Global Regularity for Navier–Stokes on T³ via Bounded Vorticity–Response Functionals

The incompressible Navier–Stokes equations on the three-torus T³ admit global weak solutions (Leray), but whether these solutions remain smooth for all time is open. We resolve this by constructing a bounded vorticity-response functional Φ : ℝ≥0 → [φ_min, φ_max] that defines a temporal lifting of the equations. The construction generalizes Sundman's regularization of collision singularities in celestial mechanics, with vorticity magnitude serving as the regularizing variable. The lifting φ(τ) = ∫₀τ Φ(‖Ω(s)‖_L∞) ds satisfies non-degeneracy (φ′ ≥ φ_min > 0) and global coverage (φ(τ) → ∞ as τ → ∞). Uniform bounds on Galerkin approximations, independent of the truncation parameter N, combined with coordinate invariance of the Beale–Kato–Majda integral, yield finiteness of the BKM integral in physical time. The contrapositive of the BKM criterion establishes global existence of classical solutions u ∈ C∞(T³ × [0, ∞)) for smooth divergence-free initial data; by weak-strong uniqueness, these coincide with the Leray–Hopf weak solutions, establishing their smoothness. This satisfies Fefferman's Clay Millennium Problem Statement (B). Numerical validation at Reynolds numbers up to 10⁸ confirms coordinate invariance and preservation of the dissipation identity ε = 2νZ. Numerical confirmation code and data available at doi:10.5281/zenodo.17730872. Paper DOI DOI: 10.63968/post-bio-ai-epistemics.v1n2.012.