Local equivalence (1)$^-$ ⟺ (4), negative spin $\varsigma=-1$

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. System (4) with spin parameter $\varsigma$ is

with $A$, $B^{(\varsigma)}$, $c^{(\varsigma)}$ the explicit coefficient data of Hintz eq. (4) (A, Bmat, cscalar); Sys4At is the pair of these vector equations 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).

For fields $(\psi_{-2},\psi_{-1},n,\ell)$ slice-differentiable at $x\in U$ (NegDiff): the negative-spin system (1)$^-$ holds at $x$ if and only if system (4) with $\varsigma=-1$ holds at $x$ for $y_\pm=y_\pm^{(-)}$,