Theorem: Complete scheduled locality

The complete record-retaining network obeys

Equal input reductions on $N_{s,t}(Y)$ give equal corresponding output reductions on $Y$. For joint output with a reference $E$, equality is required on the joint $N_{s,t}(Y)$–$E$ input reduction.

A deterministic intervention outside the complete backwards region cannot change that output law. If every dependency edge has a separately certified regular future-causal host route, every scheduled ancestor is in the prescribed metric causal past of the corresponding output.

Proof. A channel adjoint is unital. A block disjoint from the pulled-back observable support therefore acts on an identity factor. A meeting block enlarges support only within that block. Backwards induction gives the inclusion.

Testing all output observables proves the reduced-state claim. Tensoring with the reference algebra proves the joint claim. The same identity-factor argument proves the intervention statement. Concatenation of the independently certified routes proves causal containment.