Proposition: Complete scheduled support

An output observable pulls back only to its scheduled ancestors. Equality of joint ancestor–reference reductions gives equality of the joint output reductions.

Metric containment follows if every actual dependency edge has a separately certified future-causal path in the retained host.

Proof. On a disjoint block the pulled-back observable is the identity, and a channel adjoint is unital. Only a meeting block can enlarge support. Backward induction proves the support claim. Concatenating the certified edge paths proves metric containment.