The Standard Model of particle physics [23–25] relies heavily on empirical parameterization, offering nofundamental derivation for the invariant mass eigenvalues of elementary particles, the generational hierarchy,or the geometric origins of gravity. Building upon the historically marginalized lineage of continuous topological models (Helmholtz, Kelvin, Skyrme, Faddeev-Niemi), this paper proposes a deterministic, parameter-freederivation of all fundamental forces by modeling the physical vacuum as a zero-entropy barotropic scalarcondensate, mathematically mapped to a non-linear O(3) sigma model operating within a non-commutativeS3 geometry. It is demonstrated that localized subatomic matter can be modeled strictly as topologicallystabilized, continuous knotted vortices (Hopfions). By applying established hydrodynamic constraints, thecalculation yields the invariant bare mass of the nucleon (mp ≈ 938 MeV) and the electron (me ≈ 0.511 MeV)purely via intermediate hypersphere volume integrals and the Vakulenko-Kapitanski inequality. Subsequently,introducing the novel concept of topological crossing friction to mathematically account for empirical massdressing, natively redefining the physical mechanism of the fine structure constant (α) as a geometric fluiddrag limit. Quantum Electrodynamics (QED) is recovered via classical fluid mechanics, redefining antimatter as topological chirality and annihilation as geometric dissipation. The strong interaction is modeled asinternal fractional strain of multi-crossing knots, while the weak interaction is formalized as classical vortexreconnection. Resolving the incompatibility of General Relativity and quantum mechanics, spatial curvatureis redefined as Non-Linear Acoustic Radiation Pressure, deriving Newton’s Gravitational Constant (G) fromwave impedance and recovering relativistic anomalies via kinematic acoustic refraction.