  Disordered interacting spin chains that undergo a many-body localization
transition are characterized by two limiting behaviors where the dynamics are
chaotic and integrable. However, the transition region between them is not
fully understood yet. We propose here a possible finite-size precursor of a
critical point that shows a typical finite-size scaling and distinguishes
between two different dynamical phases. The kurtosis excess of the diagonal
fluctuations of the full one-dimensional momentum distribution from its
microcanonical average is maximum at this singular point in the paradigmatic
disordered $J_1$-$J_2$ model. For system sizes accessible to exact
diagonalization, both the position and the size of this maximum scale linearly
with the system size. Furthermore, we show that this singular point is found at
the same disorder strength at which the Thouless and the Heisenberg energies
coincide. Below this point, the spectral statistics follow the universal random
matrix behavior up to the Thouless energy. Above it, no traces of chaotic
behavior remain, and the spectral statistics are well described by a
generalized semi-Poissonian model, eventually leading to the integrable
Poissonian behavior. We provide, thus, an integrated scenario for the many-body
localization transition, conjecturing that the critical point in the
thermodynamic limit, if it exists, should be given by this value of disorder
strength.
