Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron

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 $-\mathrm{\infty }$. 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.