Theorem: Complete schedule locality and metric containment

Let $\Phi$ be the complete scheduled channel. Then

The output on $Y$ depends only on the initial reduction to $N_0(Y)$. With a retained reference $E$, it depends only on the joint initial reduction to $N_0(Y)E$.

Replacing a complete operation outside the backward dependency region leaves the terminal law unchanged. Every scheduled ancestor event is in the metric past of the output event.

Proof. An output observable is the identity on any block disjoint from its current support. The adjoint of a trace-preserving channel is unital, so that block remains an identity. Other blocks enlarge support only to their union. Backward induction proves the adjoint inclusion.

To obtain a reduced channel, append any fixed state on the initial complement, apply the complete channel, and reduce to $Y$. The adjoint inclusion proves independence of that appended state and agreement with the actual output even for correlated inputs. Tensoring the argument with $I_E$ proves the joint reference assertion.

A replacement outside the dependency region acts on an identity in the backward calculation. Finally, each dependency pair is future causal by (eq:block) and Lemma lem:local-causal. Concatenate the segments.