A limiting law for standardized low-temperature SK free energy
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
For Gaussian zero-field SK at every fixed $\beta >1$, does the centered free energy divided by its own standard deviation converge in distribution through all n to a nondegenerate mean-zero variance-one law with a finite exponential moment near zero, uniquely specified by the limiting moment sequence?Short English statement
For Gaussian zero-field SK at every fixed $\beta >1$, does the centered free energy divided by its own standard deviation converge in distribution through all n to a nondegenerate mean-zero variance-one law with a finite exponential moment near zero, uniquely specified by the limiting moment sequence?Sources
- S1 Original question source or mathematical context. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/build/sections/01-introduction.tex.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2911
The standardized low-temperature SK free energy is claimed to converge to a nondegenerate law, with variance asymptotic to a positive constant times $n^{1/3}$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.
A limiting law for standardized low-temperature SK free energy
OpenAI's claim
OpenAI presents a proof in the selected manuscript. The standardized low-temperature SK free energy is claimed to converge to a nondegenerate law, with variance asymptotic to a positive constant times $n^{1/3}$.Mathematical statement of the claimed result
Result 1.
Fix $\beta>1$ and independent standard Gaussian $g_{ij}$. Put \[ F_n(\beta)=\log\sum_{\sigma\in\{-1,1\}^n} \exp\left(\frac\beta{\sqrt n}\sum_{i<j}g_{ij}\sigma_i\sigma_j\right),\quad s_n^2=\operatorname{Var}F_n,\quad X_n=(F_n-\mathbb EF_n)/s_n. \] Along the entire integer sequence $n\to\infty$, $X_n\Longrightarrow\nu_\beta$ for a nondegenerate probability law of mean zero and variance one. There is a positive $c$ with $\int e^{c|x|}\nu_\beta(dx)<\infty$; its finite-system limiting moments uniquely specify the law. No external field or temperature varying with $n$ is included.Outline of the argument
Refined overlap-error covariance equations are solved uniformly over finite hierarchies. Pressure comparisons and Gaussian-history martingale moments bound route errors; signed coefficients are combined before estimation so leading polynomial terms cancel, making the rescaled increments summable and identifying the common law with a positive variance prefactor.Formal verification
Family 217 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.Status caveat
OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.Sources
- OpenAI, The low-temperature Sherrington–Kirkpatrick free-energy limiting law, 2026-09-24; citation key
OAI:The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.