  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 work of Perkins-Xu '21, the last two conjectures were also
established by proving that the partition function concentrates on an
exponential scale, under an analytical assumption on a real-valued function.
This left open the contiguity conjecture and the lognormal limit
characterization, which are established here unconditionally, with the
analytical assumption verified. 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.
