Natural ghost equations without an interacting quantum differential

This is an equation-level construction of X-G XG (:ghost). For compactly supported $\xi$, the retarded inverse ${G_{\mathrm{gh}}}^{\mathrm{R}}=-{{\mathsf{E}}_0}^{\mathrm{R}}$ therefore gives

Indeed ${\mathcal{L}}_{\xi }c$ is compactly supported and the retarded left inverse applies. This identity does not construct a nonlinear interacting differential or a quantum master operation.