The pure-trace sector of the cubic source

For the spatially homogeneous sector ${\gamma }_{0\mu }=0$, ${\gamma }_{ij}={\gamma }_{ij}(t)$, XI-F's Lemma gives ${J_3}^{0i}=0$ and ${J_3}^{00}=q(\dot{\gamma },\dot{\gamma })$, $q(a,b)=\frac{1}{4}(\mathrm{tr}a\mathrm{tr}b-a_{ij}{b}_{ij})$, so there $B_{\Sigma }={\int }_{\Sigma }{\beta }_{00}q({\dot{\gamma }}_1,{\dot{\gamma }}_2)$. The full twenty-component form of $Q$ is XI-E's ${\mathcal{H}}_2$ with ${{\partial }_t}^2\mapsto \Delta$; the witness needs only the pure-trace sector printed above.