Analytical derivation based on the 5D Einstein-Hilbert action and the Shiromizu-Maeda-Sasaki projection method. Numerical stability analysis of the radion-warp coupling in high-density regimes. We present a unified synthesis of the CCEGA (Curvature-Controlled Extra-dimensional Gravity Approach) framework. We propose a feedback mechanism in which the warp factor of a five-dimensional geometry depends dynamically on the local energy density. This coupling acts as a 'Curvature Brake' that regulates geodesic divergences, eliminating Big Bang singularities through a cosmological bounce and transforming black hole cores into regular de Sitter regions. We demonstrate that gauge interactions emerge naturally from the excess metric degrees of freedom, providing a consistent path toward geometric unification. Observational signatures in gravitational wave dispersion and high-energy cosmology are discussed.