  This paper completes the program started in arXiv:2104.11204 and
arXiv:2110.04565 aiming at providing a full rigorous justification of the wave
kinetic theory for the nonlinear Schr\"odinger (NLS) equation. Here, we cover
the full range of scaling laws for the NLS on an arbitrary periodic rectangular
box, and derive the wave kinetic equation up to small multiples of the kinetic
time.
  The proof is based on a diagrammatic expansion and a deep analysis of the
resulting Feynman diagrams. The main novelties of this work are three-fold: (1)
we present a robust way to identify arbitrarily large "bad" diagrams which
obstruct the convergence of the Feynman diagram expansion, (2) we
systematically uncover intricate cancellations among these large "bad"
diagrams, and (3) we present a new robust algorithm to bound all remaining
diagrams and prove convergence of the expansion. These ingredients are highly
robust, and constitute a powerful new approach in the geneal mathematical study
of Feynman diagrams.
