We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.

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.

We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.

We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.