Theorem: Scheduled locality

The complete network obeys

Equal input reductions on that backwards region give equal corresponding output reductions. A prescribed intervention outside it cannot change the corresponding joint output law.

If every dependency edge has a separately certified regular future-causal host route, ancestry is contained in the supplied host 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 can enlarge support only within itself. Backward induction proves the inclusion. Testing all output observables proves the reduction and intervention claims. References are retained unchanged in the same calculation.

Concatenation of the separately certified routes proves the final causal containment.