Digital shifts preserve net counts and uniform marginals

For a fixed point and coordinate, adding an independent uniform digit modulo $b$ makes the output digit uniform; the output digits are independent because the shift digits at distinct positions are independent. Thus every prescribed output prefix of length $K$ has probability $b^{-K}$. For each $m\ge 1$, the event that every output digit from position $m$ onward is $b-1$ is contained, for every $L\ge 1$, in the event that its first $L$ tail digits are all $b-1$, which has probability $b^{-L}$; [F5] makes that tail event null. Outside the countable union of these null events, each prefix places the numeric output in its corresponding half-open interval, not at its right endpoint. These intervals form a generating $\pi$-system, so [F4] extends the prefix probabilities to the uniform Borel law. The coordinate shift streams are independent, hence the same cylinder calculation for rectangles in $[0,1{)}^d$ gives each vector the uniform cube law. The half-open modification at output value $1$ changes only a null event.

For any coordinate and depth $K$, adding a fixed digit word modulo $b$ is a bijection of the finite set of length-$K$ words. The full coordinatewise map is therefore a bijection of the depth-vector digit boxes. It permutes, without splitting, all boxes of total depth $m-t$, preserving their point counts. Outside the null expansion events in step 1.1 these digit boxes are exactly the numeric half-open elementary intervals, so [F1] gives the same count $\lambda b^t$ in each shifted box.

Step 1.1 proves each shifted point uniform, and step 2.1 proves preservation of the net counts with probability one.