Probability and statistical mechanics·OpenAI Math·Result 221·Lean formalisation listed
The Mézard–Parisi formula for diluted spin glasses
One of the 372 results in OpenAI's public mathematics catalogue, filed under probability and statistical mechanics: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a proof of the Mézard–Parisi formula for diluted spin glasses (probability and statistical mechanics) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 52 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 17:42 · watch from there on YouTube →
What OpenAI claims
Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability. (Lean)
Paper
The Mézard–Parisi formula for diluted spin glasses
OpenAI's abstract We prove the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. This resolves the variational equality conjecture for that class: the limiting pressure equals the infimum of the trial functional over all finite hierarchy depths and trial laws. Only first moments of the interaction and external field are required.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability. (Lean)
- Where is the proof?
- In the paper, The Mézard–Parisi formula for diluted spin glasses (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 52).
Also in
- Chapter 7 of 10 in Benjamini–Schramm Conjecture, Mézard–Parisi Formula, Voronoi Percolation and 7 More · starts at 2:07 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 52 from 17:42 · description, sources and chapters →
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