Theorem: Scheduled locality

For the complete network,

Equal input reductions on that backwards region give equal corresponding output reductions. Interventions outside the complete backwards region do not change the corresponding joint output law.

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

Proof. A channel adjoint is unital. A block disjoint from the current support therefore acts on an identity factor. A meeting block enlarges support only within that block. Induct backwards through the ticks. Testing every output observable gives the reduction and intervention claims. References are retained in the same argument. Concatenation of the certified routes proves metric containment.