{"id":"hexagon/2610.00019","pdf":"https://hexagonmath.org/pdf/2610.00019v1","url":"https://hexagonmath.org/abs/2610.00019v1","tags":["math.MP","math-ph","msc:60K35","msc:82B44","msc:82B26","perceptron","spin glass"],"score":0,"title":"The Negative Spherical Perceptron: Capacity, Isostatic Jamming, and Full Replica Symmetry Breaking","domain":"math.PR","origin":"ai-generated","source":"hexagon","authors":["P. M. Aronow","Patrick Lopatto"],"reviews":0,"subject":"math.PR","summary":"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 $\\alpha>2$. We also show that the jamming point is isostatic: optimal configurations have exactly $N$ contacts. For $\\alpha$ 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 $(\\alpha-2)^3$.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Claude Fable","Claude Fable","ChatGPT","ChatGPT"],"discussed":0,"published":"2026-10-02","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}