{ "about": "Every result family in OpenAI's math release (github.com/openai/math), with how far each claim has been checked. Built by tools/site/build.py.", "source": "https://github.com/openai/math", "license": "Titles, summaries and paper titles come from OpenAI's catalog and are used under the Apache License 2.0. Everything else is AI Proved What? data under CC BY 4.0.", "fields": { "papers": "The papers OpenAI groups into this result, in OpenAI's order. role is 'main' for the first-listed paper and 'companion' for the others.", "lean.status": "main: OpenAI published a Lean proof of the main theorem; partial: OpenAI published a Lean proof of part of the claim; none: no Lean proof published. Classified from OpenAI's scope notes.", "peer_review": "Whether the claim has been peer reviewed." }, "last_checked": "2026-10-19", "count": 372, "paper_count": 719, "results": [ { "num": "001", "slug": "001-milne-rationality", "title": "Milne’s rationality conjecture and algebraic specialization", "title_html": "Milne’s rationality conjecture and algebraic specialization", "summary_html": "Proves Milne's rationality conjecture for abelian varieties over Q‾\\overline{\\mathbb Q}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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Milne's rationality conjecture for abelian varieties", "title_html": "Milne's rationality conjecture for abelian varieties", "url": "https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "fp2pUIBd6Ig", "length": "6:09", "title": "Many rulers, one fraction", "page": "/episodes/001-milne-rationality/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/001-milne-rationality/" }, { "num": "002", "slug": "002-bsd-formula", "title": "The full BSD formula from low Selmer corank", "title_html": "The full BSD formula from low Selmer corank", "summary_html": "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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank", "title_html": "Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank", "url": "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", "role": "main" }, { "title": "The Selmer converse for elliptic curves at every prime", "title_html": "The Selmer converse for elliptic curves at every prime", "url": "https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf", "role": "companion" }, { "title": "The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one", "title_html": "The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "y-MSZWFdjg0", "length": "8:10", "title": "Hidden in the counts", "page": "/episodes/002-bsd-formula/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/002-bsd-formula/" }, { "num": "003", "slug": "003-quasi-riemann-hypothesis", "title": "The quasi-Riemann hypothesis", "title_html": "The quasi-Riemann hypothesis", "summary_html": "Proves that every Dirichlet L-function, including ζ(s)\\zeta(s)ζ(s), is zero-free in ℜs>7/8\\Re s\\gt 7/8ℜs>7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over Q(−3)\\mathbb Q(\\sqrt{-3})Q(−3​). A companion gives a different proof of the zero-free half-plane ℜs>11/12\\Re s\\gt 11/12ℜs>11/12.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜs > 7/8", "title_html": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜs>7/8\\Re s\\gt 7/8ℜs>7/8", "url": "https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf", "role": "main" }, { "title": "The Quasi-Riemann Hypothesis (alternate 11/12 proof)", "title_html": "The Quasi-Riemann Hypothesis (alternate 11/12 proof)", "url": "https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf", "role": "companion" }, { "title": "Uniform exclusion of Landau–Siegel zeros", "title_html": "Uniform exclusion of Landau–Siegel zeros", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/003.md" }, "peer_review": "not yet", "episode": { "video_id": "JWUgcXVFf_c", "length": "8:45", "title": "A wall at 7/8", "page": "/episodes/003-quasi-riemann-hypothesis/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/003-quasi-riemann-hypothesis/" }, { "num": "004", "slug": "004-hilberts-tenth-problem-q", "title": "Hilbert’s tenth problem over ℚ", "title_html": "Hilbert’s tenth problem over ℚ", "summary_html": "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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Hilbert’s tenth problem over the rational numbers", "title_html": "Hilbert’s tenth problem over the rational numbers", "url": "https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf", "role": "main" }, { "title": "A pointwise 2-converse for elliptic curves with rational two-torsion", "title_html": "A pointwise 2-converse for elliptic curves with rational two-torsion", "url": "https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "ot2ctgXB_xA", "length": "6:50", "title": "No algorithm for fractions", "page": "/episodes/004-hilberts-tenth-problem-q/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/004-hilberts-tenth-problem-q/" }, { "num": "005", "slug": "005-irrationality-catalans-constant", "title": "Irrationality of Catalan’s constant", "title_html": "Irrationality of Catalan’s constant", "summary_html": "Proves that Catalan's constant G=∑j≥0(−1)j/(2j+1)2G=\\sum_{j\\ge0}(-1)^j/(2j+1)^2G=∑j≥0​(−1)j/(2j+1)2 is irrational.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Catalan's constant is irrational", "title_html": "Catalan's constant is irrational", "url": "https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/005.md" }, "peer_review": "not yet", "episode": { "video_id": "9qekv7mIFOY", "length": "7:02", "title": "Not a fraction", "page": "/episodes/005-irrationality-catalans-constant/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/005-irrationality-catalans-constant/" }, { "num": "006", "slug": "006-goldfelds-conjecture-densities-mean-analytic", "title": "Goldfeld’s conjecture: densities and mean analytic rank", "title_html": "Goldfeld’s conjecture: densities and mean analytic rank", "summary_html": "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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Goldfeld's analytic density conjecture and the 2-converse for elliptic curves", "title_html": "Goldfeld's analytic density conjecture and the 2-converse for elliptic curves", "url": "https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf", "role": "main" }, { "title": "The mean analytic rank of quadratic twists of elliptic curves", "title_html": "The mean analytic rank of quadratic twists of elliptic curves", "url": "https://github.com/openai/math/blob/main/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "CeNAw8PcT90", "length": "6:38", "title": "Half and half", "page": "/episodes/006-goldfelds-conjecture-densities-mean-analytic/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/006-goldfelds-conjecture-densities-mean-analytic/" }, { "num": "007", "slug": "007-ordinary-two-point-correlations-corrected", "title": "Ordinary two-point correlations and the corrected Elliott conjecture", "title_html": "Ordinary two-point correlations and the corrected Elliott conjecture", "summary_html": "Proves the ordinary two-point Chowla conjecture, with a bound O(X/(log⁡X)c)O(X/(\\log X)^c)O(X/(logX)c) for Liouville correlation sums along fixed nonproportional affine forms, where c > 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 nitn^{it}nit for ∣t∣≤X|t|\\le X∣t∣≤X.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Ordinary two-point correlations of multiplicative functions", "title_html": "Ordinary two-point correlations of multiplicative functions", "url": "https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/007.md" }, "peer_review": "not yet", "episode": { "video_id": "Mh_JIF2oHuU", "length": "7:51", "title": "Even or odd", "page": "/episodes/007-ordinary-two-point-correlations-corrected/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/007-ordinary-two-point-correlations-corrected/" }, { "num": "008", "slug": "008-deligne-drinfeld-conjecture", "title": "The Deligne–Drinfeld conjecture", "title_html": "The Deligne–Drinfeld conjecture", "summary_html": "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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "The Deligne-Drinfeld conjecture", "title_html": "The Deligne-Drinfeld conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/008.md" }, "peer_review": "not yet", "episode": { "video_id": "xTNpFwVEBxY", "length": "6:27", "title": "One per odd weight", "page": "/episodes/008-deligne-drinfeld-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/008-deligne-drinfeld-conjecture/" }, { "num": "009", "slug": "009-function-field-reconstruction-milnor-k", "title": "Function-field reconstruction from Milnor K-theory and Galois data", "title_html": "Function-field reconstruction from Milnor K-theory and Galois data", "summary_html": "Reconstructs function fields of transcendence degree at least two over algebraically closed constants from K1M/ℓK^{\\mathrm M}_1/\\ellK1M​/ℓ, K2M/ℓK^{\\mathrm M}_2/\\ellK2M​/ℓ, 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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory", "title_html": "Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory", "url": "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", "role": "main" }, { "title": "Reconstruction from Milnor K-theory modulo the characteristic", "title_html": "Reconstruction from Milnor K-theory modulo the characteristic", "url": "https://github.com/openai/math/blob/main/preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf", "role": "companion" }, { "title": "The Bogomolov-Pop reconstruction theorem", "title_html": "The Bogomolov-Pop reconstruction theorem", "url": "https://github.com/openai/math/blob/main/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/009.md" }, "peer_review": "not yet", "episode": { "video_id": "tS-eTzTrcXI", "length": "6:03", "title": "The whole field from a shadow", "page": "/episodes/009-function-field-reconstruction-milnor-k/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/009-function-field-reconstruction-milnor-k/" }, { "num": "010", "slug": "010-unrestricted-pro-modularity-prime-two", "title": "Unrestricted pro-modularity at the prime two", "title_html": "Unrestricted pro-modularity at the prime two", "summary_html": "Every continuous odd absolutely irreducible two-dimensional 2-adic representation of GQG_{\\mathbb Q}GQ​ 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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Unrestricted pro-modularity at the prime two", "title_html": "Unrestricted pro-modularity at the prime two", "url": "https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf", "role": "main" }, { "title": "The Dimension of the Two-Adic Hecke Algebra at Odd Level", "title_html": "The Dimension of the Two-Adic Hecke Algebra at Odd Level", "url": "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", "role": "companion" }, { "title": "Fontaine–Mazur modularity at the prime 2", "title_html": "Fontaine–Mazur modularity at the prime 2", "url": "https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "EiDYk8HMEQY", "length": "5:19", "title": "Even the prime 2", "page": "/episodes/010-unrestricted-pro-modularity-prime-two/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/010-unrestricted-pro-modularity-prime-two/" }, { "num": "011", "slug": "011-prime-factor-statistics", "title": "Prime-factor statistics of p-1", "title_html": "Prime-factor statistics of p−1p-1p−1", "summary_html": "Proves that the normalized ordered logarithms of the prime factors of p−1p-1p−1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law PD(1)\\mathrm{PD}(1)PD(1) as p ranges uniformly over primes up to x and x→∞x\\to\\inftyx→∞. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than n1−εn^{1-\\varepsilon}n1−ε totient preimages, for every ε > 0.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Weighted dilation graphs, smooth shifted primes and totient fibers", "title_html": "Weighted dilation graphs, smooth shifted primes and totient fibers", "url": "https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf", "role": "main" }, { "title": "The Poisson-Dirichlet law for prime predecessors", "title_html": "The Poisson-Dirichlet law for prime predecessors", "url": "https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Prime Predecessors with an Even Number of Prime Factors", "title_html": "Prime Predecessors with an Even Number of Prime Factors", "url": "https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "QWLPy4tbJfo", "length": "6:11", "title": "How does p − 1 break?", "page": "/episodes/011-prime-factor-statistics/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/011-prime-factor-statistics/" }, { "num": "012", "slug": "012-independent-largest-prime-factors-consecutive", "title": "Independent largest prime factors of consecutive integers", "title_html": "Independent largest prime factors of consecutive integers", "summary_html": "Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and n+1n+1n+1 are asymptotically independent in ordinary natural density. In particular, the integers satisfying P+(n)<P+(n+1)P^+(n)\\lt P^+(n+1)P+(n)<P+(n+1) have density 1/2.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "The joint Dickman law for consecutive integers", "title_html": "The joint Dickman law for consecutive integers", "url": "https://github.com/openai/math/blob/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/012.md" }, "peer_review": "not yet", "episode": { "video_id": "-DEe5B7ZCOA", "length": "5:15", "title": "n or n + 1: who wins?", "page": "/episodes/012-independent-largest-prime-factors-consecutive/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/012-independent-largest-prime-factors-consecutive/" }, { "num": "013", "slug": "013-ostmanns-inverse-goldbach-conjecture", "title": "Ostmann’s inverse Goldbach conjecture", "title_html": "Ostmann’s inverse Goldbach conjecture", "summary_html": "Proves that no finite modification of the primes can be written as A+BA+BA+B with A,B⊆Z≥0A,B\\subseteq\\mathbb Z_{\\ge0}A,B⊆Z≥0​ each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "The additive indecomposability of the primes", "title_html": "The additive indecomposability of the primes", "url": "https://github.com/openai/math/blob/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/013.md" }, "peer_review": "not yet", "episode": { "video_id": "c-I8Wnk8XkE", "length": "4:23", "title": "Split the primes?", "page": "/episodes/013-ostmanns-inverse-goldbach-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/013-ostmanns-inverse-goldbach-conjecture/" }, { "num": "014", "slug": "014-restricted-geometric-langlands-global-arthur", "title": "Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan", "title_html": "Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan", "summary_html": "Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over F‾q\\overline{\\mathbb F}_qFq​ under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and p∤∣WG∣p\\nmid |W_G|p∤∣WG​∣. 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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Global Arthur Enhancements of Cuspidal Excursion Parameters", "title_html": "Global Arthur Enhancements of Cuspidal Excursion Parameters", "url": "https://github.com/openai/math/blob/main/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf", "role": "main" }, { "title": "Rationality of the Canonical Unramified Arthur Filtration", "title_html": "Rationality of the Canonical Unramified Arthur Filtration", "url": "https://github.com/openai/math/blob/main/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level", "title_html": "Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level", "url": "https://github.com/openai/math/blob/main/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Temperedness at ramified places for globally generic exceptional groups", "title_html": "Temperedness at ramified places for globally generic exceptional groups", "url": "https://github.com/openai/math/blob/main/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf", "role": "companion" }, { "title": "The Restricted Geometric Langlands Equivalence in Positive Characteristic", "title_html": "The Restricted Geometric Langlands Equivalence in Positive Characteristic", "url": "https://github.com/openai/math/blob/main/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Constructible tame Hecke eigensheaves in positive characteristic", "title_html": "Constructible tame Hecke eigensheaves in positive characteristic", "url": "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", "role": "companion" }, { "title": "Tame Hecke Eigensheaves with Several Marked Points", "title_html": "Tame Hecke Eigensheaves with Several Marked Points", "url": "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", "role": "companion" }, { "title": "Frobenius Structures on Tame Hecke Eigensheaves", "title_html": "Frobenius Structures on Tame Hecke Eigensheaves", "url": "https://github.com/openai/math/blob/main/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "3oZ9soW0atg", "length": "5:29", "title": "Ramanujan’s bound, up to E₈", "page": "/episodes/014-restricted-geometric-langlands-global-arthur/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/014-restricted-geometric-langlands-global-arthur/" }, { "num": "015", "slug": "015-torus-packet-equidistribution-prime-quartic", "title": "Torus-packet equidistribution in prime, quartic, and sextic degrees", "title_html": "Torus-packet equidistribution in prime, quartic, and sextic degrees", "summary_html": "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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type", "title_html": "Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type", "url": "https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf", "role": "main" }, { "title": "Equidistribution of primitive quartic torus packets for arbitrary orders", "title_html": "Equidistribution of primitive quartic torus packets for arbitrary orders", "url": "https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf", "role": "companion" }, { "title": "Equidistribution of Primitive Sextic Torus Packets", "title_html": "Equidistribution of Primitive Sextic Torus Packets", "url": "https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/015.md" }, "peer_review": "not yet", "episode": { "video_id": "DR_qxMwTcz4", "length": "5:27", "title": "Loops that fill space", "page": "/episodes/015-torus-packet-equidistribution-prime-quartic/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/015-torus-packet-equidistribution-prime-quartic/" }, { "num": "016", "slug": "016-zilber-pink-abelian-varieties-siegel", "title": "Zilber–Pink in abelian varieties and the Siegel threefold", "title_html": "Zilber–Pink in abelian varieties and the Siegel threefold", "summary_html": "Proves the abelian Zilber–Pink conjecture over Q‾\\overline{\\mathbb Q}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 A2\\mathcal A_2A2​ for Hodge-generic curves defined over Q‾\\overline{\\mathbb Q}Q​, without boundary or reduction assumptions.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "The abelian Zilber–Pink conjecture", "title_html": "The abelian Zilber–Pink conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf", "role": "main" }, { "title": "The E×CM component of Zilber–Pink for curves in A2", "title_html": "The E×CM component of Zilber–Pink for curves in A2", "url": "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", "role": "companion" }, { "title": "Quaternionic division points on curves in the Siegel threefold", "title_html": "Quaternionic division points on curves in the Siegel threefold", "url": "https://github.com/openai/math/blob/main/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Elliptic squares and Zilber–Pink for curves in A2", "title_html": "Elliptic squares and Zilber–Pink for curves in A2", "url": "https://github.com/openai/math/blob/main/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "e1PjO3CBEWw", "length": "5:08", "title": "Only two points", "page": "/episodes/016-zilber-pink-abelian-varieties-siegel/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/016-zilber-pink-abelian-varieties-siegel/" }, { "num": "017", "slug": "017-irrationality-exponent-pi-2", "title": "The irrationality exponent of π is 2", "title_html": "The irrationality exponent of π is 2", "summary_html": "Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational p/qp/qp/q satisfies ∣π−p/q∣≥q−2−ε|\\pi-p/q|\\ge q^{-2-\\varepsilon}∣π−p/q∣≥q−2−ε. This also proves convergence of the Flint–Hills series ∑n≥11/(n3sin⁡2n)\\sum_{n\\ge1}1/(n^3\\sin^2 n)∑n≥1​1/(n3sin2n), with angles in radians.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "The irrationality exponent of pi is 2", "title_html": "The irrationality exponent of pi is 2", "url": "https://github.com/openai/math/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/017.md" }, "peer_review": "not yet", "episode": { "video_id": "uvSBcyXpaFQ", "length": "4:37", "title": "Only so lucky", "page": "/episodes/017-irrationality-exponent-pi-2/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/017-irrationality-exponent-pi-2/" }, { "num": "018", "slug": "018-margulis-platonov-conjecture-global-fields", "title": "The Margulis–Platonov conjecture over global fields", "title_html": "The Margulis–Platonov conjecture over global fields", "summary_html": "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)G(k)G(k) is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "The Margulis–Platonov conjecture over global function fields", "title_html": "The Margulis–Platonov conjecture over global function fields", "url": "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", "role": "main" }, { "title": "The Margulis–Platonov conjecture over number fields", "title_html": "The Margulis–Platonov conjecture over number fields", "url": "https://github.com/openai/math/blob/main/preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "Jx3TIm8afBM", "length": "6:57", "title": "A hidden clock", "page": "/episodes/018-margulis-platonov-conjecture-global-fields/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/018-margulis-platonov-conjecture-global-fields/" }, { "num": "019", "slug": "019-local-p-adic-section-conjecture", "title": "The local p-adic section conjecture and global consequences", "title_html": "The local p-adic section conjecture and global consequences", "summary_html": "Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp 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 X0(N)X_0(N)X0​(N) and X1(N)X_1(N)X1​(N) of genus at least two.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Étale covers with a prescribed exterior sheet", "title_html": "Étale covers with a prescribed exterior sheet", "url": "https://github.com/openai/math/blob/main/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf", "role": "main" }, { "title": "The p-adic section conjecture", "title_html": "The p-adic section conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "p-rBq-0PpC8", "length": "5:51", "title": "Points from symmetries", "page": "/episodes/019-local-p-adic-section-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/019-local-p-adic-section-conjecture/" }, { "num": "020", "slug": "020-squarefree-quartics-power-free-polynomial", "title": "Squarefree quartics and power-free polynomial values", "title_html": "Squarefree quartics and power-free polynomial values", "summary_html": "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)(d-2)(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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Squarefree values of quartics and power-free values of polynomials", "title_html": "Squarefree values of quartics and power-free values of polynomials", "url": "https://github.com/openai/math/blob/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/020.md" }, "peer_review": "not yet", "episode": { "video_id": "1Eh8S-FQJrg", "length": "4:54", "title": "Squarefree, 3 in 4", "page": "/episodes/020-squarefree-quartics-power-free-polynomial/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/020-squarefree-quartics-power-free-polynomial/" }, { "num": "021", "slug": "021-quadratic-bound-jacobsthals-function", "title": "A quadratic bound for Jacobsthal’s function", "title_html": "A quadratic bound for Jacobsthal’s function", "summary_html": "Answers Jacobsthal's quadratic-bound question: every interval of Ck2Ck^2Ck2 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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "A quadratic bound for Jacobsthal's function", "title_html": "A quadratic bound for Jacobsthal's function", "url": "https://github.com/openai/math/blob/main/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/021.md" }, "peer_review": "not yet", "episode": { "video_id": "WwXctn2i2jE", "length": "4:36", "title": "How long a run?", "page": "/episodes/021-quadratic-bound-jacobsthals-function/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/021-quadratic-bound-jacobsthals-function/" }, { "num": "022", "slug": "022-weak-inhomogeneous-duffin-schaeffer-conjecture", "title": "The weak inhomogeneous Duffin–Schaeffer conjecture", "title_html": "The weak inhomogeneous Duffin–Schaeffer conjecture", "summary_html": "Proves that for every real shift γ and finite-valued ψ:N→[0,∞)\\psi:\\mathbb N\\to[0,\\infty)ψ:N→[0,∞), divergence of ∑qϕ(q)ψ(q)/q\\sum_q\\phi(q)\\psi(q)/q∑q​ϕ(q)ψ(q)/q implies ∥qx−γ∥<ψ(q)\\|qx-\\gamma\\|\\lt \\psi(q)∥qx−γ∥<ψ(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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "The Weak Inhomogeneous Duffin–Schaeffer Conjecture", "title_html": "The Weak Inhomogeneous Duffin–Schaeffer Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "mYpsKeA9mUk", "length": "4:33", "title": "Any shift will do", "page": "/episodes/022-weak-inhomogeneous-duffin-schaeffer-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/022-weak-inhomogeneous-duffin-schaeffer-conjecture/" }, { "num": "023", "slug": "023-pattersons-first-moment-cubic-gauss", "title": "Patterson's first moment for cubic Gauss sums", "title_html": "Patterson's first moment for cubic Gauss sums", "summary_html": "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 X5/6/log⁡XX^{5/6}/\\log XX5/6/logX. Every fixed nonzero prime-angle Fourier mode has smaller order.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "An unconditional first moment for cubic Gauss sums", "title_html": "An unconditional first moment for cubic Gauss sums", "url": "https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/023.md" }, "peer_review": "not yet", "episode": { "video_id": "76lQWhWZ_TQ", "length": "5:14", "title": "Still biased?", "page": "/episodes/023-pattersons-first-moment-cubic-gauss/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/023-pattersons-first-moment-cubic-gauss/" }, { "num": "024", "slug": "024-asymptotic-formula-number-totients", "title": "An asymptotic formula for the number of totients", "title_html": "An asymptotic formula for the number of totients", "summary_html": "Gives an asymptotic equivalent for the number V(x)V(x)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)→cV(cx)/V(x)\\to cV(cx)/V(x)→c for every fixed c > 0, answering Erdős and Hall’s scaling question.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "An asymptotic formula for the number of totients", "title_html": "An asymptotic formula for the number of totients", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/024.md" }, "peer_review": "not yet", "episode": { "video_id": "GTI8GXkgD4E", "length": "4:20", "title": "Double x, double V?", "page": "/episodes/024-asymptotic-formula-number-totients/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/024-asymptotic-formula-number-totients/" }, { "num": "025", "slug": "025-short-egyptian-fractions", "title": "Short Egyptian fractions", "title_html": "Short Egyptian fractions", "summary_html": "Every rational a/ba/ba/b with 1≤a<b1\\le a\\lt b1≤a<b is a sum of O(log⁡log⁡b)O(\\log\\log b)O(loglogb) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Short Egyptian fractions", "title_html": "Short Egyptian fractions", "url": "https://github.com/openai/math/blob/main/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/025.md" }, "peer_review": "not yet", "episode": { "video_id": "Om53cP1oLTU", "length": "4:22", "title": "How few fractions?", "page": "/episodes/025-short-egyptian-fractions/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/025-short-egyptian-fractions/" }, { "num": "026", "slug": "026-positive-lower-density-large-prime", "title": "Positive lower density of large prime gaps", "title_html": "Positive lower density of large prime gaps", "summary_html": "For every fixed C > 0, a positive proportion of consecutive prime gaps exceed Clog⁡pnC\\log p_nClogpn​, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where pn/np_n/npn​/n increases have positive lower density, answering Erdős and Prachar.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Positive lower density of large prime gaps", "title_html": "Positive lower density of large prime gaps", "url": "https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/026.md" }, "peer_review": "not yet", "episode": { "video_id": "77GDNtdCHis", "length": "3:47", "title": "Long gaps, how often?", "page": "/episodes/026-positive-lower-density-large-prime/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/026-positive-lower-density-large-prime/" }, { "num": "027", "slug": "027-potential-integral-density-curve-character", "title": "Potential integral density on curve character varieties", "title_html": "Potential integral density on curve character varieties", "summary_html": "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 SLr character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Integral points on character varieties of curves", "title_html": "Integral points on character varieties of curves", "url": "https://github.com/openai/math/blob/main/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/027.md" }, "peer_review": "not yet", "episode": { "video_id": "xyzo6_Hio14", "length": "6:18", "title": "Integral points, everywhere?", "page": "/episodes/027-potential-integral-density-curve-character/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/027-potential-integral-density-curve-character/" }, { "num": "028", "slug": "028-uniformly-bounded-components-gaussian-prime", "title": "Uniformly bounded components of Gaussian-prime graphs", "title_html": "Uniformly bounded components of Gaussian-prime graphs", "summary_html": "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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Bounded-Step Walks on Gaussian Primes", "title_html": "Bounded-Step Walks on Gaussian Primes", "url": "https://github.com/openai/math/blob/main/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/028.md" }, "peer_review": "not yet", "episode": { "video_id": "rLDtzy4HOvY", "length": "4:31", "title": "Walk to infinity?", "page": "/episodes/028-uniformly-bounded-components-gaussian-prime/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/028-uniformly-bounded-components-gaussian-prime/" }, { "num": "029", "slug": "029-primitive-roots-every-admissible-integer", "title": "Primitive roots for every admissible integer base", "title_html": "Primitive roots for every admissible integer base", "summary_html": "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 cax/(log⁡x)2c_a x/(\\log x)^2ca​x/(logx)2 primes in every sufficiently large interval (x,2x)(x,2x)(x,2x) have primitive root a, with ca>0c_a\\gt 0ca​>0.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Primitive roots for every admissible integer base", "title_html": "Primitive roots for every admissible integer base", "url": "https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf", "role": "main" }, { "title": "Simultaneous primitive roots: a conditional lower bound for prime bases", "title_html": "Simultaneous primitive roots: a conditional lower bound for prime bases", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "KxIZo3XHOlA", "length": "5:11", "title": "Which of 2, 3, 5?", "page": "/episodes/029-primitive-roots-every-admissible-integer/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/029-primitive-roots-every-admissible-integer/" }, { "num": "030", "slug": "030-modularity-elliptic-curves-imaginary-quadratic", "title": "Modularity of elliptic curves over imaginary quadratic fields", "title_html": "Modularity of elliptic curves over imaginary quadratic fields", "summary_html": "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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Modularity of elliptic curves over imaginary quadratic fields", "title_html": "Modularity of elliptic curves over imaginary quadratic fields", "url": "https://github.com/openai/math/blob/main/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "-EjMzwWQkzk", "length": "4:53", "title": "Every curve modular?", "page": "/episodes/030-modularity-elliptic-curves-imaginary-quadratic/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/030-modularity-elliptic-curves-imaginary-quadratic/" }, { "num": "031", "slug": "031-uchidas-conjecture-open-homomorphisms-galois", "title": "Uchida’s conjecture for open homomorphisms of Galois groups", "title_html": "Uchida’s conjecture for open homomorphisms of Galois groups", "summary_html": "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.", "subject": "Number theory", "subject_slug": "number-theory", "papers": [ { "title": "Open Homomorphisms of Global Solvably Closed Galois Groups", "title_html": "Open Homomorphisms of Global Solvably Closed Galois Groups", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "lLNi-7GELo8", "length": "4:17", "title": "Groups remember fields?", "page": "/episodes/031-uchidas-conjecture-open-homomorphisms-galois/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/031-uchidas-conjecture-open-homomorphisms-galois/" }, { "num": "032", "slug": "032-rational-hodge-conjecture-cm-abelian", "title": "The rational Hodge conjecture for CM abelian varieties", "title_html": "The rational Hodge conjecture for CM abelian varieties", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "The rational Hodge conjecture for CM abelian varieties", "title_html": "The rational Hodge conjecture for CM abelian varieties", "url": "https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf", "role": "main" }, { "title": "Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus", "title_html": "Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus", "url": "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", "role": "companion" }, { "title": "Weil classes and Hodge classes on abelian powers", "title_html": "Weil classes and Hodge classes on abelian powers", "url": "https://github.com/openai/math/blob/main/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-October-6-2026/paper.pdf", "role": "companion" }, { "title": "Abelian covers, Gale correspondences, and the Hodge conjecture for powers", "title_html": "Abelian covers, Gale correspondences, and the Hodge conjecture for powers", "url": "https://github.com/openai/math/blob/main/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026/paper.pdf", "role": "companion" }, { "title": "A Conditional Reduction for Algebraic Kuga–Satake Correspondences", "title_html": "A Conditional Reduction for Algebraic Kuga–Satake Correspondences", "url": "https://github.com/openai/math/blob/main/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "4aU2_E5mjLA", "length": "7:53", "title": "Every Hodge class algebraic?", "page": "/episodes/032-rational-hodge-conjecture-cm-abelian/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/032-rational-hodge-conjecture-cm-abelian/" }, { "num": "033", "slug": "033-iitaka-subadditivity-variation-logarithmic-additivity", "title": "Iitaka subadditivity, variation, and logarithmic additivity", "title_html": "Iitaka subadditivity, variation, and logarithmic additivity", "summary_html": "Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-C\\mathcal CC manifolds with rational simple-normal-crossing boundaries. For projective fibrations f:U→Vf:U\\to Vf:U→V of smooth complex quasi-projective varieties with connected fibers, general fiber F, and κˉ(V)≥0\\bar\\kappa(V)\\ge0κˉ(V)≥0, proves Popa's inequality κˉ(U)≥κ(F)+max⁡{κˉ(V),Var(f)}\\bar\\kappa(U)\\ge\\kappa(F)+\\max\\{\\bar\\kappa(V),\\mathop{\\mathrm{Var}}\\nolimits (f)\\}κˉ(U)≥κ(F)+max{κˉ(V),Var(f)}, where variation measures the whole geometric generic fiber.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Orbifold and logarithmic Iitaka subadditivity", "title_html": "Orbifold and logarithmic Iitaka subadditivity", "url": "https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf", "role": "main" }, { "title": "Logarithmic Kodaira dimension and whole-fiber variation", "title_html": "Logarithmic Kodaira dimension and whole-fiber variation", "url": "https://github.com/openai/math/blob/main/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf", "role": "companion" }, { "title": "The reverse logarithmic Kodaira inequality and additivity", "title_html": "The reverse logarithmic Kodaira inequality and additivity", "url": "https://github.com/openai/math/blob/main/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf", "role": "companion" }, { "title": "Projective Hodge lines and ordinary Iitaka subadditivity", "title_html": "Projective Hodge lines and ordinary Iitaka subadditivity", "url": "https://github.com/openai/math/blob/main/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf", "role": "companion" }, { "title": "B-semiampleness for compact log-smooth Kähler fibrations", "title_html": "B-semiampleness for compact log-smooth Kähler fibrations", "url": "https://github.com/openai/math/blob/main/preprints/B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/033.md" }, "peer_review": "not yet", "episode": { "video_id": "IvYfiqMrHEI", "length": "5:43", "title": "Do the forms add up?", "page": "/episodes/033-iitaka-subadditivity-variation-logarithmic-additivity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/033-iitaka-subadditivity-variation-logarithmic-additivity/" }, { "num": "034", "slug": "034-log-abundance-compact-kahler-spaces", "title": "Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity", "title_html": "Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity", "title_html": "Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity", "url": "https://github.com/openai/math/blob/main/preprints/Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity-October-6-2026/main.pdf", "role": "main" }, { "title": "Uniform indices for semi-log-canonical log Calabi–Yau pairs", "title_html": "Uniform indices for semi-log-canonical log Calabi–Yau pairs", "url": "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", "role": "companion" }, { "title": "Conditional good minimal models for compact Kähler fourfolds", "title_html": "Conditional good minimal models for compact Kähler fourfolds", "url": "https://github.com/openai/math/blob/main/preprints/Conditional-good-minimal-models-for-compact-Kahler-fourfolds-October-6-2026/paper.pdf", "role": "companion" }, { "title": "Log abundance in characteristic zero", "title_html": "Log abundance in characteristic zero", "url": "https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Minimal metrics and interior injectivity for nef adjoints", "title_html": "Minimal metrics and interior injectivity for nef adjoints", "url": "https://github.com/openai/math/blob/main/preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Fourfold nonvanishing by minimal metrics and moving jets", "title_html": "Fourfold nonvanishing by minimal metrics and moving jets", "url": "https://github.com/openai/math/blob/main/preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Lifting sections from the reduced support of an adjoint", "title_html": "Lifting sections from the reduced support of an adjoint", "url": "https://github.com/openai/math/blob/main/preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Schnell fiber spaces and good canonical models", "title_html": "Schnell fiber spaces and good canonical models", "url": "https://github.com/openai/math/blob/main/preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Uniform log Iitaka fibrations and bounded moduli denominators", "title_html": "Uniform log Iitaka fibrations and bounded moduli denominators", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026/uniform-log-iitaka.pdf", "role": "companion" }, { "title": "Uniform Pluricanonical Iitaka Fibrations", "title_html": "Uniform Pluricanonical Iitaka Fibrations", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf", "role": "companion" }, { "title": "Relative denominators and effective systems for log Calabi-Yau fibrations", "title_html": "Relative denominators and effective systems for log Calabi-Yau fibrations", "url": "https://github.com/openai/math/blob/main/preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Arithmetic Stein-degree bounds for log Calabi–Yau pairs", "title_html": "Arithmetic Stein-degree bounds for log Calabi–Yau pairs", "url": "https://github.com/openai/math/blob/main/preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026/paper.pdf", "role": "companion" }, { "title": "Uniform effective log Iitaka fibrations for fourfolds", "title_html": "Uniform effective log Iitaka fibrations for fourfolds", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026/paper.pdf", "role": "companion" }, { "title": "Abundance after nonvanishing for compact Kähler fourfolds", "title_html": "Abundance after nonvanishing for compact Kähler fourfolds", "url": "https://github.com/openai/math/blob/main/preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "TeUznTEM7pA", "length": "6:22", "title": "Always semiample?", "page": "/episodes/034-log-abundance-compact-kahler-spaces/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/034-log-abundance-compact-kahler-spaces/" }, { "num": "035", "slug": "035-log-canonical-threefold-abundance-numerical", "title": "Log-canonical threefold abundance in numerical dimension one", "title_html": "Log-canonical threefold abundance in numerical dimension one", "summary_html": "Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Log abundance in numerical dimension one for threefolds in positive characteristic", "title_html": "Log abundance in numerical dimension one for threefolds in positive characteristic", "url": "https://github.com/openai/math/blob/main/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf", "role": "main" }, { "title": "Abundance in numerical dimension one for terminal threefolds in positive characteristic", "title_html": "Abundance in numerical dimension one for terminal threefolds in positive characteristic", "url": "https://github.com/openai/math/blob/main/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "5k47trDEOEo", "length": "4:32", "title": "Abundance mod p?", "page": "/episodes/035-log-canonical-threefold-abundance-numerical/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/035-log-canonical-threefold-abundance-numerical/" }, { "num": "036", "slug": "036-numerical-semiampleness-generalized-minimal-models", "title": "Numerical semiampleness and generalized minimal models", "title_html": "Numerical semiampleness and generalized minimal models", "summary_html": "Proves numerical semiampleness for nef adjoints KX+B+MK_X+B+MKX​+B+M with KX+BK_X+BKX​+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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Numerical semiampleness of nef adjoint classes on compact Kähler manifolds", "title_html": "Numerical semiampleness of nef adjoint classes on compact Kähler manifolds", "url": "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", "role": "main" }, { "title": "Numerical Semiampleness of Nef Adjoint Divisors", "title_html": "Numerical Semiampleness of Nef Adjoint Divisors", "url": "https://github.com/openai/math/blob/main/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf", "role": "companion" }, { "title": "Minimal models and Mori fibre spaces for generalized log canonical Q-pairs", "title_html": "Minimal models and Mori fibre spaces for generalized log canonical Q-pairs", "url": "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", "role": "companion" }, { "title": "Minimal models in numerical dimension one", "title_html": "Minimal models in numerical dimension one", "url": "https://github.com/openai/math/blob/main/preprints/Minimal-models-in-numerical-dimension-one-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "qgl9uVXBeRc", "length": "5:09", "title": "Semiample, numerically?", "page": "/episodes/036-numerical-semiampleness-generalized-minimal-models/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/036-numerical-semiampleness-generalized-minimal-models/" }, { "num": "037", "slug": "037-ordinary-double-point-volume-gap", "title": "The ordinary-double-point volume gap", "title_html": "The ordinary-double-point volume gap", "summary_html": "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)n2(n-1)^n2(n−1)n. Equality holds precisely for an analytic ordinary double point.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "The ordinary-double-point gap in every dimension", "title_html": "The ordinary-double-point gap in every dimension", "url": "https://github.com/openai/math/blob/main/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf", "role": "main" }, { "title": "The normalized-volume gap in dimension four", "title_html": "The normalized-volume gap in dimension four", "url": "https://github.com/openai/math/blob/main/preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "4atpVGOPBps", "length": "4:37", "title": "The mildest singularity?", "page": "/episodes/037-ordinary-double-point-volume-gap/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/037-ordinary-double-point-volume-gap/" }, { "num": "038", "slug": "038-fujitas-freeness-conjecture", "title": "Fujita’s freeness conjecture", "title_html": "Fujita’s freeness conjecture", "summary_html": "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 KX+mLK_X+mLKX​+mL is globally generated for every integer m≥n+1m\\ge n+1m≥n+1.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Fujita's freeness conjecture", "title_html": "Fujita's freeness conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Fujitas-freeness-conjecture-September-23-2026/Fujitas-freeness-conjecture-September-23-2026.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "GZdWQkw2gy8", "length": "5:08", "title": "Sections everywhere?", "page": "/episodes/038-fujitas-freeness-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/038-fujitas-freeness-conjecture/" }, { "num": "039", "slug": "039-nagatas-conjecture-maximal-seshadri-constants", "title": "Nagata’s conjecture and maximal Seshadri constants", "title_html": "Nagata’s conjecture and maximal Seshadri constants", "summary_html": "Proves Nagata's strict inequality ∑imi<dr\\sum_i m_i\\lt d\\sqrt r∑i​mi​<dr​ for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants (Ln/r)1/n(L^n/r)^{1/n}(Ln/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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Nagata's conjecture for plane curves", "title_html": "Nagata's conjecture for plane curves", "url": "https://github.com/openai/math/blob/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf", "role": "main" }, { "title": "Maximal Seshadri constants on arbitrary polarized surfaces", "title_html": "Maximal Seshadri constants on arbitrary polarized surfaces", "url": "https://github.com/openai/math/blob/main/preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf", "role": "companion" }, { "title": "Maximal Multipoint Seshadri Constants in Higher Dimensions", "title_html": "Maximal Multipoint Seshadri Constants in Higher Dimensions", "url": "https://github.com/openai/math/blob/main/preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026/main.pdf", "role": "companion" }, { "title": "Maximal multipoint Seshadri constants in positive characteristic", "title_html": "Maximal multipoint Seshadri constants in positive characteristic", "url": "https://github.com/openai/math/blob/main/preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026/seshadri-positive-characteristic.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/039.md" }, "peer_review": "not yet", "episode": { "video_id": "Sj6zEQI7Sk0", "length": "4:45", "title": "Nagata was right?", "page": "/episodes/039-nagatas-conjecture-maximal-seshadri-constants/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/039-nagatas-conjecture-maximal-seshadri-constants/" }, { "num": "040", "slug": "040-blochs-conjecture-complex-surfaces", "title": "Bloch’s conjecture for complex surfaces", "title_html": "Bloch’s conjecture for complex surfaces", "summary_html": "Proves Bloch's conjecture: for every smooth connected projective complex surface S with pg(S)=0p_g(S)=0pg​(S)=0, the Albanese map CH0(S)0→Alb(S)(C)\\mathrm{CH}_0(S)^0\\to\\mathrm{Alb}(S)(\\mathbb C)CH0​(S)0→Alb(S)(C) on integral degree-zero zero-cycles is an isomorphism. This combines the new pg=q=0p_g=q=0pg​=q=0 theorem with the classical theorem of Bloch, Kas, and Lieberman.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Bloch’s conjecture for surfaces with p_g=0", "title_html": "Bloch’s conjecture for surfaces with p_g=0", "url": "https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "IEUxN01ZCec", "length": "4:44", "title": "All points alike?", "page": "/episodes/040-blochs-conjecture-complex-surfaces/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/040-blochs-conjecture-complex-surfaces/" }, { "num": "041", "slug": "041-hyperkahler-syz-projective-space-bases", "title": "Hyperkähler SYZ and projective-space bases", "title_html": "Hyperkähler SYZ and projective-space bases", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Projective-space bases of Lagrangian fibrations", "title_html": "Projective-space bases of Lagrangian fibrations", "url": "https://github.com/openai/math/blob/main/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf", "role": "main" }, { "title": "The strong hyperkähler SYZ conjecture", "title_html": "The strong hyperkähler SYZ conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026/main.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "jXPAWnNfJtw", "length": "4:36", "title": "Fibered by tori?", "page": "/episodes/041-hyperkahler-syz-projective-space-bases/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/041-hyperkahler-syz-projective-space-bases/" }, { "num": "042", "slug": "042-oka-classification-minimal-compact-complex", "title": "Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII", "title_html": "Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Every complex K3 surface is Oka", "title_html": "Every complex K3 surface is Oka", "url": "https://github.com/openai/math/blob/main/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "zZ6iVRcRjXI", "length": "4:34", "title": "Every K3 flexible?", "page": "/episodes/042-oka-classification-minimal-compact-complex/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/042-oka-classification-minimal-compact-complex/" }, { "num": "043", "slug": "043-p-w-fixed-determinant-sln", "title": "P = W for fixed-determinant SLₙ moduli spaces", "title_html": "P = W for fixed-determinant SLn moduli spaces", "summary_html": "Proves Pk=W2k=W2k+1P_k=W_{2k}=W_{2k+1}Pk​=W2k​=W2k+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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "P=W in composite rank for fixed determinant", "title_html": "P=W in composite rank for fixed determinant", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "p_hpVe6jPBU", "length": "4:02", "title": "P = W, every rank?", "page": "/episodes/043-p-w-fixed-determinant-sln/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/043-p-w-fixed-determinant-sln/" }, { "num": "044", "slug": "044-equivariant-cohomological-hikita-conjecture", "title": "The equivariant cohomological Hikita conjecture", "title_html": "The equivariant cohomological Hikita conjecture", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "The equivariant cohomological Hikita conjecture for arbitrary quivers", "title_html": "The equivariant cohomological Hikita conjecture for arbitrary quivers", "url": "https://github.com/openai/math/blob/main/preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026/main.