Lemma: Nonempty scalar fibers

For smooth $k$ satisfying $\mathsf{D}k=0$, $G^{(1)}(k)=0$, and ${\varphi }_0\in {\mathcal{S}}_{\varphi }$, every pair of compact smooth scalar data determines a unique smooth spacelike-compact solution

Proof. The source is smooth and supported in the cone of ${\varphi }_0$. The forced Cauchy lemma gives the solution. Propagation from both compact initial data and the source cone gives a cone of slope one with possibly enlarged initial radius.