$(\bar h,\bar a,\xi,\bar\xi,\Upsilon,\bar\Upsilon)$-space: $$\begin{aligned}
\Big[\left(\mathcal{Z}_k\right)_{\bar h\bar h}\Big]^{\mu \nu}_{\ \ \rho\sigma} &=\frac{Z_N(k) \kappa^2}{2}(\delta^\mu_\rho \delta^\nu_\sigma+\delta^\nu_\rho \delta^\mu_\sigma-\bar g^{\mu\nu} \bar g_{\rho\sigma}) \\
\Big[\left(\mathcal Z^{\textrm{gh}}_k\right)_{\bar \xi \xi}\Big]^\mu_{\ \nu}&=\sqrt{2}\; \delta^\mu_\nu
\end{aligned}
\
\begin{aligned}
\Big[\left(\mathcal{Z}_k\right)_{\bar a\bar a}\Big]^{a \mu b}_{\ \ \ \nu} &= \frac{Z_F(k)}{\hat{g}^2}\; \delta^{ab} \delta^\mu_{\nu}\\
\Big[\left(\mathcal Z^{\textrm{gh}}_k\right)_{\bar \Upsilon \Upsilon}\Big]^{ab}&=\delta^{ab}
\end{aligned}
\label{Zfactors}$$ Note that $\mathcal Z^{\textrm{gh}}_k$ is actually $k$-independent.

In setting up eq. (\[ERGE\]) we followed ref. [@nonabavact] and opted for the complete Hessian operator $\Gamma^{(2)}_k$ to play the rôle of $\Delta$.[^1] More precisely, we set $\Delta= \mathcal Z^{-1}_k\breve{\Gamma}^{(2)}_k$ and $\Delta=\mathcal Z_{\textrm{gh}}^{-1} S^{(2)}_\textrm{gh}$ in the $(\bar h,\bar a)$- and the ghost-sectors, respectively. The multiplication by the inverse $\mathcal Z$ matrices brings $\Delta$ closer to an ordinary (covariant) Laplacian; symbolically, if $\Gamma^{(2)}_k= - \zeta_k \partial^2+ \cdots, \ \mathcal Z_k =\zeta_k$, we employ $\Delta=-\partial^2+\cdots$ rather than $\Delta=-\zeta_k \partial^2+\cdots$.

The most complicated ingredient needed in order to evaluate the traces in the FRGE is the Hessian of the bosonic action $\breve{\Gamma}^{(2)}_k$, i.e. the matrix of its second functional derivatives with respect to the dynamical fields $(\bar h, \bar a)$, or equivalently $(g, A)$, at fixed backgrounds $(\bar g, \bar A)$. This Hessian is most transparently displayed by means of the associated quadratic form $\Gamma^{\textrm{quad}}_k$ which appears in the expansion $$\breve{\Gamma}_k[\bar 