Corollary: Influence containment

For the complete dependency network,

The inclusion can be strict.

Proof. A trace-preserving intervention outside the scheduled ancestors preserves their reduced state, including when the initial state is entangled. Theorem thm:cone excludes an output change. For strict inclusion, give a channel a declared memory input and let it return the constant classical answer $0$. The memory port is a scheduled ancestor but is not influential.