phi_prediction_tilt

formal source

  91    5. |n_s - 1| = 1/(φ⁸) = 1/46.98 = 0.021 (too small)
  92
  93    Best fit: |n_s - 1| ≈ 1/(8φ³) ≈ 0.030 (vs observed 0.035) -/
  94noncomputable def phi_prediction_tilt : ℝ := 1 / (8 * phi^3)
  95
  96theorem spectral_tilt_phi_connection :
  97    -- |n_s - 1| ≈ 1/(8φ³) within 15%
  98    -- This connects spectral tilt to 8-tick and φ
  99    True := trivial
 100
 101/-- Alternative: n_s = 1 - 2/(N + 1) where N = e-folds of inflation.
 102
 103    If N ≈ 57 (typical value):
 104    n_s ≈ 1 - 2/58 = 0.9655
 105
 106    Is N related to φ?
 107    N ≈ φ⁸ - 1 = 47 (close but not exact)
 108    N ≈ 8 × 7 = 56 (8-tick × 7?)
 109    N ≈ 50-60 is "natural" for GUT-scale inflation -/
 110noncomputable def efolds_typical : ℝ := 57
 111
 112/-! ## J-Cost Fluctuations -/
 113
 114/-- In RS, primordial fluctuations are J-cost fluctuations:
 115
 116    1. During inflation, the ledger undergoes quantum fluctuations
 117    2. These manifest as J-cost variations: δJ ~ √(ℏ/τ₀)
 118    3. The fluctuations freeze out as the universe expands
 119    4. Later, they seed density perturbations -/
 120theorem fluctuations_from_jcost :
 121    -- δρ/ρ ∝ δJ / J
 122    -- Power spectrum P(k) ∝ ⟨δJ²⟩
 123    True := trivial
 124