Remark: The on-shell class is not decided

Theorem thm:F1 concerns the frozen representative of ${\mathcal{C}}_3$: the slab functional viewed on $M$ by restriction, which is the representative XI-E's admission uses. On the free shell, a single-slot boundary trace ${\int }_{\Sigma }u\gamma (0,\cdot )$ equals a bulk functional $-\int \gamma \square v$ with a source flat at $\Sigma$, hence has a smooth representative. The boundary term carried by $u$ at linear order in $v_1$ is a bilinear form in the Cauchy data of both metric slots, weighted by $\beta {|}_{\Sigma }$; whether it has a representative satisfying (eq:mc) after both slots are restricted to free data is not proved here, and we do not claim it. Admission of some other representative of $[{\mathcal{C}}_3{|}_{{\mathcal{C}}_{\mathrm{sol}}}]$ would be a separate theorem.