  We consider the Ising perceptron with gaussian disorder, which is equivalent
to the discrete cube $\{-1,+1\}^N$ intersected by $M$ random half-spaces. The
perceptron's capacity is $\alpha_N \equiv M_N/N$ for the largest integer $M_N$
such that the intersection in nonempty. It is conjectured by Krauth and
M\'ezard (1989) that the (random) ratio $\alpha_N$ converges in probability to
an explicit constant $\alpha_\star \doteq 0.83$. Kim and Roche (1998) proved
the existence of a positive constant $\gamma$ such that $\gamma \le \alpha_N
\le 1-\gamma$ with high probability; see also Talagrand (1999). In this paper
we show that the Krauth--M\'ezard conjecture $\alpha_\star$ is a lower bound
with positive probability, under the condition that an explicit univariate
function $S(\lambda)$ is maximized at $\lambda=0$. Our proof is an application
of the second moment method to a certain slice of perceptron configurations, as
selected by the so-called TAP (Thouless, Anderson, and Palmer, 1977) or AMP
(approximate message passing) iteration, whose scaling limit has been
characterized by Bayati and Montanari (2011) and Bolthausen (2012). For
verifying the condition on $S(\lambda)$ we outline one approach, which is
implemented in the current version using (nonrigorous) numerical integration
packages. In a future version of this paper we intend to complete the
verification by implementing a rigorous numerical method.
