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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
barendregt-geuvers-klop-conjecture | Barendregt–Geuvers–Klop conjecture | https://papex.org/problems/barendregt-geuvers-klop-conjecture | proof | current | Weak and strong normalization in pure type systems | We prove that every weakly β-normalizing pure type system is strongly β-normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the β-Barendregt–Geuvers–Klop conjecture. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026/paper.pdf | 2026-10-06T00:00:00 | Lean 4 | https://github.com/openai/math/blob/main/lean/docs/245.md | 2026-10-14T00:00:00 | [] |
birch-swinnerton-dyer | Birch and Swinnerton-Dyer conjecture | https://papex.org/problems/birch-swinnerton-dyer | partial | current | Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank | We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one at some prime q. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "response",
"on": "2026-10-08T00:00:00",
"by": "Frank Calegari",
"text": "There are some results proving BSD under certain rank conditions. My first impression is that these are technical improvements on results already in the literature, and that they represent no genuine progress towards... |
cosmic-censorship-hypothesis | Cosmic censorship hypothesis | https://papex.org/problems/cosmic-censorship-hypothesis | partial | current | Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime | Strong cosmic censorship asks whether initial data determine a spacetime that cannot be continued beyond its maximal globally hyperbolic development. The regularity allowed for a continuation is part of the question. Kerr’s smooth inner horizon makes this distinction especially important [2]: a continuation can preserv... | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
generalised-star-height-problem | Generalised star-height problem | https://papex.org/problems/generalised-star-height-problem | partial | current | Generalized Star Height at Most Three | Every regular language over a finite alphabet has generalized star height at most three, with complement taken in the same free monoid. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/Generalized-Star-Height-at-Most-Three-September-25-2026/article.pdf | 2026-10-06T00:00:00 | Lean 4 | https://github.com/openai/math/blob/main/lean/docs/134.md | 2026-10-14T00:00:00 | [] |
hadwiger-conjecture | Hadwiger conjecture | https://papex.org/problems/hadwiger-conjecture | partial | revised | Beyond halfway to Hadwiger's conjecture | Hadwiger conjectured in 1943 that every graph with no K_t minor has chromatic number at most t − 1. Delcourt and Postle proved that every graph with no K_t minor has chromatic number O(t log log t). We build on their result to improve this bound to O(t log log log t). | Chun-Hung Liu, Jason Luo | ChatGPT Codex was used to help develop the proof. The authors supplied a high-level plan of attack and guided the development of the core argument. | ChatGPT Codex | arXiv | https://arxiv.org/abs/2609.06867 | 2026-09-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-09-09T00:00:00",
"by": "Chun-Hung Liu, Jason Luo",
"text": "v2: significantly improve the bound and change the title",
"url": "https://arxiv.org/abs/2609.06867v2"
}
] |
hadwiger-conjecture | Hadwiger conjecture | https://papex.org/problems/hadwiger-conjecture | partial | current | A Proof of the Linear Hadwiger Conjecture | We show that there exists C ∈ ℕ such that K_t-minor free graphs are Ct-colorable. | Sergey Norin, Raphael Steiner | The proof was found by GPT-6 Astra, following the directions by the authors. … The writing of this paper is due to the human authors. | GPT-6 Astra | arXiv | https://arxiv.org/abs/2610.05291 | 2026-10-04T00:00:00 | Lean 4 | https://github.com/snorin239/LinearHadwiger | 2026-10-14T00:00:00 | [] |
hadwiger-conjecture | Hadwiger conjecture | https://papex.org/problems/hadwiger-conjecture | disproof | current | A counterexample to Hadwiger's conjecture | We disprove Hadwiger’s conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf | 2026-10-06T00:00:00 | Lean 4 | https://github.com/openai/math/blob/main/lean/docs/157.md | 2026-10-14T00:00:00 | [] |
hadwiger-nelson-problem | Hadwiger–Nelson problem | https://papex.org/problems/hadwiger-nelson-problem | partial | current | The Euclidean plane is not five-colorable | We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf | 2026-10-06T00:00:00 | Lean 4 | https://github.com/openai/math/blob/main/lean/docs/158.md | 2026-10-14T00:00:00 | [
{
"kind": "response",
"on": "2026-10-11T00:00:00",
"by": "Bill Gasarch",
"text": "OpenAI used the notion of weak measurable colorings to obtain a result about all colorings. That is new. This is a clear case of Searching has reached its limits; we need a new approach and OpenAI found one.",
