J_bit_eq_phi_minus

formal source

  85    **Note**: This is exact. 1/φ = φ - 1 (from φ² = φ + 1).
  86    So φ + 1/φ = 2φ - 1.
  87    J_bit = (2φ - 1)/2 - 1 = φ - 3/2 ≈ 1.618 - 1.5 = 0.118. -/
  88theorem J_bit_eq_phi_minus : J_bit_val = phi - 3/2 := by
  89  unfold J_bit_val J
  90  -- Key identity: 1/φ = φ - 1 (from φ² = φ + 1)
  91  have h_inv : phi⁻¹ = phi - 1 := by
  92    have hphi_ne : phi ≠ 0 := phi_pos.ne'
  93    have hsq : phi^2 = phi + 1 := phi_sq_eq
  94    have : phi * phi = phi + 1 := by rw [← sq]; exact hsq
  95    field_simp at this ⊢
  96    linarith
  97  rw [h_inv]
  98  ring
  99
 100/-- **Numerical Bound**: J_bit ≈ 0.118.
 101    Since 1.61 < φ < 1.62, we have 0.11 < J_bit < 0.12. -/
 102theorem J_bit_bounds : 0.11 < J_bit_val ∧ J_bit_val < 0.12 := by
 103  rw [J_bit_eq_phi_minus]
 104  constructor
 105  · have h := phi_gt_onePointSixOne
 106    linarith
 107  · have h := phi_lt_onePointSixTwo
 108    linarith
 109
 110/-! ## Part 2: Curvature Cost from Q₃ Geometry -/
 111
 112/-- The number of faces of the D-hypercube (D-cube). F = 2D. -/
 113def cube_faces (D : ℕ) : ℕ := 2 * D
 114
 115/-- The 3-cube Q₃ has 6 faces. -/
 116theorem Q3_faces : cube_faces 3 = 6 := rfl
 117
 118/-- The number of vertices of the D-hypercube. V = 2^D. -/