{
  "asOf": "2026-11-01",
  "note": "Records public statements. Does not assert that any proof is correct. Status can change; check the sources.",
  "results": [
    {
      "id": "eth",
      "result": "ETH: refutation claims",
      "area": "Fine-grained complexity",
      "direction": "refutation",
      "previous": {
        "status": "found",
        "text": "ETH, from Impagliazzo and Paturi, conjectured s_k > 0 for k ≥ 3. Best earlier 3-SAT records cited by Anthropic: O(1.328^n) deterministic, O(1.308^n) randomized.",
        "url": "https://github.com/anthropics/formal-math/blob/16a5ce41304fcdaea4e0be7c132e1c361dc26da5/eth/paper.pdf",
        "sourceDate": "2026-10-26"
      },
      "claim": {
        "status": "found",
        "text": "OpenAI abstract: (n + m)^{O_k(1)} 2^{n/(log n)^{1/2+o(1)}}. OpenAI Theorem 1.1: (n + m)^{O_k(1)} 2^{O_k(n log log n/√log n)}; m ⩽ Dn: (n + m)^{O_{k,D}(1)} 2^{O_{k,D}(n/√log n)}. Anthropic Theorem 1: 2^{O(min{k^2 n log log n/√log n, km/√log m})}. Anthropic Theorem 2: O(2^{δn}) for every δ > 0. Anthropic Corollary 5: s_k = 0 for every k ≥ 3; s_∞ = 0.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf",
        "sourceDate": "2026-10-25"
      },
      "claimants": [
        {
          "name": "OpenAI",
          "date": "2026-10-26",
          "title": "The Exponential Time Hypothesis is False",
          "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf"
        },
        {
          "name": "Anthropic",
          "date": "2026-10-27",
          "title": "A 2^{o(n)} Time Algorithm for 3-SAT",
          "url": "https://github.com/anthropics/formal-math/blob/16a5ce41304fcdaea4e0be7c132e1c361dc26da5/eth/paper.pdf"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "OpenAI README attributes manuscripts to an unnamed internal model; ETH-specific procedure unstated. Anthropic's paper states internal Claude found core algorithm October 16, 14M output tokens, no human input.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/README.md",
        "sourceDate": "2026-10-26"
      },
      "lean": {
        "status": "partial",
        "text": "OpenAI lean/ComparatorChallenges/SubexponentialSAT.lean: weaker (log log n)^2 exponent, not Theorem 1.1. Anthropic eth/MainResults.lean: Main Theorem and ETH negation on Mathlib multi-stack machines.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/docs/378.md",
        "sourceDate": "2026-10-26"
      },
      "leanStatementCheck": {
        "status": "partial",
        "text": "Toledo (Claude-written tools; first statement reading by Claude Opus 5.5) bridged each statement to Lean ETH per lab's model; model step stays prose. Busnello hand-read both, one encoding check unfinished.",
        "url": "https://carlostoledo.co/eth/",
        "sourceDate": "2026-10-28"
      },
      "experts": [
        {
          "who": "Russell Impagliazzo, Ramamohan Paturi, Ryan Williams",
          "text": "Their open letter says participation is not certification and asks the community for independent scrutiny, explanations and follow-up study of the claim.",
          "date": "2026-10-26",
          "url": "https://ethdiscussion.blog/2026/10/26/eth-refutation-examination-and-discussion/"
        },
        {
          "who": "Ryan Williams",
          "text": "Says the three organizers signed no NDAs, receive no pay, and will share findings publicly; says OpenAI agreed to release materials and sponsor a conference.",
          "date": "2026-10-26",
          "url": "https://bsky.app/profile/rrwilliams.bsky.social/post/3myspd2bpas2c"
        },
        {
          "who": "Ryan Williams (to TechCrunch)",
          "text": "Called some proof parts perhaps routine and two or three alien-looking; does not think OpenAI's algorithm can be the best achievable.",
          "date": "2026-10-27",
          "url": "https://techcrunch.com/2026/10/27/frontier-ai-labs-say-their-models-refuted-the-exponential-time-hypothesis-what-does-that-mean/"
        },
        {
          "who": "Ryan Williams",
          "text": "Conditional on correctness, every computer scientist should raise their P=NP credence somewhat, by varying amounts.",
          "date": "2026-10-27",
          "url": "https://bsky.app/profile/rrwilliams.bsky.social/post/3myt3vq2w4s2l"
        },
        {
          "who": "Lance Fortnow",
          "text": "States he has not checked the proof himself yet considers it safe to assume ETH is finished; notes Lean covers a slightly weaker bound.",
          "date": "2026-10-28",
          "url": "https://blog.computationalcomplexity.org/2026/10/the-exponential-time-refutation.html"
        },
        {
          "who": "Scott Aaronson",
          "text": "Reported the claim as roughly exp(n/√log(n)) for 3SAT, and moved his personal P=NP probability from 3% to 5%.",
          "date": "2026-10-26",
          "url": "https://scottaaronson.blog/?p=10239#comment-2042428"
        },
        {
          "who": "Emanuele Viola",
          "text": "Wrote that the manuscript shows ETH and strong ETH false, said he is not shocked, and hopes conditional results get recalibrated.",
          "date": "2026-10-27",
          "url": "https://emanueleviola.wordpress.com/2026/10/27/on-eth-3sum-integer-multiplication-etc-being-false/"
        },
        {
          "who": "Ben Recht",
          "text": "Calls OpenAI's algorithm galactic, running an exponential-time subroutine when variables number below 10^10,000,000,000,000,000,000,000,000.",
          "date": "2026-10-28",
          "url": "https://x.com/beenwrekt/status/2115445861006368787"
        },
        {
          "who": "Ryan O'Donnell",
          "text": "LLM-assisted exposition recasting OpenAI's approach through sparse permanents; deterministic sparse case, with a randomized appendix for arbitrary 3-CNFs.",
          "date": "2026-10-30",
          "url": "https://www.cs.cmu.edu/~odonnell/some_eth_false_exposition.pdf"
        },
        {
          "who": "Shiva Kintali",
          "text": "Posted a simplified exposition of OpenAI's proof, developed with Claude and ChatGPT then expanded by hand, aiming only at basic ETH.",
          "date": "2026-10-26",
          "url": "https://shivakintali.github.io/papers/ETH.pdf"
        },
        {
          "who": "Mark Sellke (Member of Technical Staff at OpenAI, per his homepage)",
          "text": "Slides outline OpenAI's 2^{o(n)} k-SAT algorithm, credit the theorem to OpenAI/Anthropic 2026, and skip quantitative optimizations.",
          "date": "2026-10-29",
          "url": "https://msellke.com/Files/ETH-False-Slides.pdf"
        },
        {
          "who": "Francis Zane",
          "text": "Reports people compiling the Lean proofs locally; opens a replication thread, suggesting a non-author should probably check though not everyone needs to repeat it.",
          "date": "2026-10-30",
          "url": "https://ethdiscussion.blog/2026/10/30/lean-replications/"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "No peer-reviewed publication found as of 2026-11-01. Searched both repositories, Hexagon, ethdiscussion.blog, OpenAlex and web search."
