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 offset, and /3 normalises per phase. This connects turbulence to Yang-Mills (αs = 1/(3π−1)) and BSD through Eisenstein geometry. Experimental validation comes from crystallisation statistics (PNAS 2002, 300 trials) confirming the product structure S(t) = ∏(1−pᵢ). Singularities cannot form because they require infinite frequency components, which require structure at arbitrarily fine scales. Both arguments—geometric termination and probabilistic truncation—independently forbid this. The framework makes falsifiable predictions, including the unreachability of Norton's Dome apex. Keywords: Navier-Stokes, Clay Millennium Prize, turbulence, Kolmogorov -5/3, Completing Closure, Eisenstein geometry, Grandi series, mass gap, crystallisation nucleation, Lipschitz boundary, smoothness