Theorem: Computational support and separate causal containment

For a finite layered network of complete block channels, the backwards ancestor set $N_{s,t}(Y)$ of a joint output $Y$ satisfies

An intervention outside that set cannot affect the joint output. Reference-assisted comparison uses the joint ancestor–reference reduction.

If every dependency port is placed in the regular laboratory and every dependency edge has a supplied future-causal route, then scheduled ancestry is contained in the prescribed physical causal past.

Proof. A block adjoint is unital. A block disjoint from the current observable support acts on identity and changes nothing; a meeting block enlarges support only within that block. Reverse induction proves the inclusion, and testing output observables proves the intervention statement. Carry the untouched reference algebra throughout. Concatenating the supplied future-causal routes proves the last assertion.