cube_vertices

formal source

  48def D : ℕ := 3
  49
  50/-- Number of vertices in the D-hypercube: 2^D. -/
  51def cube_vertices (d : ℕ) : ℕ := 2^d
  52
  53/-- Number of edges in the D-hypercube: D · 2^(D-1). -/
  54def cube_edges (d : ℕ) : ℕ := d * 2^(d - 1)
  55
  56/-- Number of faces in the D-hypercube: 2D. -/
  57def cube_faces (d : ℕ) : ℕ := 2 * d
  58
  59/-- For D=3: vertices = 8 -/
  60theorem vertices_at_D3 : cube_vertices D = 8 := by native_decide
  61
  62/-- For D=3: edges = 12 -/
  63theorem edges_at_D3 : cube_edges D = 12 := by native_decide
  64
  65/-- For D=3: faces = 6 -/
  66theorem faces_at_D3 : cube_faces D = 6 := by native_decide
  67
  68/-! ## Part 2: Active vs Passive Edge Counting -/
  69
  70/-- Active edges per atomic tick (one edge transition per tick by T2). -/
  71def active_edges_per_tick : ℕ := 1
  72
  73/-- Passive (field) edges: total edges minus active edge.
  74    These are the edges that "dress" the interaction. -/
  75def passive_field_edges (d : ℕ) : ℕ := cube_edges d - active_edges_per_tick
  76
  77/-- The key number: for D=3, passive edges = 11. -/
  78theorem passive_edges_at_D3 : passive_field_edges D = 11 := by native_decide
  79
  80/-! ## Part 3: Structural Derivation of 4π from Q₃ Gauss-Bonnet
  81