Lemma: The source kernel exists and is carried by v=a=z

For smooth ${q^{L,I,J}}_{\alpha \beta }$, the distribution (eq:Q) is defined on ${{\mathbb{R}}^{12}}_{(v,a,z)}$ by pullback under the diffeomorphism $\kappa (v,a,z)=(v,v-z,v-a)$ and $\mathrm{supp}Q\subseteq \{v=a=z\}$. For $\Phi \in C^{\infty }({\mathbb{R}}^{12})$ whose restriction to a neighborhood of the diagonal $\{v=a=z\}$ has compact support in $v$ there,

is finite and depends only on the $2$-jet of $\Phi$ along the diagonal.

Proof. Each term is $q_{\alpha \beta }(v)$ times the pullback of $1\otimes {\partial }^{\alpha }\delta \otimes {\partial }^{\beta }\delta$ by $\kappa$, whose support is ${\kappa }^{-1}(\{\xi =\eta =0\})=\{v=a=z\}$; (eq:Qpair) is the definition of that pullback paired with $\Phi \circ {\kappa }^{-1}$, ${\kappa }^{-1}(v,\xi ,\eta )=(v,v-\eta ,v-\xi )$, Jacobian one; the jet order is $|\alpha |+|\beta |\le 2$.