Modular success and finite recovery are not identified

At modular energy $s=0$, the fixed vector $\xi$ lies in every shrinking band. With $T=I_H$, its averaged squared norm is one and $v=1$ is a bounded witness. With $T=0$, the positive-average hypothesis fails.

Either modular choice can be placed alongside an identity finite channel or a replacement finite channel in the absence of a connecting map. Thus the modular witness by itself does not distinguish those finite recovery tasks. Failure of the positive-average hypothesis likewise does not imply finite nonrecoverability.