phi5_mul_phi5

formal source

 162  rw [← Real.rpow_add phi_pos]; norm_num
 163
 164/-- Helper: φ⁵ · φ⁵ = φ¹⁰ (using rpow_add). -/
 165private lemma phi5_mul_phi5 : phi ^ (5 : ℝ) * phi ^ (5 : ℝ) = phi ^ (10 : ℝ) := by
 166  rw [← Real.rpow_add phi_pos]; norm_num
 167
 168/-- Fibonacci form of κ: κ = 8(5φ + 3).
 169
 170    Via φ⁵ = 5φ + 3 (Fibonacci identity: F₅=5, F₄=3, F₆=8):
 171    κ = F₆ · (F₅ · φ + F₄) = 8 · (5φ + 3). -/
 172theorem kappa_fibonacci_form : kappa_einstein = 8 * (5 * phi + 3) := by
 173  rw [kappa_einstein_eq]
 174  congr 1
 175  rw [show (5 : ℝ) = ((5 : ℕ) : ℝ) by norm_cast, Real.rpow_natCast]
 176  exact phi_fifth_eq
 177
 178/-- Fibonacci form of ℏ: ℏ = 1/(5φ + 3).
 179
 180    Via ℏ = φ⁻⁵ = 1/φ⁵ = 1/(5φ + 3). -/
 181theorem hbar_fibonacci_form : hbar = 1 / (5 * phi + 3) := by
 182  rw [hbar_eq_phi_inv_fifth]
 183  have h5 : phi ^ (5 : ℝ) = 5 * phi + 3 := by
 184    rw [show (5 : ℝ) = ((5 : ℕ) : ℝ) by norm_cast, Real.rpow_natCast]
 185    exact phi_fifth_eq
 186  rw [Real.rpow_neg phi_pos.le, h5]
 187  ring
 188
 189/-- Consistency: κ · ℏ from Fibonacci forms = 8(5φ+3) · 1/(5φ+3) = 8. -/
 190theorem kappa_hbar_fibonacci_consistency :
 191    8 * (5 * phi + 3) * (1 / (5 * phi + 3)) = 8 := by
 192  have h : (5 : ℝ) * phi + 3 ≠ 0 := by linarith [one_lt_phi]
 193  field_simp [h]
 194
 195/-! ## §3  Newton's Constant and Gauss-Bonnet Closure: G · ℏ = 1/π