theorem
proved
tactic proof
cosmological_predictions_cert_exists
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formal statement (Lean)
147theorem cosmological_predictions_cert_exists : ∃ _ : CosmologicalPredictionsCert, True := by
proof body
Tactic-mode proof.
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
167 5. φ⁵ bounds: 10.8 < φ⁵ < 11.1
168
169 **Proof Methods**: `norm_num`, `nlinarith`, `positivity`, `linarith`
170 **All from 1.61 < φ < 1.62 and φ² = φ + 1.** -/