Main theorem on all of $U$ (positive spin)

Throughout, $(M,a,Q)$ is a subextremal Kerr–Newman background ($M>0$, $Q\neq 0$, $a^2+Q^2<M^2$), $\Delta=r^2-2Mr+a^2+Q^2$, $\bar\rho=r+ia\cos\theta$, $\bar\rho^*=r-ia\cos\theta$, $U=\{r>r_+,\ 0<\theta<\pi\}$, and $D^-=\sqrt{\Delta}\,\partial_r-i\,\partial_\theta,\quad D^+=\sqrt{\Delta}\,\partial_r+i\,\partial_\theta$ act through coordinate-slice derivatives; the mode parameters are $\omega,m\in\mathbb{R}$. For $\omega,m\in\mathbb R$ and $F=(\psi_2,\psi_1,k,s)$ $C^\infty$ on $U$: system (1) holds at every point of $U$ if and only if system (7) with $\varsigma=+1$ holds at every point of $U$ for $z_+=\operatorname{adj}(X)\,y_+^{(+)}$, $z_-=X^*y_-^{(+)}$,