Remainder differentiation

A bound $r_{\delta }=O(\delta )$ alone does not imply a small logarithmic scale derivative. For example, for a fixed nonzero distribution $A$,

has an $O(\delta )$ family bound, whereas

The differentiated estimate in the main theorem was therefore proved directly using the exact scale profile. No differentiated remainder theorem is inferred for the appendix merely from its $O(\delta )$ remainder.