Lemma: Coordinate coefficients for the pure-gauge test

For (eq:puregauge),

All coefficient Cauchy data vanish.

Proof. The first assertion is Lemma lem:preflight.

For the second, first consider the auxiliary pure-gauge metric

The functions

solve its harmonic equation. This follows directly from scalar naturality of the wave operator applied to ${\square }_{\eta }{x}^{\mu }=0$. Since $a$ vanishes near $\Sigma$, their data are the frozen coordinate data. Uniqueness therefore proves these are its coordinate coefficients.

The actual auxiliary jet has the same first coefficient and an additional second coefficient ${\lambda }^2{d}_{\xi }$. Equation (eq:v) is linear in that second coefficient, so it adds ${\lambda }^2{\mathsf{E}}_{0}C{d}_{\xi }$. The data statement follows from the support of $a,d_{\xi }$ and the zero-data inverse.