{"schema":"postcutoff/event@1","as_of":"2026-10-21T14:07:00+02:00","url":"https://postcutoff.com/e/2026-10-21-ai-assisted-results-october-21-listing/","md":"https://postcutoff.com/e/2026-10-21-ai-assisted-results-october-21-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":"2026-10-21-ai-assisted-results-october-21-listing","date":"2026-10-21","date_precision":"day","short_title":"AI-assisted results in the 21 Oct 2026 arXiv listing","deck":"A ChatGPT 6.1 Sol counterexample to Frankl–Ota–Tokushige, klt Fano proofs from ChatGPT, Vince's conjecture with GPT Astra, Pólya for spherical shells, a Claude-built 55-addition 3×3 scheme and a Claude Opus 5.5 lemma in reverse mathematics","takeaway":"Wednesday's arXiv announcement brought four results with their own entries: the MDS conjecture in even characteristic, the double-cap conjecture, the quasitrace/monotone completeness counterexamples and the Neumann–Zagier conjecture. An expert verification of Claude's Hadamard matrices was added to that entry. This page lists the next tier, about thirty papers.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.28029: Intersecting families with large covering number (Patkós)","url":"https://arxiv.org/abs/2610.28029","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2610.27676: Nef complexity of klt Fano varieties (Watanabe)","url":"https://arxiv.org/abs/2610.27676","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"arXiv 2610.27553: A proof of Vince's conjecture (Chen, Ma)","url":"https://arxiv.org/abs/2610.27553","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"arXiv 2610.27707: Pólya's conjecture for spherical shells and Aharonov–Bohm annuli (Li)","url":"https://arxiv.org/abs/2610.27707","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"arXiv 2610.28193: The concentration conjecture for Dirac–Coulomb operators (Ammari, Qiu)","url":"https://arxiv.org/abs/2610.28193","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"arXiv 2610.28395: Sprinkling for random regular graphs and Kim–Vu's sandwich conjecture (Portier)","url":"https://arxiv.org/abs/2610.28395","type":"paper","group":"primary","domain":"arxiv.org"},{"n":7,"title":"arXiv 2610.27643: The first-order part of the Ordered Variable Word theorem (Patey)","url":"https://arxiv.org/abs/2610.27643","type":"paper","group":"primary","domain":"arxiv.org"},{"n":8,"title":"arXiv 2610.27451: A 55-addition rank-23 scheme for 3×3 matrix multiplication (Sidebottom)","url":"https://arxiv.org/abs/2610.27451","type":"paper","group":"primary","domain":"arxiv.org"},{"n":9,"title":"arXiv 2610.27294: Counterexamples to the total-current Hohenberg–Kohn conjecture","url":"https://arxiv.org/abs/2610.27294","type":"paper","group":"primary","domain":"arxiv.org"},{"n":10,"title":"arXiv 2610.27243: On counterexamples to the unification of free entropy (Jekel)","url":"https://arxiv.org/abs/2610.27243","type":"paper","group":"primary","domain":"arxiv.org"},{"n":11,"title":"arXiv 2610.28390: A note on the cube inequality (Bi)","url":"https://arxiv.org/abs/2610.28390","type":"paper","group":"primary","domain":"arxiv.org"},{"n":12,"title":"arXiv 2610.28478: A single generation theorem for simple unital C*-algebras (Ma)","url":"https://arxiv.org/abs/2610.28478","type":"paper","group":"primary","domain":"arxiv.org"},{"n":13,"title":"arXiv 2610.27482: Rational polynomial solvability and stratifiability (Cadilhac)","url":"https://arxiv.org/abs/2610.27482","type":"paper","group":"primary","domain":"arxiv.org"},{"n":14,"title":"arXiv 2610.28536: Binding number of K_k-free graphs and the Brandt conjecture","url":"https://arxiv.org/abs/2610.28536","type":"paper","group":"primary","domain":"arxiv.org"},{"n":15,"title":"arXiv 2610.28079: Irrationality exponents of logarithmic numbers (Chudnovsky)","url":"https://arxiv.org/abs/2610.28079","type":"paper","group":"primary","domain":"arxiv.org"},{"n":16,"title":"arXiv 2610.27799: New Directions in Irrationality (Technion)","url":"https://arxiv.org/abs/2610.27799","type":"paper","group":"primary","domain":"arxiv.org"},{"n":17,"title":"arXiv 