Theorem: Complete regular route

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

The logical payload channel is the identity, including all reference correlations. All non-output route registers are fixed independent blanks.

With departure at $t_{\mathrm{d}}$, arrival is

For $\upsilon =1$, the cross-port paths are null. For $\upsilon <1$, they are timelike.

Proof. A SWAP moves the entire current payload, including its reference correlations, into the next independent blank and leaves the old factor blank. Induction proves (eq:route). Each spatial displacement has length $\ell /n$ and duration $\ell /(n\upsilon )$, so its developed slope is at most one. The supplied calibration (eq:signature) converts the duration into host units, giving (eq:arrival).