Lemma: The witness is on the complete free shell

The configuration (eq:main-fields) belongs to ${\mathcal{C}}_{\mathrm{sol}}$. Its auxiliary field and its Einstein tensor coefficient are

In particular $D^{\ast }b\ne 0$.

Proof. $\square \psi =2-2=0$, hence $\square \gamma =0$ and $\square \varphi =0$. Trace reversal gives $\mathcal{P}(\psi \eta )=-\psi \eta$, so $b^{\nu }=-{\partial }^{\nu }\psi$, which is (eq:main-Ab)'s first formula. Its divergence is zero, and

with the other components zero. Lemma lem:shell gives the second formula. The ghost equations hold because the ghosts vanish.