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Existence of the thermodynamic limit for the Hopfield model

Adriano Barra, Fabrizio Durante

For a Hopfield model with K(N)/NK(N)/N tending to any positive finite load α\alpha and independent symmetric binary patterns, the quenched statistical pressure has a finite limit at every finite inverse temperature β≥0\beta \ge 0 and uniform external field h. The limit depends on the pattern count only through α\alpha and equals the supremum of the limiting pressures of mixed models retaining finitely many binary patterns in a Gaussian bulk.

What was already known

Earlier rigorous results for extensively loaded binary-pattern Hopfield models covered high-temperature or small-load regions, or assumed replica-symmetric concentration. Camilli, Contucci and Mingione separately reported a full-parameter thermodynamic limit for Gaussian patterns on 23 October 2026; that result does not establish the binary-pattern case.

AI involvement AI-led

Barra and Durante say Claude Opus 5.5 and ChatGPT Sol 6.1 generated the proofs through extensive interaction. The authors posed the problem, guided the work and suggested approaches. The models supplied the proof derivation, supporting AI-led involvement with human direction.

Importance Notable

Establishes existence of the binary-pattern thermodynamic limit throughout the stated load and temperature range, removing the earlier replica-symmetry and concentration conditions. The limit is characterized through mixed models, but its value is not given by an explicit formula.

Technical verification Formal (human)

Barra and Durante explicitly report that they checked every statement and proof and take responsibility for the paper. This is a source-reported complete human check of the stated binary-pattern thermodynamic-limit theorem.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. The paper explains the proof route, but the authors credit the proof derivation to AI and do not identify a separate human explanation of its central mechanism. Checking the statements and proofs does not alone establish human understanding, so Few is the evidence default.

Correction and publication history

2026-11-02 · publication

Newly reviewed AI-led proof of the all-load binary-pattern thermodynamic limit with an explicit author-reported proof check.

Read this published revision