Mass Law of Leptons and Quarks based on Hypercomplex Algebra: Topological Scaling and Internal Geometry

This study presents an empirical and geometric approach to accurately modeling the mass hierarchy of leptons and quarks using a three-parameter logarithmic relation involving the fine-Structure constant (α ≈ 1/137), the mathematical constant π, and internal spinor projection geometry. The mass of each fermion is fitted to the form: log(m) = A·log(α⁻¹) + B·log(π) + C·log(D), where D represents a geometric factor derived from compactified internal spinor volumes. The coefficients A, B, and C are found to scale systematically with the generation number and Cayley Dickson algebraic embedding. Each generation corresponds to a deeper layer in the spinor structure—from complex numbers to sedenions—mirroring their increasing mass. Leptons and quarks display similar geometric patterns, with fitting errors consistently below 0.001%, supporting the hypothesis that fermion masses arise from fundamental internal symmetry projections. These coefficients align with Clifford algebra spinor spaces and suggest embeddings into grand unified symmetry groups such as SU(5), SO(10), and E₈. This model offers a potentially unifying framework linking particle mass, internal curvature, and the algebraic structure of spacetime, with implications for understanding triality, mass generation, and symmetry breaking in a geometric context. Based on the simple mass law, due to the hypercomplex-base framework, our view shares the view of Takizawa-Yosue’s theory regarding these particles as composites.