three_generations

formal source

  81deriving DecidableEq, Repr
  82
  83/-- Number of generations is exactly 3. -/
  84theorem three_generations : (List.length [Generation.first, Generation.second, Generation.third]) = 3 := rfl
  85
  86/-! ## The 3 from 8 = 2³ Argument -/
  87
  88/-- The key insight: 8 = 2³ gives us 3 "independent directions" in tick-space.
  89    Each direction corresponds to one generation. -/
  90def dimensionsFromTicks : ℕ := 3  -- log₂(8) = 3
  91
  92/-- **THEOREM**: The number of dimensions equals log₂(8). -/
  93theorem dimensions_from_log : dimensionsFromTicks = Nat.log 2 8 := by
  94  native_decide
  95
  96/-- The correspondence:
  97    - Dimension 0 (x) ↔ Generation 1 (lightest)
  98    - Dimension 1 (y) ↔ Generation 2 (medium)
  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