Local equivalence (4) ⟺ (7) in predicate form

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 (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. 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.

For any real $\varsigma,\omega,m$ and fields $y_+,y_-$ whose components are $C^\infty$ on $U$, at every $x\in U$:

This restates the platform theorem Hintz2026.system4_iff_system7 through the predicates Sys4At and Sys7At.