Theorem: Complete clock–table–transport realization

Under Assumption ass:operations, the preceding finite regular protocol realizes

Here ${\mathcal{U}}_{A\to B}$ is exact transport of the payload factor and ${\omega }_{\mathrm{blank}}$ is the returned blank payload/work state. The formula is valid with arbitrary untouched references. Its complete dependencies are causally embedded in the fixed host, and the output arrives no later than $T_0+2+3L$.

Proof. Proposition prop:table gives the subnormalized clock–catalogue–sign branches. Copying a classical record gives the displayed correlated record copies, not copies of an unknown quantum state. Lemma lem:swap proves exact payload transport in every branch. All branches are retained, so their sum is the stated complete channel.

Local gates lie on regular timelike station worldlines. The communication paths have speed $1/2$, remain inside the regular tube, and deliver every control before its next use. Theorem thm:causal applies. The arrival bound is the sum of the $n$ stage durations.