Resolution of Loschmidts Paradox via Geometric Constraints on Information Accessibility

Abstract: We resolve Loschmidt's paradox -- the apparent contradiction between time-reversible microscopic dynamics and irreversible macroscopic evolution -- including the long-standing puzzle of the thermodynamic arrow of time. The resolution: entropy increases not because dynamics are asymmetric, but because information accessibility is geometrically bounded. For Hamiltonian systems (conservative dynamics), Lyapunov exponents come in positive-negative pairs (${\lambda_i, -\lambda_i}$) due to symplectic structure. Under time reversal these pairs flip ($\lambda_i \to -\lambda_i$), but stable manifolds contract below quantum resolution $\lambda = \hbar/\sqrt{mk_BT}$, becoming physically indistinguishable. We always observe only unstable manifolds where trajectories diverge. Hence information loss proceeds at the same rate $$h_{KS} = \frac{1}{2}\sum_{\text{all } i}|\lambda_i|$$ in both time directions, resolving the arrow of time: forward'' simply meanswhere we observe expansion,'' which is universal because stable manifolds always contract below measurability. Quantitatively, for N$<em>2$ gas at STP with conservative estimates ($h</em>{KS} \sim 10<sup>{10}$ s$<sup>{-1}$), time reversal at $t = 1$ nanosecond requires momentum precision $\sim 10<sup>{-13}$ times quantum limits -- geometrically impossible. At macroscopic times, the precision requirement becomes $\sim 10<sup>{-10<sup>{10}}$ times quantum limits. This framework preserves microscopic time-reversal symmetry, requires no special initial conditions or Past Hypothesis, and extends to quantum systems (OTOCs) and black hole thermodynamics.