lambda_CP_pos

formal source

  36noncomputable def kappa_CP : ℝ := phi ^ (-(9 : ℝ))
  37
  38/-- λ_CP is positive. -/
  39theorem lambda_CP_pos : 0 < lambda_CP := Real.rpow_pos_of_pos phi_pos _
  40
  41/-- κ_CP is positive. -/
  42theorem kappa_CP_pos : 0 < kappa_CP := Real.rpow_pos_of_pos phi_pos _
  43
  44/-- Both CP couplings are positive and less than 1 (phi > 1, negative exponents).
  45    For base > 1, rpow with negative exponent < 1. -/
  46theorem lambda_CP_lt_one : lambda_CP < 1 := by
  47  unfold lambda_CP
  48  have h1 : 1 = phi ^ (0 : ℝ) := (Real.rpow_zero phi).symm
  49  rw [h1]
  50  exact Real.rpow_lt_rpow_of_exponent_lt one_lt_phi (by norm_num : (-(7 : ℝ)) < 0)
  51
  52/-- phi^(-7) > phi^(-9) because -7 > -9 and phi > 1. -/
  53theorem lambda_gt_kappa : kappa_CP < lambda_CP := by
  54  unfold lambda_CP kappa_CP
  55  exact Real.rpow_lt_rpow_of_exponent_lt one_lt_phi (by norm_num : (-(9 : ℝ)) < -(7 : ℝ))
  56
  57theorem lambda_CP_bounds : 0 < lambda_CP ∧ lambda_CP < 1 :=
  58  ⟨lambda_CP_pos, lambda_CP_lt_one⟩
  59
  60/-- κ_CP < 1 (follows from κ < λ < 1). -/
  61theorem kappa_CP_lt_one : kappa_CP < 1 :=
  62  lt_trans lambda_gt_kappa lambda_CP_lt_one
  63
  64theorem kappa_CP_bounds : 0 < kappa_CP ∧ kappa_CP < 1 :=
  65  ⟨kappa_CP_pos, kappa_CP_lt_one⟩
  66
  67/-! ## Baryon Asymmetry Prediction -/
  68
  69/-- The RS prediction for the baryon-to-photon ratio η_B.