  The problem of global-in-time regularity for the 3D Navier-Stokes equations,
i.e., the question of whether a smooth flow can exhibit spontaneous formation
of singularities, is a fundamental open problem in mathematical physics. Due to
the super-criticality of the equations, the problem has been super-critical in
the sense that there has been a scaling gap between any regularity criterion
and the corresponding \emph{a priori} bound (regardless of the functional setup
utilized). The purpose of this work is to present a mathematical
framework--based on a suitably defined scale of sparseness of the super-level
sets of the positive and negative parts of the components of the higher-order
spatial derivatives of the velocity field--in which the scaling gap between the
regularity class and the corresponding \emph{a priori} bound vanishes as the
order of the derivative goes to infinity.
