Remark: Averaging a device is not knowing its unitary

If the sampled unitary is forgotten and the accessible channel itself is Haar-averaged, then

That channel is replacement and has recovery error $a_d$. The Page average is ${\mathbb{E}}_{U}S({{\rho }_R}^U)$, not $S({\mathbb{E}}_U{{\rho }_R}^U)$. Likewise, the expression “the complement is diamond-close to Haar” has no meaning until a comparison channel, ensemble, input quantifiers, and treatment of the random control are specified.