{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-27-ai-assisted-results-october-27-listing/","md":"https://postcutoff.com/e/2026-10-27-ai-assisted-results-october-27-listing/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-27-ai-assisted-results-october-27-listing","date":"2026-10-27","date_precision":"day","short_title":"AI-assisted results in the 27 Oct 2026 sweep","deck":"Muckenhoupt's C_p conjecture, Graham's rearrangement conjecture in abelian groups, a quasi-polynomial χ-bound for P₆-free graphs, Haagerup's 1975 problem, Langberg–Médard, and proofs whose main idea came from Claude or GPT","takeaway":"Tuesday's arXiv announcement (papers submitted 23–26 Oct) was the largest so far for AI-assisted mathematics. Four results have their own entries: Golomb–Welch, 3-coloring with O(n^0.1537) colors, Balko's exposition of OpenAI's Harary–Hill proof, and Ando–Umehara on Alpöge's smooth Carathéodory counterexamples. This page lists the next tier, from 46-year-old conjectures where ChatGPT supplied an alternative argument to short papers where Claude or GPT proposed the main idea.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.34099: The C_p conjecture of Muckenhoupt (Lerner, Ombrosi)","url":"https://arxiv.org/abs/2610.34099","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2610.34424: Graham's rearrangement conjecture in finite abelian groups (Kuang, Wang)","url":"https://arxiv.org/abs/2610.34424","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"arXiv 2610.33407: A quasi-polynomial χ-bound for P₆-free graphs","url":"https://arxiv.org/abs/2610.33407","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"arXiv 2610.35774: A solution to Haagerup's problem (Blecher, Choi)","url":"https://arxiv.org/abs/2610.35774","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"arXiv 2610.33655: A Proof of the Langberg–Médard Conjecture (Liu, Li, Li)","url":"https://arxiv.org/abs/2610.33655","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"arXiv 2610.33497: Proof of the Langberg–Médard Multiple Unicast Conjecture (Cai, Han, Yang, Zang)","url":"https://arxiv.org/abs/2610.33497","type":"paper","group":"primary","domain":"arxiv.org"},{"n":7,"title":"arXiv 2610.33781: A Borsuk–Ulam theorem for adaptive continuous measurements (Ullrich)","url":"https://arxiv.org/abs/2610.33781","type":"paper","group":"primary","domain":"arxiv.org"},{"n":8,"title":"arXiv 2610.34571: Chain flaring fails on the rose","url":"https://arxiv.org/abs/2610.34571","type":"paper","group":"primary","domain":"arxiv.org"},{"n":9,"title":"arXiv 2610.34607: A proof of the Grothendieck join–meet positivity","url":"https://arxiv.org/abs/2610.34607","type":"paper","group":"primary","domain":"arxiv.org"},{"n":10,"title":"arXiv 2610.35027: On almost perfect additive complements","url":"https://arxiv.org/abs/2610.35027","type":"paper","group":"primary","domain":"arxiv.org"},{"n":11,"title":"arXiv 2610.33779: The geometry of limits in ReLU network function spaces (Lindsey)","url":"https://arxiv.org/abs/2610.33779","type":"paper","group":"primary","domain":"arxiv.org"},{"n":12,"title":"arXiv 2610.34097: A Nearly Optimal Min-Cut Algorithm for Dynamic Streams","url":"https://arxiv.org/abs/2610.34097","type":"paper","group":"primary","domain":"arxiv.org"},{"n":13,"title":"arXiv 2610.34474: Unconditional lower bounds for the Fredman–Khachiyan algorithms","url":"https://arxiv.org/abs/2610.34474","type":"paper","group":"primary","domain":"arxiv.org"},{"n":14,"title":"arXiv 2610.33229: Braided cuckoo hashing (Krapivin, Kuszmaul)","url":"https://arxiv.org/abs/2610.33229","type":"paper","group":"primary","domain":"arxiv.org"},{"n":15,"title":"arXiv 2610.34548: Inductive Grounded Arithmetic: Goodstein, the Hydra, and Γ₀ (Ford)","url":"https://arxiv.org/abs/2610.34548","type":"paper","group":"primary","domain":"arxiv.org"},{"n":16,"title":"arXiv 2610.34580: Bounds for the univalent Bloch–Landau constant (Wikström)","url":"https://arxiv.org/abs/2610.34580","type":"paper","group":"primary","domain":"arxiv.org"},{"n":17,"title":"arXiv 2610.33919: The generalized Dixmier conjecture and the two-dimensional Jacobian