phi_sq_lt_2_7

formal source

 196  nlinarith
 197
 198/-- **CALCULATED**: φ² < 2.7. -/
 199theorem phi_sq_lt_2_7 : phi^2 < (2.7 : ℝ) := by
 200  have h1 : phi^2 = phi + 1 := phi_sq_eq
 201  rw [h1]
 202  have h2 : phi < 1.62 := phi_lt_onePointSixTwo
 203  nlinarith
 204
 205/-! ## Section: Certificate -/
 206
 207/-- **CERTIFICATE**: Constants predictions with calculated bounds.
 208    
 209    **C-001**: 0 < α < 0.01
 210    **C-005**: c = 1, c > 0
 211    **C-006**: 0 < k_R < 0.5 (k_R = ln(φ))
 212    **Phi formulas**: 1/φ = φ-1, φ+1/φ = √5
 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,