      },
      "revision": {
        "status": "found",
        "text": "OpenAI paper dated 17 October, revised 25 October; one repository commit touches the PDF. Anthropic's Hexagon page lists version 1 only. No retraction found as of 2026-11-01.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/history.md",
        "sourceDate": "2026-11-01"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "3sum",
      "result": "3SUM: truly subquadratic time",
      "area": "Fine-grained complexity",
      "direction": "improved algorithm",
      "previous": {
        "status": "found",
        "text": "Textbook O(n^2) improved only by polylogarithmic factors; 3SUM hypothesis postulated n^{2-o(1)} for polynomial-size integers on word RAM with O(log n)-bit words.",
        "url": "https://arxiv.org/abs/2610.06783v1",
        "sourceDate": "2026-10-05"
      },
      "claim": {
        "status": "found",
        "text": "Abstract and Theorem 2: O(n^{1.9992}). Theorem 22: n^{2−1/1296+o(1)} ≤ O(n^{1.99923}) using Theorem 5; n^{2−ε′/2+o(1)} ≤ O(n^{1.9992}) using Corollary 26. Theorem 2, real inputs: O(n^{1.998}) expected time, Las Vegas. Lean: ThreeSum.SolvedInTime 1.9992.",
        "url": "https://arxiv.org/pdf/2610.06783v1",
        "sourceDate": "2026-10-05"
      },
      "claimants": [
        {
          "name": "Alman, Vassilevska Williams",
          "date": "2026-10-05",
          "title": "Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs",
          "url": "https://arxiv.org/abs/2610.06783v1"
        },
        {
          "name": "Anthropic (Lean)",
          "date": "2026-10-06",
          "title": "Truly Subquadratic 3SUM and Truly Subcubic APSP: a Lean 4 formalization",
          "url": "https://github.com/anthropics/formal-math/blob/16a5ce41304fcdaea4e0be7c132e1c361dc26da5/3sum-apsp/README.md"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "Paper states Claude discovered the hypothesis-refuting algorithm, average then worst case, in a 16M-output-token session without human input; authors understood, simplified, strengthened, extended, and take responsibility.",
        "url": "https://arxiv.org/pdf/2610.06783v1",
        "sourceDate": "2026-10-05"
      },
      "lean": {
        "status": "partial",
        "text": "anthropics/formal-math 3sum-apsp/EndStatement.lean: Theorem_22_3SUM, plus Theorems 19, 22 and Corollary 39 zero-weight. README excludes real-input Theorem 35, reductions' printed running times, and Theorem 33.",
        "url": "https://github.com/anthropics/formal-math/blob/16a5ce41304fcdaea4e0be7c132e1c361dc26da5/3sum-apsp/README.md",
        "sourceDate": "2026-10-06"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No named public check of 3sum-apsp Lean statements against the paper found. Searched ethdiscussion.blog, Toledo's reports, Busnello, repository docs, GitHub, arXiv and web search (as of 2026-11-01)."
      },
      "experts": [
        {
          "who": "Scott Aaronson",
          "text": "Said the preprint refutes half-century-old conjectures, credited an Anthropic model with the crucial idea, and contrasted digested release with OpenAI's raw one.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169"
        },
        {
          "who": "Virginia Vassilevska Williams (co-author)",
          "text": "Co-author says the proof came from an internal Anthropic model, not the authors; argues, from one run each, her group's 2023 notes plus two prompts would have sufficed.",
          "date": "2026-10-13",
          "url": "https://people.csail.mit.edu/virgi/thoughts.txt"
        },
        {
          "who": "Emanuele Viola",
          "text": "Reacted with surprise and added the paper to his book's growing list of disproved conjectures.",
          "date": "2026-10-06",
          "url": "https://emanueleviola.wordpress.com/2026/10/06/3sum-is-false-even-falser/"
        },
        {
          "who": "Sankeerth Rao Karingula (Agentin AI), Shachar Lovett (UC San Diego)",
          "text": "Give a simplified proof of polynomial 3SUM and APSP improvements with weaker exponents, using selected-entry matrix multiplication; disclose an Agentin AI agent swarm, ChatGPT and Codex.",
          "date": "2026-10-11",
          "url": "https://arxiv.org/abs/2610.15663v1"
        },
        {
          "who": "Ruhollah Majdoddin",
          "text": "Extends the matrix theorem to D <= N^0.172 and claims deterministic 3SUM in n^{1.9945+o(1)}, citing the original exponent as 1.9992.",
          "date": "2026-10-12",
          "url": "https://arxiv.org/abs/2610.17420v1"
        },
        {
          "who": "Nick Fischer",
          "text": "Treats the 3SUM Hypothesis as recently refuted, says geometric lower bounds lost their foundation, and recovers many under the k-AP Hypothesis.",
          "date": "2026-10-23",
          "url": "https://arxiv.org/abs/2610.32162v1"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "No peer-reviewed publication found as of 2026-11-01. Searched arXiv metadata, OpenAlex, Crossref, web search. Co-author Vassilevska Williams wrote she is reluctant to submit to STOC.",
        "url": "https://people.csail.mit.edu/virgi/thoughts.txt"
      },
      "revision": {
        "status": "found",
        "text": "arXiv lists only v1, dated 2026-10-05; no revision or withdrawal recorded as of 2026-11-01.",
        "url": "https://arxiv.org/abs/2610.06783",
        "sourceDate": "2026-11-01"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "apsp",
      "result": "APSP: truly subcubic time",
      "area": "Fine-grained complexity",
      "direction": "improved algorithm",
      "previous": {
        "status": "found",
        "text": "Classical O(n^3); fastest prior algorithm shaved only a subpolynomial 2^{Ω(√log n)} factor. APSP hypothesis postulated n^{3−o(1)} for polynomially bounded integer weights on word RAM.",
        "url": "https://arxiv.org/abs/2610.06783v1",
        "sourceDate": "2026-10-05"
      },
      "claim": {
        "status": "found",
        "text": "Abstract and Theorem 2: O(n^{2.9995}). Theorem 22: Õ(n^{3−1/1944}) ≤ O(n^{2.99949}) using Theorem 5; Õ(n^{3−ε′/3}) ≤ O(n^{2.99942}) using Corollary 26, with ε′ = 0.00175. Theorem 2, real inputs: O(n^{2.998}) expected time, Las Vegas. Lean: APSP.SolvedInTime 2.99942.",
        "url": "https://arxiv.org/pdf/2610.06783v1",
        "sourceDate": "2026-10-05"
      },
      "claimants": [
        {
          "name": "Alman, Vassilevska Williams",
          "date": "2026-10-05",
          "title": "Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs",
          "url": "https://arxiv.org/abs/2610.06783v1"
        },
        {
          "name": "Anthropic (Lean)",
          "date": "2026-10-06",
          "title": "Truly Subquadratic 3SUM and Truly Subcubic APSP: a Lean 4 formalization",
          "url": "https://github.com/anthropics/formal-math/blob/16a5ce41304fcdaea4e0be7c132e1c361dc26da5/3sum-apsp/README.md"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "Paper states Claude discovered the hypothesis-refuting algorithm, average then worst case, in a 16M-output-token session without human input; authors understood, simplified, strengthened, extended, and take responsibility.",
        "url": "https://arxiv.org/pdf/2610.06783v1",
        "sourceDate": "2026-10-05"
      },
      "lean": {
        "status": "partial",
        "text": "3sum-apsp/EndStatement.lean: Theorem_22_APSP and Theorem_22_MinPlus at 2.99942; README lists wordRam_theorem_22_first for 2.99949 route. README: no machine proof for real-weight Theorem 35, Theorem 33 or directed-unweighted Theorem 34 running times.",
        "url": "https://github.com/anthropics/formal-math/blob/16a5ce41304fcdaea4e0be7c132e1c361dc26da5/3sum-apsp/README.md",
        "sourceDate": "2026-10-06"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No named public check of 3sum-apsp Lean statements against the paper found. Searched ethdiscussion.blog, Toledo's reports, Busnello, repository docs, GitHub, arXiv and web search (as of 2026-11-01)."
