Abstract This work presents a rigorous extension of quantum mechanics to non-Markovian processes, establishing a unified theoretical framework for open quantum systems with temporal memory. We develop a mathematical formalism based on temporal convolution integrals and memory operators, providing a unified description of Markovian and non-Markovian dynamics. Original contributions: Generalization of the Lindblad equation via temporal memory kernels Rigorous classification of processes by memory degree Geometric measure of non-Markovianity based on state space geometry Five explicitly falsifiable hypotheses with experimental protocols Keywords: Quantum mechanics Non-Markovian processes Master equation Open quantum systems Falsifiability 1. Introduction 1.1 Historical context and open problem Quantum mechanics, founded by Schrödinger (1926) and Heisenberg (1925), describes the deterministic evolution of isolated systems via the Schrödinger equation. However, any real physical system interacts with its environment — this is the paradigm of open quantum systems. Since the work of Caldeira and Leggett (1983) on quantum Brownian motion, and the contributions of Lindblad (1976) on quantum semigroup generators, the open systems formalism has become fundamental in quantum theory. However, a major problem remains unsolved: the complete and operationally significant characterization of quantum non-Markovianity. Central problem: Existing measures of non-Markovianity (BLP, RHP, Š) do not provide quantitative predictions on physical observables that could be experimentally tested with given precision. Our work addresses this gap by proposing a theoretical framework with numerically testable predictions.(...) 8. Conclusion This work presented a rigorous and systematically testable extension of quantum mechanics to non-Markovian processes. Unlike previous approaches that focused on qualitative measures, we provide a framework with numerically verifiable predictions.