The Second Law Is Aliasing: A Nyquist Theorem for Statistical Mechanics

This paper proves that the Second Law of Thermodynamics is not a law of the underlying microscopic dynamics but a conservation identity of the coarse-graining channel that maps microstates to macrostates. The proof is self-contained and uses only the data processing inequality, Pinsker's inequality, concentration of measure on Riemannian manifolds, and standard measure theory. The central insight is that the exponentially many microstates consistent with a given macrostate form an aliased set—a high-dimensional degeneracy that prevents macroscopic observers from recovering microstate information. The Boltzmann entropy measures the log-cardinality of this set exactly. When an observer enriches their measurement, the aliased set absorbs the discriminability gain under self-referential closure, giving an exact conservation identity: enrichment gain equals absorption plus robust residual. Sixteen core theorems are proved. The molecular chaos assumption (Stosszahlansatz) is derived as a theorem of high-dimensional geometry via concentration of measure, eliminating it as an independent postulate. The Loschmidt paradox is resolved as an instance of enrichment absorption. The Jaynes maximum entropy principle is proved as the unique solution to an aliasing optimization problem. The framework extends to quantum decoherence via a Holevo conservation identity with a classical-quantum complementarity that has no classical analogue, and a Thermodynamic Separation Theorem establishes that the conservation identity is strictly more general than the Second Law itself. Cross-domain results include a complete proof of the Zamolodchikov c-theorem from Wilsonian coarse-graining conditions, the Page curve derived as a Nyquist crossover of the black hole partial-trace channel, and a Glassy Degradation Theorem for the Sherrington-Kirkpatrick spin glass with Tracy-Widom scaling at the glass transition. A Universal Spectral Gap Bound reduces the entire framework to a single inequality controlled by the spectral gap of the fiber Laplacian. Numerical validation across three systems—2D Ising at criticality, SK spin glass replica simulations, and Lennard-Jones molecular dynamics—falsifies the Stosszahlansatz-based prediction of zero entropy-decrease probability, with over one million entropy-decreasing events observed. All qualitative predictions of the framework are confirmed.