Lemma: Gauge kernel of the unfixed operator

The displayed operator satisfies $PK\xi =0$. It cannot therefore be used as an invertible gauge-fixed covariance operator.

Proof. The exact Euler equations transform tensorially under pullback. Differentiate $E[{\rho }^{\ast }\bar{g},{\rho }^{\ast }\bar{\varphi }]={\rho }^{\ast }E[\bar{g},\bar{\varphi }]=0$ along the flow of $\xi$. This gives $PK\xi =0$. Compact vector fields with $K\xi \ne 0$ exist, so the kernel is nontrivial.