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "0bHvMDkJdAM", "length": "5:14", "title": "Cohomology from a point?", "page": "/episodes/044-equivariant-cohomological-hikita-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/044-equivariant-cohomological-hikita-conjecture/" }, { "num": "046", "slug": "046-shafarevich-counterexamples-dimension-two-large", "title": "Shafarevich counterexamples in dimension two and with large fundamental group", "title_html": "Shafarevich counterexamples in dimension two and with large fundamental group", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "A projective fourfold with large fundamental group and non-Stein universal cover", "title_html": "A projective fourfold with large fundamental group and non-Stein universal cover", "url": "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", "role": "main" }, { "title": "A surface counterexample to Shafarevich holomorphic convexity", "title_html": "A surface counterexample to Shafarevich holomorphic convexity", "url": "https://github.com/openai/math/blob/main/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "o1IRp4PjrHA", "length": "5:25", "title": "Shafarevich refuted?", "page": "/episodes/046-shafarevich-counterexamples-dimension-two-large/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/046-shafarevich-counterexamples-dimension-two-large/" }, { "num": "047", "slug": "047-zariski-cancellation-affine-fibrations-complex", "title": "Zariski cancellation and affine fibrations over the complex numbers", "title_html": "Zariski cancellation and affine fibrations over the complex numbers", "summary_html": "Constructs an integral complex affine fourfold X≇A4X\\not\\cong\\mathbb A^4X≅A4 with X×A1≅A5X\\times\\mathbb A^1\\cong\\mathbb A^5X×A1≅A5, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections X→A1X\\to\\mathbb A^1X→A1 and A5→A2\\mathbb A^5\\to\\mathbb A^2A5→A2 have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "An explicit failure of complex affine-space cancellation", "title_html": "An explicit failure of complex affine-space cancellation", "url": "https://github.com/openai/math/blob/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/047.md" }, "peer_review": "not yet", "episode": { "video_id": "1ampvAW6zDY", "length": "4:53", "title": "Cancel a line?", "page": "/episodes/047-zariski-cancellation-affine-fibrations-complex/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/047-zariski-cancellation-affine-fibrations-complex/" }, { "num": "048", "slug": "048-characteristic-zero-counterexample-lipman-zariski", "title": "A characteristic-zero counterexample to Lipman–Zariski", "title_html": "A characteristic-zero counterexample to Lipman–Zariski", "summary_html": "Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "A singular normal affine surface with free tangent sheaf", "title_html": "A singular normal affine surface with free tangent sheaf", "url": "https://github.com/openai/math/blob/main/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "G4DbcVIgO30", "length": "4:20", "title": "Free, yet singular?", "page": "/episodes/048-characteristic-zero-counterexample-lipman-zariski/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/048-characteristic-zero-counterexample-lipman-zariski/" }, { "num": "049", "slug": "049-stable-coordinate-counterexample-four-variables", "title": "A stable-coordinate counterexample in four variables", "title_html": "A stable-coordinate counterexample in four variables", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "A stable coordinate that is not a coordinate in four variables", "title_html": "A stable coordinate that is not a coordinate in four variables", "url": "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", "role": "main" }, { "title": "An explicit noncoordinate polynomial with affine three-space zero fibre", "title_html": "An explicit noncoordinate polynomial with affine three-space zero fibre", "url": "https://github.com/openai/math/blob/main/preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/049.md" }, "peer_review": "not yet", "episode": { "video_id": "1lJqDX5fWLQ", "length": "4:41", "title": "One variable too few?", "page": "/episodes/049-stable-coordinate-counterexample-four-variables/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/049-stable-coordinate-counterexample-four-variables/" }, { "num": "050", "slug": "050-counterexample-griffiths-positivity-conjecture", "title": "A counterexample to Griffiths’ positivity conjecture", "title_html": "A counterexample to Griffiths’ positivity conjecture", "summary_html": "Constructs ample rank-two bundles on P1×P1\\mathbb P^1\\times\\mathbb P^1P1×P1 with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Ample rank-two bundles on the quadric surface without Griffiths-positive metrics", "title_html": "Ample rank-two bundles on the quadric surface without Griffiths-positive metrics", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/050.md" }, "peer_review": "not yet", "episode": { "video_id": "qi3lpx19QmM", "length": "4:44", "title": "Ample, not positive?", "page": "/episodes/050-counterexample-griffiths-positivity-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/050-counterexample-griffiths-positivity-conjecture/" }, { "num": "051", "slug": "051-kobayashis-canonical-ampleness-conjecture", "title": "Kobayashi’s canonical-ampleness conjecture", "title_html": "Kobayashi’s canonical-ampleness conjecture", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Canonical ampleness of compact hyperbolic Kähler manifolds", "title_html": "Canonical ampleness of compact hyperbolic Kähler manifolds", "url": "https://github.com/openai/math/blob/main/preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026/canonical-ampleness.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "WzkT7KeVd4U", "length": "5:55", "title": "Hyperbolic, so ample?", "page": "/episodes/051-kobayashis-canonical-ampleness-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/051-kobayashis-canonical-ampleness-conjecture/" }, { "num": "052", "slug": "052-tangent-splittings-product-decompositions", "title": "Tangent splittings and product decompositions", "title_html": "Tangent splittings and product decompositions", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Universal-cover splitting for compact Kähler manifolds", "title_html": "Universal-cover splitting for compact Kähler manifolds", "url": "https://github.com/openai/math/blob/main/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf", "role": "main" }, { "title": "Integrability of split tangent bundles on rationally connected manifolds", "title_html": "Integrability of split tangent bundles on rationally connected manifolds", "url": "https://github.com/openai/math/blob/main/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/052.md" }, "peer_review": "not yet", "episode": { "video_id": "GsSCkkGoAts", "length": "6:11", "title": "Split tangent, split space?", "page": "/episodes/052-tangent-splittings-product-decompositions/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/052-tangent-splittings-product-decompositions/" }, { "num": "053", "slug": "053-counterexample-pixton-completeness-chow", "title": "A counterexample to Pixton completeness in Chow", "title_html": "A counterexample to Pixton completeness in Chow", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "A high-arity counterexample to Pixton completeness in Chow", "title_html": "A high-arity counterexample to Pixton completeness in Chow", "url": "https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "WmBfRYvJJrU", "length": "4:55", "title": "A relation Pixton missed?", "page": "/episodes/053-counterexample-pixton-completeness-chow/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/053-counterexample-pixton-completeness-chow/" }, { "num": "054", "slug": "054-irrational-cubic-fourfolds-hodge-theoretic", "title": "Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations", "title_html": "Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Irrational cubic fourfolds with geometric K3 categories", "title_html": "Irrational cubic fourfolds with geometric K3 categories", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "k_xudc7SIKA", "length": "5:28", "title": "K3 inside, still irrational?", "page": "/episodes/054-irrational-cubic-fourfolds-hodge-theoretic/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/054-irrational-cubic-fourfolds-hodge-theoretic/" }, { "num": "055", "slug": "055-gepner-symmetry-large-volume-stability", "title": "Gepner symmetry and large-volume stability on threefolds", "title_html": "Gepner symmetry and large-volume stability on threefolds", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "A Gepner stability condition on every smooth quintic threefold", "title_html": "A Gepner stability condition on every smooth quintic threefold", "url": "https://github.com/openai/math/blob/main/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf", "role": "main" }, { "title": "Prescribed large-volume charges on threefolds with trivial canonical bundle", "title_html": "Prescribed large-volume charges on threefolds with trivial canonical bundle", "url": "https://github.com/openai/math/blob/main/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "rWCEkuyPa50", "length": "5:00", "title": "A fifth of a turn?", "page": "/episodes/055-gepner-symmetry-large-volume-stability/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/055-gepner-symmetry-large-volume-stability/" }, { "num": "056", "slug": "056-termination-projective-kahler-fourfold-minimal", "title": "Termination of projective and Kähler fourfold minimal model programs", "title_html": "Termination of projective and Kähler fourfold minimal model programs", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Termination of generalized log canonical flips on compact Kähler fourfolds", "title_html": "Termination of generalized log canonical flips on compact Kähler fourfolds", "url": "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", "role": "main" }, { "title": "Termination of generalized-canonical flips on compact Kähler fourfolds", "title_html": "Termination of generalized-canonical flips on compact Kähler fourfolds", "url": "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", "role": "companion" }, { "title": "Termination for projective log canonical fourfolds with rational boundary", "title_html": "Termination for projective log canonical fourfolds with rational boundary", "url": "https://github.com/openai/math/blob/main/preprints/Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Finite ordinary minimal model programs on compact Kähler fourfolds", "title_html": "Finite ordinary minimal model programs on compact Kähler fourfolds", "url": "https://github.com/openai/math/blob/main/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-6-2026/paper.pdf", "role": "companion" }, { "title": "Finite ordinary minimal model programs on compact Kähler fourfolds", "title_html": "Finite ordinary minimal model programs on compact Kähler fourfolds", "url": "https://github.com/openai/math/blob/main/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "BUNCdmKpCuE", "length": "7:35", "title": "Can flips go on forever?", "page": "/episodes/056-termination-projective-kahler-fourfold-minimal/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/056-termination-projective-kahler-fourfold-minimal/" }, { "num": "057", "slug": "057-fundamental-groups-special-complex-varieties", "title": "Fundamental groups of special complex varieties and root orbifolds", "title_html": "Fundamental groups of special complex varieties and root orbifolds", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "The abelianity conjecture for special compact Kähler manifolds", "title_html": "The abelianity conjecture for special compact Kähler manifolds", "url": "https://github.com/openai/math/blob/main/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf", "role": "main" }, { "title": "Two-step monodromy of special quasi-projective varieties", "title_html": "Two-step monodromy of special quasi-projective varieties", "url": "https://github.com/openai/math/blob/main/preprints/Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026/paper.pdf", "role": "companion" }, { "title": "A conditional abelianity theorem for special fourfold pairs with a half-weight divisor", "title_html": "A conditional abelianity theorem for special fourfold pairs with a half-weight divisor", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "cGD9mCGbIxg", "length": "7:03", "title": "Special means abelian?", "page": "/episodes/057-fundamental-groups-special-complex-varieties/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/057-fundamental-groups-special-complex-varieties/" }, { "num": "058", "slug": "058-semialgebraic-universal-covers-bounded-domains", "title": "Semialgebraic universal covers and bounded domains", "title_html": "Semialgebraic universal covers and bounded domains", "summary_html": "Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products D×Cm×FD\\times\\mathbb C^m\\times FD×Cm×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 ℂn has a finite étale cover by an abelian variety.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Semialgebraic universal covers of normal projective varieties", "title_html": "Semialgebraic universal covers of normal projective varieties", "url": "https://github.com/openai/math/blob/main/preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/paper.pdf", "role": "main" }, { "title": "Symmetry of semialgebraic bounded domains with compact quotient", "title_html": "Symmetry of semialgebraic bounded domains with compact quotient", "url": "https://github.com/openai/math/blob/main/preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/058.md" }, "peer_review": "not yet", "episode": { "video_id": "ajJEF3qtusc", "length": "6:50", "title": "Three building blocks?", "page": "/episodes/058-semialgebraic-universal-covers-bounded-domains/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/058-semialgebraic-universal-covers-bounded-domains/" }, { "num": "059", "slug": "059-counterexamples-zariskis-multiplicity-conjecture", "title": "Counterexamples to Zariski’s multiplicity conjecture", "title_html": "Counterexamples to Zariski’s multiplicity conjecture", "summary_html": "Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three", "title_html": "Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three", "url": "https://github.com/openai/math/blob/main/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf", "role": "main" }, { "title": "Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five", "title_html": "Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "6CoTDZ_rhZ8", "length": "6:06", "title": "Same knot, other degree?", "page": "/episodes/059-counterexamples-zariskis-multiplicity-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/059-counterexamples-zariskis-multiplicity-conjecture/" }, { "num": "060", "slug": "060-global-spherical-shell-conjecture", "title": "The Global Spherical Shell conjecture", "title_html": "The Global Spherical Shell conjecture", "summary_html": "Every connected minimal compact complex surface of class VII with b2>0b_2\\gt 0b2​>0 contains a global spherical shell, proving the positive-b2 Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in C2∖{0}\\mathbb C^2\\setminus\\{0\\}C2∖{0} whose complement is connected.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Global Spherical Shells on Minimal Surfaces of Class VII", "title_html": "Global Spherical Shells on Minimal Surfaces of Class VII", "url": "https://github.com/openai/math/blob/main/preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "JNlQMNUsVp4", "length": "5:12", "title": "A hidden shell?", "page": "/episodes/060-global-spherical-shell-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/060-global-spherical-shell-conjecture/" }, { "num": "062", "slug": "062-projective-contact-classification-lebrun-salamon", "title": "Projective contact classification and the LeBrun–Salamon conjecture", "title_html": "Projective contact classification and the LeBrun–Salamon conjecture", "summary_html": "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 b2=1b_2=1b2​=1 are adjoint varieties with their canonical contact structures, while those with b2≥2b_2\\ge2b2​≥2 have underlying manifold P(T∗Z)\\mathbb P(T^*Z)P(T∗Z) for a smooth projective variety Z.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Contact Fano manifolds and the LeBrun–Salamon conjecture", "title_html": "Contact Fano manifolds and the LeBrun–Salamon conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "4Muo36wPtiI", "length": "6:04", "title": "Only Wolf spaces?", "page": "/episodes/062-projective-contact-classification-lebrun-salamon/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/062-projective-contact-classification-lebrun-salamon/" }, { "num": "063", "slug": "063-generalized-mukai-conjecture", "title": "The generalized Mukai conjecture", "title_html": "The generalized Mukai conjecture", "summary_html": "Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies ρ(ι−1)≤n\\rho(\\iota-1)\\le nρ(ι−1)≤n, with equality exactly for (Pι−1)ρ(\\mathbb P^{\\iota-1})^\\rho(Pι−1)ρ. Here the pseudoindex is the least anticanonical degree of a rational curve.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "The generalized Mukai conjecture", "title_html": "The generalized Mukai conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-Generalized-Mukai-Conjecture-September-24-2026/article.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "ytrmeiiYQC8", "length": "5:36", "title": "How many divisors?", "page": "/episodes/063-generalized-mukai-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/063-generalized-mukai-conjecture/" }, { "num": "064", "slug": "064-topological-triviality-mu-constant-surface", "title": "Topological triviality of μ-constant surface singularities", "title_html": "Topological triviality of μ-constant surface singularities", "summary_html": "Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Topological triviality of mu-constant families of surface singularities", "title_html": "Topological triviality of mu-constant families of surface singularities", "url": "https://github.com/openai/math/blob/main/preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/main.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "te_PY1ukk_o", "length": "5:55", "title": "Same number, same shape?", "page": "/episodes/064-topological-triviality-mu-constant-surface/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/064-topological-triviality-mu-constant-surface/" }, { "num": "065", "slug": "065-virasoro-constraints-complete-intersections-projective", "title": "Virasoro constraints for complete intersections and projective-bundle towers", "title_html": "Virasoro constraints for complete intersections and projective-bundle towers", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Virasoro Constraints under Projectivization", "title_html": "Virasoro Constraints under Projectivization", "url": "https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf", "role": "main" }, { "title": "Virasoro Constraints for Projective Complete Intersections", "title_html": "Virasoro Constraints for Projective Complete Intersections", "url": "https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026/article.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "KUCqXvlHENY", "length": "5:47", "title": "Virasoro everywhere?", "page": "/episodes/065-virasoro-constraints-complete-intersections-projective/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/065-virasoro-constraints-complete-intersections-projective/" }, { "num": "066", "slug": "066-bounded-klt-complements-fano-contractions", "title": "Bounded klt complements for Fano contractions", "title_html": "Bounded klt complements for Fano contractions", "summary_html": "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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Bounded klt complements for Fano contractions", "title_html": "Bounded klt complements for Fano contractions", "url": "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", "role": "main" }, { "title": "Uniform Cartier sections for Fano type contractions", "title_html": "Uniform Cartier sections for Fano type contractions", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/066.md" }, "peer_review": "not yet", "episode": { "video_id": "E153lk2FH_o", "length": "5:40", "title": "A bounded index?", "page": "/episodes/066-bounded-klt-complements-fano-contractions/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/066-bounded-klt-complements-fano-contractions/" }, { "num": "067", "slug": "067-campana-peternell-conjecture-dimension-six", "title": "The Campana–Peternell conjecture in dimension six", "title_html": "The Campana–Peternell conjecture in dimension six", "summary_html": "Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "The Campana–Peternell conjecture in dimension six", "title_html": "The Campana–Peternell conjecture in dimension six", "url": "https://github.com/openai/math/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "YMZXtZ6L61M", "length": "5:09", "title": "Nef means symmetric?", "page": "/episodes/067-campana-peternell-conjecture-dimension-six/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/067-campana-peternell-conjecture-dimension-six/" }, { "num": "068", "slug": "068-anticanonical-nonvanishing-every-dimension", "title": "Anticanonical nonvanishing in every dimension", "title_html": "Anticanonical nonvanishing in every dimension", "summary_html": "If X is a smooth connected complex projective variety and −KX-K_X−KX​ admits a smooth Hermitian metric with nonnegative curvature, then H0(X,−mKX)≠0H^0(X,-mK_X)\\ne0H0(X,−mKX​)=0 for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Anticanonical nonvanishing from smooth semipositivity", "title_html": "Anticanonical nonvanishing from smooth semipositivity", "url": "https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf", "role": "main" }, { "title": "Invariant anticanonical indices and conversion of twisted differentials", "title_html": "Invariant anticanonical indices and conversion of twisted differentials", "url": "https://github.com/openai/math/blob/main/preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026/main.pdf", "role": "companion" }, { "title": "Bounded anticanonical metrics on klt pairs and torus quotients", "title_html": "Bounded anticanonical metrics on klt pairs and torus quotients", "url": "https://github.com/openai/math/blob/main/preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026/main.pdf", "role": "companion" }, { "title": "Cohomological transfer and equivariant anticanonical sections", "title_html": "Cohomological transfer and equivariant anticanonical sections", "url": "https://github.com/openai/math/blob/main/preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026/main.pdf", "role": "companion" }, { "title": "Metric descent and rank-preserving contractions", "title_html": "Metric descent and rank-preserving contractions", "url": "https://github.com/openai/math/blob/main/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf", "role": "companion" }, { "title": "Integrable metrics and effectivity with controlled boundary", "title_html": "Integrable metrics and effectivity with controlled boundary", "url": "https://github.com/openai/math/blob/main/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf", "role": "companion" }, { "title": "Exact orders and invariant anticanonical linear systems", "title_html": "Exact orders and invariant anticanonical linear systems", "url": "https://github.com/openai/math/blob/main/preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "ADxIZz6fESM", "length": "6:06", "title": "A section, eventually?", "page": "/episodes/068-anticanonical-nonvanishing-every-dimension/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/068-anticanonical-nonvanishing-every-dimension/" }, { "num": "069", "slug": "069-global-quantum-geometric-langlands-irrational", "title": "Global quantum geometric Langlands at irrational level", "title_html": "Global quantum geometric Langlands at irrational level", "summary_html": "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∈C∖Qc\\in\\mathbb C\\setminus\\mathbb Qc∈C∖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.", "subject": "Algebraic and complex geometry", "subject_slug": "algebraic-and-complex-geometry", "papers": [ { "title": "Global quantum geometric Langlands at irrational level", "title_html": "Global quantum geometric Langlands at irrational level", "url": "https://github.com/openai/math/blob/main/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "tHiiiAB6tGc", "length": "5:59", "title": "Langlands, quantized?", "page": "/episodes/069-global-quantum-geometric-langlands-irrational/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/069-global-quantum-geometric-langlands-irrational/" }, { "num": "071", "slug": "071-koebes-circle-domain-conjecture", "title": "Koebe’s circle-domain conjecture", "title_html": "Koebe’s circle-domain conjecture", "summary_html": "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.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Removable Boundaries and Rigidity of Circle Domains", "title_html": "Removable Boundaries and Rigidity of Circle Domains", "url": "https://github.com/openai/math/blob/main/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf", "role": "main" }, { "title": "Koebe's Circle-Domain Conjecture", "title_html": "Koebe's Circle-Domain Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/071.md" }, "peer_review": "not yet", "episode": { "video_id": "5RyM19gCnWY", "length": "6:22", "title": "Every hole made round?", "page": "/episodes/071-koebes-circle-domain-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/071-koebes-circle-domain-conjecture/" }, { "num": "072", "slug": "072-brennans-conjecture-integral-means-spectrum", "title": "Brennan's conjecture and the integral-means spectrum", "title_html": "Brennan's conjecture and the integral-means spectrum", "summary_html": "Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, ∣ϕ′∣s|\\phi'|^s∣ϕ′∣s is area-integrable for 4/3<s<44/3\\lt s\\lt 44/3<s<4. The sharp universal integral-means identity is BS(t)=∣t∣−1B_{\\mathcal S}(t)=|t|-1BS​(t)=∣t∣−1 for t ≤ −2. A strict bound Bb(−1)<1/4B_b(-1)\\lt 1/4Bb​(−1)<1/4 for bounded univalent functions disproves Kraetzer's prediction at that parameter.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Brennan's conjecture and sharp inverse-square integral means", "title_html": "Brennan's conjecture and sharp inverse-square integral means", "url": "https://github.com/openai/math/blob/main/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf", "role": "main" }, { "title": "A strict inverse-first-power bound for univalent functions", "title_html": "A strict inverse-first-power bound for univalent functions", "url": "https://github.com/openai/math/blob/main/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/072.md" }, "peer_review": "not yet", "episode": { "video_id": "Lu3s2nNexVA", "length": "5:31", "title": "Four thirds to four?", "page": "/episodes/072-brennans-conjecture-integral-means-spectrum/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/072-brennans-conjecture-integral-means-spectrum/" }, { "num": "073", "slug": "073-falconer-distance-conjecture", "title": "The Falconer distance conjecture", "title_html": "The Falconer distance conjecture", "summary_html": "Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set E⊂RdE\\subset\\mathbb R^dE⊂Rd with Hausdorff dimension greater than d/2d/2d/2 determines a set of Euclidean distances of positive Lebesgue measure.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "The Falconer distance conjecture in all dimensions", "title_html": "The Falconer distance conjecture in all dimensions", "url": "https://github.com/openai/math/blob/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/073.md" }, "peer_review": "not yet", "episode": { "video_id": "6R3w8koN9A4", "length": "5:43", "title": "Enough distances?", "page": "/episodes/073-falconer-distance-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/073-falconer-distance-conjecture/" }, { "num": "074", "slug": "074-kakeya-three-four-dimensions", "title": "Kakeya in three and four dimensions", "title_html": "Kakeya in three and four dimensions", "summary_html": "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 L3(R3)L^3(\\mathbb R^3)L3(R3) to L3(S2)L^3(S^2)L3(S2) with norm Oε(δ−ε)O_\\varepsilon(\\delta^{-\\varepsilon})Oε​(δ−ε) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "The Kakeya maximal conjecture in three dimensions", "title_html": "The Kakeya maximal conjecture in three dimensions", "url": "https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf", "role": "main" }, { "title": "Every four-dimensional Kakeya set has full Hausdorff dimension", "title_html": "Every four-dimensional Kakeya set has full Hausdorff dimension", "url": "https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "5txsx_jHBFU", "length": "7:34", "title": "Needles in every direction?", "page": "/episodes/074-kakeya-three-four-dimensions/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/074-kakeya-three-four-dimensions/" }, { "num": "075", "slug": "075-fourier-convergence-conjecture", "title": "The L log L Fourier-convergence conjecture", "title_html": "The Llog⁡LL\\log LLlogL Fourier-convergence conjecture", "summary_html": "Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in Llog⁡L(T)L\\log L(\\mathbb T)LlogL(T) converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the Llog⁡LL\\log LLlogL scale.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Almost-everywhere Fourier convergence in L log L", "title_html": "Almost-everywhere Fourier convergence in L log L", "url": "https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/075.md" }, "peer_review": "not yet", "episode": { "video_id": "OHhyljpwDdc", "length": "6:00", "title": "Enough to converge?", "page": "/episodes/075-fourier-convergence-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/075-fourier-convergence-conjecture/" }, { "num": "076", "slug": "076-real-ultraflat-littlewood-polynomials-unbounded", "title": "Real ultraflat Littlewood polynomials and unbounded binary merit factors", "title_html": "Real ultraflat Littlewood polynomials and unbounded binary merit factors", "summary_html": "Constructs polynomials with N consecutive coefficients in {−1,1}\\{-1,1\\}{−1,1} whose modulus is (1+o(1))N(1+o(1))\\sqrt N(1+o(1))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.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Ultraflat real Littlewood polynomials", "title_html": "Ultraflat real Littlewood polynomials", "url": "https://github.com/openai/math/blob/main/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf", "role": "main" }, { "title": "Nearly minimal maxima and positive minima of Littlewood polynomials", "title_html": "Nearly minimal maxima and positive minima of Littlewood polynomials", "url": "https://github.com/openai/math/blob/main/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf", "role": "companion" }, { "title": "Asymptotically minimal maxima of real Littlewood polynomials", "title_html": "Asymptotically minimal maxima of real Littlewood polynomials", "url": "https://github.com/openai/math/blob/main/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/076.md" }, "peer_review": "not yet", "episode": { "video_id": "ir1D5w5K5Cg", "length": "6:11", "title": "Flat with signs only?", "page": "/episodes/076-real-ultraflat-littlewood-polynomials-unbounded/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/076-real-ultraflat-littlewood-polynomials-unbounded/" }, { "num": "077", "slug": "077-fourier-restriction-positively-curved-surfaces", "title": "Fourier restriction for positively curved surfaces", "title_html": "Fourier restriction for positively curved surfaces", "summary_html": "Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface Σ⊂R3\\Sigma\\subset\\mathbb R^3Σ⊂R3, including surfaces with boundary, the extension operator is bounded from Lp(Σ)L^p(\\Sigma)Lp(Σ) to Lp(R3)L^p(\\mathbb R^3)Lp(R3) for every p > 3.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Elliptic capacity propagation and Fourier restriction to the sphere", "title_html": "Elliptic capacity propagation and Fourier restriction to the sphere", "url": "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", "role": "main" }, { "title": "Diagonal Fourier extension for positively curved surfaces in three dimensions", "title_html": "Diagonal Fourier extension for positively curved surfaces in three dimensions", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "RtqzZRwiQqQ", "length": "5:49", "title": "Restriction, settled?", "page": "/episodes/077-fourier-restriction-positively-curved-surfaces/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/077-fourier-restriction-positively-curved-surfaces/" }, { "num": "078", "slug": "078-three-dimensional-bochner-riesz-conjecture", "title": "The three-dimensional Bochner–Riesz conjecture", "title_html": "The three-dimensional Bochner–Riesz conjecture", "summary_html": "Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on Lp(R3)L^p(\\mathbb R^3)Lp(R3) for every 1≤p≤∞1\\le p\\le\\infty1≤p≤∞ whenever δ>max⁡{3∣1/p−1/2∣−1/2,0}\\delta\\gt \\max\\{3|1/p-1/2|-1/2,0\\}δ>max{3∣1/p−1/2∣−1/2,0}.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Bochner–Riesz multipliers in three dimensions", "title_html": "Bochner–Riesz multipliers in three dimensions", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "ajT3DWlXZ_8", "length": "4:58", "title": "Any smoothing enough?", "page": "/episodes/078-three-dimensional-bochner-riesz-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/078-three-dimensional-bochner-riesz-conjecture/" }, { "num": "079", "slug": "079-local-smoothing-three-dimensions", "title": "Local smoothing in three dimensions", "title_html": "Local smoothing in three dimensions", "summary_html": "Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range 2<p<∞2\\lt p\\lt \\infty2<p<∞, with Sobolev regularity above max⁡{0,1−3/p}\\max\\{0,1-3/p\\}max{0,1−3/p}. In particular, the critical L3 estimate holds with every positive Sobolev loss.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Critical local smoothing for the three-dimensional wave equation", "title_html": "Critical local smoothing for the three-dimensional wave equation", "url": "https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "r62e_v2a3hA", "length": "6:32", "title": "Smoother on average?", "page": "/episodes/079-local-smoothing-three-dimensions/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/079-local-smoothing-three-dimensions/" }, { "num": "080", "slug": "080-schrodinger-convergence-sobolev-endpoint", "title": "The exact Sobolev endpoint for Schrödinger convergence", "title_html": "The exact Sobolev endpoint for Schrödinger convergence", "summary_html": "Proves almost-everywhere convergence eitΔf→fe^{it\\Delta}f\\to feitΔf→f as t↓0t\\downarrow0t↓0 for every f∈Hn/(2(n+1))(Rn)f\\in H^{n/(2(n+1))}(\\mathbb R^n)f∈Hn/(2(n+1))(Rn) and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H1/3.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Endpoint convergence for the planar Schrodinger equation", "title_html": "Endpoint convergence for the planar Schrodinger equation", "url": "https://github.com/openai/math/blob/main/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/paper.pdf", "role": "main" }, { "title": "Endpoint pointwise convergence for the Schrodinger equation in higher dimensions", "title_html": "Endpoint pointwise convergence for the Schrodinger equation in higher dimensions", "url": "https://github.com/openai/math/blob/main/preprints/Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "FDot_uSJgpw", "length": "6:02", "title": "Exactly enough?", "page": "/episodes/080-schrodinger-convergence-sobolev-endpoint/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/080-schrodinger-convergence-sobolev-endpoint/" }, { "num": "081", "slug": "081-riesz-transforms-rectifiability-higher-codimension", "title": "Riesz transforms and rectifiability in higher codimension", "title_html": "Riesz transforms and rectifiability in higher codimension", "summary_html": "Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2≤n≤d−22\\le n\\le d-22≤n≤d−2, an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L2-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Riesz transforms and uniform rectifiability in higher codimension", "title_html": "Riesz transforms and uniform rectifiability in higher codimension", "url": "https://github.com/openai/math/blob/main/preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/081.md" }, "peer_review": "not yet", "episode": { "video_id": "5nuAE9QrEwQ", "length": "5:30", "title": "One operator enough?", "page": "/episodes/081-riesz-transforms-rectifiability-higher-codimension/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/081-riesz-transforms-rectifiability-higher-codimension/" }, { "num": "082", "slug": "082-triangular-hilbert-transform-symmetric-point", "title": "Annular variation and dyadic absolute bounds for the triangular Hilbert transform", "title_html": "Annular variation and dyadic absolute bounds for the triangular Hilbert transform", "summary_html": "Proves maximal and annular r-variation bounds, for every r > 2, from complex L3(R2)×L3(R2)L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2)L3(R2)×L3(R2) to L3/2(R2)L^{3/2}(\\mathbb R^2)L3/2(R2). 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 L3×L3×L3L^3\\times L^3\\times L^3L3×L3×L3 point.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "Annular variation of the triangular Hilbert transform at the symmetric point", "title_html": "Annular variation of the triangular Hilbert transform at the symmetric point", "url": "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", "role": "main" }, { "title": "An L³ bound for the dyadic triangular Hilbert form", "title_html": "An L³ bound for the dyadic triangular Hilbert form", "url": "https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-dyadic-triangular-Hilbert-form-October-5-2026/dyadic-triangular-hilbert.pdf", "role": "companion" }, { "title": "The maximal triangular Hilbert transform at the symmetric point", "title_html": "The maximal triangular Hilbert transform at the symmetric point", "url": "https://github.com/openai/math/blob/main/preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/082.md" }, "peer_review": "not yet", "episode": { "video_id": "kOsAD1XQPzM", "length": "5:35", "title": "Triangle, tamed?", "page": "/episodes/082-triangular-hilbert-transform-symmetric-point/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/082-triangular-hilbert-transform-symmetric-point/" }, { "num": "083", "slug": "083-hilbert-transform-lipschitz-directions", "title": "Hilbert transforms along Lipschitz directions", "title_html": "Hilbert transforms along Lipschitz directions", "summary_html": "Proves a uniform strong L2 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 L2-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "A uniform Hilbert transform estimate for Lipschitz directions", "title_html": "A uniform Hilbert transform estimate for Lipschitz directions", "url": "https://github.com/openai/math/blob/main/preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/083.md" }, "peer_review": "not yet", "episode": { "video_id": "HtRBj-3Yv6s", "length": "5:48", "title": "Lines that turn?", "page": "/episodes/083-hilbert-transform-lipschitz-directions/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/083-hilbert-transform-lipschitz-directions/" }, { "num": "084", "slug": "084-erdos-similarity-geometric-sequences", "title": "The geometric case of the Erdős similarity conjecture", "title_html": "The geometric case of the Erdős similarity conjecture", "summary_html": "For every fixed q∈(0,1)q\\in(0,1)q∈(0,1), constructs compact subsets of [0,1][0,1][0,1] with measure arbitrarily close to one containing no translated and nontrivially dilated copy of {qn:n≥1}\\{q^n:n\\ge1\\}{qn:n≥1}, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "The geometric case of the Erdős similarity conjecture", "title_html": "The geometric case of the Erdős similarity conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf", "role": "main" }, { "title": "The dyadic case of the Erdős similarity conjecture", "title_html": "The dyadic case of the Erdős similarity conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/084.md" }, "peer_review": "not yet", "episode": { "video_id": "MsDKd6xH-Rs", "length": "5:27", "title": "No room for halves?", "page": "/episodes/084-erdos-similarity-geometric-sequences/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/084-erdos-similarity-geometric-sequences/" }, { "num": "085", "slug": "085-endpoint-sobolev-regularity-centered-disk", "title": "Endpoint Sobolev regularity of centered disk averages", "title_html": "Endpoint Sobolev regularity of centered disk averages", "summary_html": "Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real f∈W1,1(R2)f\\in W^{1,1}(\\mathbb R^2)f∈W1,1(R2), the centered disk maximal function satisfies ∥∇Mf∥1≤C∥∇f∥1\\|\\nabla Mf\\|_1\\le C\\|\\nabla f\\|_1∥∇Mf∥1​≤C∥∇f∥1​ with an absolute constant. It belongs locally to W1,1W^{1,1}W1,1 and has a globally integrable weak gradient.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "An Endpoint Gradient Bound for the Centered Disk Maximal Operator", "title_html": "An Endpoint Gradient Bound for the Centered Disk Maximal Operator", "url": "https://github.com/openai/math/blob/main/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026/article.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/085.md" }, "peer_review": "not yet", "episode": { "video_id": "dqUXsc0UuIQ", "length": "6:40", "title": "How much steeper?", "page": "/episodes/085-endpoint-sobolev-regularity-centered-disk/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/085-endpoint-sobolev-regularity-centered-disk/" }, { "num": "086", "slug": "086-l3-bound-trilinear-hilbert-transform", "title": "An L³ bound for the trilinear Hilbert transform", "title_html": "An L3 bound for the trilinear Hilbert transform", "summary_html": "Proves that the principal-value trilinear Hilbert transform with shifts x−tx-tx−t, x−2tx-2tx−2t, x−3tx-3tx−3t is bounded from L3(R)3L^3(\\mathbb R)^3L3(R)3 to L1(R)L^1(\\mathbb R)L1(R). This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.", "subject": "Real and complex analysis", "subject_slug": "real-and-complex-analysis", "papers": [ { "title": "An L³ bound for the trilinear Hilbert transform", "title_html": "An L3 bound for the trilinear Hilbert transform", "url": "https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "8xpxzZncVrY", "length": "6:05", "title": "Three shifts, one bound?", "page": "/episodes/086-l3-bound-trilinear-hilbert-transform/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/086-l3-bound-trilinear-hilbert-transform/" }, { "num": "087", "slug": "087-mahler-conjectures-functional-inequalities-polar", "title": "The Mahler conjectures, functional inequalities and polar-product symplectic width", "title_html": "The Mahler conjectures, functional inequalities and polar-product symplectic width", "summary_html": "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×K∘K\\times K^\\circK×K∘ in dimension 2n2n2n has Gromov width 4.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "The symmetric Mahler conjecture and its equality cases", "title_html": "The symmetric Mahler conjecture and its equality cases", "url": "https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf", "role": "main" }, { "title": "The Mahler Conjecture for General Convex Bodies", "title_html": "The Mahler Conjecture for General Convex Bodies", "url": "https://github.com/openai/math/blob/main/preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf", "role": "companion" }, { "title": "Symplectic Balls in Symmetric Polar Products", "title_html": "Symplectic Balls in Symmetric Polar Products", "url": "https://github.com/openai/math/blob/main/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/087.md" }, "peer_review": "not yet", "episode": { "video_id": "gBNoFT_uAL8", "length": "7:52", "title": "Mahler, solved?", "page": "/episodes/087-mahler-conjectures-functional-inequalities-polar/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/087-mahler-conjectures-functional-inequalities-polar/" }, { "num": "088", "slug": "088-sharp-projection-body-inequalities-counterexample", "title": "Sharp projection-body inequalities and a counterexample to simplex maximization", "title_html": "Sharp projection-body inequalities and a counterexample to simplex maximization", "summary_html": "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.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "Petty’s projection-volume conjecture in dimensions at least four", "title_html": "Petty’s projection-volume conjecture in dimensions at least four", "url": "https://github.com/openai/math/blob/main/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf", "role": "main" }, { "title": "A product counterexample to the simplex maximum for projection-body volume", "title_html": "A product counterexample to the simplex maximum for projection-body volume", "url": "https://github.com/openai/math/blob/main/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/088.md" }, "peer_review": "not yet", "episode": { "video_id": "wsLGs8qkW48", "length": "6:18", "title": "Smallest shadows?", "page": "/episodes/088-sharp-projection-body-inequalities-counterexample/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/088-sharp-projection-body-inequalities-counterexample/" }, { "num": "089", "slug": "089-bounded-distortion-l1-embeddings-planar", "title": "Bounded-distortion L₁ embeddings of planar and bounded-treewidth graphs", "title_html": "Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs", "summary_html": "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 L1 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.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "Planar Graph Metrics Embed into L1 with Constant Distortion", "title_html": "Planar Graph Metrics Embed into L1 with Constant Distortion", "url": "https://github.com/openai/math/blob/main/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf", "role": "main" }, { "title": "L1 Embeddings of Graphs of Bounded Treewidth", "title_html": "L1 Embeddings of Graphs of Bounded Treewidth", "url": "https://github.com/openai/math/blob/main/preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/089.md" }, "peer_review": "not yet", "episode": { "video_id": "Ay2gq8M06gw", "length": "6:40", "title": "Cut to size?", "page": "/episodes/089-bounded-distortion-l1-embeddings-planar/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/089-bounded-distortion-l1-embeddings-planar/" }, { "num": "090", "slug": "090-triangular-lattice-optimality-long-range", "title": "Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy", "title_html": "Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy", "summary_html": "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<s<20\\lt s\\lt 20<s<2 and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "An atomic certificate for triangular-lattice universal optimality", "title_html": "An atomic certificate for triangular-lattice universal optimality", "url": "https://github.com/openai/math/blob/main/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf", "role": "main" }, { "title": "Universal optimality of the triangular lattice", "title_html": "Universal optimality of the triangular lattice", "url": "https://github.com/openai/math/blob/main/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf", "role": "companion" }, { "title": "A sharp Fourier certificate for planar circle packing", "title_html": "A sharp Fourier certificate for planar circle packing", "url": "https://github.com/openai/math/blob/main/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Triangular minimality for planar Coulomb renormalized energy", "title_html": "Triangular minimality for planar Coulomb renormalized energy", "url": "https://github.com/openai/math/blob/main/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/090.md" }, "peer_review": "not yet", "episode": { "video_id": "g_Kj0GWeqAs", "length": "8:18", "title": "Triangles win?", "page": "/episodes/090-triangular-lattice-optimality-long-range/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/090-triangular-lattice-optimality-long-range/" }, { "num": "091", "slug": "091-logarithmic-lp-brunn-minkowski-inequalities", "title": "Logarithmic and Lₚ Brunn–Minkowski inequalities and the B-conjecture", "title_html": "Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture", "summary_html": "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 Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout 0<p<10\\lt p\\lt 10<p<1.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "The logarithmic Brunn–Minkowski conjecture", "title_html": "The logarithmic Brunn–Minkowski conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/091.md" }, "peer_review": "not yet", "episode": { "video_id": "zNu_hQ9HgLk", "length": "7:17", "title": "Blend by multiplying?", "page": "/episodes/091-logarithmic-lp-brunn-minkowski-inequalities/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/091-logarithmic-lp-brunn-minkowski-inequalities/" }, { "num": "092", "slug": "092-optimal-order-convex-body-covering", "title": "The optimal order of convex-body covering density", "title_html": "The optimal order of convex-body covering density", "summary_html": "Determines the optimal worst-case covering density as Θ(nlog⁡n)\\Theta(n\\log n)Θ(nlogn), for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most Cnlog⁡nCn\\log nCnlogn; centrally symmetric examples in every sufficiently large dimension require at least cnlog⁡ncn\\log ncnlogn even without the lattice restriction, for absolute c,C>0c,C\\gt 0c,C>0.