Proposition: Pure endpoint implication

If the complete endpoint residual is a fixed pure state, then the radiation channel is an isometric encoding:

In particular, a pure code input has a pure complete radiation output.

If the fixed residual is mixed, radiation recovery is still exact, but a pure code input can have mixed radiation of the form (eq:normal-form).

Proof. In Theorem thm:exact, a rank-one residual has a one-dimensional $K$. The general mixed case retains the input-independent syndrome state ${\sigma }_K$.