We report an empirical logarithmic relation between the gravitational and electromagnetic fine-structure constants: (mP/mₚ) = 138. 27 1/ ₄₌ = 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. 19discrepancy 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_ (-0. 6\%) and m_ 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.
liang wang (Mon,) studied this question.