{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro/","md":"https://postcutoff.com/e/2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro/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":"2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro","date":"2026-11-04","date_precision":"day","short_title":"Colourings Glauber dynamics above (1+1/e)Δ (GPT-6 Astra Pro)","deck":null,"takeaway":null,"category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2611.04699: Glauber Dynamics for Sampling Colorings: Rapid Mixing above (1+1/e)Δ","url":"https://arxiv.org/abs/2611.04699","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI"],"title":"Glauber dynamics for colourings mixes rapidly at q ≥ (1+1/e+γ)Δ: preprint whose main ideas 'emerged from extensive discussions with ChatGPT-6 Astra Pro'","summary":"Xiongxin Yang (arXiv 2611.04699) claims that for large maximum degree Δ the Glauber dynamics for proper q-colourings of any graph has spectral gap Ω(1/n) whenever q ≥ (1+1/e+γ)Δ ≈ 1.368Δ, far below the long-standing general-graph thresholds near 11/6 ≈ 1.83 (Vigoda 1999 and later small improvements). The paper says the main ideas came from ChatGPT-6 Astra Pro. Unrefereed single-author preprint.","key_facts":["Claim: for every γ > 0 and sufficiently large Δ, spectral gap Ω_γ(1/n) and mixing time O_γ(n² log q) when q ≥ (1+1/e+γ)Δ","1+1/e comes from the hard-core uniqueness threshold ≈ e/Δ on the infinite Δ-regular tree; proof via a Bochner-type inequality using 'hidden hard-core models' in the colour classes","Context: the conjecture is rapid mixing for q ≥ Δ+2; for general graphs the known threshold had stayed close to 11/6 Δ since Vigoda (1999)","Disclosure: 'The main ideas of this paper emerged from extensive discussions with ChatGPT-6 Astra Pro. ChatGPT-6 Astra Pro also suggested that a finer analysis … should improve the constant 1 + 1/e further … All mathematical claims have been independently verified by the author'","Verification: preprint; no expert comment or HN thread found on 4 Nov"],"key_numbers":[],"tags":["math","computer-science","sampling","markov-chains","gpt-6-astra","openai","arxiv"],"science":{"field":"computer-science","subfield":"Markov chain Monte Carlo / combinatorics","problem":"Rapid mixing of Glauber dynamics for proper colourings (Jerrum 1995; conjecture q ≥ Δ+2)","result":"Claimed spectral gap Ω(1/n) for q ≥ (1+1/e+γ)Δ, large Δ, any graph","open_since":"1995","ai_system":["GPT-6 Astra Pro"],"human_role":"AI-originated ideas; author simplified, verified and wrote the text","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint"},"status":"pending","shock":""},"body_md":"## What happened\n\nA single-author preprint posted on 3 Nov 2026 claims a large improvement on one of the best-known problems in sampling: when Glauber dynamics for random proper colourings mixes fast on every graph of maximum degree Δ. The claimed threshold (1+1/e)Δ ≈ 1.368Δ (for large Δ) would replace general-graph bounds that had stayed near 1.8Δ for over 25 years. The paper credits ChatGPT-6 Astra Pro with the main ideas and says the author simplified the arguments and checked all claims.\n\n## Why it matters\n\nIf it holds, this is a major step towards the Δ+2 conjecture, and another case of a frontier model supplying the central idea of a theoretical computer science result. Status is unverified: no specialist reading has appeared yet.","disputed":[{"label":"Verification","text":"Unrefereed preprint","sources":[]}],"related":[{"id":"2026-11-04-ai-assisted-results-november-4-listing","url":"https://postcutoff.com/e/2026-11-04-ai-assisted-results-november-4-listing/","date":"2026-11-04","date_precision":"day","short_title":"AI-assisted results, 4 Nov: arXiv of 4 Nov and Hexagon of 2–3 Nov","deck":null,"takeaway":"The 4 Nov sweep listed 374 arXiv papers that mention AI (193 new), 104 conjecture papers and 33 uncited Hexagon works. Three got own entries: Chandran–Gajjala's NP-completeness of Huffman coding with unequal letter costs (Codex), Yang's (1+1/e)Δ colouring mixing claim (GPT-6 Astra Pro) and Akhmedova–Lam–Yost-Wolff's ChatGPT-found counterexample to the amplituhedron magic number conjecture.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":60,"official":60,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI","Anthropic","various researchers"]},{"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"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-04","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-04","run":null,"sources_read":"The arXiv PDF","updated":"2026-11-04","human_review":null,"version":{"date":"2026-11-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":188,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":127,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":218,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":157,"in_training_data":false}],"short_url":"https://postcutoff.com/s/colourings-glauber-dynamics"}