Theorem: Reference-preserving null transport

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

under the fixed register identifications. The destination channel is exactly ${\mathrm{id}}_D$, including the reference. Arrival is at $q$. The protocol uses $m$ SWAP gates, $m+1$ payload-sized registers, and $m$ ticks, and every tick-boundary dependency is metric causal.

Proof. The invariant holds initially. At tick $j$, the destination factor is blank and independent. SWAP exchanges it with the entire current payload factor, including its correlations with $E$, leaving the previous factor blank. Induction proves the invariant and the complete identity channel. Each active spatial separation is $a=c\delta$, and the segment remains in the developing ball. The total duration is $m\delta =L/c$. The resource counts follow from the construction.