prediction5

formal source

  82    = 0.618² / 2 = 0.382 / 2 = 0.191
  83
  84    Same as prediction 2. -/
  85noncomputable def prediction5 : ℝ := (phi - 1)^2 / 2
  86
  87/-- **BEST FIT**: sin²(θ_W) = (3 - φ) / 6
  88
  89    Predicted: 0.230
  90    Observed: 0.2229
  91    Error: ~3.2%
  92
  93    This is the most promising φ-connection! -/
  94noncomputable def bestPrediction : ℝ := prediction3
  95
  96theorem best_prediction_close_to_observed :
  97    |bestPrediction - sin2ThetaW_observed| < 0.01 := by
  98  unfold bestPrediction prediction3 sin2ThetaW_observed
  99  -- Need: |(3 - φ)/6 - 0.2229| < 0.01
 100  -- φ > 1.61 → (3 - φ)/6 < 1.39/6 = 0.2317
 101  -- φ < 1.62 → (3 - φ)/6 > 1.38/6 = 0.23
 102  have h_phi_gt : phi > 1.61 := phi_gt_onePointSixOne
 103  have h_phi_lt : phi < 1.62 := phi_lt_onePointSixTwo
 104  have h_pred_gt : (3 - phi) / 6 > 0.23 := by linarith
 105  have h_pred_lt : (3 - phi) / 6 < 0.232 := by linarith
 106  rw [abs_lt]
 107  constructor <;> linarith
 108
 109/-! ## 8-Tick Geometric Derivation -/
 110
 111/-- The 8-tick circle has 8 equally spaced phases at angles kπ/4 for k = 0, 1, ..., 7.
 112
 113    The electroweak embedding uses 3 of these phases for SU(2) and 1 for U(1).
 114    The mixing angle comes from the geometric arrangement.
 115