On the Global Regularity of Navier-Stokes Equations via Discrete Hyper-Geometric Tessellation

On the Global Regularity of Navier-Stokes Equations via Discrete Hyper-Geometric Tessellation ABSTRACTThe Millennium Prize problem asks for a proof of existence and smoothness for NavierStokes solutions on ℝ³. This paper provides a constructive proof of global regularity by defining the continuum solution as the strict limit of a sequence of discrete solutions on a Hyper-Geometric D4 Lattice (ΩKSO). We demonstrate that the discrete system is energetically bounded and regular for any finite lattice spacing ϵ > 0. By taking the limit ϵ → 0, the discrete difference operator T̂ converges uniformly to the differential operator ∇, recovering the classical Navier-Stokes equations on ℝ³ without the formation of singularities. Thus, the regularity of the discrete geometry enforces the regularity of the continuum limit. This proof applies the geometric framework "The Kinematic-Substantial Octaplex (KSO): A Geometric Unification of Fundamental Constants via 24-Cell Lattice Projection".