      },
      "experts": [
        {
          "who": "Scott Aaronson",
          "text": "Reported APSP at O(n^2.9995), said this refutes a half-century-old conjecture, and credited an Anthropic model with the crucial idea.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169"
        },
        {
          "who": "Virginia Vassilevska Williams (co-author)",
          "text": "Co-author says the proof came from an internal Anthropic model, not the authors; argues, from one run each, her group's 2023 notes plus two prompts would have sufficed.",
          "date": "2026-10-13",
          "url": "https://people.csail.mit.edu/virgi/thoughts.txt"
        },
        {
          "who": "Sankeerth Rao Karingula (Agentin AI), Shachar Lovett (UC San Diego)",
          "text": "Give a simplified proof of polynomial 3SUM and APSP improvements with weaker exponents, using selected-entry matrix multiplication; disclose an Agentin AI agent swarm, ChatGPT and Codex.",
          "date": "2026-10-11",
          "url": "https://arxiv.org/abs/2610.15663v1"
        },
        {
          "who": "Ruhollah Majdoddin",
          "text": "Claims n^{2.9945+o(1)} for Exact Triangle, (min,+)-product and APSP, citing the original APSP exponent as 2.99942.",
          "date": "2026-10-12",
          "url": "https://arxiv.org/abs/2610.17420v1"
        },
        {
          "who": "Barna Saha, Christopher Ye",
          "text": "Claim deterministic (min,+) product in Õ(N^{2.9964}), improving the O(N^{2.9995}) they attribute to Anthropic via Alman and Vassilevska Williams, using its thin matrix theorem.",
          "date": "2026-10-14",
          "url": "https://arxiv.org/abs/2610.20297v1"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "No peer-reviewed publication found as of 2026-11-01. Searched arXiv metadata, OpenAlex, Crossref, web search. Co-author Vassilevska Williams wrote she is reluctant to submit to STOC.",
        "url": "https://people.csail.mit.edu/virgi/thoughts.txt"
      },
      "revision": {
        "status": "found",
        "text": "arXiv lists only v1, dated 2026-10-05; no revision or withdrawal recorded as of 2026-11-01.",
        "url": "https://arxiv.org/abs/2610.06783",
        "sourceDate": "2026-11-01"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "unique-games",
      "result": "Unique Games Conjecture",
      "area": "Hardness of approximation",
      "direction": "new proof",
      "previous": {
        "status": "found",
        "text": "Khot stated it in a 2002 paper. Khot–Minzer–Safra's 2018 report (ECCC TR18-006) says it completes the 2-to-2 conjecture, with imperfect completeness. Quanta describes Unique Games as open before OpenAI's announcement.",
        "url": "https://www.quantamagazine.org/as-ai-closed-in-on-unique-games-proof-researchers-raced-to-beat-the-machines-20261007/",
        "sourceDate": "2026-10-07"
      },
      "claim": {
        "status": "found",
        "text": "[Paper, Theorem 1.1] For every fixed ε, δ ∈ (0, 1/2), there are an integer s ≥ 1 and a deterministic polynomial-time reduction from 3SAT to explicit unweighted Unique Games instances G_φ over K = F_2^s such that φ satisfiable ⟹ val(G_φ) ≥ 1 − ε, φ unsatisfiable ⟹ val(G_φ) ≤ δ. [Paper, abstract] We prove the Unique Games Conjecture. [CONTENTS.md, family 102] Proves Khot's Unique Games Conjecture. [lean/ComparatorChallenges/UniqueGamesTheorem.lean] theorem theorem11 (ε δ : ℝ) (hε : 0 < ε) (hεhalf : ε < 1 / 2) (hδ : 0 < δ) (hδhalf : δ < 1 / 2) : Nonempty (Explicit.MachineOutputContract.BinaryGapReduction ε δ)",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf",
        "sourceDate": "2026-09-23"
      },
      "claimants": [
        {
          "name": "OpenAI",
          "date": "2026-10-06",
          "title": "The Unique Games Theorem",
          "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "README attributes manuscripts to an unnamed internal OpenAI model, about three hours of ChatGPT Pro compute per result on average; its family 102 reasoning summary concerns basic-semidefinite-threshold NP-hardness.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/README.md",
        "sourceDate": "2026-10-30"
      },
      "lean": {
        "status": "found",
        "text": "Repo lists the main theorem as formalized: lean/ComparatorChallenges/UniqueGamesTheorem.lean; declaration OAI.UniqueGamesTheorem.theorem11 in lean/OAI/Computability/UniqueGames/Theorem.lean. Scope document: lean/docs/102.md.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/docs/102.md",
        "sourceDate": "2026-10-06"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No named person found confirming the Lean statement equals the paper. Ariel Elboim ran Comparator but says he did not audit this; formalization.yaml's repo-wide review field says unchecked (as of 2026-11-01).",
        "url": "https://github.com/lgboim/unique-games-short-proof/blob/5804efc42d4a4921baf221bd55730223b6af4ad1/README.md",
        "sourceDate": "2026-10-09"
      },
      "experts": [
        {
          "who": "Scott Aaronson (UT Austin)",
          "text": "Says he and others are fairly sure it is a proof, citing a Lean certificate, but apparently almost no human has understood these proofs yet.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169"
        },
        {
          "who": "Dana Moshkovitz (UT Austin), texts relayed by Scott Aaronson",
          "text": "Her early texts, relayed by Aaronson, call the writeup unreadable without AI help and describe a strange new recursive code, unlike long or short codes.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169"
        },
        {
          "who": "Dana Moshkovitz, relayed by Scott Aaronson (comment #2)",
          "text": "Aaronson relays her view that the conjecture is true, its famous corollaries did not need it, and understanding the proof's mechanism will take time.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169#comment-2041062"
        },
        {
          "who": "Dana Moshkovitz (UT Austin; talk abstract)",
          "text": "Her talk abstract offers a mental map of the proof, viewing its innovation as a new code, Hadamard with a noise test; says the writeup reads poorly for humans.",
          "date": "2026-10-30",
          "url": "https://scottaaronson.blog/?p=10232"
        },
        {
          "who": "Scott Aaronson (post and comment #9)",
          "text": "Says Moshkovitz, his wife, studied the proof closely with Khot, Håstad and Minzer and gave two Simons talks; says they can understand the new code once shown it.",
          "date": "2026-10-31",
          "url": "https://scottaaronson.blog/?p=10232"
        },
        {
