Theorem: Reference-preserving regular route

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

The logical payload channel is the identity with every reference included. Classical reports can be forwarded with their entire joint law. Arrival occurs at

and all certified routes stay in the regular part.

Proof. A SWAP moves the whole payload tensor factor, including its reference correlations, into the next blank and leaves the old factor blank. Induction gives the formula. Forwarding diagonal classical records preserves their joint law. The target-adapted station spacing and tick duration give the stated null cross-port displacements inside the regular tube.