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The free energy of the Ising random perceptron
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Abstract
We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Daniel Fu, Youngtak Sohn, Parisi formula for Ising and Spherical Perceptrons (hexagon:2610.00154v1, 2026-10-08). Independently developed working draft proving a Parisi formula for Ising and spherical perceptrons with Gaussian disorder and i.i.d. Lipschitz random potentials at every density $\alpha\in(0,\infty)$. Its disclaimer says the authors posted it after learning on 6 October 2026 that OpenAI had released overlapping results, and it cites this paper. [AI-generated per source] [family-level]
Formalization
- OpenAI math release, Lean scope note for family 222 (lean/docs/222.md). For the Ising perceptron, the release's Lean development proves only that the variational value is finite for bounded continuous log-potentials. It does not formalize convergence of the pressure or the extension to bounded Borel potentials. [family-level]
Related prior literature mentioned by others
- E. Bolthausen, N. Kistler, A quenched large deviation principle and a Parisi formula for a Perceptron version of the GREM (arXiv:1011.5686, 2010). Fu and Sohn (hexagon:2610.00154v1) say that a consequence of their perceptron Parisi formula resolves a conjecture from this paper, through a quenched large deviation principle for the empirical distribution of the Gaussian projections.
- B. Aubin, W. Perkins, L. Zdeborová, Storage capacity in symmetric binary perceptrons (arXiv:1901.00314, J. Phys. A 52 (2019) 294003). Fu and Sohn (hexagon:2610.00154v1) say they resolve this paper's sharp-threshold conjecture for the u-function binary perceptron at every positive width.
- Yitzchak Shmalo, The storage capacity of the Ising perceptron: verification of the outstanding numerical conditions (arXiv:2607.11010v1, 13 July 2026). The AI Proved What? digest distinguishes this hard-capacity claim, also cited in the manuscript, from the finite-temperature theorem.
Possibly relevant literature (found by the agent, not checked)
- Shuta Nakajima, Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron (arXiv:2512.23195, 2025-12-29). This paper gives an analytic proof of uniqueness of the replica-symmetric saddle point for the Gaussian Ising perceptron with margin $\kappa\ge0$. That bears on the zero-margin capacity conditions of Ding–Sun, which this paper's introduction says Shmalo verified with a computer.
Digests and expositions
- AI Proved What?, How much can one neuron store? Perceptrons and jamming (Episode 222, October 2026). This AI-assisted digest distinguishes the Ising pressure theorem from hard capacity and summarizes proof methods. [AI-generated per source] [family-level]
Elsewhere
- Hexagon: Parisi formula for Ising and Spherical Perceptrons (AI-generated; human understanding: some)
- Emergent Mind: family 222
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-free-energy-of-the-Ising-random-perceptron-September-24-2026,
author = {{OpenAI}},
title = {{The free energy of the Ising random perceptron}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026/The-free-energy-of-the-Ising-random-perceptron-September-24-2026.pdf}{OAI:The-free-energy-of-the-Ising-random-perceptron-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/222.md. The Lean statements have not been checked by people against the paper's claims.
Family 222: Perceptron free energies and microscopic jamming exponents
Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.
Other papers in this family:
- Microscopic jamming in the negative spherical perceptron · September 24, 2026
- The spherical perceptron with bi-orthogonally invariant disorder · September 24, 2026
- The free energy of the spherical random perceptron · September 24, 2026