Corollary: An empty named core is not an empty complete complement

A genuinely classical-output channel on a nontrivial full quantum code cannot have a complete pure dilation with one-dimensional environment.

Proof. With a one-dimensional environment, its complete dilation would be an isometry into the returned output alone. Theorem thm:global would supply an exact decoder. This contradicts the classical-output obstruction proved in Theorem thm:classical below.