Monte Carlo Assessment of Natural-Unit Base Power Law Mass–Force Hierarchy of Hydrogen on the Minkowski Z2 Lattice

The origin of the mass–force hierarchy remains an open question in fundamental physics, since the Standard Model (SM) does not provide an explanation. Therefore, we examine whether fundamental constants, analyzed as natural-unit frequency scalars, display an underlying constrained integer-scaling power structure. We use number theory (NT) and Hermann Minkowski’s Z2 lattice to approximate the log ratios of constants using simple rational exponents, ij, on a two-dimensional integer lattice. The most parsimonious first approximation convergences transform a continuous slope into ij values. ij is linked to Diophantine residuals (Dr) and conformal factor frequency (vcf), a dimensioned measure from exact power laws. We apply this to hydrogen quanta, using all constants as the reference for one another, but focus on the Rydberg frequency to compare electromagnetic and gravitational forces. All exponents, besides the reference, are near-perfect partial harmonic fractions (PHFs) in a structure based on NT. The ratio- hierarchy of two constants is encoded by i, j, vcf, and one constant scalar. Monte Carlo simulations demonstrate that the ensembles’ patterns are statistically distinct from randomized datasets, and Bayesian testing supports the validity of the mathematical hypothesis. These results indicate that part of the hierarchy among fundamental constants may be captured by simple rational scaling relations based on NT properties, without any new physics.