Local equivalence of the chiral system (4) and the transformed system (7)

Fix a subextremal Kerr–Newman background ($M>0$, $Q\neq 0$, $a^2+Q^2<M^2$), the domain $U=\{r>r_+,\ 0<\theta<\pi\}$, a real frequency $\omega$, a real azimuthal number $m$, and a real spin parameter $\varsigma$ (the paper's positive- and negative-spin systems are the instances $\varsigma=+1$ and $\varsigma=-1$). Let $y_+, y_-\colon U\to\mathbb{C}^2$ be $C^\infty$ fields and $x\in U$. Then the two statements below are equivalent.

System (4) at $x$ (Hintz eq. (4), both equations):

where $D^- = \sqrt{\Delta}\,\partial_r-i\,\partial_\theta,\quad D^+ = \sqrt{\Delta}\,\partial_r+i\,\partial_\theta$, $A$ is the Hermitian off-diagonal matrix of eq. (4), $c^{(\varsigma)} = \bigl(\varsigma\tfrac{iK}{\Delta} + \tfrac{3}{\bar\rho^*} - \tfrac{3\Delta'}{4\Delta}\bigr)\sqrt{\Delta}$ and $B^{(\varsigma)}$ is the traceless matrix of eq. (4).

System (7) at $x$ for the transformed variables $z_+ := \operatorname{adj}(X)\,y_+$ (which equals $(\det X)X^{-1}y_+$) and $z_- := X^*y_-$:

with $C^{(\varsigma)} = (\det X)X^{-1}(c^{(\varsigma)}I - B^{(\varsigma)})(X^*)^{-1}$.

The forward direction is the two transport lemmas for $z_\pm$; the backward direction applies the inverse-transformation lemmas to $z_\pm$ and then uses the round trips $(Q^2-M^2)^{-1}X\operatorname{adj}(X) = I$ and $(X^*)^{-1}X^* = I$ on $U$, together with the locality of $D^\pm$ (they depend only on values in $U$). Both spin directions are covered by the single real parameter $\varsigma$; no frequency or mode restriction beyond $\omega, m\in\mathbb{R}$ is used. Together with Cmat_eq_Csep and adjugate_Csep_eq, this yields the separated-coefficient version of (7), from which the paper writes the block form (8). This theorem itself is the (4) iff (7) statement; it is not a separate theorem about the full four-component differential operator in (8), nor about spectral separation.