Proposition: Action compatibility does not repair the coordinates

For $\xi \in {\mathfrak{g}}_C$,

as a formal variational identity. This is compatible with $c_{\epsilon }>0$.

Proof. The unswitched first-derivative gravitational representative differs from the Hilbert density by a divergence. Together with the natural scalar density, this gives local current coefficients ${n_j}^{\mu }$ satisfying

The currents are local in a finite jet of $\xi$. Their support is compact in the gauge-support region. These are the coefficient Noether identities for the fixed representative XConv (:actionidentity).

Varying (eq:action) and integrating by parts gives

The boundary terms vanish by compact support. Every remaining factor involving $1-g$ or $\mathrm{d}g$ vanishes on the gauge support.

The convolution regulator appears nowhere in the action. It appears in (eq:harmonic), so the independent action identity cannot change (eq:preflight).