Algebraic Mass Relations in the Standard Model from a Single Complex QCD Parameter

We report a system of approximate empirical formulas that relate eight Standard Model particle masses and the fine-structure constant through three inputs: the Planck mass, the number of colors Nc = 3, and a transcendental constant X = e^π + 2π². All leading numerical coefficients are rational functions of the one-loop QCD β-function coefficient β₀ = 7 and Nc = 3, encoded in a single complex number z = β₀ + iNc = 7 + 3i. The formulas are not claimed to be exact: each is presented with an explicit residual, and we make no assertion about the convergence or completeness of the higher-order terms. The best relations achieve 10.9 digits (neutron-to-electron mass ratio) and 8.4 digits (Planck-to-neutron hierarchy), while others reach 4–6 digits. Monte Carlo trials (5 × 10⁵ samples) confirm that the probability of obtaining comparable combined precision by chance is p < 2 × 10⁻⁶. The system predicts mH = 125,099 ± 1 MeV (testable by HL-LHC), excludes a light fourth boson below ~129 GeV, and determines sin²θW = 0.23120 from z-algebra alone. This work synthesizes results from a broader program including a six-mass Planck product relation and its geometric interpretation. We present these observations in the hope that they may point toward an underlying theoretical structure, while acknowledging that no derivation from first principles currently exists.