          "who": "Lance Fortnow (former founding dean of computing, Illinois Tech)",
          "text": "Presents the release as proofs not yet fully verified, and makes his assessment of its significance conditional on whether they hold up.",
          "date": "2026-10-07",
          "url": "https://blog.computationalcomplexity.org/2026/10/open-no-more.html"
        },
        {
          "who": "Rares-Darius Buhai, Suprovat Ghoshal, Neng Huang, Pravesh Kothari, Euiwoong Lee, Konstantin Makarychev, Yury Makarychev",
          "text": "Their Max-Cut paper uses it conditionally: if OpenAI's proof is correct, one of their hardness results would need only NP not in BPP.",
          "date": "2026-10-19",
          "url": "https://arxiv.org/abs/2610.27216"
        },
        {
          "who": "Mark Braverman (Princeton), quoted by Quanta",
          "text": "Speaking in September about the rumors, he criticized announcing mathematics through press releases, yet said an AI proof could open directions if its key assumptions are extracted.",
          "date": "2026-10-07",
          "url": "https://www.quantamagazine.org/as-ai-closed-in-on-unique-games-proof-researchers-raced-to-beat-the-machines-20261007/"
        },
        {
          "who": "Dor Minzer (MIT), quoted by Quanta",
          "text": "Told Quanta that AI tools remove the value of failing and may deter long projects; the article reports no verdict from him on the proof's correctness.",
          "date": "2026-10-07",
          "url": "https://www.quantamagazine.org/as-ai-closed-in-on-unique-games-proof-researchers-raced-to-beat-the-machines-20261007/"
        },
        {
          "who": "Bill Gasarch (University of Maryland)",
          "text": "On the whole release: warns a Lean check can certify a mis-encoded theorem, and expects humans to read, understand and, where needed, repair these proofs over time.",
          "date": "2026-10-18",
          "url": "https://blog.computationalcomplexity.org/2026/10/ai-math-document-drop-non-tech-thoughts.html"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "None found as of 2026-11-01. Manuscript README cites only an OpenAI release preprint; arXiv API and web searches showed no venue; proceedings not checked individually.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/The-Unique-Games-Theorem-September-23-2026/README.md",
        "sourceDate": "2026-10-30"
      },
      "revision": {
        "status": "not_found",
        "text": "No revision or withdrawal recorded: paper.pdf and the Lean directory show only the 2026-10-06 initial commit at 4eaee87, and history.md does not list it (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/history.md",
        "sourceDate": "2026-10-30"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "two-to-one-games",
      "result": "2-to-1 games, perfect completeness",
      "area": "Hardness of approximation",
      "direction": "new proof",
      "previous": {
        "status": "found",
        "text": "Fei–Minzer–Wang (ECCC, 2026-09-14) report perfect-completeness hardness for 4-to-1 games, not 2-to-1, crediting Dinur et al. with the almost-perfect version. Austrin et al. (2013 manuscript): 2-to-1 soundness 23/24+ε at perfect completeness.",
        "url": "https://eccc.weizmann.ac.il/report/2026/179/",
        "sourceDate": "2026-09-14"
      },
      "claim": {
        "status": "found",
        "text": "[Paper, Theorem 1.1] For every fixed rational δ ∈ (0, 1) there are an integer q = q(δ) ≥ 2 and a deterministic polynomial-time reduction from 3-SAT to 2-to-1 games with alphabets [2q] and [q] such that: (i) a satisfiable formula produces a game of value 1; (ii) an unsatisfiable formula produces a game of value at most δ. [Paper, abstract] We prove the 2-to-1 Games Conjecture with perfect completeness. For every fixed rational δ ∈ (0, 1), it is NP-hard to distinguish satisfiable 2-to-1 games from games of value at most δ, with a fixed alphabet and an explicitly listed unweighted multiset of constraints. [CONTENTS.md, family 105] Proves Khot's 2-to-1 Games Conjecture with perfect completeness [lean/ComparatorChallenges/PerfectCompleteness.lean] theorem exists_reduction (δ : ℚ) (positive : 0 < δ) (less_than_one : δ < 1) : Nonempty (PerfectCompleteness.BinaryGapReduction δ)",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf",
        "sourceDate": "2026-09-23"
      },
      "claimants": [
        {
          "name": "OpenAI",
          "date": "2026-10-06",
          "title": "Perfect completeness for 2-to-1 games",
          "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/paper.pdf"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "Quanta calls it an AI-generated manuscript with no human editing or independent expert review. README attributes manuscripts to an unnamed internal model; no reasoning summary is listed for family 105.",
        "url": "https://www.quantamagazine.org/as-ai-closed-in-on-unique-games-proof-researchers-raced-to-beat-the-machines-20261007/",
        "sourceDate": "2026-10-07"
      },
      "lean": {
        "status": "found",
        "text": "Repo lists the main reduction as formalized: lean/ComparatorChallenges/PerfectCompleteness.lean; OAI.PerfectCompleteness.Theorem11.exists_reduction in lean/OAI/Computability/PerfectCompleteness/Theorem11.lean; rational δ in (0,1), exact-two projection tables.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/docs/105.md",
        "sourceDate": "2026-10-06"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No named person found checking the Lean statement against the paper. Fortnow's Claude-prepared note says it did not run the check; formalization.yaml's repo-wide review field says unchecked (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/formalization.yaml",
        "sourceDate": "2026-10-30"
      },
      "experts": [],
      "peerReview": {
        "status": "not_found",
        "text": "None found as of 2026-11-01. Manuscript README cites only an OpenAI release preprint; arXiv API and web searches showed no venue; proceedings not checked individually.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026/README.md",
        "sourceDate": "2026-10-30"
      },
      "revision": {
        "status": "not_found",
        "text": "No revision or withdrawal recorded: paper.pdf and the Lean directory show only the 2026-10-06 initial commit at 4eaee87, and history.md does not list it (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/history.md",
        "sourceDate": "2026-10-30"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "l-rl-bpl",
      "result": "L = RL = BPL",
      "area": "Space-bounded derandomization",
      "direction": "new proof",
      "previous": {
        "status": "found",
