Theorem: Complete scheduled locality

The record-retaining network satisfies

Equal input reductions on that region give equal corresponding output reductions. For joint output with a reference $E$, equality must hold on the joint ancestor–reference input reduction.

Deterministic interventions outside the complete backwards region cannot change the corresponding joint output law. If every dependency edge has an independently certified regular future-causal host route, scheduled ancestry is contained in the prescribed metric causal past of the output.

Proof. A trace-preserving channel has a unital adjoint. A block disjoint from the current observable support therefore acts on an identity factor. A meeting block enlarges support only within itself. Backwards induction proves the inclusion. Testing all joint output observables proves the reduction and intervention statements. The reference algebra is retained unchanged in this argument.

Concatenating the separately certified host routes proves causal containment.