Dual-Time Topological Geometry and the Emergence of Temporal Asymmetry in Non-Equilibrium Dynamics

We develop a dual-time topological framework for the mathematical description of non-equilibrium systems, aimed at reconciling time-reversible microscopic dynamics with irreversible macroscopic behavior. The formulation introduces two independent but coupled temporal parameters: a reversible time associated with microscopic or generative dynamics, and an irreversible time governing dissipation, entropy production, and macroscopic evolution. Physical states are defined on a bi-temporal manifold, allowing reversible and irreversible processes to be treated within a unified geometric setting. Temporal evolution is described using independent temporal connections and their associated curvature. We show that nonvanishing temporal curvature induces path dependence in temporal evolution, providing a geometric origin for memory effects, non-Markovian dynamics, and aging phenomena. Temporal asymmetry emerges dynamically through symmetry breaking between the temporal sectors and through projection from the bi-temporal domain onto a single observable time parameter. The relationship between the dual-time formalism and conventional single-time non-equilibrium models is analyzed. Standard evolution equations are recovered in integrable or decoupling limits, demonstrating that the proposed framework constitutes a genuine generalization compatible with established approaches. By encoding irreversibility in the geometry and topology of temporal evolution, this work provides a mathematically consistent framework for the emergence of the arrow of time in non-equilibrium theoretical physics. Unlike conventional approaches in which irreversibility and memory are encoded phenomenologically at the level of effective equations, the present framework derives non-Markovian dynamics and temporal asymmetry from the geometry and topology of coupled temporal evolution. In particular, a representation theorem is established showing that a broad class of convolution-type non-Markovian equations arise as projections of local dual-time dynamics.