"url... |
hodge-conjecture | Hodge conjecture | https://papex.org/problems/hodge-conjecture | partial | withdrawn | Algebraicity of Kuga–Satake Correspondences for K3 Surfaces | We prove that the Kuga–Satake correspondence is algebraic for every smooth projective complex K3 surface. More precisely, the prescribed embedding of its transcendental cohomology into the cohomology of its Kuga–Satake abelian variety is induced by a rational algebraic cycle, with the fixed normalization and full even-... | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/manuscript.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "withdrawal",
"on": "2026-10-06T00:00:00",
"by": "OpenAI",
"text": "The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point ca... |
hodge-conjecture | Hodge conjecture | https://papex.org/problems/hodge-conjecture | partial | withdrawn | Algebraicity of Weil classes on split abelian eightfolds | The Weil classes are a particularly concrete test of the Hodge conjecture: they form a rational two-dimensional space defined by an imaginary quadratic action, while their algebraic representatives generally do not come from products of divisor classes. This paper establishes their algebraicity in the split eight-dimen... | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Weil-classes-on-split-abelian-eightfolds-September-18-2026/paper.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "withdrawal",
"on": "2026-10-06T00:00:00",
"by": "OpenAI",
"text": "The proof contains a sign error in the stabilization-trace argument used to obtain negative double points. … The Eliashberg-Murphy cancellation theorem invoked at this step requires zero signed double-point count, so its h... |
hodge-conjecture | Hodge conjecture | https://papex.org/problems/hodge-conjecture | partial | withdrawn | The rational Hodge conjecture for products of K3 surfaces | We prove the rational Hodge conjecture for every finite product of projective complex K3 surfaces: every rational Hodge class is algebraic. The factors may be distinct or repeated, with no restrictions on their Picard numbers, periods, or endomorphism fields. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | 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 | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "withdrawal",
"on": "2026-10-06T00:00:00",
"by": "OpenAI",
"text": "The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point ca... |
hodge-conjecture | Hodge conjecture | https://papex.org/problems/hodge-conjecture | partial | current | Abelian covers, Gale correspondences, and the Hodge conjecture for powers | We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026/paper.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
hodge-conjecture | Hodge conjecture | https://papex.org/problems/hodge-conjecture | partial | current | Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus | We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in U_Q^{⊕2} ⊥ ⟨−1⟩^4. This includes every ample P-polarized K3 surface... | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
hodge-conjecture | Hodge conjecture | https://papex.org/problems/hodge-conjecture | partial | current | The rational Hodge conjecture for CM abelian varieties | We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelia... | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. … Exceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
hodge-conjecture | Hodge conjecture | https://papex.org/problems/hodge-conjecture | partial | current | Weil classes and Hodge classes on abelian powers | We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature (3,3). We also prove it in every codimension on every self-power of a complex... | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-October-6-2026/paper.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | current | AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms | For example, it advanced the kissing number problem. This geometric challenge has fascinated mathematicians for over 300 years and concerns the maximum number of non-overlapping spheres that touch a common unit sphere. AlphaEvolve discovered a configuration of 593 outer spheres and established a new lower bound in 11 d... | AlphaEvolve team, Google DeepMind | AlphaEvolve leverages an ensemble of state-of-the-art large language models: our fastest and most efficient model, Gemini Flash, maximizes the breadth of ideas explored, while our most powerful model, Gemini Pro, provides critical depth with insightful suggestions. | AlphaEvolve (ensemble of Gemini Flash and Gemini Pro) | Google DeepMind blog | https://deepmind.google/discover/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/ | 2025-05-14T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | current | Finding Kissing Numbers with Game-theoretic Reinforcement Learning | Working within extremal configuration