Theorem: Complete scheduled locality

The network obeys

Equal input reductions on that region give equal corresponding output reductions. For joint outputs with a reference $E$, equality is required on the joint ancestor–reference input reduction.

A deterministic intervention outside the complete backwards region cannot change the corresponding joint output law. If every dependency edge has a certified regular future-causal host route, scheduled ancestry is contained in the prescribed metric causal relation.

Proof. A channel adjoint is unital. A block disjoint from the current observable support therefore acts on an identity factor. A meeting block enlarges support only within itself. Backwards induction proves the inclusion. Testing all joint output observables proves the reduction and intervention statements, with the reference carried unchanged. Concatenation of the separately certified routes proves the geometric containment.