IndisputableMonolith.Masses.ElectroweakMasses

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Masses.Anchor
   4import IndisputableMonolith.Masses.Verification
   5import IndisputableMonolith.Numerics.Interval.PhiBounds
   6
   7/-!
   8# Electroweak Boson Mass Predictions
   9
  10## Mechanism
  11
  12The Z boson sits at rung 1 in the Electroweak sector, giving:
  13
  14  m_Z = 2 × φ^51 / 10^6  [MeV]
  15
  16The W boson mass is derived from Z via the Weinberg angle:
  17
  18  m_W = m_Z × cos(θ_W)
  19
  20where sin²θ_W = (3 − φ)/6 is the RS prediction (zero parameters).
  21
  22## Differentiation from the placeholder
  23
  24The original `r_boson` mapped W, Z, H all to rung 1. This module provides
  25the physical differentiation: Z uses rung 1 directly; W uses the gauge
  26relation m_W = m_Z cos θ_W; Higgs is approximately m_W × φ.
  27-/
  28
  29namespace IndisputableMonolith.Masses.ElectroweakMasses
  30
  31open Constants Anchor Verification
  32
  33noncomputable section
  34
  35private lemma phi_eq_goldenRatio : Constants.phi = Real.goldenRatio := by
  36  unfold Constants.phi Real.goldenRatio; ring
  37
  38/-! ## PDG 2024 Experimental Values -/
  39
  40def m_W_exp : ℝ := 80369.2
  41def m_Z_exp : ℝ := 91187.6
  42def m_H_exp : ℝ := 125200
  43
  44/-! ## RS Weinberg Angle
  45
  46sin²θ_W = (3 − φ)/6 — derived from the gauge embedding geometry. -/
  47
  48noncomputable def sin2_theta_W_rs : ℝ := (3 - Constants.phi) / 6
  49noncomputable def cos2_theta_W_rs : ℝ := 1 - sin2_theta_W_rs
  50noncomputable def cos_theta_W_rs : ℝ := Real.sqrt cos2_theta_W_rs
  51
  52theorem cos2_theta_W_rs_eq : cos2_theta_W_rs = (3 + Constants.phi) / 6 := by
  53  unfold cos2_theta_W_rs sin2_theta_W_rs; ring
  54
  55theorem sin2_theta_positive : 0 < sin2_theta_W_rs := by
  56  unfold sin2_theta_W_rs
  57  have hphi : Constants.phi < 2 := by
  58    rw [phi_eq_goldenRatio]; exact Real.goldenRatio_lt_two
  59  linarith
  60
  61theorem sin2_theta_lt_half : sin2_theta_W_rs < 1/2 := by
  62  unfold sin2_theta_W_rs
  63  have hphi : 0 < Constants.phi := phi_pos
  64  linarith
  65
  66theorem cos2_theta_positive : 0 < cos2_theta_W_rs := by
  67  unfold cos2_theta_W_rs; linarith [sin2_theta_lt_half]
  68
  69/-! ## Z Boson Mass — Rung 1 in Electroweak Sector
  70
  71m_Z = rs_mass_MeV(.Electroweak, 1) = 2 × φ^51 / 10^6 -/
  72
  73noncomputable def z_pred : ℝ := rs_mass_MeV .Electroweak 1
  74
  75theorem z_pred_eq : z_pred = 2 * Constants.phi ^ (51 : ℕ) / 1000000 := by
  76  unfold z_pred rs_mass_MeV
  77  simp only [B_pow_Electroweak_eq, r0_Electroweak_eq]
  78  have hphi : Constants.phi ≠ 0 := ne_of_gt phi_pos
  79  have hphi_combine : Constants.phi ^ (-5 : ℤ) * Constants.phi ^ (55 : ℤ) * Constants.phi ^ (1 : ℤ) =
  80      Constants.phi ^ ((51 : ℕ) : ℤ) := by
  81    rw [← zpow_add₀ hphi, ← zpow_add₀ hphi]; norm_num
  82  conv_lhs =>
  83    rw [show (2 : ℝ) ^ (1 : ℤ) * Constants.phi ^ (-5 : ℤ) * Constants.phi ^ (55 : ℤ) * Constants.phi ^ (1 : ℤ)
  84      = (2 : ℝ) ^ (1 : ℤ) * (Constants.phi ^ (-5 : ℤ) * Constants.phi ^ (55 : ℤ) * Constants.phi ^ (1 : ℤ)) from by ring]
  85    rw [hphi_combine, zpow_natCast]
  86  simp only [zpow_one]
  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
 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
 140     wz_is_cos := wz_ratio_eq_cos }⟩
 141
 142end
 143
 144end IndisputableMonolith.Masses.ElectroweakMasses
 145