Lemma: Propagated coherent ambiguity

Let $\Delta {\mathfrak{R}}_{\mathrm{L}}$ be a coherent change as in Lemma lem:Lamb, with contact coefficients $C_{\lambda \gamma }$. The root-attached coefficient built from ${\mathfrak{R}}_{\mathrm{L}}^{\prime }={\mathfrak{R}}_{\mathrm{L}}+\Delta {\mathfrak{R}}_{\mathrm{L}}$ differs from ${{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}$ by

The sign is net $+1$, exactly as in XVI-K, eq:deltaCK: the chain-rule identity $({{\partial }_z}^{\gamma }\delta )(y-w)=(-1{)}^{|\gamma |}{{\partial }_w}^{\gamma }[\delta (y-w)]$ and $|\gamma |$ integrations by parts in the relative variable each contribute $(-1{)}^{|\gamma |}$, and the product is $+1$.

Proof. Copy the argument of XVI-K, Lemma lem:amb, with the sum extended over complete-source labels and with ${\mathfrak{R}}_{\mathrm{K}}$ replaced by ${\mathfrak{R}}_{\mathrm{L}}$. Form of $\Delta {\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$: Lemma lem:Lamb. Change of ${U^{{\mathfrak{R}}_{\mathrm{L}}}}_{\mathrm{L}}$: linearity of (eq:UL) in the extended kernels, each term existing by Lemma lem:Lattach. Pass to coordinates $(y,x,z)$, $z=y-w$; the product becomes the tensor product $B_{f,g}\otimes {\partial }^{\gamma }\delta$; pairing against $\tilde{a}\tilde{\psi }$ produces $(-1{)}^{|\gamma |}{{\partial }_z}^{\gamma }[\tilde{a}\tilde{\psi }]{|}_{z=0}$; the substitution ${\partial }_z=-{\partial }_w$ contributes a second $(-1{)}^{|\gamma |}$, whence net $+1$ and (eq:L-deltaCL).