Cubic Scaling in Charged Lepton Mass Ratios: A Phenomenological Note

We report a phenomenological observation: the mass ratios of charged leptons independently imply a cubic scaling exponent (αμ = 2.976, ατ = 3.011, mean 2.993±0.018) and align with small integer cubes (63 and 153) to within 3–4% fractional error. A robustness analysis demonstrates that the integers {6, 15} are unique local minima for cubic scaling errors, with neighboring integers yielding 4–10× larger deviations. Model comparison shows cubic scaling outperformsquadratic and quartic alternatives by nearly an order of magnitude (RMS errors: 3.6% vs. 34% and 27%). Under a log-uniform null distribution, the joint probability of this alignment is ∼ 4 × 10−5, though look-elsewhere effects and small sample size limit confidence to the percent level. The integers correspond to hexagonal packing numbers Hk = k(2k − 1), suggesting a geometric interpretation. We briefly discuss a speculative scalar-tensor framework that could provide a geometric origin for cubic mass scaling through volumetric field quantization, though rigorous derivation remains an open problem. This note presents a constrained empirical scaling relation requiring theoretical explanation