Proposition: Raw functional obstruction

The unsectioned functional

at a fixed free scalar is not, in general, invariant under compact metric gauge changes. In particular, some such functionals cannot be written for all admitted representatives as $\int h_{\mu \nu }{J}^{\mu \nu }$ with compact smooth symmetric ${\partial }_{\mu }{J}^{\mu \nu }=0$. The retarded harmonic section above is not a left inverse on every spacelike-compact pure gauge.

Proof. Choose $\xi$ compactly supported strictly after $t_0$. Equation (eq:Sgauge) then gives

which is nonzero for suitable data and a test supported where a scalar derivative is nonzero. A compact conserved tensor current would instead annihilate $K\xi$ by integration by parts.

For the last statement choose a nonzero vector wave $\xi$ with compact Cauchy data and $K\xi \ne 0$. It is spacelike compact and $\square \xi =0$, so ${G_{\mathrm{ret}}}^{\square }D(K\xi )=0$, not $\xi$.