Lemma: Every-prefix paced path

Let $x,y\in {\mathbb{R}}^D$, let $N\ge 1$, and reserve one residue $r$. Define

If

then successive distinct positions are connected by legal generator-pair blocks. Equal successive positions are waits.

For every prefix $k$,

The piecewise affine interpolation stays within $b_n$ of the affine target path under the common affine parametrization.

Proof. Write

Then

which proves legality. Subtracting the expression for $y_0$ gives (eq:prefixestimate). Between successive samples, the error is a convex combination of $e_k$ and $e_{k+1}$, so its norm is at most $b_n$.