A separately proved differentiated remainder

The scale derivative is evaluated directly on ${\theta }_{\delta }$, rather than obtained by differentiating (eq:explicitrem).

By (eq:chain), the homogeneous contributions to ${\Omega }_{\Phi }({\theta }_{\delta })$ are exactly the logarithmic scale derivatives of their terms in (eq:germexpansion). The degree-zero constant term has zero scale derivative. Therefore

Since $\int \theta =0$,

This proves (eq:maindrem) by an independent argument. Its bound is integrable with respect to $\mathrm{d}\delta /\delta$.