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The finite-temperature free energy of the spherical perceptron with bounded continuous potentials (Gyorgyi-Reimann continuous-RSB formula)

For xx uniform on the sphere of radius N\sqrt N in RN\mathbb R^N, M=⌊αN⌋M=\lfloor\alpha N\rfloor standard Gaussian patterns and a single-pattern potential ϕ\phi, the pressure is pN=N−1log⁡∫eβ∑aϕ(ga⋅x/N)dσN(x)p_N=N^{-1}\log\int e^{\beta\sum_a\phi(g^a\cdot x/\sqrt N)}d\sigma_N(x). Gardner (1988) began the study of such storage volumes; Shcherbina-Tirozzi proved Gardner's replica-symmetric formula for nonnegative margins in a restricted regime. Gyorgyi and Reimann (2000) derived a continuous replica-symmetry-breaking formula for the finite-temperature model, which in general is not replica-symmetric and not a Gaussian spin glass. Does pNp_N converge, for every density and temperature and nonconvex potentials, to that variational formula?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Spin glasses; perceptron models
Posed by
Geza Gyorgyi and Peter Reimann, Beyond storage capacity in a single model neuron: continuous replica symmetry breaking (J. Stat. Phys., 2000), after Elizabeth Gardner (1988)
Year posed
2000
Years open
26y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for all α,β>0\alpha,\beta>0 and ϕ∈Cb(R)\phi\in C_b(\mathbb R), EpN\mathbb Ep_N and pNp_N (in probability) converge to inf⁡m{αVβϕ(m)+S(m)}\inf_m\{\alpha V_{\beta\phi}(m)+S(m)\} over nondecreasing m:[0,1]→[0,1]m:[0,1]\to[0,1], with VV a Brownian stochastic-control value and SS the spherical entropy; this matches the Gyorgyi-Reimann continuous-RSB functional. Companions: the analogous formula for the Gaussian Ising perceptron with any bounded Borel log-potential at all densities and positive temperatures, and for spherical models with bi-orthogonally invariant disorder without outliers. Not shown: hard constraints or zero temperature (storage capacity), discontinuous potentials in the spherical case, or properties of the optimizer.

What the AI did

The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family 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 manuscripts are authored 'OpenAI' and name no human author. This entry draws on three manuscripts dated September 24, 2026: the spherical free energy (principal), the Ising perceptron free energy for bounded Borel log-potentials, and the spherical model with bi-orthogonally invariant disorder.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 of 'The free energy of the spherical random perceptron' were read against the Gyorgyi-Reimann formula as the manuscript cites it; Gyorgyi-Reimann was not read. The theorem needs a bounded continuous potential, so the discontinuous error-counting cost of the classical storage problem is not covered directly; hence partial. Lean: lean/ComparatorChallenges/PerceptronFreeEnergy.lean (solution module OAI.Probability.Perceptron.Main, present at the pinned commit) is not in the formalization catalogue; it was found through lean/docs/222.md. Its statement SphericalPerceptronFreeEnergy.main was read: for alpha, beta > 0 and bounded continuous phi, the infimum over monotone trials of alpha times a Brownian control value plus the spherical entropy is a finite real p, the expected pressure converges to p, and the pressure converges to p in probability. That is the headline of the principal paper. Not rebuilt here. The Ising companion's Lean challenge covers only finiteness of its variational value.

Sources

Changelog1 change

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