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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 formationof singularities, is a fundamental open problem in mathematical physics. Due tothe super-criticality of the equations, the problem has been super-critical inthe sense that there has been a scaling gap between any regularity criterionand the corresponding \emph{a priori} bound (regardless of the functional setuputilized). The purpose of this work is to present a mathematicalframework--based on a suitably defined scale of sparseness of the super-levelsets of the positive and negative parts of the components of the higher-orderspatial derivatives of the velocity field--in which the scaling gap between theregularity class and the corresponding \emph{a priori} bound vanishes as theorder of the derivative goes to infinity.

Asymptotic Criticality of the Navier-Stokes Regularity Problem