Proposition: Affine obstruction to definite packet selection

Let $\mathcal{S}$ be a convex state set and suppose an affine map assigns each state a probability measure concentrated at one classical packet:

If distinct packets are separated by measurable events, then $\gamma$ is constant on $\mathcal{S}$.

Consequently, if $\theta \kappa \ne 0$ and $I_f$ takes two distinct values on ${\mathcal{D}}_{\mathrm{src}}$, the sourced constructor cannot be the definite classical output of an affine channel on that domain.

Proof. If $\gamma (\varrho )\ne \gamma (\sigma )$, choose a measurable event containing one packet and not the other. For $0<t<1$, affinity gives

The chosen event has probability strictly between zero and one, so this measure is not concentrated at one packet. The mixture belongs to $\mathcal{S}$, a contradiction. For the application, distinct selected deficits distinguish the packets.