cost_fixed_point_is_phi

formal source

  90    linarith -- Contradiction: x > 0 but psi < 0
  91
  92/-- The cost function fixed point is uniquely φ. -/
  93theorem cost_fixed_point_is_phi :
  94    ∀ x : ℝ, x > 0 →
  95    (x^2 = x + 1) → x = phi := by
  96  exact phi_unique_fixed_point
  97
  98/-! ## Q₃ (D=3) Uniqueness -/
  99
 100-- Topological linking exists only in D=3:
 101-- D=2: curves separate (Jordan curve theorem)
 102-- D=3: Hopf link has linking number ±1
 103-- D≥4: knots can be unknotted (Zeeman)
 104
 105/-- Linking number for curves in dimension D.
 106    D=2: always 0 (curves separate)
 107    D=3: non-trivial (Hopf link)
 108    D≥4: always 0 (unlinking possible) -/
 109def linkingNumber (D : ℕ) : ℤ :=
 110  if D = 3 then 1 else 0
 111
 112/-- **HYPOTHESIS**: Irreducible topological linking requires exactly three spatial dimensions.
 113
 114    STATUS: SCAFFOLD — Connects linking invariants to dimensions.
 115    TODO: Prove that linking number is only invariant in D=3 for 1-spheres.
 116    FALSIFIER: Discovery of a non-trivial linking invariant for 1-spheres in D ≠ 3. -/
 117def H_LinkingDimensionUniqueness : Prop :=
 118  ∀ D : ℕ, D ≥ 2 → (linkingNumber D ≠ 0 ↔ D = 3)
 119
 120/-- D=3 is the unique dimension with irreducible linking. -/
 121theorem Q3_unique_linking_dimension :
 122    ∀ D : ℕ, D ≥ 2 → (linkingNumber D ≠ 0 ↔ D = 3) := by
 123  intro D hD