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The limiting free energy of orthogonally invariant Ising spin glasses at every temperature

Let ΛN\Lambda_N be real diagonal matrices whose empirical spectral laws converge to a compactly supported μ\mu, UNU_N Haar on O(N)O(N), and PN=N−1log⁡(2−N∑σ∈{±1}Nexp⁡{12σTUNTΛNUNσ})P_N=N^{-1}\log\big(2^{-N}\sum_{\sigma\in\{\pm1\}^N}\exp\{\tfrac12\sigma^TU_N^T\Lambda_NU_N\sigma\}\big). This family includes the random orthogonal model and Gaussian Hopfield interactions; Marinari, Parisi and Ritort (1994) and Cherrier, Dean and Lefevre (2003) gave replica-symmetric and one-step replica-symmetry-breaking predictions for its free energy in terms of the spectral law. Rigorous results covered only high temperature: Bhattacharya-Sen (zero field), Fan-Wu (external field), Fan-Misiakiewicz-Wang-Wen. Fan and Wu note that for these models replica predictions for the free energy are conjectured but not rigorously known. Does PNP_N converge at every temperature, and to what variational formula in terms of μ\mu?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Spin glasses: mean-field models with rotationally invariant couplings
Posed by
Replica predictions: E. Marinari, G. Parisi, F. Ritort (1994) and R. Cherrier, D. S. Dean, A. Lefevre (2003); stated as conjectured but unproven by Z. Fan and Y. Wu (arXiv 2105.02797, 2021; PTRF 2024)
Year posed
1994
Years open
32y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: under the stated spectral assumptions, lim⁡EPN=lim⁡PN=inf⁡p{S(p)+G1(p)}\lim\mathbb EP_N=\lim P_N=\inf_p\{S(p)+G_1(p)\} almost surely, where pp ranges over overlap quantile functions, SS is a Parisi-type one-site functional and G1G_1 involves the edge-extended RR-transform of μ\mu; no independence across NN is needed. Extensions: deterministic external fields converging in Wasserstein-1, random spectra, the zero-temperature ground-state energy, and Gaussian Hopfield consequences (limiting max and min of N−1∥Gσ∥2N^{-1}\|G\sigma\|_2). Not shown: outliers in the spectrum, explicit identification with the published 1RSB formulas, or the structure (full RSB or not) of the optimizer.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The release also supplies a Lean formalization of the main limits, produced as part of the same release.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, and the Lean statement lean/ComparatorChallenges/InvariantIsing.lean, whose solution module OAI/Probability/InvariantIsing/Unconditional.lean exists at the pinned commit. This challenge is not in the formalization catalogue; its statement was read here and was not rebuilt. Its first theorem, limiting_pressure_of_extreme_limits_unconditional, states the headline: under weak convergence of spectral laws to a compactly supported law with extreme eigenvalues converging to the support edges and Haar rotations, the pressure converges in expectation and almost surely to the variational functional. Readers should note the paper does not claim its formula coincides with the physicists' 1RSB predictions, and the no-outlier hypothesis is essential.

Sources

Changelog1 change

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