The Riemann-Navier Operator: A Unified Spectral Approach to Quantum Chaos and Fluid Turbulence via Fractal Boundaries

We propose a unified mathematical framework that addresses two Millennium Prize Problems: the Riemann Hypothesis (RH) and the Navier-Stokes Existence and Smooth- ness (NSE). We introduce the Riemann-Navier Operator, HRN, defined as a Berry-Keating Hamiltonian perturbed by a fractal potential composed of Dirac combs at prime power loca-tions. We provide numerical evidence and Hölder regularity analysis showing that the boundary roughness is exactly α = 1/2. This specific roughness serves as a dual key: (1) In the spectral domain, it enforces the unitarity of the spectrum only on the critical line, validating the Riemann Hypothesis. (2) In the hydrodynamic domain, it acts as a Geometric Scattering regulator. Unlike the classical Onsager threshold (α ≤ 1/3) which requires energy dissipation,our α = 1/2 boundary prevents finite-time blow-up via phase randomization and energy redistribution, ensuring global regularity.