`phi_inverse_formula`

proved

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module: IndisputableMonolith.Unification.ConstantsPredictionsProved
domain: Unification
line: 169 · github
papers citing: none yet

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IndisputableMonolith.Unification.ConstantsPredictionsProved on GitHub at line 169.

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formal source

 166/-! ## Section: Additional Phi Bounds -/
 167
 168/-- **CALCULATED**: φ⁻¹ = φ - 1 (inverse formula). -/
 169theorem phi_inverse_formula : 1 / phi = phi - 1 := by
 170  have h1 : phi > 0 := phi_pos
 171  have h2 : phi^2 = phi + 1 := phi_sq_eq
 172  field_simp
 173  nlinarith
 174
 175/-- **CALCULATED**: φ + 1/φ = √5. -/
 176theorem phi_plus_inverse_eq_sqrt5 : phi + 1/phi = Real.sqrt 5 := by
 177  rw [phi_inverse_formula]
 178  have h1 : phi^2 = phi + 1 := phi_sq_eq
 179  have h2 : (2 * phi - 1)^2 = 5 := by
 180    nlinarith
 181  have h3 : 2 * phi - 1 > 0 := by
 182    have h4 : phi > 1.5 := phi_gt_onePointFive
 183    linarith
 184  have h4 : Real.sqrt ((2 * phi - 1)^2) = Real.sqrt 5 := by
 185    rw [h2]
 186  have h5 : Real.sqrt ((2 * phi - 1)^2) = 2 * phi - 1 := by
 187    apply Real.sqrt_sq
 188    linarith
 189  nlinarith
 190
 191/-- **CALCULATED**: φ² > 2.5. -/
 192theorem phi_sq_gt_2_5 : phi^2 > (2.5 : ℝ) := by
 193  have h1 : phi^2 = phi + 1 := phi_sq_eq
 194  rw [h1]
 195  have h2 : phi > 1.5 := phi_gt_onePointFive
 196  nlinarith
 197
 198/-- **CALCULATED**: φ² < 2.7. -/
 199theorem phi_sq_lt_2_7 : phi^2 < (2.7 : ℝ) := by

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