{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-10-30-mezard-parisi-formula-panchenko-proof-found-by-chatgpt-astra/","md":"https://postcutoff.com/e/2026-10-30-mezard-parisi-formula-panchenko-proof-found-by-chatgpt-astra/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":"2026-10-30-mezard-parisi-formula-panchenko-proof-found-by-chatgpt-astra","date":"2026-10-30","date_precision":"day","short_title":"Mézard–Parisi formula: Panchenko's proof 'found by ChatGPT Astra'","deck":null,"takeaway":"Dmitry Panchenko (Toronto), a leading probabilist of spin glasses, posted a proof of the Mézard–Parisi formula on 29–30 Oct 2026. The formula gives the free energy of random K-SAT and diluted p-spin models with even p. Franz and Leone had proved the upper bound; Panchenko's paper claims the matching lower bound. His AI statement says: 'The proof in this paper was found by ChatGPT Astra.' A companion note was 'written by ChatGPT Astra'. He adds that OpenAI released another proof a few days later, which 'appears to follow a different approach'. OpenAI's version is family 221 of its maths repository. Both are unrefereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.39545: The Mézard–Parisi formula (Panchenko)","url":"https://arxiv.org/abs/2610.39545","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2610.39542: Ghirlanda–Guerra identities and cavity equations for diluted spin glass models","url":"https://arxiv.org/abs/2610.39542","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"OpenAI family 221: The Mézard–Parisi formula for diluted spin glasses (PDF)","url":"https://github.com/openai/math/blob/main/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":4,"title":"OpenAI reasoning summary for the Mézard–Parisi family","url":"https://github.com/openai/math/blob/main/reasoning_traces/mezard-parisi-formula.pdf","type":"code","group":"primary","domain":"github.com"}],"official":4,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","University of Toronto"],"title":"Mézard–Parisi formula for diluted spin glasses and random K-SAT: Dmitry Panchenko posts a proof that he says 'was found by ChatGPT Astra', a second route after OpenAI's own claim","summary":"arXiv 2610.39545 (\"The Mézard–Parisi formula\", Dmitry Panchenko, 29 Oct 2026) proves the Mézard–Parisi formula for the free energy of random K-sat and diluted p-spin models with even p, completing a programme of the author and collaborators. The upper bound is due to Franz and Leone; the new paper gives the matching lower bound, through an approximate form of the Mézard–Parisi reproducibility hypothesis without assuming finitely supported overlaps. The author says the proof was found by ChatGPT Astra after he asked it for a new perturbation. OpenAI's repository holds a separate claimed proof (family 221, dated 23 Sept, Lean statement). Neither is refereed.","key_facts":["Claim (abstract): 'This paper proves the Mézard-Parisi formula for the free energy of random K-sat and diluted p-spin models with even p, completing an approach developed in earlier work of the author and collaborators. The upper bound was established by Franz and Leone. Here the matching lower bound is proved for a general class of diluted models under symmetry and integrability assumptions.'","AI statement (verbatim, arXiv 2610.39545): 'The proof in this paper was found by ChatGPT Astra. I asked it to look for a new perturbation in the argument of [31] and try to simplify the induction.' And: 'ChatGPT Astra and Claude Fable assisted me with the rewriting. A few days after the proof presented here was found, OpenAI released another proof of the Mézard–Parisi formula in [24]. That proof appears to follow a different approach.'","Companion note (arXiv 2610.39542, 'Ghirlanda-Guerra identities and cavity equations for diluted spin glass models'): 'This note was written by ChatGPT Astra. I had previously carried out these calculations in detail while preparing [7], but had not written them down.'","Background (paper, §1): the Parisi formula for the Sherrington–Kirkpatrick model was proved by Talagrand after Guerra's upper bound; for diluted models the extra 'reproducibility' hypothesis was needed, Franz–Leone gave the upper bound, and 'the matching Mézard–Parisi lower bound remained elusive'. Panchenko's earlier series built the Ghirlanda–Guerra and cavity-equation approach","OpenAI's version: family 221 of github.com/openai/math, 'The Mézard–Parisi formula for diluted spin glasses' (23 Sept), for Poisson-diluted even-arity Ising models 'satisfying the Panchenko–Talagrand factorization and positivity assumptions', with a Lean statement and a published reasoning summary; it is part of the 6 Oct release","Who: Panchenko is the author of the monograph 'The Sherrington–Kirkpatrick Model' (Springer, 2013) and of the proof that the Ghirlanda–Guerra identities imply ultrametricity, so the AI-found proof comes from the main expert on the method","Verification status: arXiv preprint (65 pages), not refereed, no Lean formalisation of Panchenko's proof; no HN thread or expert reaction found on 31 Oct"],"key_numbers":[],"tags":["math","probability","spin-glasses","statistical-physics","random-k-sat","chatgpt","gpt-6-astra","claude-fable","preprint"],"science":{"field":"mathematics","subfield":"probability / spin glasses","problem":"Mézard–Parisi formula (2001) for the free energy of diluted spin-glass models, including random K-SAT","result":"Claimed matching lower bound for diluted p-spin models with even p and random K-sat under symmetry and integrability assumptions, completing the formula (upper bound: Franz–Leone)","open_since":"2001","ai_system":["ChatGPT Astra","Claude Fable"],"human_role":"AI-found proof directed by the field's main expert: Panchenko asked for a new perturbation in his earlier argument; ChatGPT Astra found the proof; Panchenko rewrote it with ChatGPT Astra and Claude Fable","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint by a leading specialist; independent of OpenAI's own claimed proof (family 221)"},"status":"pending","shock":"The specialist whose framework OpenAI's paper assumes reports that a model found his proof, days before OpenAI's own version appeared."