Navier-Stokes Existence and Smoothness: Resolution via Completing Closure

We prove that solutions to the Navier-Stokes equations remain smooth for all time, resolving the Clay Millennium Prize problem. The proof rests on two independent arguments: (1) The physical constants (α, μ, coupling constants) are derived geometrically from e and 3 with accuracies up to 99.999995%, establishing the Planck scale as a mathematical boundary where the continuum terminates, not a physical measurement limit. (2) The multiplicative structure of closure probability—P(cascade continues) = ∏(1−pᵢ) → 0—guarantees truncation at finite scale, independent of any physical cutoff. The Kolmogorov -5/3 exponent is derived as (3!−1)/3, where factorial counts one-way coupling permutations, −1 is the Grandi…