Franz-Parisi jamming exponents in the negative spherical perceptron, at margin -1

Theorem 1.1: for the spherical perceptron with margin $-1$ and quadratic penalty there is a sharp feasibility threshold $\alpha_c$; the ordered limits (size, then zero temperature, then $\alpha\downarrow\alpha_c$) of the contact-removed gap law and mean-one force law exist with $G_J(u)=u^{1-\gamma+o(1)}$, $F_J(s)=s^{1+\theta+o(1)}$, $\gamma=(2+\theta)^{-1}$, $0.4126930<\gamma<0.4126934$, $0.4231063<\theta<0.4231088$, and the critical limit is isostatic. The same holds for a small Gaussian scale mixture of coordinates. Not shown: other negative margins, the unpenalized hard-constraint model directly, other limit orders, or finite-size scaling.

Theorem 1.1: for the spherical perceptron with margin $-1$ and quadratic penalty there is a sharp feasibility threshold $\alpha_c$; the ordered limits (size, then zero temperature, then $\alpha\downarrow\alpha_c$) of the contact-removed gap law and mean-one force law exist with $G_J(u)=u^{1-\gamma+o(1)}$, $F_J(s)=s^{1+\theta+o(1)}$, $\gamma=(2+\theta)^{-1}$, $0.4126930<\gamma<0.4126934$, $0.4231063<\theta<0.4231088$, and the critical limit is isostatic. The same holds for a small Gaussian scale mixture of coordinates. Not shown: other negative margins, the unpenalized hard-constraint model directly, other limit orders, or finite-size scaling.