Theorem: Complete scheduled locality

The complete network satisfies

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

Replacing a complete channel outside the backwards dependency region cannot change the corresponding unconditional output, with the retained wiring fixed.

If every dependency edge has a certified regular future-causal host path, each scheduled ancestor lies in the prescribed metric past of the output.

Proof. A block disjoint from the pulled-back observable support acts on an identity factor. Its adjoint is unital, so that factor remains the identity. A meeting block enlarges support only within its own input interface. Backward induction proves (eq:locality).

Testing all output observables proves the reduced-state claim. Tensoring with the reference algebra gives the joint version. The same identity argument permits replacement of an excluded complete channel.

Concatenate the separately certified future-causal paths along each dependency path to obtain metric containment.