Lemma: Partition legality and vanishing footprints

Every phase is a genuine partition into finite blocks of graph diameter at most one. Every block's spatial diameter is at most $a_n$. In the calibrated spacetime norm

the input–output footprint of a firing has diameter at most $\sqrt{2}{a}_n$.

Proof. The junction cliques are disjoint and contain all endpoint sites. At transport phases, the lane chains have disjoint site labels, and the parity pairs on each chain are disjoint. All remaining sites are explicitly assigned singleton blocks.

Clique and pair blocks have graph diameter one. Their physical diameters satisfy Lemma lem:embedding. The tick duration is $h_n=a_n/c$, giving the spacetime bound.