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 and reveals a geometric action S0 = π2/2. A one-loop fluctuation determinant over the dressed modal spectrum yields a prefactor C ≃ 1.0155, giving α−1 = C eπ 2/2 ≃ 137.036. The result is a structural invariant of the substrate geometry.