Lemma: Exact reference-preserving transport

The route protocol implements

More strongly, for every untouched reference $R$ and every state ${\rho }_{MR}$, its physical state after tick $k$ is

under the fixed register identifications. Every non-output route register is therefore in a payload-independent state, jointly independent of the payload and reference, at the terminal tick.

Proof. At tick zero, (eq:state-invariant) is precisely the declared product preparation.

A wait preserves the expression. For a nonzero increment, the target register is blank and independent of every other factor. SWAP exchanges the entire current payload factor, including its correlations with $R$, with that target factor. The former payload site becomes blank. This proves the invariant by induction.

At $k=N$, the payload-reference state is ${\rho }_{A_{b}R}$, and all other factors are blank. Taking the destination reduction and identifying $A_b$ with $M$ gives

The argument is an identity of tensor-factor permutations, so it holds for arbitrary reference Hilbert spaces carrying density operators, not only for references of a particular finite size.