Theorem: Complete scheduled locality

The network satisfies

Equal input reductions there give equal corresponding output reductions. An intervention outside the complete backwards region cannot change the corresponding joint record law.

If every dependency edge has a certified regular future-causal host route, ancestry is contained in the prescribed host causal relation.

Proof. A channel adjoint is unital. A block not meeting the current observable support acts on an identity factor. A meeting block enlarges support only within itself. Backward induction proves containment. Testing all output observables proves the reduction and intervention statements, with references retained. Concatenate the independently certified routes for the geometric containment.