2610.27046: ReuF2F benchmark","url":"https://arxiv.org/abs/2610.27046","type":"paper","group":"primary","domain":"arxiv.org"},{"n":18,"title":"arXiv 2610.28124: A proof of the Janson–Sós conjecture","url":"https://arxiv.org/abs/2610.28124","type":"paper","group":"primary","domain":"arxiv.org"}],"official":18,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"],"title":"AI-assisted results in the 21 Oct 2026 arXiv listing: a ChatGPT 6.1 Sol counterexample to Frankl–Ota–Tokushige, klt Fano proofs from ChatGPT, Vince's conjecture with GPT Astra, Pólya for spherical shells, a Claude-built 55-addition 3×3 scheme and a Claude Opus 5.5 lemma in reverse mathematics","summary":"Catalogue of AI-disclosing papers in the 21 Oct 2026 arXiv announcement (submitted 19–20 Oct). Own entries: MDS conjecture in even characteristic (Chen–Huang with ChatGPT; Kumar's Lean proof with Claude), double-cap conjecture (Lean proof built with AI tools), counterexamples to the quasitrace and monotone completeness conjectures (from an unreleased OpenAI model's manuscript), and the Neumann–Zagier conjecture (first proved with ChatGPT 5.5 Pro and Opus 4.8). Enriched: Hadamard matrices below 2000 (independent verification, 16 new orders with Claude). The rest is listed here.","key_facts":["Own entries: MDS even characteristic (2610.28208), double cap (2610.27523), quasitrace/monotone completeness (2610.27184), Neumann–Zagier (2610.27095). Enriched: Hadamard below 2000 (2610.27175)","Frankl–Ota–Tokushige conjecture b(t) = 2t refuted (Balázs Patkós; 2610.28029): 'The counterexample was found by ChatGPT 6.1Sol, the lower bound proof is due to the human author'; 'ChatGPT 6.1Sol produced a lower bound with a better leading' term as well, and the author improved the multiplicative constant","Nef complexity of klt Fano varieties (Kiwamu Watanabe; 2610.27676): no value of c_X strictly between 0 and 1, full classification for c_X < 2 and the full spectrum. 'Most proofs in this paper were developed from arguments generated by OpenAI's ChatGPT through discussions initiated and guided by the author'; 'many of the central technical ideas were also proposed by the model' (GPT-5.6 Sol, later GPT-6 Sol for checking)","Vince's conjecture on the average order of connected induced subgraphs proved (Jungang Chen, Hongxia Ma; 2610.27553): 'The Danus multi-agent workflow with GPT Astra was used to assist in developing some of the proofs. GPT Sol 6.1 was used to revise and polish the language'","Pólya's conjecture for Dirichlet eigenvalues of spherical shells and Aharonov–Bohm annuli (Yutian Li; 2610.27707): 'the author used ChatGPT (OpenAI) to assist with mathematical exploration and proof development, checking intermediate arguments and calculations, literature searches, and drafting'","Concentration conjecture for 3D Dirac–Coulomb operators (Habib Ammari, Jiayu Qiu; 2610.28193): 'They used ChatGPT 6 Astra to assist in working out details, particularly the small-separation analysis in Sections 5–7. The authors then substantially simplified the arguments and rewrote the manuscript.'","Kim–Vu sandwich conjecture via sprinkling for random regular graphs (Julien Portier; 2610.28395): final proofs are the author's, but 'the author first produced a completely different proof of Theorem 1.2 which heavily relied on ChatGPT assistance', with ChatGPT proving d_TV(µ_d, ν_d) = O(d^(−1/2))","Reverse mathematics (Ludovic Patey; 2610.27643), first-order part of the Ordered Variable Word theorem: 'Claude Opus 5.5 (Anthropic) was used to prove the Π⁰₁ extension lemma (Lemma 3.13) which is a core lemma in the proof. The proof was completely rewritten by the author'","3×3 matrix multiplication with 23 products and 55 additions over any ring, one below the record of 56, correctness in Lean 4 (Greg Sidebottom; 2610.27451): 'extended interactive collaboration with Claude (Anthropic; the Fable 5 and Opus 4.8 models), which implemented the repository's search tooling, exact minimizers, the Lean