Theorem: Reference-preserving identity transport

For every fixed finite-dimensional register $Q$ and every planned route of Theorem thm:routing, there is a complete implementation whose logical sender-to-receiver channel is

For every untouched reference $R$ and every state ${\rho }_{QR}$,

The same assertion holds for a finite hybrid register.

Proof. Reserve one transport colour for $Q$, and prepare all its unused slots in a fixed blank state. At each pair firing, swap the two slots. At each planned junction transition, permute the arriving slot into the next outgoing slot or into hub memory. The timetable in Lemma lem:frame sends the unique logical factor to the next endpoint. Induct through the route.

The total transformation on that logical factor and the blank slots is a tensor-factor permutation. After identifying the returned factor as $Q$, tracing the other blank factors leaves ${\rho }_Q$. The identical permutation calculation with ${\mathrm{id}}_R$ leaves ${\rho }_{QR}$. Classical block labels are relocated with their quantum blocks, without normalization or readout.