The NXE Program: Geometric Depletion of Vortex Stretching in 3D Navier-Stokes — A Numerical Investigation and Conditional Regularity Framework

We present the NXE Program, a theoretical-numerical investigation of geometric mechanisms that may prevent finite-time blow-up in the 3D incompressible Navier-Stokes equations. The central object is the Cartan helicity scalar, an implicit variable encoding the helical structure of vorticity lines, derived explicitly from the Navier-Stokes system without any modification of the original equations. Three pseudospectral experiments (N=48³ and N=64³) consistently demonstrate a dynamical anti-correlation between Cartan torsion and vorticity magnitude (Corr ∈ [-0.44, -0.38]) throughout temporal evolution. An automated adversarial Monte Carlo Fourier search spanning 500 chaotic fields with severe intermittency yields a worst-case Besov constant of C = 0.0058, confirming that geometric coherence prevents the growth of local velocity gradients without prior alignment. The program reduces the Clay Millennium Problem to four explicitly stated analytical gaps, the central one being a dimensionally consistent conjecture linking the Biot-Savart kernel, the Cartan helicity scalar, and the velocity field. All numerical code, discrete solver pipelines, and dyadic multi-scale filters are fully documented for comprehensive reproducibility. Epistemic Status: This preprint presents a research program, not a proof. The Clay Millennium Problem remains open.