Theorem: Complete channels of the benchmark

The map $V$ is an isometry. Let ${\tau }_F={\sum }_j{p}_j|j⟩⟨j|$. The complete accessible channel, its complement, and the $E$ restriction are

The decoder

followed by the fixed relabeling $D\to M$, satisfies ${\mathcal{D}}_E{\mathcal{N}}_E={\mathrm{id}}_M$. Thus recovery from $E$ alone is exact, with every reference. The complete complement of the $E$ channel is also replacement:

Proof. Since $J$ is unitary and $\beta$ and $|0⟩$ are normalized, $V^{†}V=I_M$. Taking the specified partial traces proves (eq:modelN)–(eq:modelNE) on every matrix unit, including off-diagonal units. Applying (eq:modeldecoder) gives ${\mathrm{Tr}}_F({\tau }_F\otimes X_D)=X_D$. For the $E$ complement, trace $E$; the unitary $U$ acts only on the traced output and disappears. The remaining $BK$ state is the displayed fixed state times $\mathrm{Tr}(X)$. Every equality is a channel identity, hence is reference stable.