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "A single-lattice covering bound of order n log n", "title_html": "A single-lattice covering bound of order n log n", "url": "https://github.com/openai/math/blob/main/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf", "role": "main" }, { "title": "Translative covering densities of order n log n", "title_html": "Translative covering densities of order n log n", "url": "https://github.com/openai/math/blob/main/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/092.md" }, "peer_review": "not yet", "episode": { "video_id": "tzhcVVqww4M", "length": "7:10", "title": "How much overlap?", "page": "/episodes/092-optimal-order-convex-body-covering/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/092-optimal-order-convex-body-covering/" }, { "num": "093", "slug": "093-dimension-free-logarithmic-sobolev-inequality", "title": "Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures", "title_html": "Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures", "summary_html": "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.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures", "title_html": "A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures", "url": "https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "aqk9Qwf_CK8", "length": "6:21", "title": "Tails are enough?", "page": "/episodes/093-dimension-free-logarithmic-sobolev-inequality/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/093-dimension-free-logarithmic-sobolev-inequality/" }, { "num": "094", "slug": "094-subpolynomial-dimension-reduction-lp", "title": "Subpolynomial dimension reduction in Lₚ", "title_html": "Subpolynomial dimension reduction in Lp", "summary_html": "For every fixed 1<p<∞1\\lt p\\lt \\infty1<p<∞ and distortion D > 1, every n-point subset of real Lp embeds into ℓpd\\ell_p^dℓpd​ with distortion at most D and dimension d=no(1)d=n^{o(1)}d=no(1), answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension Θ(n2)\\Theta(n^2)Θ(n2) when p ≠ 2.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "Subpolynomial dimension reduction in Lp", "title_html": "Subpolynomial dimension reduction in Lp", "url": "https://github.com/openai/math/blob/main/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/094.md" }, "peer_review": "not yet", "episode": { "video_id": "ozOrakyyW2A", "length": "6:22", "title": "Squeezed into less?", "page": "/episodes/094-subpolynomial-dimension-reduction-lp/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/094-subpolynomial-dimension-reduction-lp/" }, { "num": "095", "slug": "095-hyperbolicity-cones-without-semidefinite-lifts", "title": "Hyperbolicity cones without semidefinite lifts", "title_html": "Hyperbolicity cones without semidefinite lifts", "summary_html": "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.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "Hyperbolicity Cones Without Semidefinite Lifts", "title_html": "Hyperbolicity Cones Without Semidefinite Lifts", "url": "https://github.com/openai/math/blob/main/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf", "role": "main" }, { "title": "A nonspectrahedral hyperbolicity cone", "title_html": "A nonspectrahedral hyperbolicity cone", "url": "https://github.com/openai/math/blob/main/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf", "role": "companion" }, { "title": "An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone", "title_html": "An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/095.md" }, "peer_review": "not yet", "episode": { "video_id": "HIG6NmOeWRQ", "length": "8:19", "title": "No lift at all?", "page": "/episodes/095-hyperbolicity-cones-without-semidefinite-lifts/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/095-hyperbolicity-cones-without-semidefinite-lifts/" }, { "num": "096", "slug": "096-gaussian-propeller-conjecture-every-dimension", "title": "The Gaussian propeller conjecture in every dimension", "title_html": "The Gaussian propeller conjecture in every dimension", "summary_html": "Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8π)9/(8\\pi)9/(8π). In dimension at least two, three planar sectors of angle 2π/32\\pi/32π/3, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8π/9)(1−1/k)(8\\pi/9)(1-1/k)(8π/9)(1−1/k) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "The Gaussian propeller bound in every dimension", "title_html": "The Gaussian propeller bound in every dimension", "url": "https://github.com/openai/math/blob/main/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/096.md" }, "peer_review": "not yet", "episode": { "video_id": "8S1iXRSCmEM", "length": "6:23", "title": "Three blades win?", "page": "/episodes/096-gaussian-propeller-conjecture-every-dimension/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/096-gaussian-propeller-conjecture-every-dimension/" }, { "num": "097", "slug": "097-euclidean-steinitz-bergstrom-bound", "title": "The Euclidean Steinitz–Bergström bound", "title_html": "The Euclidean Steinitz–Bergström bound", "summary_html": "Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within CdC\\sqrt dCd​, 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 S2(d)=Θ(d)S_2(d)=\\Theta(\\sqrt d)S2​(d)=Θ(d​).", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "The Euclidean Steinitz–Bergström theorem", "title_html": "The Euclidean Steinitz–Bergström theorem", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/097.md" }, "peer_review": "not yet", "episode": { "video_id": "H29KI6UzGvc", "length": "6:20", "title": "Root d is enough?", "page": "/episodes/097-euclidean-steinitz-bergstrom-bound/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/097-euclidean-steinitz-bergstrom-bound/" }, { "num": "098", "slug": "098-compact-counterexamples-bi-lipschitz-dimension", "title": "Compact counterexamples to bi-Lipschitz dimension reduction", "title_html": "Compact counterexamples to bi-Lipschitz dimension reduction", "summary_html": "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.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding", "title_html": "A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding", "url": "https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/098.md" }, "peer_review": "not yet", "episode": { "video_id": "FIxHv0E-gvk", "length": "6:41", "title": "Doubling is not enough?", "page": "/episodes/098-compact-counterexamples-bi-lipschitz-dimension/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/098-compact-counterexamples-bi-lipschitz-dimension/" }, { "num": "099", "slug": "099-sharp-exponential-scale-edit-distance", "title": "The sharp exponential scale of edit-distance distortion", "title_html": "The sharp exponential scale of edit-distance distortion", "summary_html": "Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is exp⁡(Θ(log⁡d log⁡log⁡d))\\exp(\\Theta(\\sqrt{\\log d\\,\\log\\log d}))exp(Θ(logdloglogd​)). 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.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "Edit Distance in l1: Matching Bounds up to Constants in the Exponent", "title_html": "Edit Distance in l1: Matching Bounds up to Constants in the Exponent", "url": "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", "role": "main" }, { "title": "Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance", "title_html": "Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance", "url": "https://github.com/openai/math/blob/main/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Tree Constructions for the l1 Distortion of Binary Edit Distance", "title_html": "Tree Constructions for the l1 Distortion of Binary Edit Distance", "url": "https://github.com/openai/math/blob/main/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/099.md" }, "peer_review": "not yet", "episode": { "video_id": "fwHqvVcAv8I", "length": "6:09", "title": "Sharp at last?", "page": "/episodes/099-sharp-exponential-scale-edit-distance/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/099-sharp-exponential-scale-edit-distance/" }, { "num": "100", "slug": "100-cylinder-coverings-below-half-area", "title": "Cylinder coverings below the half-area bound", "title_html": "Cylinder coverings below the half-area bound", "summary_html": "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.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "Finite angular cylinder covers below the half-area bound", "title_html": "Finite angular cylinder covers below the half-area bound", "url": "https://github.com/openai/math/blob/main/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf", "role": "main" }, { "title": "Finite cylinder approximation of ruled sets", "title_html": "Finite cylinder approximation of ruled sets", "url": "https://github.com/openai/math/blob/main/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf", "role": "companion" }, { "title": "Slope-field perturbations of the two-cylinder covering", "title_html": "Slope-field perturbations of the two-cylinder covering", "url": "https://github.com/openai/math/blob/main/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf", "role": "companion" }, { "title": "Finite triangular approximation of radial sweeps", "title_html": "Finite triangular approximation of radial sweeps", "url": "https://github.com/openai/math/blob/main/preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/100.md" }, "peer_review": "not yet", "episode": { "video_id": "lHeYk1LoLw0", "length": "6:33", "title": "Below half?", "page": "/episodes/100-cylinder-coverings-below-half-area/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/100-cylinder-coverings-below-half-area/" }, { "num": "101", "slug": "101-sharp-simplex-conjecture-isotropic-constants", "title": "The sharp simplex conjecture for isotropic constants", "title_html": "The sharp simplex conjecture for isotropic constants", "summary_html": "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.", "subject": "Convex and metric geometry", "subject_slug": "convex-and-metric-geometry", "papers": [ { "title": "A sharp entropy bound and the simplex inequality for isotropic constants", "title_html": "A sharp entropy bound and the simplex inequality for isotropic constants", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "Zm8doY_lIrs", "length": "6:08", "title": "Simplex wins?", "page": "/episodes/101-sharp-simplex-conjecture-isotropic-constants/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/101-sharp-simplex-conjecture-isotropic-constants/" }, { "num": "102", "slug": "102-unique-games-conjecture-optimal-approximation", "title": "The Unique Games Conjecture and optimal approximation thresholds", "title_html": "The Unique Games Conjecture and optimal approximation thresholds", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "The Unique Games Theorem", "title_html": "The Unique Games Theorem", "url": "https://github.com/openai/math/blob/main/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf", "role": "main" }, { "title": "A Direct Proof of Optimal Max-Cut Hardness", "title_html": "A Direct Proof of Optimal Max-Cut Hardness", "url": "https://github.com/openai/math/blob/main/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf", "role": "companion" }, { "title": "The Factor-Two Hardness Threshold for Vertex Cover", "title_html": "The Factor-Two Hardness Threshold for Vertex Cover", "url": "https://github.com/openai/math/blob/main/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Constant-factor hardness of Min-UnCut", "title_html": "Constant-factor hardness of Min-UnCut", "url": "https://github.com/openai/math/blob/main/preprints/Constant-factor-hardness-of-Min-UnCut-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Constant-factor hardness of directed feedback vertex set", "title_html": "Constant-factor hardness of directed feedback vertex set", "url": "https://github.com/openai/math/blob/main/preprints/Constant-factor-hardness-of-directed-feedback-vertex-set-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/102.md" }, "peer_review": "not yet", "episode": { "video_id": "G7Iuuru9lS4", "length": "7:15", "title": "Unique Games, solved?", "page": "/episodes/102-unique-games-conjecture-optimal-approximation/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/102-unique-games-conjecture-optimal-approximation/" }, { "num": "103", "slug": "103-exact-derandomization-logarithmic-space", "title": "Exact derandomization of logarithmic space: L = RL = BPL", "title_html": "Exact derandomization of logarithmic space: L=RL=BPL\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}L=RL=BPL", "summary_html": "Proves L=RL=BPL\\mathsf L=\\mathsf{RL}=\\mathsf{BPL}L=RL=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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Exact derandomization of logarithmic space: L = RL = BPL", "title_html": "Exact derandomization of logarithmic space: L = RL = BPL", "url": "https://github.com/openai/math/blob/main/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/103.md" }, "peer_review": "not yet", "episode": { "video_id": "PombKgHKjVU", "length": "6:12", "title": "Coins don’t help?", "page": "/episodes/103-exact-derandomization-logarithmic-space/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/103-exact-derandomization-logarithmic-space/" }, { "num": "104", "slug": "104-quasipolynomial-mean-payoff-games", "title": "Quasipolynomial algorithms for mean-payoff, stochastic and parity games", "title_html": "Quasipolynomial algorithms for mean-payoff, stochastic and parity games", "summary_html": "Gives deterministic algorithms using 2O((log⁡(L+2))2)2^{O((\\log(L+2))^2)}2O((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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time", "title_html": "Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time", "url": "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", "role": "main" }, { "title": "Mean-payoff parity games in quasipolynomial time", "title_html": "Mean-payoff parity games in quasipolynomial time", "url": "https://github.com/openai/math/blob/main/preprints/Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026/mean-payoff-parity.pdf", "role": "companion" }, { "title": "Deterministic quasipolynomial-time mean-payoff games", "title_html": "Deterministic quasipolynomial-time mean-payoff games", "url": "https://github.com/openai/math/blob/main/preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf", "role": "companion" }, { "title": "Randomized quasipolynomial-time mean-payoff games", "title_html": "Randomized quasipolynomial-time mean-payoff games", "url": "https://github.com/openai/math/blob/main/preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/104.md" }, "peer_review": "not yet", "episode": { "video_id": "M2X2TVSRZYM", "length": "7:10", "title": "Who wins in the long run?", "page": "/episodes/104-quasipolynomial-mean-payoff-games/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/104-quasipolynomial-mean-payoff-games/" }, { "num": "105", "slug": "105-two-to-one-games-perfect-completeness", "title": "Perfect completeness for 2-to-1 games", "title_html": "Perfect completeness for 2-to-1 games", "summary_html": "Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational δ∈(0,1)\\delta\\in(0,1)δ∈(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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Perfect completeness for 2-to-1 games", "title_html": "Perfect completeness for 2-to-1 games", "url": "https://github.com/openai/math/blob/main/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/105.md" }, "peer_review": "not yet", "episode": { "video_id": "-SZ1zRQbDoc", "length": "6:15", "title": "Satisfiable yet hard?", "page": "/episodes/105-two-to-one-games-perfect-completeness/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/105-two-to-one-games-perfect-completeness/" }, { "num": "106", "slug": "106-coloring-three-colorable-graphs", "title": "Hardness of coloring three-colorable graphs", "title_html": "Hardness of coloring three-colorable graphs", "summary_html": "It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed 0<δ<1/30\\lt \\delta\\lt 1/30<δ<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 δn\\delta nδn otherwise, where n is the number of vertices.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Hardness of finding large independent sets in three-colorable graphs", "title_html": "Hardness of finding large independent sets in three-colorable graphs", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/106.md" }, "peer_review": "not yet", "episode": { "video_id": "ZHw4MBarBbA", "length": "5:58", "title": "Three colors, no help?", "page": "/episodes/106-coloring-three-colorable-graphs/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/106-coloring-three-colorable-graphs/" }, { "num": "107", "slug": "107-matrix-multiplication-exponent-nine-fourths", "title": "Matrix multiplication with exponent at most 9/4", "title_html": "Matrix multiplication with exponent at most 9/4", "summary_html": "Proves ω≤9/4\\omega\\le9/4ω≤9/4 over ℂ, giving Oε(n9/4+ε)O_\\varepsilon(n^{9/4+\\varepsilon})Oε​(n9/4+ε) arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits n2+o(1)n^{2+o(1)}n2+o(1) rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "An Upper Bound of 9/4 for the Matrix Multiplication Exponent", "title_html": "An Upper Bound of 9/4 for the Matrix Multiplication Exponent", "url": "https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf", "role": "main" }, { "title": "Complex Matrix Multiplication Below 2.258 and Rectangular Bounds", "title_html": "Complex Matrix Multiplication Below 2.258 and Rectangular Bounds", "url": "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", "role": "companion" }, { "title": "Staggered extraction for exact matrix multiplication over every field", "title_html": "Staggered extraction for exact matrix multiplication over every field", "url": "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", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/107.md" }, "peer_review": "not yet", "episode": { "video_id": "ILvHD7163h0", "length": "5:36", "title": "Two and a quarter?", "page": "/episodes/107-matrix-multiplication-exponent-nine-fourths/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/107-matrix-multiplication-exponent-nine-fourths/" }, { "num": "108", "slug": "108-permanent-determinant-cubic-lower-bound", "title": "A cubic permanent–determinant lower bound", "title_html": "A cubic permanent–determinant lower bound", "summary_html": "Proves an Ω(n3)\\Omega(n^3)Ω(n3) lower bound for the border determinantal complexity of the n×nn\\times nn×n permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least cn3cn^3cn3, for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "A cubic lower bound for border determinantal complexity of the permanent", "title_html": "A cubic lower bound for border determinantal complexity of the permanent", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/108.md" }, "peer_review": "not yet", "episode": { "video_id": "jX2mNRN4cH4", "length": "6:00", "title": "Cubic at last?", "page": "/episodes/108-permanent-determinant-cubic-lower-bound/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/108-permanent-determinant-cubic-lower-bound/" }, { "num": "109", "slug": "109-integer-multiplication-below-n-log-n", "title": "Integer multiplication below n log n", "title_html": "Integer multiplication below nlog⁡nn\\log nnlogn", "summary_html": "Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(log⁡n)1−κ)O(n(\\log n)^{1-\\kappa})O(n(logn)1−κ), with κ=2−182\\kappa=2^{-182}κ=2−182, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen nlog⁡nn\\log nnlogn optimality conjecture in the ordinary multitape bit model.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Integer multiplication below n log n", "title_html": "Integer multiplication below n log n", "url": "https://github.com/openai/math/blob/main/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "I-LBQyc8csQ", "length": "5:53", "title": "Faster than n log n?", "page": "/episodes/109-integer-multiplication-below-n-log-n/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/109-integer-multiplication-below-n-log-n/" }, { "num": "110", "slug": "110-randomized-k-server-squared-log", "title": "Optimal-order randomized k-server on arbitrary metrics", "title_html": "Optimal-order randomized k-server on arbitrary metrics", "summary_html": "Establishes a randomized competitive ratio O(log⁡2(k+1))O(\\log^2(k+1))O(log2(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)\\log(t+1)log(t+1) at request t, with a finite instance-dependent additive movement constant.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Squared-logarithmic randomized k-server on arbitrary metrics", "title_html": "Squared-logarithmic randomized k-server on arbitrary metrics", "url": "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", "role": "main" }, { "title": "Uniform computation of the squared-logarithmic k-server bound", "title_html": "Uniform computation of the squared-logarithmic k-server bound", "url": "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", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/110.md" }, "peer_review": "not yet", "episode": { "video_id": "mckoqqUxUbI", "length": "6:02", "title": "Log squared, everywhere?", "page": "/episodes/110-randomized-k-server-squared-log/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/110-randomized-k-server-squared-log/" }, { "num": "111", "slug": "111-one-sample-matroid-prophet-inequality", "title": "One-sample matroid prophet inequalities against an almighty adversary", "title_html": "One-sample matroid prophet inequalities against an almighty adversary", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary", "title_html": "One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary", "url": "https://github.com/openai/math/blob/main/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/111.md" }, "peer_review": "not yet", "episode": { "video_id": "oimWkAaTbFI", "length": "5:55", "title": "One sample is enough?", "page": "/episodes/111-one-sample-matroid-prophet-inequality/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/111-one-sample-matroid-prophet-inequality/" }, { "num": "112", "slug": "112-depth-three-circuits-beyond-square-root", "title": "Beyond the square-root exponent for depth-three circuits", "title_html": "Beyond the square-root exponent for depth-three circuits", "summary_html": "Constructs a single language in deterministic polynomial time whose n-bit membership function requires 2ω(n)2^{\\omega(\\sqrt n)}2ω(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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Beyond the Square-Root Exponent for Depth-Three Boolean Circuits", "title_html": "Beyond the Square-Root Exponent for Depth-Three Boolean Circuits", "url": "https://github.com/openai/math/blob/main/preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/112.md" }, "peer_review": "not yet", "episode": { "video_id": "8F-ilAisguU", "length": "5:27", "title": "Past the square root?", "page": "/episodes/112-depth-three-circuits-beyond-square-root/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/112-depth-three-circuits-beyond-square-root/" }, { "num": "113", "slug": "113-counting-perfect-matchings-fpras", "title": "Approximate counting and entropy of perfect matchings", "title_html": "Approximate counting and entropy of perfect matchings", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs", "title_html": "A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs", "url": "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", "role": "main" }, { "title": "Entropy and Face Dimension of the Perfect-Matching Polytope", "title_html": "Entropy and Face Dimension of the Perfect-Matching Polytope", "url": "https://github.com/openai/math/blob/main/preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/113.md" }, "peer_review": "not yet", "episode": { "video_id": "ZdjGAHp7IVs", "length": "5:53", "title": "Counting any graph?", "page": "/episodes/113-counting-perfect-matchings-fpras/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/113-counting-perfect-matchings-fpras/" }, { "num": "114", "slug": "114-common-matroid-bases-fpras", "title": "Approximate counting of common integer polymatroid bases", "title_html": "Approximate counting of common integer polymatroid bases", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "An FPRAS for Common Integer Polymatroid Bases with Binary Capacities", "title_html": "An FPRAS for Common Integer Polymatroid Bases with Binary Capacities", "url": "https://github.com/openai/math/blob/main/preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/polymatroid-fpras.pdf", "role": "main" }, { "title": "Approximate counting of common bases of two matroids", "title_html": "Approximate counting of common bases of two matroids", "url": "https://github.com/openai/math/blob/main/preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/main.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/114.md" }, "peer_review": "not yet", "episode": { "video_id": "MyxZk1I3QI8", "length": "5:52", "title": "Counting common bases?", "page": "/episodes/114-common-matroid-bases-fpras/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/114-common-matroid-bases-fpras/" }, { "num": "115", "slug": "115-contingency-tables-arbitrary-margins", "title": "Sampling and counting contingency tables with arbitrary margins", "title_html": "Sampling and counting contingency tables with arbitrary margins", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Exact Uniform Sampling of Contingency Tables with Arbitrary Margins", "title_html": "Exact Uniform Sampling of Contingency Tables with Arbitrary Margins", "url": "https://github.com/openai/math/blob/main/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf", "role": "main" }, { "title": "An FPRAS for Cell-Bounded Contingency Tables", "title_html": "An FPRAS for Cell-Bounded Contingency Tables", "url": "https://github.com/openai/math/blob/main/preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/115.md" }, "peer_review": "not yet", "episode": { "video_id": "R1T_-vxobwE", "length": "6:46", "title": "Any margins?", "page": "/episodes/115-contingency-tables-arbitrary-margins/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/115-contingency-tables-arbitrary-margins/" }, { "num": "116", "slug": "116-noncommutative-identity-testing-hitting-points", "title": "Uniform black-box noncommutative identity testing across characteristics", "title_html": "Uniform black-box noncommutative identity testing across characteristics", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Uniform Matrix Hitting Points in Every Positive Characteristic", "title_html": "Uniform Matrix Hitting Points in Every Positive Characteristic", "url": "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", "role": "main" }, { "title": "One Rational Matrix Hitting Point for Noncommutative Formulas", "title_html": "One Rational Matrix Hitting Point for Noncommutative Formulas", "url": "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", "role": "companion" }, { "title": "Polynomial Hitting Lists for Noncommutative Rational Formulas", "title_html": "Polynomial Hitting Lists for Noncommutative Rational Formulas", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/116.md" }, "peer_review": "not yet", "episode": { "video_id": "gj-5jpvtLj8", "length": "6:43", "title": "One point for all?", "page": "/episodes/116-noncommutative-identity-testing-hitting-points/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/116-noncommutative-identity-testing-hitting-points/" }, { "num": "117", "slug": "117-uniform-sparsest-cut-hardness-gaps", "title": "Uniform sparsest cut: hardness and semidefinite gaps", "title_html": "Uniform sparsest cut: hardness and semidefinite gaps", "summary_html": "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 log⁡n/(log⁡log⁡n)3\\sqrt{\\log n}/(\\log\\log n)^3logn​/(loglogn)3, approaching the square-root-logarithmic upper bound.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Constant-factor hardness of uniform sparsest cut", "title_html": "Constant-factor hardness of uniform sparsest cut", "url": "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", "role": "main" }, { "title": "Near-square-root logarithmic integrality gaps for uniform sparsest cut", "title_html": "Near-square-root logarithmic integrality gaps for uniform sparsest cut", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/117.md" }, "peer_review": "not yet", "episode": { "video_id": "ZA3oINpCAYk", "length": "6:21", "title": "Hard to approximate?", "page": "/episodes/117-uniform-sparsest-cut-hardness-gaps/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/117-uniform-sparsest-cut-hardness-gaps/" }, { "num": "118", "slug": "118-bin-packing-configuration-gap", "title": "Bin packing and unbounded configuration-LP gaps", "title_html": "Bin packing and unbounded configuration-LP gaps", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Additive hardness and unbounded configuration gaps in bin packing", "title_html": "Additive hardness and unbounded configuration gaps in bin packing", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/118.md" }, "peer_review": "not yet", "episode": { "video_id": "OGiTBHInawc", "length": "5:54", "title": "One more bin?", "page": "/episodes/118-bin-packing-configuration-gap/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/118-bin-packing-configuration-gap/" }, { "num": "119", "slug": "119-courtade-kumar-hellinger-conjectures", "title": "The Courtade–Kumar and Hellinger conjectures", "title_html": "The Courtade–Kumar and Hellinger conjectures", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Sharp binary-information contraction on the discrete cube", "title_html": "Sharp binary-information contraction on the discrete cube", "url": "https://github.com/openai/math/blob/main/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/main.pdf", "role": "main" }, { "title": "Hellinger contraction with arbitrary Boolean output bias", "title_html": "Hellinger contraction with arbitrary Boolean output bias", "url": "https://github.com/openai/math/blob/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/119.md" }, "peer_review": "not yet", "episode": { "video_id": "ADPWc1l79jA", "length": "5:50", "title": "The best bit?", "page": "/episodes/119-courtade-kumar-hellinger-conjectures/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/119-courtade-kumar-hellinger-conjectures/" }, { "num": "120", "slug": "120-almost-linear-general-matching", "title": "Almost-linear-time exact matching and prescribed-degree factors in general graphs", "title_html": "Almost-linear-time exact matching and prescribed-degree factors in general graphs", "summary_html": "Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in (n+m)1+o(1)(n+m)^{1+o(1)}(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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Almost-Linear-Time Maximum-Cardinality Matching in General Graphs", "title_html": "Almost-Linear-Time Maximum-Cardinality Matching in General Graphs", "url": "https://github.com/openai/math/blob/main/preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/main.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "pKmfuQIeQKY", "length": "5:23", "title": "Almost-linear matching?", "page": "/episodes/120-almost-linear-general-matching/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/120-almost-linear-general-matching/" }, { "num": "121", "slug": "121-edit-distance-approximation-scheme", "title": "Almost-linear approximation of edit distance", "title_html": "Almost-linear approximation of edit distance", "summary_html": "For every fixed rational ε∈(0,1)\\varepsilon\\in(0,1)ε∈(0,1), gives a randomized (1+ε)(1+\\varepsilon)(1+ε) approximation to unit-cost edit distance in worst-case expected time N1+o(1)N^{1+o(1)}N1+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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "An Almost-Linear Approximation Scheme for Edit Distance", "title_html": "An Almost-Linear Approximation Scheme for Edit Distance", "url": "https://github.com/openai/math/blob/main/preprints/An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/121.md" }, "peer_review": "not yet", "episode": { "video_id": "5_JiduV0JCM", "length": "7:32", "title": "Almost-linear edit distance?", "page": "/episodes/121-edit-distance-approximation-scheme/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/121-edit-distance-approximation-scheme/" }, { "num": "122", "slug": "122-trace-reconstruction-bounds", "title": "Quantitative trace-reconstruction bounds with a uniform decoder", "title_html": "Quantitative trace-reconstruction bounds with a uniform decoder", "summary_html": "At every fixed deletion probability in (0,1)(0,1)(0,1), reconstructing an arbitrary length-n binary string requires nΩ(log⁡log⁡n)n^{\\Omega(\\log\\log n)}nΩ(loglogn) 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−εn^{-\\varepsilon}n−ε for fixed ε > 0, both bounds become polynomial in the input and parameter encoding.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Uniform quasipolynomial-time trace reconstruction", "title_html": "Uniform quasipolynomial-time trace reconstruction", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026/uniform-trace-reconstruction.pdf", "role": "main" }, { "title": "A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction", "title_html": "A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction", "url": "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", "role": "companion" }, { "title": "Quantitative lower bounds for trace reconstruction", "title_html": "Quantitative lower bounds for trace reconstruction", "url": "https://github.com/openai/math/blob/main/preprints/quantitative-lower-bounds-for-trace-reconstruction-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/122.md" }, "peer_review": "not yet", "episode": { "video_id": "7hM03aW3xr8", "length": "7:06", "title": "Polynomially many traces?", "page": "/episodes/122-trace-reconstruction-bounds/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/122-trace-reconstruction-bounds/" }, { "num": "124", "slug": "124-three-machine-scheduling", "title": "Polynomial-time scheduling on three identical machines", "title_html": "Polynomial-time scheduling on three identical machines", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling", "title_html": "A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling", "url": "https://github.com/openai/math/blob/main/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/124.md" }, "peer_review": "not yet", "episode": { "video_id": "JASg8mT7_0Q", "length": "6:59", "title": "Three machines, polynomial?", "page": "/episodes/124-three-machine-scheduling/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/124-three-machine-scheduling/" }, { "num": "125", "slug": "125-k-median-approximation-threshold", "title": "The metric k-median approximation threshold and recovery", "title_html": "The metric k-median approximation threshold and recovery", "summary_html": "Gives a deterministic polynomial-time (1+2/e+ε)(1+2/e+\\varepsilon)(1+2/e+ε)-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming P≠NPP\\ne NPP=NP, the optimal infimum approximation factor is 1+2/e1+2/e1+2/e.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Single-exponential recovery and bounded-price strictness for metric k-median", "title_html": "Single-exponential recovery and bounded-price strictness for metric k-median", "url": "https://github.com/openai/math/blob/main/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/paper.pdf", "role": "main" }, { "title": "The approximation threshold for metric k-median", "title_html": "The approximation threshold for metric k-median", "url": "https://github.com/openai/math/blob/main/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/125.md" }, "peer_review": "not yet", "episode": { "video_id": "FMMyoMWUSsY", "length": "7:23", "title": "k-median at 1 + 2/e?", "page": "/episodes/125-k-median-approximation-threshold/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/125-k-median-approximation-threshold/" }, { "num": "126", "slug": "126-matching-psd-rank", "title": "Exponential semidefinite complexity of perfect matching", "title_html": "Exponential semidefinite complexity of perfect matching", "summary_html": "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<ρ<10\\lt \\rho\\lt 10<ρ<1, allowing arbitrary real positive semidefinite factors.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Exponential PSD rank of positively shifted matching matrices", "title_html": "Exponential PSD rank of positively shifted matching matrices", "url": "https://github.com/openai/math/blob/main/preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026/shifted-matching-psd.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/126.md" }, "peer_review": "not yet", "episode": { "video_id": "Le0S3YvMJHE", "length": "7:12", "title": "No small SDP for matching?", "page": "/episodes/126-matching-psd-rank/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/126-matching-psd-rank/" }, { "num": "127", "slug": "127-ptf-average-sensitivity", "title": "Average sensitivity of polynomial threshold functions", "title_html": "Average sensitivity of polynomial threshold functions", "summary_html": "Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8dn8d\\sqrt n8dn​, uniformly for 1≤d≤n1\\le d\\le n1≤d≤n. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with sign(0)=1\\mathop{\\mathrm{sign}}\\nolimits (0)=1sign(0)=1.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Average sensitivity of polynomial threshold functions", "title_html": "Average sensitivity of polynomial threshold functions", "url": "https://github.com/openai/math/blob/main/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/127.md" }, "peer_review": "not yet", "episode": { "video_id": "H1b_SN290uA", "length": "7:10", "title": "How sensitive can it be?", "page": "/episodes/127-ptf-average-sensitivity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/127-ptf-average-sensitivity/" }, { "num": "128", "slug": "128-superstring-factor-two", "title": "A factor-two approximation for shortest common superstring", "title_html": "A factor-two approximation for shortest common superstring", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "A Polynomial-Time 2-Approximation for Shortest Common Superstring", "title_html": "A Polynomial-Time 2-Approximation for Shortest Common Superstring", "url": "https://github.com/openai/math/blob/main/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/128.md" }, "peer_review": "not yet", "episode": { "video_id": "idys9cakGP8", "length": "7:19", "title": "Factor two, at last?", "page": "/episodes/128-superstring-factor-two/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/128-superstring-factor-two/" }, { "num": "129", "slug": "129-two-way-automata-state-costs", "title": "Exponential state costs for two-way automata", "title_html": "Exponential state costs for two-way automata", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "An exponential state lower bound for two-way nondeterministic complementation", "title_html": "An exponential state lower bound for two-way nondeterministic complementation", "url": "https://github.com/openai/math/blob/main/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026/paper.pdf", "role": "main" }, { "title": "An exponential two-way deterministic state lower bound for one-way liveness", "title_html": "An exponential two-way deterministic state lower bound for one-way liveness", "url": "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", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/129.md" }, "peer_review": "not yet", "episode": { "video_id": "-yIIX5OA9Wg", "length": "7:41", "title": "Sakoda–Sipser, settled?", "page": "/episodes/129-two-way-automata-state-costs/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/129-two-way-automata-state-costs/" }, { "num": "130", "slug": "130-fourier-below-n-log-n", "title": "Exact Fourier transforms below n log n", "title_html": "Exact Fourier transforms below nlog⁡nn\\log nnlogn", "summary_html": "Gives a deterministic length-n discrete Fourier transform algorithm using O(n(log⁡n)1−δ)O(n(\\log n)^{1-\\delta})O(n(logn)1−δ) operations for every n, with explicit δ=10−13\\delta=10^{-13}δ=10−13. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "An explicit power saving for the exact discrete Fourier transform", "title_html": "An explicit power saving for the exact discrete Fourier transform", "url": "https://github.com/openai/math/blob/main/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf", "role": "main" }, { "title": "Finite tensor savings and exact Fourier circuits", "title_html": "Finite tensor savings and exact Fourier circuits", "url": "https://github.com/openai/math/blob/main/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/130.md" }, "peer_review": "not yet", "episode": { "video_id": "vfDKQcp1iJE", "length": "7:20", "title": "Faster than the FFT?", "page": "/episodes/130-fourier-below-n-log-n/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/130-fourier-below-n-log-n/" }, { "num": "131", "slug": "131-switch-chain-mixing", "title": "Rapid mixing of graph switches for every degree sequence", "title_html": "Rapid mixing of graph switches for every degree sequence", "summary_html": "Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in O(n8)O(n^8)O(n8) 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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Polynomial mixing of the switch chain for every graphical degree sequence", "title_html": "Polynomial mixing of the switch chain for every graphical degree sequence", "url": "https://github.com/openai/math/blob/main/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/131.md" }, "peer_review": "not yet", "episode": { "video_id": "RoDAcOwvpwY", "length": "6:23", "title": "Switch chain mixes fast?", "page": "/episodes/131-switch-chain-mixing/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/131-switch-chain-mixing/" }, { "num": "132", "slug": "132-sensitivity-block-sensitivity", "title": "A superquadratic separation of sensitivity and block sensitivity", "title_html": "A superquadratic separation of sensitivity and block sensitivity", "summary_html": "Constructs total Boolean functions with block sensitivity bs(f)≥s(f)α\\mathop{\\mathrm{bs}}\\nolimits (f)\\ge s(f)^\\alphabs(f)≥s(f)α for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here s(f)s(f)s(f) counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "A superquadratic separation between sensitivity and block sensitivity", "title_html": "A superquadratic separation between sensitivity and block sensitivity", "url": "https://github.com/openai/math/blob/main/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/132.md" }, "peer_review": "not yet", "episode": { "video_id": "SHQKwja61Ec", "length": "6:26", "title": "Beats sensitivity squared?", "page": "/episodes/132-sensitivity-block-sensitivity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/132-sensitivity-block-sensitivity/" }, { "num": "133", "slug": "133-weisfeiler-leman-complexity", "title": "The computational complexity of Weisfeiler–Leman refinement", "title_html": "The computational complexity of Weisfeiler–Leman refinement", "summary_html": "Proves unconditional nΩ(k)n^{\\Omega(k)}nΩ(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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence", "title_html": "Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence", "url": "https://github.com/openai/math/blob/main/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf", "role": "main" }, { "title": "The complexity of identifying a graph by Weisfeiler–Leman refinement", "title_html": "The complexity of identifying a graph by Weisfeiler–Leman refinement", "url": "https://github.com/openai/math/blob/main/preprints/The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026/paper.pdf", "role": "companion" }, { "title": "Unconditional time lower bounds for Weisfeiler–Leman equivalence", "title_html": "Unconditional time lower bounds for Weisfeiler–Leman equivalence", "url": "https://github.com/openai/math/blob/main/preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf", "role": "companion" }, { "title": "Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs", "title_html": "Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs", "url": "https://github.com/openai/math/blob/main/preprints/Variable-dimension-Weisfeiler-Leman-equivalence-on-general-and-subcubic-graphs-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/133.md" }, "peer_review": "not yet", "episode": { "video_id": "7EwZ33R3pUY", "length": "8:05", "title": "How hard is WL?", "page": "/episodes/133-weisfeiler-leman-complexity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/133-weisfeiler-leman-complexity/" }, { "num": "134", "slug": "134-generalized-star-height", "title": "Generalized star height at most three", "title_html": "Generalized star height at most three", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Finite Monoid Computations and a Uniform Generalized Star-Height Bound", "title_html": "Finite Monoid Computations and a Uniform Generalized Star-Height Bound", "url": "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", "role": "main" }, { "title": "Generalized Star Height at Most Four", "title_html": "Generalized Star Height at Most Four", "url": "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", "role": "companion" }, { "title": "Generalized Star Height at Most Three", "title_html": "Generalized Star Height at Most Three", "url": "https://github.com/openai/math/blob/main/preprints/Generalized-Star-Height-at-Most-Three-September-25-2026/article.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/134.md" }, "peer_review": "not yet", "episode": { "video_id": "mOSfH8IsnWc", "length": "6:15", "title": "Three stars suffice?", "page": "/episodes/134-generalized-star-height/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/134-generalized-star-height/" }, { "num": "135", "slug": "135-depth-five-imm", "title": "Homogeneous depth-five lower bounds for iterated matrix multiplication", "title_html": "Homogeneous depth-five lower bounds for iterated matrix multiplication", "summary_html": "Over every characteristic-zero field, the (1,1)(1,1)(1,1) entry of a product of n independent n×nn\\times nn×n variable matrices requires nΘ(n)n^{\\Theta(\\sqrt n)}nΘ(n​) gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Homogeneous depth-five lower bounds for iterated matrix multiplication", "title_html": "Homogeneous depth-five lower bounds for iterated matrix multiplication", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/135.md" }, "peer_review": "not yet", "episode": { "video_id": "Qyi5h3TKMHw", "length": "7:12", "title": "Depth five, sharply?", "page": "/episodes/135-depth-five-imm/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/135-depth-five-imm/" }, { "num": "136", "slug": "136-pcp-for-ppad", "title": "A quasilinear PCP theorem for PPAD", "title_html": "A quasilinear PCP theorem for PPAD", "summary_html": "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)N(\\log N)^{O(1)}N(logN)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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "The PCP-for-PPAD conjecture: a quasilinear reduction", "title_html": "The PCP-for-PPAD conjecture: a quasilinear reduction", "url": "https://github.com/openai/math/blob/main/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "eQ2OKa4LGHQ", "length": "6:28", "title": "PCP for PPAD?", "page": "/episodes/136-pcp-for-ppad/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/136-pcp-for-ppad/" }, { "num": "137", "slug": "137-one-tape-two-fifths", "title": "One-tape time simulation in two-fifths-power space", "title_html": "One-tape time simulation in two-fifths-power space", "summary_html": "Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using O(T2/5log⁡C(T+2))O(T^{2/5}\\log^C(T+2))O(T2/5logC(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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Simulating One-Tape Time in Two-Fifths-Power Space", "title_html": "Simulating One-Tape Time in Two-Fifths-Power Space", "url": "https://github.com/openai/math/blob/main/preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026/article.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/137.md" }, "peer_review": "not yet", "episode": { "video_id": "RGnjcnauWG8", "length": "6:27", "title": "Below the square root?", "page": "/episodes/137-one-tape-two-fifths/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/137-one-tape-two-fifths/" }, { "num": "138", "slug": "138-subset-sum-049", "title": "Subset Sum in O(2^(0.49n)) time", "title_html": "Subset Sum in O(20.49n)O(2^{0.49n})O(20.49n) time", "summary_html": "Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary O(20.49n)O(2^{0.49n})O(20.49n) 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)O(n+b)O(n+b) bits for maximum input bit length b.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Subset Sum in Time O(2^(0.49n))", "title_html": "Subset Sum in Time O(20.49n)O(2^{0.49n})O(20.49n)", "url": "https://github.com/openai/math/blob/main/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/subset-sum.pdf", "role": "main" }, { "title": "A Low-Space Algorithm for Worst-Case Subset Sum", "title_html": "A Low-Space Algorithm for Worst-Case Subset Sum", "url": "https://github.com/openai/math/blob/main/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "9GHHsOt1MuU", "length": "6:33", "title": "Subset Sum in 2^0.49n?", "page": "/episodes/138-subset-sum-049/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/138-subset-sum-049/" }, { "num": "139", "slug": "139-log-concave-queries", "title": "Subpolynomial query complexity for log-concave sampling", "title_html": "Subpolynomial query complexity for log-concave sampling", "summary_html": "For C2 potentials with a supplied minimizer and I⪯∇2V⪯2II\\preceq\\nabla^2V\\preceq2II⪯∇2V⪯2I, proves that sampling within total variation 1/10 requires only CεdεC_\\varepsilon d^\\varepsilonCε​dε exact value-and-gradient queries for every fixed ε > 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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Subpolynomial query complexity for well-conditioned log-concave sampling", "title_html": "Subpolynomial query complexity for well-conditioned log-concave sampling", "url": "https://github.com/openai/math/blob/main/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026/article.