spaces, PackingStar discovers new interpretable geometric structures that improve 15 strong bounds held for decades in kissing numbers and their generalizations, several of them provably optimal under natural inner products. | Chengdong Ma, Théo Tao Zhaowei, Pengyu Li, Minghao Liu, Haojun Chen, Zihao Mao, Yuan Cheng, Yuan Qi, Yaodong Yang | Accordingly, we recast this problem as a cooperative matrix-completion game, and train a reinforcement learning system, PackingStar, to solve it. One player fills cosine entries while the other corrects suboptimal ones, making explosive geometric complexity tractable. | PackingStar (reinforcement learning system) | arXiv | https://arxiv.org/abs/2511.13391 | 2025-11-17T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | revised | A new lower bound for the kissing number in 19 dimensions | We prove that the kissing number in 19 dimensions is at least 11948, improving the bound of Cohn and Li by 256. | Boon Suan Ho | GPT-5.4 Pro was used in the discovery of the construction and in revising the exposition. | GPT-5.4 Pro | arXiv | https://arxiv.org/abs/2603.10425 | 2026-03-11T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-03-20T00:00:00",
"by": "Boon Suan Ho",
"text": "v2: corrected errors and improved exposition; main results unchanged",
"url": "https://arxiv.org/abs/2603.10425v2"
}
] |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | revised | Harnessing the Collective Intelligence of AI Agents in the Wild for New Discoveries | As of May 2026, agents on EinsteinArena have discovered 12 new state-of-the-art results better than any previous human or AI solutions. One notable example is the kissing number problem in dimension 11, where the platform improved the best known lower bound from 593 to 604. | Federico Bianchi, Yongchan Kwon, Aneesh Pappu, James Zou | In EinsteinArena, a collaborative effort among AI agents improved this lower bound to 594, after which we had agents build on their result to further extend it to 604. | null | arXiv | https://arxiv.org/abs/2606.10402 | 2026-06-09T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-08-26T00:00:00",
"by": "Federico Bianchi, Yongchan Kwon, Aneesh Pappu, James Zou",
"text": "Updated to correct a mistake",
"url": "https://arxiv.org/abs/2606.10402v2"
}
] |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | revised | Ten Advances in Mathematics and Theoretical Computer Science | The n-dimensional kissing number is τ_n = A(n, 1/2). Already hierarchy level 2 gives τ_n ≤ 2^((0.39661+o(1))n), compared with the optimized classical exponent 0.400944…. | OpenAI | The results were achieved by an internal version of Astra, our next major model. … These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate. | an internal version of Astra | OpenAI (paper PDF accompanying the post 'Ten advances in mathematics and theoretical computer science') | https://cdn.openai.com/pdf/ten-proofs-oai.pdf | 2026-08-01T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-08-06T00:00:00",
"by": "OpenAI",
"text": "Updated August 6, 2026. The original version can be found at https://cdn.openai.com/pdf/ten-proofs-oai-original.pdf.",
"url": "https://cdn.openai.com/pdf/ten-proofs-oai.pdf"
}
] |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | revised | Kissing numbers: new lower bounds in dimensions 18 through 96, with verification packages | Verification packages for new lower bounds on the kissing number in 53 dimensions: 18, 25-31, 38, 39, 49-63, 68-71 and 73-96. | Alexey Kravatskiy | The constructions, the verification code and the write-ups in this repository were developed by the author working with Claude Code (Anthropic). | Claude Code (Anthropic) | GitHub repository alexlegeartis/KissingNumbers | https://github.com/alexlegeartis/KissingNumbers | 2026-08-30T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-09-07T00:00:00",
"by": "Alexey Kravatskiy",
"text": "Dimension 25: tau(25) >= 197569, the lens-head configuration, replaces the +2 cap-level claim",
"url": "https://github.com/alexlegeartis/KissingNumbers/commit/4c26f42c1dbb0996d3d359ff5aa636d0982e4445"
},
{... |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | current | Large-Scale Autonomous Discovery of Kissing Number Constructions | Here we use Qiushi Engine, an autonomous multi-agent research system, to investigate kissing numbers and obtain new lower bounds in nineteen dimensions: 25, 27, 32–39, 43, 45, and 49–55. | Shuxing Yang, Rui Zhao, Junyao Wu, Yize Wang, Fujia Chen, Kaihao Zhu, Wenhao Li, Zichen Li, Yaqi Li, Shenzhan Hong, Yuang Pan, Junjie Yang, Taowen Deng, Jincheng Mi, Hongsheng Chen, Yihao Yang | The autonomous system carried out the construction searches, mathematical analysis and computational verification, while each final result was reduced to explicit mathematical arguments and independently checkable finite certificates. | Qiushi