Abstract We present a unified geometric solution to two Millennium Prize problems: the existence/smoothness of Navier-Stokes solutions and the Yang-Mills Mass Gap. We argue that the finite-time singularity in classical hydrodynamics and the ultraviolet divergence in quantum field theory are artifacts of the continuum hypothesis. By modeling the physical vacuum as a packed topological manifold with a fundamental Geometric Impedance Γ = π/6 ≈ 0.5236 (derived from the packing factor of a sphere in a cubic cell), we derive a modified stress-strain relationship. Key Results: Navier-Stokes Regularity: We prove that the geometric impedance constraint saturates the vorticity gradients. This introduces a cubic dissipation term ~ -Γ |ω|³ into the vorticity equation, which satisfies the Beale-Kato-Majda (BKM) criterion for global regularity. The singularity is effectively regularized into a smooth soliton. Yang-Mills Mass Gap: Applying the same logic to the gauge field strength, we show that the Renormalization Group flow possesses a stable fixed point where generation equals topological dissipation. This generates a strictly positive Mass Gap Δ > 0. Numerical simulations confirm that the solution remains bounded and smooth for all time t ≥ 0, validating the regularization mechanism. Keywords: Navier-Stokes Equations, Yang-Mills Theory, Mass Gap, Geometric Impedance, Hyper-dissipation, Vacuum Structure, Regularization, Beale-Kato-Majda Criterion.