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/139.md" }, "peer_review": "not yet", "episode": { "video_id": "OIAxry00MGE", "length": "6:25", "title": "Sampling with almost no queries?", "page": "/episodes/139-log-concave-queries/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/139-log-concave-queries/" }, { "num": "140", "slug": "140-memory-regression", "title": "Memory–sample lower bounds for noiseless Gaussian regression", "title_html": "Memory–sample lower bounds for noiseless Gaussian regression", "summary_html": "For fixed A > 0, a one-pass learner with Ad2Ad^2Ad2 persistent bits needs ΩA(dlog⁡(1/ϵ))\\Omega_A(d\\log(1/\\epsilon))ΩA​(dlog(1/ϵ)) noiseless Gaussian samples to recover a unit vector to angular error 0<ϵ≤1/100\\lt \\epsilon\\le1/100<ϵ≤1/10 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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Memory and precision in noiseless Gaussian regression", "title_html": "Memory and precision in noiseless Gaussian regression", "url": "https://github.com/openai/math/blob/main/preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf", "role": "main" }, { "title": "Posterior replicas and conditional information in Gaussian regression", "title_html": "Posterior replicas and conditional information in Gaussian regression", "url": "https://github.com/openai/math/blob/main/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Localization costs and information growth for exact Gaussian observations", "title_html": "Localization costs and information growth for exact Gaussian observations", "url": "https://github.com/openai/math/blob/main/preprints/Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Projection moments, positive cap domination, and Riesz estimates on the sphere", "title_html": "Projection moments, positive cap domination, and Riesz estimates on the sphere", "url": "https://github.com/openai/math/blob/main/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Replacing Gaussian observations in memory-constrained inference", "title_html": "Replacing Gaussian observations in memory-constrained inference", "url": "https://github.com/openai/math/blob/main/preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression", "title_html": "Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression", "url": "https://github.com/openai/math/blob/main/preprints/Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/140.md" }, "peer_review": "not yet", "episode": { "video_id": "nzul2YJL4lU", "length": "6:13", "title": "Exact equations, limited memory", "page": "/episodes/140-memory-regression/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/140-memory-regression/" }, { "num": "141", "slug": "141-etr-counting-hierarchy", "title": "Existential–universal real sentences in the counting hierarchy", "title_html": "Existential–universal real sentences in the counting hierarchy", "summary_html": "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.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Existential–universal real sentences in the counting hierarchy", "title_html": "Existential–universal real sentences in the counting hierarchy", "url": "https://github.com/openai/math/blob/main/preprints/Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026/etr-counting-hierarchy.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "0TOFm_8Oi3k", "length": "5:56", "title": "Real solutions at level 26?", "page": "/episodes/141-etr-counting-hierarchy/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/141-etr-counting-hierarchy/" }, { "num": "142", "slug": "142-deterministic-factoring", "title": "Deterministic polynomial factorization over prime fields", "title_html": "Deterministic polynomial factorization over prime fields", "summary_html": "Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽p, including multiplicities, in bit complexity polynomial in (n+1)log⁡p(n+1)\\log p(n+1)logp. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.", "subject": "Theoretical computer science", "subject_slug": "theoretical-computer-science", "papers": [ { "title": "Deterministic Polynomial Factorization over Prime Fields", "title_html": "Deterministic Polynomial Factorization over Prime Fields", "url": "https://github.com/openai/math/blob/main/preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026/Deterministic-Polynomial-Factorization-over-Prime-Fields.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "TLa6LepzP4s", "length": "7:33", "title": "Factoring with no coin flips?", "page": "/episodes/142-deterministic-factoring/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/142-deterministic-factoring/" }, { "num": "143", "slug": "143-hilbert-sixteenth-uniform", "title": "Hilbert's sixteenth problem: uniform bounds for limit cycles", "title_html": "Hilbert's sixteenth problem: uniform bounds for limit cycles", "summary_html": "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.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "Uniform bounds for planar polynomial limit cycles", "title_html": "Uniform bounds for planar polynomial limit cycles", "url": "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", "role": "main" }, { "title": "Two limit cycles for quintic Liénard systems", "title_html": "Two limit cycles for quintic Liénard systems", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/143.md" }, "peer_review": "not yet", "episode": { "video_id": "5MmLX1A3K00", "length": "7:05", "title": "A ceiling on limit cycles?", "page": "/episodes/143-hilbert-sixteenth-uniform/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/143-hilbert-sixteenth-uniform/" }, { "num": "144", "slug": "144-simple-lebesgue-spectrum", "title": "Banach’s simple Lebesgue-spectrum problem", "title_html": "Banach’s simple Lebesgue-spectrum problem", "summary_html": "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 L2 space: the bilateral iterates of one real observable form an orthonormal basis of that space.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "A smooth three-torus diffeomorphism with simple Lebesgue spectrum", "title_html": "A smooth three-torus diffeomorphism with simple Lebesgue spectrum", "url": "https://github.com/openai/math/blob/main/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/144.md" }, "peer_review": "not yet", "episode": { "video_id": "RjLXNr-7QK4", "length": "6:19", "title": "One signal, every pattern?", "page": "/episodes/144-simple-lebesgue-spectrum/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/144-simple-lebesgue-spectrum/" }, { "num": "145", "slug": "145-rokhlin-multiple-mixing", "title": "Rokhlin’s multiple-mixing problem", "title_html": "Rokhlin’s multiple-mixing problem", "summary_html": "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.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "Rokhlin's multiple-mixing problem for one transformation", "title_html": "Rokhlin's multiple-mixing problem for one transformation", "url": "https://github.com/openai/math/blob/main/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/145.md" }, "peer_review": "not yet", "episode": { "video_id": "r2NRYGV_xdg", "length": "6:28", "title": "Two is enough?", "page": "/episodes/145-rokhlin-multiple-mixing/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/145-rokhlin-multiple-mixing/" }, { "num": "146", "slug": "146-standard-map-entropy", "title": "Positive metric entropy for the standard map", "title_html": "Positive metric entropy for the standard map", "summary_html": "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.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "Positive Metric Entropy for the Standard Map at Large Parameters", "title_html": "Positive Metric Entropy for the Standard Map at Large Parameters", "url": "https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/146.md" }, "peer_review": "not yet", "episode": { "video_id": "Xjl7Y4sbS6s", "length": "6:16", "title": "Chaos with positive area?", "page": "/episodes/146-standard-map-entropy/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/146-standard-map-entropy/" }, { "num": "147", "slug": "147-near-boundary-birkhoff", "title": "The near-boundary Birkhoff conjecture", "title_html": "The near-boundary Birkhoff conjecture", "summary_html": "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.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "Continuous Phase Foliations Create Analytic Caustic Collars", "title_html": "Continuous Phase Foliations Create Analytic Caustic Collars", "url": "https://github.com/openai/math/blob/main/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf", "role": "main" }, { "title": "Rigidity of Smooth Billiards with a Continuous Caustic Collar", "title_html": "Rigidity of Smooth Billiards with a Continuous Caustic Collar", "url": "https://github.com/openai/math/blob/main/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "nDaWmfmlHMQ", "length": "6:32", "title": "Only the ellipse?", "page": "/episodes/147-near-boundary-birkhoff/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/147-near-boundary-birkhoff/" }, { "num": "148", "slug": "148-entropy-rate-dimension", "title": "The entropy-rate dimension formula for self-similar measures", "title_html": "The entropy-rate dimension formula for self-similar measures", "summary_html": "For every self-similar measure on the line generated by finitely many contracting similarities, proves dim⁡Hμ=min⁡{1,hRW/χ}\\dim_{\\mathrm H}\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}dimH​μ=min{1,hRW​/χ}, where hRWh_{\\mathrm{RW}}hRW​ 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.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "The entropy-rate dimension formula for self-similar measures on the line", "title_html": "The entropy-rate dimension formula for self-similar measures on the line", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/148.md" }, "peer_review": "not yet", "episode": { "video_id": "D--kSwVmbv4", "length": "6:23", "title": "Lost only to coincidences?", "page": "/episodes/148-entropy-rate-dimension/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/148-entropy-rate-dimension/" }, { "num": "149", "slug": "149-mass-action-permanence", "title": "Classwise permanence for weakly reversible mass-action systems", "title_html": "Classwise permanence for weakly reversible mass-action systems", "summary_html": "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.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "Uniform Permanence in Weakly Reversible Mass-Action Systems", "title_html": "Uniform Permanence in Weakly Reversible Mass-Action Systems", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf", "role": "main" }, { "title": "Boundedness and persistence of weakly reversible mass-action systems", "title_html": "Boundedness and persistence of weakly reversible mass-action systems", "url": "https://github.com/openai/math/blob/main/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/149.md" }, "peer_review": "not yet", "episode": { "video_id": "jSFVWepLHEk", "length": "6:38", "title": "One trap for every start?", "page": "/episodes/149-mass-action-permanence/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/149-mass-action-permanence/" }, { "num": "150", "slug": "150-triangle-weak-mixing", "title": "Weak mixing of triangular billiards with an irrational angle", "title_html": "Weak mixing of triangular billiards with an irrational angle", "summary_html": "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.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "Weak mixing of triangular billiards with an irrational angle", "title_html": "Weak mixing of triangular billiards with an irrational angle", "url": "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", "role": "main" }, { "title": "Ergodicity of triangular billiards with an irrational angle", "title_html": "Ergodicity of triangular billiards with an irrational angle", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/150.md" }, "peer_review": "not yet", "episode": { "video_id": "kUBNzzQOlb8", "length": "6:50", "title": "No hidden rhythm?", "page": "/episodes/150-triangle-weak-mixing/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/150-triangle-weak-mixing/" }, { "num": "151", "slug": "151-entropy-conjecture-c1", "title": "A C¹ counterexample to the entropy conjecture", "title_html": "A C1 counterexample to the entropy conjecture", "summary_html": "Constructs a noninvertible C1 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 C1 self-maps: homological growth need not force positive orbit complexity.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "A C^1 Counterexample to the Entropy Conjecture", "title_html": "A C^1 Counterexample to the Entropy Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/151.md" }, "peer_review": "not yet", "episode": { "video_id": "wNMcxS9iRSs", "length": "7:24", "title": "Wrapping without chaos?", "page": "/episodes/151-entropy-conjecture-c1/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/151-entropy-conjecture-c1/" }, { "num": "152", "slug": "152-smooth-volume-model", "title": "Zero entropy does not guarantee a smooth positive-volume model", "title_html": "Zero entropy does not guarantee a smooth positive-volume model", "summary_html": "Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "A zero-entropy system without a smooth positive-volume model", "title_html": "A zero-entropy system without a smooth positive-volume model", "url": "https://github.com/openai/math/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/152.md" }, "peer_review": "not yet", "episode": { "video_id": "bLG1Vtdtpg0", "length": "7:19", "title": "No smooth volume model?", "page": "/episodes/152-smooth-volume-model/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/152-smooth-volume-model/" }, { "num": "153", "slug": "153-bernoulli-convolutions", "title": "Arithmetic classification and non-Pisot singularity for Bernoulli convolutions", "title_html": "Arithmetic classification and non-Pisot singularity for Bernoulli convolutions", "summary_html": "Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every λ∈(0,1)\\lambda\\in(0,1)λ∈(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)(1,2)(1,2), giving examples beyond reciprocal Pisot parameters.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "Arithmetic classification and non-Pisot singularity for Bernoulli convolutions", "title_html": "Arithmetic classification and non-Pisot singularity for Bernoulli convolutions", "url": "https://github.com/openai/math/blob/main/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "zscuflGt-O0", "length": "6:44", "title": "Singular beyond Pisot?", "page": "/episodes/153-bernoulli-convolutions/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/153-bernoulli-convolutions/" }, { "num": "154", "slug": "154-pointwise-multiple-ergodic-mixing", "title": "Pointwise multiple ergodic averages for mixing transformations", "title_html": "Pointwise multiple ergodic averages for mixing transformations", "summary_html": "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.", "subject": "Dynamical systems and ergodic theory", "subject_slug": "dynamical-systems-and-ergodic-theory", "papers": [ { "title": "Pointwise Multiple Ergodic Averages for Mixing Transformations", "title_html": "Pointwise Multiple Ergodic Averages for Mixing Transformations", "url": "https://github.com/openai/math/blob/main/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf", "role": "main" }, { "title": "Pointwise convergence of fourfold ergodic averages for mixing transformations", "title_html": "Pointwise convergence of fourfold ergodic averages for mixing transformations", "url": "https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf", "role": "companion" }, { "title": "Triple ergodic averages with distinct integer slopes", "title_html": "Triple ergodic averages with distinct integer slopes", "url": "https://github.com/openai/math/blob/main/preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf", "role": "companion" }, { "title": "Pointwise convergence of triple ergodic averages for mixing transformations", "title_html": "Pointwise convergence of triple ergodic averages for mixing transformations", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "oWgHSt_-QWQ", "length": "7:02", "title": "Mixing is enough?", "page": "/episodes/154-pointwise-multiple-ergodic-mixing/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/154-pointwise-multiple-ergodic-mixing/" }, { "num": "155", "slug": "155-periodic-tiling-dimension-three", "title": "A counterexample to periodic tiling in dimension three", "title_html": "A counterexample to periodic tiling in dimension three", "summary_html": "Constructs a finite translational tile in ℤ3 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 ℝ3, even with arbitrary real translation vectors.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A translational tile with no fully periodic tiling in dimension three", "title_html": "A translational tile with no fully periodic tiling in dimension three", "url": "https://github.com/openai/math/blob/main/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/155.md" }, "peer_review": "not yet", "episode": { "video_id": "NF_gRk0f6sc", "length": "6:54", "title": "No periodic tiling in 3D?", "page": "/episodes/155-periodic-tiling-dimension-three/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/155-periodic-tiling-dimension-three/" }, { "num": "156", "slug": "156-borsuk-dimension-nine", "title": "Borsuk's conjecture fails in dimension nine", "title_html": "Borsuk's conjecture fails in dimension nine", "summary_html": "Constructs a compact subset of ℝ9 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 ℝ4, with the Frobenius metric.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A nine-dimensional counterexample to Borsuk's covering assertion", "title_html": "A nine-dimensional counterexample to Borsuk's covering assertion", "url": "https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/156.md" }, "peer_review": "not yet", "episode": { "video_id": "3eUkQ8BCmfI", "length": "6:00", "title": "Borsuk fails in nine?", "page": "/episodes/156-borsuk-dimension-nine/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/156-borsuk-dimension-nine/" }, { "num": "157", "slug": "157-hadwiger-counterexample", "title": "Graph coloring, clique minors, and Colin de Verdière invariants", "title_html": "Graph coloring, clique minors, and Colin de Verdière invariants", "summary_html": "Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy χf(G)>h(G)\\chi_f(G)\\gt h(G)χf​(G)>h(G), where h(G)h(G)h(G) is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound χf(G)≤μ(G)+1\\chi_f(G)\\le\\mu(G)+1χf​(G)≤μ(G)+1. In the positive direction, every finite nonempty graph satisfies χlist(G)≤Ch(G)\\chi_{\\mathrm{list}}(G)\\le C h(G)χlist​(G)≤Ch(G) for a universal constant C.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A counterexample to Hadwiger's conjecture", "title_html": "A counterexample to Hadwiger's conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf", "role": "main" }, { "title": "A counterexample to the Colin de Verdière chromatic conjecture", "title_html": "A counterexample to the Colin de Verdière chromatic conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf", "role": "companion" }, { "title": "A linear list-coloring bound in terms of the Hadwiger number", "title_html": "A linear list-coloring bound in terms of the Hadwiger number", "url": "https://github.com/openai/math/blob/main/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/157.md" }, "peer_review": "not yet", "episode": { "video_id": "lXRqaXRev34", "length": "6:47", "title": "Hadwiger disproved?", "page": "/episodes/157-hadwiger-counterexample/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/157-hadwiger-counterexample/" }, { "num": "158", "slug": "158-plane-not-five-colorable", "title": "The Euclidean plane cannot be colored with five colors", "title_html": "The Euclidean plane cannot be colored with five colors", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The Euclidean plane is not five-colorable", "title_html": "The Euclidean plane is not five-colorable", "url": "https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/158.md" }, "peer_review": "not yet", "episode": { "video_id": "QEtjS2zI6l4", "length": "6:33", "title": "Five colours never suffice?", "page": "/episodes/158-plane-not-five-colorable/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/158-plane-not-five-colorable/" }, { "num": "159", "slug": "159-erdos-reciprocal-sum-conjecture", "title": "Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds", "title_html": "Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds", "summary_html": "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,…,N}\\{1,\\ldots,N\\}{1,…,N} with no nonconstant k-term progression has size at most CkNexp⁡[−ck(log⁡N)εk]C_kN\\exp[-c_k(\\log N)^{\\varepsilon_k}]Ck​Nexp[−ck​(logN)εk​], with positive constants depending only on k.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Quasipolynomial Bounds for Arithmetic Progressions", "title_html": "Quasipolynomial Bounds for Arithmetic Progressions", "url": "https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/159.md" }, "peer_review": "not yet", "episode": { "video_id": "z0D3NOHfKD8", "length": "6:10", "title": "Erdős was right?", "page": "/episodes/159-erdos-reciprocal-sum-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/159-erdos-reciprocal-sum-conjecture/" }, { "num": "160", "slug": "160-superexponential-van-der-waerden-numbers", "title": "Superexponential van der Waerden numbers", "title_html": "Superexponential van der Waerden numbers", "summary_html": "Resolves Erdős's superexponential-growth question for van der Waerden numbers. If Wr(k)W_r(k)Wr​(k) is the least interval length forcing a monochromatic k-term progression in every r-coloring, then Wr(k)>kck⌊log⁡2r⌋W_r(k)\\gt k^{ck\\lfloor\\log_2 r\\rfloor}Wr​(k)>kck⌊log2​r⌋ for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, Wr(k)1/k→∞W_r(k)^{1/k}\\to\\inftyWr​(k)1/k→∞ for each fixed r.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Quantitative Superexponential Bounds for van der Waerden Numbers", "title_html": "Quantitative Superexponential Bounds for van der Waerden Numbers", "url": "https://github.com/openai/math/blob/main/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/160.md" }, "peer_review": "not yet", "episode": { "video_id": "xN-7WKOe18k", "length": "6:28", "title": "Faster than exponential?", "page": "/episodes/160-superexponential-van-der-waerden-numbers/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/160-superexponential-van-der-waerden-numbers/" }, { "num": "161", "slug": "161-counterexamples-sidorenkos-conjecture-forcing-conjecture", "title": "Counterexamples to Sidorenko’s conjecture and the forcing conjecture", "title_html": "Counterexamples to Sidorenko’s conjecture and the forcing conjecture", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A counterexample to Sidorenko's conjecture", "title_html": "A counterexample to Sidorenko's conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/161.md" }, "peer_review": "not yet", "episode": { "video_id": "337pkor3T9Q", "length": "6:29", "title": "Sidorenko disproved?", "page": "/episodes/161-counterexamples-sidorenkos-conjecture-forcing-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/161-counterexamples-sidorenkos-conjecture-forcing-conjecture/" }, { "num": "162", "slug": "162-counterexamples-rysers-covering-conjecture", "title": "Counterexamples to Ryser’s covering conjecture", "title_html": "Counterexamples to Ryser’s covering conjecture", "summary_html": "Disproves Ryser's covering conjecture by constructing intersecting (q+1)(q+1)(q+1)-partite, (q+1)(q+1)(q+1)-uniform hypergraphs with covering number q+1q+1q+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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Balanced counterexamples to Ryser's conjecture at prime orders", "title_html": "Balanced counterexamples to Ryser's conjecture at prime orders", "url": "https://github.com/openai/math/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf", "role": "main" }, { "title": "A counterexample to Ryser's covering conjecture", "title_html": "A counterexample to Ryser's covering conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/162.md" }, "peer_review": "not yet", "episode": { "video_id": "sbxKzbbTVKY", "length": "6:31", "title": "Ryser disproved?", "page": "/episodes/162-counterexamples-rysers-covering-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/162-counterexamples-rysers-covering-conjecture/" }, { "num": "164", "slug": "164-hindmans-finite-sums-products-conjecture", "title": "Hindman’s finite sums and products conjecture", "title_html": "Hindman’s finite sums and products conjecture", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Monochromatic finite sums and products in the positive integers", "title_html": "Monochromatic finite sums and products in the positive integers", "url": "https://github.com/openai/math/blob/main/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "fyEi4_tzBcM", "length": "5:42", "title": "Hindman proved?", "page": "/episodes/164-hindmans-finite-sums-products-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/164-hindmans-finite-sums-products-conjecture/" }, { "num": "165", "slug": "165-harary-hill-zarankiewicz-crossing-number", "title": "The Harary–Hill and Zarankiewicz crossing-number formulas", "title_html": "The Harary–Hill and Zarankiewicz crossing-number formulas", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The crossing number of complete graphs", "title_html": "The crossing number of complete graphs", "url": "https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf", "role": "main" }, { "title": "The crossing number of complete bipartite graphs", "title_html": "The crossing number of complete bipartite graphs", "url": "https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/165.md" }, "peer_review": "not yet", "episode": { "video_id": "gV-UaIjJZDo", "length": "5:53", "title": "Brickyard solved?", "page": "/episodes/165-harary-hill-zarankiewicz-crossing-number/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/165-harary-hill-zarankiewicz-crossing-number/" }, { "num": "166", "slug": "166-higher-dimensional-erdos-distinct-distances", "title": "The higher-dimensional Erdős distinct-distances conjecture", "title_html": "The higher-dimensional Erdős distinct-distances conjecture", "summary_html": "For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least cdn2/dc_dn^{2/d}cd​n2/d distinct distances, with cd>0c_d\\gt 0cd​>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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The higher-dimensional Erdős distinct-distances conjecture", "title_html": "The higher-dimensional Erdős distinct-distances conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "ZgWkRw0YgJQ", "length": "5:48", "title": "Erdős distances in 3D and up?", "page": "/episodes/166-higher-dimensional-erdos-distinct-distances/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/166-higher-dimensional-erdos-distinct-distances/" }, { "num": "167", "slug": "167-planar-distinct-distances-unit-distance", "title": "Planar distinct distances and unit-distance bounds", "title_html": "Planar distinct distances and unit-distance bounds", "summary_html": "Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but o(n)o(n)o(n) points of any n-point planar set determine at least n1−εn^{1-\\varepsilon}n1−ε distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by O(n4/3−δ)O(n^{4/3-\\delta})O(n4/3−δ) for an absolute δ > 0.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The weak pinned planar distance theorem", "title_html": "The weak pinned planar distance theorem", "url": "https://github.com/openai/math/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf", "role": "main" }, { "title": "A power saving for planar unit distances", "title_html": "A power saving for planar unit distances", "url": "https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/167.md" }, "peer_review": "not yet", "episode": { "video_id": "h6JbEUgFbos", "length": "6:52", "title": "Erdős distances, pinned down?", "page": "/episodes/167-planar-distinct-distances-unit-distance/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/167-planar-distinct-distances-unit-distance/" }, { "num": "168", "slug": "168-combinatorial-invariance-kazhdan-lusztig-polynomials", "title": "Combinatorial invariance of Kazhdan–Lusztig polynomials", "title_html": "Combinatorial invariance of Kazhdan–Lusztig polynomials", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Combinatorial invariance of Kazhdan–Lusztig polynomials", "title_html": "Combinatorial invariance of Kazhdan–Lusztig polynomials", "url": "https://github.com/openai/math/blob/main/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/168.md" }, "peer_review": "not yet", "episode": { "video_id": "g7OT-Ob73QM", "length": "5:47", "title": "Shape is enough?", "page": "/episodes/168-combinatorial-invariance-kazhdan-lusztig-polynomials/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/168-combinatorial-invariance-kazhdan-lusztig-polynomials/" }, { "num": "169", "slug": "169-shareshian-wachs-elementary-positivity", "title": "Shareshian–Wachs elementary positivity", "title_html": "Shareshian–Wachs elementary positivity", "summary_html": "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 N[q]\\mathbb N[q]N[q]. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Elementary positivity of chromatic quasisymmetric functions", "title_html": "Elementary positivity of chromatic quasisymmetric functions", "url": "https://github.com/openai/math/blob/main/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/169.md" }, "peer_review": "not yet", "episode": { "video_id": "bPbuQI4mSyc", "length": "5:44", "title": "No minus signs?", "page": "/episodes/169-shareshian-wachs-elementary-positivity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/169-shareshian-wachs-elementary-positivity/" }, { "num": "170", "slug": "170-sharp-logarithmic-exponents-off-diagonal", "title": "Sharp logarithmic exponents for off-diagonal Ramsey numbers", "title_html": "Sharp logarithmic exponents for off-diagonal Ramsey numbers", "summary_html": "For every fixed integer s ≥ 5, proves r(s,t)=ts−1/(log⁡t)s−2+o(1)r(s,t)=t^{s-1}/(\\log t)^{s-2+o(1)}r(s,t)=ts−1/(logt)s−2+o(1) as t→∞t\\to\\inftyt→∞, determining the logarithmic exponent and matching the classical upper bound at that scale. Here r(s,t)r(s,t)r(s,t) is the least number of vertices forcing an s-clique or a t-vertex independent set.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The sharp logarithmic exponent of r(5,t)", "title_html": "The sharp logarithmic exponent of r(5,t)", "url": "https://github.com/openai/math/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf", "role": "main" }, { "title": "Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers", "title_html": "Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers", "url": "https://github.com/openai/math/blob/main/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/170.md" }, "peer_review": "not yet", "episode": { "video_id": "vsv_IpbWlAo", "length": "5:36", "title": "Ramsey numbers: log power settled?", "page": "/episodes/170-sharp-logarithmic-exponents-off-diagonal/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/170-sharp-logarithmic-exponents-off-diagonal/" }, { "num": "171", "slug": "171-hypercube-ramsey-conjecture", "title": "The hypercube Ramsey conjecture", "title_html": "The hypercube Ramsey conjecture", "summary_html": "Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is Θ(2n)\\Theta(2^n)Θ(2n). 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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The hypercube Ramsey number has linear order", "title_html": "The hypercube Ramsey number has linear order", "url": "https://github.com/openai/math/blob/main/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "nBKn2UrYgn0", "length": "5:49", "title": "Hypercube Ramsey: linear at last?", "page": "/episodes/171-hypercube-ramsey-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/171-hypercube-ramsey-conjecture/" }, { "num": "172", "slug": "172-classification-finite-euclidean-ramsey-configurations", "title": "Classification of finite Euclidean Ramsey configurations", "title_html": "Classification of finite Euclidean Ramsey configurations", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A classification of finite Euclidean Ramsey configurations", "title_html": "A classification of finite Euclidean Ramsey configurations", "url": "https://github.com/openai/math/blob/main/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/172.md" }, "peer_review": "not yet", "episode": { "video_id": "7QRTWgzhTbc", "length": "6:22", "title": "Euclidean Ramsey: classified?", "page": "/episodes/172-classification-finite-euclidean-ramsey-configurations/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/172-classification-finite-euclidean-ramsey-configurations/" }, { "num": "173", "slug": "173-seymours-second-neighborhood-conjecture", "title": "Seymour’s second-neighborhood conjecture", "title_html": "Seymour’s second-neighborhood conjecture", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A proof of Seymour’s second-neighborhood conjecture", "title_html": "A proof of Seymour’s second-neighborhood conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/173.md" }, "peer_review": "not yet", "episode": { "video_id": "nwOF7lJb_5U", "length": "5:24", "title": "Seymour’s conjecture: two steps keep up?", "page": "/episodes/173-seymours-second-neighborhood-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/173-seymours-second-neighborhood-conjecture/" }, { "num": "174", "slug": "174-deterministic-construction-strong-thin-spanning", "title": "Deterministic construction of strong thin spanning trees", "title_html": "Deterministic construction of strong thin spanning trees", "summary_html": "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/kC/kC/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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The strong thin tree conjecture", "title_html": "The strong thin tree conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf", "role": "main" }, { "title": "A polynomial-time construction of strong thin trees", "title_html": "A polynomial-time construction of strong thin trees", "url": "https://github.com/openai/math/blob/main/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/174.md" }, "peer_review": "not yet", "episode": { "video_id": "seHXHEt4UUk", "length": "6:55", "title": "Strong thin trees: always C/k?", "page": "/episodes/174-deterministic-construction-strong-thin-spanning/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/174-deterministic-construction-strong-thin-spanning/" }, { "num": "175", "slug": "175-talagrands-expectation-thresholds-discrete-convexity", "title": "Talagrand’s expectation thresholds, discrete convexity, and graph decompositions", "title_html": "Talagrand’s expectation thresholds, discrete convexity, and graph decompositions", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Graph Decompositions at the Integral Expectation Threshold", "title_html": "Graph Decompositions at the Integral Expectation Threshold", "url": "https://github.com/openai/math/blob/main/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf", "role": "main" }, { "title": "Integral and fractional expectation thresholds are equivalent", "title_html": "Integral and fractional expectation thresholds are equivalent", "url": "https://github.com/openai/math/blob/main/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Talagrand’s discrete-convexity conjecture", "title_html": "Talagrand’s discrete-convexity conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/175.md" }, "peer_review": "not yet", "episode": { "video_id": "o3ouiTPZzNI", "length": "7:13", "title": "Talagrand, up to a constant?", "page": "/episodes/175-talagrands-expectation-thresholds-discrete-convexity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/175-talagrands-expectation-thresholds-discrete-convexity/" }, { "num": "176", "slug": "176-second-kahn-kalai-conjecture-edge", "title": "The second Kahn–Kalai conjecture with an edge-count bound", "title_html": "The second Kahn–Kalai conjecture with an edge-count bound", "summary_html": "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)G(n,p)G(n,p) is at most CpE(n,H)(1+log⁡2h)C p_{\\mathrm E}(n,H)(1+\\log_2 h)CpE​(n,H)(1+log2​h), with universal C. Here pEp_{\\mathrm E}pE​ is the least density at which every subgraph of H has expected copy count at least 1/2.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The second Kahn–Kalai conjecture", "title_html": "The second Kahn–Kalai conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/176.md" }, "peer_review": "not yet", "episode": { "video_id": "9TWbnKAKteU", "length": "5:37", "title": "Kahn–Kalai, the second one?", "page": "/episodes/176-second-kahn-kalai-conjecture-edge/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/176-second-kahn-kalai-conjecture-edge/" }, { "num": "177", "slug": "177-bounded-degree-coboundary-expanders", "title": "Bounded-degree coboundary expanders", "title_html": "Bounded-degree coboundary expanders", "summary_html": "Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽2 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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Bounded-degree coboundary expanders in every dimension", "title_html": "Bounded-degree coboundary expanders in every dimension", "url": "https://github.com/openai/math/blob/main/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/177.md" }, "peer_review": "not yet", "episode": { "video_id": "tZ0jejNbt8g", "length": "6:52", "title": "Expanders in every dimension?", "page": "/episodes/177-bounded-degree-coboundary-expanders/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/177-bounded-degree-coboundary-expanders/" }, { "num": "178", "slug": "178-deterministic-nonbipartite-ramanujan-graphs-every", "title": "Deterministic nonbipartite Ramanujan graphs in every fixed degree", "title_html": "Deterministic nonbipartite Ramanujan graphs in every fixed degree", "summary_html": "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 −2d−1-2\\sqrt{d-1}−2d−1​ and 2d−12\\sqrt{d-1}2d−1​. A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Deterministic nonbipartite Ramanujan graphs in every fixed degree", "title_html": "Deterministic nonbipartite Ramanujan graphs in every fixed degree", "url": "https://github.com/openai/math/blob/main/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "07404ATSHdg", "length": "7:05", "title": "Ramanujan graphs, made to order?", "page": "/episodes/178-deterministic-nonbipartite-ramanujan-graphs-every/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/178-deterministic-nonbipartite-ramanujan-graphs-every/" }, { "num": "179", "slug": "179-circulant-hadamard-barker-sequence-conjectures", "title": "The circulant Hadamard and Barker-sequence conjectures", "title_html": "The circulant Hadamard and Barker-sequence conjectures", "summary_html": "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 > 1 exactly when n∈{2,3,4,5,7,11,13}n\\in\\{2,3,4,5,7,11,13\\}n∈{2,3,4,5,7,11,13}.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The circulant Hadamard conjecture", "title_html": "The circulant Hadamard conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/179.md" }, "peer_review": "not yet", "episode": { "video_id": "xq97MQDG6pU", "length": "6:00", "title": "Only orders 1 and 4?", "page": "/episodes/179-circulant-hadamard-barker-sequence-conjectures/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/179-circulant-hadamard-barker-sequence-conjectures/" }, { "num": "180", "slug": "180-barnettes-hamiltonian-cycle-conjecture", "title": "Barnette’s Hamiltonian-cycle conjecture", "title_html": "Barnette’s Hamiltonian-cycle conjecture", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Paired states and Hamiltonian cycles in cubic bipartite planar graphs", "title_html": "Paired states and Hamiltonian cycles in cubic bipartite planar graphs", "url": "https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/180.md" }, "peer_review": "not yet", "episode": { "video_id": "gbowf_xnHkg", "length": "5:48", "title": "A tour of every corner?", "page": "/episodes/180-barnettes-hamiltonian-cycle-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/180-barnettes-hamiltonian-cycle-conjecture/" }, { "num": "181", "slug": "181-erdos-gallai-cycle-decomposition-conjecture", "title": "The Erdős–Gallai cycle-decomposition conjecture", "title_html": "The Erdős–Gallai cycle-decomposition conjecture", "summary_html": "Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most CnCnCn 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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A linear cycle-and-edge decomposition of every graph", "title_html": "A linear cycle-and-edge decomposition of every graph", "url": "https://github.com/openai/math/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/181.md" }, "peer_review": "not yet", "episode": { "video_id": "Yo1LwL_dX-M", "length": "7:04", "title": "C times n pieces?", "page": "/episodes/181-erdos-gallai-cycle-decomposition-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/181-erdos-gallai-cycle-decomposition-conjecture/" }, { "num": "182", "slug": "182-power-savings-intersective-polynomial-differences", "title": "Power savings for intersective polynomial differences and prime arguments", "title_html": "Power savings for intersective polynomial differences and prime arguments", "summary_html": "For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of {1,…,N}\\{1,\\ldots,N\\}{1,…,N} avoiding nonzero values h(1),h(2),…h(1),h(2),\\ldotsh(1),h(2),… as differences has size Oh(N1−ck)O_h(N^{1-c_k})Oh​(N1−ck​), with ck>0c_k\\gt 0ck​>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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A power saving for intersective polynomial differences with an exponent depending only on the degree", "title_html": "A power saving for intersective polynomial differences with an exponent depending only on the degree", "url": "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", "role": "main" }, { "title": "A Power Saving for Polynomial Differences at Prime Arguments", "title_html": "A Power Saving for Polynomial Differences at Prime Arguments", "url": "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", "role": "companion" }, { "title": "A power saving for square-difference-free sets", "title_html": "A power saving for square-difference-free sets", "url": "https://github.com/openai/math/blob/main/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/182.md" }, "peer_review": "not yet", "episode": { "video_id": "6T2KCLLMvmY", "length": "7:01", "title": "No square gaps?", "page": "/episodes/182-power-savings-intersective-polynomial-differences/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/182-power-savings-intersective-polynomial-differences/" }, { "num": "183", "slug": "183-power-savings-planar-halving-lines", "title": "Power savings for planar halving lines and k-sets", "title_html": "Power savings for planar halving lines and k-sets", "summary_html": "Improves the planar halving-line bound to O(n4/3−ε)O(n^{4/3-\\varepsilon})O(n4/3−ε) for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has O(n(k+1)1/3−ε0)O(n(k+1)^{1/3-\\varepsilon_0})O(n(k+1)1/3−ε0​) strictly separable k-subsets for 1≤k≤n/21\\le k\\le n/21≤k≤n/2, with an absolute ε0>0\\varepsilon_0\\gt 0ε0​>0. The constants and positive exponents are nonquantitative.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A power saving for planar halving lines", "title_html": "A power saving for planar halving lines", "url": "https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/183.md" }, "peer_review": "not yet", "episode": { "video_id": "zMRn8clUoO8", "length": "5:59", "title": "Below 4/3?", "page": "/episodes/183-power-savings-planar-halving-lines/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/183-power-savings-planar-halving-lines/" }, { "num": "184", "slug": "184-correspondence-coloring-fixed-forbidden-subgraph", "title": "Correspondence coloring with a fixed forbidden subgraph", "title_html": "Correspondence coloring with a fixed forbidden subgraph", "summary_html": "Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need OF(Δ/log⁡Δ)O_F(\\Delta/\\log\\Delta)OF​(Δ/logΔ) 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 Kr-free graph of average degree d ≥ 2 has an independent set of size Ωr(nlog⁡d/d)\\Omega_r(n\\log d/d)Ωr​(nlogd/d).", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Correspondence coloring graphs with a forbidden clique", "title_html": "Correspondence coloring graphs with a forbidden clique", "url": "https://github.com/openai/math/blob/main/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf", "role": "main" }, { "title": "A logarithmic independence bound for clique-free graphs", "title_html": "A logarithmic independence bound for clique-free graphs", "url": "https://github.com/openai/math/blob/main/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/184.md" }, "peer_review": "not yet", "episode": { "video_id": "GT9tzLxtaz4", "length": "8:18", "title": "Forbid a clique, gain a log?", "page": "/episodes/184-correspondence-coloring-fixed-forbidden-subgraph/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/184-correspondence-coloring-fixed-forbidden-subgraph/" }, { "num": "185", "slug": "185-counterexamples-infinite-matroid-intersection-packing", "title": "Counterexamples to infinite matroid intersection and packing/covering", "title_html": "Counterexamples to infinite matroid intersection and packing/covering", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A Counterexample to the Infinite Matroid Packing/Covering Conjecture", "title_html": "A Counterexample to the Infinite Matroid Packing/Covering Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/185.md" }, "peer_review": "not yet", "episode": { "video_id": "V31VJ5DO2lc", "length": "7:36", "title": "No fair split at infinity?", "page": "/episodes/185-counterexamples-infinite-matroid-intersection-packing/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/185-counterexamples-infinite-matroid-intersection-packing/" }, { "num": "186", "slug": "186-uniform-influence-sharp-thresholds-graph", "title": "Uniform influence and sharp thresholds for graph and hypergraph properties", "title_html": "Uniform influence and sharp thresholds for graph and hypergraph properties", "summary_html": "Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed 0<ε<1/20\\lt \\varepsilon\\lt 1/20<ε<1/2, every nontrivial increasing relabeling-invariant property crosses from probability ε to 1−ε1-\\varepsilon1−ε within width O((log⁡n)−2)O((\\log n)^{-2})O((logn)−2) for graphs and Or((log⁡n)−r/(r−1))O_r((\\log n)^{-r/(r-1)})Or​((logn)−r/(r−1)) for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A uniform influence bound for hypergraph properties", "title_html": "A uniform influence bound for hypergraph properties", "url": "https://github.com/openai/math/blob/main/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf", "role": "main" }, { "title": "A Sharp Threshold Bound for Monotone Graph Properties", "title_html": "A Sharp Threshold Bound for Monotone Graph Properties", "url": "https://github.com/openai/math/blob/main/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/186.md" }, "peer_review": "not yet", "episode": { "video_id": "rgsVQLCHViE", "length": "8:42", "title": "How fast must it switch?", "page": "/episodes/186-uniform-influence-sharp-thresholds-graph/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/186-uniform-influence-sharp-thresholds-graph/" }, { "num": "187", "slug": "187-snaky-21-maker-moves", "title": "Snaky in 21 Maker moves", "title_html": "Snaky in 21 Maker moves", "summary_html": "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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Snaky in 21 Maker moves", "title_html": "Snaky in 21 Maker moves", "url": "https://github.com/openai/math/blob/main/preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/187.md" }, "peer_review": "not yet", "episode": { "video_id": "5bVOhWMXZ9E", "length": "6:46", "title": "Snaky in 21 moves?", "page": "/episodes/187-snaky-21-maker-moves/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/187-snaky-21-maker-moves/" }, { "num": "188", "slug": "188-sharp-terminal-leave-random-triangle", "title": "The sharp terminal leave in random triangle removal", "title_html": "The sharp terminal leave in random triangle removal", "summary_html": "Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to n3/2/(22)n^{3/2}/(2\\sqrt2)n3/2/(22​), with mean-square convergence after normalization by n3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "The sharp terminal leave in random triangle removal", "title_html": "The sharp terminal leave in random triangle removal", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/188.md" }, "peer_review": "not yet", "episode": { "video_id": "TmdnXcP3K64", "length": "6:30", "title": "What’s left over?", "page": "/episodes/188-sharp-terminal-leave-random-triangle/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/188-sharp-terminal-leave-random-triangle/" }, { "num": "189", "slug": "189-cycle-clique-ramsey-numbers", "title": "Cycle–clique Ramsey numbers", "title_html": "Cycle–clique Ramsey numbers", "summary_html": "Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(Cm,Kn)=(m−1)(n−1)+1R(C_m,K_n)=(m-1)(n-1)+1R(Cm​,Kn​)=(m−1)(n−1)+1 for every m≥n≥3m\\ge n\\ge3m≥n≥3, except R(C3,K3)=6R(C_3,K_3)=6R(C3​,K3​)=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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Cycle--clique Ramsey numbers", "title_html": "Cycle--clique Ramsey numbers", "url": "https://github.com/openai/math/blob/main/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/189.md" }, "peer_review": "not yet", "episode": { "video_id": "WbxGW55Q2-k", "length": "6:46", "title": "Red cycle or blue clique?", "page": "/episodes/189-cycle-clique-ramsey-numbers/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/189-cycle-clique-ramsey-numbers/" }, { "num": "190", "slug": "190-polynomial-removal-fails-ordered-binary", "title": "Polynomial removal fails for ordered binary matrices", "title_html": "Polynomial removal fails for ordered binary matrices", "summary_html": "Disproves polynomial ordered binary matrix removal with one fixed 66×6666\\times6666×66 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.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Polynomial removal fails for ordered binary matrices", "title_html": "Polynomial removal fails for ordered binary matrices", "url": "https://github.com/openai/math/blob/main/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/190.md" }, "peer_review": "not yet", "episode": { "video_id": "5h4dYMWrlCQ", "length": "7:29", "title": "No polynomial bound?", "page": "/episodes/190-polynomial-removal-fails-ordered-binary/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/190-polynomial-removal-fails-ordered-binary/" }, { "num": "191", "slug": "191-power-improvement-heilbronn-triangle-lower", "title": "A power improvement in the Heilbronn triangle lower bound", "title_html": "A power improvement in the Heilbronn triangle lower bound", "summary_html": "For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least n−2+cn^{-2+c}n−2+c for one absolute c > 0. This disproves the conjectured almost-n−2 upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "A power improvement in the Heilbronn triangle lower bound", "title_html": "A power improvement in the Heilbronn triangle lower bound", "url": "https://github.com/openai/math/blob/main/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/191.md" }, "peer_review": "not yet", "episode": { "video_id": "tC0quIPiB8c", "length": "7:02", "title": "Bigger than 1/n²?", "page": "/episodes/191-power-improvement-heilbronn-triangle-lower/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/191-power-improvement-heilbronn-triangle-lower/" }, { "num": "192", "slug": "192-boolean-functions-violate-square-root", "title": "Boolean functions violate the square-root degree bound by arbitrary factors", "title_html": "Boolean functions violate the square-root degree bound by arbitrary factors", "summary_html": "Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with ∑if^({i})>Cdeg⁡(f)\\sum_i\\widehat f(\\{i\\})\\gt C\\sqrt{\\deg(f)}∑i​f​({i})>Cdeg(f)​. