Proposition: Expectation and affine-label obstructions

1. Let $\mathcal{S}$ be a convex state set. If an affine map assigns to every $\rho \in \mathcal{S}$ a classical distribution concentrated at one label,

   $$T(\rho )={\delta }_{\gamma (\rho )},$$

   then $\gamma$ is constant on $\mathcal{S}$. The statement also holds for a classical outcome space whose distinct labels can be separated by a measurable event.

2. A circuit acting only on code registers and as the identity on matter cannot, for a fixed independent code input, make its code output depend on an unencoded matter expectation.

Requiring that a putative label-producing channel also preserve the matter input does not remove the first obstruction.

Proof. If two states have distinct labels, affinity gives, for $0<t<1$,

This is not concentrated at one label. In a general outcome space, an event separating the labels has probability strictly between zero and one. Either conclusion contradicts the assumed deterministic output on the mixture.

For the second assertion, a code-only circuit $U_{\mathrm{c}}$ acts as $U_{\mathrm{c}}\otimes I_{\Gamma }$. On an independent input $c\otimes \rho$, its code marginal depends on $c$, not on $\rho$. Different unencoded expectations therefore cannot select different code outputs. An additional matter-preservation requirement cannot invalidate either argument.