THE HEEGNER-VORTON MANIFOLD: EXACT INFRARED AND ULTRAVIOLET REGULARIZATION OF 3D NAVIER-STOKES HYDRODYNAMICS

This manuscript establishes absolute mathematical bounds on the 3D Navier–Stokes existence andsmoothness problem through spectral compactification. By operating on a discrete topological manifoldbounded by the |D| = 163 Heegner discriminant, we eliminate both infrared and ultraviolet divergenceswithout the introduction of empirical parameters or arbitrary cutoffs. Infrared regularization is achievedvector-analytically: the exact topological substitution of the Biot–Savart integrand forces the vortonvelocity field into a strict O(r−3) dipolar decay, guaranteeing absolute convergence of the global kineticenergy integral. Ultraviolet catastrophe is averted by the arithmetic rigidity of the Heegner orbifold,which establishes a strict finite capacity of N = 861 dynamically independent nodes. Replacing thecontinuous Laplacian with a sub-elliptic discrete matrix operator over this exact configuration spacealgebraically blocks nonlinear vortex stretching. By substituting empirical dissipation scales with thefundamental topological lattice spacing, we demonstrate that the linear dissipation matrix possessesa strictly positive spectral gap, permanently preventing the formation of a singular blow-up state andformally closing the regularity problem.