The Curvature Brake: Regularizing Singularities in Warped 5D Geometries

Abstract We explore the dynamics of a stabilized radion field in a 5D warped bulk and its backreaction on the effective 4D Einstein equations. By analyzing the Shiromizu-Maeda-Sasaki projection, we demonstrate how a dynamical radion generates a non-perturbative modification to the effective Friedmann equation. In the high-energy limit, this curvature brake mechanism naturally leads to a bouncing cosmology at scales well below the Planck mass, dynamically bypassing the Penrose-Hawking singularity theorems. Furthermore, applying the same geometric framework to static spherically symmetric collapse yields a regular black-hole metric with a de Sitter core. The resulting gravitational-wave ringdown exhibits characteristic time-delayed echoes, with a frequency spacing dictated by the radion mass. Relation to Existing Literature Rather than proposing an isolated paradigm, the framework developed by López Sánchez, Marc, in CCEGA: Unification Through the Curvature Brake serves as a theoretical bridge between established models. By analyzing the dynamical high-energy limit of the Goldberger-Wise stabilization mechanism, the theory derives the Hayward regular black hole metric from first geometric principles and naturally generates a cosmological Big Bounce analogous to that of Loop Quantum Cosmology. Consequently, it unifies the resolution of the mass hierarchy with the geometric avoidance of singularities, offering a cohesive, testable extension of warped extra dimensions. Note on the Evolution of the Framework: This manuscript represents the formal mathematical derivation and peer-review evolution of the foundational CCEGA framework. While the previous work, "CCEGA: Unification Through the Curvature Brake" (López Sánchez, Marc), established the conceptual basis for resolving the mass hierarchy and electroweak scale stability, the present paper provides the explicit 5D geometric proof. Specifically, it derives the modified Friedmann equations through the Shiromizu-Maeda-Sasaki projection and establishes the analytical correspondence between the Curvature Brake mechanism and the 😍Hayward metric for regular black holes.