consists of bounded, continuous functions $\pmb{f} : \mathbb{R}^3 \to \mathbb{R}^3$ subjected to the following uniformly local condition. For $\pmb{x}_0 \in \mathbb{R}^3$, select the/a component $f_i^\pm$ such that $f_i^\pm(\pmb{x}_0) = \|\pmb{f}(\pmb{x}_0)\|$ (here, the norm of a vector $\pmb{v}=(a, b, c)$, $\|\pmb{v}\|$, will be computed as $\max \{|a|, |b|, |c|\}$), and require that the super-level set $$\biggl\{ \pmb{x} \in \mathbb{R}^3: \, f_i^\pm(x) > \lambda \|\pmb{f}\|_\infty\biggr\}$$ be $\delta$-sparse around $\pmb{x}_0$ at scale $\frac{1}{c} \frac{1}{\|\pmb{f}\|_\infty^\alpha}$, for some $c, \frac{1}{c_0} \le c \le c_0$. Enforce this for all $\pmb{x}_0 \in \mathbb{R}^3$. (In short, we require sparseness of the/a *locally maximal* component only.)

In this setting, the regularity class is $Z_\frac{1}{2}$ (pointwise-in-time, in the context of a blow-up argument) [@BrFaGr18]; this is on the scaling level of all the classical regularity criteria. The *a priori* bound obtained lives in $Z_\frac{2}{5}$ (pointwise-in-time, in the context of a blow-up argument) [@BrFaGr18]; since the energy level class is $Z_\frac{1}{3}$, this represents a $40\%$ reduction of the scaling gap in the $Z_\alpha$ framework!

It is instructive to make a quick comparison between the $Z_\alpha$-classes and the weak Lebesgue classes $L^p_w$ determined solely by the rate of decay of the volume of the super-level sets, encoding no geometric information. On one 