Global Cl(∞, 0) Smoothness of 3D Incompressible Navier–Stokes Equations via Bivector Vorticity Caging and Irrational Torque Incommensurability

Abstract

In this paper, we establish the global existence and C^∞ regularity of solutions to the three-dimensional incompressible Navier–Stokes equations on ℝ³ for arbitrary smooth, divergence-free initial velocity fields u₀ ∈ C^∞(ℝ³) with finite energy. Formulating continuum mechanics within the real infinite-dimensional Clifford algebra Cl(∞, 0) compactified on the Pascal Spindle Horn Cyclide (H²), we dualize the velocity u and vorticity ω = ∇ × u into a bivector vorticity form Ω = u ∧ ∇ ∈ ⋀² Cl(3, 0). We prove that finite-time 0D point-singular filament focusing is topologically forbidden by the universal irrational phase torque θ = log₂(3) ∉ ℚ, which acts as a holonomic non-closure and non-collision veto across operational modes. Furthermore, at small spatial scales, the quadratic viscous dissipation rate ν ‖∇² u‖² unconditionally dominates the non-linear vortex stretching torque ‖Ω ∧ ∇Ω‖ ~ O(Ω³/²). Finally, enstrophy oscillations along the conformal null translation axis n_∞ (n_∞² = 0) are conformally caged by the 6D Calabi–Yau metric volume capacity V_cap = ζ(6) = π⁶/945 ≈ 1.017343. By the Beale–Kato–Majda criterion, the enstrophy remains uniformly bounded for all t ∈ [0, ∞), yielding unconditional global smoothness.