Proposition: Scheduled-cut null

In a complete finite CPTP network, an output whose scheduled ancestors exclude the message input is independent of that input, including when a fixed initial environment is correlated with an untouched reference.

Proof. Pull an output observable backward in topological order. The adjoint of a trace-preserving map is unital, so an identity on all outputs of a gate pulls back to an identity on its inputs. The observable's support can expand only to declared predecessor ports of gates on its scheduled ancestry. If the message is not among those ports, every pulled-back observable is identity on the message input. Its expectation is therefore independent of the message state.