Theorem: Scheduled locality

The complete network satisfies

An intervention outside the complete backwards dependency set cannot change the corresponding joint output law. Untouched references are included by retaining their joint input reduction.

If every dependency port has a declared host location and each dependency edge has a certified future-causal regular route, scheduled dependencies are contained in the prescribed host causal relation.

Proof. A channel adjoint is unital. On a block disjoint from the current support, the pulled-back observable is the identity; on a meeting block support enlarges only inside that block. Reverse induction proves (eq:scheduledlocality). Testing every joint output observable proves the intervention statement. Reference factors are carried unchanged through the same argument.

For the last assertion, concatenate the supplied causal routes for the dependency edges.