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

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

by
Physics Department, College of Science, United Arab Emirates University, Al-Ain 15551, United Arab Emirates
Mathematics 2026, 14(5), 853; https://doi.org/10.3390/math14050853
Submission received: 29 January 2026 / Revised: 25 February 2026 / Accepted: 28 February 2026 / Published: 2 March 2026
(This article belongs to the Section E4: Mathematical Physics)

Abstract

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 dual-time 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 geometric framework for analyzing 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 reformulates non-Markovian dynamics and temporal asymmetry in terms of 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 arises as projections of local dual-time dynamics.



Cite

MDPI and ACS Style

Haj Yousef, M. Dual-Time Topological Geometry and the Emergence of Temporal Asymmetry in Non-Equilibrium Dynamics. Mathematics 2026, 14, 853. https://doi.org/10.3390/math14050853

AMA Style

Haj Yousef M. Dual-Time Topological Geometry and the Emergence of Temporal Asymmetry in Non-Equilibrium Dynamics. Mathematics. 2026; 14(5):853. https://doi.org/10.3390/math14050853

Chicago/Turabian Style

Haj Yousef, Mohamed. 2026. "Dual-Time Topological Geometry and the Emergence of Temporal Asymmetry in Non-Equilibrium Dynamics" Mathematics 14, no. 5: 853. https://doi.org/10.3390/math14050853

APA Style

Haj Yousef, M. (2026). Dual-Time Topological Geometry and the Emergence of Temporal Asymmetry in Non-Equilibrium Dynamics. Mathematics, 14(5), 853. https://doi.org/10.3390/math14050853

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The science of Time is a noble science, that reveals the secret of Eternity. Only the Elites of Sages may ever come to know this secret. It is called the First Age, or the Age of ages, from which time is emerging.
Ibn al-Arabi [The Meccan Revelations: Volume I, page 156. - Trns. Mohamed Haj Yousef]

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I have no doubt that this is the most significant discovery in the history of mathematics, physics and philosophy, ever!

By revealing the mystery of the connection between discreteness and contintuity, this novel understanding of the complex (time-time) geometry, will cause a paradigm shift in our knowledge of the fundamental nature of the cosmos and its corporeal and incorporeal structures.

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The science of Time is a noble science, that reveals the secret of Eternity. Only the Elites of Sages may ever come to know this secret. It is called the First Age, or the Age of ages, from which time is emerging.
Ibn al-Arabi [The Meccan Revelations: Volume I, page 156. - Trns. Mohamed Haj Yousef]
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