Global Vorticity Boundedness via He-Tu Antipodal Projection: A Lattice-to-Continuum Framework for the Navier-Stokes Regularity Problem

We present a lattice-based framework for establishing vorticity boundedness in the three-dimensional Navier-Stokes equations. The framework is built upon the He-Tu Lattice Boltzmann Method (3D-HLBM), whose antipodal projection operator P satisfies two rigorously proved properties: non-expansiveness and global boundedness (P(E)_i in [0,B] for all nodes i). We prove that on the N^3-node He-Tu lattice with mesh size h = L/N, the discrete vorticity satisfies ||omega_h(t)||_{L^inf} = 0, with B independent of h. By the Beale-Kato-Majda criterion, this yields global smooth solutions at the discrete level. We further identify the precise mathematical gap separating this result from a full resolution of the Millennium Prize Problem: one must show that the He-Tu projection admits a well-defined continuum limit as h->0, and that discrete solutions converge to a weak solution satisfying the same L^inf bound. Three candidate tools are outlined: the Lax equivalence theorem, Sobolev embedding H^s -> L^inf (s > 3/2), and De Giorgi-Nash-Moser regularity theory. The connection to E8 root system geometry and U(3) gauge conservation is discussed as structural motivation. This preprint contains rigorous discrete-level results and an explicit open conjecture for the continuum. No claim is made to resolve the Millennium Prize Problem.