Corollary: Loss labels and channel errors are distinct

Let

Then the path average of these scalar losses concentrates on ${\mu }_{\mathrm{loc}}$ on macroscopic paths. Also,

and

Generally, $d_{\diamond }(\overline{\Lambda },{\mathrm{id}}_C)\ne {\mu }_{\mathrm{loc}}$.

Proof. The scalar loss lies in $[0,1]$, so the bounded path law applies. Convexity proves the upper bound. The reverse triangle inequality proves the last displayed comparison. No equality follows from these inequalities.