Two issues settled before the long proof

metric does not mean linear gauge action. The metric jet $\eta +\kappa h+{\kappa }^{2}k$ is retained with its complete second-order pullback law. The second metric coefficient $k$ transforms by ${\mathcal{L}}_{\xi }h+K\zeta +\frac{1}{2}{{\mathcal{L}}_{\xi }}^2\eta$, and this term enters the second-order coordinate coefficient. No second-order relational expansion is combined with only the first term of the metric transformation.

regulator passes its own covariance test. Smoothing the dressing is not assumed to preserve the gauge identity. The frozen regulator multiplies the deviation of the metric density from flat by a smooth time ramp ${\sigma }_{\epsilon }$ inside the divergence-form coordinate operator. For a pure gauge supported where ${\sigma }_{\epsilon }=1$ and $\mathrm{d}{\sigma }_{\epsilon }=0$, the regulated first coefficient equals the exact first coefficient and returns $\xi$. For a pure gauge supported in the ramp zone it does not. Both statements are proved in Section sec:preflight before any second-order formula is written.