Covariant Non-Markovian Quantum Memory Framework: Microscopic Foundations, Integro-Differential Field Equations, and Asymptotic Numerical Evaluation

This research develops a covariant theoretical framework investigating whether non-Markovian quantum memory effects in curved spacetime can provide a physical explanation for dark-matter-like gravitational phenomena. By combining microscopic oscillator-bath dynamics, causal memory kernels, and integro-differential field equations, the study explores the possible emergence of long-range gravitational behavior from quantum environmental interactions. The framework establishes analytical foundations, examines asymptotic behavior, and presents preliminary numerical convergence tests. It also identifies key mathematical and physical constraints required for further validation. This work is a developing theoretical proposal, not an established discovery. Its predictions are presented as testable hypotheses intended to encourage independent verification, critical evaluation, and further scientific investigation.