phi51_lt

formal source

  87
  88private lemma phi51_gt : (45537548334 : ℝ) < Constants.phi ^ (51 : ℕ) := by
  89  rw [phi_eq_goldenRatio]; exact Numerics.phi_pow51_gt
  90private lemma phi51_lt : Constants.phi ^ (51 : ℕ) < (45537549354 : ℝ) := by
  91  rw [phi_eq_goldenRatio]; exact Numerics.phi_pow51_lt
  92
  93/-- The Z boson mass prediction lies in (91075.09, 91075.10) MeV. -/
  94theorem z_mass_bounds :
  95    (91075.09 : ℝ) < z_pred ∧ z_pred < (91075.10 : ℝ) := by
  96  rw [z_pred_eq]
  97  constructor
  98  · rw [lt_div_iff₀ (by norm_num : (0 : ℝ) < 1000000)]
  99    calc (91075.09 : ℝ) * 1000000 = (91075090000 : ℝ) := by norm_num
 100      _ < 2 * (45537548334 : ℝ) := by norm_num
 101      _ < 2 * Constants.phi ^ 51 := by nlinarith [phi51_gt]
 102  · rw [div_lt_iff₀ (by norm_num : (0 : ℝ) < 1000000)]
 103    calc 2 * Constants.phi ^ 51 < 2 * (45537549354 : ℝ) := by nlinarith [phi51_lt]
 104      _ = (91075098708 : ℝ) := by norm_num
 105      _ < (91075100000 : ℝ) := by norm_num
 106      _ = (91075.10 : ℝ) * 1000000 := by norm_num
 107
 108/-- The Z boson prediction is within 0.13% of the PDG value. -/
 109theorem z_relative_error :
 110    |z_pred - m_Z_exp| / m_Z_exp < 0.0013 := by
 111  have hb := z_mass_bounds
 112  have hexp_pos : (0 : ℝ) < m_Z_exp := by unfold m_Z_exp; norm_num
 113  rw [div_lt_iff₀ hexp_pos, abs_lt]
 114  unfold m_Z_exp
 115  constructor <;> nlinarith [hb.1, hb.2]
 116
 117/-! ## W Boson Mass — Z × cos(θ_W)
 118
 119m_W = m_Z × cos(θ_W) where cos²(θ_W) = (3+φ)/6 -/
 120