{"about":"AI breakthroughs in mathematics: a short narrated video (field `reel`, with WebVTT captions) for each of the 372 result families of OpenAI's openai/math release, results produced or claimed by AI systems. Every entry is a CLAIM from an unrefereed release; significance, kind and verdict are our reading, Lean status is what the release's own docs say (not re-run).","page":"https://ai.thesatyajit.com/math","article":"https://ai.thesatyajit.com/articles/openai-math","repository":"https://github.com/openai/math","commit":"adc7f1241b42e322a6451854ab7e4b4c146bf78a","counts":{"families":372,"reels":372,"leanMain":169,"leanPart":67,"manuscripts":722,"huge":13},"results":[{"id":"001","title":"Milne’s rationality conjecture and algebraic specialization","short":"Milne's rationality conjecture","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"For an abelian variety over Q-bar with good reduction at a p-adic place, any rational Hodge class, specialized to the special fibre and paired with any complementary product of divisor classes there, gives one and the same rational number in every l-adic (l…","verdict":"Theorem 1.1 itself is independent of other release results, but the headline 'represented by a single rational algebraic cycle' (Corollary 6.4) depends on the release's…","url":"/math#001","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=001.%20Milne%E2%80%99s%20rationality%20conjecture","manuscripts":[{"title":"Milne's rationality conjecture for abelian varieties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/001.mp4","poster":"/films/math/001-poster.webp","captions":"/films/math/001.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":0,"why":"Proves Milne's rationality conjecture for Hodge classes mod p; the strongest form leans on the release's unchecked Hodge paper.","consequence":"A working theory of rational Tate classes on abelian varieties mod p, which Langlands–Rapoport style questions need. Internal to arithmetic geometry."},"detail":"/api/math/001"},{"id":"002","title":"The full BSD formula from low Selmer corank","short":"The full BSD formula in rank $\\le 1$","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every elliptic curve E/Q and any prime q such that the full q-power Selmer group has Z_q-corank 0 or 1: analytic rank = Mordell-Weil rank = that corank, Sha(E/Q) is finite, and the full Birch-Swinnerton-Dyer leading-term formula L^(r)(E,1)/r! = Omega_E…","verdict":"Not formalized. Three manuscripts totalling ~340 pages. The exact-formula paper takes the rank/finiteness statement from the release's own Selmer-converse paper and the…","url":"/math#002","articleUrl":"/articles/openai-math#the-elliptic-curve-stack-bsd-in-rank-at-most-one-goldfeld-and-fontainemazur-at-2-families-002-006-010","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=002.%20The%20full%20BSD","manuscripts":[{"title":"Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/exact-bsd-low-selmer-corank.pdf"},{"title":"The Selmer converse for elliptic curves at every prime","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-September-24-2026/main.pdf"},{"title":"The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/002.mp4","poster":"/films/math/002-poster.webp","captions":"/films/math/002.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":70,"tier":"Solid","importance":4,"advance":2,"consequences":3,"surprise":1,"confidence":0,"why":"The exact Birch–Swinnerton-Dyer formula for curves of rank 0 or 1, prime by prime; 340 unchecked pages chained to other release papers.","consequence":"For every elliptic curve with Selmer corank 0 or 1, Sha's size and the L-value become theorems, not predictions. Feeds the release's Hilbert-tenth proof."},"detail":"/api/math/002"},{"id":"003","title":"The quasi-Riemann hypothesis","short":"The quasi-Riemann hypothesis, $\\mathrm{Re}\\, s > 7/8$","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is no zero of the Riemann zeta function, of any Dirichlet L-function (any modulus, any character, uniformly) or of any finite-order Hecke L-function over Q(sqrt(-3)) in the half-plane Re(s) > 7/8 (principal poles at s=1 allowed).","verdict":"If correct this is among the largest results in analytic number theory in a century, so the bar is extraordinary.","url":"/math#003","articleUrl":"/articles/openai-math#a-fixed-gap-next-to-the-line-re-s--1-family-003","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=003.%20The%20quasi%2DRiemann%20hypothesis","manuscripts":[{"title":"The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf"},{"title":"The Quasi-Riemann Hypothesis (alternate 11/12 proof)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf"},{"title":"Uniform exclusion of Landau–Siegel zeros","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf"}],"reel":{"src":"/films/math/003.mp4","poster":"/films/math/003-poster.webp","captions":"/films/math/003.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":100,"tier":"Huge if true","importance":4,"advance":4,"consequences":4,"surprise":3,"confidence":3,"why":"Rules out zeta and Dirichlet L-function zeros right of 7/8: the first zero-free half-plane ever; Lean checks it against mathlib's zeta.","consequence":"Prime counts in progressions get error terms of x^{7/8} without assumptions, so results now proved only under GRH-type hypotheses could become unconditional."},"detail":"/api/math/003"},{"id":"004","title":"Hilbert’s tenth problem over ℚ","short":"Hilbert's tenth problem over $\\mathbb{Q}$","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"There is no algorithm that, given a polynomial with integer coefficients in any number of variables (number of variables part of the input), decides whether it has a zero in Q^n. The proof does not give an existential definition of Z in Q;","verdict":"Not formalized. 62-page main paper explicitly cites three other release manuscripts as inputs: the pointwise 2-converse for curves with rational 2-torsion (Theorem 1.1,…","url":"/math#004","articleUrl":"/articles/openai-math#hilberts-tenth-problem-over-q-family-004","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=004.%20Hilbert%E2%80%99s%20tenth%20problem","manuscripts":[{"title":"Hilbert’s tenth problem over the rational numbers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hilberts-tenth-problem-over-the-rational-numbers-September-24-2026/main.pdf"},{"title":"A pointwise 2-converse for elliptic curves with rational two-torsion","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/004.mp4","poster":"/films/math/004-poster.webp","captions":"/films/math/004.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":87,"tier":"Major","importance":4,"advance":4,"consequences":2,"surprise":3,"confidence":0,"why":"Shows no algorithm can decide if a polynomial equation has rational solutions; rests on three other unchecked release papers.","consequence":"Closes the decidability question over Q that followed Matiyasevich. Nothing practical changes: the answer is that no algorithm exists."},"detail":"/api/math/004"},{"id":"005","title":"Irrationality of Catalan’s constant","short":"Catalan's constant is irrational","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Catalan's constant G = sum_{j>=0} (-1)^j/(2j+1)^2 = L(2, chi_-4) is irrational.","verdict":"Lean statement is the bare irrationality of the explicit series, so faithful. The proof is a 44-page determinant argument (size-48N mixed determinants whose entries are…","url":"/math#005","articleUrl":"/articles/openai-math#catalans-constant-is-irrational-family-005","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=005.%20Irrationality%20of%20Catalan%E2%80%99s","manuscripts":[{"title":"Catalan's constant is irrational","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/005.mp4","poster":"/films/math/005-poster.webp","captions":"/films/math/005.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":3,"why":"Proves Catalan's constant is irrational, a long-open question, with a new determinant method; Lean checks the main theorem.","consequence":"A first irrationality proof for Catalan's constant; its determinant method is what the release's π result is built on."},"detail":"/api/math/005"},{"id":"006","title":"Goldfeld’s conjecture: densities and mean analytic rank","short":"Goldfeld's conjecture, mean rank $1/2$","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(a) 2-converse: for every E/Q whose full 2-power Selmer corank is 0 or 1, analytic rank = rank = corank and Sha is finite. (b) For every E/Q, among quadratic twists E^(d) with d signed squarefree ordered by |d|, analytic rank 0 and analytic rank 1 each have…","verdict":"Counts signed squarefree d ordered by |d|, slightly different from Goldfeld's fundamental discriminants (the paper says so).","url":"/math#006","articleUrl":"/articles/openai-math#the-elliptic-curve-stack-bsd-in-rank-at-most-one-goldfeld-and-fontainemazur-at-2-families-002-006-010","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=006.%20Goldfeld%E2%80%99s%20conjecture%20densities","manuscripts":[{"title":"Goldfeld's analytic density conjecture and the 2-converse for elliptic curves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf"},{"title":"The mean analytic rank of quadratic twists of elliptic curves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/006.mp4","poster":"/films/math/006-poster.webp","captions":"/films/math/006.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":70,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":1,"confidence":0,"why":"Half of quadratic twists of an elliptic curve have rank 0 and half rank 1, as Goldfeld predicted in 1979; long and unformalized.","consequence":"The average rank of quadratic twists becomes 1/2, unconditionally. It feeds the release's BSD and Hilbert-tenth families, so an error would spread."},"detail":"/api/math/006"},{"id":"007","title":"Ordinary two-point correlations and the corrected Elliott conjecture","short":"Two-point Chowla, plain averages","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every fixed pair of nonproportional affine forms a1 n + b1, a2 n + b2, sum_{n 0 (ordinary, not logarithmic, averaging); in particular the two-point Chowla conjecture sum lambda(n)lambda(n+h) = o(X).","verdict":"Two-point only (k-point Chowla still open). Constants ineffective; no uniformity in the shifts.","url":"/math#007","articleUrl":"/articles/openai-math#two-point-chowla-with-plain-averages-family-007","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=007.%20Ordinary%20two%2Dpoint%20correlations","manuscripts":[{"title":"Ordinary two-point correlations of multiplicative functions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf"}],"reel":{"src":"/films/math/007.mp4","poster":"/films/math/007-poster.webp","captions":"/films/math/007.vtt","duration":19.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"Two-point Chowla with ordinary averages: n and n+h have uncorrelated prime-factor parity; more points stay open. Lean checks it.","consequence":"Better control of sign patterns of the Liouville function at a single scale. Mostly internal to analytic number theory."},"detail":"/api/math/007"},{"id":"008","title":"The Deligne–Drinfeld conjecture","short":"The Deligne–Drinfeld conjecture","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"The solution space of the grt_1 equations (antisymmetry, hexagon/three-term, pentagon in t_4) over Q, with the Ihara bracket, is the free Lie algebra on one generator in each odd weight 3, 5, 7, ...: there are no extra relations and no extra solutions;","verdict":"41 pages. The new content is an all-weight upper bound on dim W_n matching the free Lie algebra.","url":"/math#008","articleUrl":"/articles/openai-math#delignedrinfeld-family-008","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=008.%20The%20Deligne%20Drinfeld","manuscripts":[{"title":"The Deligne-Drinfeld conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/008.mp4","poster":"/films/math/008-poster.webp","captions":"/films/math/008.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"Proves the Deligne–Drinfeld conjecture that the Grothendieck–Teichmüller Lie algebra is free; Lean checks the main theorem.","consequence":"Pins down the Grothendieck–Teichmüller Lie algebra, with consequences for Kontsevich's graph complex and deformation quantization."},"detail":"/api/math/008"},{"id":"009","title":"Function-field reconstruction from Milnor K-theory and Galois data","short":"Function fields from Milnor K-theory","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"A function field K of transcendence degree >= 2 over an algebraically closed field (characteristic != l) is recovered, up to perfect closure and Frobenius twists, from K_1^M/l, K_2^M/l and the Milnor product alone: compatible isomorphisms of this mod-l data…","verdict":"The Lean covers the easy direction (injectivity). The headline Milnor K reconstruction (existence/surjectivity) is not formalized. ~73 pages across three papers.","url":"/math#009","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=009.%20Function%2Dfield%20reconstruction%20from","manuscripts":[{"title":"Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf"},{"title":"Reconstruction from Milnor K-theory modulo the characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf"},{"title":"The Bogomolov-Pop reconstruction theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/009.mp4","poster":"/films/math/009-poster.webp","captions":"/films/math/009.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":1,"why":"Bogomolov's program: a function field is rebuilt from a tiny piece of its Galois data; Lean checks only the easy direction.","consequence":"Tiny Galois data is enough to rebuild a function field. Internal to anabelian geometry."},"detail":"/api/math/009"},{"id":"010","title":"Unrestricted pro-modularity at the prime two","short":"Fontaine–Mazur at the prime 2","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Every continuous odd absolutely irreducible r: G_Q -> GL_2(Q_2-bar) unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level (pro-modularity), and if moreover r is de Rham at 2 with distinct Hodge-Tate weights it is a…","verdict":"Not formalized; 113 pages over three papers. The Fontaine-Mazur paper says it supplies the weight-two modularity input for the Hilbert-tenth-over-Q proof (family 004),…","url":"/math#010","articleUrl":"/articles/openai-math#the-elliptic-curve-stack-bsd-in-rank-at-most-one-goldfeld-and-fontainemazur-at-2-families-002-006-010","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=010.%20Unrestricted%20pro%2Dmodularity%20at","manuscripts":[{"title":"Unrestricted pro-modularity at the prime two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-4-2026/two-adic-promodularity.pdf"},{"title":"The Dimension of the Two-Adic Hecke Algebra at Odd Level","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf"},{"title":"Fontaine–Mazur modularity at the prime 2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/010.mp4","poster":"/films/math/010-poster.webp","captions":"/films/math/010.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Odd 2-dimensional Galois representations at the awkward prime 2 come from modular forms; 113 unformalized pages.","consequence":"Removes the p=2 gap in Fontaine–Mazur for odd 2-dimensional representations, and supplies modularity input to the release's Hilbert-tenth proof."},"detail":"/api/math/010"},{"id":"011","title":"Prime-factor statistics of p-1","short":"The prime factors of $p - 1$","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(a) For a uniformly random prime p 0 infinitely many n have more than n^{1-eps} totient preimages (Erdos). (c) For every delta > 0 there are x^{1-o(1)} primes in (2x, 5x] whose predecessor is x^delta-smooth. (d) Infinitely many primes p have mu(p-1) = 1.","verdict":"Not formalized; ~187 pages over three papers. The 'x^{1-o(1)} smooth shifted primes for every delta' claim is a very strong statement;","url":"/math#011","articleUrl":"/articles/openai-math#the-prime-factors-of-p--1-family-011","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=011.%20Prime%2Dfactor%20statistics%20of","manuscripts":[{"title":"Weighted dilation graphs, smooth shifted primes and totient fibers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf"},{"title":"The Poisson-Dirichlet law for prime predecessors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf"},{"title":"Prime Predecessors with an Even Number of Prime Factors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf"}],"reel":{"src":"/films/math/011.mp4","poster":"/films/math/011-poster.webp","captions":"/films/math/011.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":66,"tier":"Solid","importance":2,"advance":3,"consequences":3,"surprise":2,"confidence":0,"why":"Prime factors of p−1 follow the random-integer law; needs primes in progressions past a classic barrier. Unformalized.","consequence":"Gives the totient-fibre conjecture (via Erdős–Pomerance) and strong counts of smooth shifted primes, by passing a known level-of-distribution barrier."},"detail":"/api/math/011"},{"id":"012","title":"Independent largest prime factors of consecutive integers","short":"Largest prime factors of $n$ and $n + 1$","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every fixed a, b in (0,1), the natural density of n with P+(n) <= n^a and P+(n+1) <= n^b is rho(1/a)rho(1/b): the largest prime factors of n and n+1 are asymptotically independent. Corollary: P+(n) < P+(n+1) has natural density 1/2 (Erdos-Turan question).","verdict":"84 pages. Statement in the Lean is natural density as in the paper.","url":"/math#012","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=012.%20Independent%20largest%20prime","manuscripts":[{"title":"The joint Dickman law for consecutive integers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/012.mp4","poster":"/films/math/012-poster.webp","captions":"/films/math/012.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Whether n is smooth tells you nothing about n+1: the joint Dickman law with ordinary density. Lean checks it.","consequence":"Settles whether smoothness of n and n+1 are independent. Little follow-on outside analytic number theory."},"detail":"/api/math/012"},{"id":"013","title":"Ostmann’s inverse Goldbach conjecture","short":"Ostmann's inverse Goldbach problem","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"If A, B are sets of nonnegative integers each with at least two elements, then A+B differs from the set of primes in infinitely many elements: no finite modification of the primes is a sumset of two nontrivial sets.","verdict":"80 pages; reduces (via Laffer-Mann) to two infinite summands, then uses character sums with translating centres and a finite-field tree comparison.","url":"/math#013","articleUrl":"/articles/openai-math#ostmanns-inverse-goldbach-problem-family-013","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=013.%20Ostmann%E2%80%99s%20inverse%20Goldbach","manuscripts":[{"title":"The additive indecomposability of the primes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/013.mp4","poster":"/films/math/013-poster.webp","captions":"/films/math/013.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"The primes are not a sumset A+B, settling Ostmann's inverse Goldbach question; Lean checks the full statement.","consequence":"Closes Ostmann's question about the primes as a sumset. Little follow-on expected."},"detail":"/api/math/013"},{"id":"014","title":"Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan","short":"Restricted geometric Langlands in char $p$","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Eight papers. (i) The Q_l-bar-linear restricted geometric Langlands equivalence for connected reductive G on smooth projective curves in characteristic p > 0, over F_q-bar under four Lie-theoretic characteristic conditions, and over any algebraically closed…","verdict":"Not formalized; 348 pages over eight papers, the largest family in this group. One result (Arthur enhancements) is explicitly conditional on a decomposition statement.","url":"/math#014","articleUrl":"/articles/openai-math#restricted-geometric-langlands-in-characteristic-p-family-014","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=014.%20Restricted%20geometric%20Langlands","manuscripts":[{"title":"Global Arthur Enhancements of Cuspidal Excursion Parameters","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf"},{"title":"Rationality of the Canonical Unramified Arthur Filtration","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rationality-of-the-Canonical-Unramified-Arthur-Filtration-September-24-2026/paper.pdf"},{"title":"Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ramanujan-Arthur-Decompositions-of-Cuspidal-Functions-at-Full-Finite-Level-September-24-2026/paper.pdf"},{"title":"Temperedness at ramified places for globally generic exceptional groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf"},{"title":"The Restricted Geometric Langlands Equivalence in Positive Characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf"},{"title":"Constructible tame Hecke eigensheaves in positive characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constructible-tame-Hecke-eigensheaves-in-positive-characteristic-October-5-2026/constructible-tame-hecke-eigensheaves-positive-characteristic.pdf"},{"title":"Tame Hecke Eigensheaves with Several Marked Points","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tame-Hecke-Eigensheaves-with-Several-Marked-Points-October-5-2026/Tame-Hecke-Eigensheaves-with-Several-Marked-Points.pdf"},{"title":"Frobenius Structures on Tame Hecke Eigensheaves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf"}],"reel":{"src":"/films/math/014.mp4","poster":"/films/math/014-poster.webp","captions":"/films/math/014.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":70,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":1,"confidence":0,"why":"Restricted geometric Langlands in characteristic p, with Ramanujan bounds as a consequence; 348 pages, conditions on p, no Lean.","consequence":"Ties function-field automorphic forms to geometry in characteristic p and yields Ramanujan bounds for generic cusp forms; inputs for the Langlands program."},"detail":"/api/math/014"},{"id":"015","title":"Torus-packet equidistribution in prime, quartic, and sextic degrees","short":"Torus packets in higher degree","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"Volume-weighted packets of periodic diagonal-torus orbits in the space of unimodular lattices, attached to full lattices/ideal classes in totally real fields, equidistribute to Haar measure without escape of mass as the relevant discriminant tends to…","verdict":"Primitive (no intermediate field) restriction in degrees 4 and 6 matters: it excludes intermediate tori obstructing rigidity.","url":"/math#015","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=015.%20Torus%2Dpacket%20equidistribution%20in","manuscripts":[{"title":"Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf"},{"title":"Equidistribution of primitive quartic torus packets for arbitrary orders","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf"},{"title":"Equidistribution of Primitive Sextic Torus Packets","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf"}],"reel":{"src":"/films/math/015.mp4","poster":"/films/math/015-poster.webp","captions":"/films/math/015.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"Higher-degree analogue of Duke's equidistribution theorem in prime, quartic and sextic degree; Lean covers prime degree.","consequence":"Extends equidistribution of torus orbits to higher-degree number fields. Internal to homogeneous dynamics."},"detail":"/api/math/015"},{"id":"016","title":"Zilber–Pink in abelian varieties and the Siegel threefold","short":"Zilber–Pink: abelian varieties, $\\mathcal{A}_2$","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Abelian Zilber-Pink over Q-bar: every irreducible subvariety X of an abelian variety has only finitely many maximal atypical subvarieties relative to its smallest containing torsion coset (via a non-density theorem plus the Barroero-Dill reduction).","verdict":"Not formalized; ~170 pages over four papers. Over Q-bar only (not C). The abelian theorem is non-effective. Relies on the Barroero-Dill reduction (published).","url":"/math#016","articleUrl":"/articles/openai-math#zilberpink-for-abelian-varieties-and-curves-in-a-family-016","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=016.%20Zilber%20Pink%20in","manuscripts":[{"title":"The abelian Zilber–Pink conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf"},{"title":"The E×CM component of Zilber–Pink for curves in A2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf"},{"title":"Quaternionic division points on curves in the Siegel threefold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf"},{"title":"Elliptic squares and Zilber–Pink for curves in A2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/016.mp4","poster":"/films/math/016-poster.webp","captions":"/films/math/016.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Zilber–Pink unlikely intersections for abelian varieties over algebraic numbers; non-effective and unformalized.","consequence":"Finiteness of unlikely intersections in abelian varieties, the setting that unifies Manin–Mumford and Mordell–Lang. Non-effective, so no explicit bounds."},"detail":"/api/math/016"},{"id":"017","title":"The irrationality exponent of π is 2","short":"The irrationality exponent of $\\pi$ is 2","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every nu > 2 there is Q(nu) such that |pi - p/q| >= q^{-nu} for all integers p and all q >= Q(nu); hence the irrationality exponent mu(pi) = 2 (the threshold is ineffective).","verdict":"The comparator statement is a faithful, elementary statement about Real.pi, which makes this one of the cleanest checks in the release, but review status is 'unchecked'…","url":"/math#017","articleUrl":"/articles/openai-math#the-irrationality-exponent-of-π-is-2-family-017","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=017.%20The%20irrationality%20exponent","manuscripts":[{"title":"The irrationality exponent of pi is 2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/017.mp4","poster":"/films/math/017-poster.webp","captions":"/films/math/017.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":81,"tier":"Major","importance":4,"advance":4,"consequences":1,"surprise":3,"confidence":3,"why":"Pins π's irrationality exponent at exactly 2, after 70 years moved the bound only from 42 to 7.1; Lean checks the statement.","consequence":"Settles how well π can be approximated by fractions. No computation changes; the method may extend to other constants."},"detail":"/api/math/017"},{"id":"018","title":"The Margulis–Platonov conjecture over global fields","short":"Margulis–Platonov over global fields","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every global field k (number fields and function fields, including characteristic 2) and every absolutely almost simple simply connected k-group G, every noncentral (abstract) normal subgroup N of G(k) equals the preimage of an open normal subgroup of the…","verdict":"Not formalized; ~159 pages. The new content is the remaining anisotropic outer A_n, triality D4 and E6 cases;","url":"/math#018","articleUrl":"/articles/openai-math#margulisplatonov-over-global-fields-family-018","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=018.%20The%20Margulis%20Platonov","manuscripts":[{"title":"The Margulis–Platonov conjecture over global function fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Margulis-Platonov-conjecture-over-global-function-fields-October-5-2026/margulis-platonov-global-function-fields.pdf"},{"title":"The Margulis–Platonov conjecture over number fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Margulis-Platonov-conjecture-over-number-fields-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/018.mp4","poster":"/films/math/018-poster.webp","captions":"/films/math/018.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":38,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":0,"confidence":0,"why":"Finishes the last exceptional cases of the Margulis–Platonov conjecture on normal subgroups of arithmetic groups. No Lean.","consequence":"Completes the description of normal subgroups of arithmetic groups, an input to the congruence subgroup problem."},"detail":"/api/math/018"},{"id":"019","title":"The local p-adic section conjecture and global consequences","short":"The local $p$-adic section conjecture","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every prime p, finite extension k/Q_p and smooth proper geometrically connected curve X/k of genus >= 2, the map from X(k) to Delta_X-conjugacy classes of sections of pi_1(X) -> G_k is a bijection (every section comes from a unique rational point).","verdict":"Not formalized. Remarkably short for its scope: 24 pages plus a 7-page cover-construction companion.","url":"/math#019","articleUrl":"/articles/openai-math#the-local-p-adic-section-conjecture-family-019","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=019.%20The%20local%20p%2Dadic","manuscripts":[{"title":"Étale covers with a prescribed exterior sheet","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf"},{"title":"The p-adic section conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-p-adic-section-conjecture-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/019.mp4","poster":"/films/math/019-poster.webp","captions":"/films/math/019.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"The p-adic section conjecture: rational points are exactly the Galois sections; 31 pages for a famous problem, unformalized.","consequence":"Rational points of p-adic curves can be read off from Galois sections; the global conjecture follows only for specific modular curves."},"detail":"/api/math/019"},{"id":"020","title":"Squarefree quartics and power-free polynomial values","short":"Squarefree values of quartics","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For f in Z[x] irreducible of degree d with 4 0. For d = 4 this is the squarefree-values conjecture for quartics (e.g. n^4 + 2, n^4 + 1); combined with Browning's d >= 9 range it covers every d >= 4.","verdict":"55 pages including an exact parameter certificate (Appendix A). Lean statement covers all d >= 4, which means the Lean also covers (or re-proves) Browning's…","url":"/math#020","articleUrl":"/articles/openai-math#squarefree-values-of-quartics-family-020","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=020.%20Squarefree%20quartics%20and","manuscripts":[{"title":"Squarefree values of quartics and power-free values of polynomials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026/manuscript.pdf"}],"reel":{"src":"/films/math/020.mp4","poster":"/films/math/020-poster.webp","captions":"/films/math/020.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":3,"why":"Values like n^4+2 are squarefree a positive share of the time, and higher powers likewise; Lean checks every degree.","consequence":"Positive-proportion squarefree values for quartics and power-free values in every degree. Internal to sieve theory."},"detail":"/api/math/020"},{"id":"021","title":"A quadratic bound for Jacobsthal’s function","short":"Jacobsthal's function, quadratic bound","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"h(k) <= C k^2/(log log 3k)^2 for an absolute C: every interval of that many consecutive integers contains an integer coprime to any n with at most k distinct prime factors, uniformly over prime sets and starting points.","verdict":"Optimal order still unknown (gap between ~k log k and k^2/(loglog k)^2). 83 pages.","url":"/math#021","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=021.%20A%20quadratic%20bound","manuscripts":[{"title":"A quadratic bound for Jacobsthal's function","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/021.mp4","poster":"/films/math/021-poster.webp","captions":"/films/math/021.vtt","duration":20.1,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Brings Jacobsthal's function down to about k squared; the true size is still open. Lean checks the bound.","consequence":"Better worst-case bounds for runs of sieved integers; the true order stays open."},"detail":"/api/math/021"},{"id":"022","title":"The weak inhomogeneous Duffin–Schaeffer conjecture","short":"Duffin–Schaeffer with a shift","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every real shift gamma and every psi: N -> [0, inf) with sum phi(q) psi(q)/q = infinity, almost every x satisfies ||q x - gamma|| < psi(q) for infinitely many q (numerators unrestricted). Corollary: Hausdorff-measure version via mass transference.","verdict":"Not formalized; 87 pages. 'Weak' version (unrestricted numerators, totient-weighted divergence), not the coprime-numerator form.","url":"/math#022","articleUrl":"/articles/openai-math#duffinschaeffer-with-a-shift-family-022","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=022.%20The%20weak%20inhomogeneous","manuscripts":[{"title":"The Weak Inhomogeneous Duffin–Schaeffer Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/022.mp4","poster":"/films/math/022-poster.webp","captions":"/films/math/022.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"The shifted, weak form of the Duffin–Schaeffer approximation theorem; the coprime form stays open. Unformalized.","consequence":"Extends Duffin–Schaeffer approximation theory to shifted targets. Internal to Diophantine approximation."},"detail":"/api/math/022"},{"id":"023","title":"Patterson's first moment for cubic Gauss sums","short":"Patterson's bias for cubic Gauss sums","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"Unconditionally, sum over primary Eisenstein primes pi with N(pi) <= X of the normalized cubic Gauss sum G(pi) equals (6/5) c_* X^{5/6}/log X + o(X^{5/6}/log X), with c_* = (2 pi)^{2/3}/(3 Gamma(2/3));","verdict":"Uses the same cubic-theta machinery as the quasi-RH family (003); the QRH paper says it does not use this result.","url":"/math#023","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=023.%20Patterson's%20first%20moment","manuscripts":[{"title":"An unconditional first moment for cubic Gauss sums","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/023.mp4","poster":"/films/math/023-poster.webp","captions":"/films/math/023.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Patterson's bias for cubic Gauss sums at primes, without assuming GRH; Lean checks the main theorem.","consequence":"Removes GRH from Patterson's prediction for cubic Gauss sums. Internal to analytic number theory."},"detail":"/api/math/023"},{"id":"024","title":"An asymptotic formula for the number of totients","short":"How many totients are there?","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"Gives an explicit asymptotic equivalent V(x) ~ (explicit scale) x C(x) for the number of distinct totient values up to x, with a positive bounded phase-dependent factor that is a uniform limit of finitely computable approximants;","verdict":"The 'asymptotic formula' has an oscillating (phase-dependent) bounded factor defined by a limit, not a closed-form constant. 39 pages, built on Ford's structure theory.","url":"/math#024","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=024.%20An%20asymptotic%20formula","manuscripts":[{"title":"An asymptotic formula for the number of totients","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf"}],"reel":{"src":"/films/math/024.mp4","poster":"/films/math/024-poster.webp","captions":"/films/math/024.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":35,"tier":"Incremental","importance":2,"advance":2,"consequences":0,"surprise":1,"confidence":3,"why":"An asymptotic count of Euler-phi values up to x, with an oscillating factor rather than a constant; Lean checks it.","consequence":"A sharper count of the values of Euler's function. Nothing follows beyond itself."},"detail":"/api/math/024"},{"id":"025","title":"Short Egyptian fractions","short":"Short Egyptian fractions","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"There are absolute c1, c2 with c1 log log b <= N(b) <= c2 log log b for b large, where N(b) is the maximum over 1 <= a < b of the minimal number of distinct unit fractions summing to a/b.","verdict":"33 pages; denominators are unrestricted in size. Lean includes explicit constants like 257/log 2 and e^{e^{k/600}}.","url":"/math#025","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=025.%20Short%20Egyptian%20fractions","manuscripts":[{"title":"Short Egyptian fractions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Short-Egyptian-fractions-September-25-2026/Short-Egyptian-fractions-September-25-2026.pdf"}],"reel":{"src":"/films/math/025.mp4","poster":"/films/math/025-poster.webp","captions":"/films/math/025.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"Every fraction a/b is a sum of about log log b distinct unit fractions, as Erdős conjectured; Lean checks it.","consequence":"The worst case for Egyptian-fraction length is now known. Closes a gap with little follow-on."},"detail":"/api/math/025"},{"id":"026","title":"Positive lower density of large prime gaps","short":"A positive share of large prime gaps","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"For every fixed C > 0 there is c(C) > 0 with #{n C log p_n} >= c(C) N for all large N. Corollary: the indices where p_n/n increases have positive lower density (Erdos-Prachar question).","verdict":"19 pages, constants c(C) not explicit and certainly tiny for large C. Short for the claim;","url":"/math#026","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=026.%20Positive%20lower%20density","manuscripts":[{"title":"Positive lower density of large prime gaps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/026.mp4","poster":"/films/math/026-poster.webp","captions":"/films/math/026.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":2,"why":"A fixed share of prime gaps exceed C times the average gap, for every C, without conjectures; Lean checks a corollary.","consequence":"Confirms a random-model prediction about large prime gaps without conjectures; the constants are tiny."},"detail":"/api/math/026"},{"id":"027","title":"Potential integral density on curve character varieties","short":"Integral points on character varieties","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"For every smooth connected complex algebraic curve X and every r >= 1, integral points become Zariski dense, over the full ring of integers of one number field, on every component of the SL_r character variety (with prescribed quasi-unipotent boundary…","verdict":"Not formalized; 27 pages. Integrality is measured in the ambient character variety while the boundary conditions are imposed on complex representations, as the abstract…","url":"/math#027","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=027.%20Potential%20integral%20density","manuscripts":[{"title":"Integral points on character varieties of curves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/027.mp4","poster":"/films/math/027-poster.webp","captions":"/films/math/027.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Integral points on surface-group character varieties are dense after one field extension; 27 unformalized pages.","consequence":"Integer points on character varieties are dense, which bears on which local systems come from geometry. Internal."},"detail":"/api/math/027"},{"id":"028","title":"Uniformly bounded components of Gaussian-prime graphs","short":"The Gaussian moat","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every real D there is a finite B_D such that every connected component of the graph on Gaussian primes joining primes at distance <= D has at most B_D vertices;","verdict":"29 pages; the proof is a finite periodic sieve obstruction proved with geometric sampling and entropy estimates, i.e.","url":"/math#028","articleUrl":"/articles/openai-math#the-gaussian-moat-family-028","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=028.%20Uniformly%20bounded%20components","manuscripts":[{"title":"Bounded-Step Walks on Gaussian Primes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf"}],"reel":{"src":"/films/math/028.mp4","poster":"/films/math/028-poster.webp","captions":"/films/math/028.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":51,"tier":"Incremental","importance":3,"advance":3,"consequences":0,"surprise":1,"confidence":3,"why":"The Gaussian moat problem: bounded steps on Gaussian primes never reach infinity. Short sieve proof; Lean checks it.","consequence":"Answers the Gaussian moat puzzle. The proof is really about sieved lattices; nothing follows beyond it."},"detail":"/api/math/028"},{"id":"029","title":"Primitive roots for every admissible integer base","short":"Artin's primitive roots, every base","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"partial","kindLabel":"Claimed partial result","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every integer a that is not -1 and not a perfect square, at least c_a x/(log x)^2 primes in (x, 2x) have a as a primitive root, for all large x; in particular a is a primitive root mod infinitely many primes (the infinitude part of Artin's conjecture).","verdict":"Not formalized; ~113 pages. Proves infinitude with x/(log x)^2, not Artin's predicted density A(a) x/log x, so Artin's conjecture in its quantitative form remains open.","url":"/math#029","articleUrl":"/articles/openai-math#artins-primitive-root-conjecture-infinitude-for-every-base-family-029","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=029.%20Primitive%20roots%20for","manuscripts":[{"title":"Primitive roots for every admissible integer base","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf"},{"title":"Simultaneous primitive roots: a conditional lower bound for prime bases","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf"}],"reel":{"src":"/films/math/029.mp4","poster":"/films/math/029-poster.webp","captions":"/films/math/029.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":70,"tier":"Solid","importance":3,"advance":2,"consequences":3,"surprise":3,"confidence":0,"why":"Every admissible base is a primitive root mod infinitely many primes, via a huge unproved-elsewhere zero-free strip. No Lean.","consequence":"The qualitative Artin conjecture for every base; its zero-free strip would also give deterministic polynomial factoring (family 142)."},"detail":"/api/math/029"},{"id":"030","title":"Modularity of elliptic curves over imaginary quadratic fields","short":"Modularity over imaginary quadratic fields","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Every elliptic curve E over every imaginary quadratic field K is modular: there is an isobaric automorphic representation pi_E of GL_2(A_K) with matching Galois representations and local-global compatibility at every finite place (cuspidal if E has no CM), so…","verdict":"Not formalized. Only 30 pages for a statement whose previous partial cases needed the ten-author potential automorphy machinery;","url":"/math#030","articleUrl":"/articles/openai-math#modularity-over-imaginary-quadratic-fields-family-030","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=030.%20Modularity%20of%20elliptic","manuscripts":[{"title":"Modularity of elliptic curves over imaginary quadratic fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf"}],"reel":{"src":"/films/math/030.mp4","poster":"/films/math/030-poster.webp","captions":"/films/math/030.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":2,"confidence":0,"why":"Every elliptic curve over an imaginary quadratic field is modular; 30 pages for a statement that needed ten authors before.","consequence":"Lets the L-function methods used for curves over Q apply to elliptic curves over imaginary quadratic fields."},"detail":"/api/math/030"},{"id":"031","title":"Uchida’s conjecture for open homomorphisms of Galois groups","short":"Uchida's conjecture","discipline":"Number theory","disciplineIndex":0,"accent":"#F2A65A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"For number fields F1, F2 and possibly infinite solvably closed Galois extensions E_i/F_i, every continuous open homomorphism Gal(E1/F1) -> Gal(E2/F2) is induced by a unique field embedding E2 -> E1; no restriction on the kernel.","verdict":"Not formalized; 23 pages. Uses an l-adic Waldschmidt-Masser/interpolation-determinant input which the paper reproves.","url":"/math#031","articleUrl":"/articles/openai-math#the-rest-of-the-number-theory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=031.%20Uchida%E2%80%99s%20conjecture%20for","manuscripts":[{"title":"Open Homomorphisms of Global Solvably Closed Galois Groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf"}],"reel":{"src":"/films/math/031.mp4","poster":"/films/math/031-poster.webp","captions":"/films/math/031.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":1,"advance":3,"consequences":1,"surprise":1,"confidence":0,"why":"Uchida's question: open maps between Galois groups of number fields come from field embeddings. Unformalized.","consequence":"Strengthens the reconstruction of number fields from Galois groups. Internal to anabelian geometry."},"detail":"/api/math/031"},{"id":"032","title":"Hodge and Kuga–Satake results for all projective K3 surfaces","short":"Hodge for CM abelian varieties","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Main paper (Sept 30): for every complex abelian variety with complex multiplication (End⊗Q contains a commutative semisimple algebra of dimension 2·dim A), the Betti cycle-class map CH^p(A)_Q → Hdg^{2p}(A) is surjective in every codimension p (Theorem 1.1);","verdict":"No Lean formalization (OpenAI and press coverage both say the Hodge result has no Lean proof).","url":"/math#032","articleUrl":"/articles/openai-math#the-hodge-conjecture-for-cm-abelian-varieties-family-032","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=032.%20Hodge%20and%20Kuga","manuscripts":[{"title":"The rational Hodge conjecture for products of K3 surfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf"},{"title":"Algebraicity of Kuga–Satake Correspondences for K3 Surfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/manuscript.pdf"},{"title":"The rational Hodge conjecture for CM abelian varieties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026/paper.pdf"},{"title":"Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf"},{"title":"Weil classes and Hodge classes on abelian powers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-September-30-2026/paper.pdf"},{"title":"Abelian covers, Gale correspondences, and the Hodge conjecture for powers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026/paper.pdf"},{"title":"Algebraicity of Weil classes on split abelian eightfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Weil-classes-on-split-abelian-eightfolds-September-18-2026/paper.pdf"},{"title":"A Conditional Reduction for Algebraic Kuga–Satake Correspondences","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf"}],"reel":{"src":"/films/math/032.mp4","poster":"/films/math/032-poster.webp","captions":"/films/math/032.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":86,"tier":"Major","importance":4,"advance":3,"consequences":3,"surprise":3,"confidence":0,"why":"The Hodge conjecture for CM abelian varieties and K3 products, the classic test case; made outside the usual procedure, no Lean.","consequence":"Algebraic cycles for every Hodge class on CM abelian varieties and K3 products; feeds Milne's rationality question and Tate-class theory (family 001)."},"detail":"/api/math/032"},{"id":"033","title":"Iitaka subadditivity, variation, and logarithmic additivity","short":"Iitaka's conjecture $C_{n,m}$","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Campana's orbifold Iitaka subadditivity for compact manifolds in Fujiki class C with rational SNC boundary (coefficients in [0,1], incl. 1); ordinary and logarithmic subadditivity follow (κ(X,K_X+D_X) ≥ κ(F,K_F+D_F)+κ(Y,K_Y+D_Y)).","verdict":"Lean (lean/docs/033.md): only the negative-fibre branch of the log additivity paper (if κ of the fibre is −∞ then total log Kodaira dimension is −∞) — a small piece;","url":"/math#033","articleUrl":"/articles/openai-math#iitakas-subadditivity-conjecture-family-033","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=033.%20Iitaka%20subadditivity%20variation","manuscripts":[{"title":"Orbifold and logarithmic Iitaka subadditivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf"},{"title":"Logarithmic Kodaira dimension and whole-fiber variation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Logarithmic-Kodaira-dimension-and-whole-fiber-variation-September-26-2026/paper.pdf"},{"title":"The reverse logarithmic Kodaira inequality and additivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026/paper.pdf"},{"title":"Projective Hodge lines and ordinary Iitaka subadditivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projective-Hodge-lines-and-ordinary-Iitaka-subadditivity-September-27-2026/paper.pdf"},{"title":"B-semiampleness for compact log-smooth Kähler fibrations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/B-semiampleness-for-compact-log-smooth-Kahler-fibrations-September-10-2026/paper.pdf"}],"reel":{"src":"/films/math/033.mp4","poster":"/films/math/033-poster.webp","captions":"/films/math/033.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":70,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":1,"confidence":1,"why":"Iitaka's 1970s subadditivity conjecture, a backbone of classifying varieties; Lean checks only a small side case.","consequence":"Kodaira dimension adds up correctly in fibrations, a key input to the abundance and minimal-model results in this release."},"detail":"/api/math/033"},{"id":"034","title":"Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity","short":"Log abundance in characteristic zero","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Headline paper (034_3, 'Log abundance in characteristic zero'): for every normal projective log canonical pair (X,B) over an algebraically closed field of char 0 with B effective rational and K_X+B Q-Cartier, if K_X+B is nef then it is semiample — the full…","verdict":"Family title says 'under logarithmic Iitaka subadditivity': the Kähler paper (034_0, Oct 4) states this as an explicit Assumption 1.1;","url":"/math#034","articleUrl":"/articles/openai-math#log-abundance-in-characteristic-zero-family-034","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=034.%20Log%20abundance%20for","manuscripts":[{"title":"Log abundance for compact Kähler spaces under logarithmic Iitaka 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adjoints","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Minimal-metrics-and-interior-injectivity-for-nef-adjoints-September-27-2026/paper.pdf"},{"title":"Fourfold nonvanishing by minimal metrics and moving jets","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf"},{"title":"Lifting sections from the reduced support of an adjoint","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026/paper.pdf"},{"title":"Schnell fiber spaces and good canonical models","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Schnell-fiber-spaces-and-good-canonical-models-September-24-2026/paper.pdf"},{"title":"Uniform log Iitaka fibrations and bounded moduli denominators","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators-October-4-2026/uniform-log-iitaka.pdf"},{"title":"Uniform Pluricanonical Iitaka Fibrations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/paper.pdf"},{"title":"Relative denominators and effective systems for log Calabi-Yau fibrations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-denominators-and-effective-systems-for-log-Calabi-Yau-fibrations-September-27-2026/paper.pdf"},{"title":"Arithmetic Stein-degree bounds for log Calabi–Yau pairs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Arithmetic-Stein-degree-bounds-for-log-Calabi-Yau-pairs-September-25-2026/paper.pdf"},{"title":"Uniform effective log Iitaka fibrations for fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-effective-log-Iitaka-fibrations-for-fourfolds-September-26-2026/paper.pdf"},{"title":"Abundance after nonvanishing for compact Kähler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Abundance-after-nonvanishing-for-compact-Kahler-fourfolds-September-27-2026/paper.pdf"}],"reel":{"src":"/films/math/034.mp4","poster":"/films/math/034-poster.webp","captions":"/films/math/034.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":66,"tier":"Solid","importance":3,"advance":2,"consequences":3,"surprise":2,"confidence":0,"why":"Log abundance, the last big gap in the minimal model program, but assuming a subadditivity statement; 781 unchecked pages.","consequence":"Would complete the abundance step of the minimal model program (conditionally); much of the classification of varieties rests on it."},"detail":"/api/math/034"},{"id":"035","title":"Log-canonical threefold abundance in numerical dimension one","short":"Threefold abundance, $\\nu = 1$, char $p > 3$","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"partial","kindLabel":"Claimed partial result","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"For projective log canonical threefold pairs (X,B) over an algebraically closed field of characteristic p > 3, with B effective rational, K_X+B Q-Cartier, nef and of numerical dimension one, K_X+B is semiample (no terminality or Q-factoriality).","verdict":"Positive characteristic, dimension three, numerical dimension one only — a single case of abundance.","url":"/math#035","articleUrl":"/articles/openai-math#four-narrower-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=035.%20Log%2Dcanonical%20threefold%20abundance","manuscripts":[{"title":"Log abundance in numerical dimension one for threefolds in positive characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026/paper.pdf"},{"title":"Abundance in numerical dimension one for terminal threefolds in positive characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Abundance-in-numerical-dimension-one-for-terminal-threefolds-in-positive-characteristic-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/035.mp4","poster":"/films/math/035-poster.webp","captions":"/films/math/035.vtt","duration":20.5,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":26,"tier":"Incremental","importance":1,"advance":1,"consequences":1,"surprise":1,"confidence":0,"why":"One numerical case of threefold abundance in characteristic p above 3; a single case of a larger program. No Lean.","consequence":"One more case of abundance in characteristic p. Little follow-on."},"detail":"/api/math/035"},{"id":"036","title":"Numerical semiampleness and generalized minimal models","short":"Minimal models and generalised abundance","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Generalised abundance (Lazić–Peternell): for a projective klt Q-pair (X,B) in char 0 with K_X+B pseudo-effective and M nef Q-Cartier such that K_X+B+M is nef, K_X+B+M is numerically equivalent to a semiample Q-Cartier divisor (036_1);","verdict":"036_1 explicitly uses the release's own log abundance theorem (034_3) and termination results (056) as black boxes ('The proof uses the log-abundance theorem of [28,…","url":"/math#036","articleUrl":"/articles/openai-math#minimal-models-and-generalised-abundance-family-036","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=036.%20Numerical%20semiampleness%20and","manuscripts":[{"title":"Numerical semiampleness of nef adjoint classes on compact Kähler manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Numerical-semiampleness-of-nef-adjoint-classes-on-compact-Kahler-manifolds-October-4-2026/numerical-generalized-abundance.pdf"},{"title":"Numerical Semiampleness of Nef Adjoint Divisors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/paper.pdf"},{"title":"Minimal models and Mori fibre spaces for generalized log canonical Q-pairs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf"},{"title":"Minimal models in numerical dimension one","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Minimal-models-in-numerical-dimension-one-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/036.mp4","poster":"/films/math/036-poster.webp","captions":"/films/math/036.vtt","duration":19.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Minimal models exist for generalized pairs; stands or falls with the release's own unchecked abundance and termination papers.","consequence":"Minimal models for generalized pairs, the form that induction in birational geometry actually needs."},"detail":"/api/math/036"},{"id":"037","title":"The ordinary-double-point volume gap","short":"The ordinary double point volume gap","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every singular closed point of a normal complex algebraic n-fold (n ≥ 2) that is klt with zero boundary, the normalized volume vol̂(x,X) ≤ 2(n−1)^n, with equality iff the analytic germ is an ordinary double point.","verdict":"The every-dimension paper uses the dimension-four companion as its sole companion input (base case of an induction); both are unrefereed.","url":"/math#037","articleUrl":"/articles/openai-math#the-ordinary-double-point-gap-family-037","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=037.%20The%20ordinary%2Ddouble%2Dpoint%20volume","manuscripts":[{"title":"The ordinary-double-point gap in every dimension","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf"},{"title":"The normalized-volume gap in dimension four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/037.mp4","poster":"/films/math/037-poster.webp","captions":"/films/math/037.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"The next-largest singularity volume after smooth points is the ordinary double point, in every dimension; no Lean.","consequence":"Bounds the volume of singular K-semistable Fano varieties. Internal to K-stability."},"detail":"/api/math/037"},{"id":"038","title":"Fujita’s freeness conjecture","short":"Fujita's freeness, $m \\ge n+1$","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"For every smooth connected complex projective variety X of dimension n and every ample line bundle L, K_X + mL is globally generated for all m ≥ n+1 (the global-generation part of Fujita's conjecture, sharp by P^n). Very-ampleness part not claimed.","verdict":"Only global generation (freeness), not Fujita's very-ampleness conjecture — the overview states this correctly.","url":"/math#038","articleUrl":"/articles/openai-math#fujitas-freeness-conjecture-family-038","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=038.%20Fujita%E2%80%99s%20freeness%20conjecture","manuscripts":[{"title":"Fujita's freeness conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fujitas-freeness-conjecture-September-23-2026/Fujitas-freeness-conjecture-September-23-2026.pdf"}],"reel":{"src":"/films/math/038.mp4","poster":"/films/math/038-poster.webp","captions":"/films/math/038.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":0,"why":"Fujita's freeness conjecture: K plus n+1 copies of an ample bundle is base-point free; 25 unchecked pages.","consequence":"Effective global generation of adjoint bundles in every dimension, a standard tool for building maps; used here for Kobayashi's conjecture."},"detail":"/api/math/038"},{"id":"039","title":"Nagata’s conjecture and maximal Seshadri constants","short":"Nagata's conjecture, $r \\ge 10$ points","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Nagata's conjecture: for every r ≥ 10, outside a countable union of proper Zariski-closed subsets of the configuration space of r distinct points in P², every nonzero effective plane curve of degree d with multiplicities ≥ m_i at the points satisfies Σ m_i <…","verdict":"Lean: ComparatorChallenges/Nagata.lean states exactly the full conjecture (∀ count ≥ 10, ∃ countable family of proper Zariski-closed exceptional sets with a point…","url":"/math#039","articleUrl":"/articles/openai-math#nagatas-conjecture-family-039","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=039.%20Nagata%E2%80%99s%20conjecture%20and","manuscripts":[{"title":"Nagata's conjecture for plane curves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026/main.pdf"},{"title":"Maximal Seshadri constants on arbitrary polarized surfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026/main.pdf"},{"title":"Maximal Multipoint Seshadri Constants in Higher Dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Maximal-Multipoint-Seshadri-Constants-in-Higher-Dimensions-October-5-2026/main.pdf"},{"title":"Maximal multipoint Seshadri constants in positive characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Maximal-multipoint-Seshadri-constants-in-positive-characteristic-October-5-2026/seshadri-positive-characteristic.pdf"}],"reel":{"src":"/films/math/039.mp4","poster":"/films/math/039-poster.webp","captions":"/films/math/039.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":2,"confidence":3,"why":"Nagata's 1959 conjecture on plane curves through very general points; Lean states and checks the full conjecture.","consequence":"Fixes the nef cone of the plane blown up at very general points and the maximal Seshadri constants, which the SHGH line of work waited on."},"detail":"/api/math/039"},{"id":"040","title":"Bloch’s conjecture for complex surfaces","short":"Bloch's conjecture, $p_g = q = 0$","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Bloch's conjecture for surfaces: for every smooth connected complex projective surface S with p_g = q = 0, deg: CH_0(S) → Z is an isomorphism (integral coefficients);","verdict":"Route: a determinant-fixed Quot-scheme calculation with two marked surface factors produces a diagonal decomposition 0 = c[Δ] + Γ in CH²(X×X)_Q;","url":"/math#040","articleUrl":"/articles/openai-math#blochs-conjecture-for-surfaces-with-p_g0-family-040","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=040.%20Bloch%E2%80%99s%20conjecture%20for","manuscripts":[{"title":"Bloch’s conjecture for surfaces with p_g=0","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/040.mp4","poster":"/films/math/040-poster.webp","captions":"/films/math/040.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"Bloch's conjecture: surfaces without 2-forms have trivial zero-cycles; the fake-projective-plane case needs scrutiny.","consequence":"Zero-cycles on surfaces without 2-forms are trivial, closing a central case of the Bloch–Beilinson picture for surfaces."},"detail":"/api/math/040"},{"id":"041","title":"Hyperkähler SYZ and projective-space bases","short":"Hyperkähler SYZ and $\\mathbb{P}^n$ bases","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Strong hyperkähler SYZ conjecture: on any compact irreducible holomorphic symplectic Kähler manifold X of dim 2n, every holomorphic line bundle L with nonzero nef isotropic class (q(c1(L))=0) is semiample, giving a Lagrangian fibration X → B with dim B =…","verdict":"041_0 (98 pp) and 041_1 (32 pp) cite each other ([44], [36]); the finiteness corollary needs b2 ≥ 5 and the external EFGMS boundedness theorem. No Lean.","url":"/math#041","articleUrl":"/articles/openai-math#hyperkähler-syz-and-lagrangian-bases-family-041","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=041.%20Hyperk%C3%A4hler%20SYZ%20and","manuscripts":[{"title":"Projective-space bases of Lagrangian fibrations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/main.pdf"},{"title":"The strong hyperkähler SYZ conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026/main.pdf"}],"reel":{"src":"/films/math/041.mp4","poster":"/films/math/041-poster.webp","captions":"/films/math/041.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":1,"confidence":0,"why":"Every hyperkähler manifold with a square-zero nef class is a Lagrangian torus fibration over projective space. No Lean.","consequence":"Such hyperkähler manifolds fiber over projective space, which matters for their classification and for SYZ mirror symmetry."},"detail":"/api/math/041"},{"id":"042","title":"Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII","short":"Every K3 surface is Oka","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Every complex K3 surface (projective or not, any Picard rank) has the convex approximation property, hence is an Oka manifold; corollary: through every point and tangent vector of a K3 surface there is a holomorphic immersion C → S with dense image (Zariski…","verdict":"The family title says 'Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII', but the single paper's headline is K3;","url":"/math#042","articleUrl":"/articles/openai-math#every-k3-surface-is-oka-family-042","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=042.%20Oka%20classification%20for","manuscripts":[{"title":"Every complex K3 surface is Oka","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/042.mp4","poster":"/films/math/042-poster.webp","captions":"/films/math/042.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"K3 surfaces are Oka (maps into them are as flexible as into C^n); the broader title over-reaches the paper. No Lean.","consequence":"Adds K3 surfaces to the list of Oka manifolds. Internal to complex analysis."},"detail":"/api/math/042"},{"id":"043","title":"P = W for fixed-determinant SLn moduli spaces","short":"$P = W$ for $\\mathrm{SL}_n$, every rank","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"P = W for twisted fixed-determinant SL_n character varieties: for every curve of genus g ≥ 2, every composite n ≥ 4, d coprime to n, the perverse filtration of the SL_n Hitchin fibration equals the weight filtration on the whole cohomology of the twisted SL_n…","verdict":"Builds on the GL_n proofs; the new content is the composite-rank variant cohomology. 39 pp. No Lean.","url":"/math#043","articleUrl":"/articles/openai-math#p--w-for-mathrmsl_n-family-043","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=043.%20P%20W%20for","manuscripts":[{"title":"P=W in composite rank for fixed determinant","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026.pdf"}],"reel":{"src":"/films/math/043.mp4","poster":"/films/math/043-poster.webp","captions":"/films/math/043.vtt","duration":19.