Lemma: Elementary distance bounds

For isometries into a common output space,

The same allowance holds after a common partial trace. CPTP pre- and postprocessing contract half-diamond distance.

On fixed output spaces, recovery error and distance to the replacement-channel set are one-Lipschitz:

Proof. For normalized vectors $u,v$,

Apply this to referenced isometric outputs, then use convexity and partial-trace contraction.

For each decoder, the triangle inequality changes its error by at most the channel comparison allowance. Taking infima in both directions proves recovery stability. The same argument applies to distance from the replacement-channel set.