Remark

Equivalently, at the first order of (eq:metric-euler-expanded) at which the source appears, $\frac{1}{2}{G}_{\mathrm{lin}}[k]+\frac{1}{2}{\mathcal{G}}_2[h,h]=\frac{1}{2}gT[{\varphi }_0]$ on $G_{\mathrm{lin}}[h]=0$; the divergence of the left side vanishes by the Bianchi identity, that of the right side is $({\partial }_{\mu }g)T^{\mu \nu }[{\varphi }_0]$. The obstruction lives in $\mathrm{supp}\mathrm{d}g$, outside $U$; it is not removed by any choice of ${\mathfrak{g}}_U$. The known resolution switches the entire Lagrangian and recovers gauge invariance only in the algebraic adiabatic limit; it is named and not opened here. The relational observable is untouched: it was never defined by solving the metric equation.