conjecture","url":"https://arxiv.org/abs/2610.33919","type":"paper","group":"primary","domain":"arxiv.org"},{"n":18,"title":"arXiv 2610.34979: On Brezis' open problem 2.4","url":"https://arxiv.org/abs/2610.34979","type":"paper","group":"primary","domain":"arxiv.org"},{"n":19,"title":"arXiv 2610.34372: Proof of the Erdős–Nicolas–Sárközy conjecture (no AI statement)","url":"https://arxiv.org/abs/2610.34372","type":"paper","group":"primary","domain":"arxiv.org"},{"n":20,"title":"formal-conjectures commit: Lean proof link for Erdős Problem 1202","url":"https://github.com/google-deepmind/formal-conjectures/commit/c0c538071a21b36d15f70f5998370eb6a4c31c97","type":"code","group":"primary","domain":"github.com"},{"n":21,"title":"Zenodo 23678020: On (3,10,40)-graphs (R(3,10) progress report)","url":"https://zenodo.org/records/23678020","type":"paper","group":"primary","domain":"zenodo.org"},{"n":22,"title":"Zenodo 23687327: Four-state counterexample to the affine Karpelevich conjecture (anonymous)","url":"https://zenodo.org/records/23687327","type":"paper","group":"primary","domain":"zenodo.org"}],"official":22,"filed":"2026-10-27","updated":"2026-10-27","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"],"title":"AI-assisted results in the 27 Oct 2026 sweep: Muckenhoupt's C_p conjecture, Graham's rearrangement conjecture in abelian groups, a quasi-polynomial χ-bound for P₆-free graphs, Haagerup's 1975 problem, Langberg–Médard, and proofs whose main idea came from Claude or GPT","summary":"Catalogue of AI-disclosing papers from the 27 Oct 2026 arXiv announcement (math, cs.CC, cs.DS read from the listing pages because the arXiv API failed) and new commits in result repositories. Highlights: Lerner and Ombrosi prove Muckenhoupt's C_p conjecture (open 46 years), with ChatGPT supplying the deterministic part of the final proof; Kuang and Wang prove Graham's rearrangement conjecture for all large subsets of finite abelian groups with ChatGPT 5.4–6 Pro; Huang, Zhou and Zhou give the first quasi-polynomial χ-bound for P₆-free graphs, developed \"through an interactive dialogue with Claude Opus 5.5 … and ChatGPT-6\"; Blecher and Choi solve Haagerup's 1975 problem on normal weights with proofs \"substantially obtained with the aid of GPT-6.0\".","key_facts":["Own entries: Golomb–Welch conjecture (2610.33983, 2610.34291), O(n^0.1537)-coloring of 3-colorable graphs (2610.35859), OpenAI's Harary–Hill and Zarankiewicz crossing-number proofs with Balko's exposition (2610.34444), Alpöge's smooth Carathéodory/Loewner counterexamples with Ando–Umehara's simplification (2610.34036). Enriched: OpenAI ETH (reactions, Szeider's Zenodo note), local smoothing/Kakeya (Leng), Yang–Mills/QCD, Anthropic ETH (Hexagon), oai-math-revised hub (comparator run 414/416 accepted)","Muckenhoupt's C_p conjecture (Andrei K. Lerner, Sheldy Ombrosi; 2610.34099): C_p is necessary and sufficient for the weighted L^p(w) inequality between the Hilbert transform and the maximal function, and for the weighted Fefferman–Stein inequality; 'This conjecture has remained open for 46 years.' AI statement (verbatim): 'The first version of this paper was written jointly with Fedor Nazarov … That version was prepared without the use of AI and included a complete proof of the main results … For the present version, ChatGPT was used to develop an alternative treatment of the remaining sum, replacing the Bellman-type and probabilistic arguments with the deterministic arguments presented in Sections 4–7.'","Graham's rearrangement conjecture in finite abelian groups (Peiru Kuang, Yan Wang, Shanghai Jiao Tong University; 2610.34424): every S ⊆ G∖{0} with |S| ≥ n₀ has an ordering with distinct partial sums, also settling Alspach–Liversidge for large sets (the prime cyclic case was settled for large primes by Pham–Sauermann and earlier work). 'We used ChatGPT 5.4, 5.5, 5.6 and 6 Pro for mathematical discussion, numerical optimization, assistance with drafting and checking proofs'; for Sections 7–8, 'discussions with ChatGPT 5.6 and 6 Pro helped identify adaptations of existing methods and develop preliminary proof drafts', contributing to Lemmas 7.2, 8.3, 8.5, 8.6","Quasi-polynomial χ-bound for P₆-free graphs (Shenwei Huang, Yidong Zhou, Yuwen Zhou; Nankai, UBC; 2610.33407): χ(G) ≤ ω(G)^(c·log₂ω(G)) with c < 109, the first quasi-polynomial bound (previously exponential), generalizing Scott–Seymour–Spirkl's P₅ result. 