Free energies of random perceptrons

Free energies of random perceptrons are the subject of four manuscripts dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math). This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

The original results

• Claims (summary in the collection's contents list):
  • 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, by an explicit variational formula;
  • an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential, at every positive density and inverse temperature;
  • the analogous limit for a spherical perceptron with bi-orthogonally invariant disorder whose singular values have a compact limiting distribution and no outliers;
  • for the spherical perceptron with margin −1 and quadratic penalty, a sharp feasibility threshold and limiting gap and force laws, with exponents in stated intervals supported by a finite numerical certificate.
• Sources (openai/math repository, commit 1581d4c): Ising random perceptron, spherical random perceptron, bi-orthogonally invariant disorder and microscopic jamming.
• Formalization, as described by the repository. The scope note lean/docs/222.md states that the formalization proves the variational formula for the spherical random perceptron's limiting pressure (for every positive density and inverse temperature and every bounded continuous single-pattern potential, with convergence in expectation and in probability), together with a supporting dual formula. For the Ising perceptron it proves only that the variational value is finite, a supporting well-definedness result; convergence of the finite-system pressure and the extension to bounded Borel log-potentials are not asserted.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 15 October 2026)
2026-10-08	Daniel Fu and Youngtak Sohn, Hexagon record 2610.00154v1, titled Parisi formula for Ising and Spherical Perceptrons	Parallel work	A Parisi formula for Ising and spherical perceptrons with Gaussian disorder and independent random potentials with a common Lipschitz bound, by cavity in both dimensions with the Aizenman–Sims–Starr scheme for the lower bound and Guerra's interpolation for the upper bound; the abstract also claims answers to several problems posed by Talagrand, a quenched large deviation principle (a conjecture of Bolthausen and Kistler), and the sharp-threshold conjecture of Aubin, Perkins and Zdeborová for the u-function binary perceptron. The record states that the arguments were developed with ChatGPT 5.6 Pro from July 2026, and that the authors posted their working draft after learning on 6 October 2026 of the overlapping OpenAI manuscripts.	Hexagon labels: primarily AI-generated text; human understanding: some parts.

Notes

• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.