Theorem: Changed Noether identity

For the selected action, off shell,

All ramp derivatives are displayed in (eq:actualsource) and (eq:Rampdef). The $Q_{\nu }({Z_{\sigma }}_{\epsilon })$ term is the explicitly defined first-order operator (eq:Noperators) applied to the displayed quadratic source.

Proof. For an arbitrary compact interior $\xi$, use the scaled metric–scalar variation

Compact integration by parts gives

For the unweighted Einstein–Hilbert part, naturality gives zero variation. Variation of the unchanged switched scalar action gives $\kappa \int {\xi }^{\nu }{\mathcal{B}}_{g,\nu }{\mathrm{d}}^{4}x$. This follows by expanding the flat scalar variation as

and the curved scalar variation as a density divergence, then integrating the factors $1-g$ and $g$.

The added cubic functional in (eq:selected-action) varies as

Comparison of the two variation expressions, followed by Lemma lem:divergence, proves (eq:changed-noether).