Theorem: XV-C local mathematical release

For the local objects explicitly defined in this manuscript:

1. the reconstruction-null space of ${{\mathcal{P}}_r}^{\mathrm{jet}}(T)$ is exactly the simultaneous kernel of the collected coefficient maps (eq:B);

2. the supplied supported-sector normal form has the unique collected coefficients (eq:degree3)–(eq:degree0);

3. a distribution on a closed-set complement extends distributionally exactly when it satisfies the compact-uniform finite-order test estimate (eq:extensionbound);

4. a supported normal-order-preserving realization of the supplied local normal form exists in the unrestricted distribution class exactly when its collected coefficients satisfy (eq:jetbound);

5. its unrestricted supported ambiguity is (eq:ambiguity), with contact support characterized by Proposition prop:support;

6. a sequence (eq:failuresequence) for an actual coefficient would exclude its unrestricted distributional extension.

Proof. Assertion (i) is Theorem thm:jetnull. Assertion (ii) is Proposition prop:collected. Assertion (iii) is Theorem thm:extension. Assertion (iv) is Theorem thm:supported and Corollary cor:sourceestimate. Assertion (v) is the ambiguity part of Theorem thm:supported, together with Proposition prop:support. Assertion (vi) is Proposition prop:failure.