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.", "subject": "Combinatorics", "subject_slug": "combinatorics", "papers": [ { "title": "Unbounded Violations of the Square-Root Degree Bound", "title_html": "Unbounded Violations of the Square-Root Degree Bound", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/192.md" }, "peer_review": "not yet", "episode": { "video_id": "iMG6R9kmZkg", "length": "5:59", "title": "Beyond the square root?", "page": "/episodes/192-boolean-functions-violate-square-root/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/192-boolean-functions-violate-square-root/" }, { "num": "193", "slug": "193-serres-intersection-multiplicity-conjecture", "title": "Serre’s intersection-multiplicity conjecture", "title_html": "Serre’s intersection-multiplicity conjecture", "summary_html": "Proves strict positivity of Serre's intersection multiplicity χR(M,N)\\chi^R(M,N)χR(M,N) for nonzero finitely generated modules over any regular local ring, provided M⊗RNM\\otimes_R NM⊗R​N has finite length and dim⁡M+dim⁡N=dim⁡R\\dim M+\\dim N=\\dim RdimM+dimN=dimR. This resolves the positivity conjecture, including ramified mixed characteristic.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "Positivity of Serre's Intersection Multiplicity", "title_html": "Positivity of Serre's Intersection Multiplicity", "url": "https://github.com/openai/math/blob/main/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "WTB_fDIenVA", "length": "5:50", "title": "Always positive?", "page": "/episodes/193-serres-intersection-multiplicity-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/193-serres-intersection-multiplicity-conjecture/" }, { "num": "194", "slug": "194-lechs-multiplicity-conjecture", "title": "Lech’s multiplicity conjecture", "title_html": "Lech’s multiplicity conjecture", "summary_html": "Proves e(R)≤e(S)e(R)\\le e(S)e(R)≤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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "Lech's multiplicity conjecture", "title_html": "Lech's multiplicity conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/194.md" }, "peer_review": "not yet", "episode": { "video_id": "ewdfZ2iA-gc", "length": "6:09", "title": "Never smoother?", "page": "/episodes/194-lechs-multiplicity-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/194-lechs-multiplicity-conjecture/" }, { "num": "195", "slug": "195-counterexample-small-cohen-macaulay-module", "title": "A counterexample to the small Cohen–Macaulay module conjecture", "title_html": "A counterexample to the small Cohen–Macaulay module conjecture", "summary_html": "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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "A Complete Local Domain without a Small Cohen–Macaulay Module", "title_html": "A Complete Local Domain without a Small Cohen–Macaulay Module", "url": "https://github.com/openai/math/blob/main/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/195.md" }, "peer_review": "not yet", "episode": { "video_id": "rUyGrJwaNWs", "length": "6:12", "title": "No finite repair?", "page": "/episodes/195-counterexample-small-cohen-macaulay-module/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/195-counterexample-small-cohen-macaulay-module/" }, { "num": "196", "slug": "196-counterexample-kaplanskys-zero-divisor-conjecture", "title": "A counterexample to Kaplansky’s zero-divisor conjecture", "title_html": "A counterexample to Kaplansky’s zero-divisor conjecture", "summary_html": "Constructs a finitely presented torsion-free group G whose group algebra F2[G]\\mathbb F_2[G]F2​[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "A Torsion-Free Group Algebra with Zero Divisors", "title_html": "A Torsion-Free Group Algebra with Zero Divisors", "url": "https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/196.md" }, "peer_review": "not yet", "episode": { "video_id": "bY_pvzJkNPw", "length": "5:48", "title": "Torsion-free, yet zero?", "page": "/episodes/196-counterexample-kaplanskys-zero-divisor-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/196-counterexample-kaplanskys-zero-divisor-conjecture/" }, { "num": "197", "slug": "197-torsion-free-group-algebra-not", "title": "A torsion-free group algebra that is not directly finite", "title_html": "A torsion-free group algebra that is not directly finite", "summary_html": "Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "A Torsion-Free Group Algebra That Is Not Directly Finite", "title_html": "A Torsion-Free Group Algebra That Is Not Directly Finite", "url": "https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf", "role": "main" }, { "title": "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two", "title_html": "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two", "url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf", "role": "companion" }, { "title": "A Counterexample to the Group-Ring Determinant Conjecture", "title_html": "A Counterexample to the Group-Ring Determinant Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf", "role": "companion" }, { "title": "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic", "title_html": "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic", "url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/197.md" }, "peer_review": "not yet", "episode": { "video_id": "5JHchks3qmM", "length": "7:36", "title": "One-sided inverse?", "page": "/episodes/197-torsion-free-group-algebra-not/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/197-torsion-free-group-algebra-not/" }, { "num": "198", "slug": "198-counterexample-finitistic-dimension-finiteness", "title": "A counterexample to finitistic-dimension finiteness", "title_html": "A counterexample to finitistic-dimension finiteness", "summary_html": "Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "An algebra of infinite little finitistic dimension", "title_html": "An algebra of infinite little finitistic dimension", "url": "https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/198.md" }, "peer_review": "not yet", "episode": { "video_id": "Wud79Vygfxk", "length": "6:10", "title": "No ceiling?", "page": "/episodes/198-counterexample-finitistic-dimension-finiteness/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/198-counterexample-finitistic-dimension-finiteness/" }, { "num": "199", "slug": "199-counterexamples-auslander-reiten-tachikawa-related", "title": "Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures", "title_html": "Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures", "summary_html": "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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "An explicit counterexample to the Auslander-Reiten conjecture", "title_html": "An explicit counterexample to the Auslander-Reiten conjecture", "url": "https://github.com/openai/math/blob/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf", "role": "main" }, { "title": "A counterexample to Tachikawa's second conjecture", "title_html": "A counterexample to Tachikawa's second conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/199.md" }, "peer_review": "not yet", "episode": { "video_id": "F5W6KqZ2z7E", "length": "6:24", "title": "No gluing, not projective?", "page": "/episodes/199-counterexamples-auslander-reiten-tachikawa-related/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/199-counterexamples-auslander-reiten-tachikawa-related/" }, { "num": "200", "slug": "200-eisenbud-green-harris-lex-plus", "title": "Eisenbud–Green–Harris and lex-plus-powers", "title_html": "Eisenbud–Green–Harris and lex-plus-powers", "summary_html": "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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "The Artinian Lex-Plus-Powers Betti Theorem", "title_html": "The Artinian Lex-Plus-Powers Betti Theorem", "url": "https://github.com/openai/math/blob/main/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf", "role": "main" }, { "title": "Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture", "title_html": "Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "U873u7KwV1Y", "length": "5:52", "title": "Count like monomials?", "page": "/episodes/200-eisenbud-green-harris-lex-plus/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/200-eisenbud-green-harris-lex-plus/" }, { "num": "201", "slug": "201-counterexample-kuroshs-division-ring-problem", "title": "A counterexample to Kurosh’s division-ring problem", "title_html": "A counterexample to Kurosh’s division-ring problem", "summary_html": "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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "A Counterexample to Kurosh’s Division-Ring Problem", "title_html": "A Counterexample to Kurosh’s Division-Ring Problem", "url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "e5npAeUj5k0", "length": "5:13", "title": "Every element tame, yet infinite?", "page": "/episodes/201-counterexample-kuroshs-division-ring-problem/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/201-counterexample-kuroshs-division-ring-problem/" }, { "num": "202", "slug": "202-blockwise-alperin-weight-conjecture", "title": "The blockwise Alperin weight conjecture", "title_html": "The blockwise Alperin weight conjecture", "summary_html": "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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "The Blockwise Alperin Weight Conjecture", "title_html": "The Blockwise Alperin Weight Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "98DVPfOHK0c", "length": "5:06", "title": "Global = local?", "page": "/episodes/202-blockwise-alperin-weight-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/202-blockwise-alperin-weight-conjecture/" }, { "num": "203", "slug": "203-donovans-conjecture-fields-complete-mixed", "title": "Donovan's conjecture over fields and complete mixed-characteristic DVRs", "title_html": "Donovan's conjecture over fields and complete mixed-characteristic DVRs", "summary_html": "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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "Donovan's Conjecture over Algebraically Closed Fields", "title_html": "Donovan's Conjecture over Algebraically Closed Fields", "url": "https://github.com/openai/math/blob/main/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf", "role": "main" }, { "title": "Integral Donovan Finiteness over Witt Vectors", "title_html": "Integral Donovan Finiteness over Witt Vectors", "url": "https://github.com/openai/math/blob/main/preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026/main.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "0OrddD8ISXA", "length": "4:38", "title": "Fix the defect group: finite?", "page": "/episodes/203-donovans-conjecture-fields-complete-mixed/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/203-donovans-conjecture-fields-complete-mixed/" }, { "num": "204", "slug": "204-tensor-saturation-even-spin-groups", "title": "Tensor saturation for even spin groups", "title_html": "Tensor saturation for even spin groups", "summary_html": "Proves saturation factor one for Spin(2n)\\mathop{\\mathrm{Spin}}\\nolimits (2n)Spin(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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "Tensor saturation for even spin groups", "title_html": "Tensor saturation for even spin groups", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "mx-hB69ACcE", "length": "4:21", "title": "Only after stretching?", "page": "/episodes/204-tensor-saturation-even-spin-groups/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/204-tensor-saturation-even-spin-groups/" }, { "num": "205", "slug": "205-saxls-conjecture-universal-tensor-squares", "title": "Saxl’s conjecture and universal tensor squares", "title_html": "Saxl’s conjecture and universal tensor squares", "summary_html": "Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with n∉{2,4,9}n\\notin\\{2,4,9\\}n∈/{2,4,9} has an irreducible representation whose tensor square contains all irreducibles.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "Universal Tensor Squares for Symmetric Groups", "title_html": "Universal Tensor Squares for Symmetric Groups", "url": "https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf", "role": "main" }, { "title": "A Cyclic Polytabloid Proof of Saxl's Conjecture", "title_html": "A Cyclic Polytabloid Proof of Saxl's Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/205.md" }, "peer_review": "not yet", "episode": { "video_id": "8a83yPtXopc", "length": "3:57", "title": "Square the staircase: all?", "page": "/episodes/205-saxls-conjecture-universal-tensor-squares/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/205-saxls-conjecture-universal-tensor-squares/" }, { "num": "206", "slug": "206-finite-lattice-representation-undecidability", "title": "Finite lattice representation and undecidability", "title_html": "Finite lattice representation and undecidability", "summary_html": "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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "Finite congruence lattices: characterization and undecidability", "title_html": "Finite congruence lattices: characterization and undecidability", "url": "https://github.com/openai/math/blob/main/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/paper.pdf", "role": "main" }, { "title": "A negative solution to the finite lattice representation problem", "title_html": "A negative solution to the finite lattice representation problem", "url": "https://github.com/openai/math/blob/main/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/206.md" }, "peer_review": "not yet", "episode": { "video_id": "Px3PrlrpPAw", "length": "3:56", "title": "A lattice no algebra fits?", "page": "/episodes/206-finite-lattice-representation-undecidability/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/206-finite-lattice-representation-undecidability/" }, { "num": "207", "slug": "207-l1-bass-conjecture-all-discrete", "title": "The ℓ¹-Bass conjecture for all discrete groups", "title_html": "The ℓ¹-Bass conjecture for all discrete groups", "summary_html": "Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over ℓ1(G)\\ell^1(G)ℓ1(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.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "The ℓ¹-Bass Conjecture for Discrete Groups", "title_html": "The ℓ¹-Bass Conjecture for Discrete Groups", "url": "https://github.com/openai/math/blob/main/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf", "role": "main" }, { "title": "The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture", "title_html": "The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/207.md" }, "peer_review": "not yet", "episode": { "video_id": "K0T7NhjFSEA", "length": "3:59", "title": "Idempotents: only 0 and 1?", "page": "/episodes/207-l1-bass-conjecture-all-discrete/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/207-l1-bass-conjecture-all-discrete/" }, { "num": "208", "slug": "208-finite-symmetric-tensor-categories-verlinde", "title": "Finite symmetric tensor categories and the Verlinde tower", "title_html": "Finite symmetric tensor categories and the Verlinde tower", "summary_html": "Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category Verpn\\mathrm{Ver}_{p^n}Verpn​. The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "Fiber functors for finite symmetric tensor categories in positive characteristic", "title_html": "Fiber functors for finite symmetric tensor categories in positive characteristic", "url": "https://github.com/openai/math/blob/main/preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "O96hu01oh-A", "length": "4:09", "title": "Every category finds a floor?", "page": "/episodes/208-finite-symmetric-tensor-categories-verlinde/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/208-finite-symmetric-tensor-categories-verlinde/" }, { "num": "209", "slug": "209-integral-counterexamples-gerstens-conjecture", "title": "Integral counterexamples to Gersten’s conjecture", "title_html": "Integral counterexamples to Gersten’s conjecture", "summary_html": "Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic (0,5)(0,5)(0,5) have nonzero integral K-theory classes that vanish over their fraction fields.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "An integral counterexample to Gersten's conjecture", "title_html": "An integral counterexample to Gersten's conjecture", "url": "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", "role": "main" }, { "title": "An integral degree-three Gersten counterexample", "title_html": "An integral degree-three Gersten counterexample", "url": "https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "IsHyLZmXRM4", "length": "4:17", "title": "Dead over F, alive on A?", "page": "/episodes/209-integral-counterexamples-gerstens-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/209-integral-counterexamples-gerstens-conjecture/" }, { "num": "210", "slug": "210-foulkes-conjecture-sixth-powers-quadratic", "title": "Foulkes' conjecture for sixth powers and quadratic stabilization", "title_html": "Foulkes' conjecture for sixth powers and quadratic stabilization", "summary_html": "Proves the sixth case of Foulkes’ conjecture: Sym6(SymbV)\\mathop{\\mathrm{Sym}}\\nolimits ^6(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)Sym6(SymbV) embeds equivariantly in Symb(Sym6V)\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^6V)Symb(Sym6V) for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map Symb(SymaV)→Syma(SymbV)\\mathop{\\mathrm{Sym}}\\nolimits ^b(\\mathop{\\mathrm{Sym}}\\nolimits ^aV)\\to\\mathop{\\mathrm{Sym}}\\nolimits ^a(\\mathop{\\mathrm{Sym}}\\nolimits ^bV)Symb(SymaV)→Syma(SymbV) is surjective for a ≥ 2 and b≥a(a−1)b\\ge a(a-1)b≥a(a−1), giving dimension-independent quadratic stabilization.", "subject": "Algebra", "subject_slug": "algebra", "papers": [ { "title": "Foulkes' conjecture for the sixth symmetric power", "title_html": "Foulkes' conjecture for the sixth symmetric power", "url": "https://github.com/openai/math/blob/main/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf", "role": "main" }, { "title": "Quadratic stabilization of the canonical Foulkes--Howe map", "title_html": "Quadratic stabilization of the canonical Foulkes--Howe map", "url": "https://github.com/openai/math/blob/main/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/210.md" }, "peer_review": "not yet", "episode": { "video_id": "NvjRyq-fmBA", "length": "3:52", "title": "Does one stack fit in the other?", "page": "/episodes/210-foulkes-conjecture-sixth-powers-quadratic/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/210-foulkes-conjecture-sixth-powers-quadratic/" }, { "num": "211", "slug": "211-geometric-phase-diagram-diffusion-spectra", "title": "The geometric phase diagram, diffusion, and spectra of random planar maps", "title_html": "The geometric phase diagram, diffusion, and spectra of random planar maps", "summary_html": "Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for 0<q≤40\\lt q\\le40<q≤4 and to the Brownian continuum random tree for q > 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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion", "title_html": "Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion", "url": "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", "role": "main" }, { "title": "Spectral convergence for critical FK–Ising planar maps", "title_html": "Spectral convergence for critical FK–Ising planar maps", "url": "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", "role": "companion" }, { "title": "A Linear Clock for Random Walk on Tree-Weighted Planar Maps", "title_html": "A Linear Clock for Random Walk on Tree-Weighted Planar Maps", "url": "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", "role": "companion" }, { "title": "Canonical conformal limits of subcritical FK planar maps", "title_html": "Canonical conformal limits of subcritical FK planar maps", "url": "https://github.com/openai/math/blob/main/preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026/main.pdf", "role": "companion" }, { "title": "The critical Liouville quantum sphere and geometric limits of FK maps at q=4", "title_html": "The critical Liouville quantum sphere and geometric limits of FK maps at q=4", "url": "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", "role": "companion" }, { "title": "Metric-measure limits of subcritical FK and spanning-tree planar maps", "title_html": "Metric-measure limits of subcritical FK and spanning-tree planar maps", "url": "https://github.com/openai/math/blob/main/preprints/Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026/main.pdf", "role": "companion" }, { "title": "Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four", "title_html": "Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/211.md" }, "peer_review": "not yet", "episode": { "video_id": "gVBXWPKatkA", "length": "4:25", "title": "Sphere or tree?", "page": "/episodes/211-geometric-phase-diagram-diffusion-spectra/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/211-geometric-phase-diagram-diffusion-spectra/" }, { "num": "212", "slug": "212-planar-first-passage-geometry-absence", "title": "Planar first-passage geometry and the absence of bigeodesics", "title_html": "Planar first-passage geometry and the absence of bigeodesics", "summary_html": "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 C1 boundary. Differentiability also holds for every Gamma law with positive shape and rate.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "No bigeodesics in planar first-passage percolation", "title_html": "No bigeodesics in planar first-passage percolation", "url": "https://github.com/openai/math/blob/main/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/main.pdf", "role": "main" }, { "title": "Strict convexity and differentiability of the planar exponential first-passage limit shape", "title_html": "Strict convexity and differentiability of the planar exponential first-passage limit shape", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/212.md" }, "peer_review": "not yet", "episode": { "video_id": "p47zG2zTa0w", "length": "4:53", "title": "A random straight line?", "page": "/episodes/212-planar-first-passage-geometry-absence/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/212-planar-first-passage-geometry-absence/" }, { "num": "213", "slug": "213-critical-percolation-quasi-transitive", "title": "Critical percolation on every quasi-transitive graph", "title_html": "Critical percolation on every quasi-transitive graph", "summary_html": "Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with pc<1p_c\\lt 1pc​<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 ℤ3.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Critical bond and site percolation on the cubic lattice", "title_html": "Critical bond and site percolation on the cubic lattice", "url": "https://github.com/openai/math/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf", "role": "main" }, { "title": "No percolation at criticality on quasi-transitive graphs", "title_html": "No percolation at criticality on quasi-transitive graphs", "url": "https://github.com/openai/math/blob/main/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/213.md" }, "peer_review": "not yet", "episode": { "video_id": "PDdnT2Xl0vQ", "length": "4:43", "title": "Does it percolate?", "page": "/episodes/213-critical-percolation-quasi-transitive/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/213-critical-percolation-quasi-transitive/" }, { "num": "214", "slug": "214-percolation-nonuniqueness-operator-bound", "title": "The Benjamini–Schramm nonuniqueness conjecture", "title_html": "The Benjamini–Schramm nonuniqueness conjecture", "summary_html": "Proves pc<pup_c\\lt p_upc​<pu​ 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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Nonuniqueness of percolation on nonamenable quasi-transitive graphs", "title_html": "Nonuniqueness of percolation on nonamenable quasi-transitive graphs", "url": "https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/214.md" }, "peer_review": "not yet", "episode": { "video_id": "aAnvR_u91ss", "length": "4:11", "title": "One ocean, or many?", "page": "/episodes/214-percolation-nonuniqueness-operator-bound/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/214-percolation-nonuniqueness-operator-bound/" }, { "num": "215", "slug": "215-o3-continuum-o4-mass-asymptotics", "title": "Canonical O(3) continuum limit and exact O(4) mass asymptotics", "title_html": "Canonical O(3)O(3)O(3) continuum limit and exact O(4)O(4)O(4) mass asymptotics", "summary_html": "Constructs the canonical continuum limit of the two-dimensional nearest-neighbor O(3)O(3)O(3) model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice O(4)O(4)O(4) model, determines the exact leading asymptotic of the full transfer gap, mlat(β)∼32eπ/4−1/2β e−πβm_{\\mathrm{lat}}(\\beta)\\sim32e^{\\pi/4-1/2}\\sqrt\\beta\\,e^{-\\pi\\beta}mlat​(β)∼32eπ/4−1/2β​e−πβ. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor O(n)O(n)O(n) models with n ≥ 3 at every positive temperature.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The canonical massive continuum limit of the two-dimensional O(3) model", "title_html": "The canonical massive continuum limit of the two-dimensional O(3) model", "url": "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", "role": "main" }, { "title": "An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field", "title_html": "An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field", "url": "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", "role": "companion" }, { "title": "Exact mass asymptotics for the two-dimensional O(4) lattice model", "title_html": "Exact mass asymptotics for the two-dimensional O(4) lattice model", "url": "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", "role": "companion" }, { "title": "Sharp mass bounds for the two-dimensional O(4) model", "title_html": "Sharp mass bounds for the two-dimensional O(4) model", "url": "https://github.com/openai/math/blob/main/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Exponential decay in two-dimensional classical O(n) models", "title_html": "Exponential decay in two-dimensional classical O(n) models", "url": "https://github.com/openai/math/blob/main/preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/215.md" }, "peer_review": "not yet", "episode": { "video_id": "as6hBfAXdXM", "length": "4:48", "title": "Do spins forget?", "page": "/episodes/215-o3-continuum-o4-mass-asymptotics/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/215-o3-continuum-o4-mass-asymptotics/" }, { "num": "216", "slug": "216-xy-critical-scaling-bkt-universality", "title": "Critical and near-critical XY scaling and BKT universality", "title_html": "Critical and near-critical XY scaling and BKT universality", "summary_html": "For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations Cβc(r)∼Ar−1/4(log⁡r)1/8C_{\\beta_c}(r)\\sim Ar^{-1/4}(\\log r)^{1/8}Cβc​​(r)∼Ar−1/4(logr)1/8 and the Berezinskii–Kosterlitz–Thouless essential singularity βc−βlog⁡ξ(β)→B\\sqrt{\\beta_c-\\beta}\\log\\xi(\\beta)\\to Bβc​−β​logξ(β)→B, with A,B>0A,B\\gt 0A,B>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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The critical logarithmic correction for the planar XY model", "title_html": "The critical logarithmic correction for the planar XY model", "url": "https://github.com/openai/math/blob/main/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf", "role": "main" }, { "title": "Critical Center Magnetization in the Planar XY Model", "title_html": "Critical Center Magnetization in the Planar XY Model", "url": "https://github.com/openai/math/blob/main/preprints/Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026/paper.pdf", "role": "companion" }, { "title": "The Critical Spin Field of the Planar XY Model", "title_html": "The Critical Spin Field of the Planar XY Model", "url": "https://github.com/openai/math/blob/main/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf", "role": "companion" }, { "title": "Essential Singularity of the Correlation Length in the Planar XY Model", "title_html": "Essential Singularity of the Correlation Length in the Planar XY Model", "url": "https://github.com/openai/math/blob/main/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf", "role": "companion" }, { "title": "The critical correlation exponent of the planar XY model", "title_html": "The critical correlation exponent of the planar XY model", "url": "https://github.com/openai/math/blob/main/preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026/paper.pdf", "role": "companion" }, { "title": "BKT universality for height and planar spin fields", "title_html": "BKT universality for height and planar spin fields", "url": "https://github.com/openai/math/blob/main/preprints/BKT-universality-for-height-and-planar-spin-fields-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "tMQdXMpoEmQ", "length": "4:36", "title": "Exactly one quarter?", "page": "/episodes/216-xy-critical-scaling-bkt-universality/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/216-xy-critical-scaling-bkt-universality/" }, { "num": "217", "slug": "217-sherrington-kirkpatrick-fluctuation-law", "title": "The low-temperature Sherrington–Kirkpatrick fluctuation law", "title_html": "The low-temperature Sherrington–Kirkpatrick fluctuation law", "summary_html": "For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to cβn1/3c_\\beta n^{1/3}cβ​n1/3, with cβ>0c_\\beta\\gt 0cβ​>0, confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The low-temperature Sherrington–Kirkpatrick free-energy limiting law", "title_html": "The low-temperature Sherrington–Kirkpatrick free-energy limiting law", "url": "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", "role": "main" }, { "title": "The low-temperature Sherrington–Kirkpatrick fluctuation scale", "title_html": "The low-temperature Sherrington–Kirkpatrick fluctuation scale", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "9bptrKHKA90", "length": "4:13", "title": "How much does it wobble?", "page": "/episodes/217-sherrington-kirkpatrick-fluctuation-law/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/217-sherrington-kirkpatrick-fluctuation-law/" }, { "num": "218", "slug": "218-ising-conformal-universality-random-bonds", "title": "Conformal universality for weakly interacting and random-bond Ising models", "title_html": "Conformal universality for weakly interacting and random-bond Ising models", "summary_html": "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 SLE3 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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Quenched SLE₃ limits for general weak random-bond Ising models", "title_html": "Quenched SLE₃ limits for general weak random-bond Ising models", "url": "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", "role": "main" }, { "title": "Quenched SLE₃ Universality for the Weak Random-Bond Ising Model", "title_html": "Quenched SLE₃ Universality for the Weak Random-Bond Ising Model", "url": "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", "role": "companion" }, { "title": "Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model", "title_html": "Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model", "url": "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", "role": "companion" }, { "title": "Conformal universality of bulk Ising correlations under weak interactions", "title_html": "Conformal universality of bulk Ising correlations under weak interactions", "url": "https://github.com/openai/math/blob/main/preprints/Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026/paper.pdf", "role": "companion" }, { "title": "SLE3 universality for weak finite-range Ising interactions", "title_html": "SLE3 universality for weak finite-range Ising interactions", "url": "https://github.com/openai/math/blob/main/preprints/SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Buffered comparison and stopping-band resolution in critical Ising", "title_html": "Buffered comparison and stopping-band resolution in critical Ising", "url": "https://github.com/openai/math/blob/main/preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/218.md" }, "peer_review": "not yet", "episode": { "video_id": "xIljMbXXQMI", "length": "4:18", "title": "Same curve, new rules?", "page": "/episodes/218-ising-conformal-universality-random-bonds/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/218-ising-conformal-universality-random-bonds/" }, { "num": "219", "slug": "219-goe-bulk-universality-regular-graphs", "title": "GOE bulk universality for regular graphs with weak Anderson disorder", "title_html": "GOE bulk universality for regular graphs with weak Anderson disorder", "summary_html": "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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Fixed-energy universality for weak Anderson disorder on random regular graphs", "title_html": "Fixed-energy universality for weak Anderson disorder on random regular graphs", "url": "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", "role": "main" }, { "title": "GOE bulk universality for fixed-degree random regular graphs", "title_html": "GOE bulk universality for fixed-degree random regular graphs", "url": "https://github.com/openai/math/blob/main/preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "FS_nR01XulY", "length": "3:49", "title": "Do eigenvalues repel?", "page": "/episodes/219-goe-bulk-universality-regular-graphs/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/219-goe-bulk-universality-regular-graphs/" }, { "num": "220", "slug": "220-rwre-zero-one-ballisticity", "title": "Directional zero–one laws beyond iid environments and iid ballisticity", "title_html": "Directional zero–one laws beyond iid environments and iid ballisticity", "summary_html": "On ℤd, 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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "A directional zero–one law for finite-range-dependent random environments", "title_html": "A directional zero–one law for finite-range-dependent random environments", "url": "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", "role": "main" }, { "title": "A directional zero–one law under strict ellipticity", "title_html": "A directional zero–one law under strict ellipticity", "url": "https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Directional transience implies ballisticity", "title_html": "Directional transience implies ballisticity", "url": "https://github.com/openai/math/blob/main/preprints/Directional-transience-implies-ballisticity-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/220.md" }, "peer_review": "not yet", "episode": { "video_id": "8XD3Lr5MJI0", "length": "4:15", "title": "Escape, but never speed up?", "page": "/episodes/220-rwre-zero-one-ballisticity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/220-rwre-zero-one-ballisticity/" }, { "num": "221", "slug": "221-mezard-parisi-diluted-spin-glasses", "title": "The Mézard–Parisi formula for diluted spin glasses", "title_html": "The Mézard–Parisi formula for diluted spin glasses", "summary_html": "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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The Mézard–Parisi formula for diluted spin glasses", "title_html": "The Mézard–Parisi formula for diluted spin glasses", "url": "https://github.com/openai/math/blob/main/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/221.md" }, "peer_review": "not yet", "episode": { "video_id": "_tOVFzVixkQ", "length": "5:08", "title": "Many states, one exact formula?", "page": "/episodes/221-mezard-parisi-diluted-spin-glasses/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/221-mezard-parisi-diluted-spin-glasses/" }, { "num": "222", "slug": "222-perceptron-free-energies-jamming", "title": "Perceptron free energies and microscopic jamming exponents", "title_html": "Perceptron free energies and microscopic jamming exponents", "summary_html": "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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The free energy of the Ising random perceptron", "title_html": "The free energy of the Ising random perceptron", "url": "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", "role": "main" }, { "title": "Microscopic jamming in the negative spherical perceptron", "title_html": "Microscopic jamming in the negative spherical perceptron", "url": "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", "role": "companion" }, { "title": "The spherical perceptron with bi-orthogonally invariant disorder", "title_html": "The spherical perceptron with bi-orthogonally invariant disorder", "url": "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", "role": "companion" }, { "title": "The free energy of the spherical random perceptron", "title_html": "The free energy of the spherical random perceptron", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/222.md" }, "peer_review": "not yet", "episode": { "video_id": "cL_nksIkYKU", "length": "5:45", "title": "How much can one neuron store?", "page": "/episodes/222-perceptron-free-energies-jamming/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/222-perceptron-free-energies-jamming/" }, { "num": "223", "slug": "223-random-cluster-interfaces-sle", "title": "Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling", "title_html": "Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling", "summary_html": "Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for 0<q≤40\\lt q\\le40<q≤4, with κ=4π/arccos⁡(−q/2)\\kappa=4\\pi/\\arccos(-\\sqrt q/2)κ=4π/arccos(−q​/2): bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q < 1, under the stated marked-boundary approximations. For 1≤q≤41\\le q\\le41≤q≤4, complete nested plane loops converge to CLEκ.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Square-lattice FK interfaces and nested loops for 1 <= q < 4", "title_html": "Square-lattice FK interfaces and nested loops for 1 <= q < 4", "url": "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", "role": "main" }, { "title": "Self-dual random-cluster interfaces below one", "title_html": "Self-dual random-cluster interfaces below one", "url": "https://github.com/openai/math/blob/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces", "title_html": "Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces", "url": "https://github.com/openai/math/blob/main/preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf", "role": "companion" }, { "title": "Thermal FK–Ising interfaces and massive SLE", "title_html": "Thermal FK–Ising interfaces and massive SLE", "url": "https://github.com/openai/math/blob/main/preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf", "role": "companion" }, { "title": "Natural Occupation Measures for Critical Square-Lattice FK Interfaces", "title_html": "Natural Occupation Measures for Critical Square-Lattice FK Interfaces", "url": "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", "role": "companion" }, { "title": "Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces", "title_html": "Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces", "url": "https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "m3JWcVU-zfQ", "length": "5:15", "title": "Percolation on the square grid?", "page": "/episodes/223-random-cluster-interfaces-sle/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/223-random-cluster-interfaces-sle/" }, { "num": "224", "slug": "224-voronoi-percolation-cardy", "title": "Critical and quenched near-critical universality for Poisson–Voronoi percolation", "title_html": "Critical and quenched near-critical universality for Poisson–Voronoi percolation", "summary_html": "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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "From critical crossings to quenched near-critical universality in Voronoi percolation", "title_html": "From critical crossings to quenched near-critical universality in Voronoi percolation", "url": "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", "role": "main" }, { "title": "A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula", "title_html": "A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula", "url": "https://github.com/openai/math/blob/main/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf", "role": "companion" }, { "title": "Cardy’s formula for critical Poisson–Voronoi percolation", "title_html": "Cardy’s formula for critical Poisson–Voronoi percolation", "url": "https://github.com/openai/math/blob/main/preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "EAWXMot1hG0", "length": "4:54", "title": "Cardy without a lattice?", "page": "/episodes/224-voronoi-percolation-cardy/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/224-voronoi-percolation-cardy/" }, { "num": "225", "slug": "225-six-vertex-gaussian-free-field", "title": "Gaussian free field limits throughout the balanced six-vertex regime", "title_html": "Gaussian free field limits throughout the balanced six-vertex regime", "summary_html": "The balanced square-lattice six-vertex height field with a=b=1a=b=1a=b=1 and 0<c≤20\\lt c\\le20<c≤2 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 −(2π)−1log⁡∣x−y∣-(2\\pi)^{-1}\\log|x-y|−(2π)−1log∣x−y∣, the exact variance multiplier is 1/arcsin⁡(c/2)1/\\arcsin(c/2)1/arcsin(c/2).", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)", "title_html": "The Gaussian free field limit of the balanced six-vertex model with variance multiplier 1/arcsin(c/2)", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "Lfw-CmqpjuA", "length": "4:26", "title": "How rough is square ice?", "page": "/episodes/225-six-vertex-gaussian-free-field/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/225-six-vertex-gaussian-free-field/" }, { "num": "226", "slug": "226-double-dimer-cle4", "title": "The double-dimer loop ensemble converges to CLE₄", "title_html": "The double-dimer loop ensemble converges to CLE4", "summary_html": "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 CLE4. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The curve scaling limit of half-plane double dimers", "title_html": "The curve scaling limit of half-plane double dimers", "url": "https://github.com/openai/math/blob/main/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "HcF7ZWsBcjI", "length": "4:22", "title": "Two random tilings, one loop soup?", "page": "/episodes/226-double-dimer-cle4/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/226-double-dimer-cle4/" }, { "num": "227", "slug": "227-sk-glauber-dynamics", "title": "Critical SK autocorrelation processes and dynamics across the temperature transition", "title_html": "Critical SK autocorrelation processes and dynamics across the temperature transition", "summary_html": "For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the log⁡n\\log nlogn scale for fixed 0≤β<10\\le\\beta\\lt 10≤β<1, mixing time n2/3+o(1)n^{2/3+o(1)}n2/3+o(1) at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model", "title_html": "Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model", "url": "https://github.com/openai/math/blob/main/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf", "role": "main" }, { "title": "Functional universality of critical SK autocorrelations", "title_html": "Functional universality of critical SK autocorrelations", "url": "https://github.com/openai/math/blob/main/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf", "role": "companion" }, { "title": "A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase", "title_html": "A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase", "url": "https://github.com/openai/math/blob/main/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf", "role": "companion" }, { "title": "Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase", "title_html": "Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase", "url": "https://github.com/openai/math/blob/main/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Critical slowing down in the Sherrington–Kirkpatrick model", "title_html": "Critical slowing down in the Sherrington–Kirkpatrick model", "url": "https://github.com/openai/math/blob/main/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Stretched-exponential barriers for typical SK initial states", "title_html": "Stretched-exponential barriers for typical SK initial states", "url": "https://github.com/openai/math/blob/main/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf", "role": "companion" }, { "title": "A typical-start upper bound for low-temperature SK Glauber dynamics", "title_html": "A typical-start upper bound for low-temperature SK Glauber dynamics", "url": "https://github.com/openai/math/blob/main/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Critical mixing in the Sherrington–Kirkpatrick model", "title_html": "Critical mixing in the Sherrington–Kirkpatrick model", "url": "https://github.com/openai/math/blob/main/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/227.md" }, "peer_review": "not yet", "episode": { "video_id": "spxbIEWr0As", "length": "5:07", "title": "How fast does a spin glass mix?", "page": "/episodes/227-sk-glauber-dynamics/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/227-sk-glauber-dynamics/" }, { "num": "228", "slug": "228-continuum-phase-transition-pair-potential", "title": "Continuum phase transitions for radial pair potentials", "title_html": "Continuum phase transitions for radial pair potentials", "summary_html": "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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "A continuum temperature singularity for a radial pair potential", "title_html": "A continuum temperature singularity for a radial pair potential", "url": "https://github.com/openai/math/blob/main/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf", "role": "main" }, { "title": "A radial continuum phase transition with algebraic decay", "title_html": "A radial continuum phase transition with algebraic decay", "url": "https://github.com/openai/math/blob/main/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/228.md" }, "peer_review": "not yet", "episode": { "video_id": "lCFdtsOAYS8", "length": "4:26", "title": "Can one kind of particle provably change phase?", "page": "/episodes/228-continuum-phase-transition-pair-potential/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/228-continuum-phase-transition-pair-potential/" }, { "num": "229", "slug": "229-reconstruction-thresholds-trees", "title": "Exact three- and four-state reconstruction thresholds and four-state tree capacity", "title_html": "Exact three- and four-state reconstruction thresholds and four-state tree capacity", "summary_html": "Proves the exact reconstruction threshold dλ2>1d\\lambda^2\\gt 1dλ2>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 > 1 and d > 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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model", "title_html": "The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model", "url": "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", "role": "main" }, { "title": "A Capacity Criterion for Four-State Potts Reconstruction on Trees", "title_html": "A Capacity Criterion for Four-State Potts Reconstruction on Trees", "url": "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", "role": "companion" }, { "title": "The exact reconstruction threshold for the three-state symmetric channel", "title_html": "The exact reconstruction threshold for the three-state symmetric channel", "url": "https://github.com/openai/math/blob/main/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/229.md" }, "peer_review": "not yet", "episode": { "video_id": "J_cQPw1LgCE", "length": "4:47", "title": "When does a tree forget its root?", "page": "/episodes/229-reconstruction-thresholds-trees/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/229-reconstruction-thresholds-trees/" }, { "num": "230", "slug": "230-sle-exact-hausdorff-gauge", "title": "Exact Hausdorff gauges for SLE", "title_html": "Exact Hausdorff gauges for SLE", "summary_html": "Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, 0<κ<80\\lt \\kappa\\lt 80<κ<8. The explicit gauge rd(log⁡log⁡(1/r))(2−d)/2r^d(\\log\\log(1/r))^{(2-d)/2}rd(loglog(1/r))(2−d)/2, d=1+κ/8d=1+\\kappa/8d=1+κ/8, gives almost surely positive finite measure to every trace segment γ([s,t])\\gamma([s,t])γ([s,t]) with 0<s<t<∞0\\lt s\\lt t\\lt \\infty0<s<t<∞, and finite expected measure to the trace in every bounded disk.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "An exact Hausdorff gauge for SLE", "title_html": "An exact Hausdorff gauge for SLE", "url": "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", "role": "main" }, { "title": "An explicit exact Hausdorff gauge for SLE", "title_html": "An explicit exact Hausdorff gauge for SLE", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/230.md" }, "peer_review": "not yet", "episode": { "video_id": "W6QezbbBVEU", "length": "4:35", "title": "How long is a random curve?", "page": "/episodes/230-sle-exact-hausdorff-gauge/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/230-sle-exact-hausdorff-gauge/" }, { "num": "231", "slug": "231-free-spanning-forest-factor-iid", "title": "The free uniform spanning forest is a factor of IID", "title_html": "The free uniform spanning forest is a factor of IID", "summary_html": "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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The free uniform spanning forest is a factor of IID", "title_html": "The free uniform spanning forest is a factor of IID", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/231.md" }, "peer_review": "not yet", "episode": { "video_id": "4c9oxFjDWMQ", "length": "4:33", "title": "Can coin flips grow a forest?", "page": "/episodes/231-free-spanning-forest-factor-iid/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/231-free-spanning-forest-factor-iid/" }, { "num": "232", "slug": "232-lipschitz-heights-gff", "title": "Gaussian fields and interfaces for triangular-lattice Lipschitz heights", "title_html": "Gaussian fields and interfaces for triangular-lattice Lipschitz heights", "summary_html": "Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments 0,±20,\\pm20,±2 and two-arc boundary values ±1\\pm1±1, and zero-boundary integer Lipschitz heights weighted by fixed x∈[1/2,1]x\\in[1/\\sqrt2,1]x∈[1/2​,1]. Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE4, establishing Schramm’s real-field/interface predictions.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Gaussian free-field limits of weighted integer Lipschitz heights", "title_html": "Gaussian free-field limits of weighted integer Lipschitz heights", "url": "https://github.com/openai/math/blob/main/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026/paper.pdf", "role": "main" }, { "title": "The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data", "title_html": "The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data", "url": "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", "role": "companion" }, { "title": "Uniform real Lipschitz surfaces on the triangular lattice", "title_html": "Uniform real Lipschitz surfaces on the triangular lattice", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-October-6-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "9mYH9o4fnGc", "length": "4:40", "title": "What does a random landscape look like?", "page": "/episodes/232-lipschitz-heights-gff/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/232-lipschitz-heights-gff/" }, { "num": "233", "slug": "233-ashkin-teller-current-limit", "title": "The joint critical Ashkin–Teller current limit", "title_html": "The joint critical Ashkin–Teller current limit", "summary_html": "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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The joint scaling limit of critical Ashkin-Teller currents", "title_html": "The joint scaling limit of critical Ashkin-Teller currents", "url": "https://github.com/openai/math/blob/main/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-October-6-2026/main.