The Geometric Origin of Time's Arrow: Loschmidt Resolved

We resolve Loschmidt's paradox-the 150-year-old contradiction between time-reversible microscopic dynamics and irreversible macroscopic evolution. The resolution requires both quantum mechanics and classical chaos; neither alone suffices. Quantum uncertainty without chaos produces slow, polynomial spreading-not fundamentally irreversible. Classical chaos without quantum uncertainty produces computational intractability-trajectories diverge exponentially, yet the system remains on one trajectory, reversible in principle with sufficient precision. Only together do they produce geometric impossibility: chaos exponentially amplifies irreducible hbar-scale uncertainty until stable manifolds contract below quantum resolution, rendering time-reversed trajectories physically inaccessible despite being mathematically valid and equiprobable. Information is never destroyed-it becomes geometrically inaccessible. The Kolmogorov-Sinai entropy rate is identical in both time directions, preserving microscopic symmetry while explaining macroscopic irreversibility. Three decades of Loschmidt echo experiments confirm perturbation-independent decay consistent with geometric inaccessibility. The framework unifies thermodynamic, quantum, and information-theoretic arrows of time.