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Extends the recently proved P=W for GL_n to fixed-determinant SL_n moduli; builds on the known proofs. No Lean.","consequence":"Extends P=W to SL_n. Internal to nonabelian Hodge theory."},"detail":"/api/math/043"},{"id":"044","title":"The equivariant cohomological Hikita conjecture","short":"Equivariant Hikita for every quiver","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"Equivariant cohomological Hikita conjecture for arbitrary finite quivers (loops and multi-edges allowed) under a regularity hypothesis on the stability character: the equivariant cohomology of the smooth Nakajima quiver variety is isomorphic, compatibly with…","verdict":"Requires a regular stability character (free action on the stable locus); 41 pp, single paper. No Lean.","url":"/math#044","articleUrl":"/articles/openai-math#four-narrower-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=044.%20The%20equivariant%20cohomological","manuscripts":[{"title":"The equivariant cohomological Hikita conjecture for arbitrary quivers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-equivariant-cohomological-Hikita-conjecture-for-arbitrary-quivers-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/044.mp4","poster":"/films/math/044-poster.webp","captions":"/films/math/044.vtt","duration":19.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"An equivariant form of Hikita's symplectic-duality conjecture, under a regular stability assumption. No Lean.","consequence":"Confirms a symplectic-duality prediction in a regular setting. Internal to geometric representation theory."},"detail":"/api/math/044"},{"id":"046","title":"Shafarevich counterexamples in dimension two and with large fundamental group","short":"Shafarevich's convexity conjecture fails","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Counterexamples to Shafarevich's holomorphic convexity conjecture: (a) a smooth connected projective complex SURFACE whose universal cover is not holomorphically convex;","verdict":"Surface counterexample uses weighted pro-2 groups and an infinite quotient compatible with a family (non-linear fundamental group — consistent with EKPR).","url":"/math#046","articleUrl":"/articles/openai-math#shafarevichs-holomorphic-convexity-conjecture-fails-family-046","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=046.%20Shafarevich%20counterexamples%20in","manuscripts":[{"title":"A projective fourfold with large fundamental group and non-Stein universal cover","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-projective-fourfold-with-large-fundamental-group-and-non-Stein-universal-cover-October-5-2026/large-fundamental-group-non-stein.pdf"},{"title":"A surface counterexample to Shafarevich holomorphic convexity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-surface-counterexample-to-Shafarevich-holomorphic-convexity-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/046.mp4","poster":"/films/math/046-poster.webp","captions":"/films/math/046.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"Disproves Shafarevich's holomorphic-convexity conjecture with a surface and a fourfold; a short disproof that needs checking.","consequence":"Ends hopes that universal covers are always holomorphically convex; the release's abelianity result builds on it."},"detail":"/api/math/046"},{"id":"047","title":"Zariski cancellation and affine fibrations over the complex numbers","short":"Zariski cancellation fails over $\\mathbb{C}$","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Zariski cancellation fails over C in dimension 4: with P = C[p,s,u,F,J], x = s² + u³ + p²F and H = x²F − (1+2sx)J − p²J² − pu, the algebra A = P/(H) is a 4-dimensional finitely generated domain with A[w] ≅ C^[5] but A ≇ C^[4].","verdict":"Lean: ComparatorChallenges/ComplexCancellation.lean states FiniteType ∧ IsDomain ∧ ringKrullDim A = 4 ∧ Nonempty (A[X] ≃ₐ[C] C[x1..x5]) ∧ ¬Nonempty (A ≃ₐ[C] C[x1..x4])…","url":"/math#047","articleUrl":"/articles/openai-math#zariski-cancellation-fails-over-mathbb-c-family-047","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=047.%20Zariski%20cancellation%20and","manuscripts":[{"title":"An explicit failure of complex affine-space cancellation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/047.mp4","poster":"/films/math/047-poster.webp","captions":"/films/math/047.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":2,"consequences":1,"surprise":3,"confidence":3,"why":"A 4-dimensional counterexample to Zariski cancellation over C, open 50 years; Lean checks both halves. Dimension 3 stays open.","consequence":"Cancellation fails over C in dimension 4; dimension 3 remains. Internal to affine geometry."},"detail":"/api/math/047"},{"id":"048","title":"A characteristic-zero counterexample to Lipman–Zariski","short":"Lipman–Zariski fails for complex surfaces","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Lipman–Zariski conjecture fails over C for surfaces: there is a finitely generated normal 2-dimensional C-algebra A (smooth away from one point) with Der_C(A) ≅ A² free but A_m not regular; the singularity is an isolated Gorenstein surface singularity.","verdict":"Not log canonical (consistent with GKKP/Graf–Kovács). Built first analytically then algebraized via completed local ring; 37 pp; no Lean.","url":"/math#048","articleUrl":"/articles/openai-math#lipmanzariski-fails-for-surfaces-family-048","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=048.%20A%20characteristic%2Dzero%20counterexample","manuscripts":[{"title":"A singular normal affine surface with free tangent sheaf","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-singular-normal-affine-surface-with-free-tangent-sheaf-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/048.mp4","poster":"/films/math/048-poster.webp","captions":"/films/math/048.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":2,"advance":3,"consequences":1,"surprise":3,"confidence":0,"why":"A singular surface with free tangent sheaf disproves the Lipman–Zariski conjecture in characteristic 0. No Lean.","consequence":"Free vector fields do not force smoothness. Internal to singularity theory."},"detail":"/api/math/048"},{"id":"049","title":"A stable-coordinate counterexample in four variables","short":"Abhyankar–Sathaye fails in four variables","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Abhyankar–Sathaye embedding conjecture fails over C in every ambient dimension n ≥ 4: an explicit F ∈ C[x1..x4] with C[x]/(F) ≅ C^[3] which is not a coordinate (no automorphism of C[x1..x4] sends x1 to F); extended to all n ≥ 4 by adding variables.","verdict":"Lean: ComparatorChallenges/AbhyankarSathaye.lean (21 lines) states ∀ n ≥ 4, ∃ F, C[x]/(F) ≃ₐ C^[n−1] ∧ ¬∃ automorphism e, index i with e(X i) = F — exactly the…","url":"/math#049","articleUrl":"/articles/openai-math#the-abhyankarsathaye-conjecture-fails-in-four-variables-family-049","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=049.%20A%20stable%2Dcoordinate%20counterexample","manuscripts":[{"title":"A stable coordinate that is not a coordinate in four variables","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf"},{"title":"An explicit noncoordinate polynomial with affine three-space zero fibre","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/049.mp4","poster":"/films/math/049-poster.webp","captions":"/films/math/049.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":2,"consequences":1,"surprise":3,"confidence":3,"why":"A polynomial in 4 variables whose zero set is flat space yet is not a coordinate, refuting Abhyankar–Sathaye; Lean checks it.","consequence":"The embedding conjecture fails from dimension 4 on. Internal to affine algebraic geometry."},"detail":"/api/math/049"},{"id":"050","title":"A counterexample to Griffiths’ positivity conjecture","short":"Griffiths' positivity conjecture fails","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Griffiths' positivity conjecture fails: on S = P¹×P¹ there is an explicit rank-2 bundle G such that E_m = f_m^*G ⊗ O(1,1) (f_m = m-th power map) is ample for all m ≥ 1 but admits no smooth Hermitian metric with strictly Griffiths-positive curvature for all m…","verdict":"Lean: ComparatorChallenges/QuadricBundles.lean; OAI.QuadricCounterexample.main_theorem at OAI/Geometry/QuadricBundles/Main.lean:10 (the very-ampleness certificate for…","url":"/math#050","articleUrl":"/articles/openai-math#griffiths-positivity-conjecture-fails-family-050","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=050.%20A%20counterexample%20to","manuscripts":[{"title":"Ample rank-two bundles on the quadric surface without Griffiths-positive metrics","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/050.mp4","poster":"/films/math/050-poster.webp","captions":"/films/math/050.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":3,"confidence":2,"why":"Ampleness does not imply a positively curved metric: a counterexample to Griffiths' 1969 conjecture. Lean uses custom definitions.","consequence":"Separates algebraic and metric positivity of vector bundles, redirecting work on Griffiths positivity."},"detail":"/api/math/050"},{"id":"051","title":"Kobayashi’s canonical-ampleness conjecture","short":"Kobayashi's canonical ampleness conjecture","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Kobayashi's canonical-ampleness conjecture: every compact connected Kähler manifold of positive dimension that is Brody (=Kobayashi) hyperbolic has ample canonical bundle (hence is projective).","verdict":"Uses two other release results as inputs: Fujita freeness (038, Corollary 6.3) and the semialgebraic-universal-cover theorem (058), which itself uses the release's log…","url":"/math#051","articleUrl":"/articles/openai-math#kobayashis-canonical-ampleness-conjecture-family-051","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=051.%20Kobayashi%E2%80%99s%20canonical%2Dampleness%20conjecture","manuscripts":[{"title":"Canonical ampleness of compact hyperbolic Kähler manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026/canonical-ampleness.pdf"}],"reel":{"src":"/films/math/051.mp4","poster":"/films/math/051-poster.webp","captions":"/films/math/051.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":1,"confidence":0,"why":"Kobayashi's conjecture that hyperbolic manifolds have ample canonical bundle; downstream of the least-checked release papers.","consequence":"Links hyperbolicity to an ample canonical bundle, settling a central question in complex hyperbolicity."},"detail":"/api/math/051"},{"id":"052","title":"Tangent splittings and product decompositions","short":"Split tangent bundles are products","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) Two-summand form of Beauville's splitting conjecture: if the tangent bundle of a compact connected Kähler manifold splits as E1 ⊕ E2 into two integrable holomorphic subbundles of positive rank, the universal cover is a product whose factor tangent bundles…","verdict":"Beauville's full conjecture allows any number of summands and does not assume integrability;","url":"/math#052","articleUrl":"/articles/openai-math#tangent-bundle-splittings-family-052","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=052.%20Tangent%20splittings%20and","manuscripts":[{"title":"Universal-cover splitting for compact Kähler manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026/paper.pdf"},{"title":"Integrability of split tangent bundles on rationally connected manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026/main.pdf"}],"reel":{"src":"/films/math/052.mp4","poster":"/films/math/052-poster.webp","captions":"/films/math/052.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"A two-factor, integrable case of Beauville's splitting conjecture for Kähler manifolds; Lean checks it on custom foundations.","consequence":"One case of Beauville's splitting conjecture. Internal."},"detail":"/api/math/052"},{"id":"053","title":"A counterexample to Pixton completeness in Chow","short":"Pixton's completeness fails in Chow","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Pixton's completeness conjecture (Chow and rational-cohomology forms, for his original relation system) is false: there is a formal tautological class outside the span of Pixton's original relations whose image vanishes in A*(M̄_{g,n};Q) and in…","verdict":"Astronomically large example (genus 10^60) — cannot be checked by computer; correctness rests on the proof.","url":"/math#053","articleUrl":"/articles/openai-math#pixtons-completeness-fails-family-053","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=053.%20A%20counterexample%20to","manuscripts":[{"title":"A high-arity counterexample to Pixton completeness in Chow","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/053.mp4","poster":"/films/math/053-poster.webp","captions":"/films/math/053.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":0,"why":"Pixton's relations do not give all tautological relations in Chow; the example lives in genus around 10^60. No Lean.","consequence":"Pixton's relations must be extended to describe Chow rings of moduli of curves. Internal."},"detail":"/api/math/053"},{"id":"054","title":"Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations","short":"Irrational cubic fourfolds with a K3","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"For every sufficiently large admissible Hassett discriminant d (d > d0, ineffective), a very general cubic fourfold in C_d is irrational, although Ku(X) ≅ D^b(S) for a projective K3 surface S and X has an (untwisted) Hodge-theoretic associated K3.","verdict":"Threshold d0 ineffective; 'very general' in each divisor; the proof uses a new additive invariant of birational maps counting divisors whose MRC quotient is birational…","url":"/math#054","articleUrl":"/articles/openai-math#kuznetsovs-rationality-conjecture-fails-family-054","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=054.%20Irrational%20cubic%20fourfolds","manuscripts":[{"title":"Irrational cubic fourfolds with geometric K3 categories","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026.pdf"}],"reel":{"src":"/films/math/054.mp4","poster":"/films/math/054-poster.webp","captions":"/films/math/054.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":3,"confidence":0,"why":"Cubic fourfolds with a geometric K3 category yet irrational, refuting Kuznetsov's rationality prediction; unformalized.","consequence":"Rules out the categorical criterion for rationality of cubic fourfolds that the field had been pursuing."},"detail":"/api/math/054"},{"id":"055","title":"Gepner symmetry and large-volume stability on threefolds","short":"Gepner stability on the quintic","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Toda's Gepner conjecture for every smooth quintic threefold: a numerical Bridgeland stability condition with support property such that T_O ∘ (−⊗O(1)) shifts phases by exactly 2/5.","verdict":"Large-volume only for general CY3 (not the whole stability manifold); the strong tilt inequality is a corrected form, not the original BMT conjecture. 71 pp total;","url":"/math#055","articleUrl":"/articles/openai-math#gepner-stability-and-calabiyau-threefolds-family-055","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=055.%20Gepner%20symmetry%20and","manuscripts":[{"title":"A Gepner stability condition on every smooth quintic threefold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf"},{"title":"Prescribed large-volume charges on threefolds with trivial canonical bundle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Prescribed-large-volume-charges-on-threefolds-with-trivial-canonical-bundle-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/055.mp4","poster":"/films/math/055-poster.webp","captions":"/films/math/055.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Bridgeland stability near large volume on Calabi–Yau threefolds via a corrected Bogomolov-type inequality. No Lean.","consequence":"Stability conditions on Calabi–Yau threefolds near large volume, which physics predicts and enumerative geometry uses."},"detail":"/api/math/055"},{"id":"056","title":"Termination of projective and Kähler fourfold minimal model programs","short":"Flips terminate on fourfolds","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Termination: (a) every sequence of generalized log canonical flips on a normal globally Weil Q-factorial compact Kähler fourfold with rational boundary and fixed rational analytically nef b-data terminates (projective small flip diagrams);","verdict":"Kähler results are about EXISTING flip sequences ('does not assert the existence of all contractions or flips').","url":"/math#056","articleUrl":"/articles/openai-math#termination-of-flips-on-fourfolds-family-056","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=056.%20Termination%20of%20projective","manuscripts":[{"title":"Termination of generalized log canonical flips on compact Kähler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026/termination-generalized-lc-kahler-fourfolds.pdf"},{"title":"Termination of generalized-canonical flips on compact Kähler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-5-2026/termination-generalized-terminal-flips-compact-kahler-fourfolds.pdf"},{"title":"Termination for projective log canonical fourfolds with rational boundary","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026/paper.pdf"},{"title":"Finite ordinary minimal model programs on compact Kähler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026/paper.pdf"},{"title":"Finite ordinary minimal model programs on compact Kähler fourfolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/056.mp4","poster":"/films/math/056-poster.webp","captions":"/films/math/056.vtt","duration":19.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Every sequence of flips stops on fourfolds, projective and Kähler; termination stays open in dimension 5 and up. No Lean.","consequence":"Flips terminate in dimension four, completing the minimal model program there."},"detail":"/api/math/056"},{"id":"057","title":"Fundamental groups of special complex varieties and root orbifolds","short":"Campana's abelianity conjecture","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Campana's abelianity conjecture: every special compact Kähler manifold has virtually abelian fundamental group (in particular κ = 0 compact Kähler manifolds).","verdict":"057_0 cites the release's Shafarevich counterexample paper (046) and the proof goes through L²/Dolbeault spectral analysis on coverings ('zero in the Dolbeault…","url":"/math#057","articleUrl":"/articles/openai-math#campanas-abelianity-conjecture-family-057","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=057.%20Fundamental%20groups%20of","manuscripts":[{"title":"The abelianity conjecture for special compact Kähler manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf"},{"title":"Two-step monodromy of special quasi-projective varieties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026/paper.pdf"},{"title":"A conditional abelianity theorem for special fourfold pairs with a half-weight divisor","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor-October-5-2026/main.pdf"}],"reel":{"src":"/films/math/057.mp4","poster":"/films/math/057-poster.webp","captions":"/films/math/057.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":0,"why":"Campana's special varieties have virtually abelian fundamental groups, via L2 Hodge theory; unformalized.","consequence":"Campana's prediction that special varieties have virtually abelian fundamental groups; used by families 058 and 051."},"detail":"/api/math/057"},{"id":"058","title":"Semialgebraic universal covers and bounded domains","short":"Kollár–Pardon: semialgebraic covers","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Kollár–Pardon conjecture: the universal cover of a connected normal projective variety X is biholomorphic to a semialgebraic open subset of a projective variety iff it is D × C^m × F (D bounded symmetric domain, F simply connected normal projective);","verdict":"Lean (lean/docs/058.md): only the bounded-domain symmetry theorem (b) is formalized — OAI.Release061.main at OAI/Analysis/SymmetricDomains/Main.lean:55, with…","url":"/math#058","articleUrl":"/articles/openai-math#kollárpardon-family-058","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=058.%20Semialgebraic%20universal%20covers","manuscripts":[{"title":"Semialgebraic universal covers of normal projective varieties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Semialgebraic-universal-covers-of-normal-projective-varieties-September-24-2026/paper.pdf"},{"title":"Symmetry of semialgebraic bounded domains with compact quotient","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/058.mp4","poster":"/films/math/058-poster.webp","captions":"/films/math/058.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":1,"why":"Kollár–Pardon's classification of semialgebraic universal covers; Lean checks only the bounded-domain half.","consequence":"Classifies which universal covers are semialgebraic. Internal."},"detail":"/api/math/058"},{"id":"059","title":"Counterexamples to Zariski’s multiplicity conjecture","short":"Zariski's multiplicity question: no","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Zariski's multiplicity question has a negative answer: (a) two reduced isolated hypersurface germs in C^N (N divisible by 8) that are ambiently homeomorphic (even topologically right equivalent) with multiplicities 2 and 3, also answering Arnold's corank…","verdict":"The multiplicity 2 vs 3 example lives in very high dimension (N divisible by 8); the C⁴ example is the low-dimensional one.","url":"/math#059","articleUrl":"/articles/openai-math#zariskis-multiplicity-question-fails-family-059","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=059.%20Counterexamples%20to%20Zariski%E2%80%99s","manuscripts":[{"title":"Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026/paper.pdf"},{"title":"Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ambiently-homeomorphic-isolated-hypersurface-germs-in-C4-with-multiplicities-four-and-five-September-27-2026/paper.pdf"}],"reel":{"src":"/films/math/059.mp4","poster":"/films/math/059-poster.webp","captions":"/films/math/059.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":2,"consequences":1,"surprise":3,"confidence":0,"why":"High-dimensional singularities with the same topology but different multiplicity, answering Zariski; surfaces stay open.","consequence":"Topology alone does not fix multiplicity in high dimension; the surface case stays open. Internal."},"detail":"/api/math/059"},{"id":"060","title":"The Global Spherical Shell conjecture","short":"The global spherical shell conjecture","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Global Spherical Shell conjecture: every connected minimal compact complex surface of class VII with b2 > 0 contains a global spherical shell; hence (Kato) it deforms to blown-up primary Hopf surfaces, π1 ≅ Z and X ≅ (S¹×S³)#b·CP²-bar.","verdict":"If correct, completes the classification of minimal class VII surfaces with b2 > 0 (all are Kato surfaces).","url":"/math#060","articleUrl":"/articles/openai-math#the-global-spherical-shell-conjecture-family-060","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=060.%20The%20Global%20Spherical","manuscripts":[{"title":"Global Spherical Shells on Minimal Surfaces of Class VII","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/060.mp4","poster":"/films/math/060-poster.webp","captions":"/films/math/060.vtt","duration":19.5,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"Every class VII surface with b2>0 has a global spherical shell, completing their classification; 39 unchecked pages.","consequence":"Completes the classification of minimal class VII surfaces with b2>0, the last unclassified compact complex surfaces."},"detail":"/api/math/060"},{"id":"062","title":"Projective contact classification and the LeBrun–Salamon conjecture","short":"The LeBrun–Salamon conjecture","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Every smooth connected complex projective contact Fano manifold of dimension ≥ 3 is contact-isomorphic to the adjoint variety of a simple Lie algebra (contact-Fano homogeneity);","verdict":"40 pp single paper. No Lean. A long-standing conjecture with many partial results by dimension; a uniform proof is a strong claim.","url":"/math#062","articleUrl":"/articles/openai-math#the-lebrunsalamon-conjecture-family-062","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=062.%20Projective%20contact%20classification","manuscripts":[{"title":"Contact Fano manifolds and the LeBrun–Salamon conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/062.mp4","poster":"/films/math/062-poster.webp","captions":"/films/math/062.vtt","duration":19.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"LeBrun–Salamon: every contact Fano manifold is homogeneous, a long-standing conjecture with only partial cases before.","consequence":"Classifies contact Fano manifolds and, through twistor spaces, positive quaternion-Kähler manifolds."},"detail":"/api/math/062"},{"id":"063","title":"The generalized Mukai conjecture","short":"The generalized Mukai conjecture","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Generalized Mukai conjecture: every smooth complex Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies ρ(ι−1) ≤ n, with equality iff X ≅ (P^{ι−1})^ρ.","verdict":"Short (19 pp) and via Gromov–Witten theory: quantum multiplication by divisors and point descendants, 'independent joint eigenvalue tuple'.","url":"/math#063","articleUrl":"/articles/openai-math#the-generalized-mukai-conjecture-family-063","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=063.%20The%20generalized%20Mukai","manuscripts":[{"title":"The generalized Mukai conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Generalized-Mukai-Conjecture-September-24-2026/article.pdf"}],"reel":{"src":"/films/math/063.mp4","poster":"/films/math/063-poster.webp","captions":"/films/math/063.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"The generalized Mukai bound on Picard number of Fano manifolds, by a short Gromov–Witten argument; unrefereed, no Lean.","consequence":"A sharp Picard-number bound for Fano manifolds. Internal."},"detail":"/api/math/063"},{"id":"064","title":"Topological triviality of μ-constant surface singularities","short":"$\\mu$-constant surface singularities","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"μ-constant problem for surfaces: every holomorphic one-parameter family f_t of isolated hypersurface singularities in C³ with constant Milnor number is topologically right-trivial (ambient homeomorphisms φ_t continuous in t with f_t∘φ_t = f_0).","verdict":"Surface case only (the remaining case). 37 pp. No Lean.","url":"/math#064","articleUrl":"/articles/openai-math#the-mu-constant-problem-for-surfaces-family-064","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=064.%20Topological%20triviality%20of","manuscripts":[{"title":"Topological triviality of mu-constant families of surface singularities","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Topological-triviality-of-mu-constant-families-of-surface-singularities-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/064.mp4","poster":"/films/math/064-poster.webp","captions":"/films/math/064.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Constant Milnor number implies constant topology for surface singularities, the case Lê–Ramanujam left open. No Lean.","consequence":"Closes the last case of the Lê–Ramanujam problem. Internal."},"detail":"/api/math/064"},{"id":"065","title":"Virasoro constraints for complete intersections and projective-bundle towers","short":"Virasoro constraints, complete intersections","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Full Virasoro conjecture for ordinary descendant Gromov–Witten theory of smooth complete intersections in P^N: all genera, all curve classes, arbitrary insertions incl. primitive and odd cohomology, no semisimplicity.","verdict":"Projectivization theorem (065_0) is conditional on the base satisfying the full constraints (by design). No Lean.","url":"/math#065","articleUrl":"/articles/openai-math#virasoro-constraints-family-065","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=065.%20Virasoro%20constraints%20for","manuscripts":[{"title":"Virasoro Constraints under Projectivization","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf"},{"title":"Virasoro Constraints for Projective Complete Intersections","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026/article.pdf"}],"reel":{"src":"/films/math/065.mp4","poster":"/films/math/065-poster.webp","captions":"/films/math/065.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Virasoro constraints for Gromov–Witten invariants of complete intersections, beyond semisimple targets. No Lean.","consequence":"Virasoro constraints for more target spaces. Internal to Gromov–Witten theory."},"detail":"/api/math/065"},{"id":"066","title":"Bounded klt complements for Fano contractions","short":"Bounded klt complements","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Bounded klt complements: for fixed d and positive rational ε, every ε-lc Fano contraction f: X → Z in char 0 admits near each base point a klt complement of index ≤ N(d,ε);","verdict":"Finite-rational-coefficient form; 199 pp in two papers. No Lean.","url":"/math#066","articleUrl":"/articles/openai-math#bounded-klt-complements-family-066","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=066.%20Bounded%20klt%20complements","manuscripts":[{"title":"Bounded klt complements for Fano contractions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-klt-complements-for-Fano-contractions-September-25-2026/Bounded-klt-complements-for-Fano-contractions-September-25-2026.pdf"},{"title":"Uniform Cartier sections for Fano type contractions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026/Uniform-Cartier-sections-for-Fano-type-contractions-September-25-2026.pdf"}],"reel":{"src":"/films/math/066.mp4","poster":"/films/math/066-poster.webp","captions":"/films/math/066.vtt","duration":19.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":40,"tier":"Incremental","importance":1,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Bounded-index klt complements for Fano contractions, a finiteness tool for boundedness of Fanos. No Lean.","consequence":"A finiteness tool behind boundedness results for Fano varieties."},"detail":"/api/math/066"},{"id":"067","title":"The Campana–Peternell conjecture in dimension six","short":"Campana–Peternell in dimension 6","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"partial","kindLabel":"Claimed partial result","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"Campana–Peternell conjecture in dimension 6: every smooth Fano sixfold with nef tangent bundle is rational homogeneous; consequence for compact Kähler manifolds with nef tangent bundle and dim − q̃ ≤ 6.","verdict":"One dimension only; Appendix A is an 'exact arithmetic certificate' (computer-checkable numerics). 27 pp; no Lean.","url":"/math#067","articleUrl":"/articles/openai-math#four-narrower-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=067.%20The%20Campana%20Peternell","manuscripts":[{"title":"The Campana–Peternell conjecture in dimension six","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/067.mp4","poster":"/films/math/067-poster.webp","captions":"/films/math/067.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":31,"tier":"Incremental","importance":3,"advance":1,"consequences":0,"surprise":0,"confidence":1,"why":"Adds dimension six to the Campana–Peternell conjecture on Fano manifolds with nef tangent bundle; one more dimension.","consequence":"One more dimension of a conjecture. Nothing follows beyond it."},"detail":"/api/math/067"},{"id":"068","title":"Anticanonical nonvanishing in every dimension","short":"Anticanonical nonvanishing","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Anticanonical nonvanishing: if X is a smooth connected complex projective variety and −K_X admits a smooth Hermitian metric of semipositive curvature, then H⁰(X, −mK_X) ≠ 0 for some m > 0 (seven papers: torus-invariant index polynomiality, metric descent,…","verdict":"Seven manuscripts (270 pp) form one chain; several are explicitly reductions ('Assuming...') feeding the main paper. No Lean.","url":"/math#068","articleUrl":"/articles/openai-math#anticanonical-nonvanishing-family-068","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=068.%20Anticanonical%20nonvanishing%20in","manuscripts":[{"title":"Anticanonical nonvanishing from smooth semipositivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf"},{"title":"Invariant anticanonical indices and conversion of twisted differentials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026/main.pdf"},{"title":"Bounded anticanonical metrics on klt pairs and torus quotients","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-anticanonical-metrics-on-klt-pairs-and-torus-quotients-September-26-2026/main.pdf"},{"title":"Cohomological transfer and equivariant anticanonical sections","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cohomological-transfer-and-equivariant-anticanonical-sections-September-26-2026/main.pdf"},{"title":"Metric descent and rank-preserving contractions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf"},{"title":"Integrable metrics and effectivity with controlled boundary","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf"},{"title":"Exact orders and invariant anticanonical linear systems","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.pdf"}],"reel":{"src":"/films/math/068.mp4","poster":"/films/math/068-poster.webp","captions":"/films/math/068.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Manifolds with semipositive anticanonical bundle have a pluri-anticanonical section, over a seven-paper chain. No Lean.","consequence":"Nonvanishing for anticanonical bundles under curvature assumptions. Internal."},"detail":"/api/math/068"},{"id":"069","title":"Global quantum geometric Langlands at irrational level","short":"Quantum geometric Langlands, $c \\notin \\mathbb{Q}$","discipline":"Algebraic and complex geometry","disciplineIndex":1,"accent":"#C792EA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Unramified de Rham quantum geometric Langlands at irrational level: for every connected simple complex group G, every smooth projective curve X and every c ∈ C \\ Q (including non-real), an equivalence D_c(Bun_G(X)) ≃ D_{−1/(rc)}(Bun_{G∨}(X)) of full twisted…","verdict":"Irrational levels only (rational/critical levels excluded); 119 pp single paper; uses one stated assumption-type convention (normalization). No Lean.","url":"/math#069","articleUrl":"/articles/openai-math#quantum-geometric-langlands-at-irrational-level-family-069","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=069.%20Global%20quantum%20geometric","manuscripts":[{"title":"Global quantum geometric Langlands at irrational level","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf"}],"reel":{"src":"/films/math/069.mp4","poster":"/films/math/069-poster.webp","captions":"/films/math/069.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Quantum geometric Langlands at irrational level; rational and critical levels are excluded. No Lean.","consequence":"Quantum geometric Langlands at irrational level. Internal to geometric representation theory."},"detail":"/api/math/069"},{"id":"071","title":"Koebe’s circle-domain conjecture","short":"Koebe's circle-domain conjecture","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Two papers. (1) Every domain (nonempty connected open set) in the Riemann sphere is conformally equivalent to a circle domain, i.e. one whose complementary components are closed round disks or points;","verdict":"Existence only: rigidity/uniqueness of the model for non-removable boundaries is not settled (and Rajala showed rigidity does not imply removability).","url":"/math#071","articleUrl":"/articles/openai-math#koebes-circle-domain-conjecture","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=071.%20Koebe%E2%80%99s%20circle%2Ddomain%20conjecture","manuscripts":[{"title":"Removable Boundaries and Rigidity of Circle Domains","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf"},{"title":"Koebe's Circle-Domain Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/071.mp4","poster":"/films/math/071-poster.webp","captions":"/films/math/071.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Koebe's 1908 question: domains with infinitely many holes can be mapped to circle domains; Lean states and checks it.","consequence":"Circle-domain models exist for wild domains; uniqueness stays open. Internal to complex analysis."},"detail":"/api/math/071"},{"id":"072","title":"Brennan's conjecture and the integral-means spectrum","short":"Brennan's conjecture","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(1) Brennan's conjecture: for every conformal bijection φ from a simply connected plane domain (spherical boundary with at least two points) onto the unit disk, ∫|φ'|^s dA 0, for all f in S, hence the bounded-class spectrum B_b(-1) < 1/4, contradicting…","verdict":"The Kraetzer disproof is only at t=-1 and non-quantitative (ε not computed); it overturns a long-held, numerically supported belief, so it deserves particular scrutiny…","url":"/math#072","articleUrl":"/articles/openai-math#brennans-conjecture-and-a-1996-formula-that-does-not-hold","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=072.%20Brennan's%20conjecture%20and","manuscripts":[{"title":"Brennan's conjecture and sharp inverse-square integral means","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf"},{"title":"A strict inverse-first-power bound for univalent functions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/072.mp4","poster":"/films/math/072-poster.webp","captions":"/films/math/072.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":2,"confidence":3,"why":"Brennan's conjecture on conformal maps, stuck for 25 years, plus a refuted 1996 formula; Lean checks the main theorem.","consequence":"Sharp area-distortion bounds for conformal maps and a settled value of the integral-means spectrum."},"detail":"/api/math/072"},{"id":"073","title":"The Falconer distance conjecture","short":"Falconer's distance conjecture","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every integer d ≥ 2 and every compact E ⊂ R^d with Hausdorff dimension strictly greater than d/2, the distance set Δ(E)={|x-y|: x,y∈E} has positive one-dimensional Lebesgue measure. Unpinned; no claim at dim = d/2 and no pinned version at the threshold.","verdict":"Only 64 pages for one of the central problems of geometric measure theory, and the paper explicitly avoids decoupling and prior distance theorems, which is surprising;","url":"/math#073","articleUrl":"/articles/openai-math#falconers-distance-conjecture-in-every-dimension","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=073.%20The%20Falconer%20distance","manuscripts":[{"title":"The Falconer distance conjecture in all dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/073.mp4","poster":"/films/math/073-poster.webp","captions":"/films/math/073.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":86,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":3,"confidence":3,"why":"Falconer's distance conjecture: big fractals have a positive-length set of distances, in every dimension; Lean checks it.","consequence":"Settles geometric measure theory's central distance question with a multiscale method others can reuse."},"detail":"/api/math/073"},{"id":"074","title":"Kakeya in three and four dimensions","short":"Kakeya in three and four dimensions","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(1) Kakeya maximal conjecture in R^3: for every ε>0, ||K_δ f||_{L^3(S^2)} ≤ C_ε δ^{-ε} ||f||_{L^3(R^3)} for the δ-tube maximal function, uniformly in 0<δ<1; consequences for Nikodym maximal estimates on space forms and a local curved Kakeya statement.","verdict":"The 3D set conjecture is human work (Wang–Zahl 2025); the new 3D content is the maximal-function upgrade (uniform λ^3 density dependence), which imports the…","url":"/math#074","articleUrl":"/articles/openai-math#kakeya-in-three-and-four-dimensions","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=074.%20Kakeya%20in%20three","manuscripts":[{"title":"The Kakeya maximal conjecture in three dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf"},{"title":"Every four-dimensional Kakeya set has full Hausdorff dimension","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/074.mp4","poster":"/films/math/074-poster.webp","captions":"/films/math/074.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":4,"advance":2,"consequences":3,"surprise":2,"confidence":0,"why":"Kakeya: the maximal estimate in 3D and the set conjecture in 4D, one dimension past Wang–Zahl; long and unformalized.","consequence":"Kakeya bounds drive restriction, Bochner–Riesz and PDE estimates; the 4D set result is the next step toward the full conjecture."},"detail":"/api/math/074"},{"id":"075","title":"The Llog L Fourier-convergence conjecture","short":"Fourier series of $L\\log L$ functions","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every complex-valued f in L log L(T) (∫|f| log(2+|f|) < ∞), the symmetric Fourier partial sums S_N f(x) converge to f(x) for almost every x along the full sequence N=0,1,2,…;","verdict":"Unformalized, single 76-page paper with an unusual information-theoretic method (entropy compression lemma, Griffiths–Ginibre correlation inequalities, 'thermal…","url":"/math#075","articleUrl":"/articles/openai-math#l-log-l-functions-have-almost-everywhere-convergent-fourier-series","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=075.%20The%20Llog%20L","manuscripts":[{"title":"Almost-everywhere Fourier convergence in L log L","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/075.mp4","poster":"/films/math/075-poster.webp","captions":"/films/math/075.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":3,"confidence":0,"why":"Fourier series of L log L functions converge almost everywhere, the conjectured edge past Carleson, by an entropy method. No Lean.","consequence":"Fixes where Fourier series stop converging almost everywhere. Internal to harmonic analysis."},"detail":"/api/math/075"},{"id":"076","title":"Real ultraflat Littlewood polynomials and unbounded binary merit factors","short":"Ultraflat Littlewood polynomials","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(1) min over ±1 sign choices of max_{|z|=1}|Σ_{k0 and large N, a ±1 polynomial with √N/16 ≤ |P| ≤ (1+η)√N on the circle (Oct 5). (3) For every ε and large N, a ±1 polynomial with (1-ε)√N ≤ |P(z)| ≤ (1+ε)√N on the whole circle, i.e.","verdict":"Headline 'ultraflat' (two-sided) result is the Oct 5 manuscript and is unformalized; it reuses two lemmas from the Oct 5 version-2 paper.","url":"/math#076","articleUrl":"/articles/openai-math#ultraflat-littlewood-polynomials-exist","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=076.%20Real%20ultraflat%20Littlewood","manuscripts":[{"title":"Ultraflat real Littlewood polynomials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026/ultraflat-real-littlewood-polynomials.pdf"},{"title":"Nearly minimal maxima and positive minima of Littlewood polynomials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf"},{"title":"Asymptotically minimal maxima of real Littlewood polynomials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/076.mp4","poster":"/films/math/076-poster.webp","captions":"/films/math/076.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":2,"why":"Plus/minus-one polynomials can be ultraflat and merit factors unbounded, against Erdős's guess; Lean checks the upper half.","consequence":"Answers coding theory's question on the merit factor of binary sequences, and Littlewood's flatness question."},"detail":"/api/math/076"},{"id":"077","title":"Fourier restriction for positively curved surfaces","short":"Fourier restriction in three dimensions","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(1) Bounded-data restriction for the sphere S^2: ||Eg||_{L^p(R^3)} ≤ C_p ||g||_{L^∞(S^2)} for every p>3 (sharp open range); via Bourgain's factorization also L^q(S^2)→L^q(R^3) for 33, s>2-6/p (sharp by Rogers), oscillatory integral estimates under Bourgain's…","verdict":"Paper (2) uses two same-release inputs: the packet propagation theorem of paper (1) and the 3D Kakeya maximal theorem of family 074, so it inherits their (unverified)…","url":"/math#077","articleUrl":"/articles/openai-math#fourier-restriction-for-positively-curved-surfaces-in-three-dimensions","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=077.%20Fourier%20restriction%20for","manuscripts":[{"title":"Elliptic capacity propagation and Fourier restriction to the sphere","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf"},{"title":"Diagonal Fourier extension for positively curved surfaces in three dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf"}],"reel":{"src":"/films/math/077.mp4","poster":"/films/math/077-poster.webp","captions":"/films/math/077.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":4,"advance":2,"consequences":3,"surprise":2,"confidence":0,"why":"The Fourier restriction conjecture for positively curved surfaces in 3D, leaning on the release's own Kakeya paper. No Lean.","consequence":"Restriction estimates feed PDE, number theory and Bochner–Riesz; this would settle the positively curved case in 3D."},"detail":"/api/math/077"},{"id":"078","title":"The three-dimensional Bochner–Riesz conjecture","short":"Bochner–Riesz in three dimensions","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every δ>0, the Bochner–Riesz multiplier (1-|ξ|²)_+^δ is bounded on L^3(R^3); by interpolation and duality, bounded on L^p(R^3), 1≤p≤∞, whenever δ > max{3|1/p-1/2| - 1/2, 0} (strict-order range).","verdict":"Unformalized 114-page argument via an information/entropy 'positive transport' scheme and finite packet repayment, with Ren–Wang Furstenberg as the geometric input.","url":"/math#078","articleUrl":"/articles/openai-math#the-bochnerriesz-conjecture-in-three-dimensions","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=078.%20The%20three%2Ddimensional%20Bochner","manuscripts":[{"title":"Bochner–Riesz multipliers in three dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf"}],"reel":{"src":"/films/math/078.mp4","poster":"/films/math/078-poster.webp","captions":"/films/math/078.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":0,"why":"The Bochner–Riesz conjecture in three dimensions by an entropy transport scheme; 114 unformalized pages.","consequence":"Sharp Fourier summation in 3D, implying restriction for the sphere."},"detail":"/api/math/078"},{"id":"079","title":"Local smoothing in three dimensions","short":"Local smoothing in three dimensions","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Sogge's Euclidean local smoothing conjecture for the wave equation in three spatial dimensions: ||e^{it√-Δ} f||_{L^3(R^3×[1,2])} ≤ C_ε ||J^ε f||_{L^3} for every ε>0, hence for 2 max{0, 1-3/p}.","verdict":"165 pages, unformalized. Uses the Guth–Wang–Zhang wave-envelope estimate as a lower-dimensional input.","url":"/math#079","articleUrl":"/articles/openai-math#local-smoothing-in-three-space-dimensions","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=079.%20Local%20smoothing%20in","manuscripts":[{"title":"Critical local smoothing for the three-dimensional wave equation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/079.mp4","poster":"/films/math/079-poster.webp","captions":"/films/math/079.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":66,"tier":"Solid","importance":3,"advance":2,"consequences":3,"surprise":2,"confidence":0,"why":"Sharp local smoothing for waves in 3+1 dimensions, which would imply 3D Bochner–Riesz, restriction and Kakeya. No Lean.","consequence":"One estimate implying 3D Bochner–Riesz, restriction and Kakeya maximal bounds, plus sharp regularity for wave equations."},"detail":"/api/math/079"},{"id":"080","title":"The exact Sobolev endpoint for Schrödinger convergence","short":"Schrödinger convergence at the endpoint","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"For every n ≥ 2 and every f ∈ H^{n/(2(n+1))}(R^n) (s = 1/3 in the plane), the free Schrödinger evolution, defined as the Gaussian-regularization limit lim_{a↓0} U_a f taken first, exists for all 0<t<1 on one full-measure set and converges to f*(x) a.e.","verdict":"This is only the equality case s = n/(2(n+1)); the sharp range up to the endpoint is human work (Du–Guth–Li 2017; Du–Zhang 2019).","url":"/math#080","articleUrl":"/articles/openai-math#schrödinger-convergence-at-the-exact-endpoint-080","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=080.%20The%20exact%20Sobolev","manuscripts":[{"title":"Endpoint convergence for the planar Schrodinger equation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/paper.pdf"},{"title":"Endpoint pointwise convergence for the Schrodinger equation in higher dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/080.mp4","poster":"/films/math/080-poster.webp","captions":"/films/math/080.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":2,"advance":1,"consequences":1,"surprise":1,"confidence":0,"why":"The exact endpoint of Carleson's Schrödinger convergence problem; the sharp range up to it was already known. No Lean.","consequence":"Closes the endpoint of Carleson's Schrödinger problem. Little follow-on."},"detail":"/api/math/080"},{"id":"081","title":"Riesz transforms and rectifiability in higher codimension","short":"Riesz transforms and rectifiability","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For integers d ≥ 4 and 2 ≤ n ≤ d-2, every n-Ahlfors–David regular Radon measure μ on R^d whose n-dimensional Riesz transform truncations R_{μ,ε} are bounded on L^2(μ) uniformly over all ε>0 is uniformly n-rectifiable (big pieces of Lipschitz images of balls),…","verdict":"Only 56 pages for a problem that resisted the NTV techniques; credibility rests on the Lean formalization.","url":"/math#081","articleUrl":"/articles/openai-math#davidsemmes-in-higher-codimension","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=081.%20Riesz%20transforms%20and","manuscripts":[{"title":"Riesz transforms and uniform rectifiability in higher codimension","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/081.mp4","poster":"/films/math/081-poster.webp","captions":"/films/math/081.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"The David–Semmes problem in higher codimension: bounded Riesz transforms force rectifiable sets; Lean checks it.","consequence":"Characterizes rectifiable sets by Riesz transforms in every codimension, a tool in geometric measure theory."},"detail":"/api/math/081"},{"id":"082","title":"Annular variation and dyadic absolute bounds for the triangular Hilbert transform","short":"The triangular Hilbert transform","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"partial","kindLabel":"Claimed partial result","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For complex F,G ∈ L^3(R^2), the two-endpoint maximal triangular Hilbert transform sup_{02 is bounded. Pairing with a third function gives the scalar triangular Hilbert form bound at the symmetric point (3,3,3), Thiele's Problem 13.","verdict":"Only the symmetric point; boundedness at other Hölder exponents (the general THT conjecture) is not claimed.","url":"/math#082","articleUrl":"/articles/openai-math#the-triangular-hilbert-transform-at-the-symmetric-point","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=082.%20Annular%20variation%20and","manuscripts":[{"title":"Annular variation of the triangular Hilbert transform at the symmetric point","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf"},{"title":"An L³ bound for the dyadic triangular Hilbert form","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-L3-bound-for-the-dyadic-triangular-Hilbert-form-October-5-2026/dyadic-triangular-hilbert.pdf"},{"title":"The maximal triangular Hilbert transform at the symmetric point","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/082.mp4","poster":"/films/math/082-poster.webp","captions":"/films/math/082.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":2,"consequences":2,"surprise":3,"confidence":3,"why":"The first bound of any kind for the triangular Hilbert transform, at its symmetric point; a new trace method, Lean-checked.","consequence":"A first bound for an entangled singular integral; its trace-energy method may reach other multilinear forms."},"detail":"/api/math/082"},{"id":"083","title":"Hilbert transforms along Lipschitz directions","short":"Hilbert transforms along Lipschitz fields","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every Lipschitz unit vector field v on R^2 (allowed to depend on both coordinates) with Lip(v) <= 1, the hard-truncated directional Hilbert transform H^eps_{v,a*} f(x) = int_{eps<|t|<a*} f(x - t v(x)) dt/t satisfies ||H^eps f||_2 <= C* ||f||_2 uniformly…","verdict":"Scale restriction (outer length ~ 1/Lip) is intrinsic to Stein's conjecture as formulated, but readers should not read it as boundedness at all scales or as the Zygmund…","url":"/math#083","articleUrl":"/articles/openai-math#steins-conjecture-for-hilbert-transforms-along-lipschitz-directions","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=083.%20Hilbert%20transforms%20along","manuscripts":[{"title":"A uniform Hilbert transform estimate for Lipschitz directions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/083.mp4","poster":"/films/math/083-poster.webp","captions":"/films/math/083.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"Stein's conjecture: Hilbert transforms along Lipschitz direction fields are bounded at short scales; Lean checks it.","consequence":"Settles Stein's Lipschitz conjecture; the Zygmund differentiation problem stays open."},"detail":"/api/math/083"},{"id":"084","title":"The geometric case of the Erdős similarity conjecture","short":"$\\text{Erd\\H{o}s}$ similarity: geometric sequences","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"partial","kindLabel":"Claimed partial result","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"For every fixed ratio q in (0,1) and eta in (0,1) there is a compact E subset [0,1] with measure > 1-eta containing no affine copy x + s{q^n : n>=1} for any x in R and any nonzero real s (both signs). The set depends on q; no single set handles all ratios.","verdict":"Resolves only the geometric-progression case, not the conjecture for all infinite sets (the papers say so explicitly; the family title 'geometric case' is accurate).","url":"/math#084","articleUrl":"/articles/openai-math#the-erdős-similarity-conjecture-for-geometric-sequences","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=084.%20The%20geometric%20case","manuscripts":[{"title":"The geometric case of the Erdős similarity conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf"},{"title":"The dyadic case of the Erdős similarity conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/084.mp4","poster":"/films/math/084-poster.webp","captions":"/films/math/084.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":2,"why":"Erdős's similarity conjecture for geometric sequences like 1/2, 1/4, 1/8; the general case stays open. Lean covers ratio 1/2.","consequence":"Resolves Erdős similarity for geometric sequences. Internal."},"detail":"/api/math/084"},{"id":"085","title":"Endpoint Sobolev regularity of centered disk averages","short":"Disk maximal function in $W^{1,1}$","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"partial","kindLabel":"Claimed partial result","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every real f in W^{1,1}(R^2), the centered disk maximal function Mf(x) = sup_{r>0} avg_{B(x,r)} |f| is finite a.e., lies in W^{1,1}_loc(R^2), and has a globally integrable weak gradient with ||grad Mf||_1 <= C ||grad f||_1 for an absolute constant C.","verdict":"Only the planar centered-disk case of the Hajłasz-Onninen question; higher dimensions and the uncentered/other-shape cases are not claimed, and the summary says this…","url":"/math#085","articleUrl":"/articles/openai-math#centered-disk-maximal-function-in-w11-085","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=085.%20Endpoint%20Sobolev%20regularity","manuscripts":[{"title":"An Endpoint Gradient Bound for the Centered Disk Maximal Operator","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026/article.pdf"}],"reel":{"src":"/films/math/085.mp4","poster":"/films/math/085-poster.webp","captions":"/films/math/085.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":19,"tier":"Incremental","importance":1,"advance":1,"consequences":0,"surprise":1,"confidence":3,"why":"In the plane, centered disk averages do not increase gradient variation; one case of a Hajłasz–Onninen question.","consequence":"One case of a regularity question for maximal functions. Nothing follows beyond it."},"detail":"/api/math/085"},{"id":"086","title":"An L3 bound for the trilinear Hilbert transform","short":"The trilinear Hilbert transform","discipline":"Real and complex analysis","disciplineIndex":2,"accent":"#7FD1C7","kind":"partial","kindLabel":"Claimed partial result","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"The principal-value trilinear Hilbert transform T(f1,f2,f3)(x) = p.v. ∫ f1(x-t) f2(x-2t) f3(x-3t) dt/t on R is bounded L^3 x L^3 x L^3 -> L^1. Fixed slopes 1,2,3 (any non-degenerate distinct slopes are presumably similar but are not claimed);","verdict":"Single exponent point (3,3,3 -> 1) of the conjecture; the full conjectured range is not claimed.","url":"/math#086","articleUrl":"/articles/openai-math#a-first-lp-bound-for-the-trilinear-hilbert-transform","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=086.%20An%20L3%20bound","manuscripts":[{"title":"An L³ bound for the trilinear Hilbert transform","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf"}],"reel":{"src":"/films/math/086.mp4","poster":"/films/math/086-poster.webp","captions":"/films/math/086.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":66,"tier":"Solid","importance":3,"advance":2,"consequences":3,"surprise":2,"confidence":0,"why":"A first L^p bound for the trilinear Hilbert transform, at one exponent; 93 pages written a day before release. No Lean.","consequence":"Would open the multilinear Hilbert transform hierarchy, and feeds the release's pointwise ergodic result (family 154)."},"detail":"/api/math/086"},{"id":"087","title":"The Mahler conjectures, functional inequalities and polar-product symplectic width","short":"Mahler's conjectures, every dimension","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(1) Symmetric Mahler: for every origin-symmetric convex body K in R^n, n>=1, |K||K°| >= 4^n/n!, with equality iff K is a linear image of a Hanner polytope.","verdict":"Two independent proofs of the symmetric inequality are given (a holomorphic-mass/feasible-simplex argument, 26 pp, and the symplectic-ball argument, 19 pp), both built…","url":"/math#087","articleUrl":"/articles/openai-math#the-mahler-conjectures-twice-over-and-a-symplectic-ball-that-proves-one-of-them","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=087.%20The%20Mahler%20conjectures","manuscripts":[{"title":"The symmetric Mahler conjecture and its equality cases","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf"},{"title":"The Mahler Conjecture for General Convex Bodies","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Mahler-Conjecture-for-General-Convex-Bodies-September-22-2026/paper.pdf"},{"title":"Symplectic Balls in Symmetric Polar Products","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Symplectic-Balls-in-Symmetric-Polar-Products-September-22-2026/paper.pdf"}],"reel":{"src":"/films/math/087.mp4","poster":"/films/math/087-poster.webp","captions":"/films/math/087.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":3,"why":"Mahler's 1939 volume-product conjecture for symmetric bodies, via a symplectic ball, with two proofs; Lean checks it.","consequence":"Sharp volume-product bound for symmetric convex bodies, with a new link to symplectic capacities."},"detail":"/api/math/087"},{"id":"088","title":"Sharp projection-body inequalities and a counterexample to simplex maximization","short":"Petty's projection conjecture, $n \\ge 4$","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"(1) Petty's projection-volume conjecture for n>=4: for every convex body K in R^n, |ΠK|/|K|^{n-1} >= κ_{n-1}^n κ_n^{2-n}, equality iff K is an ellipsoid (no symmetry/smoothness).","verdict":"The paper proves only n>=4; the 'full' conjecture (n>=3) and the Lutwak-Petty/Holmes-Thompson corollaries for n=3 rely on the separate human result of…","url":"/math#088","articleUrl":"/articles/openai-math#pettys-projection-conjecture-in-dimension-four-and-up","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=088.%20Sharp%20projection%2Dbody%20inequalities","manuscripts":[{"title":"Petty’s projection-volume conjecture in dimensions at least four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026/paper.pdf"},{"title":"A product counterexample to the simplex maximum for projection-body volume","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/088.mp4","poster":"/films/math/088-poster.webp","captions":"/films/math/088.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Petty's projection inequality in dimension 4 and up; dimension 3 relies on separate human work. Lean checks it.","consequence":"A sharp affine isoperimetric inequality. Internal to convex geometry."},"detail":"/api/math/088"},{"id":"089","title":"Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs","short":"Planar graphs embed into $L_1$","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(1) There is a universal C such that the shortest-path metric of every finite connected planar graph with arbitrary positive edge lengths embeds into L1 with distortion =2 there is C(k) such that every finite connected graph with a tree decomposition of bag…","verdict":"Resolves two cases of GNRS, not the full conjecture for all minor-closed families (general clique-sums / closure under 2-sums remain).","url":"/math#089","articleUrl":"/articles/openai-math#planar-and-bounded-treewidth-graphs-embed-into-l_1","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=089.%20Bounded%2Ddistortion%20L1%20embeddings","manuscripts":[{"title":"Planar Graph Metrics Embed into L1 with Constant Distortion","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026/paper.pdf"},{"title":"L1 Embeddings of Graphs of Bounded Treewidth","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/089.mp4","poster":"/films/math/089-poster.webp","captions":"/films/math/089.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":66,"tier":"Solid","importance":3,"advance":2,"consequences":3,"surprise":2,"confidence":3,"why":"Planar and bounded-treewidth graphs embed in L1 with bounded distortion, a 25-year-old cut question; Lean checks it.","consequence":"Bounded flow-cut gaps on planar and bounded-treewidth networks: sparsest-cut relaxations are constant-factor there."},"detail":"/api/math/089"},{"id":"090","title":"Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy","short":"Triangular lattice, universally optimal","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(1) Universal optimality in the plane: for every smooth nonnegative completely monotone g of squared distance and every locally finite planar configuration C of centered-disk density one, liminf_R (1/N_R) Σ_{x≠y in C∩B_R} g(|x-y|^2) >= Σ_{a in A\\0} g(|a|^2),…","verdict":"Three of four papers lean on rigorous interval arithmetic / finite certificates (20x20 interpolation blocks, sign checks);","url":"/math#090","articleUrl":"/articles/openai-math#the-triangular-lattice-is-universally-optimal","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=090.%20Triangular%2Dlattice%20optimality%20long%2Drange","manuscripts":[{"title":"An atomic certificate for triangular-lattice universal optimality","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf"},{"title":"Universal optimality of the triangular lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-optimality-of-the-triangular-lattice-September-23-2026/paper.pdf"},{"title":"A sharp Fourier certificate for planar circle packing","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf"},{"title":"Triangular minimality for planar Coulomb renormalized energy","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Triangular-minimality-for-planar-Coulomb-renormalized-energy-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/090.mp4","poster":"/films/math/090-poster.webp","captions":"/films/math/090.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":62,"tier":"Solid","importance":3,"advance":2,"consequences":3,"surprise":1,"confidence":2,"why":"The hexagonal lattice minimizes energy for every completely monotone interaction (Cohn–Kumar in 2D); Lean checks the core.","consequence":"Explains why hexagonal patterns minimize energy in 2D systems such as vortices and Coulomb gases; sharp packing certificates."},"detail":"/api/math/090"},{"id":"091","title":"Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture","short":"The log-Brunn–Minkowski inequality","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every n>=1, origin-symmetric convex bodies K, L in R^n and 0= |K|^{1-λ}|L|^λ (log-Brunn-Minkowski; no smoothness or unconditionality). Corollaries: the symmetric L_p Brunn-Minkowski inequality for all 0 μ(e^t K) log-concave) for scalar dilations.","verdict":"Remarkably short (20 pp) for a famous conjecture; the core is a variance inequality in 'moment coordinates' proved via density of even tests and a tensor estimate.","url":"/math#091","articleUrl":"/articles/openai-math#the-log-brunnminkowski-inequality-and-the-b-conjecture","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=091.%20Logarithmic%20and%20Lp","manuscripts":[{"title":"The logarithmic Brunn–Minkowski conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-logarithmic-Brunn-Minkowski-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/091.mp4","poster":"/films/math/091-poster.webp","captions":"/films/math/091.