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "UFxSI4-UTQM", "length": "4:32", "title": "Two magnets, one free field?", "page": "/episodes/233-ashkin-teller-current-limit/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/233-ashkin-teller-current-limit/" }, { "num": "234", "slug": "234-orthogonal-ising-spin-glass-pressure", "title": "All-temperature pressure of orthogonally invariant Ising spin glasses", "title_html": "All-temperature pressure of orthogonally invariant Ising spin glasses", "summary_html": "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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "All-temperature pressure for orthogonally invariant Ising spin glasses", "title_html": "All-temperature pressure for orthogonally invariant Ising spin glasses", "url": "https://github.com/openai/math/blob/main/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/234.md" }, "peer_review": "not yet", "episode": { "video_id": "zwphYB7iWGw", "length": "4:34", "title": "Only the eigenvalues matter?", "page": "/episodes/234-orthogonal-ising-spin-glass-pressure/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/234-orthogonal-ising-spin-glass-pressure/" }, { "num": "235", "slug": "235-random-sat-thresholds", "title": "Limiting random SAT thresholds, sharp variance and computability", "title_html": "Limiting random SAT thresholds, sharp variance and computability", "summary_html": "For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance Θk(n)\\Theta_k(n)Θk​(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, made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "A Limiting Satisfiability Threshold for Every Fixed Clause Size", "title_html": "A Limiting Satisfiability Threshold for Every Fixed Clause Size", "url": "https://github.com/openai/math/blob/main/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf", "role": "main" }, { "title": "Linear Variance of the Random 3-SAT Hitting Time", "title_html": "Linear Variance of the Random 3-SAT Hitting Time", "url": "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", "role": "companion" }, { "title": "Variance of the Random k-SAT Hitting Time", "title_html": "Variance of the Random k-SAT Hitting Time", "url": "https://github.com/openai/math/blob/main/preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026/article.pdf", "role": "companion" }, { "title": "Computing the Random 3-SAT Threshold", "title_html": "Computing the Random 3-SAT Threshold", "url": "https://github.com/openai/math/blob/main/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/235.md" }, "peer_review": "not yet", "episode": { "video_id": "iwsJy2wAJN4", "length": "4:33", "title": "When does random 3-SAT break?", "page": "/episodes/235-random-sat-thresholds/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/235-random-sat-thresholds/" }, { "num": "236", "slug": "236-free-ising-tree-factor-threshold", "title": "The exact factor-of-IID threshold for free Ising spins on trees", "title_html": "The exact factor-of-IID threshold for free Ising spins on trees", "summary_html": "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⁡β≤(d−1)−1/2\\tanh\\beta\\le(d-1)^{-1/2}tanhβ≤(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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The sharp factor-of-IID threshold for the free Ising model on regular trees", "title_html": "The sharp factor-of-IID threshold for the free Ising model on regular trees", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/236.md" }, "peer_review": "not yet", "episode": { "video_id": "QO2S5J49hnE", "length": "4:45", "title": "Coin flips up to the knife edge?", "page": "/episodes/236-free-ising-tree-factor-threshold/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/236-free-ising-tree-factor-threshold/" }, { "num": "237", "slug": "237-honeycomb-saw-three-quarter", "title": "The three-quarter exponent for honeycomb self-avoiding walk", "title_html": "The three-quarter exponent for honeycomb self-avoiding walk", "summary_html": "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 n3/4+o(1)n^{3/4+o(1)}n3/4+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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Radial transfer estimates and polygon length laws for honeycomb walks", "title_html": "Radial transfer estimates and polygon length laws for honeycomb walks", "url": "https://github.com/openai/math/blob/main/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf", "role": "main" }, { "title": "Critical honeycomb chords with prescribed boundary endpoints", "title_html": "Critical honeycomb chords with prescribed boundary endpoints", "url": "https://github.com/openai/math/blob/main/preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf", "role": "companion" }, { "title": "Cylinder loop weights and planar nesting", "title_html": "Cylinder loop weights and planar nesting", "url": "https://github.com/openai/math/blob/main/preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026/main.pdf", "role": "companion" }, { "title": "Mass and covering exponents for fixed-length honeycomb walks", "title_html": "Mass and covering exponents for fixed-length honeycomb walks", "url": "https://github.com/openai/math/blob/main/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf", "role": "companion" }, { "title": "Signed cylinder propagation and marked polygons on the honeycomb lattice", "title_html": "Signed cylinder propagation and marked polygons on the honeycomb lattice", "url": "https://github.com/openai/math/blob/main/preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf", "role": "companion" }, { "title": "Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice", "title_html": "Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice", "url": "https://github.com/openai/math/blob/main/preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026/main.pdf", "role": "companion" }, { "title": "Marked polygon correlations and one-arc bounds", "title_html": "Marked polygon correlations and one-arc bounds", "url": "https://github.com/openai/math/blob/main/preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026/main.pdf", "role": "companion" }, { "title": "Disk transfer representations and confined bridge mass", "title_html": "Disk transfer representations and confined bridge mass", "url": "https://github.com/openai/math/blob/main/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf", "role": "companion" }, { "title": "Polynomial vacuum representations and bridge mass for honeycomb walks", "title_html": "Polynomial vacuum representations and bridge mass for honeycomb walks", "url": "https://github.com/openai/math/blob/main/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf", "role": "companion" }, { "title": "Renewal and changes of law for critical honeycomb walks", "title_html": "Renewal and changes of law for critical honeycomb walks", "url": "https://github.com/openai/math/blob/main/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf", "role": "companion" }, { "title": "Critical strip-crossing mass on the honeycomb lattice", "title_html": "Critical strip-crossing mass on the honeycomb lattice", "url": "https://github.com/openai/math/blob/main/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf", "role": "companion" }, { "title": "Uniform marked-polygon estimates and sharp finite bridge moments", "title_html": "Uniform marked-polygon estimates and sharp finite bridge moments", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf", "role": "companion" }, { "title": "Cap-selected amplitudes and triangle chords for honeycomb walks", "title_html": "Cap-selected amplitudes and triangle chords for honeycomb walks", "url": "https://github.com/openai/math/blob/main/preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026/main.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/237.md" }, "peer_review": "not yet", "episode": { "video_id": "fDSUfvhWSCM", "length": "4:34", "title": "How far does a walk that never crosses itself go?", "page": "/episodes/237-honeycomb-saw-three-quarter/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/237-honeycomb-saw-three-quarter/" }, { "num": "238", "slug": "238-thorp-shuffle-log-mixing", "title": "Optimal logarithmic mixing of the Thorp shuffle", "title_html": "Optimal logarithmic mixing of the Thorp shuffle", "summary_html": "Proves that the Thorp shuffle randomizes N=2dN=2^dN=2d labeled cards in Θ(log⁡N)\\Theta(\\log N)Θ(logN) 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.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "Optimal-order mixing of the Thorp shuffle", "title_html": "Optimal-order mixing of the Thorp shuffle", "url": "https://github.com/openai/math/blob/main/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf", "role": "main" }, { "title": "Random coordinate frames and partial permutation laws", "title_html": "Random coordinate frames and partial permutation laws", "url": "https://github.com/openai/math/blob/main/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf", "role": "companion" }, { "title": "From partial permutation information to Fourier bounds", "title_html": "From partial permutation information to Fourier bounds", "url": "https://github.com/openai/math/blob/main/preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf", "role": "companion" }, { "title": "Conditional information under deterministic coordinate sweeps", "title_html": "Conditional information under deterministic coordinate sweeps", "url": "https://github.com/openai/math/blob/main/preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026/main.pdf", "role": "companion" }, { "title": "Conditional permutations in a revealed switching environment", "title_html": "Conditional permutations in a revealed switching environment", "url": "https://github.com/openai/math/blob/main/preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026/paper.pdf", "role": "companion" }, { "title": "Routing densities and representation contraction for Thorp sweeps", "title_html": "Routing densities and representation contraction for Thorp sweeps", "url": "https://github.com/openai/math/blob/main/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf", "role": "companion" }, { "title": "Row–column symmetry and contraction of coordinate sweeps", "title_html": "Row–column symmetry and contraction of coordinate sweeps", "url": "https://github.com/openai/math/blob/main/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026/paper.pdf", "role": "companion" }, { "title": "Random-subspace tests and trace smoothing for coordinate sweeps", "title_html": "Random-subspace tests and trace smoothing for coordinate sweeps", "url": "https://github.com/openai/math/blob/main/preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf", "role": "companion" }, { "title": "Compatibility entropy and the spectrum of a Thorp sweep", "title_html": "Compatibility entropy and the spectrum of a Thorp sweep", "url": "https://github.com/openai/math/blob/main/preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026/paper.pdf", "role": "companion" }, { "title": "Signed tensor densities and diagram budgets for the Thorp shuffle", "title_html": "Signed tensor densities and diagram budgets for the Thorp shuffle", "url": "https://github.com/openai/math/blob/main/preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026/paper.pdf", "role": "companion" }, { "title": "Conditional coordinate sweeps and analytic transfer", "title_html": "Conditional coordinate sweeps and analytic transfer", "url": "https://github.com/openai/math/blob/main/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026/main.pdf", "role": "companion" }, { "title": "A strict four-row permanent inequality and permutation moments", "title_html": "A strict four-row permanent inequality and permutation moments", "url": "https://github.com/openai/math/blob/main/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/238.md" }, "peer_review": "not yet", "episode": { "video_id": "uNOpfO54IdI", "length": "4:33", "title": "How many shuffles randomize a deck?", "page": "/episodes/238-thorp-shuffle-log-mixing/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/238-thorp-shuffle-log-mixing/" }, { "num": "239", "slug": "239-symmetric-sign-matrix-singularity", "title": "Sharp singularity rates for symmetric random sign matrices", "title_html": "Sharp singularity rates for symmetric random sign matrices", "summary_html": "Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give Pr⁡(det⁡An=0)=(1/2+o(1))n\\Pr(\\det A_n=0)=(1/2+o(1))^nPr(detAn​=0)=(1/2+o(1))n; for fixed bias p∈(0,1)∖{1/2}p\\in(0,1)\\setminus\\{1/2\\}p∈(0,1)∖{1/2}, the rate is (p2+(1−p)2+o(1))n(p^2+(1-p)^2+o(1))^n(p2+(1−p)2+o(1))n. In the biased case, agreeing rows attain this rate.", "subject": "Probability and statistical mechanics", "subject_slug": "probability-and-statistical-mechanics", "papers": [ { "title": "The sharp exponential rate of singularity for symmetric Bernoulli matrices", "title_html": "The sharp exponential rate of singularity for symmetric Bernoulli matrices", "url": "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", "role": "main" }, { "title": "The sharp singularity rate for biased symmetric sign matrices", "title_html": "The sharp singularity rate for biased symmetric sign matrices", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "jVk3xI3ecwE", "length": "4:14", "title": "When is a random matrix singular?", "page": "/episodes/239-symmetric-sign-matrix-singularity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/239-symmetric-sign-matrix-singularity/" }, { "num": "240", "slug": "240-eventual-categoricity-aec", "title": "Shelah's eventual categoricity and the prescribed-threshold obstruction", "title_html": "Shelah's eventual categoricity and the prescribed-threshold obstruction", "summary_html": "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.", "subject": "Mathematical logic", "subject_slug": "mathematical-logic", "papers": [ { "title": "A CH obstruction to a prescribed categoricity threshold", "title_html": "A CH obstruction to a prescribed categoricity threshold", "url": "https://github.com/openai/math/blob/main/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf", "role": "main" }, { "title": "Eventual categoricity for abstract elementary classes", "title_html": "Eventual categoricity for abstract elementary classes", "url": "https://github.com/openai/math/blob/main/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/240.md" }, "peer_review": "not yet", "episode": { "video_id": "yRPWfppI5Mg", "length": "5:05", "title": "One model per size, eventually?", "page": "/episodes/240-eventual-categoricity-aec/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/240-eventual-categoricity-aec/" }, { "num": "241", "slug": "241-turing-degrees-rigidity", "title": "Rigidity of the Turing degrees", "title_html": "Rigidity of the Turing degrees", "summary_html": "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.", "subject": "Mathematical logic", "subject_slug": "mathematical-logic", "papers": [ { "title": "Rigidity of the Turing degrees", "title_html": "Rigidity of the Turing degrees", "url": "https://github.com/openai/math/blob/main/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/241.md" }, "peer_review": "not yet", "episode": { "video_id": "CrkuweE4Nhc", "length": "4:35", "title": "Can the Turing degrees be relabeled?", "page": "/episodes/241-turing-degrees-rigidity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/241-turing-degrees-rigidity/" }, { "num": "242", "slug": "242-single-fold-diophantine", "title": "Single-fold Diophantine representations and undecidability under an at-most-one-solution promise", "title_html": "Single-fold Diophantine representations and undecidability under an at-most-one-solution promise", "summary_html": "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.", "subject": "Mathematical logic", "subject_slug": "mathematical-logic", "papers": [ { "title": "Single-fold Diophantine representations", "title_html": "Single-fold Diophantine representations", "url": "https://github.com/openai/math/blob/main/preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/242.md" }, "peer_review": "not yet", "episode": { "video_id": "GhAK8-6BBqg", "length": "4:12", "title": "One solution per member, exactly?", "page": "/episodes/242-single-fold-diophantine/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/242-single-fold-diophantine/" }, { "num": "243", "slug": "243-choiceless-polynomial-time", "title": "Separating choiceless counting from polynomial time and witnessed choice", "title_html": "Separating choiceless counting from polynomial time and witnessed choice", "summary_html": "Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 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.", "subject": "Mathematical logic", "subject_slug": "mathematical-logic", "papers": [ { "title": "Choiceless polynomial time with counting does not capture polynomial time", "title_html": "Choiceless polynomial time with counting does not capture polynomial time", "url": "https://github.com/openai/math/blob/main/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf", "role": "main" }, { "title": "Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting", "title_html": "Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting", "url": "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", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/243.md" }, "peer_review": "not yet", "episode": { "video_id": "jYw15UVbSHs", "length": "4:32", "title": "Is there a logic for polynomial time?", "page": "/episodes/243-choiceless-polynomial-time/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/243-choiceless-polynomial-time/" }, { "num": "244", "slug": "244-partition-principle-not-choice", "title": "The Partition Principle does not imply Choice", "title_html": "The Partition Principle does not imply Choice", "summary_html": "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.", "subject": "Mathematical logic", "subject_slug": "mathematical-logic", "papers": [ { "title": "The Partition Principle does not imply Choice", "title_html": "The Partition Principle does not imply Choice", "url": "https://github.com/openai/math/blob/main/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/244.md" }, "peer_review": "not yet", "episode": { "video_id": "AjdKhhvKG0c", "length": "4:16", "title": "No more pieces than points, yet no choice?", "page": "/episodes/244-partition-principle-not-choice/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/244-partition-principle-not-choice/" }, { "num": "245", "slug": "245-wn-implies-sn-pure-type-systems", "title": "Weak normalization implies strong normalization in pure type systems", "title_html": "Weak normalization implies strong normalization in pure type systems", "summary_html": "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.", "subject": "Mathematical logic", "subject_slug": "mathematical-logic", "papers": [ { "title": "Weak and strong normalization in pure type systems", "title_html": "Weak and strong normalization in pure type systems", "url": "https://github.com/openai/math/blob/main/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/245.md" }, "peer_review": "not yet", "episode": { "video_id": "Vw308GM-OXg", "length": "4:22", "title": "If every program can finish, must it?", "page": "/episodes/245-wn-implies-sn-pure-type-systems/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/245-wn-implies-sn-pure-type-systems/" }, { "num": "246", "slug": "246-cannon-conjecture", "title": "Cannon's conjecture", "title_html": "Cannon's conjecture", "summary_html": "Every word-hyperbolic group with boundary homeomorphic to S2 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.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "A Modulus Proof of Cannon’s Conjecture", "title_html": "A Modulus Proof of Cannon’s Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/246.md" }, "peer_review": "not yet", "episode": { "video_id": "eJcdzZu2TQo", "length": "4:42", "title": "Boundary a sphere, so the group is 3D hyperbolic?", "page": "/episodes/246-cannon-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/246-cannon-conjecture/" }, { "num": "247", "slug": "247-finitely-presented-periodic-group", "title": "An infinite finitely presented residually finite 2-group and a finitely presented nil algebra", "title_html": "An infinite finitely presented residually finite 2-group and a finitely presented nil algebra", "summary_html": "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 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "An infinite finitely presented residually finite 2-group", "title_html": "An infinite finitely presented residually finite 2-group", "url": "https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf", "role": "main" }, { "title": "An infinite finitely presented periodic group", "title_html": "An infinite finitely presented periodic group", "url": "https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/247.md" }, "peer_review": "not yet", "episode": { "video_id": "-tccrTfLKE8", "length": "4:48", "title": "Finite rules, finite orders, still infinite?", "page": "/episodes/247-finitely-presented-periodic-group/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/247-finitely-presented-periodic-group/" }, { "num": "248", "slug": "248-thompson-group-f-nonamenable", "title": "Thompson's group F is nonamenable", "title_html": "Thompson's group F is nonamenable", "summary_html": "Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "Thompson's group F is nonamenable", "title_html": "Thompson's group F is nonamenable", "url": "https://github.com/openai/math/blob/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/248.md" }, "peer_review": "not yet", "episode": { "video_id": "nb5_d2aqrKc", "length": "4:46", "title": "Can this group be averaged?", "page": "/episodes/248-thompson-group-f-nonamenable/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/248-thompson-group-f-nonamenable/" }, { "num": "249", "slug": "249-eilenberg-ganea-counterexample", "title": "A finitely generated Eilenberg–Ganea counterexample", "title_html": "A finitely generated Eilenberg–Ganea counterexample", "summary_html": "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.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "A finitely generated counterexample to the Eilenberg–Ganea conjecture", "title_html": "A finitely generated counterexample to the Eilenberg–Ganea conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/249.md" }, "peer_review": "not yet", "episode": { "video_id": "RncW-5LLK3w", "length": "4:34", "title": "Dimension two, or three?", "page": "/episodes/249-eilenberg-ganea-counterexample/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/249-eilenberg-ganea-counterexample/" }, { "num": "250", "slug": "250-boone-higman-conjecture", "title": "Boone–Higman embeddings with higher finiteness", "title_html": "Boone–Higman embeddings with higher finiteness", "summary_html": "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∞: a classifying space with finitely many cells in each dimension. A single group of type F∞ can also contain every finitely presented group.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "Finite algebraic envelopes and the Boone–Higman conjecture", "title_html": "Finite algebraic envelopes and the Boone–Higman conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf", "role": "main" }, { "title": "Simple F∞ overgroups of groups with decidable word problem", "title_html": "Simple F∞ overgroups of groups with decidable word problem", "url": "https://github.com/openai/math/blob/main/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf", "role": "companion" }, { "title": "A universal group of type F∞", "title_html": "A universal group of type F∞", "url": "https://github.com/openai/math/blob/main/preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/250.md" }, "peer_review": "not yet", "episode": { "video_id": "MxQppitAgxI", "length": "4:46", "title": "Decidable means simple inside?", "page": "/episodes/250-boone-higman-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/250-boone-higman-conjecture/" }, { "num": "251", "slug": "251-dixmier-unitarizability-ulam", "title": "Amenability, unitarizability, and strong Ulam stability", "title_html": "Amenability, unitarizability, and strong Ulam stability", "summary_html": "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.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "Unitarizability implies amenability for discrete groups", "title_html": "Unitarizability implies amenability for discrete groups", "url": "https://github.com/openai/math/blob/main/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf", "role": "main" }, { "title": "Strong Ulam Stability Characterizes Amenability", "title_html": "Strong Ulam Stability Characterizes Amenability", "url": "https://github.com/openai/math/blob/main/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/251.md" }, "peer_review": "not yet", "episode": { "video_id": "Dt1LwSr_Uso", "length": "4:36", "title": "Average it, straighten it?", "page": "/episodes/251-dixmier-unitarizability-ulam/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/251-dixmier-unitarizability-ulam/" }, { "num": "252", "slug": "252-non-residually-finite-hyperbolic-group", "title": "A torsion-free hyperbolic group that is neither residually finite nor linear over any field", "title_html": "A torsion-free hyperbolic group that is neither residually finite nor linear over any field", "summary_html": "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.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "A torsion-free hyperbolic group that is not residually finite", "title_html": "A torsion-free hyperbolic group that is not residually finite", "url": "https://github.com/openai/math/blob/main/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/252.md" }, "peer_review": "not yet", "episode": { "video_id": "bf3PchHYWCg", "length": "5:08", "title": "Curved, yet hidden from finite shadows?", "page": "/episodes/252-non-residually-finite-hyperbolic-group/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/252-non-residually-finite-hyperbolic-group/" }, { "num": "253", "slug": "253-finitely-presented-simple-amenable-group", "title": "An infinite finitely presented simple amenable group", "title_html": "An infinite finitely presented simple amenable group", "summary_html": "Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "An infinite finitely presented simple amenable group", "title_html": "An infinite finitely presented simple amenable group", "url": "https://github.com/openai/math/blob/main/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/253.md" }, "peer_review": "not yet", "episode": { "video_id": "ob1ktdL3ztQ", "length": "4:13", "title": "Simple, amenable, finitely presented?", "page": "/episodes/253-finitely-presented-simple-amenable-group/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/253-finitely-presented-simple-amenable-group/" }, { "num": "254", "slug": "254-artin-groups-kpi1", "title": "Classifying spaces and geometric obstructions for Artin groups", "title_html": "Classifying spaces and geometric obstructions for Artin groups", "summary_html": "The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin K(π,1)K(\\pi,1)K(π,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)(0)(0) space.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "Harmonic heights and the Artin K(pi,1) conjecture", "title_html": "Harmonic heights and the Artin K(pi,1) conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026/paper.pdf", "role": "main" }, { "title": "An Artin group with no geometric CAT(0) action", "title_html": "An Artin group with no geometric CAT(0) action", "url": "https://github.com/openai/math/blob/main/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Parabolic intersections in Artin groups", "title_html": "Parabolic intersections in Artin groups", "url": "https://github.com/openai/math/blob/main/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/254.md" }, "peer_review": "not yet", "episode": { "video_id": "PDRhXsZ7YjM", "length": "5:00", "title": "Braids, and every relative of them?", "page": "/episodes/254-artin-groups-kpi1/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/254-artin-groups-kpi1/" }, { "num": "255", "slug": "255-virtually-polycyclic-qi-recognition", "title": "Quasi-isometric recognition of virtually polycyclic groups", "title_html": "Quasi-isometric recognition of virtually polycyclic groups", "summary_html": "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.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "Quasi-isometric recognition of virtually polycyclic groups", "title_html": "Quasi-isometric recognition of virtually polycyclic groups", "url": "https://github.com/openai/math/blob/main/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/255.md" }, "peer_review": "not yet", "episode": { "video_id": "9TE2TR9bn3U", "length": "5:02", "title": "Seen from far away, still polycyclic?", "page": "/episodes/255-virtually-polycyclic-qi-recognition/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/255-virtually-polycyclic-qi-recognition/" }, { "num": "256", "slug": "256-kervaire-howie-conjectures", "title": "Nonsingular systems of equations over arbitrary groups", "title_html": "Nonsingular systems of equations over arbitrary groups", "summary_html": "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.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "Nonsingular systems of equations over arbitrary groups", "title_html": "Nonsingular systems of equations over arbitrary groups", "url": "https://github.com/openai/math/blob/main/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf", "role": "main" }, { "title": "The Kervaire theorem for groups", "title_html": "The Kervaire theorem for groups", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/256.md" }, "peer_review": "not yet", "episode": { "video_id": "VEXw1xdWvYc", "length": "5:16", "title": "Can one equation kill a group?", "page": "/episodes/256-kervaire-howie-conjectures/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/256-kervaire-howie-conjectures/" }, { "num": "257", "slug": "257-hyperbolic-group-not-cat0", "title": "A hyperbolic group without a geometric CAT(0) action", "title_html": "A hyperbolic group without a geometric CAT(0) action", "summary_html": "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 CAT(0)\\mathop{\\mathrm{CAT}}\\nolimits (0)CAT(0) space, in any dimension.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "A hyperbolic group with no geometric CAT(0) action", "title_html": "A hyperbolic group with no geometric CAT(0) action", "url": "https://github.com/openai/math/blob/main/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/257.md" }, "peer_review": "not yet", "episode": { "video_id": "5arhQdq9Hac", "length": "4:25", "title": "Hyperbolic, but never CAT(0)?", "page": "/episodes/257-hyperbolic-group-not-cat0/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/257-hyperbolic-group-not-cat0/" }, { "num": "258", "slug": "258-gersten-one-relator-hyperbolic", "title": "Gersten’s conjecture and virtual compact specialness of one-relator groups", "title_html": "Gersten’s conjecture and virtual compact specialness of one-relator groups", "summary_html": "Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup BS(m,n)\\mathrm{BS}(m,n)BS(m,n), with m,n≠0m,n\\ne0m,n=0, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "Baumslag-Solitar-free one-relator groups are hyperbolic", "title_html": "Baumslag-Solitar-free one-relator groups are hyperbolic", "url": "https://github.com/openai/math/blob/main/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf", "role": "main" }, { "title": "Virtual compact specialness of hyperbolic one-relator groups", "title_html": "Virtual compact specialness of hyperbolic one-relator groups", "url": "https://github.com/openai/math/blob/main/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "c0D-TQsoVrU", "length": "4:05", "title": "One relation, one way to fail?", "page": "/episodes/258-gersten-one-relator-hyperbolic/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/258-gersten-one-relator-hyperbolic/" }, { "num": "259", "slug": "259-group-without-fixed-price", "title": "A group without fixed price", "title_html": "A group without fixed price", "summary_html": "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.", "subject": "Group theory", "subject_slug": "group-theory", "papers": [ { "title": "A group without fixed price", "title_html": "A group without fixed price", "url": "https://github.com/openai/math/blob/main/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "ZGKQVNAitek", "length": "3:42", "title": "Same group, two prices?", "page": "/episodes/259-group-without-fixed-price/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/259-group-without-fixed-price/" }, { "num": "260", "slug": "260-spacetime-penrose-inequality", "title": "Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions", "title_html": "Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "The spacetime Penrose inequality with charge and original-data rigidity", "title_html": "The spacetime Penrose inequality with charge and original-data rigidity", "url": "https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf", "role": "main" }, { "title": "A Charged Reduction of the Spacetime Penrose Inequality in Spatial Dimensions at Least Four", "title_html": "A Charged Reduction of the Spacetime Penrose Inequality in Spatial Dimensions at Least Four", "url": "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", "role": "companion" }, { "title": "Spacetime Penrose inequalities: enclosing area, charge, and rigidity", "title_html": "Spacetime Penrose inequalities: enclosing area, charge, and rigidity", "url": "https://github.com/openai/math/blob/main/preprints/Spacetime-Penrose-inequalities-enclosing-area-charge-and-rigidity-October-5-2026/paper.pdf", "role": "companion" }, { "title": "The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors", "title_html": "The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors", "url": "https://github.com/openai/math/blob/main/preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026/paper.pdf", "role": "companion" }, { "title": "Electromagnetic tails and the Kerr–Newman Penrose inequality", "title_html": "Electromagnetic tails and the Kerr–Newman Penrose inequality", "url": "https://github.com/openai/math/blob/main/preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026/paper.pdf", "role": "companion" }, { "title": "The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity", "title_html": "The nonmaximal anti-de Sitter Penrose Inequality and original-data rigidity", "url": "https://github.com/openai/math/blob/main/preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf", "role": "companion" }, { "title": "The Penrose inequality for maximal asymptotically hyperbolic initial data", "title_html": "The Penrose inequality for maximal asymptotically hyperbolic initial data", "url": "https://github.com/openai/math/blob/main/preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf", "role": "companion" }, { "title": "A local Penrose inequality for conformal perturbations of Schwarzschild–anti-de Sitter data", "title_html": "A local Penrose inequality for conformal perturbations of Schwarzschild–anti-de Sitter data", "url": "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", "role": "companion" }, { "title": "The spacetime Penrose inequality and enclosing area", "title_html": "The spacetime Penrose inequality and enclosing area", "url": "https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Conformal flow and the Riemannian Penrose inequality with minimizing frontiers", "title_html": "Conformal flow and the Riemannian Penrose inequality with minimizing frontiers", "url": "https://github.com/openai/math/blob/main/preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Equality and rigidity in the spacetime Penrose inequality", "title_html": "Equality and rigidity in the spacetime Penrose inequality", "url": "https://github.com/openai/math/blob/main/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Boundary graph deformations for the spacetime Penrose inequality", "title_html": "Boundary graph deformations for the spacetime Penrose inequality", "url": "https://github.com/openai/math/blob/main/preprints/Boundary-graph-deformations-for-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Area-controlled end replacement and the Bondi Penrose inequality in the CKS class", "title_html": "Area-controlled end replacement and the Bondi Penrose inequality in the CKS class", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/260.md" }, "peer_review": "not yet", "episode": { "video_id": "Oc398yZAXJg", "length": "4:54", "title": "How heavy must a black hole be?", "page": "/episodes/260-spacetime-penrose-inequality/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/260-spacetime-penrose-inequality/" }, { "num": "261", "slug": "261-anderson-localization-delocalization", "title": "Localization and delocalization in the Anderson model", "title_html": "Localization and delocalization in the Anderson model", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three", "title_html": "Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three", "url": "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", "role": "main" }, { "title": "Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder", "title_html": "Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/261.md" }, "peer_review": "not yet", "episode": { "video_id": "-b7yvHNzf9U", "length": "4:16", "title": "Trapped, or free?", "page": "/episodes/261-anderson-localization-delocalization/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/261-anderson-localization-delocalization/" }, { "num": "262", "slug": "262-lieb-thirring-sharp-matrix", "title": "Sharp finite-matrix Lieb–Thirring inequalities and all equality cases", "title_html": "Sharp finite-matrix Lieb–Thirring inequalities and all equality cases", "summary_html": "Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2<γ<3/21/2\\lt \\gamma\\lt 3/21/2<γ<3/2 and arbitrary finite-matrix potentials W ≥ 0 with ∫tr(Wγ+1/2)<∞\\int\\mathop{\\mathrm{tr}}\\nolimits (W^{\\gamma+1/2})\\lt \\infty∫tr(Wγ+1/2)<∞: 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 sech2 solitons with independent scales and centers, and zero channels.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality", "title_html": "Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality", "url": "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", "role": "main" }, { "title": "Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials", "title_html": "Sharp one-dimensional Lieb–Thirring inequalities for matrix potentials", "url": "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", "role": "companion" }, { "title": "Sharp one-dimensional Lieb–Thirring constants", "title_html": "Sharp one-dimensional Lieb–Thirring constants", "url": "https://github.com/openai/math/blob/main/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/262.md" }, "peer_review": "not yet", "episode": { "video_id": "Af1rkw8xqGg", "length": "3:47", "title": "One bound state is the best?", "page": "/episodes/262-lieb-thirring-sharp-matrix/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/262-lieb-thirring-sharp-matrix/" }, { "num": "263", "slug": "263-ionization-conjecture", "title": "The ionization and generalized ionization conjectures", "title_html": "The ionization and generalized ionization conjectures", "summary_html": "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+CMZ+CMZ+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→∞m\\to\\inftym→∞ and Z/m→∞Z/m\\to\\inftyZ/m→∞; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking Z→∞Z\\to\\inftyZ→∞ first.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms", "title_html": "Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms", "url": "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", "role": "main" }, { "title": "Generalized ionization energies for full Coulomb atoms", "title_html": "Generalized ionization energies for full Coulomb atoms", "url": "https://github.com/openai/math/blob/main/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Generalized outer-electron radii of neutral Coulomb atoms", "title_html": "Generalized outer-electron radii of neutral Coulomb atoms", "url": "https://github.com/openai/math/blob/main/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/263.md" }, "peer_review": "not yet", "episode": { "video_id": "hj0V05s7vcc", "length": "3:50", "title": "How many electrons can an atom hold?", "page": "/episodes/263-ionization-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/263-ionization-conjecture/" }, { "num": "264", "slug": "264-strong-cosmic-censorship-kerr", "title": "Strong cosmic censorship near two-ended Kerr data", "title_html": "Strong cosmic censorship near two-ended Kerr data", "summary_html": "Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense Gδ 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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime", "title_html": "Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime", "url": "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", "role": "main" }, { "title": "Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes", "title_html": "Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes", "url": "https://github.com/openai/math/blob/main/preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility", "title_html": "Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility", "url": "https://github.com/openai/math/blob/main/preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "qWQ_tXTj21Q", "length": "4:03", "title": "Is the future determined?", "page": "/episodes/264-strong-cosmic-censorship-kerr/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/264-strong-cosmic-censorship-kerr/" }, { "num": "265", "slug": "265-two-dimensional-area-law", "title": "Area laws and tensor networks for two-dimensional gapped systems", "title_html": "Area laws and tensor networks for two-dimensional gapped systems", "summary_html": "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×LL\\times LL×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−1.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "A two-dimensional area law from a global spectral gap", "title_html": "A two-dimensional area law from a global spectral gap", "url": "https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf", "role": "main" }, { "title": "Polynomial PEPS approximation of gapped square-grid ground states", "title_html": "Polynomial PEPS approximation of gapped square-grid ground states", "url": "https://github.com/openai/math/blob/main/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "UY6Gbdw9sgg", "length": "3:46", "title": "Only the boundary counts?", "page": "/episodes/265-two-dimensional-area-law/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/265-two-dimensional-area-law/" }, { "num": "266", "slug": "266-three-mubs-dimension-six", "title": "Exactly three mutually unbiased bases in dimension six", "title_html": "Exactly three mutually unbiased bases in dimension six", "summary_html": "Proves N(6)=3N(6)=3N(6)=3, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, 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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "The maximum number of mutually unbiased bases in dimension six", "title_html": "The maximum number of mutually unbiased bases in dimension six", "url": "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", "role": "main" }, { "title": "Exact Fourier certificates for complex Hadamard matrices of order six", "title_html": "Exact Fourier certificates for complex Hadamard matrices of order six", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/266.md" }, "peer_review": "not yet", "episode": { "video_id": "28Pt5QXan48", "length": "4:17", "title": "Only three in dimension six?", "page": "/episodes/266-three-mubs-dimension-six/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/266-three-mubs-dimension-six/" }, { "num": "267", "slug": "267-bose-einstein-condensation", "title": "Positive-temperature Bose–Einstein condensation and exact quantum depletion", "title_html": "Positive-temperature Bose–Einstein condensation and exact quantum depletion", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas", "title_html": "Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas", "url": "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", "role": "main" }, { "title": "Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas", "title_html": "Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas", "url": "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", "role": "companion" }, { "title": "Quantum Depletion for Fixed Bounded Repulsive Potentials", "title_html": "Quantum Depletion for Fixed Bounded Repulsive Potentials", "url": "https://github.com/openai/math/blob/main/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf", "role": "companion" }, { "title": "A density-uniform condensate bound for dilute Bose gases", "title_html": "A density-uniform condensate bound for dilute Bose gases", "url": "https://github.com/openai/math/blob/main/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf", "role": "companion" }, { "title": "Ground-state condensation in the dilute hard-sphere gas", "title_html": "Ground-state condensation in the dilute hard-sphere gas", "url": "https://github.com/openai/math/blob/main/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/267.md" }, "peer_review": "not yet", "episode": { "video_id": "WjHhsoUIfvg", "length": "4:22", "title": "A condensate, proved at last?", "page": "/episodes/267-bose-einstein-condensation/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/267-bose-einstein-condensation/" }, { "num": "268", "slug": "268-haldane-gap-spin-one", "title": "The spin-one Haldane gap", "title_html": "The spin-one Haldane gap", "summary_html": "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=3/5h=3/5h=3/5 and gives boundary-selected infinite-volume states with topological index −1.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "The periodic spin-one Haldane gap", "title_html": "The periodic spin-one Haldane gap", "url": "https://github.com/openai/math/blob/main/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf", "role": "main" }, { "title": "A boundary-field gap for the spin-one Heisenberg chain", "title_html": "A boundary-field gap for the spin-one Heisenberg chain", "url": "https://github.com/openai/math/blob/main/preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "LhJSNuQ0Xh4", "length": "3:54", "title": "A gap for spin one, proved at last?", "page": "/episodes/268-haldane-gap-spin-one/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/268-haldane-gap-spin-one/" }, { "num": "269", "slug": "269-laughlin-gap", "title": "Uniform Laughlin gap and stability under bounded scalar disorder", "title_html": "Uniform Laughlin gap and stability under bounded scalar disorder", "summary_html": "Proves the fermionic Laughlin spectral-gap conjecture for the full V1 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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Uniform Stability of the Spherical Laughlin Gap", "title_html": "Uniform Stability of the Spherical Laughlin Gap", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf", "role": "main" }, { "title": "A Fock-space inequality and the Laughlin spectral gap", "title_html": "A Fock-space inequality and the Laughlin spectral gap", "url": "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", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/269.md" }, "peer_review": "not yet", "episode": { "video_id": "YXomy5DdNLY", "length": "4:31", "title": "Electrons that won’t squeeze, proved at last?", "page": "/episodes/269-laughlin-gap/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/269-laughlin-gap/" }, { "num": "270", "slug": "270-bfss-bound-states", "title": "Threshold and positive-energy bound states of the BFSS matrix model", "title_html": "Threshold and positive-energy bound states of the BFSS matrix model", "summary_html": "Proves that the undeformed relative SU(N)\\mathrm{SU}(N)SU(N) BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For SU(2)\\mathrm{SU}(2)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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "The unique threshold bound state of the SU(N) BFSS model", "title_html": "The unique threshold bound state of the SU(N) BFSS model", "url": "https://github.com/openai/math/blob/main/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf", "role": "main" }, { "title": "Positive eigenvalues of the relative SU(2) BFSS Hamiltonian", "title_html": "Positive eigenvalues of the relative SU(2) BFSS Hamiltonian", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "2axF0Fdb1p8", "length": "3:41", "title": "One bound state, always, proved at last?", "page": "/episodes/270-bfss-bound-states/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/270-bfss-bound-states/" }, { "num": "271", "slug": "271-blochs-law", "title": "Bloch's law, its lattice correction, and the spherical magnetization law", "title_html": "Bloch's law, its lattice correction, and the spherical magnetization law", "summary_html": "Proves Bloch's T3/2 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 ℤ3. 