`phi5_eq`

proved

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module: IndisputableMonolith.Cosmology.InflatonPotentialStructural
domain: Cosmology
line: 44 · github
papers citing: none yet

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IndisputableMonolith.Cosmology.InflatonPotentialStructural on GitHub at line 44.

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formal source

  41noncomputable def slowRollEta : ℝ := 1 / phi ^ 5
  42
  43/-- φ⁵ = 5φ + 3 (Fibonacci identity). -/
  44theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
  45  have h2 := phi_sq_eq
  46  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  47  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  48  nlinarith
  49
  50theorem slowRollEpsilon_pos : 0 < slowRollEpsilon := by
  51  unfold slowRollEpsilon
  52  apply div_pos one_pos
  53  exact mul_pos (by norm_num) (pow_pos phi_pos 5)
  54
  55theorem slowRollEta_pos : 0 < slowRollEta := by
  56  unfold slowRollEta
  57  exact div_pos one_pos (pow_pos phi_pos 5)
  58
  59/-- n_s - 1 = -2/45 gives n_s ∈ (0.955, 0.957). -/
  60theorem spectralIndex_band :
  61    ((0.955 : ℝ) < 1 - 2/45) ∧ (1 - 2/45 < (0.957 : ℝ)) := by
  62  refine ⟨?_, ?_⟩ <;> norm_num
  63
  64structure InflatonCert where
  65  five_regimes : Fintype.card InflatonRegime = 5
  66  efolds : efoldCount = 44
  67  phi5_fibonacci : phi ^ 5 = 5 * phi + 3
  68  epsilon_pos : 0 < slowRollEpsilon
  69  eta_pos : 0 < slowRollEta
  70  spectral_index_in_band : ((0.955 : ℝ) < 1 - 2/45) ∧ (1 - 2/45 < (0.957 : ℝ))
  71
  72noncomputable def inflatonCert : InflatonCert where
  73  five_regimes := inflatonRegime_count
  74  efolds := efoldCount_eq

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