THE RIEMANN HYPOTHESIS An Entropy-Minimization Proof via Harmonic Coherence

This manuscript presents a proof of the Riemann Hypothesis within the Harmonic Coherence (HC) framework, which is built upon Hanners Theorem (HT). The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. HC reformulates this as an entropy-minimization problem: the global minimum of an informational entropy functional over zero configurations is shown to be uniquely attained when all non-trivial zeros lie on the critical line. The argument uses variational calculus, spectral theory (Hermitian operator interpretation of zero ordinates, in the spirit of the Hilbert–Pólya conjecture), and a convexity condition at σ = 1/2 to establish stability and uniqueness. Numerical validation of zero spacing statistics and comparison with Gaussian Unitary Ensemble (GUE) predictions support the theory. Implications for prime-counting error bounds, cryptography, and connections to quantum chaos are noted. This document is a formal preprint submitted in fulfillment of the Clay Mathematics Institute Millennium Prize Problem Requirements for the Riemann Hypothesis and is intended for peer review. It is part of the Harmonic Coherence publication ecosystem.