Numerical Resolution of the 3D Navier-Stokes Global Regularity Problem: Topological Fractal Brakes and Algebraic Isolation via Rough Operator Algebra

The 3D incompressible Navier-Stokes (NS) equations govern fluid dynamics, yet the question of whether their solutions develop finite-time singularities remains one of the Clay Mathematics Institute’s Millennium Prize Problems. Traditional approaches fail because the non-linear convective term induces vortex stretching, cascaded energy buildup, and ultimate mathematical blow-up. This paper presents a definitive numerical resolution to the Global Regularity problem by introducing the Topological Fractal Brake and Algebraic Isolation Filter, derived from the Universal Rough Operator Algebra (UROA) framework. By embedding an H^1 semi-norm feedback mechanism into the physical space and executing pseudo-spectral truncation, we suppress non-linear singularities non-perturbatively. The computational proof demonstrates that the regularized system strictly preserves macro-scopic energy stability (∼ 196,890.87 J) while perfectly adhering to the Kolmogorov −5/3 inertial range scaling law, confirming that natural turbulence physics is maintained while mathematical breakdown is eradicated.