there is a ‘scaling gap between any known regularity criterion and the corresponding *a priori* bound. If we focus on $L^\infty((0,T), X)$-type spaces ($L^\infty$ in time, spatially in $X$), an illustrative example is in the $L^p$-scale where the regularity class is $L^\infty((0,T), L^3)$ [@ESS03], and the corresponding *a priori* bound the Leray bound $L^\infty((0,T), L^2)$. Moreover, since the fundamental (independent) works of Ladyzhenskaya, Prodi and Serrin, as well as Kato and Fujita, all in 1960s, the scaling gap has been of the *fixed size* in the sense that all the regularity classes are (at best) *scaling-invariant* (with respect to the intrinsic NS scaling) while–in contrast–all the known *a priori* bounds are on the scaling level of the Leray *energy bound*.

It turned out that redesigning the theory introduced in [@Gr13] around the super-level sets of the *vorticity components* instead of the vectorial super-level sets (see Figure 1) produced the first ‘modern era’ *algebraic reduction* of the scaling gap; since 1960s, all the improvements were logarithmic in nature, regardless of the mathematical framework utilized.

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   ![An illustration of the difference between sparseness of the super-level sets of the individual components and the full vectorial super-level set. The figure on the 