Theorem: Regular reference-preserving route

For every input ${\rho }_{A_{0}E}$, after hop $j$,

is the joint state under the register identifications. The logical channel is the identity including every reference. Classical reports can be forwarded with their entire joint law. Arrival is at

within the chosen regular time reserve.

Proof. A SWAP moves the complete payload factor, including its reference correlations, into the next blank and leaves the old factor blank. Induction gives the state formula. Forwarding diagonal classical records preserves their joint correlations. The station spacing and tick duration give the certified null cross-port routes inside the regular tube.