Theorem: XV-E local mathematical release

For the finite normal-jet module with punctured reconstruction of Section sec:jet:

1. $H={\beta }^{-1}(\mathrm{Ann}(T^{\circ }{)}^{M_r})$, $H_T={\beta }^{-1}(\mathrm{Ann}(T{)}^{M_r})$, and $H/H_T\cong (\mathrm{Ann}(T^{\circ })/\mathrm{Ann}(T){)}^{M_r}$; the null defect of ${F_0}^T$ vanishes iff $\mathrm{Ann}(T)=\mathrm{Ann}(T^{\circ })$;

2. null relations are universal, flat-annihilator, or jet-detected; the first two carry no defect for any extension, and on a compact set the defect is a functional of the $k$-jets of ${\beta }_{\gamma }(a)$ along $Z$, $k$ an order of $T$ there;

3. for a finite puncture, a descending extension exists whenever any extension exists, obtained by subtracting finitely many derivatives of the deltas at the puncture; in one dimension, if the puncture lies in the closure of $\mathrm{supp}{T}^{\circ }$, every extension descends;

4. for a general puncture, a descending extension exists iff the relative estimate (eq:relest) holds; whether it can fail is not decided.

In addition, Proposition prop:erratum corrects XV-D's Proposition prop:reduction(iii) as stated in Section sec:erratum.

Proof. (i) is Theorem thm:ann; (ii) is Definition def:kinds, Lemma lem:flat, and Theorem thm:jetdep; (iii) is Theorem thm:point and Corollary cor:1d; (iv) is Theorem thm:relative and Remark rem:open.