Remark: Why the added environment matters

At $\lambda =0$, the vectors $u_0,v_0$ are orthogonal. The complete isometry retains the message coherently in $RF$, even though discarding $F$ leaves only classical reference–accessible correlations. Keeping only the original vacuum residual $E$ would incorrectly make the noisy complement look constant.

The constant-complement criterion concerns the complement of the actual accessible channel, including every discarded noise register.