Lemma: Finite-token and fixed-copy refusals

1. A finite token with a fixed numerical decoder cannot represent exactly an expectation taking infinitely many values on its preparation class.

2. For two positive definite density matrices, no measurement on a fixed finite number of copies can deterministically output different exact labels.

Proof. A finite decoder has finite image, proving (i). For (ii), if $\rho ,\sigma >0$, then their $n$-fold tensor powers are positive definite. Every nonzero positive effect $E$ has strictly positive pairing with both:

Thus no label effect can have probability one for one input and zero for the other.