Lemma: Repeated vertices and the register count

For the route (eq:route-points), every coordinate sequence $k\mapsto (p_k{)}_i$ is monotone. If $p_k=p_l$ for $k<l$, then every intermediate vertex equals that same point. Consequently, after deleting consecutive repetitions, the route has distinct vertices. If $m$ is its number of nonzero increments and $q$ its number of distinct sites, then

Proof. The affine coordinate $k{b}_i/N$ is monotone, and the scalar rounding rule is monotone. A monotone scalar sequence with equal values at two indices has the same value at every intermediate index. Applying this to every coordinate proves the assertion about repeated vertices. Each nonzero increment then introduces one new site.