Lemma: Tree kernel and support

where $(\mathcal{P}\Theta {)}_{\alpha \beta }={\Theta }_{\alpha \beta }-\frac{1}{2}{\eta }_{\alpha \beta }{\Theta }^{\lambda }{}_{\lambda }$. The $z$-integration runs over the compact set $\{t\ge 0\}\cap J^{-}(K)\cap J^{+}(\mathrm{supp}g)$, and

Proof. Insert (eq:k1) into (eq:tree); $\mathcal{P}$ is symmetric and constant. The $x$-slot carries $f$, the $y$-slot carries $g$; ${{\mathsf{E}}_0}^{\mathrm{R}}(x,z)$ forces $z\in J^{-}(x)$, ${\sigma }_{\epsilon }(z)$ forces $t_z\ge 0$, and ${{\mathsf{E}}_0}^{\mathrm{R}}(z,y)$ forces $y\in J^{-}(z)$. $J^{-}(K)\cap J^{+}(\mathrm{supp}g)$ is compact in Minkowski space.