Lemma: Local realization of scalar second jets

At every $p\in {\mathcal{D}}_0$, every jet

with symmetric $H$ and

is realized by a member of ${\mathcal{S}}_{\varphi }$.

Proof. At time $s=s(p)$, prescribe compact polynomial-times-bump Cauchy data near $x(p)$ realizing $A,b_{\mu },H_{ij},H_{0i}$. The equation supplies $H_{00}$.

Choose the support ball so small that its backward or forward propagation to time $0$ remains inside $B_{r_{\ast }}$; the strict inequality defining ${\mathcal{D}}_0$ permits this. Cauchy uniqueness then identifies the resulting local solution with the solution evolved from those compact time-zero data.