  We prove, under an assumption on the critical points of a real-valued
function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB'
structure conjectured by Krauth and Mezard in the physics literature; that is,
typical solutions of the model lie in clusters of vanishing entropy density.
Moreover, we prove this in a very strong form conjectured by Huang, Wong, and
Kabashima: a typical solution of the model is isolated with high probability
and the Hamming distance to all other solutions is linear in the dimension. The
frozen 1-RSB scenario is part of a recent and intriguing explanation of the
performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti,
and Zecchina. We prove this structural result by comparing the symmetric Ising
perceptron model to a planted model and proving a comparison result between the
two models. Our main technical tool towards this comparison is an inductive
argument for the concentration of the logarithm of number of solutions in the
model.