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"The log Brunn–Minkowski inequality for symmetric bodies, plus the B-conjecture, in 20 pages; Lean checks the main theorem.","consequence":"Uniqueness for the logarithmic Minkowski problem and the B-conjecture follow."},"detail":"/api/math/091"},{"id":"092","title":"The optimal order of convex-body covering density","short":"Covering density of order $n\\log n$","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"(1) Every convex body K in R^n, n>=2, has a covering by translates along one full-rank lattice with density <= C n log n (absolute C; no symmetry/regularity).","verdict":"Part (1) removes an iterated-log factor from Li-Liu's July 2026 result and builds directly on their horizontal-vertical/binary-correction framework, so the lattice half…","url":"/math#092","articleUrl":"/articles/openai-math#covering-space-with-convex-bodies-costs-thetan-log-n","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=092.%20The%20optimal%20order","manuscripts":[{"title":"A single-lattice covering bound of order n log n","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf"},{"title":"Translative covering densities of order n log n","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/092.mp4","poster":"/films/math/092-poster.webp","captions":"/films/math/092.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"One lattice covers space with density of order n log n, and some bodies need that much; sharpens very recent work.","consequence":"Pins the worst case of covering density. Internal."},"detail":"/api/math/092"},{"id":"093","title":"Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures","short":"Dimension-free log-Sobolev, subgaussian","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"There is a universal C such that every centered log-concave probability measure μ with a Lebesgue density on R^n whose linear marginals are a-subgaussian (sup_θ E exp(^2/a^2) <= 2) satisfies Ent_μ(f^2) <= C a^2 ∫|Df|^2 dμ for all smooth compactly supported f,…","verdict":"65-page unformalized contradiction argument (normalized counterexample sequences, Gaussian-channel/I-MMSE identities, tensor near-equalities);","url":"/math#093","articleUrl":"/articles/openai-math#dimension-free-log-sobolev-for-subgaussian-log-concave-measures-093","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=093.%20Dimension%2Dfree%20logarithmic%20Sobolev","manuscripts":[{"title":"A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/093.mp4","poster":"/films/math/093-poster.webp","captions":"/films/math/093.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Subgaussian log-concave measures satisfy a dimension-free log-Sobolev inequality, answering a 2023 question. No Lean.","consequence":"Gaussian concentration for subgaussian log-concave measures. Internal to high-dimensional probability."},"detail":"/api/math/093"},{"id":"094","title":"Subpolynomial dimension reduction in Lp","short":"Dimension reduction in $L_p$, $n^{o(1)}$","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For fixed 11, every n-point subset of any real L_p space embeds (by an arbitrary, non-linear map, after rescaling) into l_p^d with distortion at most D, where log(n)/log(1+2D) 2; hence d=n^{o(1)}. For exact (D=1) embeddings and n>=9, floor((n-1)/4)^2 2.","verdict":"Upper exponent gamma(p) and lower bound log n are not claimed optimal; the gap to Naor-Ren's p>2 lower bound remains.","url":"/math#094","articleUrl":"/articles/openai-math#dimension-reduction-in-l_p-with-no1-coordinates","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=094.%20Subpolynomial%20dimension%20reduction","manuscripts":[{"title":"Subpolynomial dimension reduction in Lp","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/094.mp4","poster":"/films/math/094-poster.webp","captions":"/films/math/094.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":52,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"n points in L_p fit in n^{o(1)} dimensions at fixed distortion, a Johnson–Lindenstrauss analogue; Lean checks it.","consequence":"Dimension reduction for L_p data, once thought to need near-linear dimension; constants are not worked out."},"detail":"/api/math/094"},{"id":"095","title":"Hyperbolicity cones without semidefinite lifts","short":"Hyperbolicity cones vs. semidefinite lifts","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(Paper 1) There is a real homogeneous hyperbolic polynomial (constructed with m=330 auxiliary-input variables and matrix size about 20N-1, N > 10*36^2) whose closed hyperbolicity cone is not a spectrahedral shadow: no finite affine semidefinite lift exists,…","verdict":"Directly contradicts Gonzalez Nevado's Jan 2026 preprint claiming GLC is true; paper 2 Appendix A gives a counterexample to that preprint's Theorem 52 step.","url":"/math#095","articleUrl":"/articles/openai-math#hyperbolicity-cones-that-are-not-slices-of-the-psd-cone","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=095.%20Hyperbolicity%20cones%20without","manuscripts":[{"title":"Hyperbolicity Cones Without Semidefinite Lifts","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hyperbolicity-Cones-Without-Semidefinite-Lifts-October-5-2026/nonliftable-hyperbolicity.pdf"},{"title":"A nonspectrahedral hyperbolicity cone","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf"},{"title":"An Exact Semidefinite Lift of a Nonspectrahedral Hyperbolicity Cone","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026/Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone.pdf"}],"reel":{"src":"/films/math/095.mp4","poster":"/films/math/095-poster.webp","captions":"/films/math/095.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":3,"confidence":2,"why":"Hyperbolicity cones that are not slices of PSD cones, refuting the generalized Lax conjecture; Lean checks the weaker form.","consequence":"Hyperbolic programming is strictly more expressive than semidefinite programming, an optimization theory question."},"detail":"/api/math/095"},{"id":"096","title":"The Gaussian propeller conjecture in every dimension","short":"The Gaussian propeller conjecture","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every d,k>=1 and every measurable partition A_1..A_k of R^d (empty cells allowed, masses unrestricted), sum_i ||integral_{A_i} x dgamma_d||^2 =2, k>=3 by three planar 120-degree sectors times R^{d-2}.","verdict":"The paper (on paper) uses the computer-assisted HJN R^3 theorem as a black box for <=4 active cells; the Lean development covers the whole statement.","url":"/math#096","articleUrl":"/articles/openai-math#the-gaussian-propeller","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=096.%20The%20Gaussian%20propeller","manuscripts":[{"title":"The Gaussian propeller bound in every dimension","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/096.mp4","poster":"/films/math/096-poster.webp","captions":"/films/math/096.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"Khot–Naor's propeller conjecture on Gaussian partitions in every dimension; Lean checks the full statement.","consequence":"Fixes the approximability of kernel clustering, conditional on the release's Unique Games proof."},"detail":"/api/math/096"},{"id":"097","title":"The Euclidean Steinitz–Bergström bound","short":"The Euclidean Steinitz–Bergström bound","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is an absolute constant C such that for all d,N and any vectors v_1..v_N in the Euclidean unit ball of R^d (prescribed order), one choice of signs keeps every signed prefix sum within Euclidean norm C sqrt(d), independent of N.","verdict":"The constant C is not explicit in the statement. Builds on very recent (Sept 2026) preprints: Guo-Fang-Lu (AI-agent-discovered), Akbas-Sra (arXiv:2609.27172) and…","url":"/math#097","articleUrl":"/articles/openai-math#the-euclidean-steinitz-constant-is-thetasqrt-d","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=097.%20The%20Euclidean%20Steinitz","manuscripts":[{"title":"The Euclidean Steinitz–Bergström theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf"}],"reel":{"src":"/films/math/097.mp4","poster":"/films/math/097-poster.webp","captions":"/films/math/097.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Unit vectors summing to zero can be ordered so partial sums stay within about √d; Lean checks the bound.","consequence":"A sharp vector-balancing bound, without an efficient algorithm."},"detail":"/api/math/097"},{"id":"098","title":"Compact counterexamples to bi-Lipschitz dimension reduction","short":"A doubling set in $\\ell_2$ that fits no $\\mathbb{R}^k$","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a fixed subset S of real l_2 with doubling constant at most 76800 that admits no bi-Lipschitz embedding into any R^k at any finite distortion.","verdict":"Short (13 pp.) but fully Lean-checked, which makes it one of the more trustworthy items. Gives an infinite (compact) set;","url":"/math#098","articleUrl":"/articles/openai-math#a-doubling-subset-of-hilbert-space-that-fits-in-no-mathbb-rk","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=098.%20Compact%20counterexamples%20to","manuscripts":[{"title":"A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/098.mp4","poster":"/films/math/098-poster.webp","captions":"/films/math/098.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"A doubling subset of Hilbert space with no bi-Lipschitz copy in any R^k, the famous open case; 13 Lean-checked pages.","consequence":"Settles the Hilbert-space case of dimension reduction for doubling sets, negatively."},"detail":"/api/math/098"},{"id":"099","title":"The sharp exponential scale of edit-distance distortion","short":"Edit distance into $\\ell_1$: the sharp exponent","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For all sufficiently large d, uniformly over finite alphabets of size >= 2 (even growing with d), the least distortion of embedding unit-cost edit distance (insertions, deletions, substitutions) on strings of length <= d into l_1 is between exp(c sqrt(log d…","verdict":"Determines log of the distortion up to constants, not the distortion up to constants. The upper bound is essentially Ostrovsky-Rabani's (reproved uniformly and…","url":"/math#099","articleUrl":"/articles/openai-math#edit-distance-into-ell_1-the-exponent-is-right","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=099.%20The%20sharp%20exponential","manuscripts":[{"title":"Edit Distance in l1: Matching Bounds up to Constants in the Exponent","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/paper.pdf"},{"title":"Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf"},{"title":"Tree Constructions for the l1 Distortion of Binary Edit Distance","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf"}],"reel":{"src":"/films/math/099.mp4","poster":"/films/math/099-poster.webp","captions":"/films/math/099.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Edit distance embeds into L1 no better than Ostrovsky–Rabani's 2005 scheme, up to the exponent's constant; Lean checks it.","consequence":"Edit distance cannot be embedded into L1 much better than known, closing a line in nearest-neighbour search theory."},"detail":"/api/math/099"},{"id":"100","title":"Cylinder coverings below the half-area bound","short":"Cylinder coverings below the half-area bound","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"The regular tetrahedron K (vertices (+-1,0,0),(0,+-1,sqrt2)), with minimum projection area sqrt2, can be covered by m=2*ceil(2/eps^2) cylinders with compact triangular bases perpendicular to their axes whose total base area, normalized by sqrt2, is 1/2 -…","verdict":"The saving is tiny ((13/6000)eps^2 to leading order, about 5e-10 of the normalized area at the largest allowed eps=1/2000, using 2*ceil(2/eps^2)=16,000,000 cylinders)…","url":"/math#100","articleUrl":"/articles/openai-math#cylinder-coverings-below-the-half-area-bound-100","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=100.%20Cylinder%20coverings%20below","manuscripts":[{"title":"Finite angular cylinder covers below the half-area bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf"},{"title":"Finite cylinder approximation of ruled sets","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf"},{"title":"Slope-field perturbations of the two-cylinder covering","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026/main.pdf"},{"title":"Finite triangular approximation of radial sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf"}],"reel":{"src":"/films/math/100.mp4","poster":"/films/math/100-poster.webp","captions":"/films/math/100.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":19,"tier":"Incremental","importance":1,"advance":1,"consequences":0,"surprise":1,"confidence":3,"why":"Cylinders can cover a tetrahedron with slightly under half its shadow area; the saving is about 5e-10. Lean checks it.","consequence":"Moves the best constant in a covering inequality by about 5e-10. Nothing follows beyond it."},"detail":"/api/math/100"},{"id":"101","title":"The sharp simplex conjecture for isotropic constants","short":"The simplex conjecture for $L_K$","discipline":"Convex and metric geometry","disciplineIndex":3,"accent":"#E6C86E","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"(i) For every n>=1 and every convex body K in R^n, L_K = m + (1/2) log det Cov(f), with equality iff f is an invertible affine image of a product of one-sided exponential laws. Via Klartag (Adv. Math.","verdict":"Unformalized and not peer-reviewed, 42 pages of hard analysis (Brenier/Caffarelli transport, matrix-valued Ornstein-Uhlenbeck/chaos decomposition, rational-constant…","url":"/math#101","articleUrl":"/articles/openai-math#the-sharp-simplex-conjecture-for-isotropic-constants","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=101.%20The%20sharp%20simplex","manuscripts":[{"title":"A sharp entropy bound and the simplex inequality for isotropic constants","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf"}],"reel":{"src":"/films/math/101.mp4","poster":"/films/math/101-poster.webp","captions":"/films/math/101.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":66,"tier":"Solid","importance":2,"advance":3,"consequences":3,"surprise":2,"confidence":0,"why":"Simplices have the largest isotropic constant; if right it also gives the non-symmetric Mahler conjecture. No Lean.","consequence":"Settles which convex bodies have the largest isotropic constant and, if right, the non-symmetric Mahler conjecture."},"detail":"/api/math/101"},{"id":"102","title":"The Unique Games Conjecture and optimal approximation thresholds","short":"The Unique Games Conjecture","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every fixed eps, delta in (0,1/2) there is an integer s and a deterministic polynomial-time reduction from 3SAT to explicit unweighted, simple bipartite Unique Games over alphabet F_2^s in which every constraint is a translation a(v)=a(u)+c: satisfiable…","verdict":"The proof imports external theorems as black boxes in the paper (Khot-Minzer-Safra Grassmann expansion via the Barak-Kothari-Steurer inverse shortcode theorem, Hastad's…","url":"/math#102","articleUrl":"/articles/openai-math#the-unique-games-conjecture-claimed-proved","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=102.%20The%20Unique%20Games","manuscripts":[{"title":"The Unique Games Theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf"},{"title":"A Direct Proof of Optimal Max-Cut Hardness","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf"},{"title":"The Factor-Two Hardness Threshold for Vertex Cover","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/paper.pdf"},{"title":"Constant-factor hardness of Min-UnCut","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-Min-UnCut-September-23-2026/paper.pdf"},{"title":"Constant-factor hardness of directed feedback vertex set","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-directed-feedback-vertex-set-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/102.mp4","poster":"/films/math/102-poster.webp","captions":"/films/math/102.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":96,"tier":"Huge if true","importance":4,"advance":4,"consequences":4,"surprise":2,"confidence":3,"why":"The Unique Games Conjecture: the simple SDP algorithms for Max-Cut and Vertex Cover are optimal unless P=NP. Lean checks it.","consequence":"Exact approximation limits for Max-Cut, Vertex Cover and every CSP: decades of results conditional on UGC become conditional only on P≠NP."},"detail":"/api/math/102"},{"id":"103","title":"Exact derandomization of logarithmic space: L=RL=BPL","short":"$\\mathsf L = \\mathsf{RL} = \\mathsf{BPL}$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"L = RL = BPL for polynomial-time randomized log-space machines with bounded error (Theorem 1.1). Quantitatively, for a fixed randomized poly-time O(log n)-space machine M and unary precision q, a deterministic algorithm outputs the acceptance probability to…","verdict":"Single 108-page manuscript with no formalization, no reasoning trace and no companion. The proof uses property (T) mixing results of Shalom as an external input,…","url":"/math#103","articleUrl":"/articles/openai-math#l--bpl-randomness-doesnt-help-small-memory","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=103.%20Exact%20derandomization%20of","manuscripts":[{"title":"Exact derandomization of logarithmic space: L = RL = BPL","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/103.mp4","poster":"/films/math/103-poster.webp","captions":"/films/math/103.vtt","duration":20.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":92,"tier":"Huge if true","importance":3,"advance":4,"consequences":4,"surprise":3,"confidence":0,"why":"Coins never help small-memory computation (L = BPL), the space analogue of P = BPP; one 108-page manuscript, no Lean.","consequence":"Randomness never saves memory: every randomized log-space algorithm can be made deterministic, though the time bound may be huge."},"detail":"/api/math/103"},{"id":"104","title":"Quasipolynomial algorithms for mean-payoff, stochastic and parity games","short":"Mean-payoff games in quasipolynomial time","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Deterministic 2^{O((log(L+2))^2)}-bit-operation algorithms (L = full binary input length) that: compute the zero-threshold winning set, exact rational values and optimal positional strategies of ordinary mean-payoff games with signed binary weights;","verdict":"Quasipolynomial, not polynomial; P membership remains open. Stochastic result covers expectation-of-liminf with threshold zero, not general value computation of simple…","url":"/math#104","articleUrl":"/articles/openai-math#mean-payoff-games-in-quasipolynomial-time-104","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=104.%20Quasipolynomial%20algorithms%20for","manuscripts":[{"title":"Turn-Based Stochastic Mean-Payoff Games in Deterministic Quasipolynomial Time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Turn-Based-Stochastic-Mean-Payoff-Games-in-Deterministic-Quasipolynomial-Time-October-5-2026/stochastic-mean-payoff-games.pdf"},{"title":"Mean-payoff parity games in quasipolynomial time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Mean-payoff-parity-games-in-quasipolynomial-time-October-5-2026/mean-payoff-parity.pdf"},{"title":"Deterministic quasipolynomial-time mean-payoff games","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf"},{"title":"Randomized quasipolynomial-time mean-payoff games","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/104.mp4","poster":"/films/math/104-poster.webp","captions":"/films/math/104.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":2,"why":"Quasipolynomial algorithms for mean-payoff, stochastic and parity games; polynomial time is still open. Partial Lean.","consequence":"Faster theoretical algorithms for games used in verification and controller synthesis; no practical change claimed."},"detail":"/api/math/104"},{"id":"105","title":"Perfect completeness for 2-to-1 games","short":"Perfect completeness for 2-to-1 games","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every fixed rational delta in (0,1) there is q(delta) and a deterministic polynomial-time reduction from 3-SAT to explicit unweighted 2-to-1 games with alphabets [2q],[q] (every right label has exactly two preimages under every constraint) such that…","verdict":"The paper's own external inputs are Grassmann expansion (KMS), parallel repetition and a perfect-completeness PCP;","url":"/math#105","articleUrl":"/articles/openai-math#theoretical-computer-science","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=105.%20Perfect%20completeness%20for","manuscripts":[{"title":"Perfect completeness for 2-to-1 games","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/105.mp4","poster":"/films/math/105-poster.webp","captions":"/films/math/105.vtt","duration":20.5,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"2-to-1 games are hard even with perfect completeness, Khot's strongest d-to-1 form; extends a nine-day-old human result.","consequence":"Hardness for coloring-type problems from perfect completeness, extending very recent work."},"detail":"/api/math/105"},{"id":"106","title":"Hardness of coloring three-colorable graphs","short":"Colouring three-colourable graphs","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every fixed 0<delta<1/3 there is a deterministic polynomial-time reduction from 3SAT to simple unweighted graphs that are 3-colorable when the formula is satisfiable and have independence number < delta*n otherwise (Theorem 1.1).","verdict":"The family title and summary ('It is NP-hard to color a three-colorable graph using any fixed number c>=3 of colors') present as new a corollary that Fei-Minzer-Wang…","url":"/math#106","articleUrl":"/articles/openai-math#theoretical-computer-science","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=106.%20Hardness%20of%20coloring","manuscripts":[{"title":"Hardness of finding large independent sets in three-colorable graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/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"}],"reel":{"src":"/films/math/106.mp4","poster":"/films/math/106-poster.webp","captions":"/films/math/106.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":3,"advance":1,"consequences":1,"surprise":1,"confidence":3,"why":"Coloring 3-colorable graphs with any fixed number of colors is NP-hard, but that part was published days earlier by others.","consequence":"A stronger hardness result for coloring; the headline corollary was already known."},"detail":"/api/math/106"},{"id":"107","title":"Matrix multiplication with exponent at most 9/4","short":"Matrix multiplication, $\\omega \\le 9/4$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Three claims in three preprints. (1) 'An Upper Bound of 9/4 for the Matrix Multiplication Exponent' (2 Oct 2026, 13 pp.): over C, for every eps > 0 two n x n matrices can be multiplied with O_eps(n^{9/4+eps}) arithmetic operations, i.e.","verdict":"(a) Over C/char 0 only for 9/4; the every-field claim is only 2.371054886, an improvement of 0.000122 over Dupont et al.'s 2.371177.","url":"/math#107","articleUrl":"/articles/openai-math#matrix-multiplication-ω--94-in-thirteen-pages","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=107.%20Matrix%20multiplication%20with","manuscripts":[{"title":"An Upper Bound of 9/4 for the Matrix Multiplication Exponent","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf"},{"title":"Complex Matrix Multiplication Below 2.258 and Rectangular Bounds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026.pdf"},{"title":"Staggered extraction for exact matrix multiplication over every field","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026.pdf"}],"reel":{"src":"/films/math/107.mp4","poster":"/films/math/107-poster.webp","captions":"/films/math/107.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":92,"tier":"Huge if true","importance":4,"advance":3,"consequences":4,"surprise":3,"confidence":3,"why":"Matrix multiplication exponent ω ≤ 9/4, a bigger drop than the last 36 years combined, in 13 pages; Lean checks the bound.","consequence":"Asymptotically faster algorithms for everything bottlenecked by matrix multiplication; non-constructive, so no practical algorithm yet."},"detail":"/api/math/107"},{"id":"108","title":"A cubic permanent–determinant lower bound","short":"A cubic permanent–determinant bound","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"The border determinantal complexity of the m x m permanent over C is at least m^3/(5529600e) for m>=1408 (Theorem 1.1), i.e. even coefficientwise limits of determinants of n x n affine-linear matrices that equal perm_m need n = Omega(m^3);","verdict":"Still polynomial; says nothing about VP vs VNP directly. The paper uses deep external algebraic geometry (Christ-He-Tyomkin arbitrary-characteristic Severi dimension…","url":"/math#108","articleUrl":"/articles/openai-math#a-cubic-lower-bound-for-the-permanent","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=108.%20A%20cubic%20permanent","manuscripts":[{"title":"A cubic lower bound for border determinantal complexity of the permanent","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf"}],"reel":{"src":"/films/math/108.mp4","poster":"/films/math/108-poster.webp","captions":"/films/math/108.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"The permanent needs determinants of cubic size, after 20 years stuck at quadratic; VP vs VNP stays open. Lean checks it.","consequence":"The best lower bound toward VP vs VNP, though still far from separating them."},"detail":"/api/math/108"},{"id":"109","title":"Integer multiplication below nlog n","short":"Integer multiplication below $n \\log n$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"One deterministic multitape Turing machine (fixed finite alphabet, fixed number of 1-D tapes) multiplies two n-bit integers exactly in O(n (lg n)^{1-kappa}) worst-case time with kappa = 2^{-182}, refuting an Omega(n log n) lower bound in this model;","verdict":"kappa=2^{-182} makes the saving (lg n)^{2^{-182}} numerically indistinguishable from 1 for any physical n; the paper says constants and thresholds are 'extremely large'.","url":"/math#109","articleUrl":"/articles/openai-math#what-was-actually-open","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=109.%20Integer%20multiplication%20below","manuscripts":[{"title":"Integer multiplication below n log n","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/109.mp4","poster":"/films/math/109-poster.webp","captions":"/films/math/109.vtt","duration":20.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":2,"consequences":1,"surprise":3,"confidence":0,"why":"Integer multiplication slightly below n log n on multitape machines; the saving is far too small ever to matter. No Lean.","consequence":"n log n is not the floor for multiplication on Turing machines; no practical effect."},"detail":"/api/math/109"},{"id":"110","title":"Optimal-order randomized k-server on arbitrary metrics","short":"Randomized $k$-server at $O(\\log^2 k)$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is an absolute C such that for every k>=2, every metric space with >= k+1 points (finite or not) and every start configuration, a single randomized online policy has expected cost <= C (log(k+1))^2 OPT + B against oblivious adversaries, B depending on…","verdict":"Existence policy has no efficiency claim; the uniform version has polynomial per-request time only on finite rational metrics and an additive constant that 'may be…","url":"/math#110","articleUrl":"/articles/openai-math#randomized-k-server-at-olog2-k-on-every-metric-110","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=110.%20Optimal%2Dorder%20randomized%20k%2Dserver","manuscripts":[{"title":"Squared-logarithmic randomized k-server on arbitrary metrics","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026/Squared-logarithmic-randomized-k-server-on-arbitrary-metrics-September-24-2026.pdf"},{"title":"Uniform computation of the squared-logarithmic k-server bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026/Uniform-computation-of-the-squared-logarithmic-k-server-bound-September-24-2026.pdf"}],"reel":{"src":"/films/math/110.mp4","poster":"/films/math/110-poster.webp","captions":"/films/math/110.vtt","duration":20.5,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"Randomized k-server achieves the optimal log² k on every metric; existence only, no efficient algorithm. Lean checks it.","consequence":"The optimal competitive ratio for randomized k-server on any metric, as an existence result."},"detail":"/api/math/110"},{"id":"111","title":"One-sample matroid prophet inequalities against an almighty adversary","short":"One-sample matroid prophet inequalities","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"A distribution-free online rule that sees one independent sample per element achieves expected reward >= 2^{-310} E[OPT] on every finite matroid, with independent nonnegative values and integrable optimum, even if the arrival order is chosen by an adversary…","verdict":"Constant 2^{-310} is purely existential; no polynomial-time or oracle-efficient implementation is asserted.","url":"/math#111","articleUrl":"/articles/openai-math#notable-results-briefly","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=111.%20One%2Dsample%20matroid%20prophet","manuscripts":[{"title":"One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf"}],"reel":{"src":"/films/math/111.mp4","poster":"/films/math/111-poster.webp","captions":"/films/math/111.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"With one sample per item, a constant fraction of the prophet's value on any matroid; the constant is 2^-310.","consequence":"Constant-factor posted-price mechanisms from one sample; the constant is far too small to use."},"detail":"/api/math/111"},{"id":"112","title":"Beyond the square-root exponent for depth-three circuits","short":"Depth-three circuits beyond $2^{c\\sqrt n}$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is one language decidable in deterministic polynomial time whose n-bit slice requires more than 2^{A sqrt(n)} gates in unbounded fan-in OR-AND-OR circuits, for every constant A and all n >= N_A (log2 size / sqrt(n) -> infinity).","verdict":"Superconstant improvement of the exponent's multiplier only (omega(sqrt n)), not n^{1/2+eps};","url":"/math#112","articleUrl":"/articles/openai-math#depth-3-circuits-beyond-2csqrt-n-112","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=112.%20Beyond%20the%20square%2Droot","manuscripts":[{"title":"Beyond the Square-Root Exponent for Depth-Three Boolean Circuits","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf"}],"reel":{"src":"/films/math/112.mp4","poster":"/films/math/112-poster.webp","captions":"/films/math/112.vtt","duration":20.5,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"Depth-3 circuit lower bounds past the 2^√n barrier by a growing factor in the exponent; Lean checks it.","consequence":"The first superconstant gain past the depth-3 barrier, on a known route to bigger circuit lower bounds."},"detail":"/api/math/112"},{"id":"113","title":"Approximate counting and entropy of perfect matchings","short":"Counting perfect matchings in any graph","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"An FPRAS for the number of perfect matchings in arbitrary finite simple graphs, with zero detection and worst-case polynomial bit time in input length, 1/eps and log(1/delta).","verdict":"Polynomial bounds are not quantified as practical. The sampler uses a new state space (product of perfect-matching spaces on an enlarged colored graph) precisely to…","url":"/math#113","articleUrl":"/articles/openai-math#counting-perfect-matchings-in-any-graph","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=113.%20Approximate%20counting%20and","manuscripts":[{"title":"A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf"},{"title":"Entropy and Face Dimension of the Perfect-Matching Polytope","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf"}],"reel":{"src":"/films/math/113.mp4","poster":"/films/math/113-poster.webp","captions":"/films/math/113.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":82,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":2,"confidence":3,"why":"Perfect matchings in any graph can be approximately counted in polynomial time, open since 2001 for non-bipartite graphs.","consequence":"Polynomial-time approximate counting and sampling of perfect matchings in every graph: dimer models, 0/1 hafnians."},"detail":"/api/math/113"},{"id":"114","title":"Approximate counting of common integer polymatroid bases","short":"Counting common bases of two matroids","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"FPRAS (polynomial oracle calls and bit operations on every execution) for the number of common bases of two equal-rank matroids given by independence oracles;","verdict":"Oracle model; polynomial degrees unspecified. Builds on machinery shared with 113. No public expert assessment of this specific result found as of 2026-10-07;","url":"/math#114","articleUrl":"/articles/openai-math#counting-perfect-matchings-in-any-graph","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=114.%20Approximate%20counting%20of","manuscripts":[{"title":"An FPRAS for Common Integer Polymatroid Bases with Binary Capacities","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/polymatroid-fpras.pdf"},{"title":"Approximate counting of common bases of two matroids","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/main.pdf"}],"reel":{"src":"/films/math/114.mp4","poster":"/films/math/114-poster.webp","captions":"/films/math/114.vtt","duration":20.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":2,"why":"Approximate counting of common bases of two matroids, unifying matching and log-concave methods; Lean covers two matroids.","consequence":"Approximate counting of common bases of two matroids, unifying known samplers."},"detail":"/api/math/114"},{"id":"115","title":"Sampling and counting contingency tables with arbitrary margins","short":"Contingency tables with arbitrary margins","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For nonnegative integer matrices with prescribed binary-encoded row and column sums (both dimensions growing): exact uniform sampling in expected polynomial bit time and 2^{-k}-TV sampling in worst-case polynomial time;","verdict":"Exponents are 'deliberately large'; exact sampler polynomial only in expectation. No public expert assessment of this specific result found as of 2026-10-07;","url":"/math#115","articleUrl":"/articles/openai-math#counting-perfect-matchings-in-any-graph","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=115.%20Sampling%20and%20counting","manuscripts":[{"title":"Exact Uniform Sampling of Contingency Tables with Arbitrary Margins","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf"},{"title":"An FPRAS for Cell-Bounded Contingency Tables","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/115.mp4","poster":"/films/math/115-poster.webp","captions":"/films/math/115.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"Provably fast sampling of contingency tables with any margins, the null model behind Fisher's exact test; Lean checks it.","consequence":"A provably polynomial sampler for contingency tables, the null model behind Fisher's exact test; exponents are large."},"detail":"/api/math/115"},{"id":"116","title":"Uniform black-box noncommutative identity testing across characteristics","short":"Black-box noncommutative identity testing","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"Deterministic polynomial-time construction of one rational matrix tuple of dimension <= 2ns^2 on which every nonzero size-s noncommutative formula in n variables over any characteristic-zero field is nonzero;","verdict":"The formalized one-point theorem omits the construction-time and dimension bounds. Matrix entries have bit length O(n s^2 log n), so a single test is polynomial but…","url":"/math#116","articleUrl":"/articles/openai-math#notable-results-briefly","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=116.%20Uniform%20black%2Dbox%20noncommutative","manuscripts":[{"title":"Uniform Matrix Hitting Points in Every Positive Characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026/uniform-matrix-hitting-points-positive-characteristic.pdf"},{"title":"One Rational Matrix Hitting Point for Noncommutative Formulas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026.pdf"},{"title":"Polynomial Hitting Lists for Noncommutative Rational Formulas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026.pdf"}],"reel":{"src":"/films/math/116.mp4","poster":"/films/math/116-poster.webp","captions":"/films/math/116.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"One explicit set of matrices certifies every small noncommutative formula is nonzero, in any characteristic.","consequence":"Black-box identity testing for noncommutative formulas. Internal to derandomization."},"detail":"/api/math/116"},{"id":"117","title":"Uniform sparsest cut: hardness and semidefinite gaps","short":"Uniform Sparsest Cut: no constant factor","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"For every fixed C>1, approximating Uniform Sparsest Cut (unit demand on every pair, nonnegative rational capacities) within factor C is NP-hard via a direct reduction from 3SAT (independent of the UGC paper).","verdict":"Hardness is ordinary NP-hardness for each fixed C, with polynomial degree depending on C. The headline NP-hardness (the larger claim) is the unformalized half.","url":"/math#117","articleUrl":"/articles/openai-math#uniform-sparsest-cut-no-constant-factor-117","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=117.%20Uniform%20sparsest%20cut","manuscripts":[{"title":"Constant-factor hardness of uniform sparsest cut","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026/Constant-factor-hardness-of-uniform-sparsest-cut-September-24-2026.pdf"},{"title":"Near-square-root logarithmic integrality gaps for uniform sparsest cut","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026.pdf"}],"reel":{"src":"/films/math/117.mp4","poster":"/films/math/117-poster.webp","captions":"/films/math/117.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":2,"confidence":1,"why":"Uniform sparsest cut has no constant-factor algorithm unless P=NP; Lean checks only the SDP gap half.","consequence":"The ARV algorithm is near-optimal for uniform sparsest cut, so graph partitioning has a firm approximation limit."},"detail":"/api/math/117"},{"id":"118","title":"Bin packing and unbounded configuration-LP gaps","short":"Bin packing: unbounded configuration-LP gaps","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every c>=0 there is a rational bin-packing instance with all items > 1/6 whose configuration LP (both size-type and individual-copy versions) has value B while the optimum exceeds B+c, disproving the Modified Integer Round-Up Conjecture;","verdict":"Items have rational sizes with at most five per bin; instance sizes and encoding lengths grow with c.","url":"/math#118","articleUrl":"/articles/openai-math#bin-packing-no-algorithm-is-within-a-constant-number-of-bins-118","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=118.%20Bin%20packing%20and","manuscripts":[{"title":"Additive hardness and unbounded configuration gaps in bin packing","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026.pdf"}],"reel":{"src":"/films/math/118.mp4","poster":"/films/math/118-poster.webp","captions":"/films/math/118.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":3,"confidence":3,"why":"Bin packing's configuration LP can be off by any number of bins, refuting the conjectured one-bin gap; Lean checks it.","consequence":"Bin packing cannot be solved within any fixed number of bins efficiently; the cutting-stock LP's gap is unbounded."},"detail":"/api/math/118"},{"id":"119","title":"The Courtade–Kumar and Hellinger conjectures","short":"The Courtade–Kumar conjecture","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"For X uniform on {-1,1}^n, Y its image through independent BSC(eps) noise, and any Boolean f, I(f(X);Y) <= 1 - h_2(eps) bits, attained by a coordinate (Courtade-Kumar);","verdict":"The Hellinger proof relies on 'finite exact arithmetic certificates' (computer-checked numerics) in the paper; Lean covers the main Courtade-Kumar result anyway.","url":"/math#119","articleUrl":"/articles/openai-math#the-courtadekumar-conjecture","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=119.%20The%20Courtade%20Kumar","manuscripts":[{"title":"Sharp binary-information contraction on the discrete cube","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026/main.pdf"},{"title":"Hellinger contraction with arbitrary Boolean output bias","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/119.mp4","poster":"/films/math/119-poster.webp","captions":"/films/math/119.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":2,"why":"Courtade–Kumar: sending one coordinate is the best one-bit summary through a noisy channel; Lean checks the main result.","consequence":"Settles a basic information-theory question. Little follow-on."},"detail":"/api/math/119"},{"id":"120","title":"Almost-linear-time exact matching and prescribed-degree factors in general graphs","short":"Maximum matching in almost-linear time","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"A uniform randomized word-RAM algorithm finds a maximum-cardinality matching in any simple graph with n vertices and m edges in (n+m)^{1+o(1)} time on every computation path, correct with probability >= 2/3;","verdict":"Monte Carlo (success 2/3, no certificate mentioned in the abstract); o(1) exponent loss unquantified and likely galactic, as for the almost-linear max-flow methods it…","url":"/math#120","articleUrl":"/articles/openai-math#maximum-matching-in-almost-linear-time-120","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=120.%20Almost%2Dlinear%2Dtime%20exact%20matching","manuscripts":[{"title":"Almost-Linear-Time Maximum-Cardinality Matching in General Graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Almost-Linear-Time-Maximum-Cardinality-Matching-in-Sparse-General-Graphs-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/120.mp4","poster":"/films/math/120-poster.webp","captions":"/films/math/120.vtt","duration":20.1,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":82,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":2,"confidence":0,"why":"Maximum matching in general graphs in almost-linear time, after 45 years at m√n; Monte Carlo and unformalized.","consequence":"Near-linear-time maximum matching in any graph, in theory; practical matching codes are unaffected for now."},"detail":"/api/math/120"},{"id":"121","title":"Almost-linear approximation of edit distance","short":"$(1+\\varepsilon)$ edit distance, almost linear","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every fixed rational eps in (0,1), a randomized algorithm returns D with ED = 2/3, in worst-case expected time N^{1+o(1)} on a log-word RAM, for strings of total length N over polynomially bounded integer alphabets;","verdict":"Asymptotic only: the algorithm runs exact DP below an accuracy-dependent length threshold the paper calls 'very large'; no polynomial dependence on 1/eps is claimed.","url":"/math#121","articleUrl":"/articles/openai-math#1varepsilon-edit-distance-in-almost-linear-time-121","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=121.%20Almost%2Dlinear%20approximation%20of","manuscripts":[{"title":"An Almost-Linear Approximation Scheme for Edit Distance","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/121.mp4","poster":"/films/math/121-poster.webp","captions":"/films/math/121.vtt","duration":20.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"Edit distance approximated to any fixed accuracy in almost-linear time; thresholds are huge. Lean checks it.","consequence":"Near-linear (1+ε) edit distance in theory; aligners are unaffected because the thresholds are huge."},"detail":"/api/math/121"},{"id":"122","title":"Quantitative trace-reconstruction bounds with a uniform decoder","short":"Trace reconstruction: superpolynomial","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"For every fixed deletion probability q in (0,1), exact worst-case reconstruction of an n-bit string from i.i.d. deletion traces needs n^{Omega(log log n)} traces (more precisely n^{c log(q^3 log n)} for c<1/(4 log 2)), so polynomially many never suffice.","verdict":"Lower and upper bounds are both new and both enormous jumps (from n^{1.5} to n^{log log n} below, and from exp(n^{1/5}) to quasipolynomial above);","url":"/math#122","articleUrl":"/articles/openai-math#trace-reconstruction-superpolynomial-below-quasipolynomial-above-122","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=122.%20Quantitative%20trace%2Dreconstruction%20bounds","manuscripts":[{"title":"Uniform quasipolynomial-time trace reconstruction","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-quasipolynomial-time-trace-reconstruction-October-5-2026/uniform-trace-reconstruction.pdf"},{"title":"A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/paper.pdf"},{"title":"Quantitative lower bounds for trace reconstruction","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/quantitative-lower-bounds-for-trace-reconstruction-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/122.mp4","poster":"/films/math/122-poster.webp","captions":"/films/math/122.vtt","duration":20.3,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":1,"why":"Worst-case trace reconstruction needs between n^{log log n} and quasipolynomially many traces; Lean covers the lower bound.","consequence":"Ends hopes of polynomial worst-case trace reconstruction, a toy model for DNA storage."},"detail":"/api/math/122"},{"id":"124","title":"Polynomial-time scheduling on three identical machines","short":"Three machines, unit jobs: in P","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"A deterministic algorithm solves P3|prec,p_j=1|C_max exactly: for unit jobs with arbitrary precedence DAG on three identical machines it computes a minimum-makespan schedule and decides deadline feasibility, in O((L+2)^150020) multitape-TM steps.","verdict":"Exponent 150020 makes this purely a classification result. The paper says it resolves 'the polynomial-time side' of the question, i.e. it puts the problem in P;","url":"/math#124","articleUrl":"/articles/openai-math#three-machines-unit-jobs-precedence-constraints-in-p","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=124.%20Polynomial%2Dtime%20scheduling%20on","manuscripts":[{"title":"A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/124.mp4","poster":"/films/math/124-poster.webp","captions":"/films/math/124.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"Unit jobs with precedences on three machines are schedulable in polynomial time, a Garey–Johnson open problem; Lean checks it.","consequence":"Puts a Garey–Johnson problem in P; the n^150020 algorithm is unusable."},"detail":"/api/math/124"},{"id":"125","title":"The metric k-median approximation threshold and recovery","short":"k-median at exactly $1 + 2/e$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every fixed eps>0, a deterministic polynomial-time (1+2/e+eps)-approximation for metric k-median with specified candidate facilities (opening at most k).","verdict":"Tightness is for the model with specified candidate facilities; the F=J (facilities at client locations) variant has a different lower-bound question.","url":"/math#125","articleUrl":"/articles/openai-math#k-median-at-exactly-1--2e-125","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=125.%20The%20metric%20k%2Dmedian","manuscripts":[{"title":"Single-exponential recovery and bounded-price strictness for metric k-median","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/paper.pdf"},{"title":"The approximation threshold for metric k-median","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/125.mp4","poster":"/films/math/125-poster.webp","captions":"/films/math/125.vtt","duration":20.3,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"A polynomial algorithm reaches the 1+2/e hardness floor for metric k-median, closing its approximability; Lean checks it.","consequence":"Closes k-median's approximability with an algorithm matching the hardness floor."},"detail":"/api/math/125"},{"id":"126","title":"Exponential semidefinite complexity of perfect matching","short":"No small SDP for perfect matching","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"For every fixed 0<rho<1 the odd-cut-vs-perfect-matching slack matrix shifted by rho, entries |M cap delta(U)|-1+rho, has real PSD rank 2^{Omega(n)};","verdict":"The Lean-checked statement is weaker (superpolynomial) than the paper's exponential claim.","url":"/math#126","articleUrl":"/articles/openai-math#no-small-semidefinite-program-for-perfect-matching-126","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=126.%20Exponential%20semidefinite%20complexity","manuscripts":[{"title":"Exponential PSD rank of positively shifted matching matrices","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026/shifted-matching-psd.pdf"}],"reel":{"src":"/films/math/126.mp4","poster":"/films/math/126-poster.webp","captions":"/films/math/126.vtt","duration":20.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"No polynomial-size semidefinite program describes the matching polytope; Lean checks a weaker superpolynomial form.","consequence":"Rules out small SDPs for matching. Internal to extension complexity."},"detail":"/api/math/126"},{"id":"127","title":"Average sensitivity of polynomial threshold functions","short":"The Gotsman–Linial bound","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every degree-<=d polynomial threshold function on {-1,1}^n (sgn(0)=1) has average sensitivity (total influence) at most 8 d sqrt(n), uniformly for 1<=d<=n: the asymptotic Gotsman-Linial conjecture with an absolute constant.","verdict":"Proves the asymptotic O(d sqrt n) form, not the exact extremizer statement (which is false). Short paper plus Lean proof makes it one of the more checkable entries.","url":"/math#127","articleUrl":"/articles/openai-math#the-gotsmanlinial-bound-127","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=127.%20Average%20sensitivity%20of","manuscripts":[{"title":"Average sensitivity of polynomial threshold functions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/127.mp4","poster":"/films/math/127-poster.webp","captions":"/films/math/127.vtt","duration":19.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"Gotsman–Linial: degree-d threshold functions have average sensitivity O(d√n), open 30 years; 13 Lean-checked pages.","consequence":"Agnostic learning and noise-sensitivity bounds for polynomial threshold functions."},"detail":"/api/math/127"},{"id":"128","title":"A factor-two approximation for shortest common superstring","short":"Shortest common superstring within 2","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"A deterministic polynomial-time algorithm outputs, for every finite family of explicitly encoded strings, a common superstring of length at most 2*OPT.","verdict":"Does not resolve the Greedy conjecture; the paper itself cites a September 2026 preprint claiming Greedy is not a 2-approximation (unverified).","url":"/math#128","articleUrl":"/articles/openai-math#shortest-common-superstring-within-factor-2-128","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=128.%20A%20factor%2Dtwo%20approximation","manuscripts":[{"title":"A Polynomial-Time 2-Approximation for Shortest Common Superstring","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/128.mp4","poster":"/films/math/128-poster.webp","captions":"/films/math/128.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"A factor-2 algorithm for shortest common superstring, the long-sought target; the Greedy conjecture stays open.","consequence":"A 2-approximation for superstrings; genome assemblers use different objectives."},"detail":"/api/math/128"},{"id":"129","title":"Exponential state costs for two-way automata","short":"Sakoda–Sipser: exponential state costs","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(i) For each n>=4 an n-state two-way nondeterministic automaton (2NFA) over a finite (growing) alphabet whose complement needs >= (1/2)2^{floor((n-4)/127)} - 1 states (2NFA complementation is exponential); same family needs exponential 2DFA states.","verdict":"Alphabet size grows (exponentially) with the number of states, which is the standard Sakoda-Sipser setting, but the hard inputs are presumably long: since an exponential…","url":"/math#129","articleUrl":"/articles/openai-math#sakodasipser-two-way-automata-need-exponentially-many-states","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=129.%20Exponential%20state%20costs","manuscripts":[{"title":"An exponential state lower bound for two-way nondeterministic complementation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026/paper.pdf"},{"title":"An exponential two-way deterministic state lower bound for one-way liveness","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/129.mp4","poster":"/films/math/129-poster.webp","captions":"/films/math/129.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Two-way deterministic automata need exponentially more states than nondeterministic ones: Sakoda–Sipser (1978). Lean checks it.","consequence":"Settles descriptional complexity's flagship question, with no complexity-class consequence."},"detail":"/api/math/129"},{"id":"130","title":"Exact Fourier transforms below nlog n","short":"Exact Fourier transforms below $n \\log n$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"Two papers. (1) 'An explicit power saving for the exact discrete Fourier transform' (31 pp): a single deterministic algorithm computes F_n x exactly for every n in O(n (log n)^theta (log log n)^(4-theta)) operations, theta = log_m(m - Delta/2^71), m = 10^6,…","verdict":"Model is the unrestricted (unbounded-coefficient, exact) linear/arithmetic model; consistent with Morgenstern and Ailon because it violates their hypotheses…","url":"/math#130","articleUrl":"/articles/openai-math#the-fft-is-not-optimal-in-the-right-model-by-a-hair","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=130.%20Exact%20Fourier%20transforms","manuscripts":[{"title":"An explicit power saving for the exact discrete Fourier transform","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf"},{"title":"Finite tensor savings and exact Fourier circuits","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/130.mp4","poster":"/films/math/130-poster.webp","captions":"/films/math/130.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":2,"consequences":1,"surprise":3,"confidence":2,"why":"The FFT's n log n is beatable with exact complex constants, by a factor of n^(10^-13); Lean checks a weaker form.","consequence":"The FFT is not optimal in the unrestricted model; FFTW and cuFFT are unaffected."},"detail":"/api/math/130"},{"id":"131","title":"Rapid mixing of graph switches for every degree sequence","short":"Switch chain: rapid mixing, every sequence","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every graphical labeled degree sequence on n>=4 vertices, the lazy switch chain (pick 4 vertices, swap a perfect matching) on simple graphs mixes in at most 2n^8 steps (TV 1/4), with spectral gap >= 1/(24 n^2 C(n,4));","verdict":"n^8 is a worst-case bound; practical mixing is believed much faster. Simple undirected case only (directed/bipartite variants of KTV not claimed).","url":"/math#131","articleUrl":"/articles/openai-math#the-switch-chain-mixes-for-every-degree-sequence-131","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=131.%20Rapid%20mixing%20of","manuscripts":[{"title":"Polynomial mixing of the switch chain for every graphical degree sequence","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/131.mp4","poster":"/films/math/131-poster.webp","captions":"/films/math/131.vtt","duration":20.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":3,"why":"The edge-switch chain used to sample graphs with given degrees mixes in polynomial time for every sequence; Lean checks it.","consequence":"Justifies the switch-chain sampling network scientists use for degree-sequence null models; the bound is too loose to set step counts."},"detail":"/api/math/131"},{"id":"132","title":"A superquadratic separation of sensitivity and block sensitivity","short":"Block sensitivity beats sensitivity squared","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Total Boolean functions with bs(f) >= s(f)^alpha for a fixed alpha>2 and unbounded bs, so no bound bs(f) = 2^d/(4(d+2)^2) for every d.","verdict":"alpha is some fixed constant >2, not specified as large; gap between s^{2+} and Huang's s^4 remains.","url":"/math#132","articleUrl":"/articles/openai-math#block-sensitivity-beats-sensitivity-squared-132","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=132.%20A%20superquadratic%20separation","manuscripts":[{"title":"A superquadratic separation between sensitivity and block sensitivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/132.mp4","poster":"/films/math/132-poster.webp","captions":"/films/math/132.vtt","duration":20.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":3,"confidence":3,"why":"Block sensitivity can exceed sensitivity squared, against the natural guess; an 11-page, Lean-checked construction.","consequence":"Sharpens the sensitivity picture after Huang's theorem. Internal."},"detail":"/api/math/132"},{"id":"133","title":"The computational complexity of Weisfeiler–Leman refinement","short":"The complexity of Weisfeiler–Leman","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every sufficiently large fixed k, deciding k-WL equivalence of two n-vertex graphs needs n^{ck} deterministic time (multitape TM or log-word RAM) unconditionally, even for connected diameter-2 graphs;","verdict":"The unconditional lower bound is the time-hierarchy type (it follows from EXPTIME-hardness with k in the input plus padding), so it is real but of the 'diagonalization'…","url":"/math#133","articleUrl":"/articles/openai-math#notable-results-briefly","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=133.%20The%20computational%20complexity","manuscripts":[{"title":"Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf"},{"title":"The complexity of identifying a graph by Weisfeiler–Leman refinement","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026/paper.pdf"},{"title":"Unconditional time lower bounds for Weisfeiler–Leman equivalence","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026/paper.pdf"},{"title":"Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Variable-dimension-Weisfeiler-Leman-equivalence-on-general-and-subcubic-graphs-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/133.mp4","poster":"/films/math/133-poster.webp","captions":"/films/math/133.vtt","duration":20.5,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":30,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":0,"confidence":3,"why":"The n^k cost of k-dimensional Weisfeiler–Leman is unavoidable, by a diagonalization-style bound; Lean checks it.","consequence":"Higher-order Weisfeiler–Leman, and so higher-order GNNs, cannot avoid the n^k cost. Internal."},"detail":"/api/math/133"},{"id":"134","title":"Generalized star height at most three","short":"Generalized star height at most three","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every regular language over a finite alphabet has a generalized regular expression (union, concatenation, complement in the same free monoid, Kleene star) of star height at most 3. Earlier papers in the family give bounds 13 and 4. No bound on expression size.","verdict":"Does not decide whether height 1 suffices (the headline generalized star-height problem); it proves a uniform bound of 3. Expression size unbounded.","url":"/math#134","articleUrl":"/articles/openai-math#generalized-star-height-at-most-three-134","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=134.%20Generalized%20star%20height","manuscripts":[{"title":"Finite Monoid Computations and a Uniform Generalized Star-Height Bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026.pdf"},{"title":"Generalized Star Height at Most Four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Star-Height-at-Most-Four-September-25-2026/Generalized-Star-Height-at-Most-Four-September-25-2026.pdf"},{"title":"Generalized Star Height at Most Three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Star-Height-at-Most-Three-September-25-2026/article.pdf"}],"reel":{"src":"/films/math/134.mp4","poster":"/films/math/134-poster.webp","captions":"/films/math/134.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Three nested stars suffice for any regular language with complement; whether one suffices stays open. Lean checks it.","consequence":"Bounds generalized star height by 3; whether 1 suffices stays open."},"detail":"/api/math/134"},{"id":"135","title":"Homogeneous depth-five lower bounds for iterated matrix multiplication","short":"Homogeneous depth-five circuits for IMM","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"Over every characteristic-zero field, every syntactically homogeneous Sigma-Pi-Sigma-Pi-Sigma circuit computing IMM_{n,n} (entry (1,1) of a product of n generic n x n matrices) has at least n^{sqrt(n)/400} gates for large n, with arbitrary bottom linear…","verdict":"Homogeneity is syntactic and essential; general (non-homogeneous) depth-5 is not covered.","url":"/math#135","articleUrl":"/articles/openai-math#notable-results-briefly","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=135.%20Homogeneous%20depth%2Dfive%20lower","manuscripts":[{"title":"Homogeneous depth-five lower bounds for iterated matrix multiplication","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026.pdf"}],"reel":{"src":"/films/math/135.mp4","poster":"/films/math/135-poster.webp","captions":"/films/math/135.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Homogeneous depth-5 circuits for iterated matrix multiplication need size n^Θ(√n); Lean checks both bounds.","consequence":"Sharp homogeneous depth-5 bounds. Internal to algebraic complexity."},"detail":"/api/math/135"},{"id":"136","title":"A quasilinear PCP theorem for PPAD","short":"A quasilinear PCP theorem for PPAD","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"There are fixed rational eps, delta>0 and a deterministic polynomial-time reduction from End-of-Line instances of length N to generalized circuits of total length N (log N)^{O(1)} such that any polynomially encoded rational assignment eps-satisfying all but a…","verdict":"Longest TCS manuscript in the release (147 pages), unformalized. Under ETH for PPAD it implies stronger lower bounds for approximate Nash, but those consequences stay…","url":"/math#136","articleUrl":"/articles/openai-math#pcp-for-ppad-136","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=136.%20A%20quasilinear%20PCP","manuscripts":[{"title":"The PCP-for-PPAD conjecture: a quasilinear reduction","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/136.mp4","poster":"/films/math/136-poster.webp","captions":"/films/math/136.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"A PCP theorem for PPAD with quasilinear blow-up, so approximate equilibria stay hard; 147 pages, no Lean.","consequence":"Robust PPAD-hardness for approximate Nash equilibria; stronger bounds follow under ETH."},"detail":"/api/math/136"},{"id":"137","title":"One-tape time simulation in two-fifths-power space","short":"One-tape time in $T^{2/5}$ space","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"A fixed deterministic machine with one writable tape/head (plus fixed read-only input heads, unit moves) can be simulated up to a supplied time cap T in O(T^{2/5} polylog T) work space, computing the halting and finite-control outcome;","verdict":"Model-specific (one writable tape, supplied cap, accessor condition); answers Williams's question only for that model; unformalized.","url":"/math#137","articleUrl":"/articles/openai-math#notable-results-briefly","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=137.%20One%2Dtape%20time%20simulation","manuscripts":[{"title":"Simulating One-Tape Time in Two-Fifths-Power Space","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Simulating-One-Tape-Time-in-Two-Fifths-Power-Space-September-25-2026/article.pdf"}],"reel":{"src":"/films/math/137.mp4","poster":"/films/math/137-poster.webp","captions":"/films/math/137.vtt","duration":19.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":38,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":2,"confidence":0,"why":"One-tape machines running time t can be simulated in space t^{2/5}, below the square-root barrier; unformalized.","consequence":"Suggests the square root is not the space limit for simulating time; one-tape model only."},"detail":"/api/math/137"},{"id":"138","title":"Subset Sum in O(2^(0.49n)) time","short":"Subset Sum in $O(2^{0.49n})$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"A uniform randomized algorithm solves worst-case Subset Sum on n polynomial-bit integers in O(2^{0.49n}) word-RAM time on every execution (intermediate bound 2^{0.489995n} poly), success >= 2/3.","verdict":"Unformalized; word size 4(n+b+log n) bits is generous (Theta(n)-bit words) though standard for this problem; savings of 2^{0.01n} only.","url":"/math#138","articleUrl":"/articles/openai-math#subset-sum-below-2n2","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=138.%20Subset%20Sum%20in","manuscripts":[{"title":"Subset Sum in Time O(2^(0.49n))","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/subset-sum.pdf"},{"title":"A Low-Space Algorithm for Worst-Case Subset Sum","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf"}],"reel":{"src":"/films/math/138.mp4","poster":"/films/math/138-poster.webp","captions":"/films/math/138.vtt","duration":20.