Lemma: Coherent change of extension

Let ${\mathfrak{R}}_{\mathrm{L}}^{\prime }$ be any linear map ${\mathcal{V}}_{\mathrm{adm}}\to {\mathcal{D}}^{\prime }({\mathbb{R}}^4)$ that extends every $u\in {\mathcal{V}}_{\mathrm{adm}}$ with $\mathrm{sd}({\mathfrak{R}}_{\mathrm{L}}^{\prime }(u))\le \mathrm{sd}(u)$ in the graded sense of Proposition prop:RL(2), and put $\Delta {\mathfrak{R}}_{\mathrm{L}}:={\mathfrak{R}}_{\mathrm{L}}^{\prime }-{\mathfrak{R}}_{\mathrm{L}}$. Then for every numerical label $\lambda$,

and the unrooted kernel changes by the local collision distribution

Proof. (eq:L-Dq) is Proposition prop:RL(4) applied to the two extensions ${\mathfrak{R}}_{\mathrm{L}}^{\prime }({q^{\circ }}_{\lambda })$ and ${\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$. Linearity of (eq:QLext) in the extended kernels gives (eq:L-deltaQ). Each summand is a smooth coefficient times a point-supported distribution in $z=y-w$; no root is used.