Input statement: XII-C: the selected action

Density before variation, at cubic order only:

Full Einstein–Hilbert weighting is rejected: it produces a flat-background Euler term and changes the free sheet. The actual cubic source is ${\mathcal{J}}_{\mathrm{D}}={\sigma }_{\epsilon }{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }$. With $e_{\mathrm{D}}=\delta S_{\mathrm{D}}/\delta \gamma$, $E_{\varphi }=\delta S_{\mathrm{D}}/\delta \varphi$, $w={\sigma }_{\epsilon }-1$, and ${Z_{\sigma }}_{\epsilon }=w{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }$, the changed Noether identity holds off shell:

For every declared switch, the fixed-input invariant jet ${{\mathcal{J}}^{\mathrm{D}}}_{g,{\sigma }_{\epsilon }}({\varphi }_m)$ is empty: a solution-fiber obstruction, not an observable no-go. The quadratic coefficients are XI-A/XI-B's; the inverse is ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$ with coefficient $-1$ XIIC,XIB.