Lemma: No invisible plane direction

If $S_{t_0}(k){\varphi }_0=0$ for every compact-data free scalar and $k\in \mathrm{span}\{e,r\}$, then $k=0$.

Proof. The field equation gives $k^{ij}{\partial }_i{\partial }_j{\varphi }_0=0$ for every such solution. At any chosen time and point, arbitrary spatial Hessians can be prescribed using compact smooth position data there. Propagating those data to $t_0$ preserves compactness. Therefore $k$ pairs to zero with every symmetric matrix at every point.