Topological Solitons in an Electrodynamic Continuum: Derivation of Lepton Generations, Baryon Invariants, and Mass Ratios from First Principles

Modern gauge theories treat fundamental fermions as zero-dimensional point particles, outsourcing mass generation, generational hierarchies, and spatial confinement to nineteen or more empirical parameters inserted into the Standard Model Lagrangian. This paper presents an alternative framework: the Single-Field Electrodynamic Continuum (SFEC) [1], wherein all stable and metastable subatomic matter emerges as finite-energy, continuous topological vortex solitons within a physical dielectric vacuum defined by Maxwellian permittivity (ϵ_0), permeability (μ_0), characteristic impedance (Z_0≈376.73 Ω), and finite dielectric breakdown strength (E_"crit" ∼10^18 " V/m" ). By categorizing configurations according to spatial knot invariants in R^3, the subatomic spectrum cleanly bisects into two fundamental classes: The Lepton Sector (B=0): Unknotted closed-loop topologies (c=0) capable of continuous uncoiling and strand coalescence. The charged lepton generations (e^-,μ^-,τ^-) emerge as discrete, metastable toroidal winding modes T(N,1) for N=1,2,3. The mass ratios scale through mutual magnetic inductance (the relativistic Bennett pinch) governed by the vacuum coupling parameter 3/2 α^(-1) [7, 9], while their lifetimes are derived from the relativistic near-c electrostatic Coulomb repulsion barrier between parallel convective strands [14]. The non-existence of a fourth charged lepton (N≥4) is shown to be a physical consequence of the vacuum's Schwinger dielectric breakdown limit [12, 15]. Neutrinos are identified as self-screened, translating toroidal vortex pulses devoid of long-range Coulomb polarization tails, yielding sub-eV rest masses via Machian drag suppression and deterministic Euler precession [35, 36]. The Baryon Sector (B=1): Irreducible prime knots (c≥3) possessing absolute topological protection against decay into leptons. The proton is modeled as the ground-state 3_1 (T(2,3)) prime trefoil knot. Its three distinct spatial lobes naturally project the observed fractional charge scattering centers (+2/3,+2/3,-1/3) without requiring autonomous quarks or ad-hoc color confinement [25–28]. Evaluating the non-local crossing inductance, minimum ropelength (λ_D≈16.372) [20, 21], and curvature strain of the 3_1 trefoil derives the proton-to-electron mass ratio (m_p/m_e≈1836.15) from first-principles continuum geometry [40]. The neutron is formulated as an electrostatic composite (3_1⊕0_1), where a negative unknot sheath clasps the positive trefoil core [31], accounting for the 1.293" MeV" mass gap, free beta decay via loop slip-off [32], and nuclear binding via electrostatic bridging without virtual gluon or pion exchange [37–39].