Theorem: XV-F local mathematical release

1. (Source.) The fourth and fifth passes reproduce the reconstruction rule (eq:Recrule), the adapted chart (eq:chart), the kernel $R$ and its square (eq:R)–(eq:RR), the cross-line definition (eq:cross) (one symbol disputed), the mixed vertex Hessian (eq:hess), the scalar stress (eq:T0), the output convention (eq:Rconv), the role of $\mathsf{A}$ (eq:a1H), and three refusals of $U_{\mathrm{can}}=W\setminus \overline{{\mathcal{N}}_{\mathrm{att}}}$; they paraphrase $m^{\circ }$, the joint-map class, the generators and coefficient class of ${\mathcal{A}}^m(V)$, and the order bound on $q_{\alpha \beta }$; and they return candidate displays (eq:a1), (eq:a2), $f\in {C^{\infty }}_c$. Verified: $R$ is the unique past-bounded fundamental solution of ${{\partial }_t}^2-\Delta$ (Lemma lem:R); the $R\ast R$ display (Lemma lem:RR); the chart is a global chart of $\{{d_{\pm }}^0>0\}$ of dimension sixteen (Lemma lem:chart); both $K_5$ placements and $K_6$ have the form of the reconstruction rule in the charts (Proposition prop:K5chart); the reconstruction family is free over $(q,r)$-independent coefficients and dependent over chart functions, XIV-J's null relation being an instance (Lemma lem:recrule); $\mathfrak{a}=-\mathcal{P}$ (Lemma lem:fiber); ${\dot{a}}_2$ (Lemma lem:a2); the polarization factor is two-valued (Lemma lem:polar); the advanced kernel exists uniquely for all $m\ge 0$ (Lemma lem:advanced); all six internal expressions exist for every completion (Theorem thm:exist), with supports as in Proposition prop:support; the candidates reduce to seven located sentences (Proposition prop:reduction); ${\mathcal{C}}_{\mathrm{part}}$ is a contact-supported class of values whose presentation-level form and parametrization are not supplied (Proposition prop:classstatus). One check cannot be performed (Remark rem:conormal).

2. (Conditional.) Under hypotheses (H1)–(H3), any lift must send each null relation into ${\mathcal{C}}_{\mathrm{part}}$, and a defect on an open set off the collision closure or a separating functional is an obstruction (Proposition prop:conditional).

3. (Jet module.) Null relations are, disjointly, universal, flat-annihilator, or jet-detected; for extendable $T^{\circ }$, jet-detected means that some extension carries a nonzero defect (Theorem thm:jetchar); descending extensions form the affine space $T+{\mathcal{S}}_Z(T^{\circ })$, infinite-dimensional in XV-D's example (Theorem thm:affine, Proposition prop:exampleSZ); puncture-supported distributions of finite order are finite delta sums, and the puncture-supported ambiguity of bounded normal order is infinite-dimensional unless the puncture is empty, with dimension $M_r\left(\genfrac{}{}{0ex}{}{k+d}{d}\right)$ per point under a tangential bound $k$ (Theorem thm:dimension, Corollary cor:finite); the same dichotomy governs ${\mathcal{C}}_{\mathrm{part}}$ through its weight space (Proposition prop:Cpartdim).

In addition, Section sec:errata records errata (E1)–(E2) to XV-E and corrections (F1)–(F3) to the circulated draft.

Proof. Each item cites the result that proves it.