formalization, and much of this text under the author's direction and review'","Hohenberg–Kohn conjecture for total currents refuted (Laestadius, Csirik; 2610.27294): 'OpenAI Codex and Anthropic's Claude were used … The proofs in this manuscript were developed using these tools. Lean 4 was also used to check formalized versions of the main results', with stated background results as assumptions","Follow-ups to OpenAI's 6 Oct maths release: free entropy χ vs χ* (David Jekel; 2610.27243) presents 'a similar example' to the one 'OpenAI recently generated', using no AI himself apart from a Claude-generated assessment of missing attributions shared by Jonathan Shi; weighted cube inequality in all dimensions (Yuchen Bi; 2610.28390) adapts 'No. 336 of OpenAI's October 6, 2026 mathematics release' after 'several discussions with ChatGPT (GPT-6 Astra)'; single generation theorem for simple unital C*-algebras (Xin Ma; 2610.28478) assumes 'the finite factor generation theorem obtained by OpenAI' (GPT 6 Astra for references and checking)","Other substantive AI roles: rational polynomial solvability reduces to stratifiability of semilinear sets (Michaël Cadilhac; 2610.27482: 'The ideas and writing in this paper were developed in conjunction with ChatGPT-6 Astra'); Brandt conjecture and sharp binding number of K_k-free graphs (Jifu Lin, Heng Yang; 2610.28536: ChatGPT 'exploring proof strategies'); bipanchromatic colourings, counterexamples and bounds (Novaković, Jojić; 2610.27133: 'ChatGPT was used to assist with proof development'); measures of maximal entropy for meromorphic maps (Muhan Luo, Qi Zhou; 2610.28617: follows [LZ26], 'whose main idea was suggested by GPT-6 Astra'); optimal global well-posedness for semilinear heat equations (Diebou, Phuc; 2610.27365: 'ChatGPT Plus GPT-5.6 Sol was used to show the continuity in time of T(u) in the proof Theorem 4.11')","Irrationality: Gregory Chudnovsky's degree-dependent irrationality exponents of logarithmic numbers (2610.28079), 'Developed with OpenAI ChatGPT, including mathematical exposition and formalization assistance', Lean statement review through 'Checkpoint136' (see the lemniscate entry); Technion's 'New Directions in Irrationality' (Kaminer group; 2610.27799) says its proofs were found 'independently of LLM-generated mathematical arguments' and cites Lai's LLM-generated proof as 'a direct competition'","Minor roles (checking, editing, figures): Janson–Sós conjecture on balanced subgraph counts proved (Wang, Wang, Yue; 2610.28124: LLM for 'routine calculations and checks'); sharp L^p-Poincaré gradient stability (2610.27281); periodic Anderson–Bernoulli model with infinitely many gaps (Fillman, Wood; 2610.27160: theorem proved without AI, AI suggested simplifications); vanishing uniform Turán densities (Lin, Ma, Zhou; 2610.28574: polishing only)","Benchmark: ReuF2F (2610.27046), 244 formalized undergraduate research problems for coding agents; the best, Gemini 4 Argon at high effort, reaches pass@1 42.42% and pass@2 44.67%, solving 109 of 244 in at least one of two attempts","Scanned, no AI statement or 'no LLM tools': Gersho's conjecture, local BCC crystallization in 3D quantization (Aydın, Iacobelli; 2610.28547: 'No LLM tools have been used'); fractional Michael–Simon counterexample (2610.27973); Kohayakawa–Prömel–Rödl conjecture (2610.27478); Tóth–Varolin counterexample (2610.27660); Hamel–Nadirashvili for Fisher-KPP (2610.27896); Merino–Welsh matroid counterexamples (2610.27706); Alperin weight conjecture elements (2610.27988); Ho–Nguyen conjecture for Gaussian mixtures (2610.27265)"],"key_numbers":[],"tags":["math","physics","computer-science","arxiv","catalogue","chatgpt","gpt-6-astra","gpt-6-1-sol","gpt-5-6-sol","codex","claude","claude-opus-5-5","lean","counterexample"],"science":{"field":"mathematics","subfield":"multiple (combinatorics, algebraic geometry, spectral theory, operator algebras, logic, algorithms)","problem":"Frankl–Ota–Tokushige; nef complexity of klt Fanos; Vince's conjecture; Pólya for shells; Dirac–Coulomb