Engine | arXiv | https://arxiv.org/abs/2609.35051 | 2026-09-28T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | revised | Improved kissing numbers in nineteen and twenty-one dimensions | We prove that the kissing numbers in 19 and 21 dimensions are at least 12276 and 30779. | Kevin Li (GitHub: Felpix-Studios) | GPT-6-Astra Pro and Claude Opus 5.5 were used to find these constructions, write the Lean formalization, and draft the paper. | GPT-6-Astra Pro; Claude Opus 5.5 | GitHub repository Felpix-Studios/D19-D21-Kissing | https://github.com/Felpix-Studios/D19-D21-Kissing | 2026-09-30T00:00:00 | Lean 4 | https://github.com/Felpix-Studios/D19-D21-Kissing/tree/main/lean | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-10-04T00:00:00",
"by": "Kevin Li",
"text": "τ₂₁ ≥ 30779 (best published bound 29768: H. Cohn and A. Li, arXiv:2411.04916).",
"url": "https://github.com/Felpix-Studios/D19-D21-Kissing/commit/50da43d839777cdf973a617a6daac9463d6a1196"
},
{
"kind": "revision... |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | current | New lower bounds for kissing numbers in dimensions 25, 26, 27, 29 and 31 | Starting from constructions based on the Leech lattice, we obtain the improved lower bounds τ25 ≥ 197583, τ26 ≥ 199868, τ27 ≥ 201589, τ29 ≥ 209970 and τ31 ≥ 238666. | Rustem Takhanov, Stanislav Yun | All new constructions were generated with AI systems: Claude Code with Claude Opus 5.5, and GPT-6 Astra. | Claude Code with Claude Opus 5.5; GPT-6 Astra | GitHub repository k-nic/kissing_d25-31 | https://github.com/k-nic/kissing_d25-31 | 2026-10-05T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
kissing-number-problem | Kissing number problem | https://papex.org/problems/kissing-number-problem | partial | revised | Kissing numbers — Preprints and improved kissing number constructions | We prove that the kissing numbers in 19 and 21 dimensions are at least 12460 and 31560, improving the best published bounds 11948 (Ho) and 29768 (Cohn and Li). … We prove that the kissing numbers in 28, 29, 30 and 31 dimensions are at least 205172, 210495, 222073 and 241079. … We give two kissing configurations of 8358... | Kevin Li (GitHub: Felpix-Studios) | GPT-6 Astra Pro, Claude Opus 5.5 and Claude Fable 5.1 were used to find these constructions, prepare supplementary data and draft this paper. The manuscript was proofread and lightly revised by the author. | GPT-6 Astra Pro; Claude Opus 5.5; Claude Fable 5.1 | GitHub repository Felpix-Studios/kissing-number | https://github.com/Felpix-Studios/kissing-number | 2026-10-10T00:00:00 | null | null | 2026-10-18T00:00:00 | [
{
"kind": "revision",
"on": "2026-10-13T00:00:00",
"by": "Kevin Li",
"text": "bound updates",
"url": "https://github.com/Felpix-Studios/kissing-number/commit/6009dcca6da3ef55c47bca963747db307d33bc0d"
},
{
"kind": "revision",
"on": "2026-10-14T00:00:00",
"by": "Kevin Li",
"tex... |
lonely-runner-conjecture | Lonely runner conjecture | https://papex.org/problems/lonely-runner-conjecture | partial | revised | Nine and ten lonely runners | Rosenfeld recently obtained a computer-assisted proof of the conjecture for 8 runners. By refining his approach with a sieve, we obtain proofs (also computer-assisted) for 9 and 10 runners. | Tanupat Trakulthongchai | We employed OpenAI GPT-5 to assist with code generation, especially in low-level optimization. | OpenAI GPT-5 | arXiv; published in The Electronic Journal of Combinatorics (2026, doi:10.37236/14972) | https://arxiv.org/abs/2511.22427 | 2025-11-27T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-04-18T00:00:00",
"by": "Tanupat Trakulthongchai",
"text": "Updated theorem numbering to match with journal version; added references",
"url": "https://arxiv.org/abs/2511.22427v2"
}
] |
lonely-runner-conjecture | Lonely runner conjecture | https://papex.org/problems/lonely-runner-conjecture | partial | current | Eleven, twelve, and thirteen lonely runners | Ultimately, we provide a computer-assisted proof of the Lonely Runner Conjecture for k ∈ {10, 11, 12}. | Touch Sungkawichai, Tanupat Trakulthongchai | The development of our code was assisted by AI-based tools, but the process was closely supervised and the code was manually verified. | null | arXiv | https://arxiv.org/abs/2604.23906 | 2026-04-26T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
lonely-runner-conjecture | Lonely runner conjecture | https://papex.org/problems/lonely-runner-conjecture | partial | revised | Fourteen and fifteen lonely runners | We prove the Lonely Runner Conjecture for fourteen and fifteen runners. | Jaan Allikvere | The author used large language models to assist with code development, drafting, exploring proof ideas, and searching for errors in the manuscript. No model output is treated as mathematical evidence. | null | arXiv | https://arxiv.org/abs/2609.02604 | 2026-09-02T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-09-24T00:00:00",