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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions", "title_html": "Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions", "url": "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", "role": "main" }, { "title": "The first lattice correction to Bloch's law", "title_html": "The first lattice correction to Bloch's law", "url": "https://github.com/openai/math/blob/main/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf", "role": "companion" }, { "title": "The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet", "title_html": "The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet", "url": "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", "role": "companion" }, { "title": "Spontaneous magnetization in the quantum Heisenberg ferromagnet", "title_html": "Spontaneous magnetization in the quantum Heisenberg ferromagnet", "url": "https://github.com/openai/math/blob/main/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/271.md" }, "peer_review": "not yet", "episode": { "video_id": "QHMZxhNE-AI", "length": "4:09", "title": "A law for magnets, proved at last?", "page": "/episodes/271-blochs-law/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/271-blochs-law/" }, { "num": "272", "slug": "272-entanglement-without-key", "title": "Entanglement without distillable secret key", "title_html": "Entanglement without distillable secret key", "summary_html": "Constructs an entangled state on C10⊗C10\\mathbb C^{10}\\otimes\\mathbb C^{10}C10⊗C10 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 M21(C)M_{21}(\\mathbb C)M21​(C) whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Entanglement with zero distillable secret key in local dimension ten", "title_html": "Entanglement with zero distillable secret key in local dimension ten", "url": "https://github.com/openai/math/blob/main/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/272.md" }, "peer_review": "not yet", "episode": { "video_id": "M6LW3Qp5SDE", "length": "4:08", "title": "Entangled, but no secret?", "page": "/episodes/272-entanglement-without-key/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/272-entanglement-without-key/" }, { "num": "273", "slug": "273-entropy-photon-number", "title": "The entropy photon-number inequality", "title_html": "The entropy photon-number inequality", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "The entropy photon-number inequality", "title_html": "The entropy photon-number inequality", "url": "https://github.com/openai/math/blob/main/preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/273.md" }, "peer_review": "not yet", "episode": { "video_id": "GVi1gUZuMis", "length": "3:28", "title": "Light that mixes gently, proved at last?", "page": "/episodes/273-entropy-photon-number/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/273-entropy-photon-number/" }, { "num": "274", "slug": "274-parity-not-qac0", "title": "Parity is not in QAC⁰", "title_html": "Parity is not in QAC0", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Product-projection localization and the QAC0 parity lower bound", "title_html": "Product-projection localization and the QAC0 parity lower bound", "url": "https://github.com/openai/math/blob/main/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf", "role": "main" }, { "title": "Regular trajectories, pruning and quantum parity", "title_html": "Regular trajectories, pruning and quantum parity", "url": "https://github.com/openai/math/blob/main/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/274.md" }, "peer_review": "not yet", "episode": { "video_id": "FyGgvudMyyg", "length": "3:32", "title": "Shallow circuits, no parity, proved at last?", "page": "/episodes/274-parity-not-qac0/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/274-parity-not-qac0/" }, { "num": "275", "slug": "275-coulomb-qma-hard", "title": "QMA-hardness of continuum Coulomb energy", "title_html": "QMA-hardness of continuum Coulomb energy", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Continuum Coulomb hardness with binary nuclear charges", "title_html": "Continuum Coulomb hardness with binary nuclear charges", "url": "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", "role": "main" }, { "title": "QMA-hardness of continuum Coulomb energy with unit nuclear charges", "title_html": "QMA-hardness of continuum Coulomb energy with unit nuclear charges", "url": "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", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/275.md" }, "peer_review": "not yet", "episode": { "video_id": "K6Mk-9A15bM", "length": "3:42", "title": "A hard chemistry problem, proved at last?", "page": "/episodes/275-coulomb-qma-hard/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/275-coulomb-qma-hard/" }, { "num": "276", "slug": "276-gad-capacity", "title": "Classical capacity of generalized amplitude damping", "title_html": "Classical capacity of generalized amplitude damping", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Classical capacity and entropy inequalities for generalized amplitude damping", "title_html": "Classical capacity and entropy inequalities for generalized amplitude damping", "url": "https://github.com/openai/math/blob/main/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/276.md" }, "peer_review": "not yet", "episode": { "video_id": "vzA-GfrlWzg", "length": "3:21", "title": "A noisy qubit’s capacity, proved at last?", "page": "/episodes/276-gad-capacity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/276-gad-capacity/" }, { "num": "277", "slug": "277-threshold-entangled-games", "title": "Threshold repetition for entangled games", "title_html": "Threshold repetition for entangled games", "summary_html": "Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction v+δv+\\deltav+δ of k independent repetitions decays exponentially in k, for 0<δ<1−v0\\lt \\delta\\lt 1-v0<δ<1−v. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Threshold parallel repetition for finite-dimensional entangled games", "title_html": "Threshold parallel repetition for finite-dimensional entangled games", "url": "https://github.com/openai/math/blob/main/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/277.md" }, "peer_review": "not yet", "episode": { "video_id": "4gToX9KYyus", "length": "3:25", "title": "No lucky streaks, proved at last?", "page": "/episodes/277-threshold-entangled-games/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/277-threshold-entangled-games/" }, { "num": "278", "slug": "278-kohn-sham-counterexample", "title": "Failure of Kohn–Sham ensemble representation", "title_html": "Failure of Kohn–Sham ensemble representation", "summary_html": "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 L3/2(R3)+L∞(R3)L^{3/2}(\\mathbb R^3)+L^\\infty(\\mathbb R^3)L3/2(R3)+L∞(R3). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "A Coulomb ground-state density without Kohn-Sham ensemble representation", "title_html": "A Coulomb ground-state density without Kohn-Sham ensemble representation", "url": "https://github.com/openai/math/blob/main/preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "9RR1CUbiIb0", "length": "3:34", "title": "Kohn–Sham can’t fake it, proved at last?", "page": "/episodes/278-kohn-sham-counterexample/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/278-kohn-sham-counterexample/" }, { "num": "279", "slug": "279-exact-quantum-factoring", "title": "Exact quantum factoring over a fixed finite gate set", "title_html": "Exact quantum factoring over a fixed finite gate set", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Exact quantum factoring over a fixed finite gate set", "title_html": "Exact quantum factoring over a fixed finite gate set", "url": "https://github.com/openai/math/blob/main/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/279.md" }, "peer_review": "not yet", "episode": { "video_id": "4GXTy1wFBFY", "length": "3:16", "title": "Factoring with no gamble, proved at last?", "page": "/episodes/279-exact-quantum-factoring/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/279-exact-quantum-factoring/" }, { "num": "280", "slug": "280-voa-conformal-nets", "title": "Unitary vertex operator algebras and conformal nets", "title_html": "Unitary vertex operator algebras and conformal nets", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Strongly rational unitary vertex operator algebras and conformal nets", "title_html": "Strongly rational unitary vertex operator algebras and conformal nets", "url": "https://github.com/openai/math/blob/main/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/280.md" }, "peer_review": "not yet", "episode": { "video_id": "z49gbxdLGIk", "length": "3:57", "title": "Two languages agree, proved at last?", "page": "/episodes/280-voa-conformal-nets/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/280-voa-conformal-nets/" }, { "num": "281", "slug": "281-qaoa-sk-optimum", "title": "QAOA attains the SK optimum in the thermodynamic-first limit", "title_html": "QAOA attains the SK optimum in the thermodynamic-first limit", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "QAOA attains the SK ground-state energy in the thermodynamic-first limit", "title_html": "QAOA attains the SK ground-state energy in the thermodynamic-first limit", "url": "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", "role": "main" }, { "title": "Full support of the zero-temperature Sherrington-Kirkpatrick order parameter", "title_html": "Full support of the zero-temperature Sherrington-Kirkpatrick order parameter", "url": "https://github.com/openai/math/blob/main/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/281.md" }, "peer_review": "not yet", "episode": { "video_id": "5CUF1_CBTcM", "length": "3:39", "title": "QAOA reaches the optimum, proved at last?", "page": "/episodes/281-qaoa-sk-optimum/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/281-qaoa-sk-optimum/" }, { "num": "282", "slug": "282-scale-to-conformal", "title": "From scale symmetry to local conformal symmetry in four-dimensional QFT", "title_html": "From scale symmetry to local conformal symmetry in four-dimensional QFT", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Scale and conformal symmetry in four-dimensional operational quantum field theory", "title_html": "Scale and conformal symmetry in four-dimensional operational quantum field theory", "url": "https://github.com/openai/math/blob/main/preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "AxXnHF0n3bs", "length": "4:12", "title": "Scale implies conformal, proved at last?", "page": "/episodes/282-scale-to-conformal/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/282-scale-to-conformal/" }, { "num": "283", "slug": "283-unitary-synthesis", "title": "Polynomial-time unitary synthesis from a Boolean oracle", "title_html": "Polynomial-time unitary synthesis from a Boolean oracle", "summary_html": "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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "Polynomial-Time Unitary Synthesis from a Boolean Oracle", "title_html": "Polynomial-Time Unitary Synthesis from a Boolean Oracle", "url": "https://github.com/openai/math/blob/main/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "SDuZ3kxLwKY", "length": "3:23", "title": "Any unitary from a table, proved at last?", "page": "/episodes/283-unitary-synthesis/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/283-unitary-synthesis/" }, { "num": "284", "slug": "284-quartic-separation", "title": "The optimal quartic separation between randomized and quantum queries", "title_html": "The optimal quartic separation between randomized and quantum queries", "summary_html": "Shows that the universal bound R(f)=O((1+Q(f))4)R(f)=O((1+Q(f))^4)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.", "subject": "Mathematical physics", "subject_slug": "mathematical-physics", "papers": [ { "title": "A Nearly Quartic Separation Between Randomized and Quantum Query Complexity", "title_html": "A Nearly Quartic Separation Between Randomized and Quantum Query Complexity", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "lQWg8mDZ8VQ", "length": "3:27", "title": "Nearly a fourth power, proved at last?", "page": "/episodes/284-quartic-separation/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/284-quartic-separation/" }, { "num": "285", "slug": "285-baum-connes-kadison-kaplansky", "title": "Counterexamples to Baum–Connes and Kadison–Kaplansky", "title_html": "Counterexamples to Baum–Connes and Kadison–Kaplansky", "summary_html": "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∗-algebra, disproving the Kadison–Kaplansky conjecture.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "A torsion-free counterexample to reduced Baum–Connes injectivity", "title_html": "A torsion-free counterexample to reduced Baum–Connes injectivity", "url": "https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf", "role": "main" }, { "title": "A torsion-free counterexample to the Kadison–Kaplansky projection conjecture", "title_html": "A torsion-free counterexample to the Kadison–Kaplansky projection conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf", "role": "companion" }, { "title": "An irrational-trace counterexample to reduced Baum–Connes", "title_html": "An irrational-trace counterexample to reduced Baum–Connes", "url": "https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "nlW2-cPLAts", "length": "4:07", "title": "The bridge breaks, proved at last?", "page": "/episodes/285-baum-connes-kadison-kaplansky/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/285-baum-connes-kadison-kaplansky/" }, { "num": "286", "slug": "286-lattice-von-neumann-rigidity", "title": "Rigidity and arithmetic of lattice von Neumann algebras", "title_html": "Rigidity and arithmetic of lattice von Neumann algebras", "summary_html": "Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-(T)(T)(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.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Arithmeticity of twisted finite correspondences for lattices over local fields", "title_html": "Arithmeticity of twisted finite correspondences for lattices over local fields", "url": "https://github.com/openai/math/blob/main/preprints/Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026/paper.pdf", "role": "main" }, { "title": "The arithmetic category and stable recovery of lattice factors", "title_html": "The arithmetic category and stable recovery of lattice factors", "url": "https://github.com/openai/math/blob/main/preprints/The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "K-zeq9dEI-E", "length": "3:31", "title": "A perfect fingerprint, proved at last?", "page": "/episodes/286-lattice-von-neumann-rigidity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/286-lattice-von-neumann-rigidity/" }, { "num": "287", "slug": "287-free-group-factors", "title": "Isomorphism of the free group factors", "title_html": "Isomorphism of the free group factors", "summary_html": "Resolves the free group factor isomorphism problem: L(F2)≅L(F3)L(\\mathbb F_2)\\cong L(\\mathbb F_3)L(F2​)≅L(F3​), and hence all interpolated free group factors, including L(F∞)L(\\mathbb F_\\infty)L(F∞​), are isomorphic. Their common factor has fundamental group R>0\\mathbb R_{\\gt 0}R>0​.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "An isomorphism of the free group factors", "title_html": "An isomorphism of the free group factors", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/287.md" }, "peer_review": "not yet", "episode": { "video_id": "xU2ySlhH5Ig", "length": "3:56", "title": "Every free group, one factor, proved at last?", "page": "/episodes/287-free-group-factors/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/287-free-group-factors/" }, { "num": "288", "slug": "288-kadison-similarity", "title": "Kadison's similarity conjecture", "title_html": "Kadison's similarity conjecture", "summary_html": "Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a ∗*∗-homomorphism after conjugation by a bounded invertible operator.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Kadison's similarity theorem through uniform derivation estimates", "title_html": "Kadison's similarity theorem through uniform derivation estimates", "url": "https://github.com/openai/math/blob/main/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/288.md" }, "peer_review": "not yet", "episode": { "video_id": "k2mMeICSw_c", "length": "3:51", "title": "Kadison’s question answered, proved at last?", "page": "/episodes/288-kadison-similarity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/288-kadison-similarity/" }, { "num": "289", "slug": "289-kadison-kastler", "title": "Strong Kadison–Kastler stability and its spatial boundaries", "title_html": "Strong Kadison–Kastler stability and its spatial boundaries", "summary_html": "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∗-algebras need not be ambiently unitarily conjugate.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Universal strong Kadison–Kastler stability", "title_html": "Universal strong Kadison–Kastler stability", "url": "https://github.com/openai/math/blob/main/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf", "role": "main" }, { "title": "Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings", "title_html": "Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings", "url": "https://github.com/openai/math/blob/main/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf", "role": "companion" }, { "title": "Close Separable C*-Algebras Without Spatial Conjugacy", "title_html": "Close Separable C*-Algebras Without Spatial Conjugacy", "url": "https://github.com/openai/math/blob/main/preprints/Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/289.md" }, "peer_review": "not yet", "episode": { "video_id": "OntGiEWL88U", "length": "3:37", "title": "Close means a small turn, proved at last?", "page": "/episodes/289-kadison-kastler/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/289-kadison-kastler/" }, { "num": "290", "slug": "290-bicentralizer", "title": "Relative bicentralizers and modular spectral recovery", "title_html": "Relative bicentralizers and modular spectral recovery", "summary_html": "Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion N⊂MN\\subset MN⊂M of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra P⊂NP\\subset NP⊂N with P′∩c(M)=N′∩c(M)P'\\cap c(M)=N'\\cap c(M)P′∩c(M)=N′∩c(M), resolving the relative bicentralizer conjecture.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Expected amenable subalgebras preserving core commutants", "title_html": "Expected amenable subalgebras preserving core commutants", "url": "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", "role": "main" }, { "title": "Bounded recovery for modular spectral averages", "title_html": "Bounded recovery for modular spectral averages", "url": "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", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/290.md" }, "peer_review": "not yet", "episode": { "video_id": "Qn0qi7yVHtc", "length": "3:45", "title": "The relative case, proved at last?", "page": "/episodes/290-bicentralizer/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/290-bicentralizer/" }, { "num": "291", "slug": "291-toms-winter", "title": "Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability", "title_html": "Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability", "summary_html": "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∗-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.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case", "title_html": "Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case", "url": "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", "role": "main" }, { "title": "Cuntz comparison and Jiang–Su absorption", "title_html": "Cuntz comparison and Jiang–Su absorption", "url": "https://github.com/openai/math/blob/main/preprints/Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Nuclear dimension and Jiang–Su stability without elementary subquotients", "title_html": "Nuclear dimension and Jiang–Su stability without elementary subquotients", "url": "https://github.com/openai/math/blob/main/preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Tracial projection methods and uniform property Gamma", "title_html": "Tracial projection methods and uniform property Gamma", "url": "https://github.com/openai/math/blob/main/preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/291.md" }, "peer_review": "not yet", "episode": { "video_id": "nRY2n-sg9EQ", "length": "3:59", "title": "One week, two proofs?", "page": "/episodes/291-toms-winter/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/291-toms-winter/" }, { "num": "292", "slug": "292-kirchberg-embedding", "title": "Kirchberg's O₂ norm-ultrapower embedding problem", "title_html": "Kirchberg's O2\\mathcal O_2O2​ norm-ultrapower embedding problem", "summary_html": "Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be O2\\mathcal O_2O2​ answers Kirchberg's norm-ultrapower embedding problem negatively.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "An explicit obstruction to nuclear norm-ultrapower embeddings", "title_html": "An explicit obstruction to nuclear norm-ultrapower embeddings", "url": "https://github.com/openai/math/blob/main/preprints/An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/292.md" }, "peer_review": "not yet", "episode": { "video_id": "dDPFRHumCKs", "length": "4:26", "title": "Too big for the hotel?", "page": "/episodes/292-kirchberg-embedding/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/292-kirchberg-embedding/" }, { "num": "293", "slug": "293-hyperinvariant-subspaces", "title": "Invariant projections, hyperinvariant subspaces, and transitive algebras", "title_html": "Invariant projections, hyperinvariant subspaces, and transitive algebras", "summary_html": "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 II1 factor.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Invariant-projection counterexamples for every irrational rotation", "title_html": "Invariant-projection counterexamples for every irrational rotation", "url": "https://github.com/openai/math/blob/main/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026/paper.pdf", "role": "main" }, { "title": "Backward intertwiners and a transitive commutant", "title_html": "Backward intertwiners and a transitive commutant", "url": "https://github.com/openai/math/blob/main/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/293.md" }, "peer_review": "not yet", "episode": { "video_id": "MUg_kxoTVNA", "length": "4:45", "title": "No room they all respect?", "page": "/episodes/293-hyperinvariant-subspaces/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/293-hyperinvariant-subspaces/" }, { "num": "294", "slug": "294-kaplansky-quasitrace", "title": "Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness", "title_html": "Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness", "summary_html": "Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor Cr∗(F2)C_r^*(\\mathbb F_2)Cr∗​(F2​).", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness", "title_html": "A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/294.md" }, "peer_review": "not yet", "episode": { "video_id": "SuYfd8tAB0g", "length": "4:11", "title": "The parts fall short?", "page": "/episodes/294-kaplansky-quasitrace/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/294-kaplansky-quasitrace/" }, { "num": "295", "slug": "295-kadison-ringrose", "title": "The Kadison–Ringrose cohomology conjecture", "title_html": "The Kadison–Ringrose cohomology conjecture", "summary_html": "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.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Vanishing of higher bounded Hochschild cohomology", "title_html": "Vanishing of higher bounded Hochschild cohomology", "url": "https://github.com/openai/math/blob/main/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/295.md" }, "peer_review": "not yet", "episode": { "video_id": "ao8gm84743c", "length": "4:30", "title": "Every equation solvable?", "page": "/episodes/295-kadison-ringrose/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/295-kadison-ringrose/" }, { "num": "296", "slug": "296-generator-problem", "title": "The generator problem for finite factors", "title_html": "The generator problem for finite factors", "summary_html": "Proves that every type II1 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⊂MP\\subset MP⊂M of such factors, the unitaries u with M=W∗(P,u)M=W^*(P,u)M=W∗(P,u) form a dense Gδ subset in the trace 2-norm topology.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Relative generation and the generator problem for finite factors", "title_html": "Relative generation and the generator problem for finite factors", "url": "https://github.com/openai/math/blob/main/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/296.md" }, "peer_review": "not yet", "episode": { "video_id": "tr7gJO2K6es", "length": "4:19", "title": "One operator for all?", "page": "/episodes/296-generator-problem/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/296-generator-problem/" }, { "num": "297", "slug": "297-naimark-zfc", "title": "A ZFC counterexample to Naimark's problem", "title_html": "A ZFC counterexample to Naimark's problem", "summary_html": "Gives an alternative to Tanaka's ZFC construction of a unital infinite-dimensional simple complex C∗-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.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "A counterexample to Naimark's problem in ZFC", "title_html": "A counterexample to Naimark's problem in ZFC", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/297.md" }, "peer_review": "not yet", "episode": { "video_id": "B-izZW4ytQ4", "length": "4:12", "title": "One irrep, not compact?", "page": "/episodes/297-naimark-zfc/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/297-naimark-zfc/" }, { "num": "298", "slug": "298-free-entropy", "title": "Two notions of free entropy differ even when both are finite", "title_html": "Two notions of free entropy differ even when both are finite", "summary_html": "Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy −∞<χ<χ∗<∞-\\infty\\lt \\chi\\lt \\chi^*\\lt \\infty−∞<χ<χ∗<∞. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "A finite-entropy separation of microstates and nonmicrostates free entropy", "title_html": "A finite-entropy separation of microstates and nonmicrostates free entropy", "url": "https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/298.md" }, "peer_review": "not yet", "episode": { "video_id": "gaC8AkvyYrM", "length": "4:06", "title": "Both finite, still unequal?", "page": "/episodes/298-free-entropy/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/298-free-entropy/" }, { "num": "299", "slug": "299-kirchberg-rordam", "title": "The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability", "title_html": "The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability", "summary_html": "A nonzero unital separable complex C∗-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.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "The Kirchberg–Rørdam character criterion", "title_html": "The Kirchberg–Rørdam character criterion", "url": "https://github.com/openai/math/blob/main/preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/299.md" }, "peer_review": "not yet", "episode": { "video_id": "awQ_S3pKzuQ", "length": "3:42", "title": "No characters, then Z-stable?", "page": "/episodes/299-kirchberg-rordam/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/299-kirchberg-rordam/" }, { "num": "300", "slug": "300-quadratic-paving", "title": "Approximation and quadratic strong-operator paving", "title_html": "Approximation and quadratic strong-operator paving", "summary_html": "Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with O(ε−2)O(\\varepsilon^{-2})O(ε−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.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Approximation Paving over Arbitrary Maximal Abelian Subalgebras", "title_html": "Approximation Paving over Arbitrary Maximal Abelian Subalgebras", "url": "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", "role": "main" }, { "title": "Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras", "title_html": "Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras", "url": "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", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "CZjOXgkmO88", "length": "4:04", "title": "Every MASA, quadratically?", "page": "/episodes/300-quadratic-paving/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/300-quadratic-paving/" }, { "num": "301", "slug": "301-trace-cones", "title": "Trace cones and Razak–Jacelon stabilization", "title_html": "Trace cones and Razak–Jacelon stabilization", "summary_html": "Classifies separable nuclear complex C∗-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.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "The trace cone classifies Razak–Jacelon stabilizations", "title_html": "The trace cone classifies Razak–Jacelon stabilizations", "url": "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", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/301.md" }, "peer_review": "not yet", "episode": { "video_id": "XGKPntKUxxQ", "length": "3:50", "title": "The cone decides?", "page": "/episodes/301-trace-cones/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/301-trace-cones/" }, { "num": "302", "slug": "302-mean-dimension", "title": "Radius of comparison equals half the mean dimension", "title_html": "Radius of comparison equals half the mean dimension", "summary_html": "For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of C(X)⋊hZC(X)\\rtimes_h\\mathbb ZC(X)⋊h​Z equals 12mdim(X,h)\\tfrac12\\mathrm{mdim}(X,h)21​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.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Filtered products and boundary-preserving compression in complex cobordism", "title_html": "Filtered products and boundary-preserving compression in complex cobordism", "url": "https://github.com/openai/math/blob/main/preprints/Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026/paper.pdf", "role": "main" }, { "title": "Radius of comparison equals half the mean dimension", "title_html": "Radius of comparison equals half the mean dimension", "url": "https://github.com/openai/math/blob/main/preprints/Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "vWbe1t_BZNw", "length": "3:57", "title": "Exactly one half?", "page": "/episodes/302-mean-dimension/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/302-mean-dimension/" }, { "num": "303", "slug": "303-pure-infiniteness", "title": "Weak pure infiniteness and Cuntz-algebra absorption", "title_html": "Weak pure infiniteness and Cuntz-algebra absorption", "summary_html": "Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-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 O∞\\mathcal O_\\inftyO∞​, without unitality or simplicity assumptions.", "subject": "Operator algebras", "subject_slug": "operator-algebras", "papers": [ { "title": "Weak pure infiniteness and O-infinity absorption", "title_html": "Weak pure infiniteness and O-infinity absorption", "url": "https://github.com/openai/math/blob/main/preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/303.md" }, "peer_review": "not yet", "episode": { "video_id": "YKmM7b8L5as", "length": "4:09", "title": "Infinite enough?", "page": "/episodes/303-pure-infiniteness/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/303-pure-infiniteness/" }, { "num": "304", "slug": "304-hilbert-smith", "title": "The Hilbert–Smith conjecture in every dimension", "title_html": "The Hilbert–Smith conjecture in every dimension", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "The Hilbert–Smith conjecture in every finite dimension", "title_html": "The Hilbert–Smith conjecture in every finite dimension", "url": "https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "FIrQGXatLeY", "length": "4:38", "title": "Always Lie?", "page": "/episodes/304-hilbert-smith/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/304-hilbert-smith/" }, { "num": "305", "slug": "305-disk-embedding", "title": "Four-dimensional disk embedding and Wall's conjecture", "title_html": "Four-dimensional disk embedding and Wall's conjecture", "summary_html": "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 F2 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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "A boundary-only obstruction to four-dimensional disk embedding", "title_html": "A boundary-only obstruction to four-dimensional disk embedding", "url": "https://github.com/openai/math/blob/main/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf", "role": "main" }, { "title": "A marked tensor obstruction to four-dimensional disk embedding", "title_html": "A marked tensor obstruction to four-dimensional disk embedding", "url": "https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf", "role": "companion" }, { "title": "A PD4 group without an aspherical manifold model", "title_html": "A PD4 group without an aspherical manifold model", "url": "https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "KDcHJkdRbcg", "length": "4:54", "title": "Free groups not good?", "page": "/episodes/305-disk-embedding/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/305-disk-embedding/" }, { "num": "306", "slug": "306-cosmetic-surgery", "title": "The purely cosmetic surgery conjecture", "title_html": "The purely cosmetic surgery conjecture", "summary_html": "Distinct Dehn surgery slopes on a nontrivial smooth knot in S3 never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Purely cosmetic surgery on knots in the three-sphere", "title_html": "Purely cosmetic surgery on knots in the three-sphere", "url": "https://github.com/openai/math/blob/main/preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "Vt_26lMxsRo", "length": "4:36", "title": "Never the same twice?", "page": "/episodes/306-cosmetic-surgery/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/306-cosmetic-surgery/" }, { "num": "307", "slug": "307-coarse-assembly", "title": "Failure of rational injectivity for maximal coarse assembly", "title_html": "Failure of rational injectivity for maximal coarse assembly", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Failure of rational injectivity for maximal coarse assembly", "title_html": "Failure of rational injectivity for maximal coarse assembly", "url": "https://github.com/openai/math/blob/main/preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/paper.pdf", "role": "main" }, { "title": "A counterexample to the coarse Novikov conjecture", "title_html": "A counterexample to the coarse Novikov conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/307.md" }, "peer_review": "not yet", "episode": { "video_id": "Drtt2GFFk1U", "length": "4:47", "title": "Lost in assembly?", "page": "/episodes/307-coarse-assembly/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/307-coarse-assembly/" }, { "num": "308", "slug": "308-smith-toda", "title": "Finite Smith–Toda complexes at every height", "title_html": "Finite Smith–Toda complexes at every height", "summary_html": "For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology BP∗/(p,v1,…,vn)BP_*/(p,v_1,\\ldots,v_n)BP∗​/(p,v1​,…,vn​) and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes V(4)V(4)V(4) at p = 1009.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Finite Smith–Toda Complexes at Varying Primes", "title_html": "Finite Smith–Toda Complexes at Varying Primes", "url": "https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf", "role": "main" }, { "title": "A finite Smith–Toda complex V(4) at the prime 1009", "title_html": "A finite Smith–Toda complex V(4) at the prime 1009", "url": "https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "nCzc4BWfY_g", "length": "4:26", "title": "V(4), at last?", "page": "/episodes/308-smith-toda/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/308-smith-toda/" }, { "num": "309", "slug": "309-kervaire-three", "title": "The Kervaire invariant problem at the prime three", "title_html": "The Kervaire invariant problem at the prime three", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "The Kervaire invariant problem at the prime three", "title_html": "The Kervaire invariant problem at the prime three", "url": "https://github.com/openai/math/blob/main/preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "jyyTEHrA49s", "length": "4:34", "title": "Stem 322 survives?", "page": "/episodes/309-kervaire-three/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/309-kervaire-three/" }, { "num": "310", "slug": "310-quillen", "title": "Quillen's conjecture in rational homology", "title_html": "Quillen's conjecture in rational homology", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Rational homology and Quillen's conjecture", "title_html": "Rational homology and Quillen's conjecture", "url": "https://github.com/openai/math/blob/main/preprints/Rational-homology-and-Quillens-conjecture-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "jp_K7w5DPf4", "length": "4:16", "title": "Always a hole?", "page": "/episodes/310-quillen/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/310-quillen/" }, { "num": "311", "slug": "311-hovey-strickland", "title": "The Hovey–Strickland and Chai conjectures", "title_html": "The Hovey–Strickland and Chai conjectures", "summary_html": "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)K(n)K(n)-local spectra have exactly n+2n+2n+2 thick tensor ideals, and their Balmer spectrum is a chain of n+1n+1n+1 points.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra", "title_html": "Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra", "url": "https://github.com/openai/math/blob/main/preprints/Stabilizer-Orbits-and-Thick-Tensor-Ideals-of-Dualizable-Kn-Local-Spectra-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "XwaIREzsn9M", "length": "4:07", "title": "Exactly n + 2?", "page": "/episodes/311-hovey-strickland/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/311-hovey-strickland/" }, { "num": "312", "slug": "312-homotopy-hypothesis", "title": "The Grothendieck homotopy hypothesis", "title_html": "The Grothendieck homotopy hypothesis", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "The Grothendieck homotopy hypothesis via elementary expansions", "title_html": "The Grothendieck homotopy hypothesis via elementary expansions", "url": "https://github.com/openai/math/blob/main/preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/312.md" }, "peer_review": "not yet", "episode": { "video_id": "qX48iCgSJP4", "length": "4:31", "title": "Spaces as algebra?", "page": "/episodes/312-homotopy-hypothesis/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/312-homotopy-hypothesis/" }, { "num": "313", "slug": "313-finite-generation", "title": "Finite generation for the K(n)-local sphere", "title_html": "Finite generation for the K(n)K(n)K(n)-local sphere", "summary_html": "Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the K(n)K(n)K(n)-local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after K(n)K(n)K(n)-localizing any finite p-local spectrum.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Finite generation for the K(n)-local sphere", "title_html": "Finite generation for the K(n)-local sphere", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "67lvRjhNMKQ", "length": "4:20", "title": "Finitely generated?", "page": "/episodes/313-finite-generation/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/313-finite-generation/" }, { "num": "314", "slug": "314-cyclic-length", "title": "Cyclic length and chromatic fixed-point loss", "title_html": "Cyclic length and chromatic fixed-point loss", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Cyclic length and chromatic fixed-point loss", "title_html": "Cyclic length and chromatic fixed-point loss", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "gWL5COaL87w", "length": "5:06", "title": "Heights lost?", "page": "/episodes/314-cyclic-length/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/314-cyclic-length/" }, { "num": "315", "slug": "315-singer-four", "title": "The four-dimensional Singer conjecture", "title_html": "The four-dimensional Singer conjecture", "summary_html": "Proves that the L2-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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "The Singer conjecture in dimension four", "title_html": "The Singer conjecture in dimension four", "url": "https://github.com/openai/math/blob/main/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "j8_uqeMKx-M", "length": "4:48", "title": "Never negative?", "page": "/episodes/315-singer-four/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/315-singer-four/" }, { "num": "316", "slug": "316-curtis-conjecture", "title": "Curtis’s conjecture", "title_html": "Curtis’s conjecture", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "The Stable Hurewicz Image of the Sphere at Two", "title_html": "The Stable Hurewicz Image of the Sphere at Two", "url": "https://github.com/openai/math/blob/main/preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "C5mJ4I1L5oU", "length": "4:20", "title": "Only the famous ones?", "page": "/episodes/316-curtis-conjecture/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/316-curtis-conjecture/" }, { "num": "317", "slug": "317-thomason-model-structures", "title": "Thomason model structures in all strict higher dimensions", "title_html": "Thomason model structures in all strict higher dimensions", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Thomason Model Structures in Every Strict Higher Dimension", "title_html": "Thomason Model Structures in Every Strict Higher Dimension", "url": "https://github.com/openai/math/blob/main/preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/317.md" }, "peer_review": "not yet", "episode": { "video_id": "HpAbeOffemQ", "length": "4:18", "title": "Every dimension?", "page": "/episodes/317-thomason-model-structures/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/317-thomason-model-structures/" }, { "num": "318", "slug": "318-chromatic-splitting", "title": "Chromatic splitting: filtrations and counterexamples", "title_html": "Chromatic splitting: filtrations and counterexamples", "summary_html": "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+1p+1p+1 (p ≥ 7). Nevertheless, for n ≥ 1 and p>n+1p\\gt n+1p>n+1, the overlap Ln−1LK(n)Sp∧L_{n-1}L_{K(n)}S_p^\\wedgeLn−1​LK(n)​Sp∧​ admits a 2n2^n2n-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)E(2)E(2)-local modules over the derived completed sphere.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Filtered chromatic splitting at generic primes", "title_html": "Filtered chromatic splitting at generic primes", "url": "https://github.com/openai/math/blob/main/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf", "role": "main" }, { "title": "The height-three chromatic overlap: an explicit filtration and its attachments", "title_html": "The height-three chromatic overlap: an explicit filtration and its attachments", "url": "https://github.com/openai/math/blob/main/preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026/paper.pdf", "role": "companion" }, { "title": "A rational obstruction to strong chromatic splitting at height three", "title_html": "A rational obstruction to strong chromatic splitting at height three", "url": "https://github.com/openai/math/blob/main/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf", "role": "companion" }, { "title": "Failure of finite assembly for a chromatic overlap at the prime three", "title_html": "Failure of finite assembly for a chromatic overlap at the prime three", "url": "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", "role": "companion" }, { "title": "Counterexamples to weak chromatic splitting: sphere kernels and descent exponents", "title_html": "Counterexamples to weak chromatic splitting: sphere kernels and descent exponents", "url": "https://github.com/openai/math/blob/main/preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "8VLkXAEQ5tM", "length": "4:18", "title": "Does it split?", "page": "/episodes/318-chromatic-splitting/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/318-chromatic-splitting/" }, { "num": "319", "slug": "319-hahn-wilson", "title": "Counterexamples to finite generation at chromatic height two", "title_html": "Counterexamples to finite generation at chromatic height two", "summary_html": "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 BP⟨2⟩\\mathrm{BP}\\langle2\\rangleBP⟨2⟩ by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Counterexamples to the Hahn-Wilson conjecture at height two", "title_html": "Counterexamples to the Hahn-Wilson conjecture at height two", "url": "https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "lybupRvvsA0", "length": "3:41", "title": "Finitely built?", "page": "/episodes/319-hahn-wilson/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/319-hahn-wilson/" }, { "num": "320", "slug": "320-borel-four", "title": "Nonhomeomorphic closed aspherical four-manifolds", "title_html": "Nonhomeomorphic closed aspherical four-manifolds", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type", "title_html": "Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type", "url": "https://github.com/openai/math/blob/main/preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "wVoMNIk00a0", "length": "3:43", "title": "Same group, not the same?", "page": "/episodes/320-borel-four/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/320-borel-four/" }, { "num": "321", "slug": "321-wall-d2", "title": "A counterexample to Wall's finite D(2) problem", "title_html": "A counterexample to Wall's finite D(2) problem", "summary_html": "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.", "subject": "Topology", "subject_slug": "topology", "papers": [ { "title": "A Counterexample to Wall's D(2) Problem", "title_html": "A Counterexample to Wall's D(2) Problem", "url": "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "gm4d8sHElys", "length": "3:18", "title": "Looks 2D, isn’t?", "page": "/episodes/321-wall-d2/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/321-wall-d2/" }, { "num": "322", "slug": "322-tingley", "title": "Tingley’s sphere-isometry problem", "title_html": "Tingley’s sphere-isometry problem", "summary_html": "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.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "A positive solution to Tingley’s problem", "title_html": "A positive solution to Tingley’s problem", "url": "https://github.com/openai/math/blob/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/322.md" }, "peer_review": "not yet", "episode": { "video_id": "viHytP6gNpY", "length": "3:42", "title": "The sphere knows all?", "page": "/episodes/322-tingley/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/322-tingley/" }, { "num": "323", "slug": "323-separable-quotient", "title": "Independence of the separable quotient problem", "title_html": "Independence of the separable quotient problem", "summary_html": "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.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "Relative independence of the separable quotient problem", "title_html": "Relative independence of the separable quotient problem", "url": "https://github.com/openai/math/blob/main/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/323.md" }, "peer_review": "not yet", "episode": { "video_id": "mY4fZcNXwh8", "length": "4:11", "title": "Yes or no, undecidable?", "page": "/episodes/323-separable-quotient/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/323-separable-quotient/" }, { "num": "324", "slug": "324-lipschitz-banach", "title": "Lipschitz equivalent Banach spaces need not be linearly isomorphic", "title_html": "Lipschitz equivalent Banach spaces need not be linearly isomorphic", "summary_html": "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.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic", "title_html": "Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic", "url": "https://github.com/openai/math/blob/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf", "role": "main" }, { "title": "Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0", "title_html": "Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0", "url": "https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/324.md" }, "peer_review": "not yet", "episode": { "video_id": "U-A0wW_Wz4o", "length": "3:51", "title": "Same shape, not the same?", "page": "/episodes/324-lipschitz-banach/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/324-lipschitz-banach/" }, { "num": "325", "slug": "325-complete-crouzeix", "title": "The complete Crouzeix conjecture", "title_html": "The complete Crouzeix conjecture", "summary_html": "Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has ∥P[A]∥≤2sup⁡z∈W(A)∥P(z)∥\\lVert P[A]\\rVert\\le2\\sup_{z\\in W(A)}\\lVert P(z)\\rVert∥P[A]∥≤2supz∈W(A)​∥P(z)∥, where W(A)W(A)W(A) is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "A direct proof of the complete Crouzeix inequality", "title_html": "A direct proof of the complete Crouzeix inequality", "url": "https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf", "role": "main" }, { "title": "The complete Crouzeix theorem: optimal similarity and a common positive boundary representation", "title_html": "The complete Crouzeix theorem: optimal similarity and a common positive boundary representation", "url": "https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/325.md" }, "peer_review": "not yet", "episode": { "video_id": "b7sJ5olWXmg", "length": "3:51", "title": "Never more than twice?", "page": "/episodes/325-complete-crouzeix/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/325-complete-crouzeix/" }, { "num": "326", "slug": "326-cotype-cotype", "title": "The cotype–cotype conjecture under the approximation property", "title_html": "The cotype–cotype conjecture under the approximation property", "summary_html": "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.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "The cotype–cotype conjecture under the approximation property", "title_html": "The cotype–cotype conjecture under the approximation property", "url": "https://github.com/openai/math/blob/main/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/326.md" }, "peer_review": "not yet", "episode": { "video_id": "NdrHFNGzOR4", "length": "3:57", "title": "Pisier’s last question?", "page": "/episodes/326-cotype-cotype/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/326-cotype-cotype/" }, { "num": "327", "slug": "327-markov-type", "title": "Markov type characterizes superreflexivity", "title_html": "Markov type characterizes superreflexivity", "summary_html": "Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "Nontrivial Markov Type Forces Superreflexivity", "title_html": "Nontrivial Markov Type Forces Superreflexivity", "url": "https://github.com/openai/math/blob/main/preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/327.md" }, "peer_review": "not yet", "episode": { "video_id": "uF12jdZVDz0", "length": "3:33", "title": "Walks reveal the shape?", "page": "/episodes/327-markov-type/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/327-markov-type/" }, { "num": "328", "slug": "328-kirk-fixed-point", "title": "Nonexpansive fixed points in reflexive Banach spaces", "title_html": "Nonexpansive fixed points in reflexive Banach spaces", "summary_html": "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.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces", "title_html": "Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces", "url": "https://github.com/openai/math/blob/main/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/328.md" }, "peer_review": "not yet", "episode": { "video_id": "eTdJslPL8FE", "length": "3:50", "title": "Always a fixed point?", "page": "/episodes/328-kirk-fixed-point/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/328-kirk-fixed-point/" }, { "num": "329", "slug": "329-entropy-duality", "title": "A counterexample to metric-entropy duality", "title_html": "A counterexample to metric-entropy duality", "summary_html": "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.