Curvature-Threshold Surfaces in Rotating Black Hole Geometry

This paper extends the curvature-threshold scale-setting programme for regular blackhole interiors beyond spherical symmetry. In the preceding works of this series, the transitionradius was fixed by imposing the invariant Schwarzschild conditionKSch (rc) = ηKP, which yields a mass-dependent scale rc ∝ M1/3 and a globally curvature-bounded effectiveinterior in the representative spherically symmetric geometries considered. The present workaddresses the next structural question: whether an analogous threshold condition can beformulated for rotating black hole geometry, where curvature is no longer a function of asingle radial coordinate. We study the Kerr geometry as the canonical stationary and axisymmetric vacuum blackhole background. In this setting the curvature-threshold condition does not define a singletransition radius. Instead, it defines a family of threshold surfaces, Ση (M, a) = (r, θ): Keff (r, θ;M, a) = ηKP, where Keff is an invariant curvature diagnostic constructed from the scalar and pseudoscalarcurvature invariants of the Kerr spacetime. In the non-rotating limit, this constructionreduces to the spherical Schwarzschild threshold used in the previous papers. For nonzerospin, the threshold set becomes an axially deformed surface depending on the mass, therotation parameter, and the polar angle. The analysis separates three issues that coincide in the spherical case but become distinctin rotating geometry: the location of the curvature threshold, the relation of this thresholdto the outer and inner Kerr horizons, and the approach to the ring singularity. We showthat any regularization of a rotating black hole based on curvature-threshold scale settingcannot be obtained by a direct spherical replacement of the central region. It must insteadaccount for the angular structure of the Kerr curvature invariants and for the deformation ofthe transition surface by spin. The purpose of this paper is diagnostic rather than constructive. We do not proposea complete regular Kerr metric, derive a rotating effective source, or solve the dynamicalbackreaction problem. We establish the geometric threshold structure that any such constructionwould have to respect. The results identify the passage from a spherical transitionradius to an axisymmetric curvature-threshold surface as the necessary next step in extendingcurvature-bounded regular black hole interiors to astrophysically relevant rotatingconfigurations.