IndisputableMonolith.Physics.InflationEfoldsFromGap45

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Inflation E-fold Count from Gap-45 — A2 Inflation Depth
   6
   7RS prediction: N_e = gap(3) - 1 = 44 e-folds during inflation.
   8
   9This gives:
  10- n_s = 1 - 2/N_e = 1 - 2/44 = 1 - 1/22 ≈ 0.9545
  11  (compared to Planck 0.965 — 1 - 2/45 = 0.956 is closer)
  12- r = 12/N_e² = 12/44² ≈ 0.0062
  13
  14Also: tensor-to-scalar ratio r = 2/(45φ²) ∈ (0.015, 0.020) from TensorToScalarRatioFromRS.lean.
  15
  16Lean: N_e = 44 = gap-45 - 1, N_e × (N_e + 1) = 44 × 45 = 1980.
  17
  18Lean status: 0 sorry, 0 axiom.
  19-/
  20
  21namespace IndisputableMonolith.Physics.InflationEfoldsFromGap45
  22
  23/-- E-fold count from RS: N_e = 44 = gap-45 - 1. -/
  24def Nefolds : ℕ := 44
  25def gap45 : ℕ := 45
  26
  27theorem Nefolds_gap45_minus_one : Nefolds = gap45 - 1 := by decide
  28theorem Nefolds_times_gap45 : Nefolds * gap45 = 1980 := by decide
  29
  30/-- Spectral index n_s = 1 - 2/N_e. -/
  31noncomputable def nS_RS : ℝ := 1 - 2 / (Nefolds : ℝ)
  32
  33theorem nS_RS_val : nS_RS = 1 - 2 / 44 := by
  34  unfold nS_RS Nefolds; norm_cast
  35
  36/-- n_s ≈ 0.954. -/
  37theorem nS_RS_band :
  38    (0.95 : ℝ) < nS_RS ∧ nS_RS < 0.96 := by
  39  rw [nS_RS_val]; norm_num
  40
  41structure InflationEfoldCert where
  42  efolds_eq : Nefolds = gap45 - 1
  43  nS_band : (0.95 : ℝ) < nS_RS ∧ nS_RS < 0.96
  44
  45noncomputable def inflationEfoldCert : InflationEfoldCert where
  46  efolds_eq := Nefolds_gap45_minus_one
  47  nS_band := nS_RS_band
  48
  49end IndisputableMonolith.Physics.InflationEfoldsFromGap45
  50