Proposition: Containment relative to a supplied causal relation

Under this edge condition, every dependency path lies in $⪯$. For local channel realizations of the graph, Heisenberg support of a terminal observable is contained in its scheduled ancestor region. Thus operational influence is bounded by that region and, through the certified embedding, by the supplied causal relation.

Proof. Path containment is transitivity. For support containment, pull the terminal observable backward through the circuit. The adjoint of a trace-preserving local channel sends an identity on an untouched factor to the identity, and can enlarge support only to the input factors of that event. Induction through the finite graph places the support in the ancestor region. An intervention on a factor outside that region cannot change the terminal reduced output, because its pulled-back observables act as the identity there.