Lemma: Private lanes and local access

Different assigned residues never share a physical lattice site, including when $D=1$.

For a vertex $v$, let its hub be $Q_{r_v,n}(x_v)$. For an incident edge $e$, its endpoint port is $Q_{r_e,n}(x_v)$. Their distance is at most $2b_n$. For sufficiently large $n$, one legal pair-block swap transfers a register between that port and the hub.

Proof. The first coordinate of a site in ${\Lambda }_{r,n}$, divided by $h_n$, is congruent to $r$ modulo $L$. Distinct residues are therefore disjoint. This argument does not use a transverse spatial dimension.

The access distance is at most $2b_n$ by (eq:roundingbound). Since

the access displacement lies in the allowed stencil for all sufficiently large $n$.