Theorem: Reference-preserving finite report route

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

The logical payload channel is the identity including every reference. The arrival time is $\ell$ in developed units, or $\ell /c_{\mathrm{h}}$ in the calibrated host time. The route stays in the regular tube.

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

Each developed spatial step has length $\ell /n$ and is assigned the same developed duration. Its straight lift is null. The retained time calibration gives the host duration.