A Note on Singular Moduli and the Fine-Structure Constant

We report a numerical coincidence linking the inverse fine-structure constant, α−1, to a geometric overlap integral evaluated at the singular modulus |τ|=24. Using a basis of Jacobi Theta functions, the bare geometric value is found to be α−1 geom≈136.757. We observe that the discrepancy with the experimental value (α−1 exp≈137.036) is corrected with high precision (0.002%) by an additive term of the form √3/2π. We suggest that this structure may find a natural explanation within the framework of Logarithmic Conformal Field Theories (LCFTs) at singular points, interpretable as a contribution from the non-unitary sector.