4.2 Liouville Lift and Entropy Monotonicity

This article derives in detail the mathematical structure of the Liouville lift, proving that entropy monotonicity is a direct consequence of the non-principal eigenvalues of the Birkhoff Mixing Matrix having modulus less than 1. The thermodynamic arrow of time emerges naturally from the algebraic structure of the coding framework, requiring no external hypotheses. This article constructs the complete spectrum of the Liouville operator, provides a rigorous proof of entropy monotonicity, and analyzes the geometric roots and landscape dependence of the arrow of time.