This paper presents a unified field model ("The Fabric Theory") treating the physical vacuum as a superfluid medium with an order parameter described by a Ginzburg-Landau scalar field. Numerical modeling demonstrates the existence of stable topological solitons (vortices) whose energy and geometry are strictly quantized. We demonstrate that the hydrodynamic impedance of such structures (the inverse fine-structure constant α⁻¹) can be analytically expressed via a sum of geometric invariants of a 4-dimensional manifold with an accuracy of 0.005 ppm. Furthermore, based on the analysis of soliton resonant modes, we derive a relationship for nucleon masses and radii, predicting a topological neutron core radius of R_n = 0.84008 fm. The paper also introduces a recursive correction term involving the number 24 (related to bosonic string theory dimensions), which refines the geometric derivation of alpha to match experimental data with sub-ppm precision.