Nested uniform scrambles preserve net counts and uniform marginals

Fix a point $a_i$ and coordinate $j$. Its input prefixes of lengths $0,1,\dots ,K-1$ are distinct nodes, so the corresponding permutations are independent. Conditional on the fixed input digits, each output digit is uniform on $\{0,\dots ,b-1\}$, independently of the earlier output digits. For each $m\ge 1$, the event that all output digits from position $m$ onward equal $b-1$ is contained, for every $L\ge 1$, in the event that its first $L$ digits equal $b-1$, which has probability $b^{-L}$; [F5] makes that tail event null. The countable union over $m$ is null. Outside it, every specified output prefix of length $K$ puts the numeric output in the corresponding half-open base-$b$ interval, not at its right endpoint. These cylinders form a generating $\pi$-system, so [F4] extends their probabilities $b^{-K}$ to the uniform Borel law. Independent permutation families across coordinates give $X_i\sim \mathrm{Unif}([0,1{)}^d)$. The finite set of points and coordinates has a common probability-one event on which none of their expansions is eventually all $b-1$.

At any fixed digit depth $K$, the map from input words of length $K$ to output words is a bijection: it is a bijection at depth one, and at each subsequent depth every existing prefix is extended by the bijection supplied by its own permutation. Taking the coordinatewise product gives a bijection of all depth-vector digit boxes. It therefore permutes the boxes of total depth $m-t$ and preserves the number of input points in each image box. On the probability-one event from step 1.1, these digit boxes are exactly the numeric half-open elementary intervals of the scrambled coordinates. By [F1] every such output box contains $\lambda b^t$ points.

Step 1.1 proves marginal uniformity and step 2.1 proves net preservation outside the null expansion events identified in step 1.1. These are exactly the two assertions.