Microscopic jamming in the negative spherical perceptron

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