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Microscopic jamming in the negative spherical perceptron

OpenAI · September 24, 2026

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Abstract

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.

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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 that, among other consequences, claims the predicted variational formulas for the entropy and capacity of the negative spherical perceptron. Its disclaimer says an initial capacity draft dates from July 2026 and cites this paper among three overlapping OpenAI manuscripts. [AI-generated per source] [family-level]

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Cite

@misc{OAI:Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026,
  author = {{OpenAI}},
  title = {{Microscopic jamming in the negative spherical perceptron}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026/Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026.pdf}{OAI:Microscopic-jamming-in-the-negative-spherical-perceptron-September-24-2026}},
  year = {2026}
}

Lean formalization

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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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