Lemma: Scalar density covariance on the frozen plateau

For a diffeomorphism generated by ${\mathfrak{g}}_U$,

coefficientwise through the retained order. The scalar solution with fixed free Cauchy data therefore obeys

Proof. The curved density $A(q)$ is natural under pullback, and $F^{\ast }g=g$. The difference between $F^{\ast }{A}_g[q]$ and $A_g[F^{\ast }q]$ is

where $\eta$ in this expression denotes its contravariant density. On the transformation support $g=1$, and outside that support $F$ and its derivative are the identity. The difference is zero. Divergence of a vector density is natural, proving the operator identity.

The pullback solution has the same Cauchy data because $F$ is the identity near $\Sigma$. Successive uniqueness of (eq:rhooneeq)–(eq:rhotwoeq) proves the solution identity.