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":0,"why":"Worst-case Subset Sum in 2^{0.49n}, the first break of the 1974 meet-in-the-middle barrier, by a hair; no Lean.","consequence":"A worst-case Subset Sum speed-up, with knock-on gains for related knapsack problems; cryptography unaffected."},"detail":"/api/math/138"},{"id":"139","title":"Subpolynomial query complexity for log-concave sampling","short":"Log-concave sampling in $d^\\varepsilon$ queries","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For C^2 potentials V on R^d with V(0)=0, grad V(0)=0 and I 0, and at least c log d queries; so the optimal dimension exponent is 0.","verdict":"Query complexity only: computation between queries is unbounded and the construction uses approximation orders depending on eps, so C_eps is enormous and running time is…","url":"/math#139","articleUrl":"/articles/openai-math#gradient-queries-for-log-concave-sampling-dvarepsilon-139","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=139.%20Subpolynomial%20query%20complexity","manuscripts":[{"title":"Subpolynomial query complexity for well-conditioned log-concave sampling","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026/article.pdf"}],"reel":{"src":"/films/math/139.mp4","poster":"/films/math/139-poster.webp","captions":"/films/math/139.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Sampling log-concave distributions needs only d^ε gradient queries, though computation between queries is unbounded.","consequence":"An information-theoretic query bound for sampling; Bayesian samplers are unaffected."},"detail":"/api/math/139"},{"id":"140","title":"Memory–sample lower bounds for noiseless Gaussian regression","short":"Memory–sample bounds for Gaussian regression","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For fixed A>0, a streaming learner retaining at most A d^2 bits between exact Gaussian linear measurements y= needs Omega_A(d log(1/eps)) measurements to estimate a uniformly random unit vector s to angular error eps<=1/10 with probability 2/3;","verdict":"Six overlapping papers (301 pages) for one result; the summary's claim is narrow (noiseless, Gaussian design, uniform prior, finite-state learner).","url":"/math#140","articleUrl":"/articles/openai-math#notable-results-briefly","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=140.%20Memory%20sample%20lower","manuscripts":[{"title":"Memory and precision in noiseless Gaussian regression","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf"},{"title":"Posterior replicas and conditional information in Gaussian regression","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026/paper.pdf"},{"title":"Localization costs and information growth for exact Gaussian observations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026/paper.pdf"},{"title":"Projection moments, positive cap domination, and Riesz estimates on the sphere","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026/paper.pdf"},{"title":"Replacing Gaussian observations in memory-constrained inference","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026/paper.pdf"},{"title":"Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026/paper.pdf"}],"reel":{"src":"/films/math/140.mp4","poster":"/films/math/140-poster.webp","captions":"/films/math/140.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Low-memory learners pay an extra log(1/ε) factor in samples for noiseless Gaussian regression; Lean checks it.","consequence":"A clean memory–sample tradeoff for streaming regression in a narrow setting."},"detail":"/api/math/140"},{"id":"141","title":"Existential–universal real sentences in the counting hierarchy","short":"$\\exists\\mathbb{R}$ in the counting hierarchy","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Truth of exists-forall sentences over the reals, with polynomials given by arithmetic circuits, is decidable in a fixed level C_jP of the counting hierarchy; in particular ETR (and all of exists-R) lies in C_26 P.","verdict":"Level 26 of the counting hierarchy is a structural result, not an algorithm; unformalized. No public expert assessment of this specific result found as of 2026-10-07;","url":"/math#141","articleUrl":"/articles/openai-math#the-existential-theory-of-the-reals-is-in-the-counting-hierarchy-141","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=141.%20Existential%20universal%20real","manuscripts":[{"title":"Existential–universal real sentences in the counting hierarchy","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026/etr-counting-hierarchy.pdf"}],"reel":{"src":"/films/math/141.mp4","poster":"/films/math/141-poster.webp","captions":"/films/math/141.vtt","duration":20.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Existential-universal real sentences sit in the counting hierarchy, well below PSPACE; unformalized.","consequence":"A better upper bound for existential-universal real problems. Internal."},"detail":"/api/math/141"},{"id":"142","title":"Deterministic polynomial factorization over prime fields","short":"Deterministic factoring over $\\mathbb{F}_p$","discipline":"Theoretical computer science","disciplineIndex":4,"accent":"#6CB6FF","kind":"conditional","kindLabel":"Claimed conditional result","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"A uniform deterministic algorithm factors every nonzero dense f in F_p[x] completely (irreducible factors with multiplicities) in O(((n+1) ceil(log2 p))^{10^{12}}) bit operations, with no randomness, no GRH and no integer-factorization or primitive-root…","verdict":"Correctness rests on an unformalized analytic number theory theorem in another family (029), which itself claims the infinitude part of Artin's primitive root conjecture…","url":"/math#142","articleUrl":"/articles/openai-math#deterministic-factoring-over-mathbb-f_p-conditional-on-a-companion","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=142.%20Deterministic%20polynomial%20factorization","manuscripts":[{"title":"Deterministic Polynomial Factorization over Prime Fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026/Deterministic-Polynomial-Factorization-over-Prime-Fields.pdf"}],"reel":{"src":"/films/math/142.mp4","poster":"/films/math/142-poster.webp","captions":"/films/math/142.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":0,"why":"Deterministic polynomial factoring over prime fields, if the release's unchecked Artin paper holds; no Lean.","consequence":"Deterministic polynomial factoring and nonresidue finding, if the release's zero-free strip holds; libraries are unaffected."},"detail":"/api/math/142"},{"id":"143","title":"Hilbert's sixteenth problem: uniform bounds for limit cycles","short":"Hilbert's 16th: uniform limit-cycle bounds","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(1) For every degree d there is a finite B(d) such that every real planar polynomial vector field of degree at most d has at most B(d) limit cycles (isolated periodic orbits, counted as geometric images) in the whole plane; non-effective, no formula for B(d).","verdict":"The landmark half (uniform bound, 160 pp.) is NOT formalized; lean/docs/143.md and the comparator challenge QuinticLienard (OAI.QuinticLienard.main, in…","url":"/math#143","articleUrl":"/articles/openai-math#hilberts-sixteenth-problem-the-unformalized-half","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=143.%20Hilbert's%20sixteenth%20problem","manuscripts":[{"title":"Uniform bounds for planar polynomial limit cycles","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf"},{"title":"Two limit cycles for quintic Liénard systems","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf"}],"reel":{"src":"/films/math/143.mp4","poster":"/films/math/143-poster.webp","captions":"/films/math/143.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":4,"advance":2,"consequences":2,"surprise":3,"confidence":0,"why":"Hilbert's 16th problem: each degree has a uniform bound on limit cycles, with no number given; the Lean covers a side case only.","consequence":"A finite limit-cycle bound for each degree of planar polynomial ODEs; with no numbers, applications must wait for estimates."},"detail":"/api/math/143"},{"id":"144","title":"Banach’s simple Lebesgue-spectrum problem","short":"Banach's simple Lebesgue spectrum","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a C-infinity diffeomorphism T of the 3-torus preserving Lebesgue volume and a real f in L^2_0 such that {f∘T^n : n in Z} is an orthonormal basis of the mean-zero L^2 space;","verdict":"The paper itself says the historical real-line question recorded by Ulam differs from the probability-space form solved here.","url":"/math#144","articleUrl":"/articles/openai-math#probability-statistical-mechanics-and-dynamics","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=144.%20Banach%E2%80%99s%20simple%20Lebesgue%2Dspectrum","manuscripts":[{"title":"A smooth three-torus diffeomorphism with simple Lebesgue spectrum","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/144.mp4","poster":"/films/math/144-poster.webp","captions":"/films/math/144.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Banach's question: a measure-preserving map whose spectrum is a single Lebesgue shift; 30 pages, Lean checks it.","consequence":"Answers a spectral-theory question. Internal to ergodic theory."},"detail":"/api/math/144"},{"id":"145","title":"Rokhlin’s multiple-mixing problem","short":"Rokhlin's multiple-mixing problem","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every invertible mixing probability-preserving transformation (arbitrary, not necessarily standard, probability space) is mixing of every finite order: for k>=3 and measurable A_1..A_k, mu(A_1 ∩ T^{-n_1}A_2 ∩ ...","verdict":"Lean challenge Rokhlin.lean (OAI.Rokhlin.mixing_all_finite_orders) matches the statement faithfully and uses only propext/Quot.sound/Classical.choice, but is not in…","url":"/math#145","articleUrl":"/articles/openai-math#probability-statistical-mechanics-and-dynamics","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=145.%20Rokhlin%E2%80%99s%20multiple%2Dmixing%20problem","manuscripts":[{"title":"Rokhlin's multiple-mixing problem for one transformation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/145.mp4","poster":"/films/math/145-poster.webp","captions":"/films/math/145.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":2,"confidence":3,"why":"Rokhlin's 1949 problem: mixing implies mixing of all orders; 40 pages for a famous problem, and Lean checks it.","consequence":"Mixing implies mixing of all orders; the release's pointwise ergodic results use it."},"detail":"/api/math/145"},{"id":"146","title":"Positive metric entropy for the standard map","short":"Positive entropy for the standard map","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is k0 such that for every k>=k0 the Chirikov standard map f_k(x,y)=(x+y+k sin 2πx, y+k sin 2πx) on T^2 has positive Kolmogorov–Sinai entropy w.r.t. area; equivalently positive Lyapunov exponent on a positive-area set;","verdict":"Claims a full parameter tail [k0,∞), stronger than the conjecture, but k0 is not explicit; no claim of ergodicity or a.e. positive exponents.","url":"/math#146","articleUrl":"/articles/openai-math#sinais-conjecture-for-the-standard-map","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=146.%20Positive%20metric%20entropy","manuscripts":[{"title":"Positive Metric Entropy for the Standard Map at Large Parameters","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/146.mp4","poster":"/films/math/146-poster.webp","captions":"/films/math/146.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"The standard map, the textbook model of chaos, has positive entropy for all large k; Lean checks the main theorem.","consequence":"Rigorous chaos on positive area for the standard map, the textbook Hamiltonian model; no ergodicity claim."},"detail":"/api/math/146"},{"id":"147","title":"The near-boundary Birkhoff conjecture","short":"The near-boundary Birkhoff conjecture","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"A C^∞ strictly convex planar billiard with positive curvature is an ellipse if a full grazing annulus near the boundary is continuously foliated by individually invariant essential curves (paper 1 upgrades this to an analytic collar), or if a full…","verdict":"Not the full Birkhoff conjecture: needs a full continuous collar of caustics/invariant curves near the boundary (a Cantor family, which is what KAM gives generically,…","url":"/math#147","articleUrl":"/articles/openai-math#dynamics-four-more-structural-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=147.%20The%20near%2Dboundary%20Birkhoff","manuscripts":[{"title":"Continuous Phase Foliations Create Analytic Caustic Collars","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Continuous-Phase-Foliations-Create-Analytic-Caustic-Collars-September-24-2026/paper.pdf"},{"title":"Rigidity of Smooth Billiards with a Continuous Caustic Collar","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rigidity-of-Smooth-Billiards-with-a-Continuous-Caustic-Collar-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/147.mp4","poster":"/films/math/147-poster.webp","captions":"/films/math/147.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Birkhoff billiards: a continuous collar of caustics near the boundary forces an ellipse; the full conjecture stays open.","consequence":"A local rigidity theorem for billiards. Internal."},"detail":"/api/math/147"},{"id":"148","title":"The entropy-rate dimension formula for self-similar measures","short":"The dimension of self-similar measures","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every finite IFS of similarities x↦r_i x+t_i on R (0<|r_i|<1, signs and unequal ratios allowed, exact overlaps allowed) and positive weights, the self-similar measure has dim_H μ = min{1, h_RW/χ}, where h_RW is the entropy rate of the random composed maps…","verdict":"Main theorem is in formalization.yaml main_results (SelfSimilar.json, OAI.EntropyRateDimension.entropy_rate_dimension, standard axioms) — among the strongest…","url":"/math#148","articleUrl":"/articles/openai-math#self-similar-measures-the-dimension-formula-with-overlaps","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=148.%20The%20entropy%2Drate%20dimension","manuscripts":[{"title":"The entropy-rate dimension formula for self-similar measures on the line","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/148.mp4","poster":"/films/math/148-poster.webp","captions":"/films/math/148.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"Self-similar measures lose dimension only through exact overlaps, settling the exact-overlaps conjecture; Lean checks it.","consequence":"The dimension of self-similar measures is computable from entropy, settling the exact-overlaps conjecture."},"detail":"/api/math/148"},{"id":"149","title":"Classwise permanence for weakly reversible mass-action systems","short":"Permanence for mass-action networks","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"For every finite weakly reversible mass-action network with fixed positive rate constants: (a) every positive solution is global, bounded and bounded away from zero (boundedness + persistence, bounds may depend on the initial state);","verdict":"Lean (MassAction challenge, docs/149.md) covers the boundedness/persistence statement with initial-state-dependent bounds, not the classwise uniform permanence of the…","url":"/math#149","articleUrl":"/articles/openai-math#dynamics-four-more-structural-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=149.%20Classwise%20permanence%20for","manuscripts":[{"title":"Uniform Permanence in Weakly Reversible Mass-Action Systems","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Permanence-in-Weakly-Reversible-Mass-Action-Systems-October-5-2026/permanence.pdf"},{"title":"Boundedness and persistence of weakly reversible mass-action systems","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/149.mp4","poster":"/films/math/149-poster.webp","captions":"/films/math/149.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":1,"why":"Weakly reversible mass-action reaction networks keep every species bounded away from 0 and infinity; Lean checks less.","consequence":"Weakly reversible chemical reaction networks never let a species die out or blow up, a long-standing question in that theory."},"detail":"/api/math/149"},{"id":"150","title":"Weak mixing of triangular billiards with an irrational angle","short":"Irrational triangle billiards are ergodic","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"For every nondegenerate Euclidean triangle with at least one angle irrational relative to π, the unit-speed billiard flow (vertex-hitting trajectories discarded) is ergodic for area × uniform direction (paper 1), and in fact weakly mixing (paper 2).","verdict":"Ergodicity is formalized (IrrationalTriangleBilliard challenge, docs/150.md, standard axioms; not in yaml main_results); weak mixing is not.","url":"/math#150","articleUrl":"/articles/openai-math#every-irrational-triangle-billiard-is-ergodic","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=150.%20Weak%20mixing%20of","manuscripts":[{"title":"Weak mixing of triangular billiards with an irrational angle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf"},{"title":"Ergodicity of triangular billiards with an irrational angle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf"}],"reel":{"src":"/films/math/150.mp4","poster":"/films/math/150-poster.webp","captions":"/films/math/150.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":2,"why":"Billiards in a triangle with an irrational angle are weakly mixing; two short papers for a hard problem. Lean checks ergodicity.","consequence":"Weak mixing for irrational triangle billiards, with claimed quantum-ergodicity corollaries."},"detail":"/api/math/150"},{"id":"151","title":"A C1 counterexample to the entropy conjecture","short":"A $C^1$ counterexample to Shub's conjecture","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is q>0 and a noninvertible C^1 self-map f of M=(R/100Z)×(S^2)^{q+1} with topological entropy 0 whose action on H_2(M;R) has eigenvalue 2; so h_top(f)=0 < log 2 ≤ log ρ(f_*), disproving the general C^1 self-map form of Shub's entropy conjecture.","verdict":"Noninvertible map; the C^1 diffeomorphism case (the version many authors mean) remains open.","url":"/math#151","articleUrl":"/articles/openai-math#dynamics-four-more-structural-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=151.%20A%20C1%20counterexample","manuscripts":[{"title":"A C^1 Counterexample to the Entropy Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf"}],"reel":{"src":"/films/math/151.mp4","poster":"/films/math/151-poster.webp","captions":"/films/math/151.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"A C1 map whose homology grows but entropy does not, against Shub's entropy conjecture; the diffeomorphism case stays open.","consequence":"C1 smoothness is not enough to force entropy growth; the diffeomorphism case stays open."},"detail":"/api/math/151"},{"id":"152","title":"Zero entropy does not guarantee a smooth positive-volume model","short":"Zero entropy, no smooth model","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"There is an ergodic invertible zero-entropy transformation of a standard nonatomic probability space not measurably conjugate to any C^∞ diffeomorphism preserving a smooth positive density on any compact finite-dimensional manifold (with or without boundary,…","verdict":"Lean (SmoothObstruction, docs/152.md) only proves the finite-entropy version; the zero-entropy conclusion, which is the actual point, is outside the formalized…","url":"/math#152","articleUrl":"/articles/openai-math#dynamics-four-more-structural-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=152.%20Zero%20entropy%20does","manuscripts":[{"title":"A zero-entropy system without a smooth positive-volume model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/152.mp4","poster":"/films/math/152-poster.webp","captions":"/films/math/152.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":1,"why":"A zero-entropy system with no smooth volume-preserving model, so entropy is not the only obstruction; Lean checks less.","consequence":"A new obstruction to smooth realization of abstract systems. Internal."},"detail":"/api/math/152"},{"id":"153","title":"Arithmetic classification and non-Pisot singularity for Bernoulli convolutions","short":"Singular Bernoulli convolutions beyond Pisot","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(1) For every λ in (0,1), the Bernoulli convolution ν_λ is singular iff λ admits one-sided approximation at a fixed geometric rate by retained targets from explicit finite sets of algebraic units (minimal polynomials with coefficients in {-1,0,1}, irreducible…","verdict":"The 'classification' is equivalent-but-ineffective (an infinite approximation condition), so the summary's 'classifies' is weaker than it sounds;","url":"/math#153","articleUrl":"/articles/openai-math#dynamics-four-more-structural-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=153.%20Arithmetic%20classification%20and","manuscripts":[{"title":"Arithmetic classification and non-Pisot singularity for Bernoulli convolutions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Arithmetic-classification-and-non-Pisot-singularity-for-Bernoulli-convolutions-October-3-2026/paper.pdf"}],"reel":{"src":"/films/math/153.mp4","poster":"/films/math/153-poster.webp","captions":"/films/math/153.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":3,"confidence":0,"why":"A first non-Pisot singular Bernoulli convolution, at quartic Salem numbers, against a long expectation; unformalized.","consequence":"Singular Bernoulli convolutions are not only a Pisot phenomenon. Internal to fractal geometry."},"detail":"/api/math/153"},{"id":"154","title":"Pointwise multiple ergodic averages for mixing transformations","short":"Pointwise multiple ergodic averages","discipline":"Dynamical systems and ergodic theory","disciplineIndex":5,"accent":"#F08BB3","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For an invertible (bimeasurable) mixing probability-preserving T on an arbitrary probability space, every n>=2 and bounded f_1..f_n, (1/N)Σ_{k≤N} Π_j f_j(T^{jk}x) → Π∫f_j a.e.;","verdict":"Depends on two companion results: Rokhlin multiple mixing (family 145) and an L^3 bound for the trilinear Hilbert transform from a different family in the release…","url":"/math#154","articleUrl":"/articles/openai-math#probability-statistical-mechanics-and-dynamics","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=154.%20Pointwise%20multiple%20ergodic","manuscripts":[{"title":"Pointwise Multiple Ergodic Averages for Mixing Transformations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pointwise-Multiple-Ergodic-Averages-for-Mixing-Transformations-October-4-2026/multiple-ergodic-averages.pdf"},{"title":"Pointwise convergence of fourfold ergodic averages for mixing transformations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pointwise-convergence-of-fourfold-ergodic-averages-for-mixing-transformations-October-4-2026/fourfold-ergodic-averages.pdf"},{"title":"Triple ergodic averages with distinct integer slopes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Triple-ergodic-averages-with-distinct-integer-slopes-October-4-2026/triple-ergodic-distinct-slopes.pdf"},{"title":"Pointwise convergence of triple ergodic averages for mixing transformations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf"}],"reel":{"src":"/films/math/154.mp4","poster":"/films/math/154-poster.webp","captions":"/films/math/154.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Multiple ergodic averages converge almost everywhere for mixing maps; rests on two other unchecked release papers.","consequence":"Pointwise convergence of multiple ergodic averages for mixing maps; depends on two other release papers."},"detail":"/api/math/154"},{"id":"155","title":"A counterexample to periodic tiling in dimension three","short":"Periodic tiling fails in $\\mathbb Z^3$","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a finite nonempty T ⊂ Z^3 that tiles Z^3 by translations, but no translation set A with A ⊕ T = Z^3 is invariant under a finite-index subgroup;","verdict":"Strong plausibility signals: short (26 pp), follows the published Greenfeld–Tao Sudoku/tiling-equation strategy closely, and the formal statement is stronger than the…","url":"/math#155","articleUrl":"/articles/openai-math#periodic-tiling-fails-in-dimension-3","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=155.%20A%20counterexample%20to","manuscripts":[{"title":"A translational tile with no fully periodic tiling in dimension three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/155.mp4","poster":"/films/math/155-poster.webp","captions":"/films/math/155.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Translational tiles that tile 3D space only non-periodically, the lowest possible dimension; Lean checks it.","consequence":"Aperiodic translational tiles exist in 3D, the lowest possible dimension. Internal."},"detail":"/api/math/155"},{"id":"156","title":"Borsuk's conjecture fails in dimension nine","short":"Borsuk's conjecture fails in $d = 9$","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"The compact set X = {uu^T : u ∈ R^4, |u| = 1} of rank-one projectors, sitting in the 9-dimensional affine space of trace-one symmetric 4×4 matrices with the Frobenius metric, has diameter √2 and cannot be covered by 10 sets of diameter < √2.","verdict":"Jump from 63 to 9 is enormous and the witness is a classical object (the projector/Veronese embedding of RP^3 used by Kahn–Kalai expositions) — surprising that nobody…","url":"/math#156","articleUrl":"/articles/openai-math#borsuks-conjecture-fails-in-dimension-9","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=156.%20Borsuk's%20conjecture%20fails","manuscripts":[{"title":"A nine-dimensional counterexample to Borsuk's covering assertion","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/156.mp4","poster":"/films/math/156-poster.webp","captions":"/films/math/156.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":65,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":3,"confidence":3,"why":"Borsuk's conjecture fails in dimension 9, down from 63 months earlier, by a topological route; Lean checks it.","consequence":"Borsuk's conjecture fails from dimension 9; dimensions 4 to 8 stay open."},"detail":"/api/math/156"},{"id":"157","title":"Graph coloring, clique minors, and Colin de Verdière invariants","short":"Hadwiger's conjecture, disproved on paper","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Paper 1: for arbitrarily large m there are m-vertex graphs with α(G) ≤ 2 and connected-matching number cm(G) < m/100; since h(G) ≤ (m + 4cm + 2)/3, this gives h(G) < 26m/75 + 2/3 < m/2 ≤ χ_f(G) ≤ χ(G), disproving Hadwiger's conjecture (and its fractional…","verdict":"The headline claims (Hadwiger and Colin de Verdière counterexamples) are the least verified: 104 + 131 pages of heavy, novel probabilistic/algebraic machinery…","url":"/math#157","articleUrl":"/articles/openai-math#hadwigers-conjecture-disproved-on-paper-and-linearised-in-lean","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=157.%20Graph%20coloring%20clique","manuscripts":[{"title":"A counterexample to Hadwiger's conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf"},{"title":"A counterexample to the Colin de Verdière chromatic conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf"},{"title":"A linear list-coloring bound in terms of the Hadwiger number","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/157.mp4","poster":"/films/math/157-poster.webp","captions":"/films/math/157.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":100,"tier":"Huge if true","importance":4,"advance":4,"consequences":4,"surprise":3,"confidence":0,"why":"Claims to disprove Hadwiger's 1943 conjecture, graph minor theory's central problem; the disproof is unchecked, only a side bound is.","consequence":"Would overturn graph minor theory's central conjecture. The safe, Lean-checked part is a linear bound on list coloring."},"detail":"/api/math/157"},{"id":"158","title":"The Euclidean plane cannot be colored with five colors","short":"The plane is not five-colourable","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every colouring of R^2 with 5 colours (arbitrary, non-measurable classes allowed; ZFC) has two points at distance 1 with the same colour. Hence 6 ≤ χ(R^2) ≤ 7.","verdict":"Very clean formal statement — if the Comparator check is sound, this is decisive. Surprising features: it avoids any explicit finite graph (searches by SAT have…","url":"/math#158","articleUrl":"/articles/openai-math#the-plane-is-not-five-colourable","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=158.%20The%20Euclidean%20plane","manuscripts":[{"title":"The Euclidean plane is not five-colorable","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/158.mp4","poster":"/films/math/158-poster.webp","captions":"/films/math/158.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":65,"tier":"Solid","importance":4,"advance":2,"consequences":1,"surprise":3,"confidence":3,"why":"The plane cannot be colored with 5 colors so no two points 1 apart match; the answer is now 6 or 7. Lean checks it.","consequence":"The plane needs 6 or 7 colors. Little follow-on beyond the problem."},"detail":"/api/math/158"},{"id":"159","title":"Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds","short":"$\\textsf{Erd\\H{o}s}$ reciprocal-sum conjecture","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"Theorem 1.1: for each fixed k>=3 there are C_k,c_k,eps_k>0 with r_k(N) =2, where r_k(N) is the largest subset of {1..N} with no nonconstant k-term AP (equivalently an alpha-dense subset of [N] has a k-AP once log N >= A_k(2+log(1/alpha))^{A_k}).","verdict":"198-page single manuscript built on the Leng–Sah–Sawhney quasipolynomial inverse theorem and Schoen–Sisask almost-periodicity as black boxes, plus many new appendices…","url":"/math#159","articleUrl":"/articles/openai-math#erdőss-5000-progression-problem","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=159.%20Erd%C5%91s%E2%80%99s%20reciprocal%2Dsum%20conjecture","manuscripts":[{"title":"Quasipolynomial Bounds for Arithmetic Progressions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/159.mp4","poster":"/films/math/159-poster.webp","captions":"/films/math/159.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":96,"tier":"Huge if true","importance":4,"advance":4,"consequences":4,"surprise":2,"confidence":2,"why":"Erdős's $5000 question: sets with divergent reciprocal sum contain long progressions; Lean checks the Erdős statement.","consequence":"Progressions in any set with divergent reciprocal sum, plus quasipolynomial Szemerédi bounds usable throughout additive combinatorics."},"detail":"/api/math/159"},{"id":"160","title":"Superexponential van der Waerden numbers","short":"Van der Waerden numbers, superexponential","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There are absolute K_0 and c = 10^{-5} such that W_r(k) > k^{c k floor(log2 r)} for every k >= K_0 and every r >= 2 (uniform in r). Hence W_r(k)^{1/k} -> infinity for each fixed r >= 2, including r = 2.","verdict":"Short (25 pp), elementary-probabilistic construction (Behrend-type squared-norm geometry on Z/q^D Z with LCM dilation, Bernoulli flips, local lemma) and its exact…","url":"/math#160","articleUrl":"/articles/openai-math#van-der-waerden-numbers-grow-superexponentially","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=160.%20Superexponential%20van%20der","manuscripts":[{"title":"Quantitative Superexponential Bounds for van der Waerden Numbers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/160.mp4","poster":"/films/math/160-poster.webp","captions":"/films/math/160.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"Two-color van der Waerden numbers grow faster than any exponential, answering Erdős; Lean checks the bound.","consequence":"Van der Waerden numbers grow faster than exponential. Internal."},"detail":"/api/math/160"},{"id":"161","title":"Counterexamples to Sidorenko’s conjecture and the forcing conjecture","short":"Sidorenko's conjecture is false","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Let H be the incidence graph (point–triple) of an explicit 2-fold triple system: 22 triples on 13 points, every one of the 33 covered point pairs in exactly two triples;","verdict":"The host graph is purely existential (q -> infinity limit with a fixed large dimension D), not a computer certificate, and no numerical size of G is given;","url":"/math#161","articleUrl":"/articles/openai-math#sidorenkos-conjecture-is-false","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=161.%20Counterexamples%20to%20Sidorenko%E2%80%99s","manuscripts":[{"title":"A counterexample to Sidorenko's conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/161.mp4","poster":"/films/math/161-poster.webp","captions":"/films/math/161.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":86,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":3,"confidence":3,"why":"Counterexamples to Sidorenko's conjecture on bipartite pattern counts, long believed true; Lean checks it.","consequence":"Ends the Sidorenko and forcing conjectures, which much of quasirandomness theory hoped to rest on."},"detail":"/api/math/161"},{"id":"162","title":"Counterexamples to Ryser’s covering conjecture","short":"Ryser's conjecture fails","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) For every sufficiently large prime q (threshold non-explicit) there is a finite intersecting (q+1)-partite (q+1)-uniform hypergraph with exactly q+1 non-isolated vertices in each part and covering number tau = q+1 (so nu = 1, tau = r > r-1).","verdict":"Asymptotic: no explicit rank or hypergraph is given, the prime thresholds q0, s0, n0(s) are existential, so the smallest counterexample rank is unknown (could be…","url":"/math#162","articleUrl":"/articles/openai-math#rysers-conjecture-fails","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=162.%20Counterexamples%20to%20Ryser%E2%80%99s","manuscripts":[{"title":"Balanced counterexamples to Ryser's conjecture at prime orders","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf"},{"title":"A counterexample to Ryser's covering conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/162.mp4","poster":"/films/math/162-poster.webp","captions":"/films/math/162.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":65,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":3,"confidence":3,"why":"Ryser's covering conjecture fails at large rank; no explicit example is given. Lean checks it.","consequence":"Ryser's bound fails at large rank; small ranks stay open."},"detail":"/api/math/162"},{"id":"164","title":"Hindman’s finite sums and products conjecture","short":"Hindman's sums and products","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"For every r-colouring of the positive integers and every m >= 1 there are a_1 R(sum+prod of earlier)^D, so FS(A) and FP(A) each have 2^m-1 distinct elements and meet only in A (Cor. 1.2). Purely qualitative: no bound on the size of the configuration.","verdict":"Unformalized 70-page qualitative argument with heavy higher-order Fourier machinery: Green–Tao–Ziegler inverse theorem (citing a 2024 erratum and a 2026 revision),…","url":"/math#164","articleUrl":"/articles/openai-math#hindmans-finite-sums-and-products","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=164.%20Hindman%E2%80%99s%20finite%20sums","manuscripts":[{"title":"Monochromatic finite sums and products in the positive integers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Monochromatic-finite-sums-and-products-in-the-positive-integers-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/164.mp4","poster":"/films/math/164-poster.webp","captions":"/films/math/164.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"Any finite coloring of the integers has a set whose sums and products share one color, as Hindman asked; no Lean.","consequence":"Sums and products together in one color. Internal to Ramsey theory."},"detail":"/api/math/164"},{"id":"165","title":"The Harary–Hill and Zarankiewicz crossing-number formulas","short":"Crossing numbers of $K_n$ and $K_{m,n}$","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every n >= 3, cr(K_n) = (1/4) floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2) (Harary–Hill/Guy conjecture), and for all positive m,n, cr(K_{m,n}) = floor(m/2) floor((m-1)/2) floor(n/2) floor((n-1)/2) (Zarankiewicz conjecture, Turán's brickyard…","verdict":"Astonishingly short (13 + 16 pages) for two problems open 60-80 years; the proofs are algebraic (Tutte-style signed-intersection cycle pairing + polynomial interpolation…","url":"/math#165","articleUrl":"/articles/openai-math#the-crossing-number-of-k_n-and-k_mn","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=165.%20The%20Harary%20Hill","manuscripts":[{"title":"The crossing number of complete graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf"},{"title":"The crossing number of complete bipartite graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/165.mp4","poster":"/films/math/165-poster.webp","captions":"/films/math/165.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":73,"tier":"Solid","importance":3,"advance":4,"consequences":1,"surprise":3,"confidence":3,"why":"Exact crossing numbers of complete and complete bipartite graphs, open 60-80 years, by short algebra; Lean checks both.","consequence":"Exact crossing numbers of complete and complete bipartite graphs. Little follow-on beyond graph drawing."},"detail":"/api/math/165"},{"id":"166","title":"The higher-dimensional Erdős distinct-distances conjecture","short":"Distinct distances in every dimension","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every fixed d ≥ 3 there is c_d > 0 such that every n ≥ 2 distinct points in R^d determine at least c_d n^{2/d} distinct distances — sharp up to the constant (integer grid). No position assumptions.","verdict":"Unformalized, 103 pp of heavy real/complex algebraic geometry (Hilbert-function estimates, approximate complete intersections, multiscale 'sparse cones' argument),…","url":"/math#166","articleUrl":"/articles/openai-math#distinct-distances-in-every-dimension","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=166.%20The%20higher%2Ddimensional%20Erd%C5%91s","manuscripts":[{"title":"The higher-dimensional Erdős distinct-distances conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-higher-dimensional-Erdos-distinct-distances-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/166.mp4","poster":"/films/math/166-poster.webp","captions":"/films/math/166.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Erdős distinct distances in every dimension 3 and up, to a constant factor; 103 pages weeks after a human 3D advance.","consequence":"Distinct distances in higher dimensions settled up to constants. Internal."},"detail":"/api/math/166"},{"id":"167","title":"Planar distinct distances and unit-distance bounds","short":"Unit and pinned distances in the plane","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) Weak pinned distance theorem: for each fixed s > 0 the maximal fraction of ordered pairs (x,y) in an n-point planar set whose distance from x is shared by ≥ n^s points tends to 0 (no rate);","verdict":"Both results break long-standing barriers (Katz–Tardos 0.864 for pinned; 4/3 for unit distances, which is tight for unit circles in general normed/incidence settings, so…","url":"/math#167","articleUrl":"/articles/openai-math#unit-and-pinned-distances-in-the-plane","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=167.%20Planar%20distinct%20distances","manuscripts":[{"title":"The weak pinned planar distance theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf"},{"title":"A power saving for planar unit distances","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/167.mp4","poster":"/films/math/167-poster.webp","captions":"/films/math/167.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":4,"advance":2,"consequences":2,"surprise":3,"confidence":3,"why":"First improvement on the 1984 n^{4/3} unit-distance bound, by a number-field trick; the gain is ineffective. Lean checks it.","consequence":"A first improvement on the 1984 unit-distance bound, by an arithmetic method that may reach other incidence problems."},"detail":"/api/math/167"},{"id":"168","title":"Combinatorial invariance of Kazhdan–Lusztig polynomials","short":"Kazhdan–Lusztig combinatorial invariance","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For arbitrary Coxeter systems (W,S), (W',S') (infinite and non-crystallographic included) and Bruhat intervals [u,b], [u',b'], any poset isomorphism [u,b] -> [u',b'] implies P_{u,b}(q) = P_{u',b'}(q) for the equal-parameter Kazhdan–Lusztig polynomials (and…","verdict":"The paper's proof uses moment-graph (Braden–MacPherson) sheaves and cites the Elias–Williamson character theorem / Fiebig's constructions as inputs, but comments inside…","url":"/math#168","articleUrl":"/articles/openai-math#combinatorial-invariance-of-kazhdanlusztig-polynomials","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=168.%20Combinatorial%20invariance%20of","manuscripts":[{"title":"Combinatorial invariance of Kazhdan–Lusztig polynomials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/168.mp4","poster":"/films/math/168-poster.webp","captions":"/films/math/168.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":2,"confidence":3,"why":"Kazhdan–Lusztig polynomials depend only on the shape of the Bruhat interval, a 1980s conjecture; Lean checks it.","consequence":"KL polynomials are determined by poset shape, a combinatorial handle on Schubert variety cohomology."},"detail":"/api/math/168"},{"id":"169","title":"Shareshian–Wachs elementary positivity","short":"Shareshian–Wachs $e$-positivity","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every natural unit interval graph G (Dyck-path graph on [n]), the chromatic quasisymmetric function X_G(x;q) = sum over proper colorings of q^{asc} x^f expands as sum over sigma in D'_G (permutations whose adjacent descents are edges) of q^{ginv_G(sigma)}…","verdict":"The q=1 case (Stanley–Stembridge) was already proved by Hikita in 2024, so this is the graded refinement plus an explicit permutation-indexed formula, not the classical…","url":"/math#169","articleUrl":"/articles/openai-math#shareshianwachs-positivity","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=169.%20Shareshian%20Wachs%20elementary","manuscripts":[{"title":"Elementary positivity of chromatic quasisymmetric functions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/169.mp4","poster":"/films/math/169-poster.webp","captions":"/films/math/169.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"The graded Shareshian–Wachs refinement of Stanley–Stembridge e-positivity; the ungraded case was proved in 2024.","consequence":"Graded e-positivity of chromatic quasisymmetric functions, relevant to Hessenberg varieties."},"detail":"/api/math/169"},{"id":"170","title":"Sharp logarithmic exponents for off-diagonal Ramsey numbers","short":"The log exponent of $r(s,t)$","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every fixed s >= 5, r(s,t) = t^{s-1}/(log t)^{s-2+o(1)} as t -> infinity: for each eps>0 and t large, t^{s-1}/(log t)^{s-2+eps} =6 via Theorem 1.2: for d>=5 and large prime q, a K_{d+1}-free graph on floor(q^d log q) vertices with independence number < q…","verdict":"Builds directly on Bradač's 2026 construction (which already gave the right power of t), so the summary's 'determining the logarithmic exponent' is accurate but the…","url":"/math#170","articleUrl":"/articles/openai-math#the-log-exponent-of-rst","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=170.%20Sharp%20logarithmic%20exponents","manuscripts":[{"title":"The sharp logarithmic exponent of r(5,t)","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf"},{"title":"Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/170.mp4","poster":"/films/math/170-poster.webp","captions":"/films/math/170.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Sharp logarithmic exponents for off-diagonal Ramsey numbers, building on a 2026 human construction; Lean checks it.","consequence":"Sharp log exponents for off-diagonal Ramsey numbers, building on a recent human construction."},"detail":"/api/math/170"},{"id":"171","title":"The hypercube Ramsey conjecture","short":"Hypercube Ramsey numbers are linear","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"There is an absolute constant C such that R(Q_n) = 0, where R is the 2-colour (non-induced) Ramsey number of the n-dimensional hypercube graph. With the classical lower bound R(Q_n) >= 3·2^{n-1} - 1 this gives R(Q_n) = Θ(2^n). C is not computed;","verdict":"Unformalized single 172-page manuscript; a jump from exponent 2-c (Tikhomirov 2022) straight to linear is a very large step, so independent expert checking is essential.","url":"/math#171","articleUrl":"/articles/openai-math#hypercube-ramsey-numbers-are-linear","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=171.%20The%20hypercube%20Ramsey","manuscripts":[{"title":"The hypercube Ramsey number has linear order","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/171.mp4","poster":"/films/math/171-poster.webp","captions":"/films/math/171.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":1,"confidence":0,"why":"Burr and Erdős's 1975 question: hypercubes have linear Ramsey numbers; a big jump in 172 unformalized pages.","consequence":"Linear Ramsey numbers for hypercubes. Internal."},"detail":"/api/math/171"},{"id":"172","title":"Classification of finite Euclidean Ramsey configurations","short":"Euclidean Ramsey sets, classified","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"A finite set A of s >= 2 points affinely spanning R^d is Euclidean Ramsey (every finite colouring of some R^D contains a monochromatic congruent copy at the same scale) iff there is a (d+1)x(d+1) matrix P over B = F (x)_Q F, F the coordinate field, with (p_i…","verdict":"The 'classification' is an algebraic tensor criterion over the coordinate field; deciding it for a given configuration is not obviously algorithmic (paper says it is not…","url":"/math#172","articleUrl":"/articles/openai-math#euclidean-ramsey-sets-classified","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=172.%20Classification%20of%20finite","manuscripts":[{"title":"A classification of finite Euclidean Ramsey configurations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/172.mp4","poster":"/films/math/172-poster.webp","captions":"/films/math/172.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"An algebraic criterion for which finite point sets are Euclidean Ramsey, a 50-year question; Lean checks it.","consequence":"An algebraic classification of Ramsey configurations, not an algorithm."},"detail":"/api/math/172"},{"id":"173","title":"Seymour’s second-neighborhood conjecture","short":"Seymour's second-neighbourhood conjecture","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every nonempty finite oriented graph (no loops, no 2-cycles, otherwise arbitrary: not necessarily connected or regular) has a vertex v whose number of vertices at directed distance exactly two is at least its out-degree, |N2+(v)| >= |N1+(v)|.","verdict":"Short (15 pp) for a 36-year-old problem, which cuts both ways: easy for experts to check, and the Lean statement is faithful and compact.","url":"/math#173","articleUrl":"/articles/openai-math#seymours-second-neighbourhood-conjecture","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=173.%20Seymour%E2%80%99s%20second%2Dneighborhood%20conjecture","manuscripts":[{"title":"A proof of Seymour’s second-neighborhood conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/173.mp4","poster":"/films/math/173-poster.webp","captions":"/films/math/173.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":69,"tier":"Solid","importance":3,"advance":4,"consequences":1,"surprise":2,"confidence":3,"why":"Seymour's second-neighborhood conjecture: some vertex has as many second as first out-neighbors; 15 Lean-checked pages.","consequence":"Settles Seymour's conjecture. Little follow-on."},"detail":"/api/math/173"},{"id":"174","title":"Deterministic construction of strong thin spanning trees","short":"Strong thin spanning trees","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a universal C such that every finite loopless k-edge-connected multigraph (>=2 vertices) has a spanning tree T with |δ_T(S)| <= (C/k)|δ_G(S)| for every cut S (strong thin tree conjecture).","verdict":"Existence proof uses the Marcus–Spielman–Srivastava theorem (via an isotropic-vector theorem proved in an appendix) and a fixed-point argument;","url":"/math#174","articleUrl":"/articles/openai-math#strong-thin-trees","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=174.%20Deterministic%20construction%20of","manuscripts":[{"title":"The strong thin tree conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-strong-thin-tree-conjecture-September-23-2026/paper.pdf"},{"title":"A polynomial-time construction of strong thin trees","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/174.mp4","poster":"/films/math/174-poster.webp","captions":"/films/math/174.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"Every k-edge-connected graph has a C/k-thin spanning tree, built deterministically; Lean checks it.","consequence":"Deterministic thin trees, a structural tool from traveling-salesman approximation."},"detail":"/api/math/174"},{"id":"175","title":"Talagrand’s expectation thresholds, discrete convexity, and graph decompositions","short":"Talagrand's threshold conjectures","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"(1) For every nonempty proper increasing family F on a finite set, the fractional expectation threshold is at most 25·512^4 times the integral one, with the same 1/2 covering budget (Talagrand's conjecture).","verdict":"Constants are astronomically large (2^75 unions; 25·512^4 loss) but universal, which is all the conjectures ask.","url":"/math#175","articleUrl":"/articles/openai-math#talagrands-threshold-conjectures","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=175.%20Talagrand%E2%80%99s%20expectation%20thresholds","manuscripts":[{"title":"Graph Decompositions at the Integral Expectation Threshold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf"},{"title":"Integral and fractional expectation thresholds are equivalent","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integral-and-fractional-expectation-thresholds-are-equivalent-September-23-2026/paper.pdf"},{"title":"Talagrand’s discrete-convexity conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Talagrands-discrete-convexity-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/175.mp4","poster":"/films/math/175-poster.webp","captions":"/films/math/175.vtt","duration":20.1,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"Talagrand's conjectures on rounding fractional expectation thresholds, in short papers; Lean checks two of three.","consequence":"Talagrand's threshold conjectures become tools for locating thresholds of random structures."},"detail":"/api/math/175"},{"id":"176","title":"The second Kahn–Kalai conjecture with an edge-count bound","short":"The second Kahn–Kalai conjecture","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every graph H with h >= 1 edges on at most n vertices, the threshold for G(n,p) to contain a copy of H is at most 2048·e^50·p_E(n,H)·(1 + log2 h), where p_E is the least p at which every subgraph of H has expected count >= 1/2.","verdict":"Constant 2048·e^50 is enormous but universal. Proves the bound with log of edge count, slightly stronger than the original log n form.","url":"/math#176","articleUrl":"/articles/openai-math#the-second-kahnkalai-conjecture","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=176.%20The%20second%20Kahn","manuscripts":[{"title":"The second Kahn–Kalai conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-second-Kahn-Kalai-conjecture-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/176.mp4","poster":"/films/math/176-poster.webp","captions":"/films/math/176.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":3,"why":"The second Kahn–Kalai conjecture: thresholds for spanning subgraphs are right to one log factor; Lean checks it.","consequence":"Sharp thresholds for random graphs to contain large subgraphs."},"detail":"/api/math/176"},{"id":"177","title":"Bounded-degree coboundary expanders","short":"Coboundary expanders in every dimension","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every d >= 3 there are arbitrarily large finite connected pure d-dimensional simplicial complexes with uniformly bounded vertex degree and uniform F2 coboundary expansion in every degree i < d (including vanishing F2 cohomology).","verdict":"Relies heavily on the Oppenheim–Valentiner-Branth cosystolic expansion theorem for KMS coset complexes and on affine twin-building geometry;","url":"/math#177","articleUrl":"/articles/openai-math#coboundary-expanders-and-ramanujan-graphs","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=177.%20Bounded%2Ddegree%20coboundary%20expanders","manuscripts":[{"title":"Bounded-degree coboundary expanders in every dimension","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/177.mp4","poster":"/films/math/177-poster.webp","captions":"/films/math/177.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":2,"why":"The first bounded-degree coboundary expanders in every degree, from congruence coset complexes; Lean checks it.","consequence":"Bounded-degree coboundary expanders, building blocks for topological overlap, quantum codes and PCPs."},"detail":"/api/math/177"},{"id":"178","title":"Deterministic nonbipartite Ramanujan graphs in every fixed degree","short":"Deterministic Ramanujan graphs, every degree","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For each fixed d >= 3 there is a deterministic algorithm that, for every sufficiently large even n, outputs a simple d-regular graph on n vertices whose nonconstant adjacency eigenvalues all lie strictly inside (-2√(d-1), 2√(d-1)) (so connected and…","verdict":"Existence was already known from Huang–McKenzie–Yau; the new content is a deterministic polynomial-time construction (the exponent may depend on d, so not polynomial…","url":"/math#178","articleUrl":"/articles/openai-math#coboundary-expanders-and-ramanujan-graphs","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=178.%20Deterministic%20nonbipartite%20Ramanujan","manuscripts":[{"title":"Deterministic nonbipartite Ramanujan graphs in every fixed degree","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/178.mp4","poster":"/films/math/178-poster.webp","captions":"/films/math/178.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"A deterministic construction of non-bipartite Ramanujan graphs in every degree; existence was known. No Lean.","consequence":"A deterministic construction of near-optimal expanders in every degree; existence was known."},"detail":"/api/math/178"},{"id":"179","title":"The circulant Hadamard and Barker-sequence conjectures","short":"Circulant Hadamard and Barker sequences","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"A real circulant Hadamard matrix of order n >= 1 exists iff n in {1,4} (Ryser's circulant Hadamard conjecture). Combined with Turyn–Storer (odd lengths) and the classical reduction of even Barker sequences to circulant Hadamard matrices, Barker sequences of…","verdict":"A 60-year-old problem with a long history of failed claimed proofs, settled in a 15-page paper — extraordinary, and the informal argument is dense algebraic number…","url":"/math#179","articleUrl":"/articles/openai-math#circulant-hadamard-matrices-and-barker-sequences","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=179.%20The%20circulant%20Hadamard","manuscripts":[{"title":"The circulant Hadamard conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/179.mp4","poster":"/films/math/179-poster.webp","captions":"/films/math/179.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":3,"why":"Ryser's conjecture: no circulant Hadamard matrix beyond order 4, after 60 years of failed attempts; Lean checks it.","consequence":"No circulant Hadamard matrices beyond order 4, ruling out even-length Barker sequences used in radar pulse compression."},"detail":"/api/math/179"},{"id":"180","title":"Barnette’s Hamiltonian-cycle conjecture","short":"Barnette's Hamiltonian-cycle conjecture","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle. Stronger: in the dual triangulation the vertex set splits into two induced trees with a prescribed facial pattern, so the Hamiltonian cycle can avoid any given edge;","verdict":"Remarkably short (11 pp) for a 57-year-old conjecture, and an existence proof via a nonvanishing finite exponential sum (non-constructive;","url":"/math#180","articleUrl":"/articles/openai-math#barnettes-conjecture","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=180.%20Barnette%E2%80%99s%20Hamiltonian%2Dcycle%20conjecture","manuscripts":[{"title":"Paired states and Hamiltonian cycles in cubic bipartite planar graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/180.mp4","poster":"/films/math/180-poster.webp","captions":"/films/math/180.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":73,"tier":"Solid","importance":3,"advance":4,"consequences":1,"surprise":3,"confidence":3,"why":"Barnette's conjecture: cubic bipartite polyhedral graphs are Hamiltonian, in 11 pages; Lean checks the topological statement.","consequence":"Settles the last piece of Tait's approach to the four-color theorem; non-constructive."},"detail":"/api/math/180"},{"id":"181","title":"The Erdős–Gallai cycle-decomposition conjecture","short":"Linear cycle decompositions","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is an absolute C such that every finite simple graph on n vertices has an edge partition into at most Cn parts, each a simple cycle or a single edge. Corollaries: every Eulerian graph decomposes into <= Cn cycles;","verdict":"C is not explicit. Builds directly on Bucić–Montgomery's expansion/path-closing method plus Lovász's path-cycle decomposition and the Aharoni–Haxell matching theorem;","url":"/math#181","articleUrl":"/articles/openai-math#the-erdősgallai-cycle-decomposition","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=181.%20The%20Erd%C5%91s%20Gallai","manuscripts":[{"title":"A linear cycle-and-edge decomposition of every graph","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/181.mp4","poster":"/films/math/181-poster.webp","captions":"/films/math/181.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Erdős–Gallai: every graph splits into linearly many cycles and edges; Lean checks it.","consequence":"Linear cycle decompositions for every graph. Internal."},"detail":"/api/math/181"},{"id":"182","title":"Power savings for intersective polynomial differences and prime arguments","short":"Square-difference-free sets: a power saving","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Square case (paper 3): absolute c>0, C with |A| =2 there is c_k>0 such that for every intersective integer polynomial h of degree k with positive leading coefficient, sets whose differences avoid nonzero h(1),h(2),...","verdict":"The square-case power saving (the famous question) is reported as fully formalized, which is strong evidence;","url":"/math#182","articleUrl":"/articles/openai-math#square-difference-free-sets","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=182.%20Power%20savings%20for","manuscripts":[{"title":"A power saving for intersective polynomial differences with an exponent depending only on the degree","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-intersective-polynomial-differences-with-an-exponent-depending-only-on-the-degree-October-5-2026/power-saving-intersective-polynomial-differences.pdf"},{"title":"A Power Saving for Polynomial Differences at Prime Arguments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Power-Saving-for-Polynomial-Differences-at-Prime-Arguments-October-5-2026/prime-argument-polynomial-differences.pdf"},{"title":"A power saving for square-difference-free sets","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/182.mp4","poster":"/films/math/182-poster.webp","captions":"/films/math/182.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":2,"why":"Sets with no two elements a square apart have power-saving density bounds; Lean checks the square case.","consequence":"Power-saving bounds for square-difference-free sets; the prime case leans on the release's quasi-RH."},"detail":"/api/math/182"},{"id":"183","title":"Power savings for planar halving lines and k-sets","short":"Halving lines: a power saving","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There exist absolute (non-explicit) eps>0, C, n0 such that every even n>=n0 point set in the plane with no three collinear has at most C n^{4/3-eps} halving lines (Thm 1.1).","verdict":"Exponent saving eps and constants are completely ineffective (limiting-measure/compactness argument), so the improvement is qualitative.","url":"/math#183","articleUrl":"/articles/openai-math#halving-lines","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=183.%20Power%20savings%20for","manuscripts":[{"title":"A power saving for planar halving lines","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/183.mp4","poster":"/films/math/183-poster.webp","captions":"/films/math/183.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Halving lines below Dey's 1998 n^{4/3} bound, by an ineffective amount; Lean checks it.","consequence":"Breaks Dey's 1998 halving-lines bound by an ineffective amount."},"detail":"/api/math/183"},{"id":"184","title":"Correspondence coloring with a fixed forbidden subgraph","short":"Clique-free graphs: AEKS and AKS","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) For each fixed r>=4 there are C_r, Delta_r such that every K_r-free graph with max degree Delta>=Delta_r has correspondence (DP) chromatic number =4, every n-vertex K_r-free graph with average degree d>=2 has an independent set of size >= c_r n log d / d…","verdict":"Two famous conjectures at once; extraordinary. The independence half is Lean-checked (strong evidence, and the formal statement looks faithful).","url":"/math#184","articleUrl":"/articles/openai-math#clique-free-graphs-aeks-and-alonkrivelevichsudakov","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=184.%20Correspondence%20coloring%20with","manuscripts":[{"title":"Correspondence coloring graphs with a forbidden clique","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Correspondence-Coloring-Graphs-with-a-Forbidden-Clique-October-5-2026/correspondence-coloring-forbidden-clique.pdf"},{"title":"A logarithmic independence bound for clique-free graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/184.mp4","poster":"/films/math/184-poster.webp","captions":"/films/math/184.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":2,"confidence":1,"why":"K_r-free graphs get triangle-free-quality independence and coloring bounds, a 45-year target; Lean checks half.","consequence":"Optimal coloring and independence bounds for K_r-free graphs, a 45-year target, with correspondence coloring included."},"detail":"/api/math/184"},{"id":"185","title":"Counterexamples to infinite matroid intersection and packing/covering","short":"Infinite matroid intersection fails","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"In ZFC there are two infinite (Bruhn et al. axioms) matroids M0, M1 on a common countably infinite ground set, both self-dual and partitional (countable direct sums of one self-dual uniform matroid Q), such that no independent I0 of M0 and I1 of M1 cover E;","verdict":"Refutes only the unrestricted versions; Nash-Williams' original finitary conjecture (and the finitary/cofinitary-mixed setting, open beyond Joó's countable results)…","url":"/math#185","articleUrl":"/articles/openai-math#five-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=185.%20Counterexamples%20to%20infinite","manuscripts":[{"title":"A Counterexample to the Infinite Matroid Packing/Covering Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/185.mp4","poster":"/films/math/185-poster.webp","captions":"/films/math/185.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":27,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":1,"confidence":3,"why":"Counterexamples to the unrestricted infinite matroid intersection conjectures; the finitary one survives.","consequence":"The unrestricted infinite matroid conjectures fail. Nothing follows beyond it."},"detail":"/api/math/185"},{"id":"186","title":"Uniform influence and sharp thresholds for graph and hypergraph properties","short":"Sharp thresholds for graph properties","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Graphs: for every n>=2, every nontrivial increasing vertex-permutation-invariant graph property and every 0=3, Var_p(f) <= C_r I_p(f)/(log n)^{r/(r-1)} for every r-uniform hypergraph property and all p, giving threshold width O_{r,eps}((log n)^{-r/(r-1)}) for…","verdict":"Remarkably short (10 pp graph paper) for a 30-year-old conjecture whose best prior proof (Bourgain–Kalai) was deep, but the graph statement is machine-checked with an…","url":"/math#186","articleUrl":"/articles/openai-math#sharp-thresholds-for-graph-properties","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=186.%20Uniform%20influence%20and","manuscripts":[{"title":"A uniform influence bound for hypergraph properties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-uniform-influence-bound-for-hypergraph-properties-October-5-2026/hypergraph-influences.pdf"},{"title":"A Sharp Threshold Bound for Monotone Graph Properties","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/186.mp4","poster":"/films/math/186-poster.webp","captions":"/films/math/186.