Proposition: Edge partitions and relay attainment

Every generator edge extends to a partition into generator pairs. Every finite word path can therefore be relayed one edge per tick by admissible partitions.

The schedule class has $c_{\mathrm{lat}}=1$. Its maximal Euclidean signalling speed is

Proof. For a generator $s\ne 0$, the graph of edges $z\leftrightarrow z+s$ is a disjoint union of bi-infinite chains, the cosets of $\mathbb{Z}s$. Choose one representative for each coset and write its sites as $z_0+ks$. Pair $2k+p$ with $2k+p+1$, $p=0,1$. Either phase is a full partition.

Choose the phase containing each desired path edge and let other blocks act identically. Connectedness and finite balls on the infinite word graph give a vertex at every integer distance, so geodesic bit relays attain the combinatorial upper bound $1$.

Every block has physical diameter at most $h_n{\max }_s\Vert s{\Vert }_2$. The same one-layer argument bounds influence by that reach per tick. A record transport along a maximum-length edge attains the corresponding one-tick ratio. Long generators give the displayed lower bound.