A family remainder is not a derivative remainder

For any fixed nonzero distribution $A$, let

Then $r_{\delta }=O(\delta )$ in every relevant local finite-order bound, but

Thus one cannot differentiate a vanishing family remainder inequality and obtain a summable scale estimate.

The evaluated application establishes a Route B family expansion. No differentiated application remainder is claimed. The derivative statement in Proposition prop:origin was proved independently from the exact scale profile.