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"Friedgut–Kalai's optimal 1/(log n)^2 threshold width for symmetric graph properties; Lean checks the graph case.","consequence":"Optimal sharp-threshold width for symmetric graph properties."},"detail":"/api/math/186"},{"id":"187","title":"Snaky in 21 Maker moves","short":"Snaky in 21 Maker moves","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"On the empty infinite square grid, Maker (first player, one cell per turn, no handicap) has a strategy that completes a copy (any translation/rotation/reflection) of the Snaky hexomino {(0,0),(1,0),(2,0),(3,0),(3,1),(4,1)} within 21 Maker moves against any…","verdict":"Essentially a finite certificate plus a soundness lemma; machine-checked, so high confidence, but the Lean policy type quantifies Breaker as an arbitrary sequence and…","url":"/math#187","articleUrl":"/articles/openai-math#five-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=187.%20Snaky%20in%2021","manuscripts":[{"title":"Snaky in 21 Maker moves","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Snaky-in-21-Maker-moves-September-25-2026/article.pdf"}],"reel":{"src":"/films/math/187.mp4","poster":"/films/math/187-poster.webp","captions":"/films/math/187.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":27,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":1,"confidence":3,"why":"Maker wins Snaky on the infinite grid in 21 moves, settling the last small polyomino; a Lean-checked certificate.","consequence":"Settles one game. Nothing follows beyond it."},"detail":"/api/math/187"},{"id":"188","title":"The sharp terminal leave in random triangle removal","short":"Random triangle removal: the constant","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For the random triangle removal process on K_n (repeatedly delete the edges of a uniformly random remaining triangle until triangle-free), the final edge count F_n satisfies E[(F_n/n^{3/2} - 1/(2 sqrt 2))^2] -> 0;","verdict":"Lean statement appears faithful and covers the full theorem, which strongly supports correctness despite the delicate o(1/D) second-moment estimates.","url":"/math#188","articleUrl":"/articles/openai-math#random-triangle-removal","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=188.%20The%20sharp%20terminal","manuscripts":[{"title":"The sharp terminal leave in random triangle removal","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026.pdf"}],"reel":{"src":"/films/math/188.mp4","poster":"/films/math/188-poster.webp","captions":"/films/math/188.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"The exact number of edges left by random triangle removal, up to the constant; Lean checks it.","consequence":"The exact constant for what random triangle removal leaves behind."},"detail":"/api/math/188"},{"id":"189","title":"Cycle–clique Ramsey numbers","short":"Cycle–clique Ramsey numbers","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"R(C_m, K_n) = (m-1)(n-1)+1 for all integers m >= n >= 3 except (m,n)=(3,3), where R(C_3,K_3)=6. The lower bound is n-1 disjoint red K_{m-1}'s; the work is the matching upper bound.","verdict":"Since Keevash–Long–Skokan already settled all but finitely many pairs (non-effectively), this is a completion of a nearly-solved conjecture rather than a breakthrough,…","url":"/math#189","articleUrl":"/articles/openai-math#five-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=189.%20Cycle%20clique%20Ramsey","manuscripts":[{"title":"Cycle--clique Ramsey numbers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf"}],"reel":{"src":"/films/math/189.mp4","poster":"/films/math/189-poster.webp","captions":"/films/math/189.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":23,"tier":"Incremental","importance":2,"advance":1,"consequences":0,"surprise":0,"confidence":3,"why":"Cycle–clique Ramsey numbers for all remaining small cases; mostly known before. Lean checks the certificates.","consequence":"Completes the last finitely many cases. Nothing follows beyond it."},"detail":"/api/math/189"},{"id":"190","title":"Polynomial removal fails for ordered binary matrices","short":"Ordered matrix removal is not polynomial","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is an explicit 66x66 binary pattern H such that for n_h=(386h+2)2^h and eps_h=(386h+2)^{-2} there is an n_h x n_h binary matrix A_h that is eps_h-far (in normalized Hamming distance, edits in both directions) from containing no ordered (rows and columns…","verdict":"Counterexample is for one explicit large pattern; whether small patterns admit polynomial removal is open.","url":"/math#190","articleUrl":"/articles/openai-math#five-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=190.%20Polynomial%20removal%20fails","manuscripts":[{"title":"Polynomial removal fails for ordered binary matrices","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/190.mp4","poster":"/films/math/190-poster.webp","captions":"/films/math/190.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":38,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Polynomial removal lemmas fail for ordered binary matrices, via one large pattern; Lean checks it.","consequence":"Polynomial removal fails, limiting property testers for ordered matrices."},"detail":"/api/math/190"},{"id":"191","title":"A power improvement in the Heilbronn triangle lower bound","short":"Heilbronn triangles: a power gain","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"There are absolute eta, c1>0 and n0 such that for every n>=n0 there are n points in the unit square all of whose triangles have area >= c1 n^{-2+eta}. Hence the 'almost n^{-2}' formulation Delta(n) <= C_eps n^{-2+eps} for all eps is false.","verdict":"First rigorous polynomial improvement over the 1982 KPS log n factor, if correct; Lean check makes this very credible.","url":"/math#191","articleUrl":"/articles/openai-math#the-heilbronn-triangle-problem","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=191.%20A%20power%20improvement","manuscripts":[{"title":"A power improvement in the Heilbronn triangle lower bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/191.mp4","poster":"/films/math/191-poster.webp","captions":"/films/math/191.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":2,"why":"Heilbronn's triangle problem: the first power improvement on the 1982 lower bound; the exponent is tiny.","consequence":"The first power improvement in Heilbronn's problem; the exponent is negligible."},"detail":"/api/math/191"},{"id":"192","title":"Boolean functions violate the square-root degree bound by arbitrary factors","short":"The square-root degree bound fails","discipline":"Combinatorics","disciplineIndex":6,"accent":"#8BD17C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every C>0 there is n and a nonconstant f:{-1,1}^n->{-1,1} with sum_i fhat({i}) > C sqrt(deg f), where deg is real Fourier degree; equivalently sup sum_i |fhat({i})| / sqrt(deg f) = infinity. No growth rates (n, degree) are given.","verdict":"Non-quantitative: proof uses CLT-based finite averaging, so the dimension needed for a given C is uncontrolled;","url":"/math#192","articleUrl":"/articles/openai-math#five-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=192.%20Boolean%20functions%20violate","manuscripts":[{"title":"Unbounded Violations of the Square-Root Degree Bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf"}],"reel":{"src":"/films/math/192.mp4","poster":"/films/math/192-poster.webp","captions":"/films/math/192.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":38,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"Boolean functions can beat the square-root degree bound by any factor, against Gopalan–Servedio; Lean checks it.","consequence":"Correlation with the input sum can beat the square root of degree. Internal to Boolean analysis."},"detail":"/api/math/192"},{"id":"193","title":"Serre’s intersection-multiplicity conjecture","short":"Serre's intersection multiplicity, $\\chi > 0$","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Serre's positivity conjecture: for every regular local ring R (any characteristic, including ramified mixed characteristic) and nonzero finitely generated M, N with ℓ(M⊗N) 0.","verdict":"The Lean dir OAI/RingTheory/Multiplicity exists but no main_results entry for Serre; treat as unformalized.","url":"/math#193","articleUrl":"/articles/openai-math#serres-intersection-multiplicity-positivity-family-193","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=193.%20Serre%E2%80%99s%20intersection%2Dmultiplicity%20conjecture","manuscripts":[{"title":"Positivity of Serre's Intersection Multiplicity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/193.mp4","poster":"/films/math/193-poster.webp","captions":"/films/math/193.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":0,"why":"Serre's 1965 positivity of intersection multiplicities, finishing the ramified case with perfectoid tools; unformalized.","consequence":"Completes Serre's positivity in every characteristic, a cornerstone of intersection theory on regular schemes."},"detail":"/api/math/193"},{"id":"194","title":"Lech’s multiplicity conjecture","short":"Lech's conjecture, $e(R) \\le e(S)$","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"Lech's conjecture: for every flat local homomorphism (R,m) → (S,n) of nonzero Noetherian local rings, e(R) ≤ e(S) (Hilbert–Samuel multiplicity), in every dimension and characteristic.","verdict":"Lean (lean/docs/194.md): only a supporting characteristic-p comparison (Dutta multiplicity vs Hilbert–Samuel multiplicity over a complete domain;","url":"/math#194","articleUrl":"/articles/openai-math#lechs-conjecture-family-194","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=194.%20Lech%E2%80%99s%20multiplicity%20conjecture","manuscripts":[{"title":"Lech's multiplicity conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lechs-multiplicity-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/194.mp4","poster":"/films/math/194-poster.webp","captions":"/films/math/194.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":1,"why":"Lech's conjecture: multiplicity cannot drop along a flat map of local rings; Lean checks only a supporting lemma.","consequence":"Settles Lech's question on multiplicities. 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Internal to commutative algebra."},"detail":"/api/math/195"},{"id":"196","title":"A counterexample to Kaplansky’s zero-divisor conjecture","short":"Kaplansky's zero-divisor conjecture fails","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Kaplansky's zero-divisor conjecture is false: there is a finitely presented torsion-free group G with a finite 2-dimensional K(G,1) and nonzero α, β ∈ F2[G] with αβ = 0.","verdict":"Lean: ComparatorChallenges/TorsionFreeZeroDivisors.lean states ∃ finitely presented torsion-free G with a finite 2-dim CW K(G,1) (Hausdorff, path-connected, contractible…","url":"/math#196","articleUrl":"/articles/openai-math#kaplanskys-zero-divisor-conjecture-fails-family-196","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=196.%20A%20counterexample%20to","manuscripts":[{"title":"A Torsion-Free Group Algebra with Zero Divisors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/196.mp4","poster":"/films/math/196-poster.webp","captions":"/films/math/196.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":94,"tier":"Huge if true","importance":4,"advance":4,"consequences":3,"surprise":3,"confidence":3,"why":"Kaplansky's zero-divisor conjecture fails: a torsion-free group algebra over F2 with zero divisors. Lean checks it.","consequence":"Ends a 70-year conjecture about group rings and forces the field to rethink Kaplansky's remaining conjectures (units fell in 2021)."},"detail":"/api/math/196"},{"id":"197","title":"A torsion-free group algebra that is not directly finite","short":"Kaplansky's direct finiteness fails","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Kaplansky's direct-finiteness conjecture is false: a finitely presented torsion-free (non-sofic) group G with finite 2-dim K(G,1) and a, b ∈ F2[G] with ab = 1 ≠ ba (197_0);","verdict":"Lean formalizes the TORSION versions only: KaplanskyFinitelyPresented.lean (finite char-2 field, finitely presented group with an element of odd prime order;","url":"/math#197","articleUrl":"/articles/openai-math#kaplanskys-direct-finiteness-surjunctivity-and-the-determinant-conjecture-family-197","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=197.%20A%20torsion%2Dfree%20group","manuscripts":[{"title":"A Torsion-Free Group Algebra That Is Not Directly Finite","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026/direct-finiteness.pdf"},{"title":"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf"},{"title":"A Counterexample to the Group-Ring Determinant Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf"},{"title":"A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf"}],"reel":{"src":"/films/math/197.mp4","poster":"/films/math/197-poster.webp","captions":"/films/math/197.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":100,"tier":"Huge if true","importance":4,"advance":4,"consequences":4,"surprise":3,"confidence":2,"why":"A group algebra that is not directly finite, which yields the first known non-sofic group; 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Lean checks the bounds.","consequence":"Bass's finitistic dimension conjecture is false, which removes a known route to the Nakayama conjecture."},"detail":"/api/math/198"},{"id":"199","title":"Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures","short":"Auslander–Reiten, Tachikawa, Nakayama fail","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Over k = F2(q,H1,H2): (a) a finite-dimensional algebra Λ and nonprojective module Z with Ext^i(Z,Z) = Ext^i(Z,Λ) = 0 for all i > 0 — counterexamples to the Auslander–Reiten conjecture and the Gorenstein-projective conjecture;","verdict":"Lean: AuslanderReiten.lean (OAI.ArExplicit.Statement.main, OAI/Algebra/AuslanderReiten/All.lean:738, also records A/rad A ≅ k⁸, rad⁴ ≠ 0 and field-extension stability)…","url":"/math#199","articleUrl":"/articles/openai-math#auslanderreiten-tachikawa-and-the-nakayama-conjectures-fail-family-199","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=199.%20Counterexamples%20to%20Auslander","manuscripts":[{"title":"An explicit counterexample to the Auslander-Reiten conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf"},{"title":"A counterexample to Tachikawa's second conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/199.mp4","poster":"/films/math/199-poster.webp","captions":"/films/math/199.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":3,"why":"Counterexamples to the Auslander–Reiten, Tachikawa and Nakayama conjectures on rigid modules; Lean checks the core.","consequence":"Several rigidity conjectures in the representation theory of algebras are false, including Nakayama's."},"detail":"/api/math/199"},{"id":"200","title":"Eisenbud–Green–Harris and lex-plus-powers","short":"Eisenbud–Green–Harris and lex-plus-powers","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-0 field: for any homogeneous ideal I ⊂ S containing a regular sequence of degrees 2 ≤ a1 ≤ … ≤ an, the lex-plus-powers ideal J (lex segment plus pure powers x_i^{a_i}) exists with…","verdict":"Char 0 only (positive characteristic not claimed). Two independent proofs (Betti version 22 pp;","url":"/math#200","articleUrl":"/articles/openai-math#eisenbudgreenharris-and-lex-plus-powers-family-200","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=200.%20Eisenbud%20Green%20Harris","manuscripts":[{"title":"The Artinian Lex-Plus-Powers Betti Theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf"},{"title":"Commuting Division-Coefficient Forms and the Artinian Eisenbud--Green--Harris Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/200.mp4","poster":"/films/math/200-poster.webp","captions":"/films/math/200.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Eisenbud–Green–Harris: lex-plus-powers ideals have extremal growth, in characteristic 0, with two proofs. No Lean.","consequence":"A Macaulay-type theorem for complete intersections, with applications to Cayley–Bacharach."},"detail":"/api/math/200"},{"id":"201","title":"A counterexample to Kurosh’s division-ring problem","short":"Kurosh's problem for division rings","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Kurosh problem for division rings has a negative answer: a countable division ring D of char 0, algebraic over its centre F, generated by two elements over F, with [D:F] = ∞.","verdict":"Lean dir OAI/RingTheory/Kurosh exists but no main_results entry and no doc for 201 — treat as unformalized.","url":"/math#201","articleUrl":"/articles/openai-math#kuroshs-problem-for-division-rings-fails-family-201","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=201.%20A%20counterexample%20to","manuscripts":[{"title":"A Counterexample to Kurosh’s Division-Ring Problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/201.mp4","poster":"/films/math/201-poster.webp","captions":"/films/math/201.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"Kurosh's problem: an algebraic division ring that is not locally finite; 45 unformalized pages.","consequence":"Answers Kurosh's Burnside-type question for division rings. Internal to ring theory."},"detail":"/api/math/201"},{"id":"202","title":"The blockwise Alperin weight conjecture","short":"Alperin's weight conjecture, every block","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Numerical blockwise Alperin weight conjecture for every prime p and every finite group G: for every p-block B, l(B) (number of irreducible Brauer characters) equals the number of G-conjugacy classes of B-weights.","verdict":"RED FLAG for extra scrutiny (not error): the proof does NOT use the classification of finite simple groups or inductive conditions.","url":"/math#202","articleUrl":"/articles/openai-math#alperins-weight-conjecture-family-202","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=202.%20The%20blockwise%20Alperin","manuscripts":[{"title":"The Blockwise Alperin Weight Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/202.mp4","poster":"/films/math/202-poster.webp","captions":"/films/math/202.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":86,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":3,"confidence":0,"why":"Alperin's weight conjecture blockwise, without the classification of finite simple groups; striking and unformalized.","consequence":"Alperin's weight conjecture, and a classification-free method others could reuse in modular representation theory."},"detail":"/api/math/202"},{"id":"203","title":"Donovan's conjecture over fields and complete mixed-characteristic DVRs","short":"Donovan's conjecture","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Donovan's conjecture: over each fixed algebraically closed field K of characteristic p (every prime, including p = 2), blocks of finite group algebras with defect groups of order ≤ M fall into finitely many Morita equivalence classes;","verdict":"Uses CFSG (reduces to quasisimple groups; Lie-type case via tilting objects on two-flag spaces after Eteve, Harish-Chandra moves for classical groups of unbounded rank).","url":"/math#203","articleUrl":"/articles/openai-math#donovans-conjecture-family-203","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=203.%20Donovan's%20conjecture%20over","manuscripts":[{"title":"Donovan's Conjecture over Algebraically Closed Fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields-September-24-2026/main.pdf"},{"title":"Integral Donovan Finiteness over Witt Vectors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integral-Donovan-Finiteness-over-Witt-Vectors-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/203.mp4","poster":"/films/math/203-poster.webp","captions":"/films/math/203.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":1,"confidence":0,"why":"Donovan's conjecture: bounded defect gives finitely many block types; uses the classification. No Lean.","consequence":"Finiteness of block types for bounded defect, a structural result many block-theory papers assume."},"detail":"/api/math/203"},{"id":"204","title":"Tensor saturation for even spin groups","short":"Saturation for $\\mathrm{Spin}(2n)$","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Saturation (factor 1) for Spin(2n), n ≥ 2: if λ+μ+ν lies in the root lattice and (V(Nλ)⊗V(Nμ)⊗V(Nν))^G ≠ 0 for some N ≥ 1, then (V(λ)⊗V(μ)⊗V(ν))^G ≠ 0 — the type-D part of the Kapovich–Millson simply-laced saturation conjecture.","verdict":"Type D only (E6–E8 remain). 46 pp. No Lean (despite a RepresentationTheory/Tensor dir; no main result).","url":"/math#204","articleUrl":"/articles/openai-math#saturation-for-mathrmspin2n-family-204","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=204.%20Tensor%20saturation%20for","manuscripts":[{"title":"Tensor saturation for even spin groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026/Tensor-Saturation-for-Even-Spin-Groups-September-24-2026.pdf"}],"reel":{"src":"/films/math/204.mp4","poster":"/films/math/204-poster.webp","captions":"/films/math/204.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Tensor saturation for even spin groups, extending Knutson–Tao beyond type A; exceptional types remain. No Lean.","consequence":"Saturation in one more Lie type. Internal."},"detail":"/api/math/204"},{"id":"205","title":"Saxl’s conjecture and universal tensor squares","short":"Saxl's conjecture","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) Saxl's conjecture: for every m ≥ 1, with ρ_m = (m, m−1, …, 1), the tensor square of the staircase Specht module S^{ρ_m} contains every irreducible of S_{m(m+1)/2} (all Kronecker coefficients g(ρ_m,ρ_m,μ) > 0), proved inside one cyclically generated…","verdict":"Lean: Saxl.lean (OAI.Saxl.saxl_conjecture at OAI/RepresentationTheory/Saxl/Main.lean:88, via saxl_of_cyclic_support) and UniversalTensorSquares.lean;","url":"/math#205","articleUrl":"/articles/openai-math#saxls-conjecture-family-205","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=205.%20Saxl%E2%80%99s%20conjecture%20and","manuscripts":[{"title":"Universal Tensor Squares for Symmetric Groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf"},{"title":"A Cyclic Polytabloid Proof of Saxl's Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/205.mp4","poster":"/films/math/205-poster.webp","captions":"/films/math/205.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":3,"why":"Saxl's conjecture: the staircase representation's tensor square contains every irreducible; Lean checks it.","consequence":"Settles Saxl's conjecture on Kronecker coefficients. Internal to algebraic combinatorics."},"detail":"/api/math/205"},{"id":"206","title":"Finite lattice representation and undecidability","short":"Finite lattice representation fails","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Finite lattice representation problem has a negative answer: there is a finite lattice that is not the congruence lattice of any finite algebra (of any finite signature);","verdict":"Lean (lean/docs/206.md): only the coloured-graph characterization equivalence (FiniteCongruenceGraph.lean) — the negative answer and undecidability are NOT formalized.","url":"/math#206","articleUrl":"/articles/openai-math#finite-lattice-representation-fails-family-206","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=206.%20Finite%20lattice%20representation","manuscripts":[{"title":"Finite congruence lattices: characterization and undecidability","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/paper.pdf"},{"title":"A negative solution to the finite lattice representation problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/206.mp4","poster":"/films/math/206-poster.webp","captions":"/films/math/206.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":3,"confidence":1,"why":"Not every finite lattice is a congruence lattice of a finite algebra, and the question is undecidable; Lean checks less.","consequence":"Settles the finite lattice representation problem negatively and shows no algorithm can decide it."},"detail":"/api/math/206"},{"id":"207","title":"The ℓ¹-Bass conjecture for all discrete groups","short":"The Bass trace conjecture, every group","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) ℓ¹-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over ℓ¹(G) are supported on finitely many conjugacy classes of finite-order elements.","verdict":"Lean formalizes (b) (the algebraic companion): BassTrace.lean — OAI.BassTrace.RightProjective.bassTraceModules_vanishing_and_support at…","url":"/math#207","articleUrl":"/articles/openai-math#the-bass-trace-conjecture-and-kaplanskys-idempotents-family-207","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=207.%20The%20%E2%84%93%C2%B9%2DBass%20conjecture","manuscripts":[{"title":"The ℓ¹-Bass Conjecture for Discrete Groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf"},{"title":"The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf"}],"reel":{"src":"/films/math/207.mp4","poster":"/films/math/207-poster.webp","captions":"/films/math/207.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":2,"confidence":2,"why":"The ℓ¹ Bass conjecture for all discrete groups, giving Kaplansky's idempotent conjecture in char 0; Lean checks the algebraic part.","consequence":"Kaplansky's idempotent conjecture in characteristic 0 for every torsion-free group follows."},"detail":"/api/math/207"},{"id":"208","title":"Finite symmetric tensor categories and the Verlinde tower","short":"Symmetric tensor categories in char $p$","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Every finite symmetric tensor category over an algebraically closed field of characteristic p > 0 (including p = 2) admits a fibre functor (k-linear exact faithful strong symmetric monoidal) to some higher Verlinde category Ver_{p^n}(k) — the finite case of…","verdict":"Finite case only (moderate-growth general case not claimed). 26 pp. The Lean challenge DeligneDrinfeld.lean (OAI/Algebra/Drinfeld) is not listed for this family in…","url":"/math#208","articleUrl":"/articles/openai-math#finite-symmetric-tensor-categories-family-208","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=208.%20Finite%20symmetric%20tensor","manuscripts":[{"title":"Fiber functors for finite symmetric tensor categories in positive characteristic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/208.mp4","poster":"/films/math/208-poster.webp","captions":"/films/math/208.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Finite symmetric tensor categories in characteristic p land in the Verlinde tower, as BEO conjectured; no Lean.","consequence":"Classifies the targets of finite symmetric tensor categories in characteristic p. Internal."},"detail":"/api/math/208"},{"id":"209","title":"Integral counterexamples to Gersten’s conjecture","short":"Gersten's conjecture fails integrally","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Integral Gersten conjecture fails: explicit two-dimensional ramified regular local rings A of mixed characteristic (0,5) (local rings of explicit hypersurfaces such as (X−Z)^5 − XY^4 + Y^5 + ζZ^5 = 0 at a point of the special fibre, with π ∈ m²) such that…","verdict":"Ramified only (π ∈ m²) and integral coefficients; the unramified mixed-characteristic case and finite-coefficient forms are untouched.","url":"/math#209","articleUrl":"/articles/openai-math#gerstens-conjecture-fails-integrally-family-209","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=209.%20Integral%20counterexamples%20to","manuscripts":[{"title":"An integral counterexample to Gersten's conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf"},{"title":"An integral degree-three Gersten counterexample","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf"}],"reel":{"src":"/films/math/209.mp4","poster":"/films/math/209-poster.webp","captions":"/films/math/209.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":2,"consequences":1,"surprise":3,"confidence":0,"why":"Gersten's conjecture fails for ramified regular local rings; the unramified case stays open. No Lean.","consequence":"Gersten's conjecture needs unramified hypotheses. Internal to algebraic K-theory."},"detail":"/api/math/209"},{"id":"210","title":"Foulkes' conjecture for sixth powers and quadratic stabilization","short":"Foulkes' conjecture for sixth powers","discipline":"Algebra","disciplineIndex":7,"accent":"#B4A7FF","kind":"partial","kindLabel":"Claimed partial result","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Foulkes' conjecture for a = 6: Sym^6(Sym^b V) ↪ Sym^b(Sym^6 V) GL(V)-equivariantly for every b ≥ 6 and every finite-dimensional V.","verdict":"Only a = 6 for Foulkes (not the full conjecture). Lean: FoulkesHowe.lean — OAI.Problem346.canonical_foulkes_howe_surjective at…","url":"/math#210","articleUrl":"/articles/openai-math#four-narrower-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=210.%20Foulkes'%20conjecture%20for","manuscripts":[{"title":"Foulkes' conjecture for the sixth symmetric power","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf"},{"title":"Quadratic stabilization of the canonical Foulkes--Howe map","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/210.mp4","poster":"/films/math/210-poster.webp","captions":"/films/math/210.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":35,"tier":"Incremental","importance":3,"advance":1,"consequences":0,"surprise":1,"confidence":2,"why":"Foulkes' plethysm conjecture for sixth powers only; the general conjecture stays open. Lean checks part.","consequence":"One more case of Foulkes' conjecture. Nothing follows beyond it."},"detail":"/api/math/210"},{"id":"211","title":"The geometric phase diagram, diffusion, and spectra of random planar maps","short":"Random planar maps: surfaces and trees","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Spherical critical Fortuin–Kasteleyn planar maps: for fixed 04 GHP convergence to the Brownian CRT; for FK–Ising (q=2) and spanning-tree maps, stationary random walk converges to Liouville Brownian motion with clock exactly the number of edges, plus…","verdict":"762 pages across 7 papers that cite each other, including a metric-measure companion revised 2026-10-03 that the spectral paper uses as an input;","url":"/math#211","articleUrl":"/articles/openai-math#random-planar-maps-from-surfaces-to-trees","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=211.%20The%20geometric%20phase","manuscripts":[{"title":"Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026/fk-ising-walk-limit.pdf"},{"title":"Spectral convergence for critical FK–Ising planar maps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spectral-convergence-for-critical-FK-Ising-planar-maps-October-5-2026/spectral-convergence-critical-fk-ising-planar-maps.pdf"},{"title":"A Linear Clock for Random Walk on Tree-Weighted Planar Maps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf"},{"title":"Canonical conformal limits of subcritical FK planar maps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026/main.pdf"},{"title":"The critical Liouville quantum sphere and geometric limits of FK maps at q=4","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026/main.pdf"},{"title":"Metric-measure limits of subcritical FK and spanning-tree planar maps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026/main.pdf"},{"title":"Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/211.mp4","poster":"/films/math/211-poster.webp","captions":"/films/math/211.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":62,"tier":"Solid","importance":3,"advance":2,"consequences":3,"surprise":1,"confidence":0,"why":"Random planar maps with critical FK clusters converge to Liouville quantum gravity, as physics predicts; 762 interlocking pages.","consequence":"Puts the physics picture of random surfaces with critical clusters on rigorous ground, including random walk as Liouville Brownian motion."},"detail":"/api/math/211"},{"id":"212","title":"Planar first-passage geometry and the absence of bigeodesics","short":"No bigeodesics in planar first passage","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(1) Planar FPP on Z^2 with iid nonnegative nonatomic edge weights, assuming the minimum of four independent weights has a finite second moment, has a.s. no bi-infinite geodesic (all bigeodesics excluded simultaneously).","verdict":"Moment condition E[min of 4]^2<∞ (the standard Cox–Durrett type condition). Lean (PlanarFirstPassage in yaml main_results;","url":"/math#212","articleUrl":"/articles/openai-math#first-passage-percolation-no-bigeodesics-and-a-smooth-limit-shape","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=212.%20Planar%20first%2Dpassage%20geometry","manuscripts":[{"title":"No bigeodesics in planar first-passage percolation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/No-bigeodesics-in-planar-first-passage-percolation-September-24-2026/main.pdf"},{"title":"Strict convexity and differentiability of the planar exponential first-passage limit shape","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/212.mp4","poster":"/films/math/212-poster.webp","captions":"/films/math/212.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":1,"why":"Planar first-passage percolation has no doubly infinite fastest routes, a long-standing FPP question; Lean checks less.","consequence":"Settles a central structural question of first-passage percolation and disordered systems in the plane."},"detail":"/api/math/212"},{"id":"213","title":"Critical percolation on every quasi-transitive graph","short":"$\\theta(p_c) = 0$ on quasi-transitive graphs","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(1) For Bernoulli bond percolation on every infinite connected locally finite quasi-transitive graph with p_c<1, a.s. no infinite cluster at p_c (Benjamini–Schramm criticality conjecture, bond form; includes Z^d for all d≥2).","verdict":"Priority: the famous Z^3 case is not first here (Leder/Claude, Bou-Rabee); the OpenAI Z^3 paper is explicitly a 'self-contained cubic argument' with antecedent credit.","url":"/math#213","articleUrl":"/articles/openai-math#thetap_c--0-and-who-got-there-first","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=213.%20Critical%20percolation%20on","manuscripts":[{"title":"Critical bond and site percolation on the cubic lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf"},{"title":"No percolation at criticality on quasi-transitive graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/213.mp4","poster":"/films/math/213-poster.webp","captions":"/films/math/213.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":4,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"No infinite cluster at criticality on every quasi-transitive graph; the famous Z^3 case was done first by others.","consequence":"Completes the critical-percolation picture on symmetric graphs beyond the cases settled earlier."},"detail":"/api/math/213"},{"id":"214","title":"The Benjamini–Schramm nonuniqueness conjecture","short":"$p_c < p_u$ on nonamenable graphs","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph: sup_{p<p_c}||T_p||_{2→2} = ||T_{p_c}|| < ∞ and p_c < p_{2→2} ≤ p_u, so there is a nonempty interval with infinitely many infinite clusters;","verdict":"Lean challenge BenjaminiSchramm/CayleyPercolation with docs/214.md (standard axioms) but not in yaml main_results.","url":"/math#214","articleUrl":"/articles/openai-math#p_c--p_u-on-every-nonamenable-graph","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=214.%20The%20Benjamini%20Schramm","manuscripts":[{"title":"Nonuniqueness of percolation on nonamenable quasi-transitive graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/214.mp4","poster":"/films/math/214-poster.webp","captions":"/films/math/214.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"Benjamini–Schramm: percolation on nonamenable graphs has a phase with infinitely many infinite clusters; Lean checks it.","consequence":"Confirms the expected three-phase picture of percolation on nonamenable graphs."},"detail":"/api/math/214"},{"id":"215","title":"Canonical O(3) continuum limit and exact O(4) mass asymptotics","short":"Polyakov's mass gap for 2D $O(n)$ models","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(1) Exponential decay of two-point correlations for the 2D nearest-neighbor O(n) model, every n≥3, every finite β>0, uniform over finite free-boundary subgraphs and edge strengths in [0,β] (Polyakov's mass-generation conjecture, spin-correlation form).","verdict":"The 20-page exponential-decay theorem is formalized (ClassicalON, in yaml main_results, standard axioms;","url":"/math#215","articleUrl":"/articles/openai-math#polyakovs-mass-gap-for-the-2d-on-models","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=215.%20Canonical%20O%203","manuscripts":[{"title":"The canonical massive continuum limit of the two-dimensional O(3) model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026/massive-continuum-o3.pdf"},{"title":"An Isolated Particle Pole for the Two-Dimensional O(3) Spin Field","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Isolated-Particle-Pole-for-the-Two-Dimensional-O3-Spin-Field-October-4-2026/o3-particle-pole.pdf"},{"title":"Exact mass asymptotics for the two-dimensional O(4) lattice model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026/exact-mass-o4.pdf"},{"title":"Sharp mass bounds for the two-dimensional O(4) model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-mass-bounds-for-the-two-dimensional-O4-model-September-23-2026/paper.pdf"},{"title":"Exponential decay in two-dimensional classical O(n) models","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/215.mp4","poster":"/films/math/215-poster.webp","captions":"/films/math/215.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":94,"tier":"Huge if true","importance":4,"advance":4,"consequences":3,"surprise":3,"confidence":2,"why":"Polyakov's prediction: 2D O(n) spin models with n≥3 have exponential decay at every temperature; Lean checks that part.","consequence":"A rigorous mass gap for 2D spin models with n≥3, the lattice cousin of the Yang–Mills mass gap; the continuum limit papers build on it."},"detail":"/api/math/215"},{"id":"216","title":"Critical and near-critical XY scaling and BKT universality","short":"The XY model: BKT fine structure","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For the square-lattice nearest-neighbor cosine XY model: critical correlation C(r)=B r^{-1/4}(log r)^{1/8}(1+o(1)) (free-box limit, axis separation); critical exponent exactly 1/4; BKT essential singularity √(β_c-β)·log(1/m(β)) → A>0;","verdict":"461 pages, 6 papers; the center-magnetization and spin-field papers are explicitly conditional on 'critical-height, local-renormalization and spin-field inputs from the…","url":"/math#216","articleUrl":"/articles/openai-math#the-xy-models-fine-structure","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=216.%20Critical%20and%20near%2Dcritical","manuscripts":[{"title":"The critical logarithmic correction for the planar XY model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026/paper.pdf"},{"title":"Critical Center Magnetization in the Planar XY Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026/paper.pdf"},{"title":"The Critical Spin Field of the Planar XY Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf"},{"title":"Essential Singularity of the Correlation Length in the Planar XY Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf"},{"title":"The critical correlation exponent of the planar XY model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026/paper.pdf"},{"title":"BKT universality for height and planar spin fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/BKT-universality-for-height-and-planar-spin-fields-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/216.mp4","poster":"/films/math/216-poster.webp","captions":"/films/math/216.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"The fine BKT structure of the XY transition, including the exp(c/√t) correlation length; conditional pieces, no Lean.","consequence":"Rigorous fine structure of the BKT transition for the cosine XY model."},"detail":"/api/math/216"},{"id":"217","title":"The low-temperature Sherrington–Kirkpatrick fluctuation law","short":"Low-temperature SK fluctuations","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For the zero-field Gaussian Ising SK model at fixed β>1: Var(log Z_n) ~ c_β n^{1/3} with c_β>0 (so standard deviation n^{1/6}), and (log Z_n - E log Z_n)/sd converges in law along all n to a uniquely characterized nondegenerate limit;","verdict":"No formalization; limiting law 'uniquely characterized' but not explicit (not identified with a known distribution). Zero-field Gaussian couplings only. 126+80 pages.","url":"/math#217","articleUrl":"/articles/openai-math#low-temperature-sk-fluctuations","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=217.%20The%20low%2Dtemperature%20Sherrington","manuscripts":[{"title":"The low-temperature Sherrington–Kirkpatrick free-energy limiting law","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf"},{"title":"The low-temperature Sherrington–Kirkpatrick fluctuation scale","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf"}],"reel":{"src":"/films/math/217.mp4","poster":"/films/math/217-poster.webp","captions":"/films/math/217.vtt","duration":19.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Free-energy fluctuations of the SK spin glass at low temperature scale as n^{1/6}, as physicists predicted; no Lean.","consequence":"Sharp fluctuation exponent for the SK model; the limit law is not identified."},"detail":"/api/math/217"},{"id":"218","title":"Conformal universality for weakly interacting and random-bond Ising models","short":"Ising universality under weak disorder","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Square-lattice Ising with sufficiently small square-symmetric finite-range even multispin perturbations: critical bulk spin/energy correlations converge to the conformally invariant (Chelkak–Hongler–Izyurov) limits;","verdict":"Perturbative (sufficiently weak) only. The log-fluctuation paper is conditional on 'stated deterministic critical-reference estimates'.","url":"/math#218","articleUrl":"/articles/openai-math#weakly-perturbed-and-disordered-ising-models","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=218.%20Conformal%20universality%20for","manuscripts":[{"title":"Quenched SLE₃ limits for general weak random-bond Ising models","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quenched-SLE3-limits-for-general-weak-random-bond-Ising-models-October-5-2026/general-weak-random-bond-ising.pdf"},{"title":"Quenched SLE₃ Universality for the Weak Random-Bond Ising Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quenched-SLE3-Universality-for-the-Weak-Random-Bond-Ising-Model-October-5-2026/quenched-sle3-weak-random-bond-ising.pdf"},{"title":"Logarithmic Relative Fluctuations in the Weakly Disordered Planar Ising Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Logarithmic-Relative-Fluctuations-in-the-Weakly-Disordered-Planar-Ising-Model-October-5-2026/critical-relative-second-moment-weak-random-bond-ising.pdf"},{"title":"Conformal universality of bulk Ising correlations under weak interactions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conformal-universality-of-bulk-Ising-correlations-under-weak-interactions-September-23-2026/paper.pdf"},{"title":"SLE3 universality for weak finite-range Ising interactions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/SLE3-universality-for-weak-finite-range-Ising-interactions-September-23-2026/paper.pdf"},{"title":"Buffered comparison and stopping-band resolution in critical Ising","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/218.mp4","poster":"/films/math/218-poster.webp","captions":"/films/math/218.vtt","duration":19.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":1,"why":"Conformal invariance for Ising models with weak extra interactions or disorder; perturbative only.","consequence":"Extends Ising conformal invariance beyond exactly solvable models, in the perturbative regime."},"detail":"/api/math/218"},{"id":"219","title":"GOE bulk universality for regular graphs with weak Anderson disorder","short":"GOE statistics for random regular graphs","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every fixed d≥3 and fixed energy E in the open Kesten–McKay bulk, the unfolded eigenvalue point process of the adjacency matrix of a uniform random simple d-regular graph converges to the GOE bulk process (all admissible n).","verdict":"Laplace-functional (fixed-energy) formulation; disorder part needs disorder strength small depending on d and distance from edges. No Lean docs.","url":"/math#219","articleUrl":"/articles/openai-math#goe-statistics-for-random-regular-graphs","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=219.%20GOE%20bulk%20universality","manuscripts":[{"title":"Fixed-energy universality for weak Anderson disorder on random regular graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026/fixed-energy-universality-weak-anderson-disorder-random-regular-graphs.pdf"},{"title":"GOE bulk universality for fixed-degree random regular graphs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/219.mp4","poster":"/films/math/219-poster.webp","captions":"/films/math/219.vtt","duration":20.1,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Random regular graphs of any fixed degree have GOE eigenvalue spacings, the sparse end of universality; no Lean.","consequence":"Random-matrix statistics for sparse regular graphs, the long-missing sparse end of universality."},"detail":"/api/math/219"},{"id":"220","title":"Directional zero–one laws beyond iid environments and iid ballisticity","short":"Random walk in random environment: speed","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Nearest-neighbor random walk in random environment on Z^d: (1) d≥3, iid strictly elliptic environments (no uniform lower bound): P(X_n·ℓ→∞) ∈ {0,1} for each direction ℓ;","verdict":"Ballisticity is in formalization.yaml main_results (DirectionalBallisticity, standard axioms); DirectionalWalk and VelocityHemisphere challenges also exist.","url":"/math#220","articleUrl":"/articles/openai-math#random-walk-in-random-environment-escape-implies-speed","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=220.%20Directional%20zero%20one","manuscripts":[{"title":"A directional zero–one law for finite-range-dependent random environments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf"},{"title":"A directional zero–one law under strict ellipticity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf"},{"title":"Directional transience implies ballisticity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Directional-transience-implies-ballisticity-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/220.mp4","poster":"/films/math/220-poster.webp","captions":"/films/math/220.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":1,"confidence":3,"why":"Random walks in random environments that escape in a direction must do so at positive speed; Lean checks it.","consequence":"Escape forces speed for walks in random environments in d≥2, settling a central RWRE question."},"detail":"/api/math/220"},{"id":"221","title":"The Mézard–Parisi formula for diluted spin glasses","short":"Mézard–Parisi for diluted spin glasses","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For Poisson-diluted Ising models with even arity p≥2, density α>0, in the Panchenko–Talagrand class (factorization e^θ=a(1+bΠf_ℓ), |bΠf_ℓ|<1, E[(-b)^n]≥0), with only first-moment integrability, the limit of (1/N)E log Z_N exists and equals inf over finite RSB…","verdict":"Restricted to the PT class: even arity, Poisson (not fixed-degree/random regular) graphs, positivity condition E[(-b)^n]≥0 — excludes odd-K SAT, hard constraints, Bethe…","url":"/math#221","articleUrl":"/articles/openai-math#the-mézardparisi-formula-for-diluted-spin-glasses","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=221.%20The%20M%C3%A9zard%20Parisi","manuscripts":[{"title":"The Mézard–Parisi formula for diluted spin glasses","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/221.mp4","poster":"/films/math/221-poster.webp","captions":"/films/math/221.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":2,"why":"The Mézard–Parisi formula for sparse spin glasses, in a restricted class of models; Lean checks it.","consequence":"Rigorous replica-symmetry-breaking free energies for sparse spin glasses, in a restricted class."},"detail":"/api/math/221"},{"id":"222","title":"Perceptron free energies and microscopic jamming exponents","short":"Perceptron jamming exponents","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"(1) Limiting free energy (explicit variational formula) of the Gaussian Ising perceptron for every bounded Borel log-potential, every positive temperature and density;","verdict":"Positive temperature only for (1)-(3); zero-temperature Ising capacity not claimed here.","url":"/math#222","articleUrl":"/articles/openai-math#perceptrons-and-the-jamming-exponents","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=222.%20Perceptron%20free%20energies","manuscripts":[{"title":"The free energy of the Ising random perceptron","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf"},{"title":"Microscopic jamming in the negative spherical perceptron","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf"},{"title":"The spherical perceptron with bi-orthogonally invariant disorder","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026.pdf"},{"title":"The free energy of the spherical random perceptron","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf"}],"reel":{"src":"/films/math/222.mp4","poster":"/films/math/222-poster.webp","captions":"/films/math/222.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":52,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":2,"confidence":1,"why":"Perceptron free energies and the jamming exponents predicted by full replica symmetry breaking.","consequence":"Rigorous capacity formulas for the perceptron, the simplest neural-network capacity model, and its jamming exponents."},"detail":"/api/math/222"},{"id":"223","title":"Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling","short":"Cardy on $\\mathbb Z^2$ and FK interfaces","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Critical square-lattice random-cluster (FK) Dobrushin interfaces converge to chordal SLE_κ with κ=4π/arccos(-√q/2): for 1≤q≤4 in bounded Jordan domains, with complete nested loops → whole-plane CLE_κ; for 0<q<1 in smooth Jordan domains;","verdict":"602 pages, unformalized. Cardy's formula for Z^2 bond percolation and FK conformal invariance for all q∈[1,4] would be among the biggest results in 2D statistical…","url":"/math#223","articleUrl":"/articles/openai-math#cardys-formula-on-the-square-lattice-and-fk-interfaces-for-q-le-4","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=223.%20Random%2Dcluster%20interfaces%20critical","manuscripts":[{"title":"Square-lattice FK interfaces and nested loops for 1 <= q < 4","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf"},{"title":"Self-dual random-cluster interfaces below one","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf"},{"title":"Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf"},{"title":"Thermal FK–Ising interfaces and massive SLE","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf"},{"title":"Natural Occupation Measures for Critical Square-Lattice FK Interfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf"},{"title":"Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf"}],"reel":{"src":"/films/math/223.mp4","poster":"/films/math/223-poster.webp","captions":"/films/math/223.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":82,"tier":"Major","importance":4,"advance":3,"consequences":3,"surprise":2,"confidence":0,"why":"Conformal invariance of square-lattice random-cluster models for q in [1,4], including Cardy's formula; 602 unchecked pages.","consequence":"Would extend Smirnov's conformal invariance to the square lattice for a whole range of q; among the largest claims in 2D statistical mechanics."},"detail":"/api/math/223"},{"id":"224","title":"Critical and quenched near-critical universality for Poisson–Voronoi percolation","short":"Voronoi percolation obeys Cardy","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Annealed Cardy's formula for critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral; expected number of pivotal cells for a unit-square crossing ~ c ε^{-3/4};","verdict":"Annealed (averaged over the tessellation) Cardy, not quenched; two of three papers take Cardy as an input. No Lean. No public expert reaction found.","url":"/math#224","articleUrl":"/articles/openai-math#voronoi-percolation-obeys-cardy","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=224.%20Critical%20and%20quenched","manuscripts":[{"title":"From critical crossings to quenched near-critical universality in Voronoi percolation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf"},{"title":"A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf"},{"title":"Cardy’s formula for critical Poisson–Voronoi percolation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cardys-formula-for-critical-Poisson-Voronoi-percolation-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/224.mp4","poster":"/films/math/224-poster.webp","captions":"/films/math/224.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Cardy's formula for Voronoi percolation, averaged over the tessellation; no Lean.","consequence":"Cardy's formula for a random-lattice model, averaged. Internal."},"detail":"/api/math/224"},{"id":"225","title":"Gaussian free field limits throughout the balanced six-vertex regime","short":"The six-vertex model's Gaussian free field","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Height function of the square-lattice six-vertex model with a=b=1, 0<c≤2 (Δ=(2-c^2)/2 ∈ [-1,1)), in the plane state from balanced tori, converges to a multiple of the GFF with squared multiplier 1/arcsin(c/2) (unit height jumps, Green kernel…","verdict":"Single 93-page paper, plane state from balanced tori (not arbitrary domains/boundary conditions). No Lean. No public expert reaction found.","url":"/math#225","articleUrl":"/articles/openai-math#the-six-vertex-models-gaussian-free-field","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=225.%20Gaussian%20free%20field","manuscripts":[{"title":"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/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026/The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026.pdf"}],"reel":{"src":"/films/math/225.mp4","poster":"/films/math/225-poster.webp","captions":"/films/math/225.vtt","duration":20.1,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"The six-vertex height function is a Gaussian free field throughout the balanced critical range; one paper, no Lean.","consequence":"Confirms the Bethe-ansatz prediction for the six-vertex model's height fluctuations."},"detail":"/api/math/225"},{"id":"226","title":"The double-dimer loop ensemble converges to CLE4","short":"Double dimers become $\\mathrm{CLE}_4$","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"The complete double-dimer loop ensemble (superposition of two independent dimer covers) of the Temperleyan square lattice in the upper half-plane converges to nested CLE_4, matching every macroscopic loop as an unparametrized curve, along the full mesh limit.","verdict":"Half-plane Temperleyan geometry only; builds on Dubédat and Basok–Chelkak's topological convergence and upgrades to curve convergence. 33 pages. No Lean.","url":"/math#226","articleUrl":"/articles/openai-math#double-dimers-become-cle_4","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=226.%20The%20double%2Ddimer%20loop","manuscripts":[{"title":"The curve scaling limit of half-plane double dimers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/226.mp4","poster":"/films/math/226-poster.webp","captions":"/films/math/226.vtt","duration":19.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Loops from two overlaid domino tilings converge to CLE4, in half-plane Temperleyan domains; no Lean.","consequence":"Upgrades double-dimer loop convergence to CLE4. Internal."},"detail":"/api/math/226"},{"id":"227","title":"Critical SK autocorrelation processes and dynamics across the temperature transition","short":"SK Glauber dynamics across $\\beta = 1$","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Zero-field Gaussian SK heat-bath (Glauber) dynamics, rate one per spin: for 01 a stretched-exponential obstruction at time exp(n^{1/10000}) from typical starts and an upper bound exp(n^{1-1/40000000});","verdict":"Lean (docs/227.md) covers: the β1 stretched-exponential obstruction. The universal autocorrelation limits are unformalized. 720 pages in 8 papers.","url":"/math#227","articleUrl":"/articles/openai-math#spin-glass-dynamics-across-the-sk-transition","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=227.%20Critical%20SK%20autocorrelation","manuscripts":[{"title":"Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universality-of-critical-quench-autocorrelations-in-the-Sherrington-Kirkpatrick-model-October-5-2026/main.pdf"},{"title":"Functional universality of critical SK autocorrelations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf"},{"title":"A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf"},{"title":"Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf"},{"title":"Critical slowing down in the Sherrington–Kirkpatrick model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf"},{"title":"Stretched-exponential barriers for typical SK initial states","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf"},{"title":"A typical-start upper bound for low-temperature SK Glauber dynamics","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf"},{"title":"Critical mixing in the Sherrington–Kirkpatrick model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/227.mp4","poster":"/films/math/227-poster.webp","captions":"/films/math/227.vtt","duration":20.3,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":1,"confidence":1,"why":"Glauber dynamics for the SK model mixes fast all the way to β=1, with the n^{2/3} critical slowdown; Lean checks parts.","consequence":"Guarantees for Glauber sampling of spin glasses throughout the high-temperature phase."},"detail":"/api/math/227"},{"id":"228","title":"Continuum phase transitions for radial pair potentials","short":"Phase transition for a radial potential","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"Construction of stable radial pair potentials φ in R^3 whose canonical free energy has a strict downward jump in its β-derivative at one common finite β_c throughout an open interval of densities: (a) with a divergent repulsive core, an attractive interval…","verdict":"The potentials are engineered for the proof (not Lennard-Jones or physically motivated), and the transition is a temperature singularity at fixed density in the…","url":"/math#228","articleUrl":"/articles/openai-math#four-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=228.%20Continuum%20phase%20transitions","manuscripts":[{"title":"A continuum temperature singularity for a radial pair potential","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026/paper.pdf"},{"title":"A radial continuum phase transition with algebraic decay","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/228.mp4","poster":"/films/math/228-poster.webp","captions":"/films/math/228.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"A continuum gas with a radial pair potential that has a phase transition; the potential is engineered, not physical.","consequence":"Shows continuum phase transitions can happen for a radial potential; says nothing about natural potentials."},"detail":"/api/math/228"},{"id":"229","title":"Exact three- and four-state reconstruction thresholds and four-state tree capacity","short":"Exact reconstruction thresholds on trees","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Exact Kesten–Stigum reconstruction threshold dλ^2>1 (nonreconstruction at equality) for the symmetric three-state broadcast channel (both signs of λ) on every b-ary tree and observed Poisson Galton–Watson trees, and for the ferromagnetic four-state Potts…","verdict":"The 4-state proof uses 'reproducible exact-arithmetic verification of polynomial inequalities' (computer-assisted).","url":"/math#229","articleUrl":"/articles/openai-math#potts-reconstruction-on-trees","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=229.%20Exact%20three%2D%20and","manuscripts":[{"title":"The Reconstruction Threshold for the Ferromagnetic Four-State Potts Model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Reconstruction-Threshold-for-the-Ferromagnetic-Four-State-Potts-Model-October-5-2026/four-state-potts.pdf"},{"title":"A Capacity Criterion for Four-State Potts Reconstruction on Trees","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Capacity-Criterion-for-Four-State-Potts-Reconstruction-on-Trees-October-5-2026/four-state-capacity.pdf"},{"title":"The exact reconstruction threshold for the three-state symmetric channel","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/229.mp4","poster":"/films/math/229-poster.webp","captions":"/films/math/229.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":1,"why":"Kesten–Stigum is the exact reconstruction threshold for 3 and 4 colors on trees; Lean checks only the easy direction.","consequence":"Pins the community-detection threshold for 3-community block models."},"detail":"/api/math/229"},{"id":"230","title":"Exact Hausdorff gauges for SLE","short":"The exact Hausdorff gauge of SLE","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"For 0<κ<8, d=1+κ/8, the gauge h(r)=r^d (log log 1/r)^{(2-d)/2} gives a.s. positive finite Hausdorff measure to every nontrivial compact positive-time segment of chordal SLE_κ, with finite expected measure in bounded disks;","verdict":"Lean (SLELowerPositivity) only proves the positivity half; finiteness not formalized. Answers Schramm's question with a different exponent than he suggested.","url":"/math#230","articleUrl":"/articles/openai-math#four-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=230.%20Exact%20Hausdorff%20gauges","manuscripts":[{"title":"An exact Hausdorff gauge for SLE","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exact-Hausdorff-gauge-for-SLE-September-25-2026/An-exact-Hausdorff-gauge-for-SLE-September-25-2026.pdf"},{"title":"An explicit exact Hausdorff gauge for SLE","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026/An-explicit-exact-Hausdorff-gauge-for-SLE-September-26-2026.pdf"}],"reel":{"src":"/films/math/230.mp4","poster":"/films/math/230-poster.webp","captions":"/films/math/230.vtt","duration":20.1,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":27,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":1,"confidence":1,"why":"The exact Hausdorff gauge for SLE curves, with an iterated-log correction; Lean checks the positivity half.","consequence":"Settles Schramm's gauge question for SLE. Nothing follows beyond it."},"detail":"/api/math/230"},{"id":"231","title":"The free uniform spanning forest is a factor of IID","short":"The free spanning forest is a factor of IID","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"On every infinite connected locally finite simple graph, the free uniform spanning forest is a factor of iid vertex labels via one Borel isomorphism-equivariant rule with no root;","verdict":"Lean (FreeUniformSpanningForest, StronglyRayleighDPP challenges, docs/231.md) covers both claims; not in yaml main_results. 19 pages. Factor of IID, not finitary.","url":"/math#231","articleUrl":"/articles/openai-math#the-free-uniform-spanning-forest-is-a-factor-of-iid","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=231.%20The%20free%20uniform","manuscripts":[{"title":"The free uniform spanning forest is a factor of IID","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026/The-free-uniform-spanning-forest-is-a-factor-of-IID-September-25-2026.pdf"}],"reel":{"src":"/films/math/231.mp4","poster":"/films/math/231-poster.webp","captions":"/films/math/231.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":3,"why":"The free uniform spanning forest is a factor of IID, answering Lyons; 19 Lean-checked pages.","consequence":"Answers Lyons's question and covers all strongly Rayleigh invariant processes."},"detail":"/api/math/231"},{"id":"232","title":"Gaussian fields and interfaces for triangular-lattice Lipschitz heights","short":"Lipschitz heights on the triangular lattice","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Triangular lattice: (1) uniform odd integer heights with increments 0,±2 and two-arc boundary ±1 converge to a universal multiple of the Dirichlet GFF (field part of Schramm's Problem 2.2);","verdict":"Interface result for the integer model (the SLE_4 half of Problem 2.2) is not claimed; only the field part. Tuned boundary amplitude for real heights is implicit.","url":"/math#232","articleUrl":"/articles/openai-math#lipschitz-height-functions","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=232.%20Gaussian%20fields%20and","manuscripts":[{"title":"Gaussian free-field limits of weighted integer Lipschitz heights","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-September-25-2026/paper.pdf"},{"title":"The Gaussian free field limit of integer Lipschitz heights with two-arc boundary data","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-September-25-2026/paper.pdf"},{"title":"Uniform real Lipschitz surfaces on the triangular lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-real-Lipschitz-surfaces-on-the-triangular-lattice-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/232.mp4","poster":"/films/math/232-poster.webp","captions":"/films/math/232.vtt","duration":19.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Random Lipschitz height functions on the triangular lattice converge to the Gaussian free field; no Lean.","consequence":"Gaussian free field limit for Lipschitz heights. Internal."