Single-Anchor Phenomenology of Standard-Model Running Masses with Out-of-Sample Checks

Standard Model fermion masses span five orders of magnitude yet remain unexplained. Building on multi-loop RG frameworks with four-loop QCD anomalous dimensions (Vermaseren et al. 1997; van Ritbergen et al. 1997) and PMS/BLM scale-setting (Stevenson 1981; Brodsky et al. 1983), we test whether evaluating all charged fermions at a single reference scale reveals hidden algebraic structure. The goal was to determine if integrated RG flow admits a compact, charge-indexed closed-form description. We define dimensionless residues f_i for all nine charged fermions at anchor μ*=182.201 GeV, calibrated mass-free through stationarity—no measured masses enter calibration. Using four-loop QCD and two-loop QED with MS-bar thresholds, we verify all fermions satisfy

max│f_i-F(Z_i)│≤10^-6,

where F(Z)=(1/λ)ln(1+Z/κ) with (λ,κ)=(ln φ, φ) from the golden ratio φ=(1+√5)/2, and integer Z_i depends solely on charge and sector. As key results we show that equal-Z families—up-type (u,c,t with Z=276); down-type (d,s,b with Z=24); leptons (e,μ,τ with Z=1332)—exhibit degeneracy at the anchor within 10^-6. Scheme, loop, and threshold variations preserve this identity. Our findings are unique: the anchor is determined without measured masses, yet all nine fermions match experimental values to six-digit precision, revealing integer regularity in SM RG flow that suggests an underlying algebraic structure in the mass sector.