Remark: Scope: what is and is not quantified

(i) The quantifier of Theorem thm:obstruction is the copied class ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ and the free shell ${\mathcal{C}}_{\mathrm{sol}}$ of XI-E. Nothing is said about bimodules or algebras with other domains; XI-E's correction stands. (ii) The theorem obstructs universal admission of ${\mathcal{C}}_3(f,g)$ for all admissible $f$; it also obstructs every individual $f$ of F1 type, and every $(f,g,{\varphi }_0)$ caught by Proposition prop:general. A smaller observation class on which $\mathsf{b}\equiv 0$ is not declared here and not decided; by Theorem thm:criterion(b) its decision is the cone condition (eq:criterion), and the plane-wave test only certifies the leading onset coefficient. (iii) If both metric slots are restricted further to the Einstein subshell $A\gamma =0$, the pure-trace witness is not available and, by Remark rem:einstein, transverse pairs are blind to ${\beta }_{0\nu }{|}_{\Sigma }$; for the F1 data, where ${\beta }_{ij}=0$, no plane-wave pair detects the onset on that subshell. That smaller question is not decided here. XI-E's shell is the gauge-fixed one, and Theorem thm:obstruction answers the question as posed.