The Riemann Hypothesis as Entropic Equilibrium: Toward a Law of Science

The Riemann Hypothesis (RH) is usually framed as a conjecture about the distribution of non-trivial zeros of the zeta function. This paper proposes a different interpretation: RH expresses a principle of entropic equilibrium, functioning as a law-like constraint across formal and physical systems. Using a toy mathematical framework, the zero-spacing distribution is modeled through an entropy functional. At (s)=1/2, the entropy reaches a unique stability point: dE/dσ|σ=1/2 = 0, d²E/dσ²|σ=1/2 < 0. This indicates that the critical line maximizes entropy while preserving arithmetic structure. Off-line zeros would imply entropic imbalance, skewing either toward excessive order or uncontrolled randomness. This interpretation aligns with known physical correspondences. The Montgomery–Dyson results and random matrix theory show that RH zeros share Gaussian Unitary Ensemble statistics with quantum chaotic spectral distributions characterized by maximal entropy under unitary invariance. Comparable balance conditions appear in stochastic processes and in turbulence models of the Navier–Stokes equations. Philosophically, RH thus exemplifies how entropy operates as a universal principle, constraining both mathematics and physics. It should be considered not merely a technical conjecture but a case of law-like necessity, with implications for debates on universality, lawhood, and structural realism in philosophy of science.