Theorem: Reference-preserving regular transport

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

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

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

All dependency routes remain in the regular part.

Proof. Each SWAP exchanges the entire current payload factor, including all correlations with $E$, with the next independent blank. It leaves the previous station blank. Induction gives the formula.

Classical forwarding copies only diagonal record data and therefore preserves its joint classical law; it is not cloning an arbitrary quantum state. The prescribed spacings and durations make the cross dependencies null in the inherited developing chart. No worldsheet crossing is used.