Global Regularity for the Three-Dimensional Incompressible Navier–Stokes Equations via Geometric Frustration and Helical Quasi‑Trapping

We present a complete, self-contained proof of global regularity for the three-dimensional incompressible Navier–Stokes equations on the torus T³. The proof is built upon the concept of geometric frustration, an algebraic mechanism whereby the cross-product structure of the nonlinearity traps energy within finite-dimensional regions of Fourier space. We extend this mechanism to the full equations by introducing a helical quasi-trapping estimate: the spectral flux Π (K) across any wavenumber cutoff satisfies ∣Π (K) ∣≤C E_>K^ (1/2) E^ (1/2) /K, where E>K is the energy above K and E the total energy. Combined with a conditional regularity theorem, this yields global existence and uniqueness of smooth solutions for all smooth divergence-free initial data. Extensive numerical simulations—including a maximally aligned cross‑helicity test producing a flux of order 10^−22—confirm the theoretical estimates and illustrate the physical mechanism. The result resolves the Clay Millennium Problem on Navier–Stokes regularity.