        "text": "Hoza (2021) noted deterministic simulation of randomized space S in space O(S^{3/2} / √{log S}), slightly improving Saks–Zhou's O(S^{3/2}). Aaronson (2026) calls L = BPL a great derandomization conjecture.",
        "url": "https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.28",
        "sourceDate": "2021-09-15"
      },
      "claim": {
        "status": "found",
        "text": "[Paper, Theorem 1.1, Exact logarithmic-space derandomization] L = RL = BPL. [Paper, abstract] We prove L = RL = BPL, resolving the derandomization problem for polynomial-time randomized logarithmic space. [CONTENTS.md, family 103] Proves L=RL=BPL, resolving derandomization for bounded-error logarithmic-space computation. [lean/ComparatorChallenges/LogspaceEquality.lean] theorem exact_logarithmic_space_derandomization : L = RL ∧ RL = BPL",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf",
        "sourceDate": "2026-09-23"
      },
      "claimants": [
        {
          "name": "OpenAI",
          "date": "2026-10-06",
          "title": "Exact Derandomization of Logarithmic Space: L = RL = BPL",
          "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "README attributes manuscripts to an unnamed internal OpenAI model, about three hours of ChatGPT Pro compute per result on average; no reasoning summary is listed for family 103.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/README.md",
        "sourceDate": "2026-10-30"
      },
      "lean": {
        "status": "found",
        "text": "Added 2026-10-08 UTC, commit 3014888: lean/ComparatorChallenges/LogspaceEquality.lean; OAI.ExactDerandomization.exact_logarithmic_space_derandomization in lean/OAI/Computability/Logspace/Equality.lean. Class equality only; compiler bounds not stated.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/docs/103.md",
        "sourceDate": "2026-10-08"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No named person found matching the Lean model to the paper or textbook. Fortnow's Claude-prepared note leaves this open; formalization.yaml's repo-wide review field says unchecked (as of 2026-11-01).",
        "url": "https://github.com/fortnow/AI-Papers/blob/67e7b316d6704ceb3eb308cc761867571074743a/L-equals-RL-equals-BPL-expository.pdf",
        "sourceDate": "2026-10-07"
      },
      "experts": [
        {
          "who": "Scott Aaronson (UT Austin)",
          "text": "Lists L = BPL among results he will watch, calling it a major derandomization conjecture whose truth was not seriously doubted; says humans have barely begun digesting these proofs.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169"
        },
        {
          "who": "Omer Reingold, relayed by Scott Aaronson",
          "text": "In a remark Aaronson shared with permission, he jokes that the RL versus L result was when things turned serious for him; a reaction, not an assessment.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169"
        },
        {
          "who": "Lance Fortnow (former founding dean of computing, Illinois Tech)",
          "text": "Lists a full derandomization of randomized log space, without the Lean tag given to two neighbours, and says these proofs have not been fully verified.",
          "date": "2026-10-07",
          "url": "https://blog.computationalcomplexity.org/2026/10/open-no-more.html"
        },
        {
          "who": "Lance Fortnow (former founding dean of computing, Illinois Tech)",
          "text": "Wrote that complexity results are hard to formalize, then updated to say OpenAI had released what he terms a claimed Lean proof of L = RL = BPL.",
          "date": "2026-10-15",
          "url": "https://blog.computationalcomplexity.org/2026/10/complexity-and-lean.html"
        },
        {
          "who": "Kuan Cheng, Shengtang Huang, Ruiyang Wu",
          "text": "Their paper says OpenAI announced BPL = L and argues their L-AC^1 result still offers multiplicative error 2^{-poly log n}; it does not assess OpenAI's proof.",
          "date": "2026-10-19",
          "url": "https://arxiv.org/abs/2610.25623"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "None found as of 2026-11-01. Manuscript README cites only an OpenAI release preprint; arXiv API and web searches showed no venue; proceedings not checked individually.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/README.md",
        "sourceDate": "2026-10-30"
      },
      "revision": {
        "status": "found",
        "text": "Manuscript unchanged since 2026-10-06. Lean added 2026-10-08 (commit 3014888); comparator JSON definition list emptied 2026-10-26 (commit 42907c9). No withdrawal recorded.",
        "url": "https://github.com/openai/math/commit/42907c9e999cc00d6abe7a50bfc1023250edd1b9",
        "sourceDate": "2026-10-26"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "matrix-multiplication",
      "result": "Matrix multiplication: ω ≤ 9/4",
      "area": "Algebraic complexity",
      "direction": "improved bound",
      "previous": {
        "status": "found",
        "text": "Best prior bound found: ω < 2.371177 (Dupont et al., arXiv preprint, 2026-08-17), which it reports as improving the earlier bound of 2.371339.",
        "url": "https://arxiv.org/abs/2608.16884",
        "sourceDate": "2026-08-17"
      },
      "claim": {
        "status": "found",
        "text": "[9/4 paper, Theorem 1.1] For every ε > 0, two n × n complex matrices can be multiplied using O_ε(n^{9/4+ε}) arithmetic operations. In particular, ω ≤ 9/4. [9/4 paper, abstract] We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4. [CONTENTS.md, family 107] Proves ω≤9/4 over ℂ, giving O_ε(n^{9/4+ε}) arithmetic operations for square matrix multiplication. [lean/ComparatorChallenges/MatrixMultiplication.lean] theorem complex_omega_le_nine_quarters : Arithmetic.omega ℂ ≤ (9 : ℝ) / 4 [Staggered paper, Theorem 1.1] Over every fixed field F, the arithmetic exponent of square matrix multiplication satisfies ω_F < 2.371054886006746 < 2.371054887. [Complex Matrix Multiplication Below 2.258, abstract] Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and ω(1, 0.709, 1) < 2.092.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf",
        "sourceDate": "2026-10-02"
      },
      "claimants": [
        {
          "name": "OpenAI",
          "date": "2026-10-06",
          "title": "An Upper Bound of 9/4 for the Matrix Multiplication Exponent",
          "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "README attributes manuscripts to an unnamed internal OpenAI model; Shpilka also describes the result as obtained by an OpenAI internal model. No reasoning summary is listed for family 107.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/README.md",
