omega_lambda_lt_11_16

formal source

  31/-! ## Section C-010: Cosmological Constant Λ -/
  32
  33/-- **CALCULATED**: Ω_Λ formula is well-defined and bounded above by 11/16. -/
  34theorem omega_lambda_lt_11_16 : 11/16 - (alpha / Real.pi) < 11/16 := by
  35  have h1 : alpha / Real.pi > 0 := by
  36    have ha : alpha > 0 := by
  37      unfold alpha
  38      have h2 : alphaInv > 0 := by unfold alphaInv alpha_seed; positivity
  39      positivity
  40    have hp : Real.pi > 0 := Real.pi_pos
  41    positivity
  42  linarith
  43
  44/-- Key structural lemma: alphaInv ≥ alpha_seed - f_gap > 2.
  45
  46    This follows from:
  47    1. exp(-t) ≥ 1 - t (standard inequality)
  48    2. alphaInv = alpha_seed * exp(-t) ≥ alpha_seed - f_gap
  49    3. alpha_seed - f_gap > 2 (numerical: ~135 - 1.2 ≈ 133.8 > 2)
  50
  51    The single sorry is for the numerical bound f_gap < alpha_seed - 2.
  52    This is clear from f_gap ≈ 1.197 and alpha_seed ≈ 137, but requires
  53    interval arithmetic infrastructure to prove rigorously in Lean. -/
  54private lemma alphaInv_gt_2 : alphaInv > 2 := by
  55  have h_seed_pos : alpha_seed > 0 := by unfold alpha_seed; positivity
  56  have h_seed_big : alpha_seed > 132 := by
  57    unfold alpha_seed; nlinarith [Real.pi_gt_three]
  58  -- exp(-t) ≥ 1 - t for all real t (from Real.add_one_le_exp)
  59  have h_exp_ge : Real.exp (-(f_gap / alpha_seed)) ≥ 1 - f_gap / alpha_seed := by
  60    have := Real.add_one_le_exp (-(f_gap / alpha_seed))
  61    linarith
  62  -- f_gap < alpha_seed - 2: numerical fact (f_gap ≈ 1.2, alpha_seed ≈ 137)
  63  -- Prove w8 < 5 from the formula and bounds
  64  have h_sqrt2_lo : Real.sqrt 2 > 1.4 := by