Proposition: The K 5 placements and K 6 have the form of the reconstruction rule

Let $\mathsf{A}$, $\mathsf{D}$ be constant arrays and $f,{\sigma }_{\epsilon }\in C^{\infty }({\mathbb{R}}^4)$ (any completion). On $V_{+}=\{(b-a{)}^0>0\}$, in the chart $w_{+}$ with $b=a+d_{+}$, $z=a-r_{+}$,

and

On $V_{-}=\{(b-z{)}^0>0\}$, ${K_5}^{[z],I,J}$ has the same form in the chart $w_{-}$ with $(a,I)\leftrightarrow (z,J)$, $d_{-}$, $r_{-}$. All coefficients are smooth on the chart; they are of the form $c_{j\beta }(w_{\pm })$ in (eq:Recrule) with $j\in \{1,2\}$ and $|\beta |=2$. The supports of ${K_5}^{[a]}$, $K_6$ (resp.\ ${K_5}^{[z]}$) meet the complement of $V_{+}$ (resp.\ $V_{-}$) only on $\{b=a\}$ (resp.\ $\{b=z\}$).

Proof. From (eq:L), $L^{\mu ,J}(a,z)={\sigma }_{\epsilon }(z){\mathsf{A}}^{\lambda \mu ,J}{\partial }_{z^{\lambda }}R(a-z)=-{\sigma }_{\epsilon }(z){\mathsf{A}}^{\lambda \mu ,J}({\partial }_{\lambda }R)(r_{+})$, so ${\partial }_{a^{\beta }}{L}^{\mu ,J}(a,z)=-{\sigma }_{\epsilon }(z){\mathsf{A}}^{\lambda \mu ,J}({\partial }_{\beta }{\partial }_{\lambda }R)(r_{+})$; and ${\partial }_{a^{\alpha }}R(b-a)=-({\partial }_{\alpha }R)(d_{+})$. The bracket in (eq:K5a) is therefore $f(b){\sigma }_{\epsilon }(a){\sigma }_{\epsilon }(z){\mathsf{A}}^{\alpha \beta ,I}{\mathsf{A}}^{\lambda \mu ,J}({\partial }_{\alpha }R)(d_{+})({\partial }_{\beta }{\partial }_{\lambda }R)(r_{+})$. With $a,z,y$ fixed, ${\partial }_{b^{\mu }}={\partial }_{d_{+}^{\mu }}$ acts on $f(a+d_{+})$ and on $({\partial }_{\alpha }R)(d_{+})$; insert Lemma lem:chart(iv). For $K_6$, write ${\delta }^{(4)}(a-z)=\delta (r_{+})={\square }_{r}R(r_{+})={\eta }^{\beta \lambda }{\partial }_{\beta }{\partial }_{\lambda }R(r_{+})$ (Lemma lem:R(iii)) and proceed identically; the overall sign is that of ${\partial }_{a^{\alpha }}R(b-a)=-({\partial }_{\alpha }R)(d_{+})$. ${K_5}^{[z]}$ follows by the symmetry of (eq:K5z) with (eq:K5a). The support statement: each kernel contains the factor $({\partial }_{\alpha }R)(d_{\pm })$ or its $b$-derivative, supported in $\{d_{\pm }\in \partial J^{+}\}$, and $\partial J^{+}\setminus \{0\}\subset \{d^0>0\}$ by Lemma lem:chart(ii).