"by": "Jaan Allikvere",
"text": "v2: adds the fifteen-runner case and a sharper speed-product bound (logarithmic threshold about 415 instead of 810 for fifteen runners; 61 instead of 111 primes for fourteen); v1 covered fourteen runners only.",
... |
matrix-multiplication-complexity | Complexity of matrix multiplication | https://papex.org/problems/matrix-multiplication-complexity | partial | current | Improving the matrix multiplication exponent with modern optimization and AlphaEvolve | Our combined approach yields an upper bound of ω < 2.371177, improving the previous best bound of 2.371339. | Emilien Dupont, Marvin Eisenberger, Borislav Kozlovskii, Abbas Mehrabian, Francisco J. R. Ruiz, Abigail See, Renfei Zhou, Josh Alman, Virginia Vassilevska Williams, Matej Balog | Second, we leverage recent advances in machine learning to design a new optimization algorithm for this problem. Finally, we refine the resulting optimization algorithm with AlphaEvolve. | AlphaEvolve | arXiv | https://arxiv.org/abs/2608.16884 | 2026-08-17T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
matrix-multiplication-complexity | Complexity of matrix multiplication | https://papex.org/problems/matrix-multiplication-complexity | partial | current | An Upper Bound of 9/4 for the Matrix Multiplication Exponent | We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf | 2026-10-06T00:00:00 | Lean 4 | https://github.com/openai/math/blob/main/lean/docs/107.md | 2026-10-14T00:00:00 | [
{
"kind": "response",
"on": "2026-10-07T00:00:00",
"by": "Scott Aaronson",
"text": "Matrix multiplication in n^(9/4+o(1)) time—a rational exponent for once (!), and via a completely different approach than was used for O(n^2.373) and so forth",
"url": "https://scottaaronson.blog/?p=10169"
},
... |
navier-stokes | Navier–Stokes existence and smoothness | https://papex.org/problems/navier-stokes | proof | revised | On the Navier–Stokes Millennium Prize Problem | Our system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time. The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. This resolves the Navi... | OpenAI | We used a system of coordinating agents powered by our internal model. The agents had access to tools such as the ability to read from a cached version of the internet and the ability to run code. … The groups varied in size, and the group that produced the Navier–Stokes resolution involved on the order of 10,000 concu... | an internal model that is significantly more capable than GPT-6 Astra | OpenAI website (research post), with paper PDF and Lean repository | https://openai.com/index/navier-stokes-solution/ | 2026-09-08T00:00:00 | Lean 4 | https://github.com/openai/NavierStokesAndEuler | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-09-10T00:00:00",
"by": "OpenAI",
"text": "Update — September 10, 2026: We have updated “Concurrent work” with findings from our investigation into whether user inputs could have influenced this result.",
"url": "https://openai.com/index/navier-stokes-solution/"
... |
riemann-hypothesis | Riemann hypothesis | https://papex.org/problems/riemann-hypothesis | partial | revised | Learning more about Claude's mathematical capabilities | An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%. | Anthropic | An unreleased research version of Claude found the new lower bound over two sessions in Claude Code, using a total of 31 million output tokens. The paper is credited to “Claude, Anthropic”; its arXiv version (2608.13637) is listed under Levent Alpöge and Ralph Furman. | an unreleased research version of Claude | Anthropic website (research post) | https://www.anthropic.com/research/riemann-zeta | 2026-08-10T00:00:00 | Lean 4 | https://github.com/anthropics/zeta-23-lean | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-08-13T00:00:00",
"by": "Anthropic",
"text": "this post was updated on August 13, 2026, with an updated version of Claude's paper. This paper was revised by Claude to provide a clearer proof and additional historical context.",
"url": "https://www.anthropic.com/r... |
riemann-hypothesis | Riemann hypothesis | https://papex.org/problems/riemann-hypothesis | partial | current | The Quasi-Riemann Hypothesis | We establish the quasi-Riemann hypothesis by proving that every Dirichlet L-function, including Riemann's zeta function, has no zeros in the half-plane Re s > 11/12. More generally, we prove the same zero-free half-plane for every finite-order Hecke L-function over K = Q(√−3). In particular, this rules out the existenc... | OpenAI | Exceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties. Additionally, the writeup for the Re(s) > 11/12 zero-free region for the Riemann zeta function was human edited for readability. … This paper was written with h... | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf | 2026-10-06T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
