phi_from_fibonacci_ratio

formal source

 116/-! ## Part 5: φ from Fibonacci -/
 117
 118/-- **φ FROM FIBONACCI**: If r > 0 satisfies r² = r + 1, then r = φ. -/
 119theorem phi_from_fibonacci_ratio (r : ℝ) (hr_pos : 0 < r)
 120    (hfib : r ^ 2 = r + 1) : r = phi := by
 121  have hsq5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 5)
 122  have hsqrt5_gt1 : Real.sqrt 5 > 1 := by
 123    have : Real.sqrt 5 > Real.sqrt 1 :=
 124      Real.sqrt_lt_sqrt (by norm_num) (by norm_num : (1 : ℝ) < 5)
 125    simpa [Real.sqrt_one] using this
 126  have hdisc : (r - (1 + Real.sqrt 5) / 2) * (r - (1 - Real.sqrt 5) / 2) = 0 := by
 127    nlinarith [hsq5]
 128  rcases mul_eq_zero.mp hdisc with h | h
 129  · unfold phi; linarith
 130  · exfalso
 131    have h_neg : (1 - Real.sqrt 5) / 2 < 0 := by linarith
 132    linarith
 133
 134/-! ## Part 6: At Global Minimum → Holographic -/
 135
 136/-- At the J-cost global minimum (all edge costs zero), the allocation is
 137    holographic: every vertex has the same field value. -/
 138theorem optimal_at_minimum_is_holographic {V : Type*}
 139    {adj : V → V → Prop}
 140    (hconn : ∀ u v : V, Relation.ReflTransGen adj u v)
 141    {field : V → ℝ} (hpos : ∀ v, 0 < field v)
 142    (at_min : ∀ v w, adj v w → Jcost (field v / field w) = 0) :
 143    ∀ v w : V, field v = field w :=
 144  GraphRigidity.ratio_rigidity hconn hpos at_min
 145
 146/-! ## Part 7: Master Certificate -/
 147
 148/-- **LOCAL CACHE FORCING CERTIFICATE**:
 149    1. J(φ^d) strictly increasing (cost grows with distance)