Proposition: Finite engineered first-passage interpretation

If the library declares serial traversal of edge $e$ to take exactly $c_e$, with no additional waits or processing costs, then the minimum completion time among its serial path protocols is $T(x,y)$.

This is an exact statement about that library. If the costs instead price Bell pairs, entropy, or another resource, the same minimization prices that declared resource, not necessarily elapsed time.

Proof. Every serial path has its declared additive cost. A positive-cost minimizing path exists and may be chosen simple. Executing its protocol realizes the minimum.