Chapter 36: The Koide Relation and Mass Hierarchies

The three generations of fermions differ by mass—and the mass ratios are not random. The Koide formula, an unexplained empirical regularity of the Standard Model, states that (m_e + m_μ + m_τ)/(m_e + m_μ

• m_τ)^2 = 2/3 to remarkable precision. This chapter derives the Koide relation from character mode structure on 𝕃 = S¹ ∨ S¹, develops the quark mass hierarchies from ι_τ scaling, and connects CKM and PMNS mixing to the geometry of generation overlap on T².