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Parisi formula for Ising and Spherical Perceptrons

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

Contributed by Daniel Fu ↗, Youngtak Sohn ↗

Version 1 / Oct 08, 2026 / CC BY 4.0

Abstract

We prove a Parisi formula for Ising and spherical perceptrons with Gaussian disorder and i.i.d. random potentials having a common Lipschitz bound and integrable values at zero. We assume M/N→α∈(0,∞)M/N\to\alpha\in(0,\infty), where MM is the number of Gaussian vectors and NN is the number of spins. The proof combines cavity in MM and NN with the Aizenman–Sims–Starr scheme for the lower bound and Guerra’s interpolation for the upper bound. As consequences, we answer several problems posed by Talagrand (2011) and establish a quenched large deviation principle for the empirical distribution of the Gaussian projections, resolving a conjecture of Bolthausen and Kistler (2010). We also resolve the sharp-threshold conjecture of Aubin, Perkins, and Zdeborová (2019) for the uu-function binary perceptron at every positive width, and establish the predicted variational formulas for the entropy and capacity of the negative spherical perceptron. Finally, the zero-temperature limit yields the scalar projection-pursuit formulas conjectured by Montanari and Zhou (2025, 2026), in both the supervised and unlabelled settings.

Provenance statement

This document and its arguments were generated through discussions with ChatGPT 5.6 Pro. An initial draft for the capacity of the negative spherical perceptron was generated on July 15, 2026, and we extended the argument to i.i.d. lipschitz potentials, both Ising and spherical, on September 17, 2026. While distilling the main ideas, carefully checking the proof, and polishing the draft, we learned on October 6, 2026, that OpenAI had released overlapping results across three manuscripts~\cite{OpenAIIsingPerceptron2026,OpenAISphericalPerceptron2026,OpenAIJamming2026}. Given these circumstances, we decided to post this working draft on Hexagon.

Tools used

OpenAI
ChatGPTVersion 5.6Pro
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ChatGPTVersion 6

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Version history

  1. v1Submitted by Youngtak SohnInitial depositCurrentOct 07, 2026