IndisputableMonolith.Physics.SineSqThetaWFromPhiLadder

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Weinberg Angle from Phi-Ladder — A1 SM Depth
   6
   7From §XXIII.D and GaugeBosonSpectrum:
   8  sin²θ_W = (3−φ)/6 ≈ 0.230
   9  
  10PDG: sin²θ_W = 0.2312 (MS-bar scheme at M_Z).
  11RS prediction: 0.230 vs PDG 0.231 — 0.4% agreement.
  12
  13The derivation: from the (3,2,1) rank decomposition,
  14sin²θ_W = g'²/(g² + g'²) = (rank-1)/(rank-1 + rank-2) × corrections.
  15
  16RS formula: sin²θ_W = (3 - φ)/6.
  17Numerically: (3 - 1.618)/6 = 1.382/6 ≈ 0.2303.
  18
  19Lean: prove (3-φ)/6 ∈ (0.228, 0.232).
  20
  21Lean status: 0 sorry, 0 axiom.
  22-/
  23
  24namespace IndisputableMonolith.Physics.SineSqThetaWFromPhiLadder
  25open Constants
  26
  27/-- RS prediction for sin²θ_W. -/
  28noncomputable def sin2thetaW : ℝ := (3 - phi) / 6
  29
  30/-- sin²θ_W ∈ (0.228, 0.232). -/
  31theorem sin2thetaW_band :
  32    (0.228 : ℝ) < sin2thetaW ∧ sin2thetaW < 0.232 := by
  33  unfold sin2thetaW
  34  have h1 := phi_gt_onePointSixOne
  35  have h2 := phi_lt_onePointSixTwo
  36  constructor
  37  · have : (3 - phi) > 3 - 1.62 := by linarith
  38    linarith [div_lt_div_of_pos_right (show (0:ℝ) < 3 - phi by linarith) (by norm_num : (0:ℝ) < 6)]
  39  · have : (3 - phi) < 3 - 1.61 := by linarith
  40    linarith [div_lt_div_of_pos_right (show (3:ℝ) - phi < 3 - 1.61 by linarith) (by norm_num : (0:ℝ) < 6)]
  41
  42/-- PDG value 0.2312 is close to RS prediction. -/
  43def sin2thetaWPDG : ℝ := 0.2312
  44theorem rs_near_pdg : |sin2thetaW - sin2thetaWPDG| < 0.005 := by
  45  unfold sin2thetaW sin2thetaWPDG
  46  rw [abs_lt]
  47  have h1 := phi_gt_onePointSixOne
  48  have h2 := phi_lt_onePointSixTwo
  49  constructor
  50  · have : (3 - phi) / 6 > 0.228 := sin2thetaW_band.1
  51    linarith
  52  · have : (3 - phi) / 6 < 0.232 := sin2thetaW_band.2
  53    linarith
  54
  55structure SineSqThetaWCert where
  56  band : (0.228 : ℝ) < sin2thetaW ∧ sin2thetaW < 0.232
  57  near_pdg : |sin2thetaW - sin2thetaWPDG| < 0.005
  58
  59noncomputable def sineSqThetaWCert : SineSqThetaWCert where
  60  band := sin2thetaW_band
  61  near_pdg := rs_near_pdg
  62
  63end IndisputableMonolith.Physics.SineSqThetaWFromPhiLadder
  64