Electromagnetic Closure and the Fine-Structure Constant: A Geometric Derivation

A first-principles derivation of the fine-structure constant α = Z₀/2RK ≈ 1/137. 036, proceeding from the non-terminal reciprocal singularity 1/z at z = 0, completed on the Riemann sphere. The derivation uses no free parameters, no postulates, and no imported physical structure. The argument proceeds in four stages. The reciprocal singularity forces three consequences on the completed space: an irremovable source defect at p, a unique antipodal sink p* required by the Poincaré–Hopf theorem (χ (S²) = 2), and a logarithmic potential field between them. The self-similar scaling structure of this field has a unique attractor — the golden ratio φ = 1;1, 1, 1,. . . — forced by the Markov spectrum theorem and the uniform resolvent condition (Hurwitz, 1891). Rational winding is excluded by the existence of matter. The irremovable defect forces a departure δ > 0 from φ. The electromagnetic constants ε₀ and μ₀ are not introduced as physical inputs. Their own definitions coincide with the mathematical primitives already present in the structure: ε₀ is the capacity for containment (the primitive 1) and μ₀ is the openness to penetration (the primitive 0). Their ratio Z₀ = √ (μ₀/ε₀) is therefore the 0/1 geometric ratio of the vacuum, definitionally. No additional structure is required. The departure δ is assembled from four established facts: the residue of 1/z at 0 equals 1; Z₀ is the 0/1 vacuum ratio; RK = h/e², where h is the invariant of the self-similar scaling process and e is the minimum charge of the defect; and the minimum winding number is 2. The result is α = Z₀/2RK = e²/2ε₀hc ≈ 1/137. 036, derived with zero free parameters. Neither h, e, nor c is a primitive of the structure; all are derived consequences. Within this framework, the constancy of c = 1/√ (ε₀μ₀) follows as a mathematical consequence rather than a physical postulate. Empirical consequences — the universal spectroscopic gap at IE/4 confirmed across 71 atomic ions with zero falsifications, the thread law in hydrogen (r² = 1. 000 over 9 pairs), recovery of the electron rest mass and Bohr radius from the gap boundaries, and the same IE/4 structure in doubly magic nuclei — are identified as instantiations of the mathematical structure, not as premises. Related works: - Determinacy Under Quotient Representations (Heaton, 2026): https: //doi. org/10. 5281/zenodo. 18868210- Vacuum Impedance as the Organizing Principle of Atomic Spectra: A Universal Gap at IE/4 and Recovery of the Electron Mass (Heaton, 2026): https: //doi. org/10. 5281/zenodo. 19164224- Recursive Geometry, Coordinate Compression and Invariant Structure in Electromagnetism (forthcoming): https: //doi. org/10. 5281/zenodo. 19194036