A nonzero defect on free configurations

Failure of ${\mathcal{L}}_{\xi }$ to preserve the wave equation is also visible in this example, but that failure alone would not prove the proposition. The evaluated observable is the necessary witness.

The shift $\gamma \mapsto \gamma +\kappa {\mathcal{L}}_{\xi }\gamma$ does not generally preserve the free gauge-fixed solution space. It therefore cannot simply be declared an automorphism of ${\mathcal{A}}_0$ to cancel the defect. The ghost rule $s_0\gamma =\frac{1}{2}Kc$ is not that shift.