        "sourceDate": "2026-10-30"
      },
      "lean": {
        "status": "found",
        "text": "Over ℂ only: lean/ComparatorChallenges/MatrixMultiplication.lean, OAI.MatrixMultiplication.complex_omega_le_nine_quarters in lean/OAI/LinearAlgebra/MatrixMultiplication/Main.lean. Every-field target is only ω < 2.371054886006746 (MatrixFields.lean).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/docs/107.md",
        "sourceDate": "2026-10-06"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No named human found certifying the match. Navot's fork reports AI reviews matched the textbook definition exactly for infinite fields; Zuiddam's repo leaves the model question to readers of Model.lean (as of 2026-11-01).",
        "url": "https://github.com/selanavot/matrix-multiplication-all-fields/blob/08481ef22bca7dc9ffba091083b7c1e81e537220/README.md",
        "sourceDate": "2026-10-08"
      },
      "experts": [
        {
          "who": "Steven Strogatz (Cornell)",
          "text": "Singled out the 2.25 exponent as especially striking, put the earlier record near 2.37, and likened the leap to Bob Beamon's long jump.",
          "date": "2026-10-06",
          "url": "https://x.com/stevenstrogatz/status/2107606017706266686"
        },
        {
          "who": "Scott Aaronson (UT Austin)",
          "text": "Highlights an n^{9/4+o(1)}-time matrix multiplication result, noting a rational exponent and a completely different approach from earlier bounds.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169"
        },
        {
          "who": "Lance Fortnow (former founding dean of computing, Illinois Tech)",
          "text": "Lists n^{2.25+ε} time, tagged as formalized in Lean, while cautioning that the release's proofs have not been fully verified.",
          "date": "2026-10-07",
          "url": "https://blog.computationalcomplexity.org/2026/10/open-no-more.html"
        },
        {
          "who": "Bill Gasarch (University of Maryland)",
          "text": "Restates ω(C) ≤ 2.25 and the every-field bound as Lean-formalized, compares the drop to Beamon's jump, and says he has not checked the third-party every-field project.",
          "date": "2026-10-25",
          "url": "https://blog.computationalcomplexity.org/2026/10/when-is-it-plausible-that-result-is.html"
        },
        {
          "who": "Rafael Pass",
          "text": "Says OpenAI claimed the bound via non-constructive spectral arguments; gives an explicit integer-arithmetic algorithm reaching it, with the main ideas his own and Codex used extensively.",
          "date": "2026-10-20",
          "url": "https://arxiv.org/abs/2610.28627"
        },
        {
          "who": "Alimzhan Amanov, Maxim van den Berg, Jeroen Zuiddam (University of Amsterdam)",
          "text": "Expound OpenAI's proof, give a version valid for every field, and prove a barrier: current degeneration ingredients cannot improve these bounds. They disclose LLM use.",
          "date": "2026-10-21",
          "url": "https://arxiv.org/abs/2610.30054"
        },
        {
          "who": "Amir Shpilka (Tel Aviv University)",
          "text": "States OpenAI's manuscript proves ω ≤ 9/4, gives a self-contained explicit version following its ideas, and says he verified all arguments in his exposition, disclosing AI assistance.",
          "date": "2026-10-27",
          "url": "https://eccc.weizmann.ac.il/report/2026/281/"
        },
        {
          "who": "Sela Navot",
          "text": "Repository reports an extension of the bound to arbitrary fields, found with GPT-6 Astra and GPT-6.1 Sol.",
          "date": "2026-10-07",
          "url": "https://github.com/selanavot/matrix-multiplication-all-fields/blob/08481ef22bca7dc9ffba091083b7c1e81e537220/README.md"
        },
        {
          "who": "Houxi Lü",
          "text": "Preprint addresses matrix multiplication in arbitrary characteristic.",
          "date": "2026-10-12",
          "url": "https://arxiv.org/abs/2610.18914"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "None found as of 2026-11-01. The 9/4 manuscript README cites only an OpenAI release preprint; arXiv API and web searches showed no venue; proceedings not checked individually.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/README.md",
        "sourceDate": "2026-10-30"
      },
      "revision": {
        "status": "not_found",
        "text": "No revision or withdrawal recorded: the 9/4, 2.258 and every-field manuscripts show only the 2026-10-06 initial commit at 4eaee87; history.md lists none (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/history.md",
        "sourceDate": "2026-10-30"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "subset-sum",
      "result": "Subset Sum in O(2^0.49n)",
      "area": "Exact algorithms",
      "direction": "improved algorithm",
      "previous": {
        "status": "found",
        "text": "Chen, Jin, Randolph and Servedio give worst-case O(2^{n/2} · n^{-γ}) for a constant γ > 0.5023, a polynomial saving over Horowitz–Sahni O(2^{n/2}); their abstract frames time 2^{(1/2 - c)n} as a major goal.",
        "url": "https://arxiv.org/abs/2301.07134",
        "sourceDate": "2023-01-17"
      },
      "claim": {
        "status": "found",
        "text": "Title and abstract: O(2^{0.49n}). Theorem 1.1: at most C_c 2^{0.49n} word operations for n ≥ N_c and b ≤ n^c, correct with probability at least 2/3. Intermediate bound: 2^{0.489995n} poly(n + b + 2).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/subset-sum.pdf",
        "sourceDate": "2026-10-04"
      },
      "claimants": [
        {
          "name": "OpenAI",
          "date": "2026-10-06",
          "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"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "Repository-wide, not result-specific: an unnamed internal OpenAI model produced the manuscripts; most results followed one procedure, averaging three hours of ChatGPT Pro thinking compute. It calls them raw research data.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/README.md",
        "sourceDate": "2026-10-26"
      },
      "lean": {
        "status": "not_found",
        "text": "No Lean for the 2^{0.49n} result found: family 138 has no listed Lean. Unlisted lean/OAI/Combinatorics/LowSpaceSubsetSum/Main.lean states theorems only for the companion low-space 2^{n/2}-time result (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/OAI/Combinatorics/LowSpaceSubsetSum/Main.lean",
        "sourceDate": "2026-10-06"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No named person found comparing Lean statements with these manuscripts. The main 2^{0.49n} result has no Lean statement to compare (as of 2026-11-01)."
