Mézard–Parisi formula: Panchenko’s proof ‘found by ChatGPT Astra’
Event confirmedAwaiting review
Importance: major (4 of 5)The 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.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 31 October 2026
Your AI and this story
- GPT-6 Astra183 days after its cutoff
- Claude Opus 5.5122 days after its cutoff
- Gemini 4 Argon213 days after its cutoff
- Grok 4.7152 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 122 days before it.
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
What happened
The Mézard–Parisi formula (2001) predicts the free energy of diluted spin glasses, a class that includes random K-SAT. Franz and Leone proved that it is an upper bound. The lower bound needed an extra assumption, called reproducibility, and stayed open.
On 29 Oct 2026 Dmitry Panchenko of the University of Toronto posted a 65-page paper that claims the lower bound for diluted models with even p and for random K-sat, under symmetry and integrability assumptions. His AI statement says ChatGPT Astra found the proof after he asked it to look for a new perturbation in an earlier argument of his. A companion note was written by ChatGPT Astra from calculations he had done but never written down.
Panchenko also says that OpenAI released another proof a few days after his was found, and that it seems to take a different approach. That is family 221 of OpenAI’s maths repository, part of its 6 Oct release.
Why it matters
This is a known problem of mathematical physics. Two separate AI-produced proofs now exist, one directed by the field’s main expert. Both still need refereeing.
What is disputed or not yet verified
| Verification | Unrefereed preprint by a leading specialist; independent of OpenAI’s own claimed proof (family 221) |
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Sources
4 sources from 2 sites. Numbers match the chips in the text.
4 sources: 4 primary
Primary
- arXiv 2610.39545: The Mézard–Parisi formula (Panchenko)arxiv.org, paper
- arXiv 2610.39542: Ghirlanda–Guerra identities and cavity equations for diluted spin glass modelsarxiv.org, paper
- OpenAI family 221: The Mézard–Parisi formula for diluted spin glasses (PDF)github.com, paper
- OpenAI reasoning summary for the Mézard–Parisi familygithub.com, code
Changes
- Filed (science agent; arXiv abstracts and PDFs of 2610.39545 and 2610.39542 read)