Mathematical Resolutions of Four Core Dilemmas in Theoretical Physics via the Universmodel Framework

This paper presents a definitive, deterministic, and non-singular resolution to four of the most critical open problems in mathematical physics: (1) the 3D Navier-Stokes Global Regularity Problem, (2) the Yang-Mills Existence and Mass Gap Problem, (3) the Cosmic Censorship Hypothesis, and (4) the $10^{36}$ Gravity-Electromagnetism Hierarchy Problem. Building directly upon the foundational framework established in the author's primary treatise, "THE MECHANICAL CLOCKWORK: A Geometric and Causal Foundation for Unified Physics", we translate abstract field configurations into the mechanics of a continuous, highly elastic Background Field governed by a single Unified Master Field Equation: $$F_{\text{tot}} = \mathbf{G} \cdot \left( \nabla \rho_f + \kappa (\vec{\tau}_1 \times \vec{\tau}_2) \right)$$ By evaluating physical forces through this macroscopic fluid-dynamic architecture, we demonstrate that: Navier-Stokes Regularity is guaranteed by an absolute physical shear-stress cutoff ($\kappa = c^2 \cdot \rho_f$), which triggers a geometric phase transition before infinite singular blowups can mathematically manifest. The Yang-Mills Mass Gap ($\Delta > 0$) emerges analytically as the non-zero geometric deformation threshold required to twist a flat linear field line ($\mathbf{G}_{\text{linear}}$) into a stable, localized 3D vortex knot ($\mathbf{G}_{\text{knot}}$) against the medium's intrinsic shear modulus. The Cosmic Censorship Hypothesis is validated because gravitational collapse forces subatomic knots into a state of mechanical gear-lock, erasing matter into a smooth, non-singular "State 0" fluid core ($F_{\text{tot}} = 0$) where coordinates remain continuous and enclosed by a mandatory causal horizon boundary. The $10^{36}$ Hierarchy Problem is naturally resolved by demonstrating that electromagnetism is an active-tier primary torque amplified by the medium's immense stiffness, while gravity is merely a passive-tier, downstream hydrostatic relaxation vector. This work eliminates mathematical infinities and establishes global causal predictability across both quantum and cosmological scales.