Memory-Generated Transport Geometry: Curvature, Holonomy, and Irreversibility

Abstract

Memory is traditionally incorporated into transport theory as a constitutive correction acting on an already prescribed kinematic structure. Here we develop a different framework in which finite memory itself generates the geometry of transport. By reconstructing deformation from causal transport histories, the instantaneous velocity gradient is replaced by a memory-dependent transport connection whose ordered evolution gives rise to noncommutativity, curvature, and holonomy in transport-history space. We show that finite memory generates a nonvanishing geometric contribution to transport even in time-periodic, irrotational flows, providing a purely kinematic mechanism for irreversible Lagrangian transport without invoking vorticity, constitutive nonlinearities, stochastic forcing, or explicit symmetry breaking. We introduce an intrinsic curvature invariant, independent of the transport representation that measures the accumulated geometric structure generated by transport history. The framework predicts universal scaling governed by the dimensionless parameter (ωτ_m), identifies a characteristic memory scale (tau_c) separating rapid geometric accumulation from asymptotic saturation, and reveals a monotonically decreasing memory susceptibility with globally concave accumulation dynamics. Numerical simulations confirm these predictions and show that geometric irreversibility emerges through progressive curvature accumulation rather than resonance-driven amplification. These results establish finite memory as a generator of an intrinsic geometric structure rather than merely a modifier of dynamical evolution, revealing causal history as the microscopic origin of curvature, holonomy, and irreversible transport across a broad class of non-Markovian systems.