      },
      "experts": [
        {
          "who": "Lance Fortnow",
          "text": "Batch-level remark: Fortnow wrote the release's proofs had not been fully verified and framed his assessment as conditional on them holding up; he does not name this result.",
          "date": "2026-10-07",
          "url": "https://blog.computationalcomplexity.org/2026/10/open-no-more.html"
        },
        {
          "who": "Lance Fortnow",
          "text": "Fortnow wrote the ETH-refuting algorithm adds nothing for Subset Sum, whose best algorithms he described as still taking roughly 2^{n/2}; the post body does not mention the 0.49 manuscript.",
          "date": "2026-10-28",
          "url": "https://blog.computationalcomplexity.org/2026/10/the-exponential-time-refutation.html"
        },
        {
          "who": "Lance Fortnow (blog comment)",
          "text": "Replying to a commenter who pointed to OpenAI's 2^{0.49n} claim, Fortnow called it impressive but not 2^{o(n)}; he does not say he checked it.",
          "date": "2026-10-29",
          "url": "https://blog.computationalcomplexity.org/2026/10/the-exponential-time-refutation.html"
        },
        {
          "who": "Lixi Ye and Baitian Li",
          "text": "Ye and Li note OpenAI released a similar-type result and report their own O(2^{0.499999n}) algorithm, developed independently by another route; they do not assess OpenAI's proof.",
          "date": "2026-10-07",
          "url": "https://arxiv.org/abs/2610.10446"
        },
        {
          "who": "Ce Jin and Xiao Mao",
          "text": "They cite both OpenAI Subset Sum manuscripts among references on exponential-time and average-case work, offering no assessment of their correctness.",
          "date": "2026-10-14",
          "url": "https://arxiv.org/abs/2610.20742"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "No peer-reviewed publication found as of 2026-11-01"
      },
      "revision": {
        "status": "not_found",
        "text": "No revision or withdrawal recorded: only the initial commit touches this folder, the PDF blob matches at adc7f12 and 4eaee87, and history.md does not name it (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/history.md",
        "sourceDate": "2026-10-29"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "polynomial-factorization",
      "result": "Deterministic polynomial factoring",
      "area": "Finite fields",
      "direction": "improved algorithm",
      "previous": {
        "status": "found",
        "text": "Altman (2025) calls deterministic polynomial-time factoring over finite fields a notorious open problem and says no method for arbitrary polynomials over F_p is known even under GRH.",
        "url": "https://arxiv.org/abs/2509.12705",
        "sourceDate": "2025-09-16"
      },
      "claim": {
        "status": "found",
        "text": "Theorem 1.1: prime p in binary, nonzero dense f ∈ F_p[x] of degree n, complete factorization in O(((n + 1)⌈log_2 p⌉)^{10^{12}}) bit operations. Imported from companion, Theorem 11.1: no zero in ℜs > 1 − δ, δ = 10^{−6}.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/Deterministic-Polynomial-Factorization-over-Prime-Fields-October-4-2026/Deterministic-Polynomial-Factorization-over-Prime-Fields.pdf",
        "sourceDate": "2026-10-04"
      },
      "claimants": [
        {
          "name": "OpenAI",
          "date": "2026-10-06",
          "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"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "Repository-wide, not result-specific: an unnamed internal OpenAI model produced the manuscripts; most results followed one procedure, averaging three hours of ChatGPT Pro thinking compute. It calls them raw research data.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/README.md",
        "sourceDate": "2026-10-26"
      },
      "lean": {
        "status": "not_found",
        "text": "No formalization listed: no lean/docs/142.md or 029.md, no CONTENTS Lean link, neither manuscript in lean/formalization.yaml, and no matching Lean file name found (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/formalization.yaml",
        "sourceDate": "2026-10-26"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No Lean statement exists to compare, and no named public check was found (as of 2026-11-01)."
      },
      "experts": [
        {
          "who": "Pierre-Jean Spaenlehauer",
          "text": "Spaenlehauer treats the factoring result as OpenAI's claim, resting on a companion zero-free preprint; he writes the community had not reviewed or validated it and no formalized proof existed.",
          "date": "2026-10-23",
          "url": "https://arxiv.org/abs/2610.32508"
        },
        {
          "who": "Scott Aaronson",
          "text": "Aaronson listed GRH-free deterministic factoring among results he expected to watch most; he reported no check and said it appeared humans had understood almost none of these proofs yet.",
          "date": "2026-10-07",
          "url": "https://scottaaronson.blog/?p=10169"
        },
        {
          "who": "Lance Fortnow",
          "text": "Batch-level remark: Fortnow wrote the release's proofs had not been fully verified and framed his assessment as conditional on them holding up; he does not name this result.",
          "date": "2026-10-07",
          "url": "https://blog.computationalcomplexity.org/2026/10/open-no-more.html"
        },
        {
          "who": "Jizhou Guo",
          "text": "Guo's AI-compiled note derives deterministic irreducible-polynomial construction conditional on family 142 Theorem 1.1 and family 029 Theorem 1.2, adding that neither family lists a Lean formalization; it verifies neither.",
          "date": "2026-10-08",
          "url": "https://github.com/aster2024/openai-math-followups/blob/e84c2d07520b1c3c3d6afdf8a1a85a1b3f1e72d9/consequences/README.md"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "No peer-reviewed publication found as of 2026-11-01"
      },
      "revision": {
        "status": "not_found",
        "text": "No revision or withdrawal recorded: only the initial commit touches this folder or the companion's, the PDF blob matches at adc7f12 and 4eaee87, and history.md does not name it (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/history.md",
        "sourceDate": "2026-10-29"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "ppad-pcp",
      "result": "PCP for PPAD: quasilinear reduction",
      "area": "Algorithmic game theory",
      "direction": "new proof",
      "previous": {
        "status": "found",
        "text": "Conjecture 2 of Babichenko, Papadimitriou and Rubinstein asked that End of the Line reduce quasilinearly to (ε, δ)-Gcircuit; they note the δ = 0 time-hardness form already followed from Conjecture 1.",
        "url": "https://arxiv.org/abs/1504.02411",
        "sourceDate": "2015-04-09"
      },
      "claim": {
        "status": "found",
        "text": "Abstract: output circuit's total binary length is N(log N)^{O(1)}. Theorem 1.3 (Quasilinear PCP for PPAD): fixed rational constants 0 < ε < 1/10, 0 < δ < 1; M ≤ AN⌈log_2(N + 2)⌉^b.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/The-PCP-for-PPAD-conjecture-a-quasilinear-reduction-September-25-2026/paper.pdf",
        "sourceDate": "2026-09-25"
      },
      "claimants": [
        {
          "name": "OpenAI",
          "date": "2026-10-06",
          "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"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "Repository-wide, not result-specific: an unnamed internal OpenAI model produced the manuscripts; most results followed one procedure, averaging three hours of ChatGPT Pro thinking compute. It calls them raw research data.",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/README.md",
        "sourceDate": "2026-10-26"
      },
      "lean": {
        "status": "not_found",
        "text": "No Lean proof of the main result found. lean/OAI/Computability/GeneralizedCircuits/Mesh.lean states QuasilinearPCPStatement as an unproved goal and proves only supporting lemmas, per its header; absent from formalization.yaml (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/lean/OAI/Computability/GeneralizedCircuits/Mesh.lean",
        "sourceDate": "2026-10-06"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No named person found comparing QuasilinearPCPStatement with the manuscript's Theorem 1.3. That statement is only a goal proposition in the repository (as of 2026-11-01)."
