z_relative_error

formal source

 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
 121noncomputable def w_pred : ℝ := z_pred * cos_theta_W_rs
 122
 123/-- The W/Z mass ratio equals cos(θ_W) by construction. -/
 124theorem wz_ratio_eq_cos : w_pred / z_pred = cos_theta_W_rs := by
 125  unfold w_pred
 126  have hzne : z_pred ≠ 0 := ne_of_gt (by linarith [z_mass_bounds.1])
 127  exact mul_div_cancel_left₀ _ hzne
 128
 129/-! ## Summary -/
 130
 131/-- Electroweak verification certificate. -/
 132structure EWCert where
 133  z_in_range : (91075.09 : ℝ) < z_pred ∧ z_pred < 91075.10
 134  z_error : |z_pred - m_Z_exp| / m_Z_exp < 0.0013
 135  wz_is_cos : w_pred / z_pred = cos_theta_W_rs
 136
 137theorem ew_cert_exists : Nonempty EWCert :=
 138  ⟨{ z_in_range := z_mass_bounds
 139     z_error := z_relative_error