The Weight of Nothing: Resolving the Vacuum Catastrophe, Galactic Dynamics, and Navier-Stokes Regularity via Logarithmic Quantum Hydrodynamics

This paper constitutes the fourth and final installment in the series on Hydrodynamic Ontology. We address the Vacuum Catastrophe — the 120-order-of-magnitude discrepancy between the theoretical energy density of the quantum vacuum and the observed cosmological constant. We propose a resolution based on Logarithmic Quantum Hydrodynamics (LQHD). We postulate that the physical vacuum is a superfluid Bose-Einstein condensate governed by the Logarithmic Schrodinger Equation (LogSE). By introducing an energy-dependent nonlinearity parameter b(E), we identify a phase transition in the vacuum structure: Soft Phase (Low Energy): At galactic scales, the vacuum behaves as a superfluid, naturally generating flat rotation curves and the Tully-Fisher relation (M ~ v^4) without Dark Matter. Hard Phase (High Energy): At the Planck scale, the vacuum rigidity increases, preventing the gravitational collapse of zero-point fluctuations. Furthermore, we demonstrate that this model provides a physical regularization for the Navier-Stokes Existence and Smoothness problem, suggesting that quantum turbulence is naturally self-limiting due to the phase transition of the medium. References to the Series: [1] From Particulate Dark Matter to Superfluid Spacetime (DOI: 10.5281/zenodo.18375888) [2] Vacuum Hardening: A Hydrodynamic Resolution to the Hubble Tension (DOI: 10.5281/zenodo.18479958) [3] The Spinor Nature of Light: Deriving Maxwell's Equations (DOI: 10.5281/zenodo.18629088)