Theorem: Influence, schedule, and physical cone

For a complete finite network, intervention influence is contained in scheduled ancestry. If every actual input-output dependency edge has future-causal ports in $(M,\bar{g})$, scheduled ancestry is contained in the prescribed metric causal relation. Neither converse follows.

Proof. Outside the backward network, replacement trace-preserving operations act only on identity factors of the pulled-back observable and cannot change its expectation. Compose the supplied causal paths along any directed dependency path to obtain physical containment.

A replacement channel can ignore a scheduled input, showing that ancestry need not be influence. A causally related pair need not have an installed communication path, showing failure of the other converse.