'The proof presented in this paper was developed through an interactive dialogue with Claude Opus 5.5 (Anthropic) and ChatGPT-6 (OpenAI) … the AI generated the initial draft under the authors' guidance.'","Haagerup's second problem (1975) on normal weights (David P. Blecher, Ikhan Choi; 2610.35774): a weight on a C*-algebra is lower semicontinuous if it is so on commutative C*-subalgebras, hence normality of weights on von Neumann algebras is commutatively determined. 'The proofs were substantially obtained with the aid of GPT-6.0, and checked and rewritten by the authors.'","Langberg–Médard multiple-unicast conjecture (2009), two independent proofs on 24 Oct: Sirui Liu, Zongpeng Li, Baochun Li (Tsinghua, Toronto; 2610.33655): 'The authors used OpenAI's ChatGPT and Codex to explore proof strategies, generate initial drafts of the mathematical proofs in Sections V–VII and Appendix A … Google's Gemini was used primarily to check mathematical arguments'; Kai Cai, Guangyue Han, Mengxi Yang, Wenan Zang (HKU, USTC; 2610.33497): no AI statement found","Main idea credited to a model: Borsuk–Ulam theorem for adaptive continuous measurements (Mario Ullrich, JKU Linz; 2610.33781): 'The main idea of the proof was found by Claude Fable 5', with Section 5 documenting the dialogue; chain flaring fails on the rose (Jorge Andres Robinson Arrieta; 2610.34571): 'Claude proposed the main argument for the proof of Theorem 1.3'; Grothendieck join–meet positivity, a Thomas–Yong conjecture (Guo, Kang, Liu, Xiong; 2610.34607): ChatGPT 'in late September 2026, provided the main strategy for the proof of Theorem A'; Chen's conjecture on almost perfect additive complements (van Doorn, Liu, Tang; 2610.35027): 'ChatGPT 5.6 Pro identified the applicability of the result of Gabardo and Lai [9] and generated an initial proof'","Counterexamples found by models: ReLU network function spaces (Kathryn A. Lindsey; 2610.33779): 'the counterexample in Section 6 was found by Claude Opus 5.5, in a search directed by the author … an extended search lasting over an hour'; Loewner/Carathéodory (own entry); flag-algebra SDPs (2610.33850): 'All three constructions were discovered by ChatGPT-Astra'","Algorithms: nearly optimal Õ(n/ε)-space min-cut for dynamic streams (Yang P. Liu, Hoai-An Nguyen, Vihan Shah, David P. Woodruff; 2610.34097): 'while discussing several candidate hard instances and attempting to find counterexamples to them, the authors, in interaction with Sol, identified the bucketing idea underlying the algorithm'; Fredman–Khachiyan duality-testing lower bounds (2610.34474): ChatGPT 'utilized iteratively to push the initial ideas and help in completing the mathematical constructions and proofs'; braided cuckoo hashing (Krapivin, Kuszmaul; 2610.33229) disproves a Bender–Kuszmaul–Zhou conjecture, with Claude used only for grammar","Lean: Goodstein, the Hydra and Γ₀ in grounded arithmetic (Bryan Ford, EPFL; 2610.34548): machine-checked proofs and first drafts 'produced with substantial assistance from Claude models'; univalent Bloch–Landau constant bounds (Frank Wikström; 2610.34580): 'The Lean certificate for the analytic bound was generated using Opus 5.5'; nonintersecting unit diagonals (2610.33042): Codex helped with proof, manuscript and Lean; p-adic false Tate curve (2610.35006): Lean by Peking University's Archon-Horizon agent on GPT-6.1 Sol, refactored by Claude Fable 5.1; weighted minimum 3-cut (2610.34851): Claude wrote verification code and Lean","Smaller roles: generalized Dixmier conjecture vs. the 2D Jacobian conjecture (Kanel-Belov, Fedorov; 2610.33919): JC₂ ⇒ DC′₁, partial converse, counterexamples to a Kontsevich conjecture for n ≥ 3; ChatGPT 'to assist with developing and drafting mathematical arguments'; Brezis' Open Problem 2.4 answered negatively (Chen, Liu, Wei, Yang; 2610.34979; ChatGPT for brainstorming and literature); complete