Preventive Geometric Depletion and Geometric Temporal Lead in the Regularity of the 3D Navier-Stokes Equations: A Dissipative-Scale Measurable Program and Numerical Evidence from the JHTDB

This work presents a novel program toward establishing global regularity of weak solutions to the 3D incompressible Navier–Stokes equations — one of the Clay Millennium Prize Problems. We introduce a dissipation-scale measurable mechanism of preventive geometric depletion, quantified at the adaptive scale η (t) = ‖ω (t) ‖₂² / ‖∇ω (t) ‖₂², combined with a structural property termed Geometric Temporal Lead (GSUT): critical directional coherence between vorticity and the most-compressive strain eigenvector is systematically established before local intensity peaks, providing a positive temporal margin for viscous intervention. Key measurable objects include pointwise 1D sparseness on super-level sets, statistical directional coherence, a universal depletion margin δₑff > 0, and a structural safety factor Ξ ∼ 3 × 10⁵. Numerical analysis of 10, 000 Lagrangian trajectories from the Johns Hopkins Turbulence Database (JHTDB, Re_λ ≈ 418) demonstrates 100% universality of GSUT, collapse arrest at a critical radius R ≈ 1. 367η (identified as the minimal-energy Bessel–Lundgren confinement profile), and asymptotic strengthening of anisotropy in extreme regimes (mean alignment rising from 0. 517 to 0. 556 in the top 1% of |ω|). Analytically, the elliptic pressure response enforces a holonomic constraint that precludes finite-time singularity by subordinating inertial amplification to instantaneous geometric reconfiguration. The resulting visco-geometric stability margin remains uniformly large and Reynolds-independent, supporting global regularity as a structural necessity of the incompressible 3D Navier–Stokes system. Keywords: Navier-Stokes equations, global regularity, geometric depletion, vorticity alignment, Geometric Temporal Lead (GSUT), turbulence, JHTDB, filamentary structures, pressure ellipticity