cube_vertices

formal source

  60/-! ## Cube Topology Constants -/
  61
  62/-- Number of vertices in a D-cube: V = 2^D -/
  63def cube_vertices (D : ℕ) : ℕ := 2^D
  64
  65/-- Number of edges in a D-cube: E = D · 2^(D-1) -/
  66def cube_edges_count (D : ℕ) : ℕ := D * 2^(D-1)
  67
  68/-- Number of faces in a D-cube: F = D(D-1) · 2^(D-2) for D ≥ 2 -/
  69def cube_faces (D : ℕ) : ℕ :=
  70  match D with
  71  | 0 => 0
  72  | 1 => 0
  73  | 2 => 4  -- square has 4 edges (faces in 2D)
  74  | 3 => 6  -- cube has 6 faces
  75  | n+4 => (n+4) * (n+3) * 2^(n+2)
  76
  77theorem cube3_vertices : cube_vertices 3 = 8 := by native_decide
  78theorem cube3_edges : cube_edges_count 3 = 12 := by native_decide
  79theorem cube3_faces : cube_faces 3 = 6 := rfl
  80
  81/-! ## Derivation of the Coefficient 6 (Atmospheric) -/
  82
  83/-- The atmospheric correction coefficient is the face count of a 3-cube.
  84
  85    **Physical interpretation**: Each of the 6 faces of the cubic ledger
  86    contributes one unit of vacuum polarization to the atmospheric mixing.
  87    The μ-τ sector, being maximally mixed (sin²θ₂₃ ≈ 1/2), receives a
  88    symmetric correction from all faces. -/
  89def atmospheric_coefficient : ℕ := cube_faces 3
  90
  91theorem atmospheric_coefficient_eq_6 : atmospheric_coefficient = 6 := rfl
  92
  93/-- The atmospheric radiative correction is 6α. -/