  This is the second in a series of papers where we analyze the incompressible
Navier-Stokes equations in H\"older spaces. We obtain, to our knowledge, the
very first genuinely super-critical regularity criterion for this system of
equations in any dimension $d\geq3$ and in the absence of physical boundaries.
For \emph{any} $\beta\in(0,1)$, we show that $L_t^1C_x^{0,\beta}$ solutions
emanating from smooth initial data do not develop any singularities. The
novelty stems from obtaining new bounds on the fundamental solution associated
with a one-dimensional drift-diffusion equation in the presence of
destabilizing singular lower order terms. Such a bound relies heavily on the
symmetry and pointwise structure of the problem, where the drift term is shown
to ``enhance'' the parabolic nature of the equation, allowing us to break the
criticality barrier. Coupled with a subtle regularity estimate for the pressure
courtesy of Silvestre, we are able to treat the (incompressible) Navier-Stokes
equation as a perturbation of the classical drift-diffusion problem. This is
achieved by propagating moduli of continuity as was done in our previous work,
based on the elegant ideas introduced by Kiselev, Nazarov, Volberg and
Shterenberg.
