Three classical objects

Conventions: $\eta =\mathrm{diag}(1,-1,-1,-1)$, $q=\eta +\kappa h$, $a=\frac{1}{2}$, and $\mathcal{P}$ is trace reversal. Parentheses symmetrization includes the factor $\frac{1}{2}$. The predecessor gauge map is written ${R^{\mathrm{pred}}}_0$, and $R_0(c)={R^{\mathrm{pred}}}_0(ac)$. The letter $K$ denotes only the kinetic operator $-\square \mathcal{P}$. It is never the gauge map.