View source for 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 ([https://github.com/openai/math 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 &minus;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 <code>1581d4c</code>): [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026 Ising random perceptron], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026 spherical random perceptron], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026 bi-orthogonally invariant disorder] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026 microscopic jamming].
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/222.md 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 ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-08 || Daniel Fu and Youngtak Sohn, Hexagon record [https://hexagonmath.org/2610.00154v1 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.

[[Category:OpenAI results]]