Proposition: Complete pure-gauge diagnostic

The coordinate coefficients are

The relational scalar coefficients are

Proof. For the auxiliary comparison metric $e^{\lambda \kappa {\mathcal{L}}_b}\eta$, the scalar functions $e^{\lambda \kappa X_b}{x}^{\mu }$ solve the harmonic equation. They have the frozen data because $b$ vanishes near $\Sigma$. Their coefficients are $\lambda b^{\mu }$ and ${\lambda }^2{b}^{\nu }{\partial }_{\nu }{b}^{\mu }/2$.

The actual auxiliary metric has the same first coefficient and an additional second coefficient ${\lambda }^2{d}_{\xi }$. Linearity of (eq:v) in $J$ adds ${\lambda }^2{\mathsf{E}}_0\mathsf{C}{d}_{\xi }$. This proves the coordinate formulas.

Substitute the scalar pullback expansion and these coordinates into the formal composition. The quadratic operator is

Since $[X_{\xi },X_b]=X_{[\xi ,b]}$, this is (eq:puretwo).