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2026
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| Acceso en línea: | https://doi.org/10.5281/zenodo.19625947 |
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- <p><strong>Title:</strong> The Fine Structure Constant as a Soliton Aspect Ratio</p> <p><strong>Author:</strong> Alexander Novickis (alex.novickis@gmail.com)</p> <p>We construct a five-term hybrid electromagnetic Lagrangian admitting toroidal solitons with unit Hopf charge and spin ħ/2, and prove a no-go theorem: an exact scaling identity C · J_E = K renders all profile integrals independent of the E/B amplitude ratio κ, so no mechanism acting through the profile ODE can fix this free parameter. Of 16 independent constraint mechanisms investigated (24 individual analyses), 18 fail due to this identity. The asymmetric soliton branch exists only on the line 8δ + 2λ = 0, and yields a closed-form expression 1/α = Q₀(1+κ²)³/(κ³|δ|) that reproduces 1/α = 137.04 along a one-parameter family. Q₀ = 0.459 is a pure ODE constant — no electron mass, charge, ħ, or c appear.</p> <p>Key original results include:</p> <ul> <li>No-go theorem (scaling identity): C · J_E = K = constant exactly, proved from ODE symmetry — all profile integrals are κ-independent, blocking 18 of 24 constraint mechanisms (§7.8)</li> <li>Closed-form α expression: 1/α = Q₀(1+κ²)³/(κ³|δ|) where Q₀ = 0.459 depends only on three universal ODE numbers (K = −12.075, Ξ = 6.282, r_eff = 2.1558) (§7.8)</li> <li>Existence constraint: the asymmetric soliton branch is a 1D manifold on 8δ + 2λ = 0, proven by continuation, Gauss-Seidel, and eigenvalue methods (§7.1–7.7)</li> <li>Exhaustive constraint programme: 24 analyses across 16 independent mechanisms — Coulomb charge, flux quantization, toroidal curvature, (g−2)/2, Euler-Heisenberg, Casimir energy, vibrational spectrum, vacuum birefringence, SU(2) Hopfion, moduli quantization, and others — all catalogued with explicit failure modes (§8.4)</li> <li>Two-loop duality escape: the E×B cross-energy escapes the scaling identity at two-loop order; duality symmetry V(κ) = V(1/κ) centers the effective potential at κ = 1 (§8.4.21–22)</li> <li>Wilson loop holonomy: at κ = 1, both toroidal and poloidal holonomies give W = −1 (fermion sign); κ = 1 uniquely satisfies both holonomy conditions (analytic proof) (§8.4.24)</li> <li>Moduli quantization: Schrödinger equation on the moduli space gives ⟨κ⟩ = 1.04 (sharply peaked ground state) with the duality-derived effective potential (§8.4.24)</li> <li>Parameter scan: 1/α varies monotonically from 172 (δ = −0.50) to 97 (δ = −0.80), crossing 137.036 at δ = −0.601; the existence line determines all solutions (§6)</li> </ul> <p>The paper establishes the precise mathematical boundary between what the soliton profile ODE can and cannot determine, identifying δ as the remaining free parameter. This parameter is subsequently derived as |δ| = 3/112 from Kaluza-Klein Gauss-Bonnet normalization (Paper VII, Paper XIV), yielding 1/α = 137.09 with zero free parameters.</p> <p><strong>Keywords:</strong> fine structure constant, topological soliton, Hopf charge, nonlinear electrodynamics, soliton aspect ratio, scaling identity, boundary value problem, electromagnetic duality, ODE constants, Faddeev-Niemi model, moduli space, two-loop effective potential</p>