coverings of Lee spheres (2610.34805): ChatGPT 5.6 Pro improved upper bounds in Section IV-C, constructions human-made; Robin eigenvalue (2610.34320): 'This idea of distance-dependent angular phase was suggested by ChatGPT 6 Astra'; Volume of PGL₃-Hitchin moduli (2610.33417): GPT-6 Astra worked out technical details; Morse index of the Veronese plane (2610.35820): GPT-6 Astra 'suggested the approach'; Polynomial residual networks (2610.33738): one proposition 'due to Claude Fable 5'","Checked, no AI statement found in the PDF: Erdős–Nicolas–Sárközy conjecture on admissible sets, log a(N) ≍ √N (Guoqing Wang, Wenkai Yang; 2610.34372); Langberg–Médard (Cai–Han–Yang–Zang). The Carmichael totient note (2610.34808) cites a GPT-5.5 Pro solution of Erdős problem #694 on erdosproblems.com as 'a crowd-maintained website claim rather than a peer-reviewed publication'","Scale: of 859 uncited new and cross-listed papers in today's math, cs.CC and cs.DS listings, about 380 contain a sentence naming an AI model or AI tools near a disclosure verb. That is a loose count: it includes statements like 'without the use of AI' and cross-listed ML papers","Repositories: google-deepmind/formal-conjectures linked a Lean proof for Erdős Problem 1202 (#7638, 27 Oct). The commit says it 'reproduces the existing Price/GPT-5.4 Pro counterexample formalized by Codex/GPT-5.6 Sol in plby/lean-proofs; no new mathematical solution is claimed', and that Codex helped with the audit, the bridge and the PR. Further 25–26 Oct commits add Erdős problems 284, 577, 782, 805, 823, 854, 856, 934, 1114, 1162, 1189 and link proofs for solved variants of 6, 33, 58, 627, 649, 698, 918. anthropics/formal-math added the ETH paper (own entry)","Zenodo (unrefereed): Stefan Szeider's ETH threshold note (in the ETH entry); Zhiyang Chen's R(3,10) progress report (a (3,10,40)-graph would make R(3,10) = 41; partial results checked in Lean, SAT certificates; 'R(3,10) remains open'); anonymous 'unrefereed research candidate' packages refuting Conjecture 2 of Ganesh–Hauert–Valla on noisy voter models and the affine Karpelevich conjecture of Xu–Kolchinsky–Delvenne–Ito (four-state counterexample), with AI-assisted internal review. Skipped as unsupported: the 'inconsistency of ZFC and PA' and E₈×E₈ neutrino records"],"key_numbers":[],"tags":["math","computer-science","arxiv","zenodo","catalogue","chatgpt","gpt-6-astra","gpt-6-pro","gpt-6-1-sol","claude","claude-opus-5-5","claude-fable-5","codex","gemini","lean"],"science":{"field":"mathematics","subfield":"multiple (harmonic analysis, additive combinatorics, structural graph theory, operator algebras, network coding, algorithms, logic)","problem":"See list","result":"Catalogue of AI-assisted results in the 27 Oct 2026 arXiv announcement and result repositories, beyond those with their own entries","open_since":"","ai_system":["ChatGPT","GPT-6 Astra","GPT-6 Pro","GPT-6.0","ChatGPT 5.6 Sol","GPT-6.1 Sol","Codex","Claude Opus 5.5","Claude Fable 5","Claude Fable 5.1","Gemini"],"human_role":"From model-found main ideas, counterexamples and Lean proofs to AI checking and editing","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed arXiv and Zenodo preprints; Lean formalizations where stated"},"status":"pending","shock":"","verified_by":["preprint","lean"],"kind":"batch"},"body_md":"## What happened\n\nThe arXiv API failed after its first query on 27 Oct, so Tuesday's announcement (submissions of 23–26 Oct) was read from the\nlisting pages for math, cs.CC and cs.DS. All 859 papers not yet cited here were downloaded and scanned with pdftotext. The quotes\nabove come from the PDFs. The four strongest results have their own entries. The day's repository commits were mostly\nbookkeeping: Lean proof links in DeepMind's formal-conjectures, and the community comparator run on OpenAI's Lean challenges.