Proposition: Recalled from XV-F-patch

{XVFpatch (Thm. thm:exist)} For any constant arrays $\mathsf{A}$, $\mathsf{D}$, $\mathsf{P}$, $\mathfrak{b}$, any $m\in \mathbb{C}$, and any $f,g,F,{\sigma }_{\epsilon },q_{\alpha \beta }\in C^{\infty }({\mathbb{R}}^4)$, each of (eq:K5a)–(eq:K9) is a distribution on ${{\mathbb{R}}^{12}}_{(a,b,z)}$; the $v$-integrals in (eq:T), (eq:K8) are the distributions ${\mathcal{T}}_0$, ${\mathcal{T}}_{\mathcal{B}}$ with $⟨{\mathcal{T}}_0,\varphi ⟩=⟨Q,{{\Phi }^R}_{\varphi }⟩$, ${{\Phi }^R}_{\varphi }(v,a,z)=⟨R,\varphi (a,v+\cdot ,z)⟩$, and likewise for $\mathcal{B}$.

{XVFpatch (Lemma lem:Q)} For smooth $\Phi$ with compact $v$-support near the diagonal, $⟨Q,\Phi ⟩={\sum }_{\alpha ,\beta }\int q_{\alpha \beta }(v)({{\partial }_z}^{\alpha }{{\partial }_a}^{\beta }\Phi )(v,v,v){\mathrm{d}}^{4}v$.

{XVFpatch (Prop. prop:support)} $\mathrm{supp}{K}_6,\mathrm{supp}{K}_9\subseteq S_{\partial }$, $\mathrm{supp}{K}_8\subseteq S_{J^{+}}$.

{XVFpatch (Lemma lem:chart), source Ichart} With $u=a$, $d=b-a$, $q=d^2$, $\zeta =\mathbf{d}\in {\mathbb{R}}^3$, $r=a-z$, the map $w_{+}=(u,\zeta ,q,r,y)$ is a global chart of $V_{+}:=\{(b-a{)}^0>0\}\subset W$ onto ${\mathbb{R}}^4\times {\Omega }_0\times {\mathbb{R}}^4\times {\mathbb{R}}^4$, ${\Omega }_0=\{q+|\zeta {|}^2>0\}$, $d^0=\sqrt{q+|\zeta {|}^2}$; the second placement $w_{-}$ uses $u=z$, $d=b-z$, $r=z-a$ on $V_{-}=\{(b-z{)}^0>0\}$. On $V_{+}$, $R(d)=\frac{1}{2\pi }\delta (q)$, $({\partial }_{\alpha }R)(d)=\frac{1}{\pi }{d}_{\alpha }{\delta }^{\prime }(q)$, and $\mathcal{B}(d)=\frac{1}{8\pi }\Theta (q)$.

{XVFpatch (Lemma lem:recrule), source Iamodule,Jnull} XIV-I's module ${\mathcal{A}}^m(V)$ on the chart $w_{\pm }$ is reconstructed by $\mathsf{Rec}(A)={\sum }_{j,\beta }{c}_{j\beta }(w){\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)$; XIV-J's values $\mathsf{Rec}(M_{\mathrm{R}})=\delta (q)R(r)$, $\mathsf{Rec}(i\theta M_{\mathrm{R}})={\delta }^{\prime }(q)R(r)$ and null relation $(qi\theta +1)M_{\mathrm{R}}$ are instances; the generating family is free over $(q,r)$-independent coefficients and dependent over chart functions.

{XVFpatch (Prop. prop:K5chart)} On $V_{+}$, ${K_5}^{[a]}$ and $K_6$ have the form of the rule in (P5) with $j\in \{1,2\}$, $|\beta |=2$; on $V_{-}$ so does ${K_5}^{[z]}$; $K_{47}$ has this form in neither chart.