Errata to XV-E and corrections to the circulated draft

(E1) Disjoint classification. XV-E's Definition def:kinds calls $a\in H$ universal if $\beta (a)=0$, a flat-annihilator relation if each ${\beta }_{\gamma }(a)$ is flat on $Z$, and jet-detected otherwise. As worded, every universal relation is also flat-annihilator. The disjoint convention is Definition def:kinds; every theorem of XV-E holds under either reading, since each statement about the first two classes is a vanishing statement.

(E2) Jet-detection is necessary, not sufficient. The sentence in XV-E preceding its Lemma lem:cutoff, “jet-detected relations produce a defect,” overstates. What XV-E's Theorem thm:jetdep(ii) proves is that a nonzero defect is produced only by jet-detected relations. Theorem thm:jetchar settles the exact relationship.

(E3) Multiplier statement retained. XV-E's Proposition prop:erratum is retained unchanged: for $v\in C^{\infty }(S_0)$, nonextendability of $vc$ implies nonextendability of $c$; a failure sequence for $vc$ transfers to a test family for $c$ that excludes extension but need not remain normalized in every increasing seminorm.

(F1) The argument of the reconstruction coefficients. The circulated draft glossed the symbol $w$ in XIV-I's reconstruction rule (eq:Recrule) as “the remaining variables.” The fifth pass Ichart shows that $w_{\pm }$ is XIV-I's name for the full adapted chart point $(u_{\pm },{\zeta }_{\pm },q_{\pm },r_{\pm },y)$. The coefficients $c_{j\beta }(w)$ are therefore smooth functions on the chart, including of $q_{\pm }$ and $r_{\pm }$, unless XIV-I's unreproduced coefficient class restricts them.

(F2) The free-module claim. The circulated draft's Lemma lem:recrule(iii) asserted that ${\mathcal{K}}_{\mathrm{int}}$ is free on the family $\{{\delta }^{(j)}(q){{\partial }_r}^{\beta }R(r)\}$ and that $\ker \mathsf{Rec}$ is the coefficient kernel. This is correct only for coefficients independent of $(q,r)$. Over chart functions the family is dependent—$q{\delta }^{(j)}(q)=-j{\delta }^{(j-1)}(q)$ and $r^{2}R(r)=0$—and XIV-J's null relation $(qi\theta +1)M_{\mathrm{R}}$ is the instance $j=1$ of the first identity. Lemma lem:recrule states both cases; manifest items D0.3 and D0.4 are revised.

{(F3) The symbol $\hat{H}$.} The fourth pass returned XIV-J's grouped cross-line operator with hatted replacement blocks ${\hat{H}}_{\gamma \gamma }$, ${\hat{H}}_{\varphi \varphi }$. The fifth pass reports that no manuscript in the corpus contains a symbol $\hat{H}$ (the only hatted symbols in XIV-I are Fourier transforms and ${\hat{s}}_0$, which XIV-I excludes from the present construction). The content of (eq:cross)—pullback by $\pi$ of a difference of two tensor products of blocks—is retained; the replaced blocks are written ${H^{\prime }}_{\gamma \gamma }$, ${H^{\prime }}_{\varphi \varphi }$ as this document's placeholder, and D0.11 carries the annotated status T${}^{\ast }$\ (display reproduced in content; one symbol disputed between passes).