Theorem: Reference-preserving regular route

For any initial payload–reference state ${\rho }_{A_{0}E}$, after hop $j$ the complete route state is

The logical payload channel is the identity, including every reference. Arrival occurs after developed time $\ell$, or host time $\ell /c_{\mathrm{h}}$. All route registers remain in the complete resource ledger; the non-output ones are fixed independent blanks.

Proof. At each hop, SWAP exchanges the entire current payload, including its reference correlations, with the next independent blank. Induction proves the tensor identity.

Each adjacent separation and developed duration equals $\ell /n$, giving a straight null certificate in the developed chart. The retained signature and calibration (eq:signature) give the host time. The chosen tube remains disjoint from the worldsheet.