The storage capacity of the Ising perceptron: verification of the outstanding numerical conditions

Abstract

Krauth and Mézard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant $α_\star\approx0.8330786$. Let $M_N$ be the largest number of random patterns that can be stored by an $N$-dimensional Ising perceptron. We give a computer-assisted proof that \[ \frac{M_N}{N}\xrightarrow{\mathbb P}α_\star, \qquad α_\star\in[0.833078599,0.833078600]. \] Previous work established matching conditional lower and upper bounds, subject respectively to a one-variable global sign condition of Ding--Sun and a two-variable global sign condition of Huang. We rigorously verify both conditions using Arb ball arithmetic. For Huang's condition, a moment-coordinate reparametrization compresses the unbounded parameter plane onto a compact convex body. Convex duality and certified adaptive sweeps control its bulk, while a ray-concavity argument treats the degenerate maximizer. We also re-establish the shared parameter rectangle and verify the full Ding--Sun condition, including its curvature and endpoint requirements. Combining these verifications with the existing sharp-threshold and universality theorems proves the result for Gaussian disorder and for every fixed i.i.d.\ mean-zero, unit-variance subgaussian disorder law, including Bernoulli disorder. The complete verification programs, certificates, and raw records accompany the paper.