Theorem: Gauge intersection

If $\xi \in {\mathcal{G}}_{\mathcal{S}}$ and $K\xi$ is spatial TT, then $\xi =0$. In particular,

Proof. The spatial trace gives ${\partial }_i{\xi }_i=0$. The spatial divergence then gives $\Delta {\xi }_j=0$. A harmonic Schwartz function is zero, so ${\xi }_j=0$. The condition $(K\xi {)}_{0j}=0$ implies ${\partial }_j{\xi }_0=0$, and spatial decay gives ${\xi }_0=0$.