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Math-2911-A1 · revision r1 · on Math-2911

OpenAI claim summary for Math-2911

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. A limiting law for standardized low-temperature SK free energy
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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