Proposition: Exact observable survival without full recovery

There is a qubit channel preserving the observable $Z$ exactly, with nonconstant complete complement and optimal full-code half-diamond recovery error $1/2$.

Proof. Use

The accessible and complementary channels are both computational dephasing

Hence ${\Delta }_Z^{\ast }(Z)=Z$, but the complement distinguishes $|0⟩$ and $|1⟩$.

The Bell-input output is separable, and remains separable after every accessible decoder. A separable two-qubit state has Bell overlap at most $1/2$. The Bell-projector effect therefore gives a recovery lower bound $1/2$. The identity decoder attains $1/2$, by the displayed Pauli mixture and its orthogonal Bell errors.