Corollary: Descending natural lift for the point-punctured module

In the module of Section sec:relations, with $Z=\{0\}$, if $T^{\circ }$ has any distributional extension, it has an extension $T$ for which $F_T$ annihilates all of $H^{\circ }$. Hence $F_T$ descends to a map on $\mathrm{im}{\mathsf{Rec}}^{\circ }$ extending every reconstructed value.

Relative to $F_{T_0}$, the correction is explicitly

where $D$ is given by (eq:pointD). Its values are supported at $(0,0)$, with response-normal order at most $r$ and tangential order at most $M$.

Proof. For $a\in H^{\circ }$, equation (eq:puncturednull) says that every $f_{\gamma }(a)$ belongs to $I$. By Theorem thm:pointcorrection, $f_{\gamma }(a)T=0$ for every $\gamma$. Theorem thm:jetnull therefore gives $F_T(a)=0$. Proposition prop:descent gives the factorization.

Equation (eq:pointmapcorrection) follows from $T=T_0-D$. The support and order bounds follow from the insertion formula and smooth multiplication.