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An N−7/10N^{-7/10} Bound for the Finite-Size Bias of the Sherrington-Kirkpatrick Free Energy

Primarily AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.MP — Mathematical Physicsmath-ph — Mathematical Physics

Contributed by P. M. Aronow ↗, Patrick Lopatto ↗

Version 1 / Oct 12, 2026 / CC BY 4.0

Abstract

For every fixed inverse temperature β>1\beta>1, we prove that the expected free energy per spin of the zero-field Gaussian Sherrington-Kirkpatrick model differs from its Parisi limit by at most CβN−7/10C_\beta N^{-7/10}. This improves the exponent 2/32/3 of our earlier work (arXiv:2609.09103). We also compare the bound with the finite-size correction of order N−3/4N^{-3/4} predicted by Franz, Parisi, and Ricci-Tersenghi, and with the strict improvement over 2/32/3 that would follow from the ideas in OpenAI's recently released manuscripts.

Provenance statement

Proof discovery and verification were done with GPT 5.6 Sol, GPT 6 Sol, GPT 6 Astra, GPT 6.1 Sol, Claude Fable 5, Claude Fable 5.1, and Claude Opus 5.5. Lean 4 formalizations were produced by GPT 6.1 Sol as part of verification. Final revisions to the writing were done with Opus 5.5 and GPT 6.1 Sol.

Tools used

OpenAI
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Version history

  1. v1Submitted by P. M. AronowInitial depositCurrentOct 10, 2026