-nearby-lagrangian-conjecture-counterexample/"],"name_hits":["2026-10-06-hodge-conjecture-cm-abelian-varieties-openai","2026-10-08-holomorphic-cancellation-problem-counterexample","2026-10-08-hyperkahler-syz-conjecture-engel-mauri","2026-10-13-k-median-better-than-2-approximation-epfl"],"tier":"A","note":"Counterexample to Arnold's nearby Lagrangian conjecture in T*(S⁹×S^{N−1}). No Lean. Wenyuan Li's seminar talk presents it 'assuming the counterexample is correct'.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf","https://zuma-talks.github.io/gt/index.html"]},{"family":"341","title":"Donaldson's hypersymplectic deformation conjecture","summary":"Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by \\int\\omega_i\\wedge\\omega_j=\\delta_{ij}, deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.","papers":["https://github.com/openai/math/blob/main/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Donaldson's hypersymplectic deformation conjecture, cohomology-preserving form. Revised 7 Oct (criterion clarified, cone citation removed). No Lean.","lean_scope":"","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf","https://raw.githubusercontent.com/openai/math/main/history.md"]},{"family":"342","title":"Donaldson's tamed-to-compatible conjecture","summary":"Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change. ([Lean](lean/docs/342.md))","papers":["https://github.com/openai/math/blob/main/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Donaldson's tamed-to-compatible conjecture on closed 4-manifolds. Revised 7 Oct (cone-equality claim corrected). Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/342.md","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/342.md","https://raw.githubusercontent.com/openai/math/main/history.md","https://gautamparab.com/writing/one-minus-sign-three-papers/"]},{"family":"343","title":"Symplectic ball packing in higher dimensions","summary":"Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension 2n\\ge6. Finitely many closed symplectic balls of capacities R_1,\\ldots,R_k embed disjointly into an open ball of capacity R exactly when \\sum_iR_i^n\\lt R^n and R_i+R_j\\lt R for every distinct pair i, j. ([Lean](lean/docs/343.md))","papers":["https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-06-hodge-conjecture-cm-abelian-varieties-openai"],"tier":"C","note":"Siegel–Yao symplectic ball-packing criterion in dimension ≥6. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/343.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/343.md"]},{"family":"344","title":"The metric Blaschke conjecture","summary":"Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.","papers":["https://github.com/openai/math/blob/main/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra"],"tier":"B","note":"Metric Blaschke conjecture: injectivity radius equal to diameter forces a compact rank-one symmetric space. No Lean; no reactions found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf"]},{"family":"345","title":"Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds","summary":"Proves that every smooth Riemannian metric on S<sup>n</sup>, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientability or nondegeneracy restrictions.","papers":["https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-closed-geodesics-spheres-infinitely-many/"],"name_hits":[],"tier":"A","note":"Infinitely many closed geodesics for every metric on every sphere and every closed 3-manifold. Chen–Deng–Zhang give an independent, Codex-assisted proof, extended to projective spaces.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf","https://arxiv.org/abs/2610.21054"]},{"family":"346","title":"Sharp singular-set bounds for stationary integral varifolds","summary":"Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most m-1, a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero m-dimensional Hausdorff measure.","papers":["https://github.com/openai/math/blob/main/preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf","https://github.com/openai/math/blob/main/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Stationary integral varifolds: singular set of dimension ≤ m−1 and almost-everywhere regularity (Brena–Decio–De Lellis conjectures). No Lean; no reactions found.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf","https://github.com/openai/math/blob/main/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf"]},{"family":"347","title":"Counterexamples to stable-Morse and strong Arnold fixed-point bounds","summary":"Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function. ([Lean](lean/docs/347.md))","papers":["https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf","https://github.com/openai/math/blob/main/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf","https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Counterexamples to stable-Morse lower bounds and the strong Arnold conjecture (three fixed points on the quadric threefold). Lean covers the strong Arnold case.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/347.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf","https://github.com/openai/math/blob/main/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf","https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/347.md"]},{"family":"348","title":"Nonnegative-curvature Einstein classification and an L<sup>2</sup> topological gap","summary":"Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S<sup>4</sup>, Fubini–Study \\mathbb{CP}^2, or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L<sup>2</sup> norm of its trace-free Ricci curvature lies below a universal positive constant. ([Lean](lean/docs/348.md))","papers":["https://github.com/openai/math/blob/main/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf","https://github.com/openai/math/blob/main/preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf","https://github.com/openai/math/blob/main/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Einstein 4-manifolds with nonnegative sectional curvature classified, with an L² gap theorem. Lean covers only strictly positive curvature.