OpenAI · 222-02 · 2026-09-24 · Probability and statistical mechanics

Microscopic jamming in the negative spherical perceptron

preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026

Family 222. Perceptron free energies and microscopic jamming.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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.

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 $1 - ε$ and $1 + ε$ , for every sufficiently small fixed ε. The contact-removed gap cumulative law and the mean-one force cumulative law satisfy $G_{J} (u) = u^{1 - γ + o (1)}, F_{J} (s) = s^{1 + θ + o (1)},$ as $u ↓ 0$ and $s ↓ 0$ , with $γ = (2 + θ)^{- 1}$ , $0.4126930 < γ < 0.4126934$ , and $0.4231063 < θ < 0.4231088$ . A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.

Preprint source and Lean scope

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A Lean scope note is stored for family 222, but it does not name this preprint. It names 222-01, 222-04. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.