Corollary: Pure complete residual

If the exact endpoint complement is one fixed pure state, then ${\mathcal{N}}_{\infty }$ is an isometric encoding.

Proof. In Theorem thm:exact, the residual rank is one. Thus $K$ is one dimensional and ${\mathcal{N}}_{\infty }(X)=AX{A}^{†}$ for an isometry $A:C\to R_{\infty }$.