It is shown that, if the vorticity magnitude associated with a (presumed singular) three-dimensional incompressible Navier-Stokes flow blows-up in a manner exhibiting certain { \em time dependent local structure}, then { \em time independent} estimates on the $L^1$ norm of $|\omega|\log\sqrt{ 1+ |\omega|^2}$ follow. The implication is that the volume of the region of high vorticity decays at a rate of greater order than a rate connected to the critical scaling of one-dimensional local sparseness and, consequently, the solution becomes sub-critical.
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