Theorem: Scheduled locality

For the complete network,

Equal initial reductions on the complete backwards region give equal corresponding output reductions. Interventions outside that region cannot change the joint law of those retained outputs.

If every scheduled dependency edge is separately assigned a regular future-causal host route, scheduled ancestry is contained in the supplied host causal relation.

Proof. A channel adjoint preserves the identity. Blocks disjoint from the current observable support therefore see only identity factors. A meeting block can enlarge support only within itself. Backward induction proves the support inclusion. Testing every output observable gives the reduction and intervention statements. Untouched references are retained in the same argument.

Concatenating the individually certified host routes gives the stated causal containment.