Proposition: The physical residual remains the complement

If $\mathcal{C}={\mathcal{T}}_{{\tau }_B}$, then for every residual observable $Z$,

Therefore complement constancy does not imply faithful reconstruction of an independently variable full matrix algebra $\mathcal{B}(B)$ from $A$.

Proof. Evaluate the residual expectation on every logical matrix unit. The constant-channel equation gives the displayed compression. It is scalar on the logical code. Recovering all states of an additional variable $B$ would be a different input and a different task.