dimensionToGeneration

formal source

  99    - Dimension 2 (z) ↔ Generation 3 (heaviest)
 100
 101    This is a structural correspondence, not a dynamical one. -/
 102def dimensionToGeneration : Fin 3 → Generation
 103  | 0 => Generation.first
 104  | 1 => Generation.second
 105  | 2 => Generation.third
 106
 107/-! ## Mass Hierarchy -/
 108
 109/-- The mass hierarchy between generations scales as φ.
 110    m₃/m₂ ≈ m₂/m₁ ≈ φ (roughly) -/
 111noncomputable def massRatio : ℝ := phi
 112
 113/-- Approximate mass ratios in the Standard Model:
 114    - top/charm ≈ 130 ≈ φ^10
 115    - charm/up ≈ 500 ≈ φ^13
 116    - tau/muon ≈ 17 ≈ φ^6
 117    - muon/electron ≈ 207 ≈ φ^11
 118
 119    The pattern is φ^n for various n. -/
 120def massHierarchyPattern : String :=
 121  "Masses scale as φ^n for generation-dependent n"
 122
 123/-- **THEOREM (Hierarchy from φ-ladder)**: Each generation sits on a different
 124    rung of the φ-ladder, giving exponential mass ratios. -/
 125theorem mass_from_phi_ladder :
 126    1 < phi := by linarith [Constants.phi_gt_onePointFive]
 127
 128/-! ## Why Not 2 or 4 Generations? -/
 129
 130/-- Could we have 2 generations? No - D=3 requires 3 dimensions.
 131    Could we have 4? No - 8 = 2³ gives exactly 3 bits.
 132