\n\n## Why it matters\n\nThe share of maths papers with an AI statement keeps rising. The roles now include famous problems: in Muckenhoupt's C_p\nconjecture ChatGPT replaced part of an already complete human proof, and in Haagerup's problem the proofs were \"substantially\nobtained\" with GPT-6.0. In several papers a model proposed the main idea or found the counterexample.","disputed":[{"label":"Verification","text":"Unrefereed arXiv and Zenodo preprints; Lean formalizations where stated","sources":[]}],"related":[{"id":"2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential","url":"https://postcutoff.com/e/2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential/","date":"2026-10-27","date_precision":"day","short_title":"Anthropic's Claude also refutes ETH","deck":null,"takeaway":"Hours after OpenAI's ETH announcement, Anthropic posted 'A 2^o(n) Time Algorithm for 3-SAT', credited to Claude. It is a deterministic #k-SAT algorithm built on tensor-network contraction, with a different approach from OpenAI's paper, a near-identical bound and a Lean 4 proof that uses only the standard axioms. The paper says the core algorithm was found on 16 Oct with 14M output tokens and no human input. OpenAI reports its model found its disproof that same day. Neither proof has had expert review.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":6,"filed":"2026-10-27","updated":"2026-10-29","orgs":["Anthropic"]},{"id":"2026-10-26-openai-eth-false-subexponential-sat-claim","url":"https://postcutoff.com/e/2026-10-26-openai-eth-false-subexponential-sat-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims ETH is false","deck":null,"takeaway":"Anthropic independently posted a different Lean-checked refutation the next day (separate entry). OpenAI's model output claims a deterministic algorithm that counts the satisfying assignments of any k-CNF in time 2^(n/(log n)^(1/2+o(1))). If correct, it refutes ETH, SETH, Gap-ETH and NSETH, the assumptions behind hundreds of fine-grained hardness results. OpenAI says the main theorem is formalized in Lean. A working group led by ETH's proposers, Russell Impagliazzo and Ramamohan Paturi, with Ryan Williams, will examine the proof. No expert has assessed it in public yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":32,"official":10,"filed":"2026-10-26","updated":"2026-10-30","orgs":["OpenAI"]},{"id":"2026-10-26-three-coloring-n-0-1537-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-26-three-coloring-n-0-1537-gpt-6-astra/","date":"2026-10-26","date_precision":"day","short_title":"3-colorable graphs: O(n^0.1537) colors with GPT-6 Astra","deck":null,"takeaway":"Ken-ichi Kawarabayashi, Kaisei Deguchi, Hirotaka Yoneda and Masataka Yoneda (NII / University of Tokyo) give a polynomial-time algorithm that colors any 3-colorable graph with O(n^0.1537) colors. The previous best was O(n^0.1954) (Bansal–Huang–Lee, SODA '27). By the paper's account, GPT-6 Astra and GPT-6 Pro generated each intermediate algorithm (n^0.1736, n^0.1712, n^(1/6+ε), n^0.16, then n^0.1537), with the authors suggesting ideas and simplifying the analyses. The humans wrote the text.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-27","updated":"2026-10-27","orgs":["National Institute of Informatics","University of Tokyo","OpenAI"]},{"id":"2026-10-26-ai-assisted-results-october-26-listing","url":"https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/","date":"2026-10-26","date_precision":"day","short_title":"AI-assisted results in the 26 Oct 2026 sweep","deck":"Claude-written Lean proof of a frame-indifferent rank-one convex function that is not quasiconvex, a sculptor and Claude Code prove Takemura's conjecture, Claude's contribution to deterministic root-finding, and Zenodo records","takeaway":"Monday's arXiv announcement (papers submitted 22–23 Oct) brought four results with their own entries: OpenAI's propeller conjecture reproved by Heilman, Tokareva's conjecture disproved by MiniMax agents, the triangle lower tail with GPT-6 Astra and Claude, and graph-connectivity communication complexity with ChatGPT. This page lists the next tier, from Lean proofs written entirely by Claude to papers where the AI only checked the text.