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The n1/6n^{1/6} free-energy fluctuation scale of the low-temperature Sherrington-Kirkpatrick model, with a limiting law

In the zero-field Gaussian Sherrington-Kirkpatrick model Hn(σ)=1n∑i<jgijσiσjH_n(\sigma)=\frac{1}{\sqrt n}\sum_{i<j}g_{ij}\sigma_i\sigma_j, let Fn=log⁡∑σeβHn(σ)F_n=\log\sum_\sigma e^{\beta H_n(\sigma)}. At high temperature β<1\beta<1 the fluctuations of FnF_n are of order one (Aizenman-Lebowitz-Ruelle). For fixed β>1\beta>1, Crisanti, Paladin, Sommers and Vulpiani predicted that the free energy per spin fluctuates on the scale n−5/6n^{-5/6}, that is sd(Fn)≍n1/6\mathrm{sd}(F_n)\asymp n^{1/6}, and Parisi and Rizzo developed the related large-deviation calculation. Is the standard deviation of FnF_n of order n1/6n^{1/6} at every fixed β>1\beta>1, and does the centered, rescaled free energy converge to a limiting law?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Spin glasses; Sherrington-Kirkpatrick model
Posed by
A. Crisanti, G. Paladin, H.-J. Sommers and A. Vulpiani (J. Phys. I France 2, 1992); developed by Parisi and Rizzo (2008-2009)
Year posed
1992
Years open
34y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Companion (fluctuation scale): for every fixed β>1\beta>1, sd(Fn)=n1/6+o(1)\mathrm{sd}(F_n)=n^{1/6+o(1)} and ∣Fn−EFn∣|F_n-\mathbb EF_n| is typically between n1/6−ηn^{1/6-\eta} and n1/6+ηn^{1/6+\eta}. Principal (limiting law): Var(Fn)/n1/3\mathrm{Var}(F_n)/n^{1/3} converges to a constant in (0,∞)(0,\infty), every moment of the n−1/6n^{-1/6}-rescaled completed free energy converges with a polynomial rate, exponential moments are uniformly bounded, and (Fn−EFn)/sd(Fn)(F_n-\mathbb EF_n)/\mathrm{sd}(F_n) converges along all nn to a nondegenerate law with mean 0 and variance 1, uniquely determined by its moments. The law is not identified with a named distribution and no formula for the variance constant is given.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. 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). Both manuscripts are authored 'OpenAI' and name no human author. The limiting-law paper takes its finite-tree toolkit and the exponent from the companion fluctuation-scale paper of the same date.

Verification

No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts were read against the CPSV prediction. The family has no Lean formalization (lean/docs/217.md does not exist at the pinned commit). The limiting-law paper depends on precisely listed inputs from its companion (stated in a companion-interface appendix) and on Lopatto's 2026 interval-support theorem for the Parisi measure. Results are for fixed beta > 1, zero field, Gaussian couplings and Ising spins; nothing is claimed uniformly as beta tends to 1 or infinity, and the limiting law is characterized only by its moments, not identified with a named distribution.

Sources

Changelog1 change

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