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

Title: The Toroidal Electron: A Unified Geometric Theory of Electromagnetic Structure, Mass, and the Fine Structure Constant Author: Alexander Novickis (alex.novickis@gmail.com) We propose that the electron is a topologically stabilized electromagnetic soliton — a Hopf-fibred toroidal field configuration — within the Faddeev-Niemi nonlinear sigma model, derived from SU(2) gauge theory via the Cho-Faddeev-Niemi decomposition. Charge quantization corresponds to the Hopf linking number (H = ±1), spin-1/2 arises via the Finkelstein-Rubinstein mechanism (π₁ = ℤ₂), and the de Broglie wavelength derives from Lorentz-boosted internal circulation. The soliton energy matches mₑc² with coupling κ₂ = 1/α under the identification g² = α, and the electric form factor is exactly point-like via a topological Ward identity. A 2D axially symmetric solver achieves soliton energy E = 193.5 (1.005× the Battye-Sutcliffe minimum) with near-virial convergence. Key original results include: Charge quantization from topology: the Hopf linking number H = ±1 maps to electric charge ±e, with antiparticles as orientation-reversed solitons (§3–4) Spin-1/2 from the Finkelstein-Rubinstein mechanism: π₁(SO(3)) = ℤ₂ gives fermionic exchange statistics without invoking the Dirac equation (§4.3) De Broglie wavelength derived as Lorentz-boosted internal Villarceau circulation at frequency mₑc²/h (§10) Electric form factor Fₑ(q²) = 1 exactly via a topological Ward identity, with spectral one-loop confirming |δFₑ| ~ 6.6×10⁻¹⁴ at q ~ 1 GeV — seven orders below Bhabha scattering bounds (§15.5) CFN electromagnetic field from rotation: F₀ᵢ = −ω∂ᵢn₃ reproduces both electric and magnetic structure of the classical electron (§13.6) The 56× coupling gap decomposes exactly as 4π/sin²θ_W (the electroweak mixing factor), resolved by identifying g² = α as the natural coupling for an electromagnetic soliton; six independent non-perturbative methods confirm the gap cannot be closed dynamically (§13.3.1) Two independent estimates of the anomalous magnetic moment coefficient bracket the QED value: geometric C₂ ≈ −0.30 (9% from QED) and self-interaction C₂ ≈ −0.33 (0.5% agreement) (§14) The G₂ Casimir ratio δ = C₂(3)/C₂(Sym³(3)) = (4/3)/6 = 2/9, combined with Q = 2/3 from soliton topology, reproduces all three charged lepton mass ratios to 0.01% accuracy: mμ/mₑ = 206.770 (0.001%), mτ/mₑ = 3477.5 (0.009%) (§16.5) N = 3 uniqueness theorem: (N+1)/(2N²) = 2/9 has only the solution N = 3, providing a group-theoretic reason for exactly three charged lepton generations (§16.5) η-invariant computation on the Hopf-twisted S³: Chern-Simons invariant CS = 1/2, spectral asymmetry n₊ − n₋ = 1, selecting fermionic statistics from topology (§20.1) UV-finiteness: soliton energy integral converges with δM/M ~ 2×10⁻⁵, an improvement of 10³⁷ over the Standard Model quadratic divergence (§D) Lamb shift prediction: δE(2S) ~ 0.024 Hz from magnetic form factor structure, with κ-suppression resolving the r_M tension (§15.4) The paper includes a QED-soliton correspondence table mapping soliton degrees of freedom to QED perturbative structure (§15.9), and identifies the computation of (g−2)/2 from the soliton one-loop fluctuation determinant as the critical remaining test. Twenty-one open problems are tracked with honest status assessment.