II_ Cosmological-Quantum Unification Formula (UCQ): A Unified Framework for Quantum Mechanics, Gravitation, and Cosmological Dynamics

The Cosmological-Quantum Unification Formula (UCQ) introduces a theoretical framework designed to integrate quantum mechanics and general relativity within a coherent cosmological model. The objective of this work is to explore a structural path toward the unification of quantum fluctuations and gravitational geometry while maintaining numerical stability and empirical testability. The framework models the evolution of physical systems toward stable or privileged states by combining several interacting components: quantum fields representing matter fluctuations, the spacetime metric describing gravitational geometry, a dynamic memory structure capturing non-local temporal effects, and an observable layer linking theoretical dynamics to measurable quantities. This architecture allows the model to describe complex interactions between quantum and gravitational regimes within a unified structure. A central contribution of the UCQ framework is the introduction of mechanisms that regularize gravitational singularities and stabilize cosmological dynamics. By integrating complementary terms representing gravitational, quantum, and contextual contributions, the framework aims to reduce divergences that appear in conventional cosmological models. The inclusion of a dynamic memory structure also captures cumulative historical effects in system evolution, enabling the modeling of oscillatory and multi-scale behavior. The theoretical structure is accompanied by numerical simulations designed to evaluate the stability and convergence of the model. These simulations demonstrate strong numerical consistency and indicate that the framework can produce stable solutions across cosmological and quantum scenarios, including gravitational singularities and quantum decoherence processes. An extended version of the model further explores large-scale cosmological dynamics, particularly the late-time expansion of the universe. This extension proposes a mechanism capable of reconciling discrepancies between early-universe and local measurements of cosmic expansion, offering a potential avenue for addressing the Hubble tension within a unified theoretical framework. Overall, the UCQ framework presents a structured approach toward cosmological quantum unification, combining theoretical construction, algorithmic implementation, and numerical validation. The model aims to provide a foundation for future empirical tests and interdisciplinary research connecting cosmology, quantum physics, and gravitational theory.