Lemma: Microscopic speed and directional refinement

Every register hop has displacement at most $a_n$ and duration ${\tau }_n$, hence speed at most $c$. Block footprints have $d_c$-diameter at most $\sqrt{2}{a}_n$.

The convex hulls

converge in Hausdorff distance to the Euclidean ball ${\overline{B}}_c(0)$.

Proof. The speed and footprint statements follow from (eq:scales)–(eq:stencil).

The inclusion $V_n\subseteq {\overline{B}}_c(0)$ is immediate. For a unit vector $u$, round $(n^2-\sqrt{D})u$ coordinatewise to an integer vector $s$. For sufficiently large $n$,

and

Thus every point on the sphere of radius $c$ is uniformly close to an allowed velocity. Convexity and the origin give the claim for the full ball.