Proposition: Robust optimal-decoder exclusions

For channels $\mathcal{N},\tilde{\mathcal{N}}$ with common input and accessible output interfaces, define

Then

In particular, if $|A|\le 2$ and $d_{\diamond }({\mathcal{N}}_{0,A},{\mathcal{N}}_A)\le {\eta }_A$, every deterministic decoder obeys

Proof. For every fixed $D$, contraction and the reverse triangle inequality bound the difference of the two errors by the channel distance. Taking the infimum gives one inequality; interchanging the channels gives the other.