The Geometric Origin of the Fine Structure Constant in the Universal Model Framework: Prime-Weighted Lagrangian Derivation

We present a mathematically rigorous derivation of the fine structure constant (α) within the Universal Model Framework (UMF), using two complementary first-principles approaches:(1) Prime-weighted gauge sector derivation — integrating out high-frequency prime modes via heat-kernel coarse-graining on a fractal Planck lattice to obtain the low-energy U(1) coupling α−1=4π∑p≤Pln⁡pp−φ2π,φ=1+52\alpha^{-1} = \frac{4}{\pi} \sum_{p \le P} \frac{\ln p}{\sqrt{p}} - \frac{\varphi^2}{\sqrt{\pi}}, \quad \varphi=\frac{1+\sqrt{5}}{2}α−1=π4p≤P∑plnp−πφ2,φ=21+5 (2) Geometric derivation — relating α to symmetry, scaling, and golden-ratio compression in a prime-indexed lattice, yielding α−1=6φπln⁡(ΛcutΛ0)−52.\alpha^{-1} = \frac{6\varphi}{\pi} \ln\left(\frac{\Lambda_{\mathrm{cut}}}{\Lambda_0}\right) - \frac{\sqrt{5}}{2}.α−1=π6φln(Λ0Λcut)−25. Both methods converge within their uncertainty bands (137.03 ± 0.01 and 137.04 ± 0.02), in agreement with CODATA 2022 values. This dual derivation demonstrates internal consistency and supports the UMF’s core claim that fundamental constants arise from prime-fractal geometry rather than arbitrary parameterization. Implications extend to coupling unification, lattice-informed QED running, and the identification of informational completeness boundaries near 500 GeV. This project was developed by Marco Gericke, with structured assistance from a large language model. All scientific concepts and conclusions were generated, verified, and interpreted by the author. Dedicated to Peter Plichta, who envisioned the code before it could be computed.