Theorem: Complete scheduled locality

The network satisfies

Equal joint reductions on the ancestor input and an untouched reference give equal joint output reductions. A deterministic intervention outside the complete backwards region cannot change that output.

If every dependency edge has a separately certified future-causal regular host route, scheduled ancestry is contained in the prescribed host causal relation.

Proof. An excluded block acts on an identity factor in the pulled-back observable. Its adjoint is unital. A meeting block enlarges support only within itself. Backward induction proves the inclusion. Testing all output observables, including reference observables, proves the reduction and intervention claims. Concatenation of the supplied causal routes proves the geometric statement.