Theorem: Scheduled support bounds actual influence

For the channel $\mathcal{C}$ of a complete finite network,

Consequently,

and every intervention on inputs disjoint from ${\mathsf{Sch}}^{-}(B)$ has zero signalling to $B$, including with an untouched reference.

Proof. Pull an output observable backwards through a topological order of the network. If it is the identity on all outputs of a gate, unitality of the adjoint makes it the identity on that gate's inputs. Otherwise, its possible support expands only to the declared predecessors of the observed gate outputs.

For the conservative all-to-all gate graph, these are exactly all gate inputs. Tensoring with untouched factors does not add other support. Fixed preparations introduce no unknown input. A discard pulls a scalar identity backwards; a reset is treated as its stated channel. Measurement records and controlled continuations are included as ordinary channel ports.

Repeating this process leaves support only on the scheduled ancestors. This proves (eq:schedule-support). Theorem thm:nosignal gives the remaining assertions.