Foundations of Momentum in Non-Inertial Systems establishes a rigorous correction to a long-standing conceptual error in classical mechanics: the mistaken use of inertial linear momentum formulations inside non-inertial rotating systems. This document establishes the first complete mathematical and physical foundation for Momentum in Non-Inertial Dynamics (MIND)—a new framework describing how momentum emerges, transforms, and is harvested when a system transitions between non-inertial and inertial regimes. It formalizes the core components of the theory: • NIS — Non-Inertial Momentum Generation • TIME — Theorem of Instantaneous Momentum Emergence • IS — Inertial Momentum Harvesting through ICP (Impulse-on-Capture) • CCC — Closed Concomitant Cycle enabling energy regeneration and mass return Together, these results demonstrate that a closed mechanical system can generate real, net linear impulse without expelling reaction mass to the external world, provided the momentum exchange occurs across two reference-frames with different dynamical constraints. This mathematical structure forms the physical backbone of the ABÏON Drive, the first closed-cycle impulse engine operating entirely within its own mass-energy budget, enabling scalable thrust, vectorization, regeneration, and independence from atmospheric or environmental media. The document includes axioms, derivations, consistency proofs, variational formulations (Lagrangian & Hamiltonian), and the full temporal framework required to model NIS→TIME→IS transitions with correct Δt > 0 treatment.