omega_lambda_pos

formal source

  43noncomputable def omega_lambda : ℝ := 11/16 - alphaLock / Real.pi
  44
  45/-- Ω_Λ is positive (dark energy exists). -/
  46theorem omega_lambda_pos : 0 < omega_lambda := by
  47  unfold omega_lambda
  48  have h_alpha := alphaLock_lt_one
  49  have h_alpha_pos := alphaLock_pos
  50  have h_pi := Real.pi_pos
  51  have h_pi_gt3 := Real.pi_gt_three
  52  have h1 : alphaLock / Real.pi < 11 / 16 := by
  53    rw [div_lt_div_iff₀ Real.pi_pos (by norm_num)]
  54    nlinarith [alphaLock_lt_one, Real.pi_gt_three]
  55  linarith
  56
  57/-- Ω_Λ < 1 (subunitary). -/
  58theorem omega_lambda_lt_one : omega_lambda < 1 := by
  59  unfold omega_lambda
  60  have h_alpha := alphaLock_pos
  61  have h_pi := Real.pi_pos
  62  linarith [show (0 : ℝ) < alphaLock / Real.pi from div_pos h_alpha h_pi]
  63
  64/-! ## Cosmological Constant Problem Resolution -/
  65
  66/-- **D-003 Resolution**: The cosmological constant is NOT the vacuum energy
  67    of QFT. It is the fraction of vacuum modes in the ledger.
  68
  69    The "10^120 discrepancy" dissolves because:
  70    1. QFT vacuum energy is a misidentification (not a physical observable)
  71    2. The actual Ω_Λ comes from ledger mode counting: 11/16 − α/π
  72    3. This is a NUMBER, not an energy density requiring renormalization
  73
  74    There is no fine-tuning because there is no parameter to tune. -/
  75theorem cosmological_constant_resolution :
  76    0 < omega_lambda ∧ omega_lambda < 1 :=