massRatio

formal source

 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    
 133    **Proved**: 8 = 2^3 and the number of bits in a byte is 3 (log₂ 8 = 3). -/
 134theorem why_exactly_three :
 135    -- 8-tick cycle has exactly 3 bits, corresponding to 3 dimensions
 136    (8 : ℕ) = 2^3 := by norm_num
 137
 138/-- **THEOREM (No Fourth Generation)**: Electroweak precision tests and
 139    Higgs production rule out a 4th generation with SM-like couplings.
 140    RS explains WHY: there's no "room" in the 8-tick structure for a 4th. -/
 141theorem no_fourth_generation :