to the intrinsic NS scaling transformation, and all the *a priori* bounds have been on the scaling-level of Leray’s original energy bound. Recently, however, a new mathematical framework [@BrFaGr18] enabled the first *scaling reduction* of the NS super-criticality (more precisely, the *a priori* bound derived represents a $40\%$ improvement over the energy level bounds) since the 1960s; the key concept in the theory is a suitably defined ‘scale of sparseness’ (a precise definition resides in the following section) of the super-level sets of the positive and the negative parts of the vorticity components. At this point, a natural question arises of whether there might be an intrinsic obstruction to advancing this method, or whether this new framework might have more to offer (a further reduction of the scaling gap).

In this work we present the first numerical analysis of this novel geometric scale applied to two conventional scenarios i) a Kida-vortex initialized simulation: a model flow for the study of possible singular events in the 3D incompressible fluid flows (away from the boundary), in particular because it exhibits a ‘burst’ of vorticity maximum, resembling a finite singularity [@BP94] ii) a simulation in the realm of fully developed, homogeneous, isotropic turbulence 