concentration; Kim–Vu sandwich; 3×3 matrix multiplication additions; see list","result":"Catalogue of AI-assisted results in the 21 Oct 2026 arXiv announcement beyond those with their own entries","open_since":"","ai_system":["ChatGPT 6.1 Sol","GPT-6 Astra","GPT-5.6 Sol","ChatGPT","Codex","Claude Opus 5.5","Claude Opus 4.8","Claude Fable 5","Gemini 4 Argon"],"human_role":"From AI-found counterexamples and proofs (Frankl–Ota–Tokushige, klt Fano, Patey's lemma) to human-led papers with AI checking or editing","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed arXiv preprints; Lean for the 55-addition scheme, Hohenberg–Kohn (with assumptions) and Chudnovsky's statements"},"status":"pending","shock":"","verified_by":["preprint","lean"],"kind":"batch"},"body_md":"## What happened\n\nWednesday's arXiv announcement (21 Oct; submissions of 19–20 Oct) had about 40 new papers that disclose or mention AI use. We read\nthe full text of the uncited mathematics papers among them and skipped generic ML papers. Four results got their own entries:\n- the MDS conjecture in even characteristic, with two independent proofs, one Lean-checked with Claude's help;\n- Witsenhausen's double-cap problem, with a separate Lean proof built with AI tools;\n- counterexamples to the quasitrace and monotone completeness conjectures, from an unreleased OpenAI model's manuscript;\n- the Neumann–Zagier conjecture, first proved with ChatGPT 5.5 Pro and Opus 4.8.\n\nA four-author survey also independently verified the twelve Hadamard matrices below 2000 and, with Claude, built sixteen more.\n\nIn the papers listed here, AI plays roles of three kinds:\n- **The AI found the key object or proof**: Patkós's counterexample (ChatGPT 6.1 Sol), Watanabe's klt Fano classification\n  (\"most proofs … developed from arguments generated by OpenAI's ChatGPT\"), and Patey's core lemma (Claude Opus 5.5).\n- **AI developed parts of the argument**: Vince's conjecture (GPT Astra workflow), Pólya for shells, Dirac–Coulomb concentration,\n  Hohenberg–Kohn and Cadilhac.\n- **Follow-ups to OpenAI's 6 Oct release**: free entropy, the cube inequality, and the C*-algebra generation theorem.\n\nThe previous weekday's leftovers (13–17 Oct ML and physics papers that mention LLMs) are out of scope here.\n\nNo Hacker News thread or high-reach X post on these papers was found on 21 Oct.\n\n## Why it matters\n\nTwo classical problems (Segre's MDS conjecture in even characteristic, Witsenhausen's double cap) arrived with Lean proofs built\nby AI agents on the day the preprints appeared. Mathematicians also keep building directly on OpenAI's model-generated\nmanuscripts, including one that has not been published.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprints; Lean for the 55-addition scheme, Hohenberg–Kohn (with assumptions) and Chudnovsky's statements","sources":[]}],"related":[{"id":"2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean","url":"https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/","date":"2026-10-20","date_precision":"day","short_title":"MDS conjecture settled in even characteristic","deck":"Two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT","takeaway":"If it holds, this is the biggest advance on the conjecture since Ball's prime-field theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-21","updated":"2026-10-21","orgs":["South China University of Technology","Guangzhou University","OpenAI","Anthropic"]},{"id":"2026-10-20-double-cap-conjecture-witsenhausen-lean-ai","url":"https://postcutoff.com/e/2026-10-20-double-cap-conjecture-witsenhausen-lean-ai/","date":"2026-10-20","date_precision":"day","short_title":"Double-cap conjecture claimed in all dimensions, with a separate AI-built Lean proof","deck":null,"takeaway":"If confirmed, a 52-year-old geometry problem falls in every dimension, and a machine-checked proof of the statement already exists.