Proposition: Complete restoration does not identify a residual

Every finite-dimensional isometry $V:C\to R\otimes B$ has a CPTP left inverse on its complete output. This does not imply recovery from $R$, recovery from $B$, or constancy of either complement.

Proof. Choose a state ${\tau }_C$ and define

Both summands are completely positive and their traces sum to $\mathrm{Tr}X$. Thus $\mathcal{L}$ is CPTP and $\mathcal{L}(V\rho V^{†})=\rho$, including references.

For a counterexample to residual constancy, take $V|\psi ⟩=|0{⟩}_R\otimes |\psi {⟩}_B$. The accessible output is constant and the residual is the identity channel. Complete restoration uses $RB$ and returns a new code output; it does not assert that the code was relocated into either chosen output by the restore map.