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. (Lean)
Manuscripts (4)
The free energy of the Ising random perceptron
September 24, 2026 · source · cite as \cite{OAI:The-free-energy-of-the-Ising-random-perceptron-September-24-2026}
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.
Microscopic jamming in the negative spherical perceptron
September 24, 2026 · source · cite as \cite{OAI:Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026}
We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances 1-\varepsilon and 1+\varepsilon, for every sufficiently small fixed ε. The contact-removed gap cumulative law and the mean-one force cumulative law satisfy
\displaystyle G_J(u)=u^{1-\gamma+o(1)},\qquad F_J(s)=s^{1+\theta+o(1)},
as u\downarrow0 and s\downarrow0, with \gamma=(2+\theta)^{-1}, 0.4126930\lt \gamma\lt 0.4126934, and 0.4231063\lt \theta\lt 0.4231088. A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.
The spherical perceptron with bi-orthogonally invariant disorder
September 24, 2026 · source · cite as \cite{OAI:The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026}
We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.
The free energy of the spherical random perceptron
September 24, 2026 · source · cite as \cite{OAI:The-free-energy-of-the-spherical-random-perceptron-September-24-2026}
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.
Result 222 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md