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-21","updated":"2026-10-21","orgs":["Universidade Federal do Ceará","Universidade de São Paulo","OpenAI"]},{"id":"2026-10-20-ai-assisted-results-october-20-listing","url":"https://postcutoff.com/e/2026-10-20-ai-assisted-results-october-20-listing/","date":"2026-10-20","date_precision":"day","short_title":"AI-assisted results in the 20 Oct 2026 arXiv listing","deck":"Poset dimension proofs found by GPT-6 Astra, Hitchin domination from ChatGPT 5.6 Sol's ideas, the section conjecture for generic curves, Yau's geometric conjecture, Erdős–Gyárfás r = 5, a Claude-built Gray-code reduction, and a Lean harness solving 6/6 IMO 2026","takeaway":"Tuesday's arXiv announcement was the strongest maths batch yet. Seven results have their own entries, among them the Erdős–Szemerédi sunflower conjecture, Erdős–Hajnal for all paths and a non-sofic hyperbolic group. This page catalogues the next tier: about forty papers whose AI disclosures run from 'the proofs were found by ChatGPT-6 Astra' to proofreading.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI","Anthropic","DeepSeek","AWS","various mathematicians"]},{"id":"2026-10-19-quasitrace-monotone-completeness-counterexamples-openai","url":"https://postcutoff.com/e/2026-10-19-quasitrace-monotone-completeness-counterexamples-openai/","date":"2026-10-19","date_precision":"day","short_title":"Kaplansky's quasitrace conjecture reported disproved by an unreleased OpenAI model","deck":"Experts reproduce it and refute the monotone completeness conjecture","takeaway":"If it holds, an AI system has settled one of the oldest problems in operator algebras, and specialists rebuilt the result within weeks and pushed it to the companion conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI"]},{"id":"2026-10-19-neumann-zagier-conjecture-tree-lattices-ai-proof","url":"https://postcutoff.com/e/2026-10-19-neumann-zagier-conjecture-tree-lattices-ai-proof/","date":"2026-10-19","date_precision":"day","short_title":"Neumann–Zagier 1985 conjecture on tree lattices settled after an AI proof with ChatGPT 5.5 Pro and Opus 4.8","deck":null,"takeaway":"A problem posted as a benchmark task was solved with frontier models and then confirmed and improved by the experts who posed it.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt","url":"https://postcutoff.com/e/2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt/","date":"2026-10-19","date_precision":"day","short_title":"Gregory Chudnovsky, with ChatGPT, claims the lemniscate period has irrationality exponent 2, with a Lean 4 proof built on OpenAI's π-exponent library","deck":null,"takeaway":"After OpenAI's claim that π has irrationality exponent 2, a well-known transcendence theorist carries the method to the lemniscate period Ω = 2∫₀¹(1−t⁴)^(−1/2) dt. He says ChatGPT helped with the exposition and the formalization, and he ships a complete Lean 4 proof built on OpenAI's library. Like the π claim, it has not been reviewed independently.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":84,"official":13,"filed":"2026-10-07","updated":"2026-10-20","orgs":["OpenAI"]},{"id":"2026-08-12-claude-hadamard-matrices-below-2000","url":"https://postcutoff.com/e/2026-08-12-claude-hadamard-matrices-below-2000/","date":"2026-08-12","date_precision":"day","short_title":"Claude-assisted constructions complete Hadamard matrices for every order below 2000","deck":"Including 668","takeaway":"It closed a famous \"smallest unknown case\" that had stood for two decades, with an easily verified result.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-10-21","orgs":["Anthropic"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-21","type":"filed","text":"Created (32 PDFs read in full text)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-21","run":null,"sources_read":"32 PDFs read in full text","updated":"2026-10-21","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":174,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":113,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":204,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":143,"in_training_data":false}],"short_url":null}