Theorem: Scheduled locality

For the complete record-retaining network,

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

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

Proof. A channel adjoint is unital. A block disjoint from the current observable support therefore sees only an identity factor and cannot enlarge that support. A meeting block can enlarge it only inside that block. Backward induction proves (eq:scheduled). Testing all output observables proves the reduction and intervention claims. Reference factors are retained unchanged in the same argument.

Concatenating the separately certified dependency routes proves the final containment.