Six corrections to the production sheet

The sheet proposed the Cauchy pairing $B_{\Sigma }$ as the remainder to be tested. The class invariant is the trisolution ${\mathsf{W}}_u=({\mathsf{E}}_0\otimes {\mathsf{E}}_0\otimes {\mathsf{E}}_m)u$; bulk equation terms are exactly the kernel of the map $u\mapsto {\mathsf{W}}_u$ on compact kernels (Lemma lem:kernel). $B_{\Sigma }$ is the leading onset coefficient of ${\mathsf{W}}_u$ in the counter-propagating null directions (Proposition prop:planewave), not the invariant itself.

The sheet expected the forbidden covector of the class to be $\pm \mathrm{d}t$ at $\Sigma$. Pairings with solutions see only null covectors (Lemma lem:transport). The conormal $\pm \mathrm{d}t$ is the sum $k_1+k_2$ of two co-oriented null covectors, one per metric slot; that pair is what every representative inherits (Corollary cor:covector).

The witness is a pair of exact null plane waves on the gauge-fixed shell, not a Cauchy-data expansion. The tools are XI-F's (Definition 8.1.2, Proposition 8.1.3, Theorem 8.2.12 of Hormander) together with Theorems 8.2.4, 8.2.9, 8.2.13 and 8.3.1. No restriction theorem to $\Sigma$ is used.

Transverse plane-wave pairs, which lie on the Einstein shell $A\gamma =0$, do not detect XI-F's data: the Bianchi identity makes the leading symbol conserved, and the total covector $2\mathrm{d}t$ annihilates its $0\nu$-components (Remark rem:einstein). The detecting polarization is pure trace, $ϵ=\eta$, which is on XI-E's gauge-fixed shell and off the Einstein shell.

XI-F's one-slot rewriting of a boundary trace as a bulk functional is the $n=1$ case of the kernel lemma $\ker {\mathsf{E}}_{\mu }\cap {\mathcal{E}}^{\prime }=(\square +{\mu }^2){\mathcal{E}}^{\prime }$ (Corollary cor:oneslot). It is a corollary of G2, not the two-slot proof.

The constructed branch of G3 is characterized by an iff criterion on $\mathrm{WF}({\mathsf{W}}_u)$ (Theorem thm:criterion) but is not instantiated: the criterion fails for XI-F's data (Theorem thm:obstruction), and for all data whose boundary trace of $\beta$ pairs nontrivially with the counter-propagating symbol (Proposition prop:general).