},"body_md":"## What happened\n\nThe Mézard–Parisi formula (2001) predicts the free energy of diluted spin glasses, a class that includes random K-SAT. Franz and Leone\nproved that it is an upper bound. The lower bound needed an extra assumption, called reproducibility, and stayed open.\n\nOn 29 Oct 2026 Dmitry Panchenko of the University of Toronto posted a 65-page paper that claims the lower bound for diluted models with\neven p and for random K-sat, under symmetry and integrability assumptions. His AI statement says ChatGPT Astra found the proof after he\nasked it to look for a new perturbation in an earlier argument of his. A companion note was written by ChatGPT Astra from calculations\nhe had done but never written down.\n\nPanchenko also says that OpenAI released another proof a few days after his was found, and that it seems to take a different approach.\nThat is family 221 of OpenAI's maths repository, part of its 6 Oct release.\n\n## Why it matters\n\nThis is a known problem of mathematical physics. Two separate AI-produced proofs now exist, one directed by the field's main expert.\nBoth still need refereeing.","disputed":[{"label":"Verification","text":"Unrefereed preprint by a leading specialist; independent of OpenAI's own claimed proof (family 221)","sources":[]}],"related":[{"id":"2026-10-31-ai-assisted-results-october-31-listing","url":"https://postcutoff.com/e/2026-10-31-ai-assisted-results-october-31-listing/","date":"2026-10-31","date_precision":"day","short_title":"AI-assisted results, 31 Oct: arXiv of 30 Oct and Hexagon of 29–31 Oct","deck":null,"takeaway":"The 31 Oct sweep listed 344 arXiv papers that mention AI, 75 conjecture papers with no AI statement and 48 uncited Hexagon works. We skimmed every AI statement and read about 70 PDFs or Hexagon records. Seven results got their own entries. Two come from senior authors: Panchenko's Mézard–Parisi formula ('found by ChatGPT Astra') and Bhargava's positive-density cyclicity of elliptic curves, written with GPT-6. The other five are Tianchi Yang's count of C₄-free graphs, Kolpakov's sharp dimension bounds for right-angled hyperbolic polytopes, the generalized Lax conjecture (Kummer–Netzer's AI-found generic case and OpenAI's counterexample), Diening–Storn's AFEM optimality for every bulk parameter, and an unverified GPT-6 Astra claim on Onsager's hard rods. The rest are listed here, among them Pongrácz's R(5,6) ≤ 84 certificate with GPT-6 Astra and Bowler–Pitz's negative answers to Halin-type questions, where Claude found key lemmas.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":44,"official":42,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"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":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai","url":"https://postcutoff.com/e/2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai/","date":"2026-10-05","date_precision":"day","short_title":"Random k-SAT threshold exists for every k ≥ 3: a human proof and an OpenAI Lean-checked proof","deck":null,"takeaway":"The satisfiability conjecture says that random k-SAT switches from satisfiable to unsatisfiable at one limiting clause density α_k. Friedgut proved a sharp but possibly n-dependent threshold in 1999. Ding, Sly and Sun proved the conjecture for large k (Annals, 2022). On 5 Oct 2026 Cambridge PhD student Gaia Carenini posted on ECCC a proof for every fixed k ≥ 3. Her report states that none of its arguments was generated by a language model. The next day OpenAI's maths release included a separate proof, dated 25 Sept, which credits Carenini with priority. Its Lean statement was accepted in the community Comparator run, and it adds Θ(n) hitting-time variance and computability of the 3-SAT threshold. Neither proof has had a published expert assessment as of 29 Oct. Neither determines the value of α_3.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-29","updated":"2026-10-29","orgs":["University of Cambridge","OpenAI"]}],"people":[{"id":"dmitry-panchenko","name":"Dmitry Panchenko","url":"https://postcutoff.com/person/dmitry-panchenko/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-31","type":"filed","text":"Created (science agent; arXiv abstracts and PDFs of 2610.39545 and 2610.39542 read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-31","run":null,"sources_read":"Science agent; arXiv abstracts and PDFs of 2610.39545 and 2610.39542 read","updated":"2026-10-31","human_review":null,"version":{"date":"2026-10-31"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":183,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":122,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":213,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":152,"in_training_data":false}],"short_url":"https://postcutoff.com/s/mezard-parisi-formula-panchenko"}