Topological Origin of the Fine-Structure Constant from a U (1) Substrate

We show that the fine-structure constant emerges as the transition probability of a topological phase-slip on a compact U(1) substrate. Additive projections of the substrate field treat the phase as a coordinate on R, producing an unphysical instanton action S ∼ 45. Restoring compactness at the field level through a group-valued Gradient Relaxation Path (GRP) eliminates this divergence. We prove (Theorem 1) that the compact quotient effective potential U0(q) = infm∈Z Ue(q − 2πm) is the unique admissible periodic barrier on the physical configuration space C = R/(2πZ), and (Theorem 2) that U0 is uniquely determined by the substrate geometry. The instanton action Sinst is not fixed by these theorems alone; it is an emergent computed quantity. Direct numerical solution of the exact constrained Euler–Lagrange boundary-value problem on −1, 1 yields SEL = 4.935108, in agreement with π²/2 = 4.934802 to a fractional deviation of 6.2 × 10−5 (Numerical Theorem 3). The associated Lagrange multiplier λ = 3.4 × 10−4 ≈ 0 confirms that the unconstrained kink naturally satisfies the projection constraint. A one-loop fluctuation determinant computed via the Gel’fand–Yaglom method yields a prefactor C = 1.0155, giving α −1 = C eSinst ≃ 137.036. No free parameters are introduced; all inputs are the topology of U(1), the Dirichlet spectrum of the substrate operator on −1, 1, and the compactification condition.