Chiralisation identity, negative spin: $D^-$ equation of (4), component 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 (1) (positive spin, Hintz eqs (1a)–(1d)) for the four fields $(\psi_2,\psi_1,k,s)$ reads, with $\mathcal D_j=\partial_r+\tfrac{iK}{\Delta}+j\tfrac{\Delta'}{\Delta}$, $\mathcal L_j=\partial_\theta+\hat Q+j\cot\theta$, their daggers (sign of $iK/\Delta$, resp. $\hat Q$, flipped), $K=-(r^2+a^2)\omega+am$, $\hat Q=-a\omega\sin\theta+m/\sin\theta$, $F_\pm=3(M-Q^2/\bar\rho)\pm Q^2\bar\rho^*/\bar\rho^2$:

In Lean the four equations are encoded as the vanishing of the residuals res1a, res1b, res1c, res1d (left side minus right side), and Sys1PosAt 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).

For any fields that are slice-differentiable at $x\in U$ (hypothesis PosDiff/NegDiff), component 1 of the $D^-$ equation of system (4) with $\varsigma=-1$, evaluated on the chiral variables, equals the prefactor times a fixed linear combination of the residuals of system (1):

This is one of the eight scalar identities that make the chiralisation (1)→(4) an identity for arbitrary fields, not only for solutions; the proof clears all denominators ($\bar\rho,\bar\rho^*,\Delta,\sqrt{\Delta},\sin\theta,Q$) and reduces powers of $\sqrt{\Delta}$, $\cos\theta$ and $i$.