Theorem: Complete scheduled support

The network satisfies

Equal input reductions on $N_{s,t}(Y)$ give equal output reductions on $Y$. A joint referenced conclusion requires equal joint input reductions on $N_{s,t}(Y)$ and the untouched reference.

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

Proof. A block disjoint from the pulled-back support acts on an identity factor. Its adjoint is unital and preserves that factor. Other blocks enlarge support only inside their own interfaces. Backwards induction gives the inclusion.

Pairing with all output observables proves the reduced-state statement. Tensoring with the reference algebra proves the referenced version. The same unitality argument handles an excluded deterministic intervention. Concatenating the independently certified causal routes proves the geometric containment.