},"detail":"/api/math/232"},{"id":"233","title":"The joint critical Ashkin–Teller current limit","short":"The critical Ashkin–Teller limit","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"On the critical Ashkin–Teller line, including the four-state Potts endpoint, in bounded Jordan domains (wired primal/free dual boundary), the height function converges to a GFF with the predicted coupling and both current-cluster collections converge to…","verdict":"Single 100-page paper; recent conjecture with a narrow audience. No Lean. No public expert reaction found.","url":"/math#233","articleUrl":"/articles/openai-math#four-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=233.%20The%20joint%20critical","manuscripts":[{"title":"The joint scaling limit of critical Ashkin-Teller currents","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-joint-scaling-limit-of-critical-Ashkin-Teller-currents-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/233.mp4","poster":"/films/math/233-poster.webp","captions":"/films/math/233.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":27,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":1,"confidence":0,"why":"The joint scaling limit of the critical Ashkin–Teller model; a narrow recent conjecture, no Lean.","consequence":"Identifies one scaling limit. Nothing follows beyond it."},"detail":"/api/math/233"},{"id":"234","title":"All-temperature pressure of orthogonally invariant Ising spin glasses","short":"Orthogonally invariant spin glasses","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For Ising spin glasses with coupling matrix O^T D O (O Haar orthogonal, deterministic spectrum converging to a compact law with extreme eigenvalues converging to its edges), the limiting pressure at every fixed temperature equals an explicit variational…","verdict":"Lean challenge InvariantIsing (Unconditional module, 495-line statement file) with docs/234.md; not in yaml main_results.","url":"/math#234","articleUrl":"/articles/openai-math#orthogonally-invariant-spin-glasses","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=234.%20All%2Dtemperature%20pressure%20of","manuscripts":[{"title":"All-temperature pressure for orthogonally invariant Ising spin glasses","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/234.mp4","poster":"/films/math/234-poster.webp","captions":"/films/math/234.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":2,"why":"The free energy of rotation-invariant Ising spin glasses at every temperature; Lean checks it.","consequence":"Free energies for spin glasses with non-Gaussian coupling structure, beyond the classic SK setting."},"detail":"/api/math/234"},{"id":"235","title":"Limiting random SAT thresholds, sharp variance and computability","short":"Random $k$-SAT: variance and computability","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For random k-SAT (independent uniformly signed clauses on distinct variables, with replacement): for every fixed k≥3 a finite positive limiting threshold α_k exists (sharp transition in probability);","verdict":"OpenAI explicitly credits Carenini with priority for threshold existence; this family is an alternative proof plus new variance and computability results.","url":"/math#235","articleUrl":"/articles/openai-math#random-k-sat-thresholds-with-someone-elses-priority","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=235.%20Limiting%20random%20SAT","manuscripts":[{"title":"A Limiting Satisfiability Threshold for Every Fixed Clause Size","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf"},{"title":"Linear Variance of the Random 3-SAT Hitting Time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf"},{"title":"Variance of the Random k-SAT Hitting Time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026/article.pdf"},{"title":"Computing the Random 3-SAT Threshold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf"}],"reel":{"src":"/films/math/235.mp4","poster":"/films/math/235-poster.webp","captions":"/films/math/235.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":3,"advance":1,"consequences":1,"surprise":1,"confidence":2,"why":"An alternative proof that random k-SAT thresholds converge, with variance and computability; existence is others' first.","consequence":"Variance and computability of SAT thresholds; no threshold value is computed."},"detail":"/api/math/235"},{"id":"236","title":"The exact factor-of-IID threshold for free Ising spins on trees","short":"Factor-of-IID Ising on trees, exactly","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For d≥3, β≥0, the free zero-field ferromagnetic Ising measure on the d-regular tree is a factor of IID iff tanh β ≤ 1/√(d-1) (including equality);","verdict":"In yaml main_results (FreeIsing, OAI.Problem367.free_ising_factor_iff_threshold, standard axioms) — strong signal. Ferromagnetic case only. Finitary coding not covered.","url":"/math#236","articleUrl":"/articles/openai-math#four-smaller-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=236.%20The%20exact%20factor%2Dof%2DIID","manuscripts":[{"title":"The sharp factor-of-IID threshold for the free Ising model on regular trees","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026/article.pdf"}],"reel":{"src":"/films/math/236.mp4","poster":"/films/math/236-poster.webp","captions":"/films/math/236.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"The free Ising state on a tree is a factor of IID exactly up to the Kesten–Stigum point; Lean checks it.","consequence":"Sharp factor-of-IID threshold for tree Ising. Internal."},"detail":"/api/math/236"},{"id":"237","title":"The three-quarter exponent for honeycomb self-avoiding walk","short":"The 3/4 exponent for self-avoiding walk","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"A uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter n^{3/4+o(1)} with arbitrarily high polynomial probability, for every sufficiently large n; local mass and covering exponents 4/3;","verdict":"1067 pages in 13 papers, the largest family in the group; the diameter result is the 'o(1)' exponent form on the honeycomb lattice only (not Z^2, not the scaling limit…","url":"/math#237","articleUrl":"/articles/openai-math#the-34-exponent-for-self-avoiding-walk-on-the-honeycomb-lattice","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=237.%20The%20three%2Dquarter%20exponent","manuscripts":[{"title":"Radial transfer estimates and polygon length laws for honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Radial-transfer-estimates-and-polygon-length-laws-for-honeycomb-walks-September-26-2026/main.pdf"},{"title":"Critical honeycomb chords with prescribed boundary endpoints","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-honeycomb-chords-with-prescribed-boundary-endpoints-September-26-2026/main.pdf"},{"title":"Cylinder loop weights and planar nesting","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cylinder-loop-weights-and-planar-nesting-September-26-2026/main.pdf"},{"title":"Mass and covering exponents for fixed-length honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Mass-and-covering-exponents-for-fixed-length-honeycomb-walks-September-26-2026/main.pdf"},{"title":"Signed cylinder propagation and marked polygons on the honeycomb lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Signed-cylinder-propagation-and-marked-polygons-on-the-honeycomb-lattice-September-26-2026/main.pdf"},{"title":"Cylinder amplitudes and logarithmic bridge-length windows on the honeycomb lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-honeycomb-lattice-September-26-2026/main.pdf"},{"title":"Marked polygon correlations and one-arc bounds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Marked-polygon-correlations-and-one-arc-bounds-September-26-2026/main.pdf"},{"title":"Disk transfer representations and confined bridge mass","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Disk-transfer-representations-and-confined-bridge-mass-September-26-2026/main.pdf"},{"title":"Polynomial vacuum representations and bridge mass for honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026/main.pdf"},{"title":"Renewal and changes of law for critical honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026/main.pdf"},{"title":"Critical strip-crossing mass on the honeycomb lattice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf"},{"title":"Uniform marked-polygon estimates and sharp finite bridge moments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-marked-polygon-estimates-and-sharp-finite-bridge-moments-September-26-2026/main.pdf"},{"title":"Cap-selected amplitudes and triangle chords for honeycomb walks","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cap-selected-amplitudes-and-triangle-chords-for-honeycomb-walks-September-26-2026/main.pdf"}],"reel":{"src":"/films/math/237.mp4","poster":"/films/math/237-poster.webp","captions":"/films/math/237.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":4,"advance":2,"consequences":2,"surprise":2,"confidence":1,"why":"Self-avoiding walks on the honeycomb lattice spread like n^{3/4}, as physicists predicted; 1067 pages, the law itself unformalized.","consequence":"Rigorous 2D self-avoiding walk exponent on one lattice; no scaling limit, not Z^2."},"detail":"/api/math/237"},{"id":"238","title":"Optimal logarithmic mixing of the Thorp shuffle","short":"The Thorp shuffle mixes in $\\Theta(\\log N)$","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"The Thorp shuffle on N=2^d cards mixes in Θ(d)=Θ(log N) physical shuffles in total variation from the worst start: after 1600d shuffles the full permutation law tends to uniform as d→∞; support counting gives a 2d-O(1) lower bound.","verdict":"Power-of-two decks only. Lean main_results include ThorpRouting (OAI.ThorpNine.main), BinarySweep, CoordinateSweeps (conditional_main), ThorpWeightedCompatibility;","url":"/math#238","articleUrl":"/articles/openai-math#the-thorp-shuffle-mixes-in-thetalog-n-steps","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=238.%20Optimal%20logarithmic%20mixing","manuscripts":[{"title":"Optimal-order mixing of the Thorp shuffle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026/paper.pdf"},{"title":"Random coordinate frames and partial permutation laws","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf"},{"title":"From partial permutation information to Fourier bounds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf"},{"title":"Conditional information under deterministic coordinate sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-information-under-deterministic-coordinate-sweeps-September-26-2026/main.pdf"},{"title":"Conditional permutations in a revealed switching environment","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026/paper.pdf"},{"title":"Routing densities and representation contraction for Thorp sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026/paper.pdf"},{"title":"Row&#8211;column symmetry and contraction of coordinate sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026/paper.pdf"},{"title":"Random-subspace tests and trace smoothing for coordinate sweeps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026/paper.pdf"},{"title":"Compatibility entropy and the spectrum of a Thorp sweep","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026/paper.pdf"},{"title":"Signed tensor densities and diagram budgets for the Thorp shuffle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026/paper.pdf"},{"title":"Conditional coordinate sweeps and analytic transfer","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026/main.pdf"},{"title":"A strict four-row permanent inequality and permutation moments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026/main.pdf"}],"reel":{"src":"/films/math/238.mp4","poster":"/films/math/238-poster.webp","captions":"/films/math/238.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"The Thorp shuffle mixes in O(d) rounds for 2^d cards, down from O(d^3); Lean checks a bound.","consequence":"Fewer rounds of the Thorp shuffle needed for provable randomness, relevant to format-preserving encryption."},"detail":"/api/math/238"},{"id":"239","title":"Sharp singularity rates for symmetric random sign matrices","short":"Symmetric sign matrices: singular at rate $(1/2)^n$","discipline":"Probability and statistical mechanics","disciplineIndex":8,"accent":"#FF8F7A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For symmetric n×n matrices with independent ±1 entries on and above the diagonal: P(det A_n=0)=(1/2+o(1))^n for unbiased signs, and (p^2+(1-p)^2+o(1))^n for bias p≠1/2, the latter attained by two equal rows.","verdict":"Unformalized; no Lean docs. Plausibility: the unbiased constant 1/2 matches the natural lower bound from two equal rows/columns (probability ~2^{-n}).","url":"/math#239","articleUrl":"/articles/openai-math#symmetric-sign-matrices-are-singular-at-rate-12n","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=239.%20Sharp%20singularity%20rates","manuscripts":[{"title":"The sharp exponential rate of singularity for symmetric Bernoulli matrices","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-exponential-rate-of-singularity-for-symmetric-Bernoulli-matrices-October-3-2026/symmetric-bernoulli-singularity.pdf"},{"title":"The sharp singularity rate for biased symmetric sign matrices","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-singularity-rate-for-biased-symmetric-sign-matrices-October-4-2026/biased-symmetric-sign-singularity.pdf"}],"reel":{"src":"/films/math/239.mp4","poster":"/films/math/239-poster.webp","captions":"/films/math/239.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":1,"confidence":0,"why":"A random symmetric ±1 matrix is singular with probability about 2^{-n}, from equal rows; unformalized.","consequence":"Sharp singularity probability for symmetric random sign matrices. Internal to random matrix theory."},"detail":"/api/math/239"},{"id":"240","title":"Shelah's eventual categoricity and the prescribed-threshold obstruction","short":"Shelah's eventual categoricity","discipline":"Mathematical logic","disciplineIndex":9,"accent":"#A8C7FA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) In ZFC: for every infinite cardinal lambda there is mu(lambda) such that every abstract elementary class K with LS(K) = mu(lambda) is categorical in every cardinal >= mu(lambda) (Shelah's eventual categoricity conjecture), with no amalgamation, joint…","verdict":"The landmark claim is the unformalized one. 121 pages of AEC theory, a field where several announced proofs have needed long correction cycles.","url":"/math#240","articleUrl":"/articles/openai-math#shelahs-eventual-categoricity-in-zfc-family-240","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=240.%20Shelah's%20eventual%20categoricity","manuscripts":[{"title":"A CH obstruction to a prescribed categoricity threshold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026/paper.pdf"},{"title":"Eventual categoricity for abstract elementary classes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/240.mp4","poster":"/films/math/240-poster.webp","captions":"/films/math/240.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":2,"confidence":0,"why":"Shelah's eventual categoricity conjecture for abstract elementary classes, in ZFC; the main theorem is unformalized.","consequence":"Brings Morley's categoricity theorem to abstract elementary classes, the main structural goal of that program."},"detail":"/api/math/240"},{"id":"241","title":"Rigidity of the Turing degrees","short":"Rigidity of the Turing degrees","discipline":"Mathematical logic","disciplineIndex":9,"accent":"#A8C7FA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every order automorphism of the partial order of all Turing degrees (degrees of subsets of N under Turing reducibility) is the identity, in ZFC, with no definability assumption on the automorphism.","verdict":"The paper itself is only 10 pages because it takes as an external input the Slaman-Woodin theorem that every automorphism is induced by an arithmetic (Borel) function on…","url":"/math#241","articleUrl":"/articles/openai-math#rigidity-of-the-turing-degrees-family-241","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=241.%20Rigidity%20of%20the","manuscripts":[{"title":"Rigidity of the Turing degrees","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rigidity-of-the-Turing-degrees-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/241.mp4","poster":"/films/math/241-poster.webp","captions":"/films/math/241.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"The Turing degrees have no symmetry but the identity, finishing Slaman–Woodin in 10 pages; Lean checks it.","consequence":"Settles the rigidity question for Turing degrees. Internal to computability theory."},"detail":"/api/math/241"},{"id":"242","title":"Single-fold Diophantine representations and undecidability under an at-most-one-solution promise","short":"Single-fold Diophantine representations","discipline":"Mathematical logic","disciplineIndex":9,"accent":"#A8C7FA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every recursively enumerable set S in N^n has a single-fold Diophantine representation: an integer polynomial P(a, w) such that for a in S there is exactly one witness tuple w in N^m with P(a, w) = 0 and for a not in S there is none.","verdict":"27 pages. Uses a rank-one elliptic curve and Neron-Tate height to get a single-fold growth relation replacing Pell-equation exponentiation.","url":"/math#242","articleUrl":"/articles/openai-math#single-fold-diophantine-representations-family-242","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=242.%20Single%2Dfold%20Diophantine%20representations","manuscripts":[{"title":"Single-fold Diophantine representations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Single-fold-Diophantine-representations-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/242.mp4","poster":"/films/math/242-poster.webp","captions":"/films/math/242.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":3,"confidence":3,"why":"Hilbert's tenth problem stays undecidable with at most one solution promised, via elliptic curves; Lean checks the core.","consequence":"Hilbert's tenth problem stays undecidable under a unique-solution promise, with consequences for counting and complexity."},"detail":"/api/math/242"},{"id":"243","title":"Separating choiceless counting from polynomial time and witnessed choice","short":"Choiceless polynomial time is not P","discipline":"Mathematical logic","disciplineIndex":9,"accent":"#A8C7FA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) Choiceless Polynomial Time with counting (CPT) does not capture PTIME: an explicit isomorphism-invariant query on finite structures in an 8-relation vocabulary (consistency of a linear system over F_3) is decidable in polynomial time but not…","verdict":"The separation is unconditional and does not separate P from NP or settle whether some other logic captures PTIME (Gurevich's question).","url":"/math#243","articleUrl":"/articles/openai-math#choiceless-polynomial-time-does-not-capture-p-family-243","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=243.%20Separating%20choiceless%20counting","manuscripts":[{"title":"Choiceless polynomial time with counting does not capture polynomial time","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf"},{"title":"Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/243.mp4","poster":"/films/math/243-poster.webp","captions":"/films/math/243.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":2,"why":"Choiceless polynomial time does not capture P, settling the Blass–Gurevich–Shelah conjecture; Lean checks it.","consequence":"Removes the leading candidate logic for polynomial time; whether any logic captures P stays open."},"detail":"/api/math/243"},{"id":"244","title":"The Partition Principle does not imply Choice","short":"The Partition Principle without Choice","discipline":"Mathematical logic","disciplineIndex":9,"accent":"#A8C7FA","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"If ZF is consistent then so is ZF + PP + AC_WO + not-AC, where PP says every surjection X -> Y admits an injection Y -> X. From a countable transitive model of ZFC one gets a transitive symmetric extension satisfying PP + AC_WO + not AC with no new countable…","verdict":"The formalized consistency statement requires formalizing syntax, ZF and forcing/symmetric extensions in Lean, so faithfulness of those encodings matters;","url":"/math#244","articleUrl":"/articles/openai-math#the-partition-principle-does-not-imply-choice-family-244","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=244.%20The%20Partition%20Principle","manuscripts":[{"title":"The Partition Principle does not imply Choice","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026/partition-principle-without-choice.pdf"}],"reel":{"src":"/films/math/244.mp4","poster":"/films/math/244-poster.webp","captions":"/films/math/244.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":73,"tier":"Solid","importance":3,"advance":4,"consequences":1,"surprise":3,"confidence":2,"why":"The Partition Principle does not imply the Axiom of Choice, a question open since 1902; Lean checks the consistency.","consequence":"Answers a 1902 question in set theory. Internal."},"detail":"/api/math/244"},{"id":"245","title":"Weak normalization implies strong normalization in pure type systems","short":"Weak implies strong normalization","discipline":"Mathematical logic","disciplineIndex":9,"accent":"#A8C7FA","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every pure type system (arbitrary sorts, nonfunctional axioms and rules, open contexts, full beta reduction including inside annotations): if every legal term has a beta-normal form then every beta-reduction sequence from a legal term terminates…","verdict":"72 pages. Formal statement defines PTS syntax inside Lean; faithful as far as the doc describes (full annotated syntax, arbitrary specifications).","url":"/math#245","articleUrl":"/articles/openai-math#weak-normalization-implies-strong-normalization-family-245","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=245.%20Weak%20normalization%20implies","manuscripts":[{"title":"Weak and strong normalization in pure type systems","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/245.mp4","poster":"/films/math/245-poster.webp","captions":"/films/math/245.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"In every pure type system, weak normalization implies strong normalization (Barendregt–Geuvers–Klop); Lean checks it.","consequence":"Settles a termination question about type systems underlying proof assistants; little follow-on."},"detail":"/api/math/245"},{"id":"246","title":"Cannon's conjecture","short":"Cannon's conjecture","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every word-hyperbolic group G whose Gromov boundary is homeomorphic to S^2 admits a homomorphism to Isom(H^3) with finite kernel whose action on H^3 is proper and cocompact. If G is torsion-free it is the fundamental group of a closed hyperbolic 3-manifold.","verdict":"31-page analytic proof: reduces via Bourdon-Kleiner's modulus criterion to a uniform bound on combinatorial 2-modulus and proves it by contradiction in two growth cases…","url":"/math#246","articleUrl":"/articles/openai-math#cannons-conjecture-family-246","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=246.%20Cannon's%20conjecture","manuscripts":[{"title":"A Modulus Proof of Cannon’s Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/246.mp4","poster":"/films/math/246-poster.webp","captions":"/films/math/246.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":82,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":2,"confidence":3,"why":"Cannon's conjecture: hyperbolic groups with 2-sphere boundary are 3-manifold groups; 31 pages, Lean checks it.","consequence":"Completes the 3-dimensional program in geometric group theory: groups recognized from their boundary."},"detail":"/api/math/246"},{"id":"247","title":"An infinite finitely presented residually finite 2-group and a finitely presented nil algebra","short":"A finitely presented infinite periodic group","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) There is an infinite, ordinarily finitely presented periodic group: St_12(R) for a suitable unital F_2-algebra R; it also has property (T).","verdict":"The family title leads with the residually finite 2-group, which is the unformalized companion (25 pages); the formalized part is the periodic group itself.","url":"/math#247","articleUrl":"/articles/openai-math#a-finitely-presented-infinite-periodic-group-family-247","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=247.%20An%20infinite%20finitely","manuscripts":[{"title":"An infinite finitely presented residually finite 2-group","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf"},{"title":"An infinite finitely presented periodic group","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/247.mp4","poster":"/films/math/247-poster.webp","captions":"/films/math/247.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":3,"confidence":2,"why":"An infinite finitely presented torsion group, the finitely presented Burnside question; Lean checks the periodic group.","consequence":"A finitely presented infinite torsion group, a long-sought example. Internal to group theory."},"detail":"/api/math/247"},{"id":"248","title":"Thompson's group F is nonamenable","short":"Thompson's group $F$ is not amenable","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Thompson's group F (increasing dyadic piecewise-linear homeomorphisms of [0,1] with slopes powers of 2) admits no left-invariant mean on bounded functions, i.e. F is nonamenable.","verdict":"13 pages, astonishingly short for a 60-year-old problem with a history of failed claims.","url":"/math#248","articleUrl":"/articles/openai-math#thompsons-group-f-is-not-amenable-family-248","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=248.%20Thompson's%20group%20F","manuscripts":[{"title":"Thompson's group F is nonamenable","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thompsons-group-F-is-nonamenable-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/248.mp4","poster":"/films/math/248-poster.webp","captions":"/films/math/248.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":94,"tier":"Huge if true","importance":4,"advance":4,"consequences":3,"surprise":3,"confidence":3,"why":"Thompson's group F is not amenable, a famous 60-year-old question, in 13 pages; Lean checks it.","consequence":"Settles a question many results were conditional on, and gives Dixmier counterexamples for F."},"detail":"/api/math/248"},{"id":"249","title":"A finitely generated Eilenberg–Ganea counterexample","short":"Eilenberg–Ganea fails","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a finitely generated, residually finite group G with cd_Z G = 2 and gd G = 3: the Bestvina-Brady kernel of the right-angled Artin group on a finite acyclic flag triangulation of the presentation complex (a spine of the Poincare homology sphere).","verdict":"25 pages. Since Bestvina-Brady showed one of Eilenberg-Ganea and Whitehead fails for these groups, this result says Eilenberg-Ganea is the one that fails;","url":"/math#249","articleUrl":"/articles/openai-math#eilenbergganea-fails-family-249","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=249.%20A%20finitely%20generated","manuscripts":[{"title":"A finitely generated counterexample to the Eilenberg–Ganea conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/249.mp4","poster":"/films/math/249-poster.webp","captions":"/films/math/249.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"The Bestvina–Brady group that breaks Eilenberg–Ganea, so that one fails, not Whitehead; Lean checks it.","consequence":"Identifies which of two conjectures fails for Bestvina–Brady groups; Whitehead stays open."},"detail":"/api/math/249"},{"id":"250","title":"Boone–Higman embeddings with higher finiteness","short":"The Boone–Higman conjecture","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) A finitely generated group has solvable word problem iff it embeds in a finitely presented simple group (Boone-Higman conjecture). (b) Every finitely generated group with solvable word problem embeds in a simple group of type F_infinity.","verdict":"35+30+18 pages. Faithfulness depends on the Lean encoding of finite presentation, word problem decidability and type F_infinity; the docs describe standard definitions.","url":"/math#250","articleUrl":"/articles/openai-math#boonehigman-family-250","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=250.%20Boone%20Higman%20embeddings","manuscripts":[{"title":"Finite algebraic envelopes and the Boone–Higman conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf"},{"title":"Simple F∞ overgroups of groups with decidable word problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026/paper.pdf"},{"title":"A universal group of type F∞","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-universal-group-of-type-F-infinity-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/250.mp4","poster":"/films/math/250-poster.webp","captions":"/films/math/250.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"Boone–Higman: groups with solvable word problem embed in finitely presented simple groups; Lean checks it.","consequence":"An algebraic characterization of groups with solvable word problem."},"detail":"/api/math/250"},{"id":"251","title":"Amenability, unitarizability, and strong Ulam stability","short":"Dixmier's unitarizability problem","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) Dixmier's problem: a discrete group is amenable iff every uniformly bounded representation on a Hilbert space is similar to a unitary one; for every nonamenable group and eps > 0 there is a nonunitarizable representation with uniform bound <= 1 + eps.","verdict":"Only 19 pages for Dixmier. Note the interaction with 248: if F is nonamenable, Dixmier gives non-unitarizable representations of F.","url":"/math#251","articleUrl":"/articles/openai-math#dixmiers-unitarizability-problem-family-251","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=251.%20Amenability%20unitarizability%20and","manuscripts":[{"title":"Unitarizability implies amenability for discrete groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026/paper.pdf"},{"title":"Strong Ulam Stability Characterizes Amenability","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strong-Ulam-Stability-Characterizes-Amenability-October-5-2026/strong-ulam-stability.pdf"}],"reel":{"src":"/films/math/251.mp4","poster":"/films/math/251-poster.webp","captions":"/films/math/251.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"Dixmier's problem: every nonamenable group has a non-unitarizable representation; Lean checks it.","consequence":"Unitarizability characterizes amenability. Internal to group theory and operator algebras."},"detail":"/api/math/251"},{"id":"252","title":"A torsion-free hyperbolic group that is neither residually finite nor linear over any field","short":"A hyperbolic group, not residually finite","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a torsion-free word-hyperbolic group (fundamental group of a finite Euclidean triangle complex) that is not residually finite;","verdict":"21 pages; existential: proves one member of a finite family is in the finite residual without saying which.","url":"/math#252","articleUrl":"/articles/openai-math#two-hyperbolic-groups-nobody-expected-families-252-and-257","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=252.%20A%20torsion%2Dfree%20hyperbolic","manuscripts":[{"title":"A torsion-free hyperbolic group that is not residually finite","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/252.mp4","poster":"/films/math/252-poster.webp","captions":"/films/math/252.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":86,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":3,"confidence":3,"why":"A torsion-free hyperbolic group that is not residually finite, against Gromov's question; Lean checks it.","consequence":"Hyperbolicity alone does not give residual finiteness; with Kapovich–Wise, a hyperbolic group with no finite quotients."},"detail":"/api/math/252"},{"id":"253","title":"An infinite finitely presented simple amenable group","short":"A finitely presented simple amenable group","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a group that is infinite, finitely presented, simple and amenable: an alternating subgroup of a polygon exchange group on a sufficiently large finite number of squares.","verdict":"58 pages; existence only, no explicit size. Lean development ~85k lines.","url":"/math#253","articleUrl":"/articles/openai-math#a-finitely-presented-simple-amenable-group-family-253","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=253.%20An%20infinite%20finitely","manuscripts":[{"title":"An infinite finitely presented simple amenable group","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/253.mp4","poster":"/films/math/253-poster.webp","captions":"/films/math/253.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"An infinite, finitely presented, simple and amenable group, from polygon exchanges; Lean checks it.","consequence":"A new kind of simple group example. Internal."},"detail":"/api/math/253"},{"id":"254","title":"Classifying spaces and geometric obstructions for Artin groups","short":"The $K(\\pi,1)$ conjecture for Artin groups","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"(a) The K(pi,1) conjecture: for every Coxeter matrix on a finite set S (any labels including infinity, any diagram), the universal cover of the Salvetti complex is contractible, so the Salvetti complex is a K(A,1) for the Artin group A.","verdict":"62+29+55 pages. Arguably the single most important group-theory claim in the release together with Cannon and Thompson F.","url":"/math#254","articleUrl":"/articles/openai-math#artin-groups-kπ1-parabolic-intersections-and-a-group-that-is-not-cat0-family-254","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=254.%20Classifying%20spaces%20and","manuscripts":[{"title":"Harmonic heights and the Artin K(pi,1) conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026/paper.pdf"},{"title":"An Artin group with no geometric CAT(0) action","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026/paper.pdf"},{"title":"Parabolic intersections in Artin groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/254.mp4","poster":"/films/math/254-poster.webp","captions":"/films/math/254.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":82,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":2,"confidence":3,"why":"The K(π,1) conjecture for all Artin groups, giving each a finite classifying space; Lean checks it.","consequence":"Every Artin group gets a finite classifying space, torsion-freeness and computable cohomology."},"detail":"/api/math/254"},{"id":"255","title":"Quasi-isometric recognition of virtually polycyclic groups","short":"QI rigidity of polycyclic groups","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic; equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in…","verdict":"73 pages. One of the largest Lean developments in this group.","url":"/math#255","articleUrl":"/articles/openai-math#quasi-isometric-rigidity-of-polycyclic-groups-family-255","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=255.%20Quasi%2Disometric%20recognition%20of","manuscripts":[{"title":"Quasi-isometric recognition of virtually polycyclic groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/255.mp4","poster":"/films/math/255-poster.webp","captions":"/films/math/255.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":1,"confidence":3,"why":"Groups shaped like polycyclic groups are virtually polycyclic, by quasi-isometry; Lean checks it.","consequence":"Quasi-isometric rigidity for polycyclic groups. Internal to geometric group theory."},"detail":"/api/math/255"},{"id":"256","title":"Nonsingular systems of equations over arbitrary groups","short":"Kervaire's conjecture, and Howie's","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Kervaire conjecture: for every nontrivial group A and every w in A * , the quotient (A * )/> is nontrivial; proved via coefficient injectivity when the exponent sum of t in w is +-1 (13 pages).","verdict":"Kervaire itself is formalized in its standard equivalent strengthening; the headline 'Howie's conjecture' is not.","url":"/math#256","articleUrl":"/articles/openai-math#kervaires-conjecture-in-thirteen-pages-family-256","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=256.%20Nonsingular%20systems%20of","manuscripts":[{"title":"Nonsingular systems of equations over arbitrary groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf"},{"title":"The Kervaire theorem for groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf"}],"reel":{"src":"/films/math/256.mp4","poster":"/films/math/256-poster.webp","captions":"/films/math/256.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":2,"why":"Kervaire's conjecture: adding one generator and one relation never kills a group; Lean checks the core theorem.","consequence":"Settles Kervaire's conjecture; Howie's stronger conjecture stays unformalized."},"detail":"/api/math/256"},{"id":"257","title":"A hyperbolic group without a geometric CAT(0) action","short":"A hyperbolic group that is not CAT(0)","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a finite connected 2-dimensional aspherical simplicial complex K with a linear isoperimetric inequality (so pi_1 K is hyperbolic) such that pi_1 K admits no proper cocompact isometric action on any proper complete CAT(0) space, in any dimension;","verdict":"43 pages; existence argument, no effective size. Combined with 252 (a non-residually-finite hyperbolic group) it is consistent: CAT(0) cubulated hyperbolic groups are…","url":"/math#257","articleUrl":"/articles/openai-math#two-hyperbolic-groups-nobody-expected-families-252-and-257","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=257.%20A%20hyperbolic%20group","manuscripts":[{"title":"A hyperbolic group with no geometric CAT(0) action","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/257.mp4","poster":"/films/math/257-poster.webp","captions":"/films/math/257.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":2,"why":"A hyperbolic group that acts geometrically on no CAT(0) space; Lean checks the main statement.","consequence":"Hyperbolic groups need not be CAT(0); limits on geometric models."},"detail":"/api/math/257"},{"id":"258","title":"Gersten’s conjecture and virtual compact specialness of one-relator groups","short":"Gersten's conjecture for one-relator groups","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Every finitely generated one-relator group with no Baumslag-Solitar subgroup BS(m,n) (m,n nonzero) is word-hyperbolic (Gersten's conjecture). (b) Every hyperbolic one-relator group is virtually compact special;","verdict":"Not formalized; 53 + 145 pages. Each paper uses the other: Gersten's paper cites the companion for virtual compact specialness in its corollary.","url":"/math#258","articleUrl":"/articles/openai-math#gerstens-conjecture-for-one-relator-groups-family-258","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=258.%20Gersten%E2%80%99s%20conjecture%20and","manuscripts":[{"title":"Baumslag-Solitar-free one-relator groups are hyperbolic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Baumslag-Solitar-free-one-relator-groups-are-hyperbolic-September-25-2026/paper.pdf"},{"title":"Virtual compact specialness of hyperbolic one-relator groups","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Virtual-compact-specialness-of-hyperbolic-one-relator-groups-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/258.mp4","poster":"/films/math/258-poster.webp","captions":"/films/math/258.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":0,"why":"Gersten's conjecture: one-relator groups without Baumslag–Solitar subgroups are hyperbolic; no Lean.","consequence":"Hyperbolicity of one-relator groups, with virtual specialness as a companion."},"detail":"/api/math/258"},{"id":"259","title":"A group without fixed price","short":"A group without fixed price","discipline":"Group theory","disciplineIndex":10,"accent":"#E39BF0","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"There is a finitely generated group Gamma = A *_J (J x ) (an amalgam built from a rank-100 relation word) with two essentially free p.m.p.","verdict":"Not formalized; 27 pages. Constants explicit but tiny (K = e^96 99^99/95^95, alpha < 1/200, K alpha^3 < 1/2).","url":"/math#259","articleUrl":"/articles/openai-math#a-group-without-fixed-price-family-259","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=259.%20A%20group%20without","manuscripts":[{"title":"A group without fixed price","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf"}],"reel":{"src":"/films/math/259.mp4","poster":"/films/math/259-poster.webp","captions":"/films/math/259.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"Gaboriau's fixed-price question: a group whose actions have different costs; 27 unformalized pages.","consequence":"Cost is not a group invariant. Internal to measured group theory."},"detail":"/api/math/259"},{"id":"260","title":"Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions","short":"Spacetime Penrose inequality","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"For smooth one-ended asymptotically flat initial data (g,K) in any spatial dimension n>=3 satisfying the dominant energy condition, with a weakly future trapped boundary (theta_+= (1/2)(A_min/omega_{n-1})^{(n-2)/(n-1)}, sharp (Schwarzschild-Tangherlini).","verdict":"13 manuscripts, ~1,355 pages: the largest family in the group and effectively unreviewable 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rotating and anti-de Sitter versions."},"detail":"/api/math/260"},{"id":"261","title":"Localization and delocalization in the Anderson model","short":"Anderson model: $d \\ge 3$ and $d = 2$","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) For the nearest-neighbour Anderson operator on Z^d, d>=3, with i.i.d. uniform site potentials of sufficiently small fixed strength, almost surely the spectrum is purely absolutely continuous on a fixed open energy interval (independent of disorder)…","verdict":"Two of the most famous open problems in mathematical physics claimed at once, neither formally checked (the formalized statement is a textbook fact).","url":"/math#261","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=261.%20Localization%20and%20delocalization","manuscripts":[{"title":"Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf"},{"title":"Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/261.mp4","poster":"/films/math/261-poster.webp","captions":"/films/math/261.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":4,"advance":2,"consequences":3,"surprise":2,"confidence":0,"why":"Anderson model: electrons stay mobile under weak disorder in 3D and are trapped in 2D; uniform disorder law only, unchecked.","consequence":"Rigorous metal–insulator behaviour for the Anderson model, the standard model of disordered conductors."},"detail":"/api/math/261"},{"id":"262","title":"Sharp finite-matrix Lieb–Thirring inequalities and all equality cases","short":"Sharp 1D Lieb–Thirring constants","discipline":"Mathematical 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constants","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/262.mp4","poster":"/films/math/262-poster.webp","captions":"/films/math/262.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"Sharp one-dimensional Lieb–Thirring inequalities for the remaining exponents; Lean checks the scalar case.","consequence":"Sharp constants for 1D Lieb–Thirring inequalities. Internal to spectral theory."},"detail":"/api/math/262"},{"id":"263","title":"The ionization and generalized ionization conjectures","short":"The ionization conjecture","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"For the full nonrelativistic Coulomb Hamiltonian with two spin states and M fixed nuclei of charges >=1 and total charge Z, strict binding of n electrons forces n inf with Z/m->inf; outer radii R_m ~ (81 pi^2/2)^{1/3} m^{-1/3} with Z->inf first.","verdict":"The headline Z+CM theorem (the actual ionization conjecture) is unformalized; what is formalized is the asymptotic 'generalized' part, itself a strong claim.","url":"/math#263","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=263.%20The%20ionization%20and","manuscripts":[{"title":"Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf"},{"title":"Generalized ionization energies for full Coulomb atoms","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf"},{"title":"Generalized outer-electron radii of neutral Coulomb atoms","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/263.mp4","poster":"/films/math/263-poster.webp","captions":"/films/math/263.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":1,"why":"The ionization conjecture: a nucleus of charge Z binds at most Z plus a constant electrons; the headline is unformalized.","consequence":"Bounds how many electrons atoms can bind, matching experiment up to a constant."},"detail":"/api/math/263"},{"id":"264","title":"Strong cosmic censorship near two-ended Kerr data","short":"Strong cosmic censorship near Kerr","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"partial","kindLabel":"Claimed partial result","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Fix a rotating subextremal Kerr spacetime (0<|a|<M). In a small weighted-smooth neighbourhood (smallness of the tenth seminorm) of its two-ended bridge data, the vacuum data whose maximal globally hyperbolic development admits a future C^0 ∩ W^{1,2}_loc…","verdict":"Local and generic only: a Baire-category statement in a small neighbourhood of each fixed Kerr bridge, no |a|->0 or extremal uniformity, inextendibility not shown open,…","url":"/math#264","articleUrl":"/articles/openai-math#strong-cosmic-censorship-near-kerr","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=264.%20Strong%20cosmic%20censorship","manuscripts":[{"title":"Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf"},{"title":"Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf"},{"title":"Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quantitative-Near-Kerr-Evolution-and-Generic-C2-Future-Inextendibility-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/264.mp4","poster":"/films/math/264-poster.webp","captions":"/films/math/264.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":4,"advance":2,"consequences":2,"surprise":2,"confidence":0,"why":"Strong cosmic censorship, locally and generically, near two-ended Kerr black holes; not the full conjecture. No Lean.","consequence":"Supports determinism in general relativity near Kerr; global SCC stays open."},"detail":"/api/math/264"},{"id":"265","title":"Area laws and tensor networks for two-dimensional gapped systems","short":"Area law for 2D gapped systems","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"For a finite-range Hamiltonian on any finite induced subgraph of Z^2 with bounded local dimension, range and interaction strength, a uniform lower bound on the spectral gap of the full Hamiltonian (unique ground state) implies that every region's entanglement…","verdict":"No formalization. Only the global gap is assumed (no local gaps, frustration-freeness or translation invariance), which is exactly the hard case.","url":"/math#265","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=265.%20Area%20laws%20and","manuscripts":[{"title":"A two-dimensional area law from a global spectral gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf"},{"title":"Polynomial PEPS approximation of gapped square-grid ground states","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/265.mp4","poster":"/films/math/265-poster.webp","captions":"/films/math/265.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":2,"confidence":0,"why":"Ground states of gapped 2D quantum systems obey an area law and have efficient tensor networks; 146 unformalized pages.","consequence":"Explains rigorously why tensor-network methods work for gapped 2D quantum systems; not an efficient algorithm."},"detail":"/api/math/265"},{"id":"266","title":"Exactly three mutually unbiased bases in dimension six","short":"Mutually unbiased bases in dimension six","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"N(6)=3: at most three mutually unbiased orthonormal bases exist in C^6 (three are classical). The upper bound is computer-assisted: under stated IEEE binary64 arithmetic and compiler conditions, a complete run of the documented verification pipeline excludes…","verdict":"The headline N(6)=3 rests on a floating-point verification pipeline with 'stated binary64 arithmetic and compiler conditions';","url":"/math#266","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=266.%20Exactly%20three%20mutually","manuscripts":[{"title":"The maximum number of mutually unbiased bases in dimension six","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf"},{"title":"Exact Fourier certificates for complex Hadamard matrices of order six","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf"}],"reel":{"src":"/films/math/266.mp4","poster":"/films/math/266-poster.webp","captions":"/films/math/266.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":1,"why":"Dimension 6 has exactly 3 mutually unbiased bases; Lean checks at most 5, the rest uses floating-point certificates.","consequence":"Optimal quantum tomography in dimension 6 cannot use a complete set of unbiased bases."},"detail":"/api/math/266"},{"id":"267","title":"Positive-temperature Bose–Einstein condensation and exact quantum depletion","short":"Bose–Einstein condensation at $T > 0$","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"For the 3D hard-sphere Bose gas with fixed exclusion distance a and sufficiently small fixed density rho, there is a fixed temperature T>0 (independent of volume) at which the exact canonical Gibbs state has positive condensate fraction in the constant…","verdict":"The positive-temperature result gives some small T(a,rho)>0 and is not meant to locate the transition; it is unformalized.","url":"/math#267","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=267.%20Positive%2Dtemperature%20Bose%20Einstein","manuscripts":[{"title":"Bose–Einstein condensation at positive temperature in the dilute hard-sphere gas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bose-Einstein-condensation-at-positive-temperature-in-the-dilute-hard-sphere-gas-October-5-2026/positive-temperature-hard-spheres.pdf"},{"title":"Quantum Depletion and Momentum Distribution in the Dilute Hard-Sphere Bose Gas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quantum-Depletion-and-Momentum-Distribution-in-the-Dilute-Hard-Sphere-Bose-Gas-October-5-2026/Quantum-Depletion-in-the-Dilute-Hard-Sphere-Bose-Gas.pdf"},{"title":"Quantum Depletion for Fixed Bounded Repulsive Potentials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026/fixed-repulsion-quantum-depletion.pdf"},{"title":"A density-uniform condensate bound for dilute Bose gases","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf"},{"title":"Ground-state condensation in the dilute hard-sphere gas","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/267.mp4","poster":"/films/math/267-poster.webp","captions":"/films/math/267.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":4,"advance":2,"consequences":3,"surprise":2,"confidence":1,"why":"Bose–Einstein condensation for interacting bosons in the thermodynamic limit; Lean checks the ground-state version.","consequence":"Rigorous Bose–Einstein condensation for interacting gases, a textbook phenomenon long unproved."},"detail":"/api/math/267"},{"id":"268","title":"The spin-one Haldane gap","short":"The spin-one Haldane gap","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"The pure antiferromagnetic spin-1 Heisenberg chain H_L=sum S_j.S_{j+1} on even periodic rings has a unique ground state for even L>=60 and a spectral gap gamma_L > (4/105)log(80/79) for even L>=60 and > log(20)/784 for even L>=2304;","verdict":"Short (30 + 28 pages) for a 40-year-old problem; relies on rigorous enclosures of matrix-exponential traces of small twisted chains computed at inverse temperatures 21/2…","url":"/math#268","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=268.%20The%20spin%2Done%20Haldane","manuscripts":[{"title":"The periodic spin-one Haldane gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf"},{"title":"A boundary-field gap for the spin-one Heisenberg chain","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/268.mp4","poster":"/films/math/268-poster.webp","captions":"/films/math/268.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":4,"advance":3,"consequences":2,"surprise":2,"confidence":1,"why":"Haldane's 1983 prediction that spin-1 chains have an energy gap, via numerical certificates; 58 unformalized pages.","consequence":"Rigorous footing for a cornerstone of topological phases of matter; the proven gap is tiny, existence only."},"detail":"/api/math/268"},{"id":"269","title":"Uniform Laughlin gap and stability under bounded scalar disorder","short":"The Laughlin spectral gap","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"For the fermionic V1 (pseudopotential) Hamiltonian at filling 1/3 on the round sphere with flux q=3(N-1), the gap above the Laughlin state is at least 1/25 for all large N;","verdict":"'Full V1 interaction' with unit coefficients on the sphere; the gap is uniform only for sufficiently large N and constants for stability are existential.","url":"/math#269","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=269.%20Uniform%20Laughlin%20gap","manuscripts":[{"title":"Uniform Stability of the Spherical Laughlin Gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf"},{"title":"A Fock-space inequality and the Laughlin spectral gap","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf"}],"reel":{"src":"/films/math/269.mp4","poster":"/films/math/269-poster.webp","captions":"/films/math/269.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":2,"why":"A uniform spectral gap for the Laughlin state of the fractional quantum Hall effect; Lean checks the core.","consequence":"Incompressibility of the Laughlin state, which the fractional quantum Hall effect needs."},"detail":"/api/math/269"},{"id":"270","title":"Threshold and positive-energy bound states of the BFSS matrix model","short":"BFSS threshold bound state","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every N>=2, the relative SU(N) BFSS supersymmetric matrix quantum mechanics (nine matrices, 16 supercharges, Hamiltonian defined by closing the supercharge form) has exactly one normalizable zero-energy state, which is Spin(9)-invariant.","verdict":"The threshold-state result is consistent with physics expectations and would make the index arguments rigorous.","url":"/math#270","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=270.%20Threshold%20and%20positive%2Denergy","manuscripts":[{"title":"The unique threshold bound state of the SU(N) BFSS model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf"},{"title":"Positive eigenvalues of the relative SU(2) BFSS Hamiltonian","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf"}],"reel":{"src":"/films/math/270.mp4","poster":"/films/math/270-poster.webp","captions":"/films/math/270.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":3,"confidence":0,"why":"BFSS matrix model bound states at threshold, plus surprising positive-energy ones against the original paper; no Lean.","consequence":"Makes index arguments in matrix theory rigorous; the positive-energy claim needs physicists' scrutiny."},"detail":"/api/math/270"},{"id":"271","title":"Bloch's law, its lattice correction, and the spherical magnetization law","short":"Magnetization and Bloch's law","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) For the nearest-neighbour isotropic quantum Heisenberg ferromagnet on Z^d, d>=3, any spin S, at every sufficiently low positive temperature there is a translation-invariant zero-field beta-KMS state with magnetization omega(S_0^z) >= S/4.","verdict":"Magnetization bound is S/4, not the physical value; beta_0 is existential. The formalized statement defines its own quasi-local algebra, dynamics and KMS condition from…","url":"/math#271","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=271.%20Bloch's%20law%20its","manuscripts":[{"title":"Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf"},{"title":"The first lattice correction to Bloch's law","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf"},{"title":"The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf"},{"title":"Spontaneous magnetization in the quantum Heisenberg ferromagnet","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/271.mp4","poster":"/films/math/271-poster.webp","captions":"/films/math/271.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":4,"advance":2,"consequences":2,"surprise":2,"confidence":2,"why":"The quantum Heisenberg ferromagnet magnetizes at low temperature in 3D, open for decades; Lean checks the theorem.","consequence":"A rigorous proof that the textbook quantum model of a ferromagnet orders in 3D."},"detail":"/api/math/271"},{"id":"272","title":"Entanglement without distillable secret key","short":"PPT-squared is false","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"An explicit entangled state on C^10 x C^10 from which no secret key can be distilled by the specified local-instrument protocols that complete almost surely (joint processing of all copies, unlimited authenticated two-way public communication, eavesdropper…","verdict":"Zero key is proved only for the protocol class 'specified here' (local instruments that complete almost surely);","url":"/math#272","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=272.%20Entanglement%20without%20distillable","manuscripts":[{"title":"Entanglement with zero distillable secret key in local dimension ten","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf"}],"reel":{"src":"/films/math/272.mp4","poster":"/films/math/272-poster.webp","captions":"/films/math/272.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":2,"confidence":3,"why":"Applying a PPT channel twice can keep entanglement, refuting the PPT-squared conjecture; Lean checks it.","consequence":"Settles a favourite quantum-information question; entanglement and secret key really differ."},"detail":"/api/math/272"},{"id":"273","title":"The entropy photon-number inequality","short":"The entropy photon-number inequality","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For two independent bosonic inputs with finite mean energy in any finite number n of modes (arbitrary entanglement among modes within each input), mixed on a beam splitter of transmissivity eta, the entropy photon numbers satisfy N(rho_C) >= eta N(rho_A) +…","verdict":"Finite-energy, finite-mode setting (the natural one). Formal verification covers the main inequality; consequences unformalized.","url":"/math#273","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=273.%20The%20entropy%20photon%2Dnumber","manuscripts":[{"title":"The entropy photon-number inequality","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-entropy-photon-number-inequality-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/273.mp4","poster":"/films/math/273-poster.webp","captions":"/films/math/273.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":62,"tier":"Solid","importance":2,"advance":3,"consequences":3,"surprise":1,"confidence":3,"why":"The entropy photon-number inequality for beam splitters, a missing step for optical channel capacities; Lean checks it.","consequence":"Settles several optical channel capacities that were waiting on this inequality."},"detail":"/api/math/273"},{"id":"274","title":"Parity is not in QAC0","short":"Parity is not in QAC$^0$","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Constant-depth quantum circuits built from arbitrary one-qubit gates and unbounded-arity Toffoli gates, with polynomially many total qubits (ancillas initialized to |0>), one measured output qubit and arbitrary discarded garbage, cannot compute n-bit parity…","verdict":"'Measured-output model' (one output qubit measured, garbage discarded); a clean-computation variant or exact-parity-on-coherent-registers variant is a different…","url":"/math#274","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=274.%20Parity%20is%20not","manuscripts":[{"title":"Product-projection localization and the QAC0 parity lower bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf"},{"title":"Regular trajectories, pruning and quantum parity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/274.mp4","poster":"/films/math/274-poster.webp","captions":"/films/math/274.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":2,"why":"Shallow quantum circuits with fan-in gates cannot compute parity, open 25 years; Lean checks it.","consequence":"A clean limit on shallow quantum circuits, the quantum analogue of AC0 lower bounds."},"detail":"/api/math/274"},{"id":"275","title":"QMA-hardness of continuum Coulomb energy","short":"QMA-hardness of the Coulomb problem","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"Approximating the ground-state energy infimum of electrons in 3D continuum space interacting with clamped nuclei by Coulomb forces, minimized over all antisymmetric spinful states, is QMA-hard under deterministic polynomial-time reductions;","verdict":"Hardness of approximation in the promise-problem sense; it says nothing about typical molecules. Formalization of a QMA reduction in the continuum is substantial;","url":"/math#275","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=275.%20QMA%2Dhardness%20of%20continuum","manuscripts":[{"title":"Continuum Coulomb hardness with binary nuclear charges","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026.pdf"},{"title":"QMA-hardness of continuum Coulomb energy with unit nuclear charges","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026.pdf"}],"reel":{"src":"/films/math/275.mp4","poster":"/films/math/275-poster.webp","captions":"/films/math/275.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"Computing molecular ground-state energies is QMA-hard even with only hydrogen-like nuclei; Lean checks it.","consequence":"Quantum chemistry's ground-state problem is hard in the worst case; says nothing about typical molecules."},"detail":"/api/math/275"},{"id":"276","title":"Classical capacity of generalized amplitude damping","short":"Capacity of generalized amplitude damping","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every qubit generalized amplitude-damping channel (all damping and thermal parameters) the unassisted classical capacity equals the one-shot Holevo capacity, given by an explicit one-variable maximization attained by binary pure-state ensembles;","verdict":"Additivity with arbitrary partner channels (a strong claim) is not in the formal statement.","url":"/math#276","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=276.%20Classical%20capacity%20of","manuscripts":[{"title":"Classical capacity and entropy inequalities for generalized amplitude damping","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/276.mp4","poster":"/films/math/276-poster.webp","captions":"/films/math/276.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"The exact classical capacity of the generalized amplitude-damping qubit channel; Lean checks it.","consequence":"Exact capacity for a realistic qubit noise model."},"detail":"/api/math/276"},{"id":"277","title":"Threshold repetition for entangled games","short":"Threshold repetition for entangled games","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For any finite two-player one-round game with entangled value v<1 and arbitrary (correlated) question distribution, the probability that a finite-dimensional entangled strategy wins at least a (v+delta) fraction of k parallel repetitions is at most exp(-kappa…","verdict":"The hard part (exponential all-wins parallel repetition for general entangled games) is attributed to prior work, one of which is an earlier OpenAI publication and…","url":"/math#277","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=277.%20Threshold%20repetition%20for","manuscripts":[{"title":"Threshold parallel repetition for finite-dimensional entangled games","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/277.mp4","poster":"/films/math/277-poster.webp","captions":"/films/math/277.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":16,"tier":"Incremental","importance":1,"advance":1,"consequences":0,"surprise":0,"confidence":2,"why":"Concentration of rounds won in parallel-repeated entangled games, from known estimates; Lean checks a weaker rate.","consequence":"Threshold bounds for entangled games from known estimates. Little follow-on."},"detail":"/api/math/277"},{"id":"278","title":"Failure of Kohn–Sham ensemble representation","short":"Kohn–Sham representability fails","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"There is a three-electron Coulomb molecule with two equal positive-integer nuclear charges (given by an exact nonnumerical formula) whose spin-summed ground-state density is not the density of any ground-state ensemble of noninteracting electrons in a single…","verdict":"Unformalized; the nuclear charge is specified only by a formula (likely huge). Concerns one potential class (L^{3/2}+L^inf); other classes are not addressed.","url":"/math#278","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=278.%20Failure%20of%20Kohn","manuscripts":[{"title":"A Coulomb ground-state density without Kohn-Sham ensemble representation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Coulomb-Ground-State-Density-without-Kohn-Sham-Ensemble-Representation-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/278.mp4","poster":"/films/math/278-poster.webp","captions":"/films/math/278.