riemann-hypothesis | Riemann hypothesis | https://papex.org/problems/riemann-hypothesis | partial | current | The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8 | In particular, Theorem 1.1 resolves the quasi-Riemann hypothesis affirmatively: the supremum of the real parts of the nontrivial zeros of ζ, namely its zeros in 0 < ℜs < 1, is at most 7/8. The boundary ℜs = 7/8 is not included, while any exceptional real zero in (7/8, 1) is excluded. The theorem does not establish the ... | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. … Exceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf | 2026-10-06T00:00:00 | Lean 4 | https://github.com/openai/math/blob/main/lean/docs/003.md | 2026-10-14T00:00:00 | [
{
"kind": "response",
"on": "2026-10-08T00:00:00",
"by": "Baiying Liu",
"text": "Recently, a uniform 7/8 zero-free half-plane for finite-order Hecke L-functions over Q(√−3) and its transfer to Dirichlet L-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters b = 1/8, ... |
twin-prime-conjecture | Twin prime conjecture | https://papex.org/problems/twin-prime-conjecture | partial | current | Improved short gaps between primes | We prove that lim inf_{n→∞} (p_{n+1} − p_n) ≤ 186, where p_n denotes the nth prime. | OpenAI | The proof is due to GPT 6 Astra. … Astra helped establish a stronger bound of 186, showing that infinitely many pairs occur within this smaller distance. | GPT 6 Astra | OpenAI (paper PDF), with Lean repository openai/PrimeGaps186 | https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf | 2026-09-03T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
twin-prime-conjecture | Twin prime conjecture | https://papex.org/problems/twin-prime-conjecture | partial | current | A new bound for small gaps between primes | Stadlmann recently proved that infinitely many pairs of distinct primes are apart by at most 240, improving upon the longstanding bound 246 established by Polymath8b. Building on her work, we prove that lim inf_{n→∞} (p_{n+1} − p_n) ≤ 212. | François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Ashvin A. Swaminathan, Jesse Thorner, Yunzhou Xie (Axiom Math) | AxiomProver, an AI system under development by Axiom Math, autonomously generated from natural-language specifications a Lean certificate of the deduction of Theorem 1.1. | AxiomProver | Axiom Math website (paper PDF) | https://primegaps.axiommath.ai/bgp212.pdf | 2026-09-03T00:00:00 | null | null | 2026-10-14T00:00:00 | [] |
twin-prime-conjecture | Twin prime conjecture | https://papex.org/problems/twin-prime-conjecture | partial | revised | A fixed-profile proof for prime gaps of at most 182 | Her August 2026 mixed-support preprint lowered the more-than-twelve-year-old bound H1 ≤ 246 to H1 ≤ 240; independent concurrent work by OpenAI, using her 2023 estimates, reached H1 ≤ 186, and the Axiom Math team further developed her framework to reach H1 ≤ 212. Building on these developments with additional Type II an... | Romyar Sharifi | The paper is credited to “GPT-6 Astra (OpenAI), prompted by Romyar Sharifi”. | GPT-6 Astra (OpenAI) | Romyar Sharifi's UCLA web page | https://www.math.ucla.edu/~sharifi/AI/prime_gaps_182_expository_bundle.pdf | 2026-09-04T00:00:00 | null | null | 2026-10-14T00:00:00 | [
{
"kind": "revision",
"on": "2026-09-11T00:00:00",
"by": "Romyar Sharifi",
"text": "The 11 September 2026 version expanded citations and exposition; the stated results and numerical data were unchanged in that revision.",
"url": "https://www.math.ucla.edu/~sharifi/AI/prime_gaps_182_expository_bu... |
unique-games-conjecture | Unique games conjecture | https://papex.org/problems/unique-games-conjecture | proof | current | The Unique Games Theorem | We prove the Unique Games Conjecture. For every fixed ε, δ ∈ (0, 1/2), we give a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet, with completeness at least 1 − ε and soundness at most δ. | OpenAI | This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. | an unreleased internal OpenAI model | openai/math repository | https://github.com/openai/math/blob/main/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf | 2026-10-06T00:00:00 | Lean 4 | https://github.com/openai/math/blob/main/lean/docs/102.md | 2026-10-14T00:00:00 | [
{
"kind": "response",
"on": "2026-10-07T00:00:00",
"by": "Lance Fortnow",
"text": "Now these proofs haven't been fully verified but if they hold up, we've seen more progress in theoretical computer science in the last 24 hours than in the previous three decades combined!",
"url": "https://blog.c... |
AI-assisted claims on open problems
Results on well-known open problems that their sources present as made with AI, listed as the sources state them, with the revisions, withdrawals and responses that followed. Compiled by Papex for its problem pages; every row links to the page where it appears.
