We present a complete derivation of the fine structure constant α ≈ 1/137 from the geometry of the braid group B3 embedded on a tip-fused bicone manifold. We construct the explicit energy functional for a triple-helix configuration, impose the topological constraints from braid closure and spin-statistics, and perform the variational minimization. The critical pitch angle ψ_crit emerges from the balance between strand tension, inter-strand repulsion, and the holonomy constraint on the bicone. We show that α = sin(ψ_crit) represents the geometric projection factor coupling helical flow to linear propagation. All steps are derived from first principles with no free parameters.