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The Negative Spherical Perceptron: Capacity, Isostatic Jamming, and Full Replica Symmetry Breaking

Primarily AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.MP — Mathematical Physicsmath-ph — Mathematical Physics

Contributed by P. M. Aronow ↗, Patrick Lopatto ↗

Version 1 / Oct 02, 2026 / CC BY 4.0

Abstract

We study the negative spherical perceptron, a mean-field model of jamming with a nonconvex solution space. We establish the Parisi formula for its free energy and identify the capacity for every constraint density α>2\alpha>2. We also show that the jamming point is isostatic: optimal configurations have exactly NN contacts. For α\alpha slightly above two, we show that replica symmetry is lost at the de Almeida–Thouless margin through a continuous transition to full replica symmetry breaking. Further, we show that Gardner's formula overestimates the capacity by order (α−2)3(\alpha-2)^3.

Provenance statement

Proof discovery and verification were done with a combination of GPT 5.6 Sol, GPT 6 Sol, GPT 6 Astra, Claude Fable 5 and 5.1, and Claude Opus 5.5. Final revisions to the writing were done with Opus 5.5 and GPT 6 Astra.

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Version history

  1. v1Initial depositCurrentOct 02, 2026Submitted by Patrick Lopatto