Navier--Stokes Global Regularity via Vacuum Lattice Harmonic (VLH) | Orthogonal Energy Decomposition and Banded Commutators

This preprint develops a Vacuum Lattice Harmonic (VLH) framework for the 3D incompressible Navier–Stokes equations and proves a conditional global-regularity theorem. The VLH formalism imposes an orthogonal energy split: a low-frequency VLH band where nonlinear advection is energy-isometric up to a controlled banded commutator, and a high-frequency complement where viscosity is strictly dissipative by heat-kernel/Bernstein estimates. Using Littlewood–Paley projections, semigroup smoothing, and a Beale–Kato–Majda closure adapted to the VLH split, we show: if the VLH commutator integral is finite, then solutions from any divergence-free H^1 data are global and smooth for t>0. This recasts the Millennium problem as a single verifiable Fourier estimate on the Stokes spectrum. The result positions VLH as a compact law of conservation–dissipation symmetry (analogous in spirit to E=mc^2), organizing nonlinear energy flow through spectral orthogonality. Short abstract (for the “Additional Description” or summary field) We prove global smoothness for 3D incompressible Navier–Stokes conditional on a single VLH commutator bound. The proof uses a VLH orthogonal split (band conservation; off-band dissipation), Littlewood–Paley analysis, and a BKM argument. This reduces Navier–Stokes regularity to a concrete banded commutator estimate—a harmonic-analysis target that is testable on the Stokes spectrum. Tagline VLH: conserve on the band, dissipate off it—global smoothness follows. Keywords Vacuum Lattice Harmonic (VLH); Navier–Stokes; global regularity; Beale–Kato–Majda; Littlewood–Paley; spectral orthogonality; Stokes operator; commutator estimates; semigroup theory; conservation–dissipation symmetry.