Theorem: Reference-preserving endpoint transport

For every input ${\rho }_{A_{0}E}$, after tick $j$ the state is

Therefore:

1. the logical payload channel is exactly ${\mathrm{id}}_X$, with every untouched reference;

2. all non-output route registers finish jointly blank and payload independent;

3. arrival is at $T_{\mathrm{d}}+L/c$;

4. all tick-boundary dependencies are metric causal;

5. the protocol uses $n$ register SWAPs, $n$ ticks, and $n+1$ three-qubit payload registers.

Proof. At each step the next register is blank and independent. SWAP transfers the entire current payload factor, including all reference correlations, to that register and leaves the previous one blank. Induction gives the state identity.

Each separation is $L/n=c\Delta T$, and the developed segment is inside the convex patch. Lemma lem:causal certifies the dependencies. The total time and resource counts are immediate.