Proposition: Switched scalar intertwiner

For every $(h,k)$, every free ${\varphi }_0$, and every $F\in {\mathcal{G}}_U$,

Coefficientwise,

Proof. By Lemma lem:plateaueuler, $F^{\ast }{\rho }_g[q;{\varphi }_0]$ solves the switched equation for $F^{\ast }q$. Its zeroth coefficient is ${\varphi }_0$, and its positive-order coefficients ${\rho }_{1,g}+X_{\xi }{\varphi }_0$, ${\rho }_{2,g}+X_{\xi }{\rho }_{1,g}+X_{\zeta }{\varphi }_0+\frac{1}{2}{X_{\xi }}^2{\varphi }_0$ have zero Cauchy data because $\xi ,\zeta$ vanish near $\Sigma$. Lemma lem:rhounique gives (eq:scalarintertwiner).