The Mezard-Parisi formula for diluted spin glasses: the Panchenko-Talagrand equality conjecture
In diluted spin glasses such as the Viana-Bray model each spin has a finite mean number of interactions. Mezard and Parisi proposed a hierarchical cavity description of the limiting free energy by nested laws of local fields, with replica symmetry breaking. Franz-Leone obtained replica-symmetric and one-step bounds for even arity by Guerra's interpolation, and Panchenko and Talagrand proved the arbitrary finite-level hierarchical upper bounds for Poisson even-arity Ising models with factorized interactions under a positivity condition, conjecturing equality. In that class, is the limiting pressure equal to the infimum of the Mezard-Parisi trial functional over all finite hierarchy depths and trial laws?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Spin glass theory; diluted mean-field models
- Posed by
- Dmitry Panchenko and Michel Talagrand, 'Bounds for diluted mean-fields spin glass models' (PTRF 2004), conjecture after Theorem 4
- Year posed
- 2004
- Years open
- 22y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 32 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Main theorem: for Poisson-diluted even-arity Ising models in the Panchenko-Talagrand class (factorized interactions with the stated independence and positivity of moments of ), with only first moments of the interaction and field, the limiting pressure exists and equals of the infimum of the Mezard-Parisi hierarchical functional of depth . Covers Viana-Bray, symmetric diluted even-spin models and soft random even- SAT at positive temperature. Not covered: odd arity (for example K-SAT with odd K), models outside the positivity class, fixed-degree random regular graphs, identification of the optimal hierarchy depth, or zero temperature beyond the stated ground-state limits.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript; its main theorem is formalized in Lean (see verification).
Verification
No independent mathematician has checked this yet. Checked here: the main theorem was read against the Panchenko-Talagrand conjecture as the paper quotes it; the paper claims it for their Poisson even-arity Ising class with factorized interactions and their positivity condition, needing only first moments. lean/docs/221.md points to ComparatorChallenges/DilutedSpin.json (theorem OAI.DilutedSpinGlass.mezard_parisi, solution module OAI.Probability.DilutedSpin.Main); the solution file exists at the pinned commit and the challenge is not in the formalization catalogue formalization.yaml. The statement was read here: for every even arity , positive density, integrable interactions and field, and the factorization, independence, finite-moment and positivity hypotheses, the finite- pressure converges to the infimum over depths of the infimum of the explicit finite-depth cavity functional. That is the headline claim. Not rebuilt here.