  We consider the symmetric binary perceptron model, a simple model of neural
networks that has gathered significant attention in the statistical physics,
information theory and probability theory communities, with recent connections
made to the performance of learning algorithms in Baldassi et al. '15.
  We establish that the partition function of this model, normalized by its
expected value, converges to a lognormal distribution. As a consequence, this
allows us to establish several conjectures for this model: (i) it proves the
contiguity conjecture of Aubin et al. '19 between the planted and unplanted
models in the satisfiable regime; (ii) it establishes the sharp threshold
conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case,
conjectured first by Krauth-M\'ezard '89 in the asymmetric case.
  In a recent concurrent work of Perkins-Xu [PX21], the last two conjectures
were also established by proving that the partition function concentrates on an
exponential scale. This left open the contiguity conjecture and the lognormal
limit characterization, which are established here. In particular, our proof
technique relies on a dense counter-part of the small graph conditioning
method, which was developed for sparse models in the celebrated work of
Robinson and Wormald.
