Proposition: Affine deterministic-packet obstruction

An affine map on a convex state set that assigns a probability distribution concentrated at one classical label to every state must have a constant label. The statement applies to any label space in which two distinct labels can be separated by a measurable event.

Proof. For two different labels, the output on a nontrivial convex mixture is the same mixture of two different point masses. It is not concentrated at one label. A separating event has probability strictly between zero and one.