why_exactly_three

formal source

 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 :
 142    List.length [Generation.first, Generation.second, Generation.third] ≠ 4 := by decide
 143
 144/-! ## Mixing Between Generations -/
 145
 146/-- The CKM matrix elements encode how generations "talk" to each other.
 147    In RS, this comes from the overlap of 8-tick phases. -/
 148structure CKMElement where
 149  /-- From generation (1, 2, or 3). -/
 150  fromGen : Fin 3
 151  /-- To generation (1, 2, or 3). -/
 152  toGen : Fin 3
 153  /-- Mixing amplitude (complex). -/
 154  amplitude : ℂ
 155
 156/-- **THEOREM (CKM from Phase Overlap)**: The CKM matrix elements come from
 157    the overlap of 8-tick phase patterns between generations. -/
 158theorem ckm_from_phase_overlap :
 159    Fintype.card (Fin 3 × Fin 3) = 9 := by decide
 160
 161/-- The Cabibbo angle θ_C ≈ 13° emerges from φ-structure. -/
 162noncomputable def cabibboAngle : ℝ := Real.arcsin (1/phi^2)  -- Approximately correct
 163
 164/-! ## Neutrino Generations -/