The Toroidal Electron: A Unified Geometric Theory of Electromagnetic Structure, Mass, and the Fine Structure Constant

Description: This paper presents a comprehensive theoretical framework describing the electron as a topologically trapped electromagnetic field in a toroidal configuration, stabilized by Hopf fibration topology. The model provides unified geometric explanations for fundamental electron properties that are typically treated as empirical inputs in standard physics. Key Results: Fine structure constant identity: α = Z₀/ (2Rₖ), expressing the fine structure constant as the ratio of vacuum impedance to twice the quantum of resistance Electron mass formula: mₑ = mₚ × α^ (21/2 − 15α/4), achieving 0. 008% accuracy where 21 = 3 × 7 arises from Hopf fibration dimensions and 15 = dim (SO (4, 2) ) from the conformal group Charge quantization from topological linking number (Hopf invariant H = ±1) Spin-½ from SU (2) ≅ S³ structure requiring 4π rotation for identity Anomalous magnetic moment: First-order Schwinger term α/ (2π) = r/ (2πR), the ratio of minor to major torus circumference Revision 3 Expansions: Physical anomalies addressed: g-2 anomaly, self-energy divergence, zitterbewegung, Lamb shift, scattering form factors Gravity and vacuum connections: entropic gravity, hierarchy problem resolution, bootstrap hypothesis for self-determining constants Detailed experimental proposals for testing electron structure via precision Lamb shift measurements in hydrogen, muonic hydrogen, positronium, and antihydrogen systems The framework suggests the electron is "structured light" — electromagnetic field energy organized by Hopf topology — with mass, charge, and spin emerging from geometry rather than being fundamental inputs. Keywords: electron structure, fine structure constant, Hopf fibration, toroidal electron, conformal group, Wyler formula, anomalous magnetic moment, Lamb shift, entropic gravity, bootstrap hypothesis, vacuum permittivity