The Electromagnetic Running Coupling from Foam Geometry

The fine structure constant α was previously derived from the geometry of a Planck-scale foam with truncated octahedral cell structure (DOI: 10.5281/zenodo.19011758). That derivation used three structural ingredients: two endpoints per displacement event (Axiom Zero), d = 3 spatial dimensions, and the Gaussian return weight π. We show here that these identical three ingredients, applied to virtual loop processes rather than static coupling, produce the one-loop QED beta function coefficient b₀^EM = 4/3 and the complete running formula β(α) = 2α²/(3π). The running coupling α(M_Z) = 1/128.95 is recovered to 0.7% from α(0) = 1/137 using only foam-derived inputs. The 2-loop coefficient b₁^EM = 352/27 is derived from the full charged sector (Parts XXXI, XXXV of the Core Framework v2). The physical mapping from foam sectors to Standard Model fields is a hypothesis confirmed numerically; the Lorentz invariance emergence from discrete O_h symmetry remains an open problem.