Bergmann Bounce Relativity (BBR) and Multi-Cycle Cosmology (BMC): Addressing Unresolved Challenges in Relativity Experiments

The theory of relativity, encompassing special relativity (SRT) and general relativity (ART), has been one of the most rigorously tested frameworks in physics since its inception. Countless experiments have confirmed its predictions with extraordinary precision, from time dilation in particle accelerators to gravitational lensing observed by the Hubble Space Telescope. Yet, several key experiments and observational programs designed to probe relativity's limits have stalled, yielding null results, precision ceilings, or persistent inconsistencies that resist explanation within standard relativity. These unresolved challenges often arise at the frontiers where relativity intersects quantum mechanics, cosmology, or extreme gravitational regimes, such as singularities or the Planck scale. This preprint reviews a selection of such stalled experiments, detailing their objectives, the reasons progress has halted, and how Bergmann Bounce Relativity (BBR) and Bergmann Multi-Cycle (BMC) cosmology can offer resolutions or meaningful complements. BBR introduces stochastic mini-bounces to regularize gravitational singularities in ART, while BMC extends this to an eternal cyclic cosmology in a closed 3-sphere, eliminating the need for a singular origin. By integrating these models, we derive mathematical extensions to stalled tests, provide quantitative estimates of potential deviations, and highlight novel predictions testable with current or near-future facilities. The work emphasizes the novelty of BBR/BMC in bridging relativity's empirical successes with its theoretical limitations and discusses open challenges, such as the quantization of bounce terms.