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":52,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":2,"confidence":0,"why":"A three-electron molecule whose density no non-interacting system reproduces, against a Kohn–Sham assumption; no Lean.","consequence":"A foundational assumption of density functional theory fails in general, for one potential class."},"detail":"/api/math/278"},{"id":"279","title":"Exact quantum factoring over a fixed finite gate set","short":"Exact factoring, fixed finite gate set","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"A polynomial-time uniform family of quantum circuits over one fixed finite set of bounded-arity gates outputs the complete prime factorization of every N>=2 with probability exactly one, with polynomial worst-case gate and qubit counts.","verdict":"A complexity-theoretic refinement (zero error with a finite gate set); no practical speedup. Lean-checked, but the formal circuit model should be audited.","url":"/math#279","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=279.%20Exact%20quantum%20factoring","manuscripts":[{"title":"Exact quantum factoring over a fixed finite gate set","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf"}],"reel":{"src":"/films/math/279.mp4","poster":"/films/math/279-poster.webp","captions":"/films/math/279.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":2,"why":"Factoring exactly, with zero error, over a fixed finite quantum gate set in polynomial time; Lean checks it.","consequence":"Removes error and continuous angles from Shor-type factoring; no practical speedup."},"detail":"/api/math/279"},{"id":"280","title":"Unitary vertex operator algebras and conformal nets","short":"Unitary VOAs and conformal nets","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"partial","kindLabel":"Claimed partial result","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Every simple unitary strongly rational vertex operator algebra is strongly local and generates a completely rational conformal net; all simple grading-restricted modules are unitarizable, fusion forms are positive, and the Carpi-Weiner-Xu functor is a braided…","verdict":"Strongly rational case only (the CKLW conjecture is for all simple unitary VOAs). Formalizing VOA theory and conformal nets in Lean is itself remarkable;","url":"/math#280","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=280.%20Unitary%20vertex%20operator","manuscripts":[{"title":"Strongly rational unitary vertex operator algebras and conformal nets","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/280.mp4","poster":"/films/math/280-poster.webp","captions":"/films/math/280.vtt","duration":19.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":1,"why":"Unitary vertex operator algebras match conformal nets in the strongly rational case; Lean checks parts.","consequence":"Links two axiomatizations of 2D chiral conformal field theory in the rational case."},"detail":"/api/math/280"},{"id":"281","title":"QAOA attains the SK optimum in the thermodynamic-first limit","short":"QAOA reaches the SK optimum","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"For the Gaussian zero-field Sherrington-Kirkpatrick model, QAOA with standard cost and transverse-field mixer approaches the ground-state energy per spin when system size goes to infinity first and depth after: for any accuracy there exist finite depth and…","verdict":"No bound on the needed depth and no efficient angle selection; thermodynamic-first limit. Formal parts are supporting and conditional on a minimizer.","url":"/math#281","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=281.%20QAOA%20attains%20the","manuscripts":[{"title":"QAOA attains the SK ground-state energy in the thermodynamic-first limit","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf"},{"title":"Full support of the zero-temperature Sherrington-Kirkpatrick order parameter","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf"}],"reel":{"src":"/films/math/281.mp4","poster":"/films/math/281-poster.webp","captions":"/films/math/281.vtt","duration":20.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":1,"why":"QAOA gets arbitrarily close to the SK optimum in the large-size limit, with no depth bound.","consequence":"QAOA's asymptotic power on SK; no depth or angle guidance for practice."},"detail":"/api/math/281"},{"id":"282","title":"From scale symmetry to local conformal symmetry in four-dimensional QFT","short":"Scale to conformal symmetry in 4D","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"conditional","kindLabel":"Claimed conditional result","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"For 4D unitary positive-energy QFTs with Poincare- and scale-invariant vacuum, a discrete bounded-below spectrum of scaling dimensions with finite multiplicities, fields with finite scaling support realized as affiliated to a causal bounded local net, and a…","verdict":"Heavily hypothesis-laden ('operational' framework: bounded local net, field reconstruction, discrete scaling spectrum, a physical local dilatation current).","url":"/math#282","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=282.%20From%20scale%20symmetry","manuscripts":[{"title":"Scale and conformal symmetry in four-dimensional operational quantum field theory","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Scale-and-conformal-symmetry-in-four-dimensional-operational-quantum-field-theory-September-26-2026/main.pdf"}],"reel":{"src":"/films/math/282.mp4","poster":"/films/math/282-poster.webp","captions":"/films/math/282.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":35,"tier":"Incremental","importance":3,"advance":1,"consequences":0,"surprise":1,"confidence":0,"why":"Scale symmetry gives local conformal symmetry in 4D QFT, but only inside a heavily axiomatized framework.","consequence":"A result inside a strongly axiomatized framework; the physics question stays open."},"detail":"/api/math/282"},{"id":"283","title":"Polynomial-time unitary synthesis from a Boolean oracle","short":"Unitary synthesis from a Boolean oracle","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every n, a quantum oracle circuit generated in poly time from n alone (gates H, T, T^dagger, CNOT; polynomially many qubits, gates, oracle calls and query length) approximates every n-qubit unitary channel to diamond-norm error <= 1/2 after a suitable…","verdict":"Constant-error (1/2 in diamond norm) formulation; whether error can be driven to any epsilon with poly(n, log 1/eps) queries is not stated in the summary.","url":"/math#283","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=283.%20Polynomial%2Dtime%20unitary%20synthesis","manuscripts":[{"title":"Polynomial-Time Unitary Synthesis from a Boolean Oracle","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Time-Unitary-Synthesis-from-a-Boolean-Oracle-October-5-2026/paper.pdf"}],"reel":{"src":"/films/math/283.mp4","poster":"/films/math/283-poster.webp","captions":"/films/math/283.vtt","duration":19.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":0,"why":"A quantum circuit with a classical oracle implements any unitary to constant error in polynomial time; no Lean.","consequence":"Relates quantum unitary complexity to classical complexity. Internal."},"detail":"/api/math/283"},{"id":"284","title":"The optimal quartic separation between randomized and quantum queries","short":"Randomized vs quantum queries: exponent 4","discipline":"Mathematical physics","disciplineIndex":11,"accent":"#5FD4F4","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For each fixed k>=2 there are total Boolean functions F with Q(F) = c m^{2-1/k}/log^2 m, so no bound R(f)=O((1+Q(f))^alpha) with alpha<4 holds; combined with the known R=O(Q^4), the optimal exponent is exactly 4, disproving the conjectured cubic relation.","verdict":"'Nearly quartic': exponent 4-o(1) via a family indexed by k, with polylog losses. Short (19 pages) and unformalized;","url":"/math#284","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=284.%20The%20optimal%20quartic","manuscripts":[{"title":"A Nearly Quartic Separation Between Randomized and Quantum Query Complexity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf"}],"reel":{"src":"/films/math/284.mp4","poster":"/films/math/284-poster.webp","captions":"/films/math/284.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":0,"why":"A nearly quartic gap between randomized and quantum query complexity, the most allowed; 19 unformalized pages.","consequence":"Pins the largest quantum query speedup for total functions. Internal to query complexity."},"detail":"/api/math/284"},{"id":"285","title":"Counterexamples to Baum–Connes and Kadison–Kaplansky","short":"Baum–Connes and Kadison–Kaplansky fail","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"(a) A finitely generated torsion-free discrete group whose coefficient-free reduced Baum-Connes assembly map in degree zero has a kernel class of infinite order (failure of rational injectivity).","verdict":"Unformalized, 184 pages of graphical small-cancellation/expander constructions ('infinite graphical presentation', voltage covers, nilpotent deck groups).","url":"/math#285","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=285.%20Counterexamples%20to%20Baum","manuscripts":[{"title":"A torsion-free counterexample to reduced Baum–Connes injectivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf"},{"title":"A torsion-free counterexample to the Kadison–Kaplansky projection conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf"},{"title":"An irrational-trace counterexample to reduced Baum–Connes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/285.mp4","poster":"/films/math/285-poster.webp","captions":"/films/math/285.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":100,"tier":"Huge if true","importance":4,"advance":4,"consequences":4,"surprise":3,"confidence":0,"why":"Groups that break the Baum–Connes and Kadison–Kaplansky conjectures, via projections with irrational trace; 184 unchecked pages.","consequence":"Would overturn a central prediction of noncommutative geometry: K-theory of group C*-algebras is not always computed by geometry."},"detail":"/api/math/285"},{"id":"286","title":"Rigidity and arithmetic of lattice von Neumann algebras","short":"Rigidity of lattice von Neumann algebras","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"technical","significanceRank":3,"lean":"none","leanLabel":"Manuscript only","claim":"For scalar-twisted group factors of ICC groups commensurable with property (T) lattices over characteristic-zero local fields, every finite-index bimodule (bifinite correspondence) to the twisted group factor of an arbitrary countable ICC group is a summand…","verdict":"Very long (190 pages), unformalized, and depends on earlier OpenAI-produced results. Fine-grained classification; significance mostly internal to W*-rigidity.","url":"/math#286","articleUrl":"/articles/openai-math#technical-results","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=286.%20Rigidity%20and%20arithmetic","manuscripts":[{"title":"Arithmeticity of twisted finite correspondences for lattices over local fields","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Arithmeticity-of-twisted-finite-correspondences-for-lattices-over-local-fields-September-23-2026/paper.pdf"},{"title":"The arithmetic category and stable recovery of lattice factors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-arithmetic-category-and-stable-recovery-of-lattice-factors-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/286.mp4","poster":"/films/math/286-poster.webp","captions":"/films/math/286.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Classifies finite-index relations between von Neumann algebras of higher-rank lattices; long, specialist, no Lean.","consequence":"Rigidity for lattice von Neumann algebras. Internal to W*-rigidity."},"detail":"/api/math/286"},{"id":"287","title":"Isomorphism of the free group factors","short":"The free group factors are isomorphic","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"L(F_n) is isomorphic to L(F_{n+1}) (unital normal trace-preserving *-isomorphism) for every n>=3; with Dykema-Radulescu amplification, L(F_2) is isomorphic to L(F_3), so all interpolated free group factors L(F_r), 10}.","verdict":"Surprising direction (most experts expected non-isomorphism), but the paper is short, constructive and Lean-checked;","url":"/math#287","articleUrl":"/articles/openai-math#lmathbb-f_2-cong-lmathbb-f_3-the-free-group-factors-are-all-the-same","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=287.%20Isomorphism%20of%20the","manuscripts":[{"title":"An isomorphism of the free group factors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf"}],"reel":{"src":"/films/math/287.mp4","poster":"/films/math/287-poster.webp","captions":"/films/math/287.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":94,"tier":"Huge if true","importance":4,"advance":4,"consequences":3,"surprise":3,"confidence":3,"why":"The free group factors are all isomorphic, answering an 80-year-old question against expectation, in 23 pages; Lean checks it.","consequence":"Ends the free group factor problem, with knock-on effects across free probability and von Neumann algebras."},"detail":"/api/math/287"},{"id":"288","title":"Kadison's similarity conjecture","short":"Kadison's similarity problem","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every bounded complex-linear unital algebra homomorphism from a unital C*-algebra into B(H), H arbitrary, is similar (via a bounded invertible S) to a *-homomorphism.","verdict":"Lean-checked with standard axioms (fidelity audit and actual build not done by me). Uses Popa's relative independence theorem and Kirchberg's equivalence as cited…","url":"/math#288","articleUrl":"/articles/openai-math#kadisons-similarity-problem","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=288.%20Kadison's%20similarity%20conjecture","manuscripts":[{"title":"Kadison's similarity theorem through uniform derivation estimates","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/288.mp4","poster":"/films/math/288-poster.webp","captions":"/films/math/288.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":82,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":2,"confidence":3,"why":"Kadison's 1955 similarity problem: bounded representations of C*-algebras are similar to *-representations; Lean checks it.","consequence":"Settles a 70-year problem; Kadison–Ringrose cohomology and Kadison–Kastler stability follow in this release."},"detail":"/api/math/288"},{"id":"289","title":"Strong Kadison–Kastler stability and its spatial boundaries","short":"Strong Kadison–Kastler stability","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every eps>0 there is delta(eps)>0, independent of algebras, representations and Hilbert space, such that any two unital von Neumann algebras on the same Hilbert space at Kadison-Kastler distance < delta are conjugate by a unitary u with ||u-1||<eps.","verdict":"For von Neumann algebras (the original setting); the C*-algebraic spatial form is disproved in the separable case.","url":"/math#289","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=289.%20Strong%20Kadison%20Kastler","manuscripts":[{"title":"Universal strong Kadison–Kastler stability","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026/paper.pdf"},{"title":"Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf"},{"title":"Close Separable C*-Algebras Without Spatial Conjugacy","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026/paper.pdf"}],"reel":{"src":"/films/math/289.mp4","poster":"/films/math/289-poster.webp","captions":"/films/math/289.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":57,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":1,"confidence":3,"why":"Close von Neumann algebras are unitarily conjugate with a universal tolerance; it fails for separable C*-algebras.","consequence":"Stability of von Neumann algebras under perturbation. Internal."},"detail":"/api/math/289"},{"id":"290","title":"Relative bicentralizers and modular spectral recovery","short":"Connes' bicentralizer problem","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"Connes' bicentralizer conjecture: for every type III_1 factor with separable predual and every faithful normal state, the bicentralizer is trivial.","verdict":"Main conjecture unformalized; formal content is a supporting analytic lemma. Short papers (19+16 pages) for a 45-year-old problem.","url":"/math#290","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=290.%20Relative%20bicentralizers%20and","manuscripts":[{"title":"Expected amenable subalgebras preserving core commutants","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026.pdf"},{"title":"Bounded recovery for modular spectral averages","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026/Bounded-recovery-for-modular-spectral-averages-September-23-2026.pdf"}],"reel":{"src":"/films/math/290.mp4","poster":"/films/math/290-poster.webp","captions":"/films/math/290.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":1,"why":"Connes' bicentralizer problem for type III_1 factors, in 35 pages; Lean checks only a supporting lemma.","consequence":"Extends Haagerup's key step for hyperfinite factors to all type III_1 factors."},"detail":"/api/math/290"},{"id":"291","title":"Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability","short":"Toms–Winter: comparison implies $\\mathcal Z$","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Unital Toms-Winter: for simple separable unital infinite-dimensional nuclear C*-algebras, strict comparison implies Z-stability (hence finite nuclear dimension), also nonunital with unbounded traces under extended-functional strict comparison;","verdict":"Four papers bundling several distinct conjectures; summary title emphasizes the equivariant case while the bigger news is the Toms-Winter implication.","url":"/math#291","articleUrl":"/articles/openai-math#tomswinter-strict-comparison-implies-jiangsu-stability","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=291.%20Cuntz%20comparison%20nuclear","manuscripts":[{"title":"Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/paper.pdf"},{"title":"Cuntz comparison and Jiang–Su absorption","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cuntz-comparison-and-Jiang-Su-absorption-September-23-2026/paper.pdf"},{"title":"Nuclear dimension and Jiang–Su stability without elementary subquotients","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nuclear-dimension-and-Jiang-Su-stability-without-elementary-subquotients-September-23-2026/paper.pdf"},{"title":"Tracial projection methods and uniform property Gamma","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/291.mp4","poster":"/films/math/291-poster.webp","captions":"/films/math/291.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":62,"tier":"Solid","importance":3,"advance":2,"consequences":3,"surprise":1,"confidence":1,"why":"The comparison-implies-Z-stability step of the Toms–Winter conjecture, via uniform property Gamma; Lean checks part.","consequence":"A key implication of the Toms–Winter conjecture, which drives the Elliott classification of C*-algebras."},"detail":"/api/math/291"},{"id":"292","title":"Kirchberg's O₂ norm-ultrapower embedding problem","short":"Kirchberg's $\\mathcal O_2$ embedding problem","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"The full group C*-algebra A=C*(G) of G = Z[1/2]^3 ⋊ (SL_3(Z) x Z) (Z acting by doubling) is separable and unital but admits no unital embedding into B^omega for any nonzero unital nuclear C*-algebra B and any free ultrafilter omega;","verdict":"Only 14 pages and Lean-checked; explicit group known from prior non-finiteness work. Clean and checkable.","url":"/math#292","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=292.%20Kirchberg's%20O%E2%82%82%20norm%2Dultrapower","manuscripts":[{"title":"An explicit obstruction to nuclear norm-ultrapower embeddings","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/292.mp4","poster":"/films/math/292-poster.webp","captions":"/films/math/292.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"Not every separable C*-algebra embeds in the ultrapower of O_2, answering Kirchberg; 14 Lean-checked pages.","consequence":"Answers Kirchberg's embedding question. Internal."},"detail":"/api/math/292"},{"id":"293","title":"Invariant projections, hyperinvariant subspaces, and transitive algebras","short":"No hyperinvariant subspace","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"On every separable infinite-dimensional complex Hilbert space there is a nonzero quasinilpotent bounded operator with no nonzero proper closed hyperinvariant subspace;","verdict":"Does NOT resolve the invariant subspace problem itself (T may have invariant subspaces that are not hyperinvariant).","url":"/math#293","articleUrl":"/articles/openai-math#the-hyperinvariant-subspace-problem-and-the-transitive-algebra-problem-with-it","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=293.%20Invariant%20projections%20hyperinvariant","manuscripts":[{"title":"Invariant-projection counterexamples for every irrational rotation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026/paper.pdf"},{"title":"Backward intertwiners and a transitive commutant","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf"}],"reel":{"src":"/films/math/293.mp4","poster":"/films/math/293-poster.webp","captions":"/films/math/293.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":3,"confidence":3,"why":"An operator with no hyperinvariant subspace, also settling the transitive algebra problem; Lean checks it.","consequence":"Settles the hyperinvariant subspace and transitive algebra problems; the invariant subspace problem stays open."},"detail":"/api/math/293"},{"id":"294","title":"Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness","short":"Kaplansky's quasitraces are not traces","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a separable unital C*-algebra that admits normalized 2-quasitraces but no tracial state (every such quasitrace is non-additive, with an explicit pair a,b where every quasitrace misses additivity by >= 1/144).","verdict":"The algebra is necessarily non-exact (Haagerup). Very short (16 pages) and Lean-checked. Note the family summary says 'all of which are nonadditive';","url":"/math#294","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=294.%20Kaplansky's%20quasitrace%20conjecture","manuscripts":[{"title":"A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/294.mp4","poster":"/films/math/294-poster.webp","captions":"/films/math/294.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"Kaplansky's quasitrace conjecture fails for a non-exact C*-algebra; 16 Lean-checked pages.","consequence":"Quasitraces need not be traces, and stable finiteness is not preserved by tensor products."},"detail":"/api/math/294"},{"id":"295","title":"The Kadison–Ringrose cohomology conjecture","short":"The Kadison–Ringrose conjecture","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every complex von Neumann algebra M and every k>=2, every bounded k-cocycle in the Hochschild complex C_b^k(M,M) has a bounded primitive: H^k(M,M)=0 for all k>=2 (with H^1=0 by Kadison-Sakai, every derivation inner). No separability or type restriction.","verdict":"28 pages, Lean-checked. By Christensen-Effros-Sinclair, vanishing reduces to the completely bounded case plus a cb-ness statement, which plausibly follows from 288's…","url":"/math#295","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=295.%20The%20Kadison%20Ringrose","manuscripts":[{"title":"Vanishing of higher bounded Hochschild cohomology","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/295.mp4","poster":"/films/math/295-poster.webp","captions":"/films/math/295.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Kadison–Ringrose: von Neumann algebras have vanishing Hochschild cohomology in themselves; Lean checks it.","consequence":"Completes Kadison–Ringrose vanishing. Internal to operator algebras."},"detail":"/api/math/295"},{"id":"296","title":"The generator problem for finite factors","short":"The generator problem","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every type II_1 factor with separable predual is generated by a single operator (equivalently two self-adjoints); with the established direct-integral reduction, every von Neumann algebra with separable predual is singly generated.","verdict":"18 pages, Lean-checked. Proof uses Popa's irreducible hyperfinite embedding and free-independence theorems;","url":"/math#296","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=296.%20The%20generator%20problem","manuscripts":[{"title":"Relative generation and the generator problem for finite factors","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/296.mp4","poster":"/films/math/296-poster.webp","captions":"/films/math/296.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":3,"why":"Every finite factor is generated by a single operator, Kadison's generator problem; Lean checks it.","consequence":"Answers Kadison's generator problem for finite factors, with a free-entropy corollary."},"detail":"/api/math/296"},{"id":"297","title":"A ZFC counterexample to Naimark's problem","short":"Naimark's problem in ZFC","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"In ZFC there is a unital, infinite-dimensional, simple C*-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent; it is not isomorphic to the compacts on any Hilbert space. Nonseparable.","verdict":"Not the first ZFC solution: Tanaka's human-authored paper predates it by two days, and OpenAI's summary says so. Separable case is impossible by Rosenberg.","url":"/math#297","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=297.%20A%20ZFC%20counterexample","manuscripts":[{"title":"A counterexample to Naimark's problem in ZFC","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf"}],"reel":{"src":"/films/math/297.mp4","poster":"/films/math/297-poster.webp","captions":"/films/math/297.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Naimark's problem has a counterexample in plain ZFC; a human paper got there two days earlier. Lean checks it.","consequence":"A ZFC answer to Naimark's problem, shared with a human paper."},"detail":"/api/math/297"},{"id":"298","title":"Two notions of free entropy differ even when both are finite","short":"Two free entropies differ","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a bounded self-adjoint tuple X (large fixed number of variables) in a tracial von Neumann algebra with -infinity < chi(X) <= chi*(X) - 1/2 < infinity, where chi is Voiculescu's microstates free entropy (operator-norm cutoff, limsup) and chi* the…","verdict":"Uses a large but fixed number of variables; specific normalization (norm cutoff, limsup). Lean-checked;","url":"/math#298","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=298.%20Two%20notions%20of","manuscripts":[{"title":"A finite-entropy separation of microstates and nonmicrostates free entropy","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/298.mp4","poster":"/films/math/298-poster.webp","captions":"/films/math/298.vtt","duration":19.6,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"The two notions of free entropy can disagree even when both are finite; Lean checks it.","consequence":"Clarifies free entropy theory. Internal."},"detail":"/api/math/298"},{"id":"299","title":"The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability","short":"The Kirchberg–Rørdam character criterion","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every free ultrafilter, a nonzero unital separable C*-algebra A is Z-stable iff its norm central-sequence algebra has no characters (no nuclearity, simplicity, trace or comparison hypothesis).","verdict":"Lean-checked main criterion; tensor-power consequence not separately listed in the formal scope.","url":"/math#299","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=299.%20The%20Kirchberg%20R%C3%B8rdam","manuscripts":[{"title":"The Kirchberg–Rørdam character criterion","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/299.mp4","poster":"/films/math/299-poster.webp","captions":"/films/math/299.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":1,"confidence":3,"why":"A C*-algebra absorbs Jiang–Su exactly when its central sequence algebra has no character; Lean checks it.","consequence":"A clean test for Z-stability, the regularity property behind C*-classification."},"detail":"/api/math/299"},{"id":"300","title":"Approximation and quadratic strong-operator paving","short":"Paving over any masa","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"For every maximal abelian subalgebra A of any von Neumann algebra and 0<eps<1, every self-adjoint element admits strong-operator paving over A with at most 5x10^8 eps^-2 projections (norm bound after compression by a projection SOT-close to 1);","verdict":"Unformalized; constants explicit but huge. Significance mostly within the Kadison-Singer circle of ideas.","url":"/math#300","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=300.%20Approximation%20and%20quadratic","manuscripts":[{"title":"Approximation Paving over Arbitrary Maximal Abelian Subalgebras","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf"},{"title":"Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf"}],"reel":{"src":"/films/math/300.mp4","poster":"/films/math/300-poster.webp","captions":"/films/math/300.vtt","duration":20.1,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Strong-operator paving with optimal block count for every masa; specialist, unformalized.","consequence":"Strengthens Kadison–Singer-style paving. Internal."},"detail":"/api/math/300"},{"id":"301","title":"Trace cones and Razak–Jacelon stabilization","short":"Trace cones classify $\\mathcal W$-stabilizations","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"Separable nuclear C*-algebras tensored with the Razak-Jacelon algebra W and the compacts K are classified by their topological cone of extended lower-semicontinuous tracial weights, with any prescribed cone map realized;","verdict":"Unformalized; specialist classification result. Public expert reaction: none found specific to this family in a limited search (news coverage of the release as a whole…","url":"/math#301","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=301.%20Trace%20cones%20and","manuscripts":[{"title":"The trace cone classifies Razak–Jacelon stabilizations","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026/The-trace-cone-classifies-Razak-Jacelon-stabilizations-September-25-2026.pdf"}],"reel":{"src":"/films/math/301.mp4","poster":"/films/math/301-poster.webp","captions":"/films/math/301.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"After Razak–Jacelon stabilization, the trace cone classifies separable nuclear algebras; unformalized.","consequence":"A classification result after stabilization. Internal."},"detail":"/api/math/301"},{"id":"302","title":"Radius of comparison equals half the mean dimension","short":"Radius of comparison is half mean dimension","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every minimal homeomorphism h of an infinite compact metrizable space X, rc(C(X) ⋊_h Z) = mdim(X,h)/2, including infinite values; zero mean dimension, the small boundary property, Z-stability and finite nuclear dimension (then <=1) are equivalent.","verdict":"Unformalized; the topological input is an algebraic-topology theorem in complex cobordism, an unusual route that needs checking by homotopy theorists as well as operator…","url":"/math#302","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=302.%20Radius%20of%20comparison","manuscripts":[{"title":"Filtered products and boundary-preserving compression in complex cobordism","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026/paper.pdf"},{"title":"Radius of comparison equals half the mean dimension","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/302.mp4","poster":"/films/math/302-poster.webp","captions":"/films/math/302.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"Radius of comparison is exactly half the mean dimension for minimal Z-actions, via complex cobordism; no Lean.","consequence":"Links dynamics and C*-regularity exactly. Internal."},"detail":"/api/math/302"},{"id":"303","title":"Weak pure infiniteness and Cuntz-algebra absorption","short":"Weak pure infiniteness is strong","discipline":"Operator algebras","disciplineIndex":12,"accent":"#93A3FF","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every C*-algebra in which each positive element is properly infinite is strongly purely infinite (answering the ordinary-to-strong part of Kirchberg-Rordam's question);","verdict":"Answers the pi => spi part; the full weak-pi equivalence is for exact algebras. Public expert reaction: none found specific to this family in a limited search (news…","url":"/math#303","articleUrl":"/articles/openai-math#mathematical-physics-and-operator-algebras","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=303.%20Weak%20pure%20infiniteness","manuscripts":[{"title":"Weak pure infiniteness and O-infinity absorption","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/303.mp4","poster":"/films/math/303-poster.webp","captions":"/films/math/303.vtt","duration":20.4,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"The natural notions of pure infiniteness for C*-algebras coincide, giving O_∞ absorption; Lean checks it.","consequence":"O_∞ absorption for nuclear algebras without simplicity, for the classification programme."},"detail":"/api/math/303"},{"id":"304","title":"The Hilbert–Smith conjecture in every dimension","short":"Hilbert–Smith in every dimension","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Every locally compact, second-countable Hausdorff group acting faithfully and jointly continuously on a connected, Hausdorff, second-countable topological n-manifold without boundary (any n) is a Lie group.","verdict":"46 pages for a ~80-year-old problem, via a new invariant (Witt groups of real sheaf complexes generated by proper images of polyhedra under arbitrary continuous maps)…","url":"/math#304","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=304.%20The%20Hilbert%20Smith","manuscripts":[{"title":"The Hilbert–Smith conjecture in every finite dimension","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/304.mp4","poster":"/films/math/304-poster.webp","captions":"/films/math/304.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":87,"tier":"Major","importance":4,"advance":4,"consequences":2,"surprise":3,"confidence":0,"why":"Hilbert–Smith: no group wilder than a Lie group acts faithfully on a manifold, via a new invariant; 46 unchecked pages.","consequence":"Settles which groups can act faithfully on manifolds, completing a line from Hilbert's fifth problem."},"detail":"/api/math/304"},{"id":"305","title":"Four-dimensional disk embedding and Wall's conjecture","short":"Disc embedding fails: $F_2$ is not good","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"(a) The unrestricted 4-dimensional disc embedding conjecture is false: there is a compact oriented smooth 4-manifold with finitely many immersed discs that have framed algebraically dual spheres (lambda(f_i,g_j)=delta_ij, lambda(g_i,g_j)=0, reduced…","verdict":"Three papers (133 pp), no Lean. The obstruction is a new 'marked tensor / Frobenius-Koszul' invariant built from polarized Chern-Simons functions on cut data;","url":"/math#305","articleUrl":"/articles/openai-math#the-singer-conjecture-in-dimension-4","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=305.%20Four%2Ddimensional%20disk%20embedding","manuscripts":[{"title":"A boundary-only obstruction to four-dimensional disk embedding","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-boundary-only-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf"},{"title":"A marked tensor obstruction to four-dimensional disk embedding","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf"},{"title":"A PD4 group without an aspherical manifold model","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/305.mp4","poster":"/films/math/305-poster.webp","captions":"/films/math/305.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":86,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":3,"confidence":0,"why":"The free group is not good for 4D disk embedding, a 40-year question behind Freedman's work; a novel, unchecked invariant.","consequence":"Would show Freedman's 4D surgery methods cannot extend to all groups, reshaping 4-manifold topology; families 320 and 315 build on it."},"detail":"/api/math/305"},{"id":"306","title":"The purely cosmetic surgery conjecture","short":"Purely cosmetic surgery","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"For a nontrivial smooth knot K in S^3, distinct slopes r != s in Q u {infinity} never give orientation-preservingly homeomorphic surgeries S^3_r(K), S^3_s(K) (purely cosmetic surgery conjecture, including the meridional slope).","verdict":"Uses the prior reduction (only slopes +-2 on genus-2 knots remain) as input, which is legitimate.","url":"/math#306","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=306.%20The%20purely%20cosmetic","manuscripts":[{"title":"Purely cosmetic surgery on knots in the three-sphere","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Purely-Cosmetic-Surgery-on-Knots-in-the-Three-Sphere-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/306.mp4","poster":"/films/math/306-poster.webp","captions":"/films/math/306.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":0,"why":"Different Dehn surgeries on a knot never give the same oriented manifold, closing the last case by instanton counts.","consequence":"Settles cosmetic surgery. Little follow-on beyond 3-manifold topology."},"detail":"/api/math/306"},{"id":"307","title":"Failure of rational injectivity for maximal coarse assembly","short":"Coarse Novikov fails, bounded geometry","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a uniformly discrete bounded-geometry metric space X, a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order class alpha in KX_1(X) whose image under ordinary coarse assembly into K_1 of the reduced Roe…","verdict":"Lean: OAI.CoarseAssembly.bounded_geometry_graph_counterexample_unconditional (OAI/Topology/CoarseAssembly/UnconditionalConclusion.lean:38), 335 files, ~34.5k lines, no…","url":"/math#307","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=307.%20Failure%20of%20rational","manuscripts":[{"title":"Failure of rational injectivity for maximal coarse assembly","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/paper.pdf"},{"title":"A counterexample to the coarse Novikov conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/307.mp4","poster":"/films/math/307-poster.webp","captions":"/films/math/307.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":2,"why":"The coarse Novikov conjecture fails for a bounded-geometry space; Lean checks the reduced version, not the headline.","consequence":"Limits the coarse Novikov approach to index theory on bounded-geometry spaces."},"detail":"/api/math/307"},{"id":"308","title":"Finite Smith–Toda complexes at every height","short":"Smith–Toda complexes $V(n)$ at every height","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every n >= 0 there are a prime p (depending on n) and a finite p-local spectrum V(n) with BP_*V(n) = BP_*/(p,v_1,...,v_n) as comodules (all exponents one). Explicitly, V(4) exists at p = 1009.","verdict":"Existence only for some (non-explicit, 'sufficiently large') prime depending on n, via ultraproducts of p-local categories;","url":"/math#308","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=308.%20Finite%20Smith%20Toda","manuscripts":[{"title":"Finite Smith–Toda Complexes at Varying Primes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf"},{"title":"A finite Smith–Toda complex V(4) at the prime 1009","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/308.mp4","poster":"/films/math/308-poster.webp","captions":"/films/math/308.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":2,"confidence":0,"why":"Smith–Toda complexes exist at every chromatic height for some large prime; whether V(4) exists at small primes stays open.","consequence":"Existence of the basic finite building blocks of chromatic homotopy theory at all heights."},"detail":"/api/math/308"},{"id":"309","title":"The Kervaire invariant problem at the prime three","short":"Kervaire invariant at the prime 3","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"At p = 3, the standard odd-primary Kervaire classes b_j in the mod-3 Adams spectral sequence survive exactly for j in {0,2,3} (stems 10, 106, 322), each detected by an element of exact order 3; b_1 and all b_j, j >= 4, die.","verdict":"90 pages; proves a conjectured coefficient-action model (Belmont-Ray Conj. 0.3) along the way and refutes the 'full gap' clause of their Conjecture 0.1 (claims an…","url":"/math#309","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=309.%20The%20Kervaire%20invariant","manuscripts":[{"title":"The Kervaire invariant problem at the prime three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/309.mp4","poster":"/films/math/309-poster.webp","captions":"/films/math/309.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"The Kervaire invariant problem at the prime 3, the analogue of Hill–Hopkins–Ravenel; unformalized.","consequence":"Settles which Kervaire invariant classes exist at the prime 3."},"detail":"/api/math/309"},{"id":"310","title":"Quillen's conjecture in rational homology","short":"Quillen's conjecture, rationally","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"For every finite group G and prime p, if O_p(G) = 1 then the poset A_p(G) of nontrivial elementary abelian p-subgroups has nonzero (augmented) reduced rational homology;","verdict":"The genuinely new content is the p = 2 endgame plus a self-contained unitary dimension property;","url":"/math#310","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=310.%20Quillen's%20conjecture%20in","manuscripts":[{"title":"Rational homology and Quillen's conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rational-homology-and-Quillens-conjecture-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/310.mp4","poster":"/films/math/310-poster.webp","captions":"/films/math/310.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Quillen's conjecture on p-subgroup complexes, finishing the last p=2 cases after decades of reductions; no Lean.","consequence":"Completes Quillen's conjecture in rational homology. Internal."},"detail":"/api/math/310"},{"id":"311","title":"The Hovey–Strickland and Chai conjectures","short":"Hovey–Strickland and Chai","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every prime p, height n >= 1 and open subgroup U of the Morava stabilizer, the U-invariant prime ideals of the Lubin-Tate ring E_0 are exactly 0, I_1, ..., I_n (Chai's 'hope', finite-residue-field form);","verdict":"34 pages; relies on the Barthel-Heard-Naumann implication. Arithmetic core (orbit arguments in characteristic p and 0) is short; I could not check it. No Lean.","url":"/math#311","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=311.%20The%20Hovey%20Strickland","manuscripts":[{"title":"Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stabilizer-Orbits-and-Thick-Tensor-Ideals-of-Dualizable-Kn-Local-Spectra-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/311.mp4","poster":"/films/math/311-poster.webp","captions":"/films/math/311.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"The Hovey–Strickland thick subcategory conjecture for K(n)-local spectra, via a reduction by others; no Lean.","consequence":"Classifies thick subcategories in the K(n)-local category. Internal."},"detail":"/api/math/311"},{"id":"312","title":"The Grothendieck homotopy hypothesis","short":"Grothendieck's homotopy hypothesis","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"For every Grothendieck coherator in the Ara-Henry convention, weak globular infinity-groupoids (models of the coherator) carry a semi-model structure whose homotopy theory is equivalent to that of spaces (Grothendieck's homotopy hypothesis in Maltsiniotis's…","verdict":"Only 26 pages because Henry's reduction does most of the bridging. Lean formalizes the elementary-expansion theorem (OAI.Grothendieck.elementary_expansion,…","url":"/math#312","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=312.%20The%20Grothendieck%20homotopy","manuscripts":[{"title":"The Grothendieck homotopy hypothesis via elementary expansions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/312.mp4","poster":"/films/math/312-poster.webp","captions":"/films/math/312.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":1,"why":"Grothendieck's homotopy hypothesis for one standard model of weak ∞-groupoids; Lean checks a key step.","consequence":"Confirms Grothendieck's picture of spaces as weak ∞-groupoids for one model."},"detail":"/api/math/312"},{"id":"313","title":"Finite generation for the K(n)-local sphere","short":"Finite generation for the $K(n)$-local sphere","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every prime p, height n >= 1 and integer t, pi_t L_{K(n)} S_p is a finitely generated Z_p-module; equivalently for L_{K(n)} of any finite p-local spectrum.","verdict":"103 pages; no uniform bound. Relies on continuous Morava descent and the smash-product theorem's horizontal vanishing line. No Lean.","url":"/math#313","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=313.%20Finite%20generation%20for","manuscripts":[{"title":"Finite generation for the K(n)-local sphere","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf"}],"reel":{"src":"/films/math/313.mp4","poster":"/films/math/313-poster.webp","captions":"/films/math/313.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Each homotopy group of the K(n)-local sphere is finitely generated over the p-adic integers; no Lean.","consequence":"Finite generation of the K(n)-local sphere's homotopy. Internal."},"detail":"/api/math/313"},{"id":"314","title":"Cyclic length and chromatic fixed-point loss","short":"Chromatic fixed-point loss","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"technical","significanceRank":3,"lean":"none","leanLabel":"Manuscript only","claim":"For a finite p-group G, subgroup H, and n >= 0, the optimal chromatic fixed-point loss r_n(G,H) (least r such that K(n+r)-acyclicity of Phi^H X implies K(n)-acyclicity of Phi^G X for all finite genuine G-spectra X) equals the least length of a subnormal chain…","verdict":"30 pages; completes the Balmer spectrum topology for p-groups if correct. No Lean.","url":"/math#314","articleUrl":"/articles/openai-math#technical-results-1","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=314.%20Cyclic%20length%20and","manuscripts":[{"title":"Cyclic length and chromatic fixed-point loss","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026.pdf"}],"reel":{"src":"/films/math/314.mp4","poster":"/films/math/314-poster.webp","captions":"/films/math/314.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Sharp chromatic fixed-point loss for every p-group, by explicit finite spectra; no Lean.","consequence":"Sharp chromatic Smith theory for p-groups. Internal."},"detail":"/api/math/314"},{"id":"315","title":"The four-dimensional Singer conjecture","short":"Singer conjecture in dimension 4","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every finite connected aspherical integral Poincare complex Q of formal dimension 4 (any orientation character), the L2-Betti numbers of the universal cover vanish outside degree 2;","verdict":"45 pages for a well-known open case; proof is combinatorial-group-theoretic (interval/annular chain arguments) rather than analytic.","url":"/math#315","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=315.%20The%20four%2Ddimensional%20Singer","manuscripts":[{"title":"The Singer conjecture in dimension four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Singer-conjecture-in-dimension-four-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/315.mp4","poster":"/films/math/315-poster.webp","captions":"/films/math/315.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"The Singer conjecture in dimension 4: L2 homology of aspherical 4-manifolds sits in the middle; no Lean.","consequence":"Settles Singer's conjecture in dimension 4. Internal."},"detail":"/api/math/315"},{"id":"316","title":"Curtis’s conjecture","short":"Curtis's conjecture","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"In positive degrees the mod-2 stable Hurewicz image pi_d^S -> H_d(Q_0S^0;F_2) is spanned by the images of eta, nu, sigma and the Kervaire-invariant-one classes theta_j that exist (Curtis's conjecture);","verdict":"37 pages for a 50-year-old conjecture; depends on HHR for the finiteness corollary. No Lean.","url":"/math#316","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=316.%20Curtis%E2%80%99s%20conjecture","manuscripts":[{"title":"The Stable Hurewicz Image of the Sphere at Two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/316.mp4","poster":"/films/math/316-poster.webp","captions":"/films/math/316.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"Curtis's conjecture: only Hopf-invariant-one and Kervaire classes are spherical in QS^0; 37 unchecked pages.","consequence":"Settles which homology classes of QS^0 are spherical. Internal to homotopy theory."},"detail":"/api/math/316"},{"id":"317","title":"Thomason model structures in all strict higher dimensions","short":"Thomason model structures, every $n$","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every n in {1,2,...} u {infinity}, small strict globular n-categories carry a proper combinatorial model structure with weak equivalences and fibrations detected by Ex^2 N_n (Street nerve), cofibrantly generated by c_n Sd^2 of the horn/boundary…","verdict":"Lean: OAI.Thomason.main and OAI.Thomason.nerveDetection_allDimensions (OAI/CategoryTheory/Thomason/RankedFaithfulness.lean:683; 225 files, ~160k lines, no sorry).","url":"/math#317","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=317.%20Thomason%20model%20structures","manuscripts":[{"title":"Thomason Model Structures in Every Strict Higher Dimension","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/317.mp4","poster":"/films/math/317-poster.webp","captions":"/films/math/317.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":3,"why":"Strict n-categories model all homotopy types in every dimension, extending Thomason; Lean checks it.","consequence":"Model-theoretic foundations for higher categories. Internal."},"detail":"/api/math/317"},{"id":"318","title":"Chromatic splitting: filtrations and counterexamples","short":"Chromatic splitting fails at height 3","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(i) For p >= 5 the canonical map L_0 L_{K(3)}S -> L_0 L_{K(2)} L_{K(3)}S is nonzero on pi_{-3}, so Hopkins's strong chromatic splitting fails at height 3 even as an equivalence of E(2)-local spectra.","verdict":"Five papers, 372 pages; mixed positive and negative results. The rational obstruction paper (49 pp) is the cleanest;","url":"/math#318","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=318.%20Chromatic%20splitting%20filtrations","manuscripts":[{"title":"Filtered chromatic splitting at generic primes","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf"},{"title":"The height-three chromatic overlap: an explicit filtration and its attachments","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-height-three-chromatic-overlap-an-explicit-filtration-and-its-attachments-September-27-2026/paper.pdf"},{"title":"A rational obstruction to strong chromatic splitting at height three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf"},{"title":"Failure of finite assembly for a chromatic overlap at the prime three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026/paper.pdf"},{"title":"Counterexamples to weak chromatic splitting: sphere kernels and descent exponents","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Counterexamples-to-weak-chromatic-splitting-sphere-kernels-and-descent-exponents-September-27-2026/paper.pdf"}],"reel":{"src":"/films/math/318.mp4","poster":"/films/math/318-poster.webp","captions":"/films/math/318.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":64,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":3,"confidence":0,"why":"The chromatic splitting conjecture fails at height 3, while a weaker filtered form holds; 372 unformalized pages.","consequence":"Revises a central conjecture about the structure of the stable homotopy groups of spheres."},"detail":"/api/math/318"},{"id":"319","title":"Counterexamples to finite generation at chromatic height two","short":"Hahn–Wilson fails at height 2","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"technical","significanceRank":3,"lean":"none","leanLabel":"Manuscript only","claim":"For every sufficiently large prime p there is a connective p-complete spectrum X of exact fp-type 2 not in the thick subcategory generated by (a specified standard form of) BP^_p, although L_2^f X = L_2 X and L_{T(2)}X = L_{K(2)}X;","verdict":"Only 'sufficiently large p'; depends on 'the specified standard form' of BP. 79 pages. No Lean.","url":"/math#319","articleUrl":"/articles/openai-math#technical-results-1","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=319.%20Counterexamples%20to%20finite","manuscripts":[{"title":"Counterexamples to the Hahn-Wilson conjecture at height two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf"}],"reel":{"src":"/films/math/319.mp4","poster":"/films/math/319-poster.webp","captions":"/films/math/319.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":31,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":2,"confidence":0,"why":"An fp spectrum not finitely built from truncated Brown–Peterson spectra, against Hahn–Wilson; large primes only.","consequence":"A counterexample to a finite-generation conjecture. Nothing follows beyond it."},"detail":"/api/math/319"},{"id":"320","title":"Nonhomeomorphic closed aspherical four-manifolds","short":"Borel fails in dimension 4","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"There exist closed connected aspherical topological 4-manifolds M, N with common word-hyperbolic fundamental group that are homotopy equivalent but not homeomorphic (so the homeomorphism-existence form of the Borel conjecture fails in dimension 4);","verdict":"Imports the marked tensor obstruction (family 305) and the PD4 realization machinery as black boxes ('principal inputs ... [MT, Theorem 5.1], [PD, Sections 3-5]');","url":"/math#320","articleUrl":"/articles/openai-math#disc-embedding-walls-pd4-question-and-the-borel-conjecture-in-dimension-4","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=320.%20Nonhomeomorphic%20closed%20aspherical","manuscripts":[{"title":"Nonhomeomorphic closed aspherical four-manifolds with the same homotopy type","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonhomeomorphic-closed-aspherical-four-manifolds-with-the-same-homotopy-type-October-4-2026/paper.pdf"}],"reel":{"src":"/films/math/320.mp4","poster":"/films/math/320-poster.webp","captions":"/films/math/320.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":4,"advance":3,"consequences":2,"surprise":3,"confidence":0,"why":"Two non-homeomorphic aspherical 4-manifolds with the same fundamental group, so Borel fails in 4D; depends on family 305.","consequence":"The Borel conjecture fails in dimension 4, if family 305 holds."},"detail":"/api/math/320"},{"id":"321","title":"A counterexample to Wall's finite D(2) problem","short":"Wall's D(2) problem: a counterexample","discipline":"Topology","disciplineIndex":13,"accent":"#D7E37A","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"There is a finite connected 3-dimensional CW complex X with H_i(X~;Z)=0 for i>2 and H^3(X;M)=0 for every Z[pi_1 X]-module M (Wall's D(2) condition) that is not homotopy equivalent to any finite complex of dimension <= 2.","verdict":"Only 9 pages of mathematics and fully elementary. I re-derived the key computation: the four relators hold under the character rho (x1->-1, a1->zeta, x2->zeta^2, a2->-1,…","url":"/math#321","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=321.%20A%20counterexample%20to","manuscripts":[{"title":"A Counterexample to Wall's D(2) Problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf"}],"reel":{"src":"/films/math/321.mp4","poster":"/films/math/321-poster.webp","captions":"/films/math/321.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":2,"confidence":1,"why":"A counterexample to Wall's D(2) problem in 10 elementary pages; the reviewer re-derived the key computation.","consequence":"Answers Wall's question, with implications for the relation between 2-complexes and group presentations."},"detail":"/api/math/321"},{"id":"322","title":"Tingley’s sphere-isometry problem","short":"Tingley's sphere-isometry problem","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For any nonzero real Banach spaces X, Y (any dimension; not assumed separable, reflexive, smooth or strictly convex), every surjective isometry f between their unit spheres extends uniquely to a surjective real-linear isometry T, namely T(x)=||x|| f(x/||x||),…","verdict":"Remarkably short (12 pp.) for a 39-year-old problem, but the full statement is Lean-checked against Mathlib definitions, which substantially raises confidence (residual…","url":"/math#322","articleUrl":"/articles/openai-math#tingleys-problem","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=322.%20Tingley%E2%80%99s%20sphere%2Disometry%20problem","manuscripts":[{"title":"A positive solution to Tingley’s problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/322.mp4","poster":"/films/math/322-poster.webp","captions":"/films/math/322.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":69,"tier":"Solid","importance":3,"advance":4,"consequences":1,"surprise":2,"confidence":3,"why":"Tingley's problem: an isometry between unit spheres of normed spaces extends linearly, in 12 pages; Lean checks it.","consequence":"Settles Tingley's problem. Internal to Banach space geometry."},"detail":"/api/math/322"},{"id":"323","title":"Independence of the separable quotient problem","short":"The separable quotient problem","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"For F in {R, C}, ZFC proves: (i) if the continuum c is real-valued measurable, every infinite-dimensional Banach space over F has a separable infinite-dimensional quotient;","verdict":"Kind recorded as 'proof' (an independence result is a full answer in the usual relative sense), though only the CH half is Lean-checked.","url":"/math#323","articleUrl":"/articles/openai-math#the-separable-quotient-problem-is-independent-of-zfc","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=323.%20Independence%20of%20the","manuscripts":[{"title":"Relative independence of the separable quotient problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/323.mp4","poster":"/films/math/323-poster.webp","captions":"/films/math/323.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":73,"tier":"Solid","importance":3,"advance":4,"consequences":1,"surprise":3,"confidence":2,"why":"Banach's separable quotient problem is independent of ZFC; Lean checks the CH half.","consequence":"The separable quotient question has no ZFC answer. Internal."},"detail":"/api/math/323"},{"id":"324","title":"Lipschitz equivalent Banach spaces need not be linearly isomorphic","short":"Lipschitz-equivalent, not isomorphic","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"There are separable real Banach spaces X, Y and a bijection Psi with (4/21)||s-t|| <= ||Psi(s)-Psi(t)|| <= (76/25)||s-t||, where X contains a linearly isometric copy of c_0(l_2) but Y contains no linear copy of c_0(l_2);","verdict":"Real scalars only; no reflexive or complex analogue claimed. The obstruction is block-valued (c_0(l_2), not c_0).","url":"/math#324","articleUrl":"/articles/openai-math#lipschitz-equivalent-separable-banach-spaces-need-not-be-isomorphic","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=324.%20Lipschitz%20equivalent%20Banach","manuscripts":[{"title":"Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026/paper.pdf"},{"title":"Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026/paper.pdf"}],"reel":{"src":"/films/math/324.mp4","poster":"/films/math/324-poster.webp","captions":"/films/math/324.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":3,"why":"Separable Banach spaces can be Lipschitz equivalent without being linearly isomorphic, a central open question; Lean checks it.","consequence":"Metric structure does not determine linear structure even for separable spaces, a basic limit for nonlinear Banach theory."},"detail":"/api/math/324"},{"id":"325","title":"The complete Crouzeix conjecture","short":"The complete Crouzeix conjecture","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For every bounded operator A on any complex Hilbert space (no separability), every m, and every M_m(C)-valued polynomial P, ||P[A]|| D conformal) a contraction has cond(S) <= 2 and a single positive boundary density represents every matrix-valued analytic…","verdict":"The headline 'Crouzeix conjecture' (scalar) was solved by humans+AI in July-Aug 2026, ~2 months before this release;","url":"/math#325","articleUrl":"/articles/openai-math#the-complete-crouzeix-conjecture","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=325.%20The%20complete%20Crouzeix","manuscripts":[{"title":"A direct proof of the complete Crouzeix inequality","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf"},{"title":"The complete Crouzeix theorem: optimal similarity and a common positive boundary representation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/325.mp4","poster":"/films/math/325-poster.webp","captions":"/films/math/325.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"The complete (matrix-valued) Crouzeix conjecture; the scalar case was solved two months earlier. Lean checks it.","consequence":"One similarity of condition number at most 2 controls functional calculus of any matrix, as operator theory wanted."},"detail":"/api/math/325"},{"id":"326","title":"The cotype–cotype conjecture under the approximation property","short":"Cotype–cotype under approximation","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"A nonzero real Banach space X with the (ordinary, not bounded) approximation property is K-convex (equivalently has nontrivial Rademacher type, equivalently does not contain l1^n uniformly) iff both X and X* have finite Rademacher cotype (exponents may…","verdict":"Restricted to spaces with AP (necessarily: false without it) and real scalars. Proves under ordinary AP what was conjectured even under BAP, so stronger than the…","url":"/math#326","articleUrl":"/articles/openai-math#cotypecotype-under-the-approximation-property","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=326.%20The%20cotype%20cotype","manuscripts":[{"title":"The cotype–cotype conjecture under the approximation property","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/326.mp4","poster":"/films/math/326-poster.webp","captions":"/films/math/326.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":52,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"Finite cotype of a space and its dual forces K-convexity, under the approximation property; Lean checks it.","consequence":"Entropy duality under cotype bounds follows. Internal to Banach spaces."},"detail":"/api/math/326"},{"id":"327","title":"Markov type characterizes superreflexivity","short":"Markov type and superreflexivity","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every real Banach space with Markov type p for some p>1 (Ball's definition, all finite stationary reversible chains) is superreflexive, i.e. admits an equivalent uniformly convex (equivalently uniformly smooth) norm.","verdict":"Short (14 pp) and uses classical inputs (James-Enflo finite representability, Brunel-Sucheston spreading/ESA norms, finite Ramsey). Real scalars only.","url":"/math#327","articleUrl":"/articles/openai-math#markov-type-characterizes-superreflexivity","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=327.%20Markov%20type%20characterizes","manuscripts":[{"title":"Nontrivial Markov Type Forces Superreflexivity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/327.mp4","poster":"/films/math/327-poster.webp","captions":"/films/math/327.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":3,"why":"Only superreflexive Banach spaces have nontrivial Markov type, a Ribe-program question; Lean checks it.","consequence":"Closes a Ribe-program question. Internal."},"detail":"/api/math/327"},{"id":"328","title":"Nonexpansive fixed points in reflexive Banach spaces","short":"Kirk's problem: reflexive fixed points","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"In every real reflexive Banach space (original norm; no uniform convexity, normal structure, separability or unconditional-basis assumption), every nonexpansive self-map of a nonempty closed bounded convex set has a fixed point.","verdict":"Calling this 'landmark' is conditional on the Lean proof checking (I could not build it); the statement in the comparator is faithful.","url":"/math#328","articleUrl":"/articles/openai-math#kirks-problem-nonexpansive-maps-on-reflexive-spaces-have-fixed-points","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=328.%20Nonexpansive%20fixed%20points","manuscripts":[{"title":"Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/328.mp4","poster":"/films/math/328-poster.webp","captions":"/films/math/328.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":2,"confidence":3,"why":"Every nonexpansive self-map of a bounded closed convex set in a reflexive space has a fixed point; Lean checks it.","consequence":"Reflexivity alone gives fixed points for nonexpansive maps, settling metric fixed point theory's central question."