Geometric Relation Between the Cosmological Constant and Electromagnetic Coupling

This work presents a standalone, falsifiable prediction linking the cosmological constant Λ\LambdaΛ to the fine-structure constant α0\alpha_0α0 through a geometric surface-projection relation. The proposed relation is ΩΛ≡Λc23H02=23N2α02,\Omega_\Lambda \equiv \frac{\Lambda c^2}{3H_0^2} = \frac{2}{3}N^2\alpha_0^2,ΩΛ≡3H02Λc2=32N2α02, where N=ln⁡(1061)N=\ln(10^{61})N=ln(1061) represents the logarithmic hierarchy between Planck and cosmological scales. This yields the numerical prediction: ΩΛ,pred≈0.700,\Omega_{\Lambda,\mathrm{pred}} \approx 0.700,ΩΛ,pred≈0.700, corresponding to Λpred≈1.12×10−52 m−2forH0≈67.4 km s−1 Mpc−1.\Lambda_{\mathrm{pred}} \approx 1.12\times10^{-52}\,\mathrm{m^{-2}} \quad \text{for} \quad H_0 \approx 67.4\,\mathrm{km\,s^{-1}\,Mpc^{-1}}.Λpred≈1.12×10−52m−2forH0≈67.4kms−1Mpc−1. The relation is presented independently of any specific cosmological model, dark-energy field, or ΛCDM parameter fitting, and is formulated as a purely geometric predictive statement. The prediction is directly falsifiable by future precision measurements of ΩΛ\Omega_\LambdaΩΛ, α0\alpha_0α0, and H0H_0H0. Any statistically significant deviation from the above relation would rule out the proposed correspondence. This record establishes priority for the proposed relation and provides a testable bridge between microphysical coupling and cosmological vacuum geometry.