The Kerr–Newman Penrose Inequality for Axisymmetric Electrovacuum Exteriors

We prove the sharp Kerr–Newman Penrose inequality $m^2\ge \frac{A}{16\pi }+\frac{Q^2}{2}+\frac{\pi (Q^4+4J^2)}{A}.$ Here $Q^2=Q_e^2+Q_b^2$, and J is the conserved total angular momentum, including its electromagnetic contribution. The result applies to smooth axisymmetric electrovacuum exteriors in the stated one-ended topological and decay class, with a connected outermost, outer-area-minimizing future marginally outer trapped boundary, Coulomb electromagnetic asymptotics, zero ADM momentum, and the explicitly assumed physical-area condition $A\ge 4\pi \sqrt{Q^4+4J^2}$. No maximality assumption is made. On the strict area branch, equality within this class characterizes the original data as an admissible spacelike exterior slice of a subextremal dyonic Kerr–Newman spacetime, with the boundary mapped smoothly to a future-horizon cross-section or the bifurcation sphere.