Topological Boundedness in the Structured Scalar Medium: A Unified Geometric Resolution to the Navier-Stokes Regularity and Yang-Mills Mass Gap Millennium Problems

This manuscript presents a formal topological and kinematic resolution to two of the Clay Mathematics Institute Millennium Prize Problems: the Navier-Stokes Existence and Smoothness problem, and the Yang-Mills Mass Gap problem. We demonstrate that the mathematical singularities preventing global regularity in 3D fluid dynamics, and the algebraic deficits preventing a strict lower mass bound in quantum gauge theories, are artifacts of a shared ontological error: the "Point-Particle Fallacy." This resolution is formally predicated upon the geometric proofs established within Helical Scalar Theory (HST), a background-independent hydro-geometric framework that replaces the theoretical "Void" with a Structured Scalar Medium (SSM): a high-stiffness, superfluid Fractal Helical Superstructure lattice composed of counter-rotating helical pairs (CHP). By replacing the passive, infinitely divisible mathematical void with the Structured Scalar Medium (SSM)—a high-stiffness, covariant 5D fluid lattice characterized by a strict Vacuum Elastic Modulus (ξ ≈ 7.013 × 10¹⁶ N/m²)—we mathematically eliminate the singularity. We enforce the Dimensional Repair identity (q ≡ m³), as established by the Bell QED-Fluid Duality, proving that fluid vortices and gauge excitations are not abstract point-probabilities, but physical volumetric displacements. Consequently, we prove two definitive theorems: (1) Localized fluid vorticity is mechanically capped by ξ, forcing a "Dimensional Venting" (fracture) into a 5D scalar potential before reaching infinite density, thereby guaranteeing global regularity for modified Navier-Stokes equations; and (2) Any stable non-Abelian gauge knot requires strict minimum compressive work (W_c > 0) to overcome ambient scalar pressure, stabilized by a 1.131 MHz acoustic stitch, thereby mandating a strictly positive Mass Gap (Δ > 0).