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The free energy of the spherical random perceptron

OpenAI · September 24, 2026

Lean

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Abstract

We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in 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 $\alpha\in(0,\infty)$, with a spherical entropy term of Crisanti–Sommers type. 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). According to this note, the release's Lean development proves the main theorem for every $\alpha,\beta>0$ and every bounded continuous potential: the variational value is finite, and the pressure converges to it in expectation and in probability. It also proves a supporting dual formula and approximation result for the spherical linear field. [family-level]

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:The-free-energy-of-the-spherical-random-perceptron-September-24-2026,
  author = {{OpenAI}},
  title = {{The free energy of the spherical random perceptron}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf}{OAI:The-free-energy-of-the-spherical-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.

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