Lemma: Propagated scale variation

Under the same root hypotheses, the scale derivative of the root-attached coefficient is

i.e.\ (eq:L-deltaCL) with the free coefficients $C_{\lambda \gamma }$ replaced by the scale-variation coefficients $a_{\mu }({q^{\circ }}_{\lambda },\gamma )$ of (eq:L-Amu-exp).

Proof. Differentiate ${{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L},\mu }}}_{\mathrm{L}}$ under the pairing (justified by Lemma lem:Ldiffmu and compact support, Lemma lem:Lattach(3)): the $\mu$-derivative hits only ${\mathfrak{R}}_{\mathrm{L},\mu }({q^{\circ }}_{\lambda })$, producing $A_{\mu }({q^{\circ }}_{\lambda })$. Apply Lemma lem:Lambprop to the coherent infinitesimal change $\epsilon A_{\mu }$ and take $\epsilon \to 0$, or repeat the relative-variable computation with ${\partial }^{\gamma }\delta$ coefficients $a_{\mu }({q^{\circ }}_{\lambda },\gamma )$.