The positive-spin chiral variables are $C^\infty$ on $U$

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 chiral variables of eq. (3) are $y_\pm^{(+)}=\Delta^{5/4}\sqrt{\sin\theta}\,\bigl(\psi_2\pm i\Delta^{-1/2}\psi_1,\ -2Q(k\pm i\Delta^{-1/2}s)\bigr)$ (yPos p (±1) F) and $y_\pm^{(-)}=\Delta^{-3/4}\sqrt{\sin\theta}\,\bigl(\psi_{-2}\mp i\Delta^{1/2}\psi_{-1},\ 2Q(n\mp i\Delta^{1/2}\ell)\bigr)$ (yNeg p (±1) G).

If $\psi_2,\psi_1,k,s$ are $C^\infty$ on $U$ (PosSmooth), then for every real $e$ and each component $j\in\{0,1\}$ the function $x\mapsto y_e^{(+)}(x)_j$ is $C^\infty$ on $U$. This uses $\Delta>0$ and $\sin\theta>0$ on $U$, so that $\Delta^{5/4}$, $\sqrt{\sin\theta}$ and $1/\sqrt{\Delta}$ are smooth there.