Remark: 222-02. Microscopic jamming in the negative spherical perceptron

Authors: OpenAI. Family 222.

Title: Microscopic jamming in the negative spherical perceptron.

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.

This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.