Theorem: Complete locality and causal containment

For the complete scheduled channel ${\Phi }_{\mathrm{sch}}$,

The terminal reduction to $Y$ factors through the initial reduction to $N_0(Y)$. With a retained reference $E$, the joint output factors through the initial reduction to $N_0(Y)E$.

Replacing a complete channel outside the backward dependency region does not change the terminal law. Every scheduled ancestor event maps into the terminal metric past.

Proof. An output observable is the identity on each block disjoint from its current support. A channel adjoint is unital, so it preserves that identity. Other blocks enlarge support only to their union. Backward induction proves the inclusion.

Append any fixed state on the unused initial complement, apply the channel, and reduce to $Y$. The adjoint inclusion proves that this defines a reduced channel with the same action even on initially correlated inputs. Tensoring the argument with $I_E$ proves the reference statement.

An operation outside the dependency region acts on an identity in the backward calculation and is invisible. Each dependency edge is causally embedded, and concatenation proves the final containment.