Main theorem (negative spin): (1)$^-$ ⟺ (7) at each point of $U$ for $C^\infty$ fields

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 system (1)$^-$ is obtained from (1a)–(1d) by Hintz's replacement rules $\psi_1\to-\psi_{-1}$, $\psi_2\to\psi_{-2}$, $k\to-n$, $s\to\ell$, $\mathcal L_{-1}^\dagger\to\mathcal L_{-1}$, $\mathcal L_2\to\mathcal L_2^\dagger$, $(\mathcal D_0+\tfrac{3}{\bar\rho^*})\to\Delta(\mathcal D_{-1}^\dagger+\tfrac{3}{\bar\rho^*})$, $\Delta(\mathcal D_2^\dagger-\tfrac{3}{\bar\rho^*})\to(\mathcal D_0-\tfrac{3}{\bar\rho^*})$; its residuals are res1aN…res1dN and Sys1NegAt is their conjunction at a point. 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). System (7) for the transformed variables $z_+=\operatorname{adj}(X)\,y_+$ and $z_-=X^*y_-$ (with $X$ the matrix integrating factor of eq. (5) and $\operatorname{adj}X=\det X\cdot X^{-1}$ since $\det X=Q^2-M^2$ is constant) is

Sys7At is this pair of equations at a point.

Statement. Let $\omega,m\in\mathbb R$ and let $G=(\psi_{-2},\psi_{-1},n,\ell)$ be $C^\infty$ on $U$ (NegSmooth). Then for every $x\in U$,

with $z_+=\operatorname{adj}(X)\,y_+^{(-)}$, $z_-=X^*y_-^{(-)}$; the map is explicit and invertible on $U$ (invNeg). The two spins are the instances $\varsigma=\pm1$ of the same (4)⟺(7) step.