Corollary: Two readings of the supported-sector presentation

1. If XIV-J's presentation space for an entry $e\in \{6,8,9\}$ is the space of coefficient tuples $(b_{\gamma }{)}_{|\gamma |\le 3}$, $b_{\gamma }\in {\mathcal{D}}^{\prime }(S_{\mathrm{resp}})$, with ${\mathsf{Rec}}_e={\sum }_{\gamma }{\mathsf{E}}_{\gamma }$, then ${\mathsf{Rec}}_e$ is injective: $\ker {\mathsf{Rec}}_e=0$.

2. If it is instead a module of smooth functions acting on fixed insertions, as in XIV-I's chart module (P5), then $\ker {\mathsf{Rec}}_e$ consists of the Taylor relations of Lemma lem:taylor5(i) and is nonzero.

3. In either reading, ${\mathcal{K}}_6,{\mathcal{K}}_8,{\mathcal{K}}_9\subseteq {\mathcal{J}}^3$ on $V_{+}$, and the three images are not in direct sum as soon as each contains ${\mathsf{E}}_0(b)$ for $b$ in a common coefficient class; then Lemma lem:directsum produces cross-entry null relations.

The record does not display which reading XIV-J adopts Jreference; the traces $c_{D,\beta }$ are named, not displayed.

Proof. (i) is Lemma lem:inject; (ii) is Lemma lem:taylor5(i) read as a relation; (iii) is Theorem thm:normalorder and Lemma lem:directsum.