Lemma: A rounded straight hub route

For $k,k^{\prime }\in {\mathbb{Z}}^2$, put $L=\Vert {\delta }_n(k^{\prime }-k){\Vert }_2$. There is a hub route using $m$ stencil steps, with

Every hub on the route is within ${\rho }_n$ of the straight segment joining the two hubs.

Proof. If $k=k^{\prime }$, use no steps. Otherwise put $w=k^{\prime }-k$ and choose

Round each $jw/m$, $j=0,\dots ,m$, to a nearest integer vector $k_j-k$, retaining the exact endpoints. Consecutive differences satisfy

Delete zero differences. Every remaining difference belongs to $P_n$. The number of steps does not increase. The rounding error at each hub is at most ${\rho }_n$ in physical units.