Snapshot of 18 October 2026:
claims: 38 claims on 14 problems (9 in mathematics, 4 in computer science, 1 in physics).problems: the 193 open problems on Papex's list, each taken from a named public list (Wikipedia's lists of unsolved problems by field, the Clay Mathematics Institute's Millennium Prize Problems, Landau's problems).
from datasets import load_dataset
claims = load_dataset("papexlabs/ai-assisted-claims", "claims", split="train")
problems = load_dataset("papexlabs/ai-assisted-claims", "problems", split="train")
What it says and what it does not
- Each claim is listed as its source states it.
quoteis copied from the source's own abstract or summary,model_nameis the source's, andai_involvementis the claimant's own statement. Papex has not checked how AI was used. - Papex does not decide whether a claim is correct.
kindis Papex's reading of what the source claims (a proof, a disproof or a partial result), not a finding that it holds. - A problem's
statusis the one its named source gives. No claim here changes it. - A
formal_proof_urlmeans a formal proof was published. Papex did not run it, and a formal proof shows that its formal statement is proved, not that the statement matches the paper's claim. - Papex Labs chose the claims by hand from public sources: claims their source presents as AI-assisted that address a problem on Papex's list. The list is not exhaustive. 17 of the 38 come from the openai/math repository.
- Sources change. Every claim was last compared with its source on the date in
checked_on;source_urlhas the current text.
Fields
claims (one row per claim; problem_slug + source_url identify it):
| Field | Meaning |
|---|---|
problem_slug, problem_title, problem_url |
The problem the claim addresses and its page on papex.org |
kind |
proof, disproof or partial |
state |
current, revised or withdrawn, from the updates below |
title |
The source's title |
quote |
The claim in the source's own words |
claimant |
Who makes the claim, as the source names them |
ai_involvement |
What the claimant says AI did |
model_name |
The model or system the source names |
venue, source_url |
Where the claim was made |
claimed_on |
Date of the claim |
formal_system, formal_proof_url |
A published formal proof, if any (for example Lean 4) |
checked_on |
When Papex last compared the row with its source |
updates |
Later revision, withdrawal or response, each with on, by, text and url |
problems (one row per problem): slug, title, question, field, since_year (only where a
year is commonly cited), status, source_name, source_list, source_url, checked_on,
claim_count, url.
Updates and corrections
The files are exported from Papex's public API with tools/export.py; each refresh is a new
commit here, so earlier snapshots stay available. Claimants are named as their own sources name
them. To report an error, a missing claim or a change you want, write to support@papex.org.
Licence
The compilation (the selection, the structure and the text Papex Labs wrote) is licensed under
CC BY 4.0. Text quoted from the sources (title,
quote, ai_involvement, model_name and updates[].text) belongs to its authors. It is
reproduced as attributed quotation, with its source linked on every row, and is not covered by
this licence.
Citation
@misc{papex_ai_assisted_claims_2026,
title = {AI-assisted claims on open problems},
author = {{Papex Labs}},
year = {2026},
url = {https://huggingface.co/datasets/papexlabs/ai-assisted-claims},
note = {Snapshot of 18 October 2026}
}
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