The Toroidal Electron: A Topological Soliton Model for Electromagnetic Structure, Mass, and the Fine Structure Constant

Title: The Toroidal Electron: A Topological Soliton Model for 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 (§4.2) De Broglie wavelength derived as Lorentz-boosted internal Villarceau circulation at frequency mₑc²/h (§4.4) 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 (§6.3) g = 2 from topology: the single U(1) Hopf fiber forces the Dirac magnetic-moment-to-angular-momentum ratio, in contrast with the Skyrmion's g ≈ 5.6 from two charge channels (§7.1) The coupling gap is the electroweak mixing factor 4π/sin²θ_W = 54.3, an exact algebraic identity matching the measured ~56× to a few percent, 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 (§2.4) The Schwinger coefficient from the soliton profile is C₂ = −0.33 to −0.34 depending on the extraction, against QED's −0.3285 — agreement of 0.5% to 3.5% (§7.2) 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%) (§8.1, §8.4) 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 (§8.3) η-invariant computation on the Hopf-twisted S³: Chern-Simons invariant CS = 1/2, spectral asymmetry n₊ − n₋ = 1, selecting fermionic statistics from topology (§4.2) UV-finiteness: the soliton mass is finite by Derrick stability together with the Vakulenko-Kapitansky bound, with no counterterm required (§1, Appendix B) Lamb shift: the soliton's magnetic structure shifts the 2S level by δE_mag(2S) ≈ 21 Hz, three orders below the 20 kHz measurement precision, so the model is not currently distinguishable from QED here (§13.1) The paper derives Maxwell's equations, mass-energy equivalence, the Dirac equation and the Einstein field equations from the same geometry (§12), and states plainly that at the level of the electron sector alone it makes no sharp prediction distinguishing it from QED at currently accessible energies (§13.3). Open problems are tracked with explicit status, the principal one being the weak-interaction embedding (§14). Keywords: topological soliton, Hopf fibration, Faddeev-Niemi model, electron structure, Cho-Faddeev-Niemi decomposition, fine structure constant, anomalous magnetic moment, Finkelstein-Rubinstein mechanism, Koide formula, lepton generations, topological field theory, knot soliton, Lamb shift