Theorem: Mathematical conclusions of XV-B

For the interfaces explicitly defined in this paper:

1. a presentation-level linear construction descends exactly when it annihilates the reconstruction kernel;

2. within a specified affine correction family, a descending candidate exists exactly when its relative obstruction vanishes, and all descending candidates have the stated affine ambiguity;

3. contact-supported corrections cannot cancel a null defect detected on an open set disjoint from their support;

4. in the specified finite normal-jet module, the coefficient equations (eq:completekernel) give the complete null test;

5. supported extension preserving a fixed normal-order bound is equivalent to coefficient extension, and its full unrestricted supported ambiguity is (eq:jetambiguity);

6. coefficient extension across a relatively closed set is equivalent to the uniform local finite-order estimates (eq:closedestimate);

7. for natural lifts across a smooth missing locus, universal and flat-annihilator relations contribute no defect, and the remaining defect is locally determined by finite transverse jets;

8. at a point puncture, an existing coefficient extension can be corrected by finitely many delta derivatives to preserve all smooth annihilator relations simultaneously;

9. finite normal order and flat multiplication alone do not guarantee extendability.

Proof. Assertion (i) is Proposition prop:descent. Assertion (ii) is Theorem thm:lifting. Assertion (iii) is Theorem thm:contact. Assertion (iv) is Theorem thm:jetnull. Assertion (v) is Theorem thm:jetextension. Assertion (vi) is Theorem thm:closedextension. Assertion (vii) follows from Lemma lem:jetdefect and Corollary cor:relationclassification. Assertion (viii) is Theorem thm:pointcorrection and Corollary cor:pointdescent. Assertion (ix) follows from Corollary cor:normalnotextension and Proposition prop:flatcounterexample.