Theorem: Order-theoretic domain of dependence

For a finite complete wired instrument network, the output reduction on terminal ports $Y$ factors through the initial reduction on their scheduled ancestors:

Changing complete gates outside the ancestor subnetwork does not change that reduction, provided the wiring and the included gates agree.

For a partitioned schedule, the initial ancestor region is $N_{t_0}(Y)$ from (eq:scheduledpast). No metric is required for the network statement.

Proof. Pull an output observable backward in reverse topological order. If it is the identity on all outputs of a gate, unitality makes it the identity on that gate's inputs. Otherwise its possible support expands only to the declared input predecessors. Include all classical control, outcome, and resource ports in this graph.

At the input cut the support lies in the scheduled ancestors. Lemma lem:comparison gives the channel factorization. Excluded gates act on identity factors, so replacing them by any complete gates on the same interfaces does not change the pullback. For a partitioned schedule the same induction is precisely (eq:scheduledpast).