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Anthropic","OpenAI","various researchers"]},{"id":"2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt","url":"https://postcutoff.com/e/2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt/","date":"2026-10-25","date_precision":"day","short_title":"Golomb–Welch conjecture (perfect Lee codes) claimed, with ChatGPT","deck":null,"takeaway":"Bocong Chen, Jing Huang and Gaojun Luo posted two arXiv papers on 25 Oct 2026. They claim that Lee balls of radius r ≥ 2 never tile ℤⁿ for n ≥ 3, first for lattice tilings and then for arbitrary centre sets. That is the full Golomb–Welch conjecture from around 1970, the central open problem on perfect Lee codes. Both papers say ChatGPT helped explore proof strategies and check calculations. They are unrefereed, and no expert has commented yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-27","updated":"2026-10-27","orgs":["South China University of Technology","Guangzhou University","Nanjing University of Aeronautics and Astronautics","OpenAI"]},{"id":"2026-10-24-oai-math-revised-community-hub","url":"https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/","date":"2026-10-24","date_precision":"day","short_title":"oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)","deck":null,"takeaway":"This is the first organized attempt by research mathematicians to digest and credit the OpenAI catalogue.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":6,"official":5,"filed":"2026-10-25","updated":"2026-10-27","orgs":["OpenAI","mathcommunity"]},{"id":"2026-10-06-harary-hill-zarankiewicz-crossing-numbers-openai","url":"https://postcutoff.com/e/2026-10-06-harary-hill-zarankiewicz-crossing-numbers-openai/","date":"2026-10-06","date_precision":"day","short_title":"Harary–Hill and Zarankiewicz crossing-number conjectures (OpenAI, Lean)","deck":null,"takeaway":"Two short manuscripts credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claim the exact crossing numbers of complete graphs and complete bipartite graphs. Those are the Harary–Hill conjecture (late 1950s) and Zarankiewicz's 1954 formula for Turán's 1944 brick-factory problem. Lean proves both formulas, and a community comparator run accepted both Lean challenges. On 25 Oct the graph-drawing specialist Martin Balko posted a simplified exposition of the Harary–Hill proof, which he says is based on the polynomial method. Gelasio Salazar is preparing another.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-27","updated":"2026-10-28","orgs":["OpenAI","Charles University"]},{"id":"2026-08-19-caratheodory-loewner-smooth-counterexamples-alpoge-claude","url":"https://postcutoff.com/e/2026-08-19-caratheodory-loewner-smooth-counterexamples-alpoge-claude/","date":"2026-08-19","date_precision":"day","short_title":"Smooth Carathéodory and Loewner conjectures disproved with Claude","deck":null,"takeaway":"On 19 Aug 2026 Levent Alpöge (Anthropic) posted an explicit formula on X: a smooth strictly convex closed surface in ℝ³ with exactly one umbilic point, against Carathéodory's conjecture that every such surface has at least two, and umbilics of index 1 + k/2, against Loewner's index ≤ 1. He credited John-Paul Smith 'and of course claude'. On 25 Oct the umbilic specialists Naoya Ando and Masaaki Umehara posted a simpler, self-contained construction on arXiv, found with GPT-6.1's help. The analytic case, the classical form of both conjectures, is not affected.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-27","updated":"2026-10-27","orgs":["Anthropic","Kumamoto University","Institute of Science Tokyo","OpenAI"]}],"people":[{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-27","type":"filed","text":"Created (science agent; arXiv listing pages and PDFs; formal-conjectures commit read through the GitHub API; Zenodo API)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-27","run":null,"sources_read":"Science agent; arXiv listing pages and PDFs; formal-conjectures commit read through the GitHub API; Zenodo API","updated":"2026-10-27","human_review":null,"version":{"date":"2026-10-27"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":180,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":119,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":210,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":149,"in_training_data":false}],"short_url":null}