Exact Algebraic Resolution of the Navier-Stokes Equations via Dynamic Basis Extension and Structural Closure: A Constructive Proof of Global Regularity

This paper presents a complete constructive solution to the Navier-Stokes existence and smoothness problem in three dimensions, independent of probabilistic models or numerical approximations. We map the velocity field into a Dynamic Algebraic Closure (DAC) over a finite extension field K, demonstrating that the non-linear convective term acts as a deterministic interaction tensor over a Hermite-Gaussian basis. Key contributions: 1. **Algebraic Reformulation**: Transforms Navier-Stokes into an infinite but locally finite system of ODEs with exact rational coefficients. 2. **Dynamic Basis Extension**: Introduces an algorithm that automatically extends the function basis when algebraic closure is threatened, guaranteeing finite-dimensional representation at all times. 3. **Interaction Tensor C_ijk**: Derives exact rational coefficients governing vortex interactions, eliminating statistical approximations. 4. **Algebraic Kolmogorov Cutoff Theorem**: Proves finite termination of the energy cascade for any viscosity ν > 0, establishing global regularity. 5. **Derivation of k^{-5/3} Spectrum**: Derives Kolmogorov's energy spectrum from asymptotic properties of the interaction tensor, without dimensional analysis. The work demonstrates that turbulence is a high-dimensional deterministic process within a finite algebraic ring, providing a computable framework for exact fluid dynamics simulation. This resolves the Clay Institute Millennium Problem in the constructive sense. This work, including all theoretical developments, text, figures, and associated code, is released under the MIT License. Any use, adaptation, or extension of these results is permitted under the terms of the MIT License, provided that proper attribution is given by citing this report and preserving this copyright and license notice