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/348.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf","https://github.com/openai/math/blob/main/preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf","https://github.com/openai/math/blob/main/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/348.md"]},{"family":"349","title":"The Solomon–Yau least-volume conjecture","summary":"Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every m ≥ 2, a closed connected minimal immersion into the unit sphere S^{m+1} with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.","papers":["https://github.com/openai/math/blob/main/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-06-openai-math-release-722-manuscripts","2026-10-14-ai-assisted-results-mid-october-2026"],"tier":"B","note":"Solomon–Yau least-volume conjecture for minimal hypersurfaces in spheres. No Lean. Jingwen Chen uses it (arXiv 2610.20453).","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf","https://arxiv.org/abs/2610.20453"]},{"family":"350","title":"Yau’s nodal bounds: surfaces and higher dimensions","summary":"Proves the sharp C\\sqrt\\lambda upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S<sup>3</sup> arbitrarily close to round. In dimension five, nodal measure can grow faster than \\lambda^{1/2+\\varepsilon_0} for some fixed \\varepsilon_0\\gt 0, ruling out even arbitrarily small power losses. ([Lean](lean/docs/350.md))","papers":["https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-yau-nodal-conjecture-surfaces-counterexamples/"],"name_hits":["2026-09-30-ai-assisted-conjecture-wave-summer-2026","2026-10-11-yau-math-ai-lab-alliance","2026-10-13-ai-assisted-results-october-2026","2026-10-20-ai-assisted-results-october-20-listing"],"tier":"A","note":"Yau's nodal upper bound proved on smooth surfaces, disproved for smooth metrics in dimensions 3–5. Lean for all three Comparator-accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/350.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/350.md","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv"]},{"family":"351","title":"Scalar curvature and finite-time Ricci-flow singularities","summary":"Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.","papers":["https://github.com/openai/math/blob/main/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Four-dimensional Ricci flow extends while scalar curvature stays bounded; counterexample in higher dimension. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf"]},{"family":"352","title":"A finite-time singularity of Calabi flow","summary":"Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth U(10)-invariant Kähler metric on \\mathbb{CP}^{10} whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.","papers":["https://github.com/openai/math/blob/main/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-10-09-arxiv-ai-disclosed-results-early-october-2026","2026-10-13-ai-assisted-results-october-2026","2026-10-26-ai-assisted-results-october-26-listing","2026-10-29-ai-assisted-results-october-29-listing"],"tier":"B","note":"Chen's long-time existence conjecture for Calabi flow disproved: finite-time singularity on CP¹⁰ in the Fubini–Study class. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf"]},{"family":"353","title":"Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample","summary":"Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric. ([Lean](lean/docs/353.md))","papers":["https://github.com/openai/math/blob/main/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf","https://github.com/openai/math/blob/main/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Affine Bernstein problem: rigidity in dimensions 3–9, smooth counterexample in 10. Trudinger–Wang independently give n≥10 examples. Lean covers graph case.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/353.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf","https://github.com/openai/math/blob/main/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/353.md","https://arxiv.org/abs/2610.09385"]},{"family":"354","title":"The isoperimetric profile of the cubic three-torus","summary":"Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are 4\\pi/81 and 1/\\pi, with exactly the adjacent two types minimizing at each transition. ([Lean](lean/docs/354.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Isoperimetric profile and all minimizers of the cubic flat 3-torus. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/354.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/354.md"]},{"family":"355","title":"Unique tangent flows at the first surface singularity","summary":"Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ<sup>3</sup>. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.","papers":["https://github.com/openai/math/blob/main/preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-21-openai-100-open-problems-claim","2026-10-06-openai-math-release-722-manuscripts","2026-10-06-openai-release-3d-restriction-bochner-riesz-claims","2026-10-06-openai-spin-one-haldane-gap-proof","2026-10-06-unique-games-conjecture-openai","2026-10-23-propeller-conjecture-openai-proof-heilman-simplification"],"tier":"B","note":"Uniqueness of tangent flows at the first singular time for embedded surfaces under mean curvature flow. