Packing selector to coherent finite-scale sources

Let $E^4=\mathbb R^4$ with its Euclidean norm and inner product, and let a marked line be a triple $\ell=((\theta,b),m)\in(E^4\times E^4)\times\mathbb R$, with direction $\theta$, offset $b$, and mark $m$. Let $S$ be a measurable set of marked lines such that every $\ell=((\theta,b),m)\in S$ satisfies $\|\theta\|=1$ and $\langle b,\theta\rangle=0$; for every unit $\theta\in E^4$ there exists exactly one $\ell\in S$ with that direction; and the custom packing dimension of the unmarked carrier $\{(\theta,b):\exists m,((\theta,b),m)\in S\}$ is exactly $3$. The custom packing dimension is the infimum of the bounds on upper Minkowski dimensions over countable covers, and upper Minkowski dimension is defined by finite open-ball covering numbers satisfying $N(P,r)\le K(\max\{r,0\})^{-e}$ for all sufficiently small positive $r$, with finite extended-nonnegative-real $e,K$. Then, for every real $\varepsilon>0$, there exist $C\in[0,\infty]$, a measure $\mu$ on $E^4$, and $\delta_0>0$ such that $C\ne0,\infty$, $\mu$ is a probability measure supported on

and, for every $0<\delta\le\delta_0$, there exist a natural number $n$ (possibly $0$) and finite-scale data $D$ indexed by $\operatorname{Fin}(n)$. These data consist of thickness $h=\delta$, marked lines $\ell_i=((\theta_i,b_i),m_i)\in S$, measurable shadings $Y_i\subseteq E^4$, weights $w_i\in[0,\infty]$, recorded fibre marks $q_i\in\mathbb R$, and a nested carrier tree whose child levels are larger, child cells are contained in parent cells, and whose $i$th cell contains $(\theta_i,b_i)$. They satisfy $0<h<1$, $w_i\le1$, $q_i=m_i$, $\|\theta_i\|=1$, $\langle b_i,\theta_i\rangle=0$, every $y\in Y_i$ is within distance $h$ of the marked unit segment of $\ell_i$, and for $h\le r\le1$,

Writing $f_D(y)=\sum_i w_i\mathbf1_{Y_i}(y)$ and $M(D)=\int f_D\,d\operatorname{vol}$, one has $C^{-1}\le M(D)\le C$. Finally, for every $x\in E^4$ there exist data $R$ with the same index set, thickness, marked lines, fibre marks, and entire tree as $D$, with measurable $Y_i^R\subseteq Y_i$ and $w_i^R\le w_i$, such that, for $f_R(y)=\sum_iw_i^R\mathbf1_{Y_i^R}(y)$ and $M(R)=\int f_R\,d\operatorname{vol}$,