Local equivalence (1) ⟺ (4), positive 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 $\omega,m\in\mathbb{R}$. 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. 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 $(\psi_2,\psi_1,k,s)$ slice-differentiable at $x\in U$ (PosDiff): the four equations of system (1) hold at $x$ if and only if system (4) with $\varsigma=+1$ holds at $x$ for $y_\pm=y_\pm^{(+)}$,

The forward direction substitutes the four residual identities; the converse uses that the prefactor $\Delta^{5/4}\sqrt{\sin\theta}$, $\sqrt{\Delta}$ and $Q$ are nonzero on $U$ and inverts the $2\times2$ combinations $(-i\,a+b/\sqrt{\Delta},\ i\,a+b/\sqrt{\Delta})$.