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THE CRITICAL CAPACITY OF THE HOPFIELD MODEL

Primarily AI-generated textHuman understanding: some partsmath.PR — Probability

Contributed by Yuwen Zhou ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

Abstract. We prove that the critical capacity of the Hopfield model at zero temperature lies in [0.138198129, 0.13820106]. The best rigorous lower bound was 0.113, and our lower end exceeds the published one-step and two-step replica symmetry breaking values. The capacity is the supremum of the loads at which the free energy at fixed overlap with one pattern has a well. At every positive temperature, we give a variational formula for this free energy and show that binary patterns and Gaussian rows have the same wells, which characterizes the capacity. The proof builds on the Hamilton–Jacobi solution of the model by Ali, Boursier, Chen, Ho, Issa and Mourrat, and on a

Provenance statement

The statements, proofs, code and text were generated mainly by AI systems: Claude Opus 5.5 (Anthropic) derived the proofs and wrote the rest, and GPT-6.1 Sol (OpenAI) wrote an independent report on each result. A final audit by fresh AI readers covered the longer version in the repository. Parts checked and understood by a human: Yuwen Zhou Code and data: https://github.com/OctopusTide/hopfield-capacity

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Version history

  1. v1 · 57 pagesSubmitted by Yuwen ZhouInitial depositCurrentNov 04, 2026