Lemma: Second jets of compact-data free solutions

At any $p\in M$, let $a\in \mathbb{R}$, $b_{\mu }\in {\mathbb{R}}^4$, and symmetric $H_{\mu \nu }$ satisfy

There is $\varphi \in {\mathcal{S}}_{\varphi }$ with

Proof. Translate $p$ to $(0,0)$. Choose compact smooth data $q,\pi$ with

Finite polynomials multiplied by a bump equal to one near the origin realize these data jets. The free equation supplies

which is the remaining required component.