Corollary: Ancestry upper-bounds influence

An unconditioned trace-preserving local intervention outside the scheduled ancestors of $Y$ cannot change its output law. Thus

Equality need not hold.

Proof. Such an intervention preserves the ancestors' reduced state, also jointly with an untouched reference. Apply Lemma lem:locality. For strict inclusion, a channel may have a declared input port but return a constant classical answer independently of that input.