Finite Curvature Transition Hypothesis: A Phenomenological Model for Singularity Avoidance in Black Holes

This preprint presents the Finite Curvature Transition Hypothesis (FCTH), a speculative phenomenological model for avoiding infinite spacetime curvature inside black holes. In classical General Relativity, the curvature associated with a Schwarzschild black hole increases without bound as the radial coordinate approaches zero. This paper explores the possibility that physical spacetime may instead possess a maximum attainable curvature, Kmax, beyond which the classical description of smooth spacetime is replaced by a different high-curvature regime. A bounded toy model is introduced to illustrate how ordinary General Relativity can be recovered at low curvature while preventing the curvature from diverging to infinity. The paper also discusses possible implications for black-hole singularities, geodesic completeness, quantum gravity, black-hole evaporation, gravitational waves, and early-universe cosmology. The proposed model is not presented as a proven theory or a complete theory of quantum gravity. Its purpose is to develop a testable theoretical framework and identify the mathematical and observational work needed to determine whether a limiting-curvature description of spacetime could be physically realistic.