Recalled definitions

Let $X\subset {\mathbb{R}}^d$ and $V_q\subset {\mathbb{R}}^m$ be open with $0\in V_q$, let $Z\subset X$ be relatively closed, and put $X^{\circ }=X\setminus Z$, $U=X\times V_q$, $U^{\circ }=X^{\circ }\times V_q$. For $b\in {\mathcal{D}}^{\prime }(X)$, $\varphi \in \mathcal{D}(U)$,

and ${{\mathsf{E}}^{\circ }}_{\gamma }:{\mathcal{D}}^{\prime }(X^{\circ })\to {\mathcal{D}}^{\prime }(U^{\circ })$ is defined by the same formula. Fix $r\ge 0$, let $M_r=\#\{\gamma \in {\mathbb{N}}^m:|\gamma |\le r\}$, and set ${{\mathcal{P}}_r}^{\mathrm{jet}}={⨁}_{|\alpha |\le r}{C}^{\infty }(U)$. For $a\in {{\mathcal{P}}_r}^{\mathrm{jet}}$ put

Fix $T^{\circ }\in {\mathcal{D}}^{\prime }(X^{\circ })$. The punctured reconstruction and, for an extension $T\in {\mathcal{D}}^{\prime }(X)$ of $T^{\circ }$, the full reconstruction are

with kernels $H=\ker {\mathsf{Rec}}^{\circ }$, $H_T=\ker {\mathsf{Rec}}_T$. The extension lift is ${F_0}^T={\mathsf{Rec}}_T$ and its null defect is ${d_0}^T={F_0}^T{|}_H$. In XV-B's interface, $O=U^{\circ }$, $m^{\circ }=\mathrm{id}$, $r$ is restriction, and $Y=\{t\in {\mathcal{D}}^{\prime }(U):\mathrm{supp}t\subset Z\times V_q\}$.