The low-temperature Sherrington–Kirkpatrick fluctuation law

The low-temperature Sherrington–Kirkpatrick fluctuation law concerns the sample-to-sample fluctuations of the log partition function of the zero-field Sherrington–Kirkpatrick spin glass below its critical temperature. Two manuscripts dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim that at every fixed inverse temperature β > 1 the standard deviation is of order n^1/6, as predicted in the physics literature, and that the standardized fluctuations converge to a nondegenerate law. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

• As recounted in the introductions of the two manuscripts:
  • The prediction. Crisanti, Paladin, Sommers and Vulpiani (1992) predicted fluctuations of order n^−5/6 for the free energy per spin in the low-temperature zero-field phase; at fixed β this is the scale n^1/6 for the log partition function. Parisi and Rizzo developed the associated large-deviation calculation through the low-temperature phase. The manuscripts note that inferring typical fluctuations from it needs a matching assumption, discussed in Section VI of Parisi and Rizzo's 2009 paper, and call that step heuristic.
  • Other temperatures. Order-one Gaussian fluctuations at high temperature go back to Aizenman, Lebowitz and Ruelle (1987). Chatterjee proved that the variance is at most of order n divided by the logarithm of n. At the critical temperature, Du and Huang proved that the variance is one sixth of the logarithm of n up to a bounded error, with a Gaussian limit, in arXiv:2607.02172.
  • Fixed low temperature. For each fixed β > 1, Aronow and Lopatto proved that the variance lies between constant multiples of n^4/15 and n^7/15, in Theorem 1.4 of arXiv:2609.09103v1. The manuscripts describe this work as a direct antecedent of their argument.
  • The Parisi measure. After Zhou's interval-support result near the critical temperature (Communications on Pure and Applied Mathematics, 2026), Lopatto proved that at every fixed β > 1 the Parisi measure is supported on an interval from 0 to some q_β, in Theorem 1.1 of arXiv:2607.11756, titled Full replica symmetry breaking in the Sherrington–Kirkpatrick model. The scale manuscript uses this theorem together with Lemma 2.2 of the same preprint, citing its third version; the preprint's fourth version is dated 5 October 2026.

The original results

• Fluctuation scale (24 September 2026). Theorem 1.1 of the first manuscript states that for every fixed 1 < β < ∞ the log partition function F_n of the zero-field Gaussian model has standard deviation n^1/6+o(1) and variance n^1/3+o(1), and that the typical centered absolute fluctuation has the same order with probability tending to one. The manuscript stresses that this concerns power exponents only: it gives neither a limiting law nor a constant prefactor, and nothing uniform as β approaches 1 or infinity.
  • Method, as described: a completed covariance with an independent common Gaussian, a positive Laplace window for the centered free energy, and comparison of transforms by revealing the Gaussian covariance in stages along finite trees, in the lineage of the Parisi recursion, Ruelle's cascades, Guerra–Toninelli interpolation and the Aizenman–Sims–Starr cavity scheme. A strictly positive initial density of the Parisi measure is derived from scalar diffusion identities and a fourth-derivative estimate.
• Limiting law (24 September 2026). The second manuscript states that for every fixed β > 1 the free energy, centered by its mean and divided by its standard deviation, converges in distribution through all integers to a nondegenerate law with mean zero, variance one and a finite exponential moment near the origin, uniquely determined by its moments. It also states that the variance divided by n^1/3 converges to a finite positive constant. The law is characterized by a finite-system moment formula and is not identified with a named distribution. The scale manuscript supplies the exponent and the comparison toolkit.
• Sources. Commit 1581d4c of the openai/math repository, folders The low-temperature Sherrington–Kirkpatrick fluctuation scale and The low-temperature Sherrington–Kirkpatrick free-energy limiting law.
• Formalization. The collection has no Lean scope note in lean/docs/ or Comparator challenge for these two manuscripts.
  • The repository's Lean library contains a theorem OAI.ParisiFinite.full_support_minimizer in lean/OAI/Probability/ParisiFinite/ParisiEntropyDecomposition.lean. For β > 1 it states that every probability measure on the unit interval minimizing the zero-field Parisi functional defined in the library has support equal to an interval from 0 to some q with 0 < q < 1. This is the support part of the Parisi-measure input recalled above; the density and atom statements of Lopatto's Theorem 1.1 have no Lean statement there.
  • The fluctuation theorems themselves have no Lean statement in the collection.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 17 October 2026)
2026-10-12	P. M. Aronow and Patrick Lopatto, Hexagon record 2610.00177v1, titled An N−7/10 Bound for the Finite-Size Bias of the Sherrington-Kirkpatrick Free Energy	Remark	For every fixed β > 1, the expected free energy per spin of the zero-field model differs from its Parisi limit by at most a constant times N−7/10, improving the exponent 2/3 of the authors' arXiv:2609.09103. The abstract compares this with the correction of order N−3/4 predicted by Franz, Parisi and Ricci-Tersenghi, and with the strict improvement over 2/3 that would follow from the ideas in OpenAI's recently released manuscripts; the references include both manuscripts on this page.	Hexagon labels: primarily AI-generated text; human understanding: some parts. The provenance statement names GPT and Claude models and Lean 4 formalizations produced during verification.

Notes

• The library theorem above is not among the Comparator challenges of the collection, so the collection-wide kernel checks of the Comparator challenges do not cover it.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.