The inverse fine-structure constant, α−1 ≈ 137.035999177, is one of the most precisely measured dimensionless numbers in physics, but within the Standard Model its value is not derived from deeper first principles; it is measured and inserted as an input. This paper develops a numerical proof-of-principle inside the Fractal Consistency Law (FCL/LCF) program: a weak coupling of the observed order of α can emerge as the probability of non-trivial topological linking between closed random loops in an effective fractal-geometric substrate. The model represents elementary interaction events as topological engagement events between two closed loops. The Gauss linking number Lk is estimated numerically, and the effective coupling is defined as αLCF(s) = P(|Lk| > θ), with the separation parameter fixed as the dimensionless geometric ratio s = d/Rg, where d is the distance between loop centers and Rg is the loop radius of gyration. An initial sweep shows a monotonic decay of link probability with increasing s, from α ≈ 0.409 at s = 1.2 to α ≈ 0.0075 at s = 4.0. A higher-statistics grid search reports s* = 3.950, αLCF = 0.007304 ± 0.000109, and ⟨|Lk|⟩ = 0.038166, compatible with α = 0.0072973525643 within the reported statistical uncertainty. The result is not claimed as a derivation of QED or as evidence for new physics; rather, it establishes a reproducible numerical mechanism by which a small electromagnetic-scale coupling can arise as topological rarity. The paper identifies the technical obligations required for a stronger claim: deriving s* from the Principle of Minimum Inconsistency, controlling threshold and ensemble dependence, replacing the Monte Carlo linking estimator by an exact polygonal linking algorithm or certified error bounds, and producing an independent second prediction without recalibration.
César Daniel Reyna Ugarriza (Wed,) studied this question.