J_bit_bounds

formal source

  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. -/
 119def cube_vertices (D : ℕ) : ℕ := 2^D
 120
 121/-- The 3-cube Q₃ has 8 vertices (= 8 ticks in the Gray cycle). -/
 122theorem Q3_vertices : cube_vertices 3 = 8 := rfl
 123
 124/-- **Curvature Packet Axiom** (PHYSICAL HYPOTHESIS):
 125
 126A ±4 curvature packet is distributed over the 8 vertices of Q₃.
 127The curvature cost per vertex is proportional to λ²/4.
 128
 129The total curvature cost is then 8 × (λ²/4) = 2λ².
 130
 131This is the curvature functional J_curv(λ). -/
 132noncomputable def J_curv (lam : ℝ) : ℝ := 2 * lam^2