Theorem: Scheduled support upper-bounds influence

Let $\mathcal{C}$ be the complete channel of a network in Definition def:network. For terminal outputs $Y$,

Consequently:

1. two initial states with equal reductions to ${\mathsf{Sch}}^{-}(Y)$ give equal output reductions on $Y$;

2. a trace-preserving intervention acting only on external inputs outside ${\mathsf{Sch}}^{-}(Y)$ has no unconditioned signalling effect on $Y$;

3. the same assertion holds jointly with an untouched reference, provided the intervention does not act on that reference.

Proof. Take an output observable supported on $Y$. Pull it backwards through a reverse topological order of the network.

If its support is disjoint from all outputs of an operation, the observable is the identity on those outputs. The adjoint of a trace-preserving map is unital, so that operation contributes an identity on its inputs.

Otherwise the observable may acquire support on the operation's declared input predecessors, but on no undeclared input. Under the conservative graph these are all of its input ports. Untouched tensor factors remain untouched. Fixed preparations introduce no unknown external input.

The recorded instrument is a single trace-preserving channel, so the same argument includes measurements, their records, and classically controlled continuations. Iterating leaves support only on ${\mathsf{Sch}}^{-}(Y)$, proving (eq:scheduledsupport).

Equal input reductions on that support give equal expectations for every output observable on $Y$. A trace-preserving local intervention outside the support preserves its reduction, even for initially correlated states.

For an untouched reference $R$, tensor the network with ${\mathrm{id}}_R$. The pulled-back support is contained in ${\mathsf{Sch}}^{-}(Y)\cup R$, and the same local-intervention argument applies.