Capacity Threshold for the Ising Perceptron

We show that the capacity of the Ising perceptron is with high probability upper bounded by the constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha_{\star}\approx 0.833$</tex> conjectured by Krauth and Mézard, under the condition that an explicit two-variable function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathscr{S}_{\star}\left(\lambda_1, \lambda_2\right)$</tex> is maximized at (1,0). The earlier work of Ding and Sun [1] proves the matching lower bound subject to a similar numerical condition, and together these results give a conditional proof of the conjecture of Krauth and Mézard,