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf"]},{"family":"356","title":"Gigli’s characterization of Alexandrov curvature","summary":"Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support \\mathop{\\mathrm{RCD}}\\nolimits ((n-1)\\kappa,n) condition with reference measure \\mathcal H^n and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced. ([Lean](lean/docs/356.md))","papers":["https://github.com/openai/math/blob/main/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Gigli's characterization of Alexandrov curvature through RCD conditions. Lean covers a supporting Hessian lemma.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/356.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf","https://github.com/openai/math/blob/main/lean/docs/356.md"]},{"family":"357","title":"Bi-Lipschitz coordinates at every regular RCD point","summary":"Proves that every regular point of a noncollapsed \\mathop{\\mathrm{RCD}}\\nolimits (K,n) space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝ<sup>n</sup>. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly \\mathcal H^n, and the chart compares ambient distances with a point-dependent finite constant.","papers":["https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"C","note":"Bi-Lipschitz charts at every regular point of noncollapsed RCD spaces. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf"]},{"family":"358","title":"A three-manifold without conjugate points or nonpositive curvature","summary":"Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second. ([Lean](lean/docs/358.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Closed 3-manifold with a no-conjugate-points metric but no nonpositively curved metric. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/358.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/358.md"]},{"family":"359","title":"Negative Kähler curvature without bounded holomorphic coordinates","summary":"Constructs a contractible domain in ℂ<sup>3</sup> with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below. ([Lean](lean/docs/359.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Contractible domain in ℂ³ with negatively pinched Kähler metric but no bounded holomorphic coordinates. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/359.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/359.md"]},{"family":"360","title":"Weak MTW curvature gives convexity and regular optimal transport","summary":"On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous. ([Lean](lean/docs/360.md))","papers":["https://github.com/openai/math/blob/main/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Villani's conjecture: weak MTW curvature gives convex injectivity domains and Hölder optimal transport. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/360.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/360.md"]},{"family":"361","title":"Failure of integer-degree harmonic dimension comparison","summary":"Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ<sup>3</sup> has at least (k+2)^2 independent harmonic functions of growth at most k, exceeding the Euclidean count (k+1)^2. The metric may depend on k. ([Lean](lean/docs/361.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-13-ai-assisted-results-october-2026","2026-10-14-first-proof-third-batch"],"tier":"B","note":"Yau's harmonic-function dimension comparison fails for integer growth. Lean covers a 16-dimensional instance, not the paper's ℝ³ claim.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/361.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/361.md"]},{"family":"362","title":"Global smoothness for relativistic Vlasov–Maxwell","summary":"Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval. ([Lean](lean/docs/362.md))","papers":["https://github.com/openai/math/blob/main/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-relativistic-vlasov-maxwell-global-solutions-lean/"],"name_hits":[],"tier":"A","note":"Large-data global smooth solutions of 3D relativistic Vlasov–Maxwell; Lean full statement accepted in 2 runs. Reasoning summary; Xie: small exponent margin. No expert reaction. Own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/362.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/362.md","https://github.com/MColbrook/AIM/pull/35","https://github.com/openai/math/blob/main/reasoning_traces/relativistic-vlasov-maxwell.pdf","https://arxiv.org/abs/2610.17090","https://github.com/MColbrook/AIM/blob/main/problems/073-vlasov-maxwell-global-classical.md","https://math.vet/openai-math/"]},{"family":"363","title":"Nonuniqueness with local conservation for the hard-sphere Boltzmann equation","summary":"Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L<sup>1</sup> and satisfy exact local conservation of mass, momentum and kinetic energy. ([Lean](lean/docs/363.md))","papers":["https://github.com/openai/math/blob/main/preprints/Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Two distinct global entropy solutions of the hard-sphere Boltzmann equation from the same data. Lean accepted.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/363.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/363.md"]},{"family":"364","title":"Kinetic limits and fluctuations over the Boltzmann lifespan","summary":"Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and gradient bounds are assumed. A companion gives finite-dimensional hard-sphere Gaussian fluctuations, centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation.","papers":["https://github.com/openai/math/blob/main/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Boltzmann equation derived from Newtonian gases over the full kinetic lifespan, with Gaussian fluctuations. