face_solid_angle_sum

formal source

 142  simp only [per_face_solid_angle, solid_angle_Q3_eq, h6]; ring
 143
 144/-- Face solid angles reconstruct the total: 6 × (2π/3) = 4π. -/
 145theorem face_solid_angle_sum :
 146    (cube_faces D : ℝ) * per_face_solid_angle = 4 * Real.pi := by
 147  have h6 : (cube_faces D : ℝ) = