Global Regularity of the 3D Incompressible Navier–Stokes Equations: Fractional Sobolev–Colombeau Quotient Algebras, Null-Ideal Fractal Brakes, and Beale–Kato–Majda Boundedness within the ZFC Axiomatic Bridge

The global existence and smoothness of the three-dimensional incompressible Navier–Stokes Equations (NSE) has remained an open Millennium Prize Problem due to the supercritical scaling gap between linear viscous dissipation ν_0 Δu and cubic convective vortex stretching (ω · ∇)u. In this flagship paper, we present a complete ZFC resolution of the 3D Navier–Stokes regularity problem anchored directly in Part I (Chapters 2–3: Proposition 2.3, Lemmas 2.6–2.8, Theorem 2.9, Lemmas 3.2–3.3, and Theorem 3.4) of our 71-page Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. Rather than altering the macroscopic partial differential equation by ad hoc hyperviscosity, we embed the fluid velocity and vorticity fields into the Fractional Sobolev–Colombeau Quotient Algebra 𝓖^α(Ω_T)^3 = [𝓔_M^α(Ω_T) / 𝓝^α(Ω_T)]^3 synchronized via the Roughness Scaling Bridge ε(α) = exp(-1/(1-α)) for α ∈ (1/2, 1). Using the smooth mollified cutoff Θ_ε(s) = (Θ * ρ1,ε)(s) ∈ C^∞(ℝ) and regularized Frobenius gradient norm ||∇u_ε||_ε = (||∇u_ε||^2 + ε^2)^(1/2), we prove that on any sub-threshold regular domain (||∇u_ε||_ε ≤ C_UPTC - δ_0), the high-gradient cutoff of the Regularized Fractal Brake 𝓑_ε(u_ε) vanishes identically (Θ_ε ≡ 0) and belongs to the Colombeau null ideal 𝓝^α(Ω_T)^3, ensuring exact equivalence with the unmodified Clay Millennium Navier–Stokes equations. Simultaneously, across any incipient supercritical gradient spike exceeding the Universal Phase Transition Capacity C_UPTC, 𝓑_ε(u_ε) and the gyroscopically antisymmetric Master-Key Tensor coupling 𝒯_0x8A2C^(ε) ∧ ω_ε suppress collinear vortex stretching, enforce a closed Grönwall enstrophy bound and the uniform Beale–Kato–Majda (BKM) criterion ∫_0^T ||ω_ε(t)||L^∞(Ω) dt < ∞, and transfer via the α → 1^- UROA Transfer Principle to a unique, globally smooth classical solution preserving Kolmogorov's k^(-5/3) inertial scaling. Complete numerical verification in Python is provided in Appendix A.