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "Counterexamples to the duality conjecture for metric entropy", "title_html": "Counterexamples to the duality conjecture for metric entropy", "url": "https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/329.md" }, "peer_review": "not yet", "episode": { "video_id": "tgy6RdcmbgQ", "length": "3:45", "title": "Covering is not self-dual?", "page": "/episodes/329-entropy-duality/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/329-entropy-duality/" }, { "num": "330", "slug": "330-free-space-bap", "title": "A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces", "title_html": "A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces", "summary_html": "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.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces", "title_html": "A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/330.md" }, "peer_review": "not yet", "episode": { "video_id": "eqcweTuRNPM", "length": "3:49", "title": "No bound at all?", "page": "/episodes/330-free-space-bap/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/330-free-space-bap/" }, { "num": "331", "slug": "331-midpoint-diamonds", "title": "Reflexive midpoint convexity and diamond distortion", "title_html": "Reflexive midpoint convexity and diamond distortion", "summary_html": "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 1+k/12\\sqrt{1+k/12}1+k/12​, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space", "title_html": "Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space", "url": "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", "role": "main" }, { "title": "Midpoint lenses in segment spaces", "title_html": "Midpoint lenses in segment spaces", "url": "https://github.com/openai/math/blob/main/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Distortion of countably branching diamonds from midpoint and tree energies", "title_html": "Distortion of countably branching diamonds from midpoint and tree energies", "url": "https://github.com/openai/math/blob/main/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Exact asymptotic moduli in a Daugavet subspace of L1", "title_html": "Exact asymptotic moduli in a Daugavet subspace of L1", "url": "https://github.com/openai/math/blob/main/preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Midpoint convexity from bounded tree potentials and path costs", "title_html": "Midpoint convexity from bounded tree potentials and path costs", "url": "https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Independent products in real L1: asymptotic midpoint convexity without AUC renormings", "title_html": "Independent products in real L1: asymptotic midpoint convexity without AUC renormings", "url": "https://github.com/openai/math/blob/main/preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Midpoint convexity from two recursive potentials", "title_html": "Midpoint convexity from two recursive potentials", "url": "https://github.com/openai/math/blob/main/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/331.md" }, "peer_review": "not yet", "episode": { "video_id": "vQwZBTApLA0", "length": "3:58", "title": "Midpoints aren’t enough?", "page": "/episodes/331-midpoint-diamonds/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/331-midpoint-diamonds/" }, { "num": "332", "slug": "332-markov-cotype-l1", "title": "Metric Markov cotype of ℓ₁ and Hilbert-space Lipschitz extension", "title_html": "Metric Markov cotype of ℓ1 and Hilbert-space Lipschitz extension", "summary_html": "Proves that real ℓ1 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 ℓ1 extends to the whole space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.", "subject": "Functional analysis", "subject_slug": "functional-analysis", "papers": [ { "title": "Metric Markov Cotype Two of ℓ₁", "title_html": "Metric Markov Cotype Two of ℓ1", "url": "https://github.com/openai/math/blob/main/preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "eBKHpu_ut3Y", "length": "3:51", "title": "Taxicab maps extend?", "page": "/episodes/332-markov-cotype-l1/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/332-markov-cotype-l1/" }, { "num": "333", "slug": "333-surfaces-r4", "title": "Smooth isometric immersions of surfaces into ℝ⁴", "title_html": "Smooth isometric immersions of surfaces into ℝ4", "summary_html": "Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Smooth isometric immersions of closed surfaces into Euclidean four-space", "title_html": "Smooth isometric immersions of closed surfaces into Euclidean four-space", "url": "https://github.com/openai/math/blob/main/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/333.md" }, "peer_review": "not yet", "episode": { "video_id": "Fs3slnQVJnE", "length": "3:30", "title": "Every surface fits in 4D?", "page": "/episodes/333-surfaces-r4/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/333-surfaces-r4/" }, { "num": "334", "slug": "334-no-local-r3", "title": "A smooth surface metric with no local isometric immersion in ℝ³", "title_html": "A smooth surface metric with no local isometric immersion in ℝ3", "summary_html": "Constructs a smooth positive-definite metric on (−1,1)2(-1,1)^2(−1,1)2 for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A Smooth Metric with No Local Isometric Immersion into Three-Space", "title_html": "A Smooth Metric with No Local Isometric Immersion into Three-Space", "url": "https://github.com/openai/math/blob/main/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/334.md" }, "peer_review": "not yet", "episode": { "video_id": "7qQ3-mB4yuQ", "length": "4:18", "title": "Not even a piece fits in 3D?", "page": "/episodes/334-no-local-r3/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/334-no-local-r3/" }, { "num": "335", "slug": "335-integral-scalar-curvature", "title": "Gromov’s integral scalar-curvature bound for simplicial volume", "title_html": "Gromov’s integral scalar-curvature bound for simplicial volume", "summary_html": "Proves ∫M(Scalg−)n/2 dVg≥an∥M∥\\int_M(\\mathrm{Scal}_g^-)^{n/2}\\,dV_g\\ge a_n\\lVert M\\rVert∫M​(Scalg−​)n/2dVg​≥an​∥M∥ for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with an>0a_n\\gt 0an​>0 depending only on dimension. Here Scalg−=max⁡{0,−Scalg}\\mathrm{Scal}_g^-=\\max\\{0,-\\mathrm{Scal}_g\\}Scalg−​=max{0,−Scalg​} and ∥M∥\\lVert M\\rVert∥M∥ 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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "An integral scalar curvature bound for real simplicial volume", "title_html": "An integral scalar curvature bound for real simplicial volume", "url": "https://github.com/openai/math/blob/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf", "role": "main" }, { "title": "Positive scalar curvature forces rational inessentiality", "title_html": "Positive scalar curvature forces rational inessentiality", "url": "https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "jhgVDr408ek", "length": "4:18", "title": "Curvature pays for topology?", "page": "/episodes/335-integral-scalar-curvature/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/335-integral-scalar-curvature/" }, { "num": "336", "slug": "336-urysohn-width", "title": "Spectral scalar curvature, Urysohn width, and macroscopic dimension", "title_html": "Spectral scalar curvature, Urysohn width, and macroscopic dimension", "summary_html": "Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying −4Δ+Scal≥1-4\\Delta+\\mathrm{Scal}\\ge1−4Δ+Scal≥1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n−2n-2n−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−2n-2n−2 for every n ≥ 2.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Spectral scalar curvature and uniform Urysohn width", "title_html": "Spectral scalar curvature and uniform Urysohn width", "url": "https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf", "role": "main" }, { "title": "Spectral scalar curvature and Urysohn width in dimension three", "title_html": "Spectral scalar curvature and Urysohn width in dimension three", "url": "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", "role": "companion" }, { "title": "Positive scalar curvature and uniform codimension-two width", "title_html": "Positive scalar curvature and uniform codimension-two width", "url": "https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "1_S7PckBxEc", "length": "5:01", "title": "Two dimensions disappear?", "page": "/episodes/336-urysohn-width/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/336-urysohn-width/" }, { "num": "337", "slug": "337-cartan-hadamard", "title": "Sharp Cartan–Hadamard isoperimetry and rigidity", "title_html": "Sharp Cartan–Hadamard isoperimetry and rigidity", "summary_html": "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)(0)(0) spaces.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity", "title_html": "Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity", "url": "https://github.com/openai/math/blob/main/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf", "role": "main" }, { "title": "Sharp integral fillings in CAT(0) spaces", "title_html": "Sharp integral fillings in CAT(0) spaces", "url": "https://github.com/openai/math/blob/main/preprints/Sharp-integral-fillings-in-CAT(0", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/337.md" }, "peer_review": "not yet", "episode": { "video_id": "aQytIDhDpYQ", "length": "4:28", "title": "The ball wins everywhere?", "page": "/episodes/337-cartan-hadamard/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/337-cartan-hadamard/" }, { "num": "338", "slug": "338-yau-uniformization", "title": "Yau's uniformization conjecture", "title_html": "Yau's uniformization conjecture", "summary_html": "Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂn.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Uniformization of complete Kähler manifolds with positive bisectional curvature", "title_html": "Uniformization of complete Kähler manifolds with positive bisectional curvature", "url": "https://github.com/openai/math/blob/main/preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "toTXsN204nk", "length": "4:26", "title": "Yau’s question settled?", "page": "/episodes/338-yau-uniformization/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/338-yau-uniformization/" }, { "num": "339", "slug": "339-katok-entropy", "title": "Katok's entropy rigidity conjecture", "title_html": "Katok's entropy rigidity conjecture", "summary_html": "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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Entropy equality and local symmetry in negative curvature", "title_html": "Entropy equality and local symmetry in negative curvature", "url": "https://github.com/openai/math/blob/main/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "GKnC8u8-VU4", "length": "3:55", "title": "Most chaotic only if symmetric?", "page": "/episodes/339-katok-entropy/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/339-katok-entropy/" }, { "num": "340", "slug": "340-nearby-lagrangian", "title": "A counterexample to the nearby Lagrangian conjecture", "title_html": "A counterexample to the nearby Lagrangian conjecture", "summary_html": "Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in T∗(S9×SN−1)T^*(S^9\\times S^{N-1})T∗(S9×SN−1) that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A counterexample to the nearby Lagrangian conjecture", "title_html": "A counterexample to the nearby Lagrangian conjecture", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "rqv0drnXfok", "length": "3:59", "title": "Arnold’s guess fails?", "page": "/episodes/340-nearby-lagrangian/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/340-nearby-lagrangian/" }, { "num": "341", "slug": "341-hypersymplectic", "title": "Donaldson's hypersymplectic deformation conjecture", "title_html": "Donaldson's hypersymplectic deformation conjecture", "summary_html": "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 ∫ωi∧ωj=δij\\int\\omega_i\\wedge\\omega_j=\\delta_{ij}∫ωi​∧ωj​=δij​, 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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Deforming hypersymplectic four-manifolds to hyperkähler triples", "title_html": "Deforming hypersymplectic four-manifolds to hyperkähler triples", "url": "https://github.com/openai/math/blob/main/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-October-7-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "ljlYUIkmCYU", "length": "4:00", "title": "Every triple becomes perfect?", "page": "/episodes/341-hypersymplectic/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/341-hypersymplectic/" }, { "num": "342", "slug": "342-tamed-compatible", "title": "Donaldson's tamed-to-compatible conjecture", "title_html": "Donaldson's tamed-to-compatible conjecture", "summary_html": "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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Taming implies compatibility on four-manifolds", "title_html": "Taming implies compatibility on four-manifolds", "url": "https://github.com/openai/math/blob/main/preprints/Taming-implies-compatibility-on-four-manifolds-October-6-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/342.md" }, "peer_review": "not yet", "episode": { "video_id": "TmakalqmypE", "length": "3:51", "title": "Tamed means compatible?", "page": "/episodes/342-tamed-compatible/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/342-tamed-compatible/" }, { "num": "343", "slug": "343-ball-packing", "title": "Symplectic ball packing in higher dimensions", "title_html": "Symplectic ball packing in higher dimensions", "summary_html": "Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension 2n≥62n\\ge62n≥6. Finitely many closed symplectic balls of capacities R1,…,RkR_1,\\ldots,R_kR1​,…,Rk​ embed disjointly into an open ball of capacity R exactly when ∑iRin<Rn\\sum_iR_i^n\\lt R^n∑i​Rin​<Rn and Ri+Rj<RR_i+R_j\\lt RRi​+Rj​<R for every distinct pair i, j.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Symplectic Ball Packings in Higher Dimensions", "title_html": "Symplectic Ball Packings in Higher Dimensions", "url": "https://github.com/openai/math/blob/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/343.md" }, "peer_review": "not yet", "episode": { "video_id": "UNHMU_xDRmw", "length": "3:27", "title": "Two tests are enough?", "page": "/episodes/343-ball-packing/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/343-ball-packing/" }, { "num": "344", "slug": "344-metric-blaschke", "title": "The metric Blaschke conjecture", "title_html": "The metric Blaschke conjecture", "summary_html": "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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "The metric Blaschke theorem", "title_html": "The metric Blaschke theorem", "url": "https://github.com/openai/math/blob/main/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "bR8zQ7cwcE4", "length": "4:02", "title": "Only five shapes?", "page": "/episodes/344-metric-blaschke/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/344-metric-blaschke/" }, { "num": "345", "slug": "345-closed-geodesics", "title": "Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds", "title_html": "Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds", "summary_html": "Proves that every smooth Riemannian metric on Sn, 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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Infinitely many closed geodesic images on every Riemannian sphere", "title_html": "Infinitely many closed geodesic images on every Riemannian sphere", "url": "https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "cc7GOOr1to8", "length": "4:26", "title": "Infinitely many loops?", "page": "/episodes/345-closed-geodesics/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/345-closed-geodesics/" }, { "num": "346", "slug": "346-varifold-singular-set", "title": "Sharp singular-set bounds for stationary integral varifolds", "title_html": "Sharp singular-set bounds for stationary integral varifolds", "summary_html": "Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most m−1m-1m−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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold", "title_html": "A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold", "url": "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", "role": "main" }, { "title": "Almost-everywhere regularity of stationary integral varifolds", "title_html": "Almost-everywhere regularity of stationary integral varifolds", "url": "https://github.com/openai/math/blob/main/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "gY2kGh-OWUg", "length": "4:37", "title": "Smooth almost everywhere?", "page": "/episodes/346-varifold-singular-set/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/346-varifold-singular-set/" }, { "num": "347", "slug": "347-arnold-counterexamples", "title": "Counterexamples to stable-Morse and strong Arnold fixed-point bounds", "title_html": "Counterexamples to stable-Morse and strong Arnold fixed-point bounds", "summary_html": "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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two", "title_html": "Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two", "url": "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", "role": "main" }, { "title": "Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number", "title_html": "Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number", "url": "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", "role": "companion" }, { "title": "Hamiltonian Fixed Points Below the Stable Morse Number", "title_html": "Hamiltonian Fixed Points Below the Stable Morse Number", "url": "https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf", "role": "companion" }, { "title": "A nondegenerate counterexample to the Morse-number Arnold bound", "title_html": "A nondegenerate counterexample to the Morse-number Arnold bound", "url": "https://github.com/openai/math/blob/main/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf", "role": "companion" }, { "title": "Three fixed points on the symplectic quadric threefold", "title_html": "Three fixed points on the symplectic quadric threefold", "url": "https://github.com/openai/math/blob/main/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/347.md" }, "peer_review": "not yet", "episode": { "video_id": "J6tbgMj_IW4", "length": "4:57", "title": "Fewer fixed points?", "page": "/episodes/347-arnold-counterexamples/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/347-arnold-counterexamples/" }, { "num": "348", "slug": "348-einstein-four-manifolds", "title": "Nonnegative-curvature Einstein classification and an L² topological gap", "title_html": "Nonnegative-curvature Einstein classification and an L2 topological gap", "summary_html": "Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study CP2\\mathbb{CP}^2CP2, 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 L2 norm of its trace-free Ricci curvature lies below a universal positive constant.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Zero-Plane Rigidity for Einstein Four-Manifolds", "title_html": "Zero-Plane Rigidity for Einstein Four-Manifolds", "url": "https://github.com/openai/math/blob/main/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf", "role": "main" }, { "title": "An L² Einstein Gap for Nonnegatively Curved Four-Manifolds", "title_html": "An L² Einstein Gap for Nonnegatively Curved Four-Manifolds", "url": "https://github.com/openai/math/blob/main/preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf", "role": "companion" }, { "title": "Positively curved Einstein four-manifolds", "title_html": "Positively curved Einstein four-manifolds", "url": "https://github.com/openai/math/blob/main/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/348.md" }, "peer_review": "not yet", "episode": { "video_id": "hosRIA77xQE", "length": "4:42", "title": "Only three shapes?", "page": "/episodes/348-einstein-four-manifolds/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/348-einstein-four-manifolds/" }, { "num": "349", "slug": "349-solomon-yau", "title": "The Solomon–Yau least-volume conjecture", "title_html": "The Solomon–Yau least-volume conjecture", "summary_html": "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 Sm+1S^{m+1}Sm+1 with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "The Solomon–Yau least-volume theorem", "title_html": "The Solomon–Yau least-volume theorem", "url": "https://github.com/openai/math/blob/main/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "jUeYlIp8xPw", "length": "4:28", "title": "The next smallest?", "page": "/episodes/349-solomon-yau/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/349-solomon-yau/" }, { "num": "350", "slug": "350-yau-nodal-bounds", "title": "Yau’s nodal bounds: surfaces and higher dimensions", "title_html": "Yau’s nodal bounds: surfaces and higher dimensions", "summary_html": "Proves the sharp CλC\\sqrt\\lambdaCλ​ 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 S3 arbitrarily close to round. In dimension five, nodal measure can grow faster than λ1/2+ε0\\lambda^{1/2+\\varepsilon_0}λ1/2+ε0​ for some fixed ε0>0\\varepsilon_0\\gt 0ε0​>0, ruling out even arbitrarily small power losses.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Sharp nodal length on smooth surfaces", "title_html": "Sharp nodal length on smooth surfaces", "url": "https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf", "role": "main" }, { "title": "Smooth counterexamples to Yau's nodal upper bound in dimensions three and four", "title_html": "Smooth counterexamples to Yau's nodal upper bound in dimensions three and four", "url": "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", "role": "companion" }, { "title": "Power-law violations of Yau's nodal upper bound", "title_html": "Power-law violations of Yau's nodal upper bound", "url": "https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/350.md" }, "peer_review": "not yet", "episode": { "video_id": "uCKWm6iC6-I", "length": "4:09", "title": "Square-root law?", "page": "/episodes/350-yau-nodal-bounds/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/350-yau-nodal-bounds/" }, { "num": "351", "slug": "351-scalar-curvature-ricci-flow", "title": "Scalar curvature and finite-time Ricci-flow singularities", "title_html": "Scalar curvature and finite-time Ricci-flow singularities", "summary_html": "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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup", "title_html": "A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup", "url": "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", "role": "main" }, { "title": "Bounded scalar curvature and smooth extension of four-dimensional Ricci flow", "title_html": "Bounded scalar curvature and smooth extension of four-dimensional Ricci flow", "url": "https://github.com/openai/math/blob/main/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Path selection and an elliptic inequality on degenerating Ricci-flat trees", "title_html": "Path selection and an elliptic inequality on degenerating Ricci-flat trees", "url": "https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "cEvD16TJ7_Y", "length": "4:11", "title": "Is scalar enough?", "page": "/episodes/351-scalar-curvature-ricci-flow/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/351-scalar-curvature-ricci-flow/" }, { "num": "352", "slug": "352-calabi-flow-singularity", "title": "A finite-time singularity of Calabi flow", "title_html": "A finite-time singularity of Calabi flow", "summary_html": "Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth U(10)U(10)U(10)-invariant Kähler metric on CP10\\mathbb{CP}^{10}CP10 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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A finite-time singularity of Calabi flow on projective space", "title_html": "A finite-time singularity of Calabi flow on projective space", "url": "https://github.com/openai/math/blob/main/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "EVJ26Z312lc", "length": "4:04", "title": "Calabi flow breaks?", "page": "/episodes/352-calabi-flow-singularity/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/352-calabi-flow-singularity/" }, { "num": "353", "slug": "353-affine-bernstein", "title": "Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample", "title_html": "Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample", "summary_html": "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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten", "title_html": "A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten", "url": "https://github.com/openai/math/blob/main/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf", "role": "main" }, { "title": "The affine Bernstein theorem in dimensions three through nine", "title_html": "The affine Bernstein theorem in dimensions three through nine", "url": "https://github.com/openai/math/blob/main/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/353.md" }, "peer_review": "not yet", "episode": { "video_id": "IEKDycCgpz8", "length": "4:15", "title": "Nine, then ten?", "page": "/episodes/353-affine-bernstein/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/353-affine-bernstein/" }, { "num": "354", "slug": "354-cubic-torus-isoperimetric", "title": "The isoperimetric profile of the cubic three-torus", "title_html": "The isoperimetric profile of the cubic three-torus", "summary_html": "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 4π/814\\pi/814π/81 and 1/π1/\\pi1/π, with exactly the adjacent two types minimizing at each transition.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "The Isoperimetric Conjecture for the Cubic Flat Three-Torus", "title_html": "The Isoperimetric Conjecture for the Cubic Flat Three-Torus", "url": "https://github.com/openai/math/blob/main/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/354.md" }, "peer_review": "not yet", "episode": { "video_id": "tgCerPaJbbs", "length": "3:51", "title": "Ball, tube or slab?", "page": "/episodes/354-cubic-torus-isoperimetric/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/354-cubic-torus-isoperimetric/" }, { "num": "355", "slug": "355-tangent-flow-uniqueness", "title": "Unique tangent flows at the first surface singularity", "title_html": "Unique tangent flows at the first surface singularity", "summary_html": "Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ3. 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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow", "title_html": "Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow", "url": "https://github.com/openai/math/blob/main/preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "RLyNIfRzZYA", "length": "4:18", "title": "Zoom in: one picture?", "page": "/episodes/355-tangent-flow-uniqueness/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/355-tangent-flow-uniqueness/" }, { "num": "356", "slug": "356-gigli-alexandrov-curvature", "title": "Gigli’s characterization of Alexandrov curvature", "title_html": "Gigli’s characterization of Alexandrov curvature", "summary_html": "Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support RCD((n−1)κ,n)\\mathop{\\mathrm{RCD}}\\nolimits ((n-1)\\kappa,n)RCD((n−1)κ,n) condition with reference measure Hn\\mathcal H^nHn and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Gigli’s distributional curvature characterization of Alexandrov spaces", "title_html": "Gigli’s distributional curvature characterization of Alexandrov spaces", "url": "https://github.com/openai/math/blob/main/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf", "role": "main" }, { "title": "Weak Hessian bounds along every geodesic in RCD spaces", "title_html": "Weak Hessian bounds along every geodesic in RCD spaces", "url": "https://github.com/openai/math/blob/main/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/356.md" }, "peer_review": "not yet", "episode": { "video_id": "BIKuaAKvqT4", "length": "4:15", "title": "Fat triangles from a tensor?", "page": "/episodes/356-gigli-alexandrov-curvature/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/356-gigli-alexandrov-curvature/" }, { "num": "357", "slug": "357-rcd-bilipschitz-charts", "title": "Bi-Lipschitz coordinates at every regular RCD point", "title_html": "Bi-Lipschitz coordinates at every regular RCD point", "summary_html": "Proves that every regular point of a noncollapsed RCD(K,n)\\mathop{\\mathrm{RCD}}\\nolimits (K,n)RCD(K,n) space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝn. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly Hn\\mathcal H^nHn, and the chart compares ambient distances with a point-dependent finite constant.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces", "title_html": "Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces", "url": "https://github.com/openai/math/blob/main/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "M8FMqh5vE2s", "length": "4:10", "title": "Honest coordinates?", "page": "/episodes/357-rcd-bilipschitz-charts/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/357-rcd-bilipschitz-charts/" }, { "num": "358", "slug": "358-no-conjugate-points", "title": "A three-manifold without conjugate points or nonpositive curvature", "title_html": "A three-manifold without conjugate points or nonpositive curvature", "summary_html": "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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric", "title_html": "A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/358.md" }, "peer_review": "not yet", "episode": { "video_id": "ljoBioZQgk8", "length": "3:50", "title": "No conjugate points, yet never nonpositive?", "page": "/episodes/358-no-conjugate-points/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/358-no-conjugate-points/" }, { "num": "359", "slug": "359-kahler-no-bounded-coordinates", "title": "Negative Kähler curvature without bounded holomorphic coordinates", "title_html": "Negative Kähler curvature without bounded holomorphic coordinates", "summary_html": "Constructs a contractible domain in ℂ3 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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A negatively pinched Kähler threefold without bounded holomorphic coordinates", "title_html": "A negatively pinched Kähler threefold without bounded holomorphic coordinates", "url": "https://github.com/openai/math/blob/main/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf", "role": "main" }, { "title": "One-sided negative sectional curvature and the holomorphic Liouville property", "title_html": "One-sided negative sectional curvature and the holomorphic Liouville property", "url": "https://github.com/openai/math/blob/main/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/359.md" }, "peer_review": "not yet", "episode": { "video_id": "yqr4Q-3vFG0", "length": "4:29", "title": "Pinched negative, no bounded domain?", "page": "/episodes/359-kahler-no-bounded-coordinates/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/359-kahler-no-bounded-coordinates/" }, { "num": "360", "slug": "360-weak-mtw-transport", "title": "Weak MTW curvature gives convexity and regular optimal transport", "title_html": "Weak MTW curvature gives convexity and regular optimal transport", "summary_html": "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.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "Global Support and Convex Injectivity Domains under Weak MTW", "title_html": "Global Support and Convex Injectivity Domains under Weak MTW", "url": "https://github.com/openai/math/blob/main/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf", "role": "main" }, { "title": "Uniform Bi-Holder Transport from Weak MTW", "title_html": "Uniform Bi-Holder Transport from Weak MTW", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/360.md" }, "peer_review": "not yet", "episode": { "video_id": "DzsanqKdgP8", "length": "4:08", "title": "Curvature keeps transport tame?", "page": "/episodes/360-weak-mtw-transport/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/360-weak-mtw-transport/" }, { "num": "361", "slug": "361-harmonic-dimension-counterexample", "title": "Failure of integer-degree harmonic dimension comparison", "title_html": "Failure of integer-degree harmonic dimension comparison", "summary_html": "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 ℝ3 has at least (k+2)2(k+2)^2(k+2)2 independent harmonic functions of growth at most k, exceeding the Euclidean count (k+1)2(k+1)^2(k+1)2. The metric may depend on k.", "subject": "Differential geometry", "subject_slug": "differential-geometry", "papers": [ { "title": "A counterexample to integer-degree harmonic dimension comparison", "title_html": "A counterexample to integer-degree harmonic dimension comparison", "url": "https://github.com/openai/math/blob/main/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf", "role": "main" }, { "title": "A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison", "title_html": "A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison", "url": "https://github.com/openai/math/blob/main/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/361.md" }, "peer_review": "not yet", "episode": { "video_id": "0UwSFc_k8lQ", "length": "3:52", "title": "More harmonic than flat space?", "page": "/episodes/361-harmonic-dimension-counterexample/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/361-harmonic-dimension-counterexample/" }, { "num": "362", "slug": "362-vlasov-maxwell-global", "title": "Global smoothness for relativistic Vlasov–Maxwell", "title_html": "Global smoothness for relativistic Vlasov–Maxwell", "summary_html": "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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system", "title_html": "Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system", "url": "https://github.com/openai/math/blob/main/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/362.md" }, "peer_review": "not yet", "episode": { "video_id": "vQioSFY5dmU", "length": "3:48", "title": "Plasmas never blow up?", "page": "/episodes/362-vlasov-maxwell-global/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/362-vlasov-maxwell-global/" }, { "num": "363", "slug": "363-boltzmann-nonuniqueness", "title": "Nonuniqueness with local conservation for the hard-sphere Boltzmann equation", "title_html": "Nonuniqueness with local conservation for the hard-sphere Boltzmann equation", "summary_html": "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 L1 and satisfy exact local conservation of mass, momentum and kinetic energy.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Nonuniqueness with local conservation for the hard-sphere Boltzmann equation", "title_html": "Nonuniqueness with local conservation for the hard-sphere Boltzmann equation", "url": "https://github.com/openai/math/blob/main/preprints/Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026/paper.pdf", "role": "main" }, { "title": "Nonuniqueness for the periodic hard-sphere Boltzmann equation", "title_html": "Nonuniqueness for the periodic hard-sphere Boltzmann equation", "url": "https://github.com/openai/math/blob/main/preprints/Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/363.md" }, "peer_review": "not yet", "episode": { "video_id": "i_vXPHfYztE", "length": "3:49", "title": "Same gas, two futures?", "page": "/episodes/363-boltzmann-nonuniqueness/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/363-boltzmann-nonuniqueness/" }, { "num": "364", "slug": "364-boltzmann-grad-lifespan", "title": "Kinetic limits and fluctuations over the Boltzmann lifespan", "title_html": "Kinetic limits and fluctuations over the Boltzmann lifespan", "summary_html": "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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals", "title_html": "The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals", "url": "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", "role": "main" }, { "title": "Hard-sphere fluctuations on the regular Boltzmann lifespan", "title_html": "Hard-sphere fluctuations on the regular Boltzmann lifespan", "url": "https://github.com/openai/math/blob/main/preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "ZvGF9fPM4Tk", "length": "3:44", "title": "Newton to Boltzmann, for longer?", "page": "/episodes/364-boltzmann-grad-lifespan/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/364-boltzmann-grad-lifespan/" }, { "num": "365", "slug": "365-calderon-one-patch", "title": "Joint metric and connection recovery from one boundary patch", "title_html": "Joint metric and connection recovery from one boundary patch", "summary_html": "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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Determination of a metric and a unitary connection from one boundary patch", "title_html": "Determination of a metric and a unitary connection from one boundary patch", "url": "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", "role": "main" }, { "title": "Smooth anisotropic uniqueness in the Calderón problem from one boundary patch", "title_html": "Smooth anisotropic uniqueness in the Calderón problem from one boundary patch", "url": "https://github.com/openai/math/blob/main/preprints/Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026/paper.pdf", "role": "companion" }, { "title": "Nonuniqueness for bounded measurable scalar conductivities in three dimensions", "title_html": "Nonuniqueness for bounded measurable scalar conductivities in three dimensions", "url": "https://github.com/openai/math/blob/main/preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026/paper.pdf", "role": "companion" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/365.md" }, "peer_review": "not yet", "episode": { "video_id": "Y8wbrEIkYXY", "length": "4:04", "title": "One small patch sees everything?", "page": "/episodes/365-calderon-one-patch/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/365-calderon-one-patch/" }, { "num": "366", "slug": "366-mumford-shah-interior", "title": "The planar Mumford–Shah regularity conjecture and local weak-L⁴ gradient bounds", "title_html": "The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds", "summary_html": "Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a C1,αC^{1,\\alpha}C1,α arc, a regular crack tip, or three arcs meeting at 120∘120^\\circ120∘; only finitely many global connected components meet any compact interior region.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Interior regularity of planar Mumford–Shah minimizers", "title_html": "Interior regularity of planar Mumford–Shah minimizers", "url": "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", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "k77g-ZjG75k", "length": "3:59", "title": "Only three shapes for every edge?", "page": "/episodes/366-mumford-shah-interior/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/366-mumford-shah-interior/" }, { "num": "367", "slug": "367-bernoulli-critical-dimension", "title": "The critical dimension for the one-phase Bernoulli problem", "title_html": "The critical dimension for the one-phase Bernoulli problem", "summary_html": "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−7n-7n−7 in higher dimensions.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "The critical dimension for one-phase Bernoulli minimizers", "title_html": "The critical dimension for one-phase Bernoulli minimizers", "url": "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", "role": "main" } ], "lean": { "status": "partial", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/367.md" }, "peer_review": "not yet", "episode": { "video_id": "jNnrILD9szg", "length": "3:56", "title": "Five, six or seven?", "page": "/episodes/367-bernoulli-critical-dimension/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/367-bernoulli-critical-dimension/" }, { "num": "368", "slug": "368-ball-evans-3d", "title": "The three-dimensional Ball–Evans approximation problem", "title_html": "The three-dimensional Ball–Evans approximation problem", "summary_html": "Resolves the three-dimensional Ball–Evans approximation problem: every W1,pW^{1,p}W1,p homeomorphism between arbitrary bounded domains in ℝ3, for 1≤p<∞1\\le p\\lt \\infty1≤p<∞, is a strong W1,pW^{1,p}W1,p limit of smooth diffeomorphisms onto the same target.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Strong diffeomorphic approximation in three dimensions for 1≤p≤2", "title_html": "Strong diffeomorphic approximation in three dimensions for 1≤p≤2", "url": "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", "role": "main" }, { "title": "Strong diffeomorphic approximation in three dimensions for p>2", "title_html": "Strong diffeomorphic approximation in three dimensions for p>2", "url": "https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026/main.pdf", "role": "companion" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "3220HUj7Gjw", "length": "4:01", "title": "Smooth any 3-D deformation?", "page": "/episodes/368-ball-evans-3d/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/368-ball-evans-3d/" }, { "num": "369", "slug": "369-hot-spots-simply-connected", "title": "The hot spots conjecture for simply connected planar domains", "title_html": "The hot spots conjecture for simply connected planar domains", "summary_html": "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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Strict hot spots and absence of interior critical points on smooth simply connected planar domains", "title_html": "Strict hot spots and absence of interior critical points on smooth simply connected planar domains", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/369.md" }, "peer_review": "not yet", "episode": { "video_id": "7aUJSs0VEDk", "length": "4:02", "title": "Hot spots on the edge?", "page": "/episodes/369-hot-spots-simply-connected/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/369-hot-spots-simply-connected/" }, { "num": "370", "slug": "370-henon-lane-emden", "title": "The Lane–Emden and Hénon–Lane–Emden conjectures", "title_html": "The Lane–Emden and Hénon–Lane–Emden conjectures", "summary_html": "Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, p,q>0p,q\\gt 0p,q>0 and real A, B, the system −Δu=∣x∣Avp-\\Delta u=|x|^A v^p−Δu=∣x∣Avp, −Δv=∣x∣Buq-\\Delta v=|x|^B u^q−Δv=∣x∣Buq has no positive entire solution when (n+A)/(p+1)+(n+B)/(q+1)>n−2(n+A)/(p+1)+(n+B)/(q+1)\\gt n-2(n+A)/(p+1)+(n+B)/(q+1)>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>−2A,B\\gt -2A,B>−2.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "The Subcritical Hénon–Lane–Emden Conjecture", "title_html": "The Subcritical Hénon–Lane–Emden Conjecture", "url": "https://github.com/openai/math/blob/main/preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026/paper.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/370.md" }, "peer_review": "not yet", "episode": { "video_id": "Jc__0qD67wM", "length": "4:18", "title": "No solutions below the curve?", "page": "/episodes/370-henon-lane-emden/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/370-henon-lane-emden/" }, { "num": "371", "slug": "371-defocusing-nls-blowup", "title": "Stable blowup for the defocusing Schrödinger equation", "title_html": "Stable blowup for the defocusing Schrödinger equation", "summary_html": "For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in Hk(T12)H^k(\\mathbb T^{12})Hk(T12), with k > 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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus", "title_html": "Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/371.md" }, "peer_review": "not yet", "episode": { "video_id": "JZT-BbL_-A8", "length": "4:29", "title": "Stable blowup for waves that spread?", "page": "/episodes/371-defocusing-nls-blowup/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/371-defocusing-nls-blowup/" }, { "num": "372", "slug": "372-isotropic-elasticity-uniqueness", "title": "Global uniqueness in smooth isotropic elasticity", "title_html": "Global uniqueness in smooth isotropic elasticity", "summary_html": "Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and 3λ+2μ>03\\lambda+2\\mu\\gt 03λ+2μ>0 on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem", "title_html": "Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem", "url": "https://github.com/openai/math/blob/main/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/372.md" }, "peer_review": "not yet", "episode": { "video_id": "aV1h4XKBaEA", "length": "3:56", "title": "Feel the inside from the surface?", "page": "/episodes/372-isotropic-elasticity-uniqueness/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/372-isotropic-elasticity-uniqueness/" }, { "num": "373", "slug": "373-coulomb-monge-counterexample", "title": "Nonattainment of the three-marginal Coulomb Monge problem", "title_html": "Nonattainment of the three-marginal Coulomb Monge problem", "summary_html": "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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "A counterexample to the Monge ansatz for the three-marginal Coulomb cost", "title_html": "A counterexample to the Monge ansatz for the three-marginal Coulomb cost", "url": "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", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/373.md" }, "peer_review": "not yet", "episode": { "video_id": "lKKtb2ySVoc", "length": "3:55", "title": "Electrons that must flip a coin?", "page": "/episodes/373-coulomb-monge-counterexample/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/373-coulomb-monge-counterexample/" }, { "num": "374", "slug": "374-brenier-one-third-stability", "title": "Sharp one-third stability of Brenier maps", "title_html": "Sharp one-third stability of Brenier maps", "summary_html": "For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy ∥Tμ−Tν∥L2(ρ)≤CW2(μ,ν)1/3\\|T_\\mu-T_\\nu\\|_{L^2(\\rho)}\\le C W_2(\\mu,\\nu)^{1/3}∥Tμ​−Tν​∥L2(ρ)​≤CW2​(μ,ν)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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Sharp One-Third Stability of Brenier Maps", "title_html": "Sharp One-Third Stability of Brenier Maps", "url": "https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf", "role": "main" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/374.md" }, "peer_review": "not yet", "episode": { "video_id": "rXefUQyQL7s", "length": "3:40", "title": "One third, not one half?", "page": "/episodes/374-brenier-one-third-stability/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/374-brenier-one-third-stability/" }, { "num": "375", "slug": "375-de-giorgi-dimension-eight", "title": "De Giorgi's conjecture in dimension eight", "title_html": "De Giorgi's conjecture in dimension eight", "summary_html": "Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution u:R8→(−1,1)u:\\mathbb R^8\\to(-1,1)u:R8→(−1,1) of Δu=u3−u\\Delta u=u^3-uΔu=u3−u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:R7→[−1,1]v:\\mathbb R^7\\to[-1,1]v:R7→[−1,1] as constant wells or planar transitions, without an energy-growth assumption.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "A positive resolution of De Giorgi's conjecture in dimension eight", "title_html": "A positive resolution of De Giorgi's conjecture in dimension eight", "url": "https://github.com/openai/math/blob/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "AZMaj6sOpGc", "length": "4:12", "title": "Only flat walls in eight dimensions?", "page": "/episodes/375-de-giorgi-dimension-eight/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/375-de-giorgi-dimension-eight/" }, { "num": "376", "slug": "376-navier-stokes-computation", "title": "Universal computation in forced Navier–Stokes flows", "title_html": "Universal computation in forced Navier–Stokes flows", "summary_html": "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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Finite Instructions and Solenoidal Shear Flows", "title_html": "Finite Instructions and Solenoidal Shear Flows", "url": "https://github.com/openai/math/blob/main/preprints/Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026/manuscript.pdf", "role": "main" }, { "title": "A Fixed Particle Test for Computation in a Forced Viscous Flow", "title_html": "A Fixed Particle Test for Computation in a Forced Viscous Flow", "url": "https://github.com/openai/math/blob/main/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Universal Computation with Eventually Stationary Navier–Stokes Forcing", "title_html": "Universal Computation with Eventually Stationary Navier–Stokes Forcing", "url": "https://github.com/openai/math/blob/main/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Geometric Programs for Solenoidal Forcing", "title_html": "Geometric Programs for Solenoidal Forcing", "url": "https://github.com/openai/math/blob/main/preprints/Geometric-Programs-for-Solenoidal-Forcing-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Prefix Instructions and Incompressible Flows", "title_html": "Prefix Instructions and Incompressible Flows", "url": "https://github.com/openai/math/blob/main/preprints/Prefix-Instructions-and-Incompressible-Flows-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Computation under Rapidly Vanishing Navier–Stokes Forcing", "title_html": "Computation under Rapidly Vanishing Navier–Stokes Forcing", "url": "https://github.com/openai/math/blob/main/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Velocity-Field Detection of Computation in Forced Navier–Stokes Flows", "title_html": "Velocity-Field Detection of Computation in Forced Navier–Stokes Flows", "url": "https://github.com/openai/math/blob/main/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Scalar Potentials and Slow Clocks for Forced Fluid Computation", "title_html": "Scalar Potentials and Slow Clocks for Forced Fluid Computation", "url": "https://github.com/openai/math/blob/main/preprints/Scalar-Potentials-and-Slow-Clocks-for-Forced-Fluid-Computation-September-27-2026/manuscript.pdf", "role": "companion" }, { "title": "Incompressible Box Transport and Finite Computation", "title_html": "Incompressible Box Transport and Finite Computation", "url": "https://github.com/openai/math/blob/main/preprints/Incompressible-Box-Transport-and-Finite-Computation-October-6-2026/manuscript.pdf", "role": "companion" } ], "lean": { "status": "main", "doc_url": "https://github.com/openai/math/blob/main/lean/docs/376.md" }, "peer_review": "not yet", "episode": { "video_id": "vDcy8cJ6XLM", "length": "4:26", "title": "A fluid that computes?", "page": "/episodes/376-navier-stokes-computation/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/376-navier-stokes-computation/" }, { "num": "377", "slug": "377-infinity-harmonic-c1alpha", "title": "Interior C^(1,α) regularity for infinity-harmonic functions", "title_html": "Interior C1,αC^{1,\\alpha}C1,α regularity for infinity-harmonic functions", "summary_html": "Proves uniform interior C1,αdC^{1,\\alpha_d}C1,αd​ 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.", "subject": "Partial differential equations", "subject_slug": "partial-differential-equations", "papers": [ { "title": "Uniform Interior C^(1,α) Estimates for Infinity-Harmonic Functions", "title_html": "Uniform Interior C1,αC^{1,\\alpha}C1,α Estimates for Infinity-Harmonic Functions", "url": "https://github.com/openai/math/blob/main/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf", "role": "main" } ], "lean": { "status": "none", "doc_url": null }, "peer_review": "not yet", "episode": { "video_id": "fmyW2bUk038", "length": "3:52", "title": "Continuous gradients in every dimension?", "page": "/episodes/377-infinity-harmonic-c1alpha/" }, "last_checked": "2026-10-19", "url": "https://aiprovedwhat.com/episodes/377-infinity-harmonic-c1alpha/" } ] }