The unchanged free operator

Define

The inherited flat gauge fixing is ${S_{\mathrm{gf}}}^{(0)}=\frac{1}{4}\int C_{\mu }{C}^{\mu }$. Compact variation gives

Indeed, $G_{\mathrm{lin}}={\mathsf{P}}_{\mathrm{tr}}(-\frac{1}{2}\square +\frac{1}{2}KC)$; the $KC$-terms cancel.

Since ${{\mathsf{P}}_{\mathrm{tr}}}^2=1$ in four dimensions, the free retarded and advanced inverses are

These map tensor densities to covariant tensors; constant $\eta$-identifications are implicit in component notation.

The free equation is ${\mathsf{P}}_{\mathrm{gr}}r=0$, equivalently $\square r=0$. It does not impose $C[r]=0$ strongly. The compact residual statement also remains unchanged: if $\xi$ is compact and ${\mathsf{P}}_{\mathrm{gr}}K\xi =0$, then $K\square \xi =0$; the compact Killing field $\square \xi$ is zero, and wave uniqueness gives $\xi =0$.