Navier-Stokes Global Regularity via Discrete Volumetric Projection

We resolve the Clay Millennium Problem on the existence and smoothness of solutions to the incompressible Navier-Stokes equations in three spatial dimensions. The resolution proceeds in two stages. Stage 1 (Diagnostic): We demonstrate that the classical formulation inherits a structural deficiency from its dependence on infinite-dimensional functional analysis (Sobolev spaces, distributional derivatives, Lebesgue integration). The 2D-to-3D transition is modeled as an accumulation of infinitesimal slices — a procedure that violates the Completeness Principle and creates artificial singularities absent from the physical fluid. We prove that the vortex stretching term (ω · ∇u), which has no analogue in 2D and is the sole obstruction to global regularity, arises from the failure to project dimensionally rather than accumulate. Stage 2 (Resolution): We introduce the Discrete Volumetric Projection (DVP) reformulation, in which the dimensional transition is governed by a single structural constant k_arc = 4.5/9 = 1/2, derived from the unique barycenter of the involution φ(x) = 9 − x on the digit system {1,...,9}. Under DVP, the vortex stretching term is bounded by the projection cost rather than growing without limit. We prove that: (i) smooth solutions exist globally for all smooth, divergence-free initial data with finite energy; (ii) the L² energy inequality is strict; and (iii) the Reynolds number transition from laminar to turbulent flow is a discrete projection event, not a continuous bifurcation. Seven independent convergences to k_arc = 1/2 establish the structural robustness of the constant.

MSC 2020: 35Q30, 76D05, 76F02, 11A63

Keywords: Navier-Stokes, global regularity, vortex stretching, discrete projection, barycenter, Completeness Principle, turbulence.