Nested uniform scrambling of a finite point set

Fix a base $b\ge 2$ and points $a_1,\dots ,a_n\in [0,1{)}^d$. Write each coordinate in its terminating base-$b$ expansion when it has two expansions, and otherwise in its unique expansion: $a_{i,j}={\sum }_{k\ge 1}{a}_{i,j,k}{b}^{-k},a_{i,j,k}\in \{0,\dots ,b-1\}\mathrm{.}$ Enumerate the countable pairs $(j,w)$ by increasing word length, then coordinate, then lexicographic word order. For every such pair use a uniform random permutation ${\pi }_{j,w}$ of $\{0,\dots ,b-1\}$, mutually independent over all pairs; A single uniform variable supplies countably many independent base-$b$ digits supplies this array. Define $x_{i,j,k}={\pi }_{j,(a_{i,j,1},\dots ,a_{i,j,k-1})}(a_{i,j,k}),X_{i,j}={\sum }_{k\ge 1}{x}_{i,j,k}{b}^{-k}\mathrm{.}$ The same permutations are used for all points in a coordinate; different coordinates use independent permutation families. The resulting random point array is the nested uniform scramble of the input array. Null events on which an output expansion is eventually all $b-1$ do not alter its law; choose the usual half-open representative there.