Theorem: Complete schedule locality

The schedule channel obeys

Hence its output on $Y$, together with any untouched reference $E$, depends only on the input reduction to $N_0(Y)E$.

If every scheduled dependency edge has future-causal ports in $(M,\bar{g})$, scheduled ancestors lie in the output's metric past. Operational intervention influence is contained in scheduled ancestry, but need not equal it.

Proof. On a block disjoint from the current backward support, the observable is the identity. A trace-preserving channel's adjoint fixes that identity. A block that meets the support can enlarge it only within its declared union. Backward induction proves the algebra inclusion.

Equal input reductions therefore give equal expectations for every output observable, including referenced observables. Causal transitivity gives metric containment along each dependency path.

Replacing an operation outside the backward region acts only on identity factors and cannot change the output. Strictness is possible: a scheduled input to a replacement channel is an ancestor even though that channel ignores it.