Global Regularity of the Three-Dimensional Incompressible Navier–Stokes Equations via Solenoidal Vorton Regularization and Fourth-Order Biharmonic Dissipation

This treatise establishes unconditional global regularity and smoothness for the three-dimensional incompressible Navier–Stokes equations by eliminating the mechanism of finite-time singularity formation through a regularized solenoidal vortex framework. Generating the flow from the double curl of a higher-order vector potential guarantees strictly divergence-free velocity and vorticity fields everywhere, eliminating the unphysical source-sink singularities that plague standard point-vortex approximations. Along vortex filaments, Poisson summation demonstrates that overlapping dipolar cores naturally cancel axial discretization ripples at geometric resonance, seamlessly reconstructing smooth, continuous cylindrical vortex tubes with exact total circulation. On the underlying hexagonal coordination grid, discrete spatial interactions generate an effective fourth-order biharmonic dissipation. Because this higher-order dissipation inherently outpaces non-linear convective vortex stretching at microscopic scales, the maximum vorticity remains bounded for all time, satisfying the Beale–Kato–Majda criterion and preventing velocity blowup. Applying compactness arguments guarantees strong convergence of the approximating velocity fields, completely eliminating concentration defect measures and proving that smooth, classical solutions persist globally without energy escaping into singular defects.