An operator audit of the density terms

Let $V,W$ and $V^{\prime },W^{\prime }$ be the operators before and after (eq:metric-transform). Put

Expanding (eq:plateaucov) gives the off-shell identities

These retain the density divergence in $D$ and the terms proportional to $P_m$.

One can verify the second scalar law without appealing further to a covariance slogan. Set $a=u_{1,g}$, $b=u_{2,g}$, and $b_{\mathrm{cand}}=b+Xa+Y{\varphi }_0$. Using $P_{m}a=-V{\varphi }_0$, $P_{m}b=-Va-W{\varphi }_0$, and (eq:operator-audit),

The candidate has zero Cauchy data because the gauge generators vanish near ${\Sigma }_0$. Uniqueness gives the second scalar law.