  Necessary and sufficient conditions for the third order Kolmogorov universal
scaling flux laws are derived for the stochastically forced incompressible
Navier-Stokes equations on the torus in 2d and 3d. This paper rigorously
generalizes the result of \cite{bedrossian2019sufficient} to functions which
are heavy-tailed in Fourier space or have local finite time singularities in
the inviscid limit. In other words we have rigorously derived the well known
physical relationship the direct cascade is a local process and is non-trivial
if and only if energy moves toward the small scales or singularities have
occurred. Similarly, an inverse cascade occurs if and only if energy moves
towards the $k = 0$ Fourier mode in the invisicid limit.
