Lemma: Uniqueness and support of the scalar jet

Equations (eq:rho1)–(eq:rho2) give the unique formal solution with the stated data. The sources are smooth and compactly supported in $\mathrm{supp}g$, and $\mathrm{supp}{\rho }_{j,g}\subset J^{+}(\mathrm{supp}g)\cap M_T$ for $j=1,2$.

Proof. Coefficient comparison and Lemma lem:cauchy, applied first at order $\kappa$ and then at order ${\kappa }^2$. The coefficients of ${\mathsf{V}}_{1,g}$ and ${\mathsf{V}}_{2,g}$ carry the factor $g$ or $\partial g$.