 213    **Phi bounds**: 2.5 < φ² < 2.7
 214    
 215    **All from φ with rigorous bounds.** -/
 216structure ConstantsPredictionsCert where
 217  alpha_pos : alpha > 0
 218  alpha_lt : alpha < (0.1 : ℝ)
 219  c_eq_one : c = 1
 220  c_pos : c > 0
 221  k_R_pos : Real.log phi > 0
 222  k_R_lt : Real.log phi < (0.5 : ℝ)
 223  phi_inv : 1 / phi = phi - 1
 224  phi_sqrt5 : phi + 1/phi = Real.sqrt 5
 225  phi_sq_lower : phi^2 > (2.5 : ℝ)
 226  phi_sq_upper : phi^2 < (2.7 : ℝ)
 227
 228theorem constants_predictions_cert_exists : ∃ _ : ConstantsPredictionsCert, True := by
 229  refine ⟨⟨alpha_positive, alpha_lt_0_1,
 230          c_eq_one, c_positive,
 231          boltzmann_analog_positive, boltzmann_analog_lt_half,
 232          phi_inverse_formula, phi_plus_inverse_eq_sqrt5,
 233          phi_sq_gt_2_5, phi_sq_lt_2_7⟩, trivial⟩
 234
 235/-! ## Summary -/
 236
 237/-- **SUMMARY**: Constants with calculated proofs:
 238    
 239    1. C-001: 0 < α < 0.1 (fine-structure constant)
 240    2. C-005: c = 1 (RS-native speed of light)
 241    3. C-006: 0 < ln(φ) < 0.5 (Boltzmann analog)
 242    4. Phi formulas: 1/φ = φ-1, φ+1/φ = √5
 243    5. Phi bounds: 2.5 < φ² < 2.7
 244    
 245    **Proof Methods**: `norm_num`, `nlinarith`, `positivity`, `field_simp`, `Real.log_lt_log`
 246    **All from 1.5 < φ < 1.62 and φ² = φ + 1.** -/

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