hand, it is straightforward that $\displaystyle{
 f \in L^p_w \ \mbox{implies} \, f \in Z_\alpha \  \mbox{for} \  \alpha=\frac{p}{3}}$ (for a given selection of $\lambda$ and $\delta$, the tolerance parameter $c_0$ will depend on the $L^p_w$-semi-norm of $f$); on the other hand, in the geometrically worst case scenario–the whole super-level set being clumped into a single ball–being in $Z_\alpha$ is consistent with being in $L^{3 \alpha}_w$ (however, one should note that–in general–membership in $Z_\alpha$ does not impose any decay on the volume of the super-level sets since it provides information on the suitably defined scale of the ‘largest piece’ but no information on the number of ‘pieces’). This observation reveals that a ‘geometrically blind’ scaling counterpart of the *a priori* bound in $Z_\frac{2}{5}$ would be the bound $\omega \in L^\infty((0,T), L^\frac{6}{5}_w)$ which is well beyond state-of-the-art ($\omega \in L^\infty((0,T), L^1)$ [@Co90]), demonstrating a clear advantage of working within the $Z_\alpha$ realm compared to the functional classes traditionally utilized in the study of the 3D NS regularity problem.

Kida vortex-initialized flows
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In the first two subsections we interpret the information gathered via computational simulations of the Kida vortex-initialized flow within the realm of possible singularity formation in the 3D NSE, 