Verified rotating electric KN physical model

Given: $G_N,c,{\mu }_0>0$, $m_{\ast }>0$, SI parameters $J,Q_e$, zero magnetic flux, the stated coordinate domains, and the zero-source equation-based specialization.

SI and geometric solution parameter identifications: $M=G_N{m}_{\ast }\mathrm{/}{c}^2$, $a=J\mathrm{/}(m_{\ast }c)$, $q_e=\sqrt{G_N\mathrm{/}(4\pi {\epsilon }_0{c}^4)}{Q}_e$, $\mathcal{F}=\alpha F$, and $\alpha =\sqrt{4\pi G_N\mathrm{/}({\mu }_0{c}^4)}>0$.

Kerr-Newman metric and potential on the exact Boyer-Lindquist domain: on a pole-free chart, $\mathrm{\Sigma }=r^2+a^2{\cos }^2\theta$, $\mathrm{\Delta }=r^2-2Mr+a^2+q^2$, $g=-\frac{\mathrm{\Delta }}{\mathrm{\Sigma }}(d{x}^0-a{\sin }^2\theta d\varphi {)}^2+\frac{\mathrm{\Sigma }}{\mathrm{\Delta }}d{r}^2+\mathrm{\Sigma }d{\theta }^2+\frac{{\sin }^2\theta }{\mathrm{\Sigma }}((r^2+a^2)d\varphi -ad{x}^0{)}^2$, and $\mathcal{A}=-qr{\mathrm{\Sigma }}^{-1}(d{x}^0-a{\sin }^2\theta d\varphi )$ with $\mathcal{F}=d\mathcal{A}$.

Full-sphere exterior extension of the Kerr-Newman datum: for $R_0>4\mathrm{\mid }M\mathrm{\mid }$, the full-sphere exterior has a smooth global metric and potential agreeing with [F2] away from the axes, with $r>R_0$, $\mathrm{\Sigma }>0$, and $\mathrm{\Delta }>0$.

Equation-based Einstein–Maxwell model: select its zero-matter instance ${\mathcal{M}}_0$, with configuration set ${\mathcal{C}}_0[g,F]=\{\ast \}$, no matter-field variables, equations or additional constraints, and maps ${\mathsf{J}}_0[g,F;\ast ]=0$ and ${\mathsf{T}}_0[g,F;\ast ]=0$ to the smooth zero vector field and smooth zero symmetric covariant two-tensor on each stated domain. Thus $\mathrm{\Psi }=\ast$ is admissible and $j=0$, $T^m=0$. With $\mathrm{\Lambda }=0$, the adopted equations are $dF=0$, ${\mathrm{\nabla }}_a^g{F}^{ab}=-{\mu }_0{j}^b$, and $G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m)$.

Fully verified electric Kerr-Newman electrovacuum: for the exact pole-free KN datum and its stated full-sphere exterior, the current mathematical Statement gives $d\mathcal{F}=d({\star }_g\mathcal{F})=0$, $\mathrm{Ric}=2Q_g(\mathcal{F})$, and, for $F=\mathcal{F}\mathrm{/}\alpha$, $G_{ab}={\kappa }_E{T}_{ab}^{\mathrm{E}\mathrm{M}}$. Its current proof derives the Hodge-divergence identity locally and uses the exact present observer-stress interface; its Step-1 record is authoring-ready, but the mathematical theorem remains draft and has no independent proof acceptance, so the conclusion is used only conditionally.

Sourced Maxwell sign and current conservation: its current Statement assumes the sourced divergence equation before stating the form-equation equivalence. Its Facts [F1]–[F3] specify the smooth oriented Lorentzian geometry, torsion-free metric connection and Hodge conventions. Its Proof 2.1 derives the pointwise identity $d({\star }_{g}F)=-{\iota }_v{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, $v^b={\mathrm{\nabla }}_a^g{F}^{ab}$, using only smoothness and those geometric conventions; the equivalence clause is not used as a black-box implication here. The supplier is a draft proof without independent acceptance.

Lorentzian exterior calculus and Hodge star: in signature $(-,+,+,+)$ and the stated orientation, $({\star }_{g}F{)}_{ab}=\frac{1}{2}({\mathrm{v}\mathrm{o}\mathrm{l}}_g{)}_{abcd}{F}^{cd}$ for smooth two-forms. The metric-selected connection is torsion-free and metric-compatible under the geometric conventions of [L2]. The identity $\mathrm{\nabla }{\mathrm{v}\mathrm{o}\mathrm{l}}_g=0$ used below is derived locally in step 3.1.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the geometric flux convention inherited through [F1] uses only ${\mathrm{A}\mathrm{C}}_{\omega }$, meaning each specified countable family of nonempty sets has a choice function. The finite field-equation and tensor-rescaling argument below uses no further choice and does not use full AC.