Bergmann-Bounce Relativity: A Hybrid Theory Integrating Stochastic Mini-Bounces with General Relativity

General relativity (GR) provides an exceptionally accurate description of gravity on macroscopic scales but fails at singularities and Planck lengths, where quantum effects become dominant. The Bergmann Multi-Cycle (BMC) model introduces stochastic mini-bounces as quantum-gravity corrections to avoid singularities and generate emergent expansion. This paper proposes Bergmann-Bounce Relativity (BBR), a hybrid framework that embeds BMC into GR: Mini-bounces serve as localized quantum modifications to the Einstein equations at high densities, while preserving GR's predictions on larger scales. Detailed mathematical derivations of the bounce-modified metric and field equations are provided, including covariance discussions and stability analyses. Predictions for observable effects, such as spacetime perturbations in neutron stars, gravitational wave anomalies, and resolutions to the black hole information paradox, are derived. Implications for quantum gravity unification and technological applications are explored. The theory is falsifiable through high-precision experiments and observations.