Theorem: Scheduled influence bound

The pulled-back terminal algebra satisfies

If every block dependency is certified metric causal, then each scheduled ancestor event belongs to the metric past of a terminal event. Equal input reductions on the scheduled past give equal output reductions. With references, equality of the corresponding joint input reductions gives equality of the joint output reductions.

Proof. The adjoint of a channel is unital. A block disjoint from the current observable support therefore acts on an identity and cannot enlarge that support. Repeating this argument backwards proves the algebra inclusion. Equal reductions give equal expectations of every pulled-back observable. Tensoring with a reference proves the complete statement. Composing the certified causal dependencies gives a piecewise causal curve, proving metric containment.