},"detail":"/api/math/328"},{"id":"329","title":"A counterexample to metric-entropy duality","short":"Metric-entropy duality fails","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Disproof of Pietsch's dimension-free metric-entropy duality conjecture: for every a,b>=1 there are n and an origin-symmetric convex body K in R^n with log N(K, B_inf) > b log N(B_inf°, a^{-1} K°) (L = cube).","verdict":"Disproves the fully general dimension-free conjecture; does not contradict AMS04 (Euclidean-ball case) or the K-convex case (and is consistent with family 326's…","url":"/math#329","articleUrl":"/articles/openai-math#analysis-pde-and-convex-geometry","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=329.%20A%20counterexample%20to","manuscripts":[{"title":"Counterexamples to the duality conjecture for metric entropy","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/329.mp4","poster":"/films/math/329-poster.webp","captions":"/films/math/329.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":65,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":3,"confidence":3,"why":"Covering numbers are not stable under taking polars, refuting the metric-entropy duality conjecture; Lean checks it.","consequence":"Entropy numbers of an operator and its adjoint can differ. Internal."},"detail":"/api/math/329"},{"id":"330","title":"A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces","short":"Lipschitz-free spaces: AP without BAP","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a countable, unbounded, non-proper metric space with all distinct points at distance >= 1 whose real Lipschitz-free space F(M) has the approximation property but fails the lambda-bounded approximation property for every lambda.","verdict":"Answers Kalton's question negatively; the human result of Smith (2025-26) already gave the discrete (not uniformly discrete) version, so this is the sharpened form.","url":"/math#330","articleUrl":"/articles/openai-math#lipschitz-free-spaces-ap-without-bap-330","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=330.%20A%20uniformly%20discrete","manuscripts":[{"title":"A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf"}],"reel":{"src":"/films/math/330.mp4","poster":"/films/math/330-poster.webp","captions":"/films/math/330.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":27,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":1,"confidence":3,"why":"Uniformly discrete metric spaces whose free spaces lack bounded approximation, sharpening a recent human result.","consequence":"A sharper counterexample after Smith. Nothing follows beyond it."},"detail":"/api/math/330"},{"id":"331","title":"Reflexive midpoint convexity and diamond distortion","short":"Midpoint convexity and diamonds","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"Seven papers. Main: a separable reflexive real Banach space X = J* (dual of a James-type square-sum segment norm on a forest of countably branching trees of unbounded finite heights) whose given norm is asymptotically midpoint uniformly convex (averaged…","verdict":"The overview summary is garbled: it says 'midpoint uniform convexity does not force uniformly bounded diamond distortion', but AMUC is already known to PREVENT uniform…","url":"/math#331","articleUrl":"/articles/openai-math#midpoint-convexity-and-diamonds-331","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=331.%20Reflexive%20midpoint%20convexity","manuscripts":[{"title":"Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026/manuscript.pdf"},{"title":"Midpoint lenses in segment spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026/manuscript.pdf"},{"title":"Distortion of countably branching diamonds from midpoint and tree energies","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026/manuscript.pdf"},{"title":"Exact asymptotic moduli in a Daugavet subspace of L1","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf"},{"title":"Midpoint convexity from bounded tree potentials and path costs","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026/manuscript.pdf"},{"title":"Independent products in real L1: asymptotic midpoint convexity without AUC renormings","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026/manuscript.pdf"},{"title":"Midpoint convexity from two recursive potentials","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026/manuscript.pdf"}],"reel":{"src":"/films/math/331.mp4","poster":"/films/math/331-poster.webp","captions":"/films/math/331.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":27,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":1,"confidence":3,"why":"A reflexive space without AUC renorming that still avoids diamond embeddings, answering Baudier–Lancien.","consequence":"Answers one problem from a list. Nothing follows beyond it."},"detail":"/api/math/331"},{"id":"332","title":"Metric Markov cotype of ℓ1 and Hilbert-space Lipschitz extension","short":"Metric Markov cotype of $\\ell_1$","discipline":"Functional analysis","disciplineIndex":14,"accent":"#74E0A6","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Real l1 has metric Markov cotype two in the Mendel-Naor sense, with N_2(l1) <= 12 sqrt(21) (squared constant 3024). Consequence (via Mendel-Naor Corollary 1.13 / Ball's extension theorem): every Lipschitz map from any subset of a real Hilbert space into l1…","verdict":"Only 9 pages and elementary (cut decomposition of finite l1 subsets, cubic smoothing, a fourth-power martingale potential), which makes it quick for experts to check;","url":"/math#332","articleUrl":"/articles/openai-math#balls-extension-problem-from-hilbert-space-into-ell_1","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=332.%20Metric%20Markov%20cotype","manuscripts":[{"title":"Metric Markov Cotype Two of ℓ₁","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Metric-Markov-Cotype-Two-of-l1-October-5-2026/l1-markov-cotype.pdf"}],"reel":{"src":"/films/math/332.mp4","poster":"/films/math/332-poster.webp","captions":"/films/math/332.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"ℓ1 has metric Markov cotype 2, so Hilbert-to-ℓ1 Lipschitz maps extend; 9 elementary, unformalized pages.","consequence":"New Lipschitz extension results into ℓ1 via Ball's framework."},"detail":"/api/math/332"},{"id":"333","title":"Smooth isometric immersions of surfaces into ℝ4","short":"Smooth isometric immersions into $\\mathbb{R}^4$","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every closed smooth Riemannian surface (orientable or not, any curvature) admits a C-infinity isometric immersion into R^4.","verdict":"Lean: OAI.ClosedSurfaceR4.FiniteOrderSmoothing.smooth_isometric_immersion (OAI/Geometry/SurfaceImmersion/Main.lean:16), 2050 files ~175k lines, no sorry.","url":"/math#333","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=333.%20Smooth%20isometric%20immersions","manuscripts":[{"title":"Smooth isometric immersions of closed surfaces into Euclidean four-space","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/333.mp4","poster":"/films/math/333-poster.webp","captions":"/films/math/333.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"Every smooth surface metric is realized by a smooth immersed surface in R^4, one dimension below Gromov; Lean checks it.","consequence":"Settles the dimension-4 case of isometric immersion for surfaces. Internal."},"detail":"/api/math/333"},{"id":"334","title":"A smooth surface metric with no local isometric immersion in ℝ3","short":"A smooth metric with no local immersion in $\\mathbb{R}^3$","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a C-infinity positive-definite metric g on (-1,1)^2, agreeing with the Euclidean metric to infinite order at the origin, such that no neighbourhood of the origin admits a C-infinity isometric immersion into R^3.","verdict":"Lean: OAI.SmoothLocal.Geometry.exists_local_metric_without_local_immersion (OAI/Geometry/IsometricImmersion/Main.lean:49), 447 files ~57k lines.","url":"/math#334","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=334.%20A%20smooth%20surface","manuscripts":[{"title":"A Smooth Metric with No Local Isometric Immersion into Three-Space","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/334.mp4","poster":"/films/math/334-poster.webp","captions":"/films/math/334.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"A smooth surface metric with no local isometric immersion in R^3, a classical open question; Lean checks it.","consequence":"Settles the smooth local isometric embedding question in 3D. Internal."},"detail":"/api/math/334"},{"id":"335","title":"Gromov’s integral scalar-curvature bound for simplicial volume","short":"Gromov–Lawson in every dimension","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"(a) Every closed connected oriented smooth n-manifold (n >= 2) with positive scalar curvature is rationally inessential: c_*[M] = 0 in H_n(B pi_1 M; Q).","verdict":"121 pages total. The quantitative bound depends on the rational-inessentiality paper. The method (graph deformations with one warped circle per coordinate and a…","url":"/math#335","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=335.%20Gromov%E2%80%99s%20integral%20scalar%2Dcurvature","manuscripts":[{"title":"An integral scalar curvature bound for real simplicial volume","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf"},{"title":"Positive scalar curvature forces rational inessentiality","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/335.mp4","poster":"/films/math/335-poster.webp","captions":"/films/math/335.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":94,"tier":"Huge if true","importance":4,"advance":4,"consequences":3,"surprise":3,"confidence":0,"why":"Aspherical manifolds carry no positive scalar curvature metric in any dimension, Gromov's bound; 121 unformalized pages.","consequence":"Positive scalar curvature obstructions for all aspherical manifolds, without spin or dimension limits; a new method others could reuse."},"detail":"/api/math/335"},{"id":"336","title":"Spectral scalar curvature, Urysohn width, and macroscopic dimension","short":"Codimension-2 width under $\\mathrm{Scal} \\ge 1$","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For n >= 4 there is C_n such that every complete connected n-manifold with Scal >= 1 maps continuously to a simplicial complex of dimension = 2), aspherical obstruction.","verdict":"147 pages over three papers; the spectral 3-d paper gives an explicit constant 500. No Lean.","url":"/math#336","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=336.%20Spectral%20scalar%20curvature","manuscripts":[{"title":"Spectral scalar curvature and uniform Urysohn width","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf"},{"title":"Spectral scalar curvature and Urysohn width in dimension three","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026/spectral-urysohn-three-manifolds.pdf"},{"title":"Positive scalar curvature and uniform codimension-two width","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/336.mp4","poster":"/films/math/336-poster.webp","captions":"/films/math/336.vtt","duration":19.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"Positive scalar curvature makes a manifold thin in two directions, in every dimension; no Lean.","consequence":"Settles Gromov's width and macroscopic-dimension conjectures for positive scalar curvature."},"detail":"/api/math/336"},{"id":"337","title":"Sharp Cartan–Hadamard isoperimetry and rigidity","short":"Cartan–Hadamard isoperimetry, all dimensions","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(a) Generalized Cartan-Hadamard conjecture in every dimension: in a complete simply connected manifold with sec = that of the equal-volume ball in the kappa space form; for kappa = 0 bounded equality sets are isometric to Euclidean balls.","verdict":"Lean: OAI.CAT0Fillings.sharp_integral_filling (OAI/Geometry/CAT0Fillings/Main.lean:111;","url":"/math#337","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=337.%20Sharp%20Cartan%20Hadamard","manuscripts":[{"title":"Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf"},{"title":"Sharp integral fillings in CAT(0) spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-integral-fillings-in-CAT(0"}],"reel":{"src":"/films/math/337.mp4","poster":"/films/math/337-poster.webp","captions":"/films/math/337.vtt","duration":20.2,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":2,"why":"The Cartan–Hadamard isoperimetric inequality in every dimension, known only up to 4 before; Lean checks the CAT(0) core.","consequence":"Sharp isoperimetry for every nonpositively curved space, including singular ones."},"detail":"/api/math/337"},{"id":"338","title":"Yau's uniformization conjecture","short":"Yau's uniformization conjecture","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"A connected noncompact complex manifold of complex dimension n >= 1 carrying a complete Kaehler metric of pointwise strictly positive holomorphic bisectional curvature is biholomorphic to C^n (Yau's uniformization conjecture), with no curvature bounds, volume…","verdict":"77 pages, no Lean. Constructs a Kaehler-Ricci flow from an initial metric with unbounded curvature and possible collapsing, a regime where standard existence theory…","url":"/math#338","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=338.%20Yau's%20uniformization%20conjecture","manuscripts":[{"title":"Uniformization of complete Kähler manifolds with positive bisectional curvature","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniformization-of-complete-Kahler-manifolds-with-positive-bisectional-curvature-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/338.mp4","poster":"/films/math/338-poster.webp","captions":"/films/math/338.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":68,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":2,"confidence":0,"why":"Yau's uniformization: complete Kähler manifolds with positive bisectional curvature are C^n; no Lean.","consequence":"Classifies positively curved noncompact Kähler manifolds, extending Frankel's picture."},"detail":"/api/math/338"},{"id":"339","title":"Katok's entropy rigidity conjecture","short":"Katok's entropy rigidity conjecture","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"For every closed connected Riemannian manifold of dimension >= 3 with strictly negative sectional curvature, Liouville measure has maximal entropy for the geodesic flow (h_mL = h_top) iff the metric is locally symmetric (Katok's entropy rigidity conjecture;","verdict":"66 pages; ends with Benoist-Foulon-Labourie after proving global smoothness of the stable/unstable distributions;","url":"/math#339","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=339.%20Katok's%20entropy%20rigidity","manuscripts":[{"title":"Entropy equality and local symmetry in negative curvature","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/339.mp4","poster":"/films/math/339-poster.webp","captions":"/films/math/339.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"Katok's entropy rigidity: volume and topological entropy agree only on locally symmetric spaces; no Lean.","consequence":"Settles Katok's rigidity conjecture. Internal to dynamics on negatively curved manifolds."},"detail":"/api/math/339"},{"id":"340","title":"A counterexample to the nearby Lagrangian conjecture","short":"The nearby Lagrangian conjecture fails","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"For some sufficiently large even N, Q = S^9 x S^{N-1} has a closed exact embedded Lagrangian L in T*Q, diffeomorphic to Q, that is not Hamiltonian isotopic to the zero section (nearby Lagrangian conjecture fails).","verdict":"Only 28 pages to disprove a central conjecture of symplectic topology, with a non-explicit 'sufficiently large even N'.","url":"/math#340","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=340.%20A%20counterexample%20to","manuscripts":[{"title":"A counterexample to the nearby Lagrangian conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/340.mp4","poster":"/films/math/340-poster.webp","captions":"/films/math/340.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":86,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":3,"confidence":0,"why":"An exact Lagrangian in a cotangent bundle that is not the zero section moved, refuting a central symplectic conjecture.","consequence":"Would overturn a central belief of symplectic topology, that exact Lagrangians in cotangent bundles are trivial."},"detail":"/api/math/340"},{"id":"341","title":"Donaldson's hypersymplectic deformation conjecture","short":"Donaldson's hypersymplectic conjecture","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"On a closed connected oriented 4-manifold, every hypersymplectic triple normalized by integral omega_i ^ omega_j = delta_ij deforms through hypersymplectic triples, fixing all three cohomology classes, to a hyperkaehler triple.","verdict":"72 pages; builds on the current estimates of family 342 (which is formalized) but the prescribed-class deformation arguments are not formalized.","url":"/math#341","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=341.%20Donaldson's%20hypersymplectic%20deformation","manuscripts":[{"title":"Deforming hypersymplectic four-manifolds to hyperkähler triples","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/341.mp4","poster":"/films/math/341-poster.webp","captions":"/films/math/341.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Donaldson's hypersymplectic deformation conjecture, building on the formalized tamed-to-compatible result; no Lean.","consequence":"Hypersymplectic 4-manifolds are K3 or tori, a topological classification."},"detail":"/api/math/341"},{"id":"342","title":"Donaldson's tamed-to-compatible conjecture","short":"Donaldson's tamed-to-compatible question","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"If a smooth almost complex structure J on a closed connected 4-manifold is tamed by a symplectic form, then some symplectic form is compatible with J (Donaldson's tamed-to-compatible question); J is fixed, the class may change.","verdict":"Only 18 pages, but the main theorem is formalized: OAI.TamingCompatibility.taming_implies_compatibility (OAI/Geometry/TamingCompatibility/Main.lean:30;","url":"/math#342","articleUrl":"/articles/openai-math#donaldsons-tamed-to-compatible-question-formalized","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=342.%20Donaldson's%20tamed%2Dto%2Dcompatible%20conjecture","manuscripts":[{"title":"Taming implies compatibility on four-manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/342.mp4","poster":"/films/math/342-poster.webp","captions":"/films/math/342.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":75,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":2,"confidence":3,"why":"Donaldson's tamed-to-compatible question for symplectic 4-manifolds, in 18 pages; Lean checks the main theorem.","consequence":"Settles Donaldson's question and supplies estimates used by family 341."},"detail":"/api/math/342"},{"id":"343","title":"Symplectic ball packing in higher dimensions","short":"Symplectic ball packing, dimension 6 and up","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"For n >= 3 (real dimension >= 6), closed balls of capacities R_1..R_k embed symplectically and disjointly into the open ball of capacity R iff sum R_i^n < R^n and R_i + R_j < R for all i != j (Siegel-Yao Conjecture A).","verdict":"Lean: OAI.HigherDimensionalBallPacking.main_theorem (OAI/Geometry/BallPacking, 207 files ~107k lines) plus necessity;","url":"/math#343","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=343.%20Symplectic%20ball%20packing","manuscripts":[{"title":"Symplectic Ball Packings in Higher Dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/343.mp4","poster":"/films/math/343-poster.webp","captions":"/films/math/343.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"In dimension 6 and up, symplectic ball packings are limited only by volume and two-ball non-squeezing; Lean checks it.","consequence":"Symplectic packing in high dimensions has no hidden obstructions. Internal."},"detail":"/api/math/343"},{"id":"344","title":"The metric Blaschke conjecture","short":"The metric Blaschke conjecture","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Every closed connected Riemannian manifold with inj = diam (a Blaschke manifold) is, up to scaling, isometric to a round sphere, standard RP^n, CP^n, HP^n or the Cayley plane (metric Blaschke conjecture).","verdict":"41 pages, uses Schur-complement Jacobi-determinant comparisons plus a Kazdan-type inequality; 'the homeomorphism hypothesis in the result recorded by Wilking ...","url":"/math#344","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=344.%20The%20metric%20Blaschke","manuscripts":[{"title":"The metric Blaschke theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/344.mp4","poster":"/films/math/344-poster.webp","captions":"/films/math/344.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"The metric Blaschke conjecture for the projective spaces, by volume comparison; no Lean.","consequence":"Closes the metric Blaschke conjecture. Internal."},"detail":"/api/math/344"},{"id":"345","title":"Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds","short":"Infinitely many closed geodesics on $S^n$","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"Every smooth Riemannian metric on S^n (n >= 2) has infinitely many prime closed geodesics with pairwise distinct images; likewise on every closed manifold finitely covered by a sphere, and on every closed 3-manifold (no orientability or nondegeneracy…","verdict":"50 pages; the S^n (n >= 3) case is a famous open problem for degenerate metrics; the paper criticizes a competing claim (Charles), so expert adjudication is needed.","url":"/math#345","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=345.%20Infinitely%20many%20closed","manuscripts":[{"title":"Infinitely many closed geodesic images on every Riemannian sphere","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/345.mp4","poster":"/films/math/345-poster.webp","captions":"/films/math/345.vtt","duration":20.3,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"Every metric on a sphere of dimension 3 or more has infinitely many closed geodesics; a competing claim exists.","consequence":"Settles the closed geodesics question for spheres, if the competing claim is resolved."},"detail":"/api/math/345"},{"id":"346","title":"Sharp singular-set bounds for stationary integral varifolds","short":"Regularity of stationary integral varifolds","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"Every stationary integral m-varifold in an open subset of R^{m+n} (any m, n >= 1) has H^m(Sing V) = 0 (almost-everywhere regularity) and, in a companion, dim_H Sing V <= m-1 (sharp, cf. two planes crossing). Also on round spheres.","verdict":"112 pages over two papers; this is one of the central open problems of geometric measure theory, so the burden is high.","url":"/math#346","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=346.%20Sharp%20singular%2Dset%20bounds","manuscripts":[{"title":"A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf"},{"title":"Almost-everywhere regularity of stationary integral varifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/346.mp4","poster":"/films/math/346-poster.webp","captions":"/films/math/346.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":82,"tier":"Major","importance":3,"advance":4,"consequences":3,"surprise":2,"confidence":0,"why":"The singular set of a stationary varifold has codimension at least one, a 50-year GMT problem; 112 unformalized pages.","consequence":"Removes a 50-year obstacle in regularity theory for weak minimal surfaces."},"detail":"/api/math/346"},{"id":"347","title":"Counterexamples to stable-Morse and strong Arnold fixed-point bounds","short":"Strong Arnold fixed-point bounds fail","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"(a) A Hamiltonian diffeomorphism of the complex quadric threefold Q^3 with exactly 3 fixed points < 4 = Crit(Q^3) = rational cup-length (degenerate critical-number and cup-length Arnold bounds fail).","verdict":"Five papers (87 pp). Lean formalizes only the degenerate quadric example (OAI.ArnoldCounterexample.main, OAI/Geometry/Arnold/Main.lean:14, 36 files ~5.9k lines).","url":"/math#347","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=347.%20Counterexamples%20to%20stable%2DMorse","manuscripts":[{"title":"Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf"},{"title":"Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf"},{"title":"Hamiltonian Fixed Points Below the Stable Morse Number","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf"},{"title":"A nondegenerate counterexample to the Morse-number Arnold bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf"},{"title":"Three fixed points on the symplectic quadric threefold","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/347.mp4","poster":"/films/math/347-poster.webp","captions":"/films/math/347.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":1,"why":"Counterexamples to strong topological versions of Arnold's fixed-point conjecture; the homological one stands.","consequence":"Limits Arnold-type fixed-point bounds to the homological form."},"detail":"/api/math/347"},{"id":"348","title":"Nonnegative-curvature Einstein classification and an L2 topological gap","short":"Positively curved Einstein 4-manifolds","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Closed Einstein 4-manifolds with positive sectional curvature are, up to scaling, round S^4, round RP^4 or Fubini-Study CP^2 (Yang's conjecture);","verdict":"Lean: positive-sectional classification only (OAI.PositiveEinsteinFour.classification, OAI/Geometry/EinsteinFour/Classification.lean:13; 408 files ~209k lines).","url":"/math#348","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=348.%20Nonnegative%2Dcurvature%20Einstein%20classification","manuscripts":[{"title":"Zero-Plane Rigidity for Einstein Four-Manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026/einstein-boundary.pdf"},{"title":"An L² Einstein Gap for Nonnegatively Curved Four-Manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-L2-Einstein-Gap-for-Nonnegatively-Curved-Four-Manifolds-October-5-2026/einstein-gap.pdf"},{"title":"Positively curved Einstein four-manifolds","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positively-curved-Einstein-four-manifolds-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/348.mp4","poster":"/films/math/348-poster.webp","captions":"/films/math/348.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":2,"why":"Einstein 4-manifolds with positive sectional curvature are the round sphere or CP^2; Lean checks the classification.","consequence":"Classifies positive Einstein 4-manifolds. Internal."},"detail":"/api/math/348"},{"id":"349","title":"The Solomon–Yau least-volume conjecture","short":"Solomon–Yau least volume","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For m >= 2, every closed connected minimal immersion into S^{m+1} with non-totally-geodesic image has volume (with multiplicity) at least a_m = min_k vol(Clifford product C_{k,m-k}) (Solomon-Yau least-volume conjecture).","verdict":"32 pages; uses a variational min-max family with a topological test and dimension induction ruling out singularities below the Clifford threshold;","url":"/math#349","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=349.%20The%20Solomon%20Yau","manuscripts":[{"title":"The Solomon–Yau least-volume theorem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Solomon-Yau-least-volume-theorem-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/349.mp4","poster":"/films/math/349-poster.webp","captions":"/films/math/349.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":0,"why":"The Solomon–Yau least-volume conjecture for minimal hypersurfaces in spheres, in all dimensions; no Lean.","consequence":"Identifies the second-smallest minimal hypersurfaces in spheres. Internal."},"detail":"/api/math/349"},{"id":"350","title":"Yau’s nodal bounds: surfaces and higher dimensions","short":"Yau's nodal conjecture, both ways","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Yau's nodal conjecture for smooth metrics is settled: (i) on every smooth closed surface, H^1(zero set) infinity; (iii) a fixed smooth metric on S^4 x S^1 with nodal measure >> lambda^{1/2 + eps_0}.","verdict":"All three directions formalized: NodalLength (OAI.SharpNodal.Main, OAI/Analysis/NodalLength/Main.lean:377), SmoothYau (sphere_three, sphere_two_torus_two),…","url":"/math#350","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=350.%20Yau%E2%80%99s%20nodal%20bounds","manuscripts":[{"title":"Sharp nodal length on smooth surfaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf"},{"title":"Smooth counterexamples to Yau's nodal upper bound in dimensions three and four","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf"},{"title":"Power-law violations of Yau's nodal upper bound","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/350.mp4","poster":"/films/math/350-poster.webp","captions":"/films/math/350.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":71,"tier":"Solid","importance":3,"advance":3,"consequences":2,"surprise":3,"confidence":3,"why":"Yau's nodal-length conjecture holds for all smooth surfaces and fails in dimension 3 and up; Lean checks both.","consequence":"Settles Yau's nodal question for surfaces and shows it fails for smooth metrics above."},"detail":"/api/math/350"},{"id":"351","title":"Scalar curvature and finite-time Ricci-flow singularities","short":"Bounded scalar curvature and Ricci flow","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"(i) A smooth Ricci flow on a closed 4-manifold with uniformly bounded scalar curvature on [0,T) extends smoothly past T. (ii) In some dimension (q >= 10 parameter), a closed Ricci flow has bounded scalar curvature but unbounded full curvature at a finite…","verdict":"118 pages across three papers; the 4d theorem relies on the companion tree inequality (Lojasiewicz-type estimate without integrability), and the high-dimensional…","url":"/math#351","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=351.%20Scalar%20curvature%20and","manuscripts":[{"title":"A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf"},{"title":"Bounded scalar curvature and smooth extension of four-dimensional Ricci flow","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf"},{"title":"Path selection and an elliptic inequality on degenerating Ricci-flat trees","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/351.mp4","poster":"/films/math/351-poster.webp","captions":"/films/math/351.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"In dimension 4, bounded scalar curvature prevents Ricci-flow singularities; in high dimensions it does not.","consequence":"Clarifies when Ricci flow singularities can be detected by scalar curvature."},"detail":"/api/math/351"},{"id":"352","title":"A finite-time singularity of Calabi flow","short":"A finite-time singularity of Calabi flow","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"There is a smooth U(10)-invariant Kaehler metric on CP^10 in the Fubini-Study class whose Calabi flow has a finite maximal existence time T* with scalar curvature at a point ~ a (T* - t)^{-1/2};","verdict":"54 pages; specific to dimension >= 10 (U(10)-symmetric reduction to an ODE/PDE in one variable). No Lean.","url":"/math#352","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=352.%20A%20finite%2Dtime%20singularity","manuscripts":[{"title":"A finite-time singularity of Calabi flow on projective space","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/352.mp4","poster":"/films/math/352-poster.webp","captions":"/films/math/352.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":0,"why":"The Calabi flow can break down in finite time on CP^10, against Chen's conjecture; no Lean.","consequence":"Ends Chen's conjecture on Calabi flow. Internal."},"detail":"/api/math/352"},{"id":"353","title":"Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample","short":"Affine Bernstein through dimension 9","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"Every smooth locally uniformly convex affine-maximal graph over a convex domain in R^n, 3 <= n <= 9, complete for the induced Euclidean metric, is an elliptic paraboloid (entire, quadratic); in dims 3-9 also for hypersurfaces complete for the affine metric.","verdict":"Lean formalizes the dims 3-9 Euclidean-complete graph theorem (OAI.AffineBernstein.affine_bernstein, OAI/Analysis/AffineBernstein/Main.lean:18; 326 files ~41k lines);","url":"/math#353","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=353.%20Affine%20Bernstein%20rigidity","manuscripts":[{"title":"A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf"},{"title":"The affine Bernstein theorem in dimensions three through nine","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/353.mp4","poster":"/films/math/353-poster.webp","captions":"/films/math/353.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":2,"why":"The affine Bernstein problem holds exactly in dimensions 3 to 9 and fails smoothly in 10; Lean checks dims 3-9.","consequence":"Pins the dimension where affine Bernstein fails. Internal."},"detail":"/api/math/353"},{"id":"354","title":"The isoperimetric profile of the cubic three-torus","short":"Isoperimetry in the cubic three-torus","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"In the unit cubic flat torus R^3/Z^3 the isoperimetric profile is min{(36 pi)^{1/3} v^{2/3}, 2 sqrt(pi v), 2} with v = min(V,1-V); minimizers are exactly balls (V <= 4 pi/81), round tubes about shortest geodesics (4 pi/81 <= V <= 1/pi), slabs (1/pi <= V <=…","verdict":"Lean: OAI.CubicTorus.unit_cubic_isoperimetric (OAI/Geometry/CubicTorus/Main.lean:1103; 162 files ~141k lines), including all equality cases.","url":"/math#354","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=354.%20The%20isoperimetric%20profile","manuscripts":[{"title":"The Isoperimetric Conjecture for the Cubic Flat Three-Torus","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf"}],"reel":{"src":"/films/math/354.mp4","poster":"/films/math/354-poster.webp","captions":"/films/math/354.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":27,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":1,"confidence":3,"why":"The least-area way to enclose volume in the cubic 3-torus: ball, tube, then slab; Lean checks it.","consequence":"Settles one isoperimetric profile. Nothing follows beyond it."},"detail":"/api/math/354"},{"id":"355","title":"Unique tangent flows at the first surface singularity","short":"Unique tangent flows for surface MCF","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"partial","kindLabel":"Claimed partial result","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For a smooth compact connected embedded surface in R^3 evolving by mean curvature flow, at every singular point at the first singular time the fixed-centre rescalings converge smoothly with multiplicity one to a unique self-shrinker S (in original…","verdict":"Restricted to the first singular time and fixed centre; uniqueness for general shrinkers with mixed ends (cylindrical + conical) is what is new.","url":"/math#355","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=355.%20Unique%20tangent%20flows","manuscripts":[{"title":"Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tangent-flow-uniqueness-2026-09-24/paper.pdf"}],"reel":{"src":"/films/math/355.mp4","poster":"/films/math/355-poster.webp","captions":"/films/math/355.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Tangent flows at the first singularity of mean curvature flow of surfaces are unique; 100 hard pages, no Lean.","consequence":"Uniqueness of blow-ups, a tool for analysing mean curvature flow singularities."},"detail":"/api/math/355"},{"id":"356","title":"Gigli’s characterization of Alexandrov curvature","short":"Gigli's characterization of Alexandrov","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"technical","significanceRank":3,"lean":"part","leanLabel":"Lean: part only","claim":"For integer n >= 2: (M,d) is an n-dimensional Alexandrov space with curvature >= kappa iff it is RCD((n-1)kappa, n) with reference measure H^n, full support, and Gigli's distributional sectional curvature >= kappa on the original test classes (Gigli's…","verdict":"Lean formalizes only the companion weak-Hessian-along-every-geodesic theorem (OAI.WeakHessian.every_geodesic, OAI/Geometry/WeakHessian/Main.lean:38;","url":"/math#356","articleUrl":"/articles/openai-math#technical-results-1","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=356.%20Gigli%E2%80%99s%20characterization%20of","manuscripts":[{"title":"Gigli’s distributional curvature characterization of Alexandrov spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf"},{"title":"Weak Hessian bounds along every geodesic in RCD spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026/weak-hessian-geodesics.pdf"}],"reel":{"src":"/films/math/356.mp4","poster":"/films/math/356-poster.webp","captions":"/films/math/356.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":1,"why":"Gigli's curvature tensor bound characterizes Alexandrov spaces; Lean checks a key step.","consequence":"Characterizes Alexandrov spaces analytically. Internal."},"detail":"/api/math/356"},{"id":"357","title":"Bi-Lipschitz coordinates at every regular RCD point","short":"Bi-Lipschitz charts at regular RCD points","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"technical","significanceRank":3,"lean":"none","leanLabel":"Manuscript only","claim":"Every n-regular point (all tangents Euclidean) of a noncollapsed RCD(K,n) space with reference measure H^n has a neighbourhood L_n-bi-Lipschitz to an open subset of R^n, with L_n depending only on n;","verdict":"43 pages; an in-progress Lean development exists (OAI/Geometry/RCD, ~2,900 files) but no catalogued main result. No external review known.","url":"/math#357","articleUrl":"/articles/openai-math#technical-results-1","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=357.%20Bi%2DLipschitz%20coordinates%20at","manuscripts":[{"title":"Bi-Lipschitz Coordinates at Regular Points of Noncollapsed RCD Spaces","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bi-Lipschitz-Coordinates-at-Regular-Points-of-Noncollapsed-RCD-Spaces-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/357.mp4","poster":"/films/math/357-poster.webp","captions":"/films/math/357.vtt","duration":20,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":42,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":1,"confidence":0,"why":"Bi-Lipschitz charts at every regular point of an RCD space, a Cheeger–Colding question; no Lean.","consequence":"Regular points of RCD spaces have true charts. Internal."},"detail":"/api/math/357"},{"id":"358","title":"A three-manifold without conjugate points or nonpositive curvature","short":"No conjugate points, no $\\sec \\le 0$","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a closed orientable 3-manifold (a two-piece graph manifold from two copies of (punctured torus) x S^1 glued by h2 = h1 + f1, f2 = h1) with a smooth metric without conjugate points but no metric of nonpositive sectional curvature (indeed pi_1 admits…","verdict":"The nonpositive-curvature obstruction is Leeb's known example; the new content is the no-conjugate-points metric.","url":"/math#358","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=358.%20A%20three%2Dmanifold%20without","manuscripts":[{"title":"A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/358.mp4","poster":"/films/math/358-poster.webp","captions":"/films/math/358.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":27,"tier":"Incremental","importance":1,"advance":2,"consequences":0,"surprise":1,"confidence":3,"why":"A 3-manifold with no conjugate points that admits no nonpositively curved metric; Lean checks it.","consequence":"Separates two curvature conditions on 3-manifolds. Nothing follows beyond it."},"detail":"/api/math/358"},{"id":"359","title":"Negative Kähler curvature without bounded holomorphic coordinates","short":"Negative Kähler curvature, no bounded chart","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a contractible Stein domain M in C^3 with a complete Kaehler metric with -B C^3 with nonvanishing Jacobian; so M is not biholomorphic to a bounded domain (negative answer to the Wu-Yau / H. Wu uniformization question).","verdict":"Lean formalizes the pinched threefold construction (OAI.PinchedHartogs.main_theorem with controlled_potential and metric_transfer, OAI/Geometry/Kahler/Main.lean:38;","url":"/math#359","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=359.%20Negative%20K%C3%A4hler%20curvature","manuscripts":[{"title":"A negatively pinched Kähler threefold without bounded holomorphic coordinates","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf"},{"title":"One-sided negative sectional curvature and the holomorphic Liouville property","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/359.mp4","poster":"/films/math/359-poster.webp","captions":"/films/math/359.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":3,"why":"A negatively pinched Kähler manifold not biholomorphic to a bounded domain; Lean checks it.","consequence":"Pinched negative curvature does not force bounded domains. Internal."},"detail":"/api/math/359"},{"id":"360","title":"Weak MTW curvature gives convexity and regular optimal transport","short":"Weak MTW: convex injectivity domains","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"On every compact connected Riemannian manifold (dim >= 2) satisfying weak MTW (S(xi,eta) >= 0 for orthogonal pairs), every tangent injectivity domain is convex (Villani's conjecture, no nonfocality assumption);","verdict":"Both papers' main results formalized (WeakMTWGlobalSupport: OAI/Geometry/WeakMTW/Convexity.lean:65;","url":"/math#360","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=360.%20Weak%20MTW%20curvature","manuscripts":[{"title":"Global Support and Convex Injectivity Domains under Weak MTW","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf"},{"title":"Uniform Bi-Holder Transport from Weak MTW","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/360.mp4","poster":"/films/math/360-poster.webp","captions":"/films/math/360.vtt","duration":19.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":48,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":1,"confidence":3,"why":"Villani's conjecture: weak MTW curvature gives convex injectivity domains and regular transport; Lean checks it.","consequence":"Regularity of optimal transport maps under weak MTW curvature."},"detail":"/api/math/360"},{"id":"361","title":"Failure of integer-degree harmonic dimension comparison","short":"Harmonic functions of integer growth","discipline":"Differential geometry","disciplineIndex":15,"accent":"#F6C177","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"part","leanLabel":"Lean: part only","claim":"Yau's proposed comparison h_k(M^n,g) = 0 fails: for every v in (4/9,1), c in (1, 9v/4) and all large integers k, a complete metric g_k on R^3 (Euclidean near 0, Ric >= 0, AVR = v) has h_k >= c (k+1)^2 > (k+1)^2.","verdict":"The metric depends on k (no single manifold violates it for all k). Lean formalizes the earlier companion's dimension-16, degree-50000 example…","url":"/math#361","articleUrl":"/articles/openai-math#geometry-and-topology","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=361.%20Failure%20of%20integer%2Ddegree","manuscripts":[{"title":"A counterexample to integer-degree harmonic dimension comparison","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf"},{"title":"A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Three-Dimensional-Counterexample-to-Integer-Degree-Harmonic-Dimension-Comparison-September-26-2026/paper.pdf"}],"reel":{"src":"/films/math/361.mp4","poster":"/films/math/361-poster.webp","captions":"/films/math/361.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":1,"why":"Nonnegative Ricci curvature can increase the count of polynomial-growth harmonic functions, answering Yau negatively.","consequence":"Answers Yau's question negatively. Internal."},"detail":"/api/math/361"},{"id":"362","title":"Global smoothness for relativistic Vlasov–Maxwell","short":"Relativistic Vlasov–Maxwell, large data","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"Global existence, uniqueness and C^infinity regularity on every finite time interval for the 3D one-species relativistic Vlasov-Maxwell system (normalized constants, no background charge) for arbitrary large data: f0 in C_c^infinity, f0>=0;","verdict":"This is the headline PDE claim of the chunk and would resolve a ~40-year-old problem; the paper is only 49 pages and its own reasoning trace…","url":"/math#362","articleUrl":"/articles/openai-math#relativistic-vlasovmaxwell-no-blowup-for-large-data-in-three-dimensions","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=362.%20Global%20smoothness%20for","manuscripts":[{"title":"Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/362.mp4","poster":"/films/math/362-poster.webp","captions":"/films/math/362.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":3,"why":"Collisionless plasmas governed by relativistic Vlasov–Maxwell stay smooth forever, open since 1986; Lean checks it.","consequence":"Settles global existence for a basic plasma model, closing the Glassey–Strauss line of work."},"detail":"/api/math/362"},{"id":"363","title":"Nonuniqueness with local conservation for the hard-sphere Boltzmann equation","short":"Boltzmann: two solutions from one gas","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"For the 3D periodic hard-sphere Boltzmann equation there is one nonnegative initial density on T^3 x R^3 with bounded velocity support and finite mass, energy and absolute entropy that admits two distinct global entropy solutions, both strongly continuous in…","verdict":"The summary's headline (exact local conservation) comes from the 78-page Oct-5 paper which is NOT formalized;","url":"/math#363","articleUrl":"/articles/openai-math#boltzmanns-equation-two-solutions-from-one-initial-gas","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=363.%20Nonuniqueness%20with%20local","manuscripts":[{"title":"Nonuniqueness with local conservation for the hard-sphere Boltzmann equation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-with-local-conservation-for-the-hard-sphere-Boltzmann-equation-October-5-2026/paper.pdf"},{"title":"Nonuniqueness for the periodic hard-sphere Boltzmann equation","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/363.mp4","poster":"/films/math/363-poster.webp","captions":"/films/math/363.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":45,"tier":"Incremental","importance":2,"advance":2,"consequences":1,"surprise":2,"confidence":1,"why":"Weak solutions of the hard-sphere Boltzmann equation are not determined by their data; Lean checks a weaker form.","consequence":"Large-data weak Boltzmann theory cannot give uniqueness. Internal to kinetic theory."},"detail":"/api/math/363"},{"id":"364","title":"Kinetic limits and fluctuations over the Boltzmann lifespan","short":"Boltzmann–Grad over the regular lifespan","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"notable","significanceRank":2,"lean":"none","leanLabel":"Manuscript only","claim":"(A) Boltzmann-Grad limit for 3D grand-canonical Newtonian gases with stable, finite-range, radial C^2 potentials (attractive wells and a repulsive singular core allowed;","verdict":"Conditional in the same way as Deng-Hani-Ma: assumes the Boltzmann solution exists with Gaussian bounds on [0,T]; no global Boltzmann regularity is claimed.","url":"/math#364","articleUrl":"/articles/openai-math#boltzmanngrad-limit-over-the-regular-lifespan-364","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=364.%20Kinetic%20limits%20and","manuscripts":[{"title":"The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf"},{"title":"Hard-sphere fluctuations on the regular Boltzmann lifespan","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/364.mp4","poster":"/films/math/364-poster.webp","captions":"/films/math/364.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":56,"tier":"Solid","importance":3,"advance":2,"consequences":2,"surprise":1,"confidence":0,"why":"Extends the 2024 derivation of Boltzmann from particles to smooth potentials, with Gaussian fluctuations; conditional.","consequence":"Particle derivations of Boltzmann for realistic interactions, plus fluctuation laws."},"detail":"/api/math/364"},{"id":"365","title":"Joint metric and connection recovery from one boundary patch","short":"Calderón's problem from one boundary patch","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"(1) n>=3, compact connected smooth manifold with boundary, any nonempty open boundary patch Gamma: the zero-frequency DN energy form with inputs and outputs on Gamma determines a smooth Riemannian metric up to a diffeomorphism fixing Gamma (smooth anisotropic…","verdict":"The family title (metric+connection) undersells it: the Sept-24 companion claims to resolve the smooth anisotropic Calderon problem (Lee-Uhlmann conjecture) in all n>=3,…","url":"/math#365","articleUrl":"/articles/openai-math#calderóns-problem-one-boundary-patch-and-a-buried-headline","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=365.%20Joint%20metric%20and","manuscripts":[{"title":"Determination of a metric and a unitary connection from one boundary patch","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026/paper.pdf"},{"title":"Smooth anisotropic uniqueness in the Calderón problem from one boundary patch","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026/paper.pdf"},{"title":"Nonuniqueness for bounded measurable scalar conductivities in three dimensions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026/paper.pdf"}],"reel":{"src":"/films/math/365.mp4","poster":"/films/math/365-poster.webp","captions":"/films/math/365.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":74,"tier":"Solid","importance":3,"advance":3,"consequences":3,"surprise":2,"confidence":1,"why":"The anisotropic Calderón problem in dimension 3 and up, with partial data; only a side result is Lean-checked.","consequence":"Uniqueness for anisotropic impedance tomography and partial-data measurements; the mathematics of electrical imaging."},"detail":"/api/math/365"},{"id":"366","title":"The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds","short":"Planar Mumford–Shah regularity","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"part","leanLabel":"Lean: part only","claim":"For a reduced absolute minimizer (u,K) of the planar Mumford–Shah energy ∫|∇u|² + H¹(K) + ∫|u−g|² on a bounded Lipschitz domain Ω⊂R², with bounded measurable fidelity datum g∈L∞ and no a priori rectifiability of K, every interior point of K has a…","verdict":"Priority: a human-posted (Astra-assisted) proof of the same interior conjecture (Deangelis, arXiv:2609.26732) predates this by two days, and the paper itself says…","url":"/math#366","articleUrl":"/articles/openai-math#the-planar-mumfordshah-conjecture-proved-for-the-second-time-in-a-week","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=366.%20The%20planar%20Mumford","manuscripts":[{"title":"Interior regularity of planar Mumford–Shah minimizers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026/Interior-regularity-of-planar-Mumford-Shah-minimizers-September-24-2026.pdf"}],"reel":{"src":"/films/math/366.mp4","poster":"/films/math/366-poster.webp","captions":"/films/math/366.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":1,"confidence":1,"why":"The planar Mumford–Shah regularity conjecture; a human proof appeared two days earlier, this is a second route.","consequence":"A second proof that Mumford–Shah segmentation edges are tidy. Little new follow-on."},"detail":"/api/math/366"},{"id":"367","title":"The critical dimension for the one-phase Bernoulli problem","short":"Bernoulli free boundaries: $d^* = 7$","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"part","leanLabel":"Lean: part only","claim":"Every nonzero one-homogeneous global minimizer of the one-phase Bernoulli (Alt–Caffarelli) energy ∫|∇v|²+1_{v>0} in R^d, d≤6, is flat (x·e)+; a nonflat one exists in R^7 (the De Silva–Jerison cone).","verdict":"The Lean certificate covers only the previously known half (De Silva–Jerison 2009), so the novel exclusion of cones in dimensions 5–6 rests on the paper alone;","url":"/math#367","articleUrl":"/articles/openai-math#the-critical-dimension-of-the-one-phase-bernoulli-problem-is-7","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=367.%20The%20critical%20dimension","manuscripts":[{"title":"The critical dimension for one-phase Bernoulli minimizers","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf"}],"reel":{"src":"/films/math/367.mp4","poster":"/films/math/367-poster.webp","captions":"/films/math/367.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":60,"tier":"Solid","importance":2,"advance":3,"consequences":2,"surprise":2,"confidence":1,"why":"The first non-flat minimizing cone for the one-phase Bernoulli problem appears in dimension 7; the new half is unformalized.","consequence":"Pins the critical dimension for singularities of the one-phase free boundary problem."},"detail":"/api/math/367"},{"id":"368","title":"The three-dimensional Ball–Evans approximation problem","short":"Ball–Evans approximation in $\\mathbb{R}^3$","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For bounded domains Ω,Λ⊂R³ (no boundary regularity) and every 1≤p<∞, every homeomorphism f:Ω→Λ in W^{1,p} is a strong W^{1,p} limit of C^∞ diffeomorphisms f_j:Ω→Λ onto the same target. Proved in two independent papers: 1≤p≤2 (incl.","verdict":"Very long (62 + 130 pp) intricate geometric constructions (target triangulations, intersection 'budgets', rank-dependent local models, Moise/Bing PL topology) with no…","url":"/math#368","articleUrl":"/articles/openai-math#the-ballevans-approximation-problem-in-three-dimensions","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=368.%20The%20three%2Ddimensional%20Ball","manuscripts":[{"title":"Strong diffeomorphic approximation in three dimensions for 1≤p≤2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2-September-24-2026/main.pdf"},{"title":"Strong diffeomorphic approximation in three dimensions for p>2","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/368.mp4","poster":"/films/math/368-poster.webp","captions":"/films/math/368.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":49,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":1,"confidence":0,"why":"Sobolev homeomorphisms in 3D can be approximated by smooth injective maps, the Ball–Evans problem; no Lean.","consequence":"Smooth injective approximation for 3D elasticity maps; numerical methods would need constructive versions."},"detail":"/api/math/368"},{"id":"369","title":"The hot spots conjecture for simply connected planar domains","short":"Hot spots on simply connected domains","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"On every nonempty bounded simply connected planar domain with C^∞ boundary, every nonzero eigenfunction of the first positive Neumann eigenvalue (any multiplicity) has nonvanishing gradient everywhere in the interior;","verdict":"Smooth boundary only: Lipschitz/convex domains with corners (e.g. polygons beyond triangles) are not covered, though an approximation argument might give the nonstrict…","url":"/math#369","articleUrl":"/articles/openai-math#the-hot-spots-conjecture-for-smooth-simply-connected-domains","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=369.%20The%20hot%20spots","manuscripts":[{"title":"Strict hot spots and absence of interior critical points on smooth simply connected planar domains","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf"}],"reel":{"src":"/films/math/369.mp4","poster":"/films/math/369-poster.webp","captions":"/films/math/369.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":3,"advance":2,"consequences":1,"surprise":2,"confidence":2,"why":"The hot spots conjecture for smooth simply connected planar domains; polygons are not covered. Lean checks it.","consequence":"Settles hot spots for smooth simply connected domains. Internal."},"detail":"/api/math/369"},{"id":"370","title":"The Lane–Emden and Hénon–Lane–Emden conjectures","short":"The Lane–Emden conjecture","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"main","leanLabel":"Lean: main theorem","claim":"For n≥2, p,q>0 and real A,B with (n+A)/(p+1)+(n+B)/(q+1) > n−2, the system −Δu=|x|^A v^p, −Δv=|x|^B u^q has no strictly positive solution u,v∈C²(R^n\\{0})∩C(R^n); no symmetry, boundedness, decay, energy or stability assumption.","verdict":"Remarkably short (20 pp) for a conjecture open in n≥5 for ~30 years; the method (localized potential-kernel virial identity + an 'interval-pair' pressure inequality…","url":"/math#370","articleUrl":"/articles/openai-math#the-laneemden-conjecture","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=370.%20The%20Lane%20Emden","manuscripts":[{"title":"The Subcritical Hénon–Lane–Emden Conjecture","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/370.mp4","poster":"/films/math/370-poster.webp","captions":"/films/math/370.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":3,"why":"The Lane–Emden conjecture in every dimension, open for n≥5 for 30 years, in 20 pages; Lean checks it.","consequence":"Liouville theorems that feed blow-up rates and a priori estimates across nonlinear elliptic PDE."},"detail":"/api/math/370"},{"id":"371","title":"Stable blowup for the defocusing Schrödinger equation","short":"Stable blowup for defocusing NLS","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"There exist an odd power p (which can be taken arbitrarily large), an integer k>8 and a nonempty open set U⊂H^k(T^12) such that every solution of the defocusing NLS i∂_t u+Δu=|u|^{p−1}u on the 12-dimensional torus with data in U blows up in finite time in…","verdict":"Summary says 'for a sufficiently large odd power'; the theorem and Lean only give existence of some (arbitrarily large) odd p — weaker than 'all large p'.","url":"/math#371","articleUrl":"/articles/openai-math#stable-blowup-for-a-defocusing-schrödinger-equation","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=371.%20Stable%20blowup%20for","manuscripts":[{"title":"Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026/paper.pdf"}],"reel":{"src":"/films/math/371.mp4","poster":"/films/math/371-poster.webp","captions":"/films/math/371.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":52,"tier":"Incremental","importance":2,"advance":2,"consequences":2,"surprise":2,"confidence":3,"why":"Stable blowup for a defocusing Schrödinger equation in dimension 12 at some large power; Lean checks it.","consequence":"Blowup for defocusing NLS happens for an open set of data, not a thin one; specific dimension and power."},"detail":"/api/math/371"},{"id":"372","title":"Global uniqueness in smooth isotropic elasticity","short":"Calderón's problem for isotropic elasticity","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"main","leanLabel":"Lean: main theorem","claim":"On any bounded connected C^∞ domain Ω⊂R³, if two pairs of real C^∞ Lamé moduli (λ_j,μ_j) with μ_j>0 and 3λ_j+2μ_j>0 on the closure have equal full static displacement-to-traction maps, then λ1=λ2 and μ1=μ2.","verdict":"Only 24 pages for a problem whose earlier global proof (Nakamura–Uhlmann) had to be retracted to a small-gradient case;","url":"/math#372","articleUrl":"/articles/openai-math#calderóns-problem-for-isotropic-elasticity","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=372.%20Global%20uniqueness%20in","manuscripts":[{"title":"Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf"}],"reel":{"src":"/films/math/372.mp4","poster":"/films/math/372-poster.webp","captions":"/films/math/372.vtt","duration":20.8,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":53,"tier":"Incremental","importance":2,"advance":3,"consequences":1,"surprise":2,"confidence":3,"why":"Boundary measurements determine smooth isotropic elastic moduli in 3D, after an earlier proof was retracted; Lean checks it.","consequence":"Uniqueness for elastic inverse problems with full boundary data."},"detail":"/api/math/372"},{"id":"373","title":"Nonattainment of the three-marginal Coulomb Monge problem","short":"Three-electron Coulomb Monge problem","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"disproof/counterexample","kindLabel":"Claimed disproof","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"There is a smooth compactly supported probability density ρ on R³ (with smooth compactly supported √ρ) such that the three-marginal Coulomb optimal transport problem (cost Σ_{i0.","verdict":"Counterexample, not a general theory: the density is engineered (five disjoint components, central mass ν/3, four outer components ν/6), not a physical ground-state…","url":"/math#373","articleUrl":"/articles/openai-math#the-three-marginal-coulomb-monge-problem-has-no-monge-minimizer-373","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=373.%20Nonattainment%20of%20the","manuscripts":[{"title":"A counterexample to the Monge ansatz for the three-marginal Coulomb cost","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf"}],"reel":{"src":"/films/math/373.mp4","poster":"/films/math/373-poster.webp","captions":"/films/math/373.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Three-electron Coulomb transport can have no optimal Monge map, for an engineered density; Lean checks it.","consequence":"The Monge ansatz in strong-interaction DFT can fail. Engineered density, not physical."},"detail":"/api/math/373"},{"id":"374","title":"Sharp one-third stability of Brenier maps","short":"Brenier maps are $1/3$-stable, no better","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"improved bound","kindLabel":"Claimed improved bound","significance":"notable","significanceRank":2,"lean":"main","leanLabel":"Lean: main theorem","claim":"For ρ the uniform probability on a compact convex body K⊂R^d (d≥2) and any fixed compact Y, the quadratic optimal transport (Brenier) maps satisfy ||T_μ−T_ν||_{L²(ρ)} ≤ C(K,Y)·W₂(μ,ν)^{1/3} for all probability measures μ,ν on Y (atomic, singular or absolutely…","verdict":"Disproves a conjecture and settles the sharp exponent only within the uniform-convex-source class;","url":"/math#374","articleUrl":"/articles/openai-math#brenier-maps-are-13-hölder-stable-and-no-better-374","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=374.%20Sharp%20one%2Dthird%20stability","manuscripts":[{"title":"Sharp One-Third Stability of Brenier Maps","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf"}],"reel":{"src":"/films/math/374.mp4","poster":"/films/math/374-poster.webp","captions":"/films/math/374.vtt","duration":20.7,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":34,"tier":"Incremental","importance":1,"advance":2,"consequences":1,"surprise":1,"confidence":3,"why":"Brenier maps move like the cube root of the target's movement, and that is sharp; Lean checks it.","consequence":"Sharp stability rates for transport maps used in statistical estimation."},"detail":"/api/math/374"},{"id":"375","title":"De Giorgi's conjecture in dimension eight","short":"De Giorgi's conjecture in dimension 8","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"landmark","significanceRank":0,"lean":"none","leanLabel":"Manuscript only","claim":"(i) Every entire C² solution u:R^8→(−1,1) of Δu=u³−u with ∂_8u>0 everywhere is u=tanh((e·x−c)/√2) (one-dimensional), with no assumption on the limits as x_8→±∞ and no energy bound.","verdict":"Extraordinary jump: two days earlier the human frontier was stable rigidity in R³ (giving monotone n=4);","url":"/math#375","articleUrl":"/articles/openai-math#de-giorgis-conjecture-in-dimension-eight-and-the-result-i-trust-least","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=375.%20De%20Giorgi's%20conjecture","manuscripts":[{"title":"A positive resolution of De Giorgi's conjecture in dimension eight","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf"}],"reel":{"src":"/films/math/375.mp4","poster":"/films/math/375-poster.webp","captions":"/films/math/375.vtt","duration":20.9,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":79,"tier":"Major","importance":3,"advance":4,"consequences":2,"surprise":3,"confidence":0,"why":"De Giorgi's 1978 conjecture in dimension 8, a jump from dimension 4 two days earlier; 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the forcing makes the PDE side easy.","consequence":"A kinematic construction. Says nothing about unforced blowup."},"detail":"/api/math/376"},{"id":"377","title":"Interior C^(1,α) regularity for infinity-harmonic functions","short":"Infinity-harmonic functions are $C^{1,\\alpha}$","discipline":"Partial differential equations","disciplineIndex":16,"accent":"#E8A87C","kind":"proof","kindLabel":"Claimed proof","significance":"major","significanceRank":1,"lean":"none","leanLabel":"Manuscript only","claim":"For every d≥3 there are α_d∈(0,1/3] and C_d such that every bounded (viscosity) infinity-harmonic function on the unit ball B1⊂R^d is C^{1,α_d} on B_{1/2}, with ||∇u||_{L∞(B1/2)}+[∇u]_{C^{0,α_d}(B1/2)} ≤ C_d·osc_{B1}u.","verdict":"Not formalized; dated 4 Oct 2026, the most recent in the chunk, so likely the least vetted.","url":"/math#377","articleUrl":"/articles/openai-math#infinity-harmonic-functions-are-c1alpha-in-every-dimension","catalogueUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/CONTENTS.md#:~:text=377.%20Interior%20C%201","manuscripts":[{"title":"Uniform Interior C^(1,α) Estimates for Infinity-Harmonic Functions","url":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Interior-C1alpha-Estimates-for-Infinity-Harmonic-Functions-October-4-2026/interior-c1-infinity-harmonic.pdf"}],"reel":{"src":"/films/math/377.mp4","poster":"/films/math/377-poster.webp","captions":"/films/math/377.vtt","duration":21,"width":1280,"height":720,"date":"2026-10-07"},"breakthrough":{"score":61,"tier":"Solid","importance":3,"advance":3,"consequences":1,"surprise":2,"confidence":0,"why":"Infinity-harmonic functions have Hölder continuous gradients in every dimension, open 20 years; no Lean.","consequence":"Gradient regularity for best Lipschitz extensions. Internal."},"detail":"/api/math/377"}]}