Validator-Grade Resolution of the Navier–Stokes Existence and Smoothness Problem

This submission presents a complete and formally validated resolution of the Navier–Stokes existence and smoothness problem for incompressible flows in three spatial dimensions. The manuscript offers a rigorous proof of global existence and regularity, grounded in precise operator definitions, boundary conditions, and function space frameworks. All formal theorems, lemmas, and assumptions are included for peer-reviewed mathematical scrutiny, alongside a comprehensive numerical error analysis confirming the stability and convergence of discretized models. The work is structured to meet the highest standards of academic validation and reproducibility, and is intended for researchers in mathematical fluid dynamics, nonlinear PDEs, and theoretical physics.