Theorem: Complete scheduled-support bound

For the channel $\Phi$ of a complete network,

Inputs outside ${\mathrm{Sch}}^{-}(B)$ cannot signal to $B$, including when the observer also retains an arbitrary untouched reference. Replacing a gate outside the backward region by any CPTP map on the same ports does not change that output reduction.

Proof. Pull an output observable backward in reverse topological order. A gate whose outputs all carry identity is invisible, because its adjoint is unital. Otherwise the conservative graph adds only that gate's input ports. Certified reduced graphs use their joint-output support hypothesis instead. A fixed preparation creates no unknown model-input predecessor. Measurements, classical feed-forward, and resets are ordinary complete channel nodes in this argument. Induction gives (eq:schedule). Apply Theorem thm:nosignal. The same reasoning with reference observables proves complete equality. On an excluded gate the pulled-back observable is identity, also preserved by any replacement channel.