Lemma: Scalar covariance on the frozen plateau

For the formal transformations generated by ${\mathfrak{g}}_C$,

Proof. Before truncation the scalar density coefficient can be written

with $\eta$ interpreted here as a contravariant density. The curved density is natural under pullback. The only possible mismatch is $(1-g)(F_{\kappa }^{\ast }\eta -\eta )$. It vanishes because $g=1$ wherever the transformation acts, and the transformation is the identity elsewhere. Also $F_{\kappa }^{\ast }g=g$.

Divergence of a vector density is natural. The pulled-back solution has the same Cauchy data because the transformation is the identity near $\Sigma$. Successive Cauchy uniqueness proves the assertion.