Corollary: The supported sector is carried by the null incidence

Under the hypotheses of (P1), $\mathrm{supp}{K}_8\subseteq S_{\partial }$. Hence $\mathrm{supp}{K}_{\mathrm{supp}}\subseteq S_{\partial }=\{a=z,b-a\in \partial J^{+}\}\subseteq \{(b-a{)}^2=0,a=z\}$, and $K_{\mathrm{supp}}$ vanishes on the interior of the solid cone.

Proof. ${\partial }_{b^{\kappa }}{\mathcal{T}}_{\mathcal{B}}$ is the distribution with $⟨{\partial }_{b^{\kappa }}{\mathcal{T}}_{\mathcal{B}},\varphi ⟩=-⟨Q,{{\Phi }^{\mathcal{B}}}_{{\partial }_{b^{\kappa }}\varphi }⟩$, and ${{\Phi }^{\mathcal{B}}}_{{\partial }_{b^{\kappa }}\varphi }(v,a,z)=\int \mathcal{B}(u)({\partial }_{b^{\kappa }}\varphi )(a,v+u,z)\mathrm{d}u=-⟨{\partial }_{\kappa }\mathcal{B},\varphi (a,v+\cdot ,z)⟩$. Thus ${\partial }_{b^{\kappa }}{\mathcal{T}}_{\mathcal{B}}$ is the $\mathcal{T}$-construction of (P1) with $R$ replaced by ${\partial }_{\kappa }\mathcal{B}=\frac{1}{2}{x}_{\kappa }R$, and the proof of (P3) for $K_9$ applies verbatim with $\mathrm{supp}{\partial }_{\kappa }\mathcal{B}=\partial J^{+}$. Multiplication by smooth functions and the outer ${\partial }_{b^{\mu }}$ preserve the support statement.