Theorem: Exact serial route reduction

For a fixed simple route $\gamma$, the completion time in Specification spec:repeater is

The complete corrected teleportation protocol implements the identity message channel at completion, including every untouched reference. Minimizing over the advertised routes gives

Proof. The first edge service starts at time zero. The next starts only after the preceding teleportation completes. The zero-duration convention for the other steps makes completion times additive.

For one perfect teleportation step, the four Bell outcomes have uncorrected path maps $\frac{1}{2}{P}_j$ on the message, up to the fixed Bell-basis convention. Applying the corresponding inverse Pauli gives four corrected path maps $\frac{1}{2}I$. Their complete sum is

The identity holds after tensoring with any reference. Composition proves the route claim; records are retained with their complete probabilities.

There are finitely many simple routes in the finite graph, and positive costs allow loop erasure. The advertised controller can therefore select a minimizing route with a fixed tie rule. Its cost is precisely the finite FPP passage time.