Theorem: Regular reference-preserving transport

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

under the declared register identifications. The logical payload channel is the identity, including all reference correlations.

The classical report can be forwarded with its full joint law. Arrival is at

and every dependency route stays in the regular part.

Proof. A SWAP moves the entire payload factor, including its correlations with $E$, into the next independent blank and leaves the previous factor blank. Induction proves the formula.

Classical forwarding preserves diagonal records and their joint correlations; it is not cloning an unknown quantum state. The inherited chart and selected tick durations certify the null cross-port routes.