Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron

Abstract

We study the replica-symmetric saddle point equations for the Ising perceptron with Gaussian disorder and margin $\kappa\ge 0$. We prove that for each $\kappa\ge 0$ there is a critical capacity $\alpha_c(\kappa)=\frac{2}{\pi\,\mathbb E[(\kappa-Z)_+^2]}$, where $Z$ is a standard normal and $(x)_+=\max\{x,0\}$, such that the saddle point equation has a unique solution for $\alpha\in(0,\alpha_c(\kappa))$ and has no solution when $\alpha\ge \alpha_c(\kappa)$. When $\alpha\uparrow \alpha_c(\kappa)$ and $\kappa>0$, the replica-symmetric free energy at this solution diverges to $-\infty$. In the zero-margin case $\kappa=0$, Ding and Sun obtained a conditional uniqueness result, with one step verified numerically. Our argument gives a fully analytic proof without computer assistance. We used GPT-5 to help develop intermediate proof steps and to perform sanity-check computations.