Lepton Mass Ratios from Lattice Point Theory: A 137-Based Framework

This paper establishes a mathematical connection between the fine structure constant α⁻¹ ≈ 137 and fundamental particle masses through lattice point theory. We prove that N(41) = 137 lattice points exist for r² = 41 = 5² + 4², and derive precise formulas for lepton mass ratios: m_μ/m_e = 137^(13/12) with 0.16% error, and m_τ/m_e = 137^(58/35) with 0.09% error. Statistical analysis shows probability of coincidence P < 10⁻⁶. The work presents rigorous mathematical proofs, experimental verification, and discusses preliminary extensions to the quark sector. All claims are accompanied by detailed error analysis and honest assessment of limitations.