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf"]},{"family":"365","title":"Joint metric and connection recovery from one boundary patch","summary":"Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data. ([Lean](lean/docs/365.md))","papers":["https://github.com/openai/math/blob/main/preprints/Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-24-openai-agent-medicare-breach-australia","2026-09-28-mathathon-joint-statement","2026-09-29-white-house-ai-summit","2026-10-01-anthropic-australia-copyright-opt-out-submission","2026-10-01-openai-agent-nsw-npws-fire-data-breach","2026-10-06-australia-ai-committee-kwon-anthropic-testimony"],"tier":"B","note":"One-patch recovery of metric and connection; L∞ conductivities in 3D with identical Calderón data (Lean). Mikko Salo flags direct interest.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/365.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/365.md","https://fameflagship.fi/blog/mathematics-and-ai/"]},{"family":"366","title":"The planar Mumford–Shah regularity conjecture and local weak-L<sup>4</sup> gradient bounds","summary":"Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a C^{1,\\alpha} arc, a regular crack tip, or three arcs meeting at 120^\\circ; only finitely many global connected components meet any compact interior region.","papers":["https://github.com/openai/math/blob/main/preprints/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026.pdf"],"lean":false,"entries":[],"name_hits":["2026-09-21-openai-100-open-problems-claim","2026-09-22-mumford-shah-conjecture-astra-claim","2026-10-06-openai-math-release-722-manuscripts"],"tier":"B","note":"Planar Mumford–Shah regularity conjecture. No Lean. A separate Astra-generated claim (Deangelis, 22 Sept) has our entry; OpenAI's version has none.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026.pdf"]},{"family":"367","title":"The critical dimension for the one-phase Bernoulli problem","summary":"Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most n-7 in higher dimensions. ([Lean](lean/docs/367.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf"],"lean":true,"entries":[],"name_hits":["2026-09-03-dying-percolation-theta-pc-zero","2026-09-22-odlyzko-poonen-conjecture-proved","2026-10-07-exact-overlaps-conjecture-gpt-6-astra","2026-10-21-ai-assisted-results-october-21-listing"],"tier":"B","note":"Critical dimension seven for one-phase Bernoulli minimizers. OpenAI Lean covers the dimension-7 example; William Feldman's Lean proves flatness up to dimension six.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/367.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf","https://github.com/openai/math/blob/main/lean/docs/367.md","https://github.com/willmfeldman/one-phase-critical-dim"]},{"family":"368","title":"The three-dimensional Ball–Evans approximation problem","summary":"Resolves the three-dimensional Ball–Evans approximation problem: every W^{1,p} homeomorphism between arbitrary bounded domains in ℝ<sup>3</sup>, for 1\\le p\\lt \\infty, is a strong W^{1,p} limit of smooth diffeomorphisms onto the same target.","papers":["https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026/main.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Ball–Evans approximation problem in 3D: Sobolev homeomorphisms are limits of diffeomorphisms. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026/main.pdf"]},{"family":"369","title":"The hot spots conjecture for simply connected planar domains","summary":"Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed. ([Lean](lean/docs/369.md))","papers":["https://github.com/openai/math/blob/main/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-06-openai-hot-spots-conjecture-simply-connected-lean/"],"name_hits":[],"tier":"A","note":"Strict hot spots for smooth simply connected planar domains; Lean accepted. Burdzy and Bañuelos (ChatGPT) extend it to rough domains, unchecked; Atar–Burdzy 3D counterexample. Own entry.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/369.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf","https://github.com/openai/math/blob/main/lean/docs/369.md","https://arxiv.org/abs/2610.12622","https://arxiv.org/abs/2610.19908","https://arxiv.org/abs/2610.32920","https://arxiv.org/abs/2610.18546","https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","https://github.com/MColbrook/AIM/blob/main/problems/003-planar-convex-hot-spots.md"]},{"family":"370","title":"The Lane–Emden and Hénon–Lane–Emden conjectures","summary":"Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, p,q\\gt 0 and real A, B, the system -\\Delta u=|x|^A v^p, -\\Delta v=|x|^B u^q has no positive entire solution when (n+A)/(p+1)+(n+B)/(q+1)\\gt n-2, with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and A,B\\gt -2. ([Lean](lean/docs/370.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/"],"name_hits":[],"tier":"B","note":"Subcritical Lane–Emden and Hénon–Lane–Emden conjectures. Lean accepted. Mei–Wei and Li–Lu–Zhang extend it to fractional cases (listing entry).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/370.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/370.md","https://arxiv.org/abs/2610.10188","https://arxiv.org/abs/2610.22802"]},{"family":"371","title":"Stable blowup for the defocusing Schrödinger equation","summary":"For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in H^k(\\mathbb T^{12}), with k &gt; 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime. ([Lean](lean/docs/371.md))","papers":["https://github.com/openai/math/blob/main/preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Stable finite-time blowup for defocusing NLS on 𝕋¹². Lean accepted with definition-body caveat; covers arbitrarily large odd powers, not all.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/371.