$D^+$ of the normalised chiral component $g_1$ (negative 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 $\omega,m\in\mathbb{R}$. The negative-spin chiral variables of eq. (3) are $y_e^{(-)}=\Delta^{-3/4}\sqrt{\sin\theta}\,(g_1^{(e)},g_2^{(e)})$ with $g_1^{(e)}=\psi_{-2}-e\,i\sqrt{\Delta}\,\psi_{-1}$, $e=\pm1$. For fields $(\psi_{-2},\psi_{-1},n,\ell)$ slice-differentiable at $x\in U$ (NegDiff) and any real $e$,