Theorem: Exact target-adapted null transport

Prepare $A_1,\dots ,A_m$ in the fixed blank $\sigma =|0⟩⟨0|$, independently of the payload and any reference. For every state ${\rho }_{A_{0}E}$, after tick $j$ the state is

under the fixed register identifications. Consequently:

1. the logical source-to-destination channel is exactly ${\mathrm{id}}_d$, including every untouched reference;

2. all non-output route registers end in a jointly payload-independent blank state;

3. the arrival event is exactly $q$;

4. every scheduled dependency is metric causal;

5. the protocol uses $m$ two-register SWAP gates, $m+1$ payload-sized registers, and $m$ ticks.

Proof. At $j=0$, (eq:swap-invariant) is the preparation. At tick $j$, register $A_j$ is blank and independent. SWAP exchanges it with the entire current payload factor $A_{j-1}$, including all correlations with $E$. The previous site becomes blank, proving the invariant by induction.

At $j=m$, the destination-reference state is the original payload state under the fixed identification. The remaining registers are blank, proving the complete-channel assertions.

Each active pair has separation $a=c\delta$. Hence its input-output dependencies satisfy the local causal certificate. The payload positions at successive ticks trace the prescribed null segment. Finally $m\delta =L/c$, and the gate and register counts follow from the explicit schedule.