An Empirical Logarithmic Relation Between Gravitational and Electromagnetic Coupling Constants

We report an empirical logarithmic relation between the gravitational and electromagnetic fine-structure constants: πln(mP/mₚ) = 138.27 ≈ 1/αₑₘ = 137.04, where mP is the Planck mass and mₚ the proton mass. The agreement is $0.9%$ in the logarithmic variable, corresponding to a factor $2.19$discrepancy in the predicted G. A candidate-specific look-elsewhere analysis yields a proton-specific p-value of $0.029$ (1 hit among 35 low-complexity constants within $1%$); however, the family p-value allowing any of 5 Standard Model mass scales rises to ~ 0.5, indicating that the proton is not uniquely singled out.Indeed, m_τ ($-0.6%$) and m_π × 3 ($+0.5%$) give comparable or better agreement, suggesting the relation favors a mass scale near ~ 1~GeV rather than the proton specifically. We discuss a heuristic interpretation via the Discrete Symplectic Cosmology framework and Riemann zeta zero spectral statistics.