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/371.md","https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv"]},{"family":"372","title":"Global uniqueness in smooth isotropic elasticity","summary":"Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ<sup>3</sup>, provided μ &gt; 0 and 3\\lambda+2\\mu\\gt 0 on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. ([Lean](lean/docs/372.md))","papers":["https://github.com/openai/math/blob/main/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf"],"lean":true,"entries":[],"name_hits":["2025-09-30-periodic-labs-300m-seed","2026-02-03-international-ai-safety-report-2026","2026-09-09-openai-ai-policy-window","2026-09-21-openai-building-standards-rsi","2026-09-29-bain-6-trillion-ai-revenue-gap","2026-10-05-cohere-north-2"],"tier":"B","note":"Global uniqueness of smooth Lamé parameters from boundary data in 3D. Lean accepted. Diao–Durdiev claim the same theorem (arXiv 2610.14741).","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/372.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/372.md","https://arxiv.org/abs/2610.14741"]},{"family":"373","title":"Nonattainment of the three-marginal Coulomb Monge problem","summary":"An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case. ([Lean](lean/docs/373.md))","papers":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"C","note":"Monge non-attainment for three-marginal Coulomb optimal transport. Lean accepted for the 3D case.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/373.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/373.md"]},{"family":"374","title":"Sharp one-third stability of Brenier maps","summary":"For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy \\|T_\\mu-T_\\nu\\|_{L^2(\\rho)}\\le C W_2(\\mu,\\nu)^{1/3} uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound. ([Lean](lean/docs/374.md))","papers":["https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf"],"lean":true,"entries":[],"name_hits":["2026-10-29-ai-assisted-results-october-29-listing"],"tier":"C","note":"Sharp one-third stability of Brenier maps, refuting Letrouit's square-root conjecture. Lean accepted with definition caveat; concurrent human papers.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/374.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf","https://github.com/openai/math/blob/main/lean/docs/374.md","https://arxiv.org/abs/2610.09688"]},{"family":"375","title":"De Giorgi's conjecture in dimension eight","summary":"Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C<sup>2</sup> solution u:\\mathbb R^8\\to(-1,1) of \\Delta u=u^3-u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:\\mathbb R^7\\to[-1,1] as constant wells or planar transitions, without an energy-growth assumption.","papers":["https://github.com/openai/math/blob/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf"],"lean":false,"entries":["https://postcutoff.com/e/2026-10-06-openai-de-giorgi-conjecture-dimension-eight/","https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/"],"name_hits":[],"tier":"A","note":"De Giorgi's conjecture in dimension eight and stable Allen–Cahn solutions in ℝ⁷. No Lean; Long–Zou build on it. Own entry.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf","https://arxiv.org/abs/2610.28080"]},{"family":"376","title":"Universal computation in forced Navier–Stokes flows","summary":"Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable. ([Lean](lean/docs/376.md))","papers":["https://github.com/openai/math/blob/main/preprints/Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Geometric-Programs-for-Solenoidal-Forcing-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Scalar-Potentials-and-Slow-Clocks-for-Forced-Fluid-Computation-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf"],"lean":true,"entries":[],"name_hits":[],"tier":"B","note":"Forced Navier–Stokes flows perform universal computation. Revised 7 Oct (decay estimates). Lean partial. Anders Sandberg called it his favourite family.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/376.md","status":"updated","sources":["https://github.com/openai/math/blob/main/preprints/Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf","https://github.com/openai/math/blob/main/lean/docs/376.md","https://raw.githubusercontent.com/openai/math/main/history.md","https://x.com/anderssandberg/status/2111120089437151292"]},{"family":"377","title":"Interior $`C^{1,\\alpha}`$ regularity for infinity-harmonic functions","summary":"Proves uniform interior C^{1,\\alpha_d} regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.","papers":["https://github.com/openai/math/blob/main/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf"],"lean":false,"entries":[],"name_hits":[],"tier":"B","note":"Interior C^{1,α} regularity for infinity-harmonic functions in every dimension d≥3. No Lean.","lean_scope":"","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf"]},{"family":"378","title":"Disproof of the Exponential Time Hypothesis","summary":"For every fixed k\\ge2, a deterministic algorithm counts the satisfying assignments of a k-CNF with n variables and m clauses in time (n+m)^{O_k(1)}2^{n/(\\log n)^{1/2+o(1)}}. This yields superlinear circuit lower bounds for a language in \\mathsf{E}^{\\mathsf{NP}} on infinitely many input lengths. ([Lean](lean/docs/378.md))","papers":["https://github.com/openai/math/blob/main/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf"],"lean":true,"entries":["https://postcutoff.com/e/2026-10-26-openai-eth-false-subexponential-sat-claim/","https://postcutoff.com/e/2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential/"],"name_hits":[],"tier":"A","note":"ETH refuted: subexponential exact #k-SAT, added 26 Oct. Lean for a weaker bound; Impagliazzo–Paturi–Williams working group. Own entries.","lean_scope":"https://github.com/openai/math/blob/main/lean/docs/378.md","status":"pending","sources":["https://github.com/openai/math/blob/main/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf","https://github.com/openai/math/blob/main/lean/docs/378.md","https://www.cs.cmu.edu/~odonnell/some_eth_false_exposition.pdf"]}]}