Theorem: Containment in the prescribed cone

Suppose each actual input-output dependency edge of the complete finite network is assigned future-causal ports in $(M,\bar{g})$. Then scheduled ancestors of a terminal port lie in its metric past.

Operational intervention influence is contained in scheduled ancestry. Neither converse is asserted.

Proof. Compose future-causal paths along any directed dependency path. Transitivity gives metric containment.

An intervention outside the backward dependency region changes maps acting only on identity factors in every pulled-back terminal observable; its effect is zero. Conversely, a scheduled input to a replacement channel can be ignored by that channel, so ancestry need not imply influence. A metric-causal pair need not be connected by an installed network.