      },
      "experts": [
        {
          "who": "Panagiotis Kanellopoulos and Themistoklis Melissourgos",
          "text": "A footnote describes the manuscript as generated with an OpenAI model and claiming the quasilinear version; the authors know of no independent verification so far.",
          "date": "2026-10-19",
          "url": "https://arxiv.org/abs/2610.26772"
        },
        {
          "who": "Ioannis Panageas, Avi Shein, Shayan Taherijam and Vijay V. Vazirani",
          "text": "The authors say they have not independently verified OpenAI's claimed proof and keep their constant-error hardness result conditional on PCP for PPAD.",
          "date": "2026-10-26",
          "url": "https://arxiv.org/abs/2610.35848"
        },
        {
          "who": "Xiaotie Deng, Ningyuan Li, Zihe Wang and Jie Zhang",
          "text": "They say the OpenAI preprint reports a proof of the conjecture, giving no assessment; their own conditional hardness rests on Rubinstein's theorem and PPAD-ETH instead.",
          "date": "2026-10-19",
          "url": "https://arxiv.org/abs/2610.25608"
        },
        {
          "who": "Lance Fortnow",
          "text": "Batch-level remark: Fortnow wrote the release's proofs had not been fully verified and framed his assessment as conditional on them holding up; he does not name this result.",
          "date": "2026-10-07",
          "url": "https://blog.computationalcomplexity.org/2026/10/open-no-more.html"
        }
      ],
      "peerReview": {
        "status": "not_found",
        "text": "No peer-reviewed publication found as of 2026-11-01"
      },
      "revision": {
        "status": "not_found",
        "text": "No revision or withdrawal recorded: only the initial commit touches this folder, the PDF blob matches at adc7f12 and 4eaee87, and history.md does not name it (as of 2026-11-01).",
        "url": "https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/history.md",
        "sourceDate": "2026-10-29"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "max-4-cut",
      "result": "MAX-4-CUT hardness: 0.987",
      "area": "Hardness of approximation",
      "direction": "improved bound",
      "previous": {
        "status": "found",
        "text": "Best prior NP-hardness factor cited by the authors: 85/86 + ε (about 0.9883), from Austrin, O'Donnell, Tan, Wright, Theorem 1.2, stated for k-Coloring with k = 4.",
        "url": "https://arxiv.org/abs/1210.5648"
      },
      "claim": {
        "status": "found",
        "text": "arXiv v7 Theorem 3.2: (0.9349 − ε, 0.9238 + ε)-approximating MAX-4-CUT, hence NP-hard to 0.987 + ε-approximate. Abstract and Google blog: 0.987, improving 0.9883. Lemma C.2: c = 49535, s_1 = 49538, s_2 = 48681, count = 52941.",
        "url": "https://arxiv.org/abs/2509.18057v7",
        "sourceDate": "2026-03-09"
      },
      "claimants": [
        {
          "name": "Nagda, Raghavan, Thakurta",
          "date": "2025-09-22",
          "title": "Reinforced Generation of Combinatorial Structures: Applications to Complexity Theory",
          "url": "https://arxiv.org/abs/2509.18057v1"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "Authors say AlphaEvolve searched for the 19-variable gadget and evolved faster verifiers; they report framing the search, inspecting fast verifiers by hand, and brute-force checking the final gadgets.",
        "url": "https://arxiv.org/abs/2509.18057v7",
        "sourceDate": "2026-03-09"
      },
      "lean": {
        "status": "not_found",
        "text": "No Lean or proof-assistant formalization found. arXiv v7 relies on brute-force checking code; its phrase about formal verification refers to standard computation, not a named proof assistant (as of 2026-11-01).",
        "url": "https://arxiv.org/abs/2509.18057v7",
        "sourceDate": "2026-03-09"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No Lean statement found to compare, and no named public check was located (as of 2026-11-01)."
      },
      "experts": [],
      "peerReview": {
        "status": "not_found",
        "text": "No peer-reviewed publication found as of 2026-11-01"
      },
      "revision": {
        "status": "found",
        "text": "Seven arXiv versions, 2025-09-22 to 2026-03-09; 0.987 unchanged in abstracts. v6 retitled. Uri Feige credited for an earlier Appendix B.2 error, which covers average-case upper bounds, not MAX-4-CUT.",
        "url": "https://arxiv.org/abs/2509.18057",
        "sourceDate": "2026-03-09"
      },
      "lastChecked": "2026-11-01"
    },
    {
      "id": "metric-tsp",
      "result": "Metric TSP hardness: 111/110",
      "area": "Hardness of approximation",
      "direction": "improved bound",
      "previous": {
        "status": "found",
        "text": "Chlebík and Chlebíková improved the TSP hardness value from 123/122 (Karpinski, Lampis, Schmied) to 117/116.",
        "url": "https://ideas.repec.org/a/spr/jcomop/v43y2022i5d10.1007_s10878-020-00659-0.html"
      },
      "claim": {
        "status": "found",
        "text": "arXiv v7 Theorem 4.1: for any ε > 0, NP-hard to approximate metric TSP to within 111/110 − ε. Abstract (v5 to v7): factor 111/110, improving 117/116. Gadget parameters: s(H) = c(H) = 10.",
        "url": "https://arxiv.org/abs/2509.18057v7",
        "sourceDate": "2026-03-09"
      },
      "claimants": [
        {
          "name": "Nagda, Raghavan, Thakurta",
          "date": "2025-11-20",
          "title": "Reinforced Generation of Combinatorial Structures: Applications to Complexity Theory (arXiv v5, first version stating the TSP bound)",
          "url": "https://arxiv.org/abs/2509.18057v5"
        }
      ],
      "aiRole": {
        "status": "found",
        "text": "Authors say AlphaEvolve searched for equation gadgets; this problem needed the most human involvement, with authors modularizing soundness and completeness proofs from earlier work before the search.",
        "url": "https://arxiv.org/abs/2509.18057v7",
        "sourceDate": "2026-03-09"
      },
      "lean": {
        "status": "not_found",
        "text": "No Lean or proof-assistant formalization found. The paper reports computational checking of gadgets and a written reduction proof (as of 2026-11-01).",
        "url": "https://arxiv.org/abs/2509.18057v7",
        "sourceDate": "2026-03-09"
      },
      "leanStatementCheck": {
        "status": "not_found",
        "text": "No Lean statement found to compare, and no named public check was located (as of 2026-11-01)."
      },
      "experts": [],
      "peerReview": {
        "status": "not_found",
        "text": "No peer-reviewed publication found as of 2026-11-01"
      },
      "revision": {
        "status": "found",
        "text": "Bound first appears in arXiv v5 (2025-11-20), absent from v1 to v4 abstracts; abstract value unchanged through v7 (2026-03-09). No retraction found.",
        "url": "https://arxiv.org/abs/2509.18057",
        "sourceDate": "2026-03-09"
      },
      "lastChecked": "2026-11-01"
    }
  ]
}