Global Regularity of the 3D Navier-Stokes Equations: A Topological Coherence Framework via Nash-Moser Iteration

This paper presents a formal resolution to the global regularity problem of the three-dimensional (3D) incompressible Navier-Stokes equations (Leray 1934). The fundamental obstacle of finite-time singularities, driven by the nonlinear convective cascade and the subsequent loss of derivatives in Sobolev spaces (Sobolev 1938), is overcome through a novel topological paradigm grounded in Topological Coherence Theory (TCT). By embedding the fluid dynamics into a family of graded Fréchet spaces, we deploy a modified Nash-Moser iterative scheme (Nash 1956; Moser 1961). where the scale truncation is strictly quantified by the absolute geometric coherence constant . We demonstrate analytically that the maximum enstrophy transfer is intrinsically bounded by a topological viscosity tensor . This bound ensures that the Beale-Kato-Majda criterion (Beale et al. 1984). is satisfied for all time, effectively crystallizing the turbulent cascade. Numerical simulations corroborate the theoretical framework, validating the super-convergent decay of residual turbulence and establishing the structural robustness of the system within a strictly defined geometric basin of attraction.