Lemma: No ancestral path gives a constant receiver channel

Under Specification spec:bounded-step, if there is no dependency path from the message source to the receiver interface, its logical channel is a replacement channel. Its error from identity satisfies

Proof. Pull a receiver observable backwards through the acyclic network. A channel whose outputs carry only the identity observable contributes the input identity, because its adjoint is unital. Including classical and quantum predecessors gives support only on the receiver's scheduled ancestors. If the message input is absent from those ancestors, the receiver law is independent of every message state. This is the scheduled-support argument of VII-J and En06-A CheungVIIJ CheungEn06A (thm:order).

Linearity then gives

for one state $\sigma$. On a maximally entangled message–reference input, the output is $\sigma \otimes I_R/d$, whose Bell overlap is $1/d^2$. The Bell-projector test gives (eq:constant-floor).