Vacuum Media — Topological Solitons, Fermion Generation, and Emergent Electromagnetism in Compressible Substrates

Abstract In the Standard Model, elementary particles are conventionally introduced as fundamental point-like excitations of abstract quantum fields, leading to well-known divergences and self-energy infinities. Within the Dynamic Substrate Theory (DST) framework, elementary particles are instead modeled as localized, stable topological defects of the physical vacuum substrate. This work derives fermion mass generation and classical electrodynamics directly from substrate topology and vortical dynamics. Using homotopy theory, we classify stable knot structures governed by non-linear strain configurations. Furthermore, by isolating the substrate velocity curl field under Maxwell-Oldroyd-B rheology, we recover exact Maxwell equations, with the fine-structure constant (\alpha_{\text{em}}) emerging naturally as a mechanical impedance ratio between transverse shear and longitudinal bulk waves. Keywords: Topological Solitons, Homotopy Theory, Emergent Electromagnetism, Viscoelastic Vacuum, Fermion Generation.