Theorem: XV-D local mathematical release

For the finite normal-jet module with punctured reconstruction defined in Section sec:nulljet:

1. every extension $T$ of the punctured coefficient $T^{\circ }$ defines a preliminary lift ${F_0}^T={\mathsf{Rec}}_T$ with $r{F}_0^T={\mathsf{Rec}}^{\circ }$;

2. the two reconstruction kernels are (eq:Hpunct) and (eq:Hfull), with $H_T\subseteq H$;

3. the null defect is exactly ${d_0}^T(a)={\sum }_{\gamma }{\mathsf{E}}_{\gamma }({\beta }_{\gamma }(a)T)$, is supported in $Z\times \{0\}$, and induces an injection $H/H_T\to Y$;

4. the defect can be nonzero (Proposition prop:example), depends on the extension, and is cancelled by a change of extension exactly when some extension $T^{\prime }$ has $H_{T^{\prime }}=H$;

5. the relative obstruction in the unrestricted puncture-supported class vanishes, and no separated contact witness exists;

6. universal relations, including the jet analogue (eq:ajet) of the inherited identity (eq:inheritednull), carry no defect for any extension.

In addition, under hypotheses (H1)–(H2), extendability of the cross-line traces $c_{D,\beta }$ implies XV-C's supported-sector estimates order by order (Proposition prop:reduction).

Proof. (i)–(ii) are Theorem thm:kernels. (iii) is Theorem thm:defect. (iv) is Proposition prop:example and Corollary cor:extension. (v) is Proposition prop:unrestricted and its corollary. (vi) is Proposition prop:universal. The last sentence is Proposition prop:reduction.