Theorem: Complete-network locality

For a finite layered block-channel network and a joint output $Y$, the recursively defined backwards ancestor set $N_{s,t}(Y)$ satisfies

An intervention outside that complete ancestor set cannot alter the joint output law. Reference-assisted comparison retains the joint ancestor–reference reduction.

If every dependency port has a regular laboratory placement and every dependency edge has a supplied future-causal route, then those dependencies are contained in the prescribed host causal relation.

Proof. Every channel adjoint is unital. A block disjoint from the current support acts on an identity factor; a meeting block enlarges support only inside that block. Reverse induction proves the inclusion. Testing all output observables gives the intervention statement, with references carried throughout. Concatenating the supplied causal routes proves physical containment.