Dynamics, Stability, and Rotating Extension of the Bühring Interior Metric Deformation Model: Toward a Mathematically Structured Completion of a Regular Black Hole Framework

This paper develops the third step of the Bühring regular black hole framework by addressing dynamical consistency, linear stability, and rotating generalization of the interior metric deformation model. Paper I introduced a regular static black hole interior with exact Schwarzschild exterior matching. Paper II provided a collapse-based phenomenological origin of the deformation factor, leading to bounded core density and cubic mass scaling m (r) ~ r³ with corresponding deformation scaling β (r) ~ r^-3/2. In the present work, a time-dependent spherical framework is formulated using the Misner–Sharp mass and static-endpoint hypothesis. It is shown that any bounded-density collapse endpoint approaching a static configuration necessarily converges to a de Sitter-type regular core. A linearized perturbation structure is then derived, and positivity of the axial core potential is proven, excluding central blow-up instabilities. A Kerr-like rotating extension is introduced through the replacement Δ (r) = r² F (r) + a² which reduces exactly to the static Bühring solution for vanishing rotation and reproduces the Kerr geometry outside the horizon. The classical Kerr ring singularity mechanism is removed whenever the static seed satisfies the regular-core condition F (r) = 1 − Λ r² + O (r³). The resulting framework provides a dynamically embedded, perturbatively structured, and rotating-compatible regular black hole model. Remaining open questions, including nonlinear collapse, full spectral stability, and microscopic matter origin, are clearly identified.