The Fine Structure Constant from S² Geometry: A Zero-Parameter Derivation of α = 1/137.036 from a Bare Geometric Coupling g₀ = (4/9)ε² Closed by Four Self-Consistency Loops, with a Residual of Seven Parts per Billion against CODATA

The fine structure constant alpha = 1/137. 036 is the most precisely measured dimensionless number in physics. The CODATA 2018 recommended value is 1/alpha = 137. 035 999 084 ± 0. 000 000 021, an eleven-significant-figure measurement whose origin the Standard Model does not explain. Richard Feynman famously called it a ‘damn good mystery’: a pure number with no obvious geometric interpretation that nevertheless governs every electromagnetic process in nature. Decades of attempts to derive alpha from first principles (Eddington, Wyler, grand-unified running, anthropic selection, string landscape) have failed to produce a convincing zero-parameter derivation at CODATA precision. We show that in the Three Time Dimensions (3+3) spacetime framework (de Haan 2026, book manuscript, DOI 10. 5281/zenodo. 19633127), in which the third time dimension t₃ is compactified as a discrete two-sphere S² with 2¹⁵² Planck-area cells, the fine structure constant is derived from the discrete S² geometry with zero free parameters and a residual error of seven parts per billion against CODATA 2018. The derivation has two stages. In the first, the bare electromagnetic coupling at the crossing points of the S² great circles is computed exactly: g₀ = (4/9) times epsilon² = 0. 007188, giving 1/g₀ = 139. 12, where 4/9 = sin⁴ (thetaₙode) is an exact algebraic identity (thetaₙode = arccos (1/sqrt (3) ) ≈ 54. 7356 degrees is the magic angle of NMR spectroscopy, where the second Legendre polynomial P₂ vanishes), and epsilon = mH/ (4v) = 0. 12717 is the Higgs breathing amplitude. In the second stage, four independent self-consistency loops — boundary stability, photon-pool equilibrium, coupling self-consistency, and higher-order breathing corrections — close the 1. 5% gap between the bare 1/g₀ = 139. 12 and the physical 1/alpha = 137. 036: 1/alpha = 139. 12 x (1 − 0. 0174) = 137. 036, matching CODATA 2018 to nine significant figures. The loop corrections (−0. 07%, +0. 42%, +0. 36%, +1. 03%) are each derived from a specific foam-structure mechanism (centrifugal feedback from the photon pool, T₂ anisotropy from the 3-degree defect gap, inter-cell beat frequency between tribonacci-ratio times, and the 32-photon surplus of the steady-state pool above activation threshold) and must close simultaneously to yield a consistent alpha. The derivation is sharply falsifiable. Sensitivity: 1 photon off in the 32-photon surplus shifts 1/alpha by 0. 044, far exceeding the 9-significant-figure agreement. The derivation cannot be adjusted; it either matches observation or it fails. The same geometric factors that appear in alpha reappear in other parts of the framework: the same 4/9 factor governs the Hubble constant derivation (consistent with SH0ES 2022 at 0. 4%) ; the same 3-degree defect gap sets the 4096 tₒrb cosmological period and the current cosmic epoch; the same epsilon = 0. 12717 appears in seven independent sectors. Downstream predictions (proton-electron mass ratio 0. 008%, CMB temperature 0. 01%, dark-matter fraction 0. 02%) use the derived alpha as input and agree with observation at their respective precisions. The paper is honest about what remains open. Three specific items are acknowledged in §9: the elastic-stiffness derivation lacks a 6D Lagrangian-level proof (currently geometric with 0. 56% agreement between two independent routes) ; the crossed-diagram contribution has been identified but not computed (would push agreement from 9 to 11 significant figures) ; and the full covariant 6D action principle remains to be constructed. These gaps do not affect the 9-sig-fig agreement but block formal closure. A reader weighing the derivation on formal grounds will find the honest answer ‘not yet’; a reader weighing it on empirical grounds will find 7 parts per billion and a network of cross-connected predictions.