Theorem: Complete schedule locality

The complete channel obeys

Thus the output reduction on $Y$ depends only on the input reduction to $N_0(Y)$. With an untouched reference $E$, the output on $YE$ depends only on the input on $N_0(Y)E$.

If every scheduled input-output dependency maps to a future-causal pair in $(M,\bar{g})$, every scheduled ancestor of an output lies in its metric past. Operational intervention influence is contained in that ancestry.

Proof. On a block disjoint from the current backward support, the observable is the identity. A trace-preserving channel adjoint preserves it. Other blocks can enlarge support only within their declared union. Reverse-tick induction gives the inclusion.

Equality of all pulled-back expectations gives equality of output reductions. Appending a reference identity gives the referenced statement. Causal transitivity gives geometric containment along a dependency path. Changing a complete operation outside that path acts only on an identity factor in the backward calculation and cannot affect the output.