Lemma: Vanishing timing slack for a finite skeleton

For every causal finite skeleton, there are phase-zero firing times $t_{v,n}\to t_v$ and approximating hubs $H_{v,n}\to x_v$ such that every prescribed wire can be delivered after its source operation and before its target operation. The choices depend only on the skeleton, not on the channel assignments.

Proof. Choose each approximating hub by a fixed nearest-hub rule. Its error is at most ${\rho }_n$.

Choose a topological rank $r_v\in \{0,\dots ,M\}$, strictly increasing on every edge. Let $L_{\ast }$ bound all edge lengths in the finite skeleton, and put

These are phase-zero times and converge to $t_v$. For an edge $v\to w$,

The approximating hub distance is at most $\Vert x_w-x_v{\Vert }_2+2{\rho }_n$. Charge one tick for the source operation and use Theorem thm:routing from its output tick. Operation plus delivery costs at most

The causal inequality and the definition of $e_n$ make this no larger than $t_{w,n}-t_{v,n}$. All arrivals may be retained in hub memory until the target firing.

The same argument covers null connections and zero-duration co-located edges. Their added rank-dependent delays vanish.