CosmologicalPredictionsCert

formal source

 133    **Phi powers**: φ², φ⁴, φ⁵ bounds for various predictions
 134    
 135    **All from φ with rigorous bounds.** -/
 136structure CosmologicalPredictionsCert where
 137  spectral_index_gt : (0.5 : ℝ) < 1 - 2 / (phi ^ 3)
 138  spectral_index_lt : 1 - 2 / (phi ^ 3) < 1
 139  hubble_pos : Real.log phi / 8 > 0
 140  phi_sq_lower : (2.59 : ℝ) < phi^2
 141  phi_sq_upper : phi^2 < (2.62 : ℝ)
 142  phi_fourth_lower : (6.7 : ℝ) < (phi : ℝ)^(4 : ℕ)
 143  phi_fourth_upper : (phi : ℝ)^(4 : ℕ) < (6.9 : ℝ)
 144  phi_fifth_lower : (10.7 : ℝ) < (phi : ℝ)^(5 : ℕ)
 145  phi_fifth_upper : (phi : ℝ)^(5 : ℕ) < (11.3 : ℝ)
 146
 147theorem cosmological_predictions_cert_exists : ∃ _ : CosmologicalPredictionsCert, True := by
 148  have h_hubble : Real.log phi / 8 > 0 := by
 149    have h1 : Real.log phi > 0 := by
 150      apply Real.log_pos
 151      exact one_lt_phi
 152    positivity
 153  refine ⟨⟨spectral_index_gt_half, spectral_index_lt_one,
 154          h_hubble,
 155          phi_squared_bounds.1, phi_squared_bounds.2,
 156          phi_fourth_bounds.1, phi_fourth_bounds.2,
 157          phi_fifth_bounds.1, phi_fifth_bounds.2⟩, trivial⟩
 158
 159/-! ## Summary -/
 160
 161/-- **SUMMARY**: Cosmological predictions with calculated proofs:
 162    
 163    1. EU-003: n_s = 1 - 2/φ³ with 0.5 < n_s < 1
 164    2. T-001: H₀ > 0 from ln(φ)/8
 165    3. φ² bounds: 2.59 < φ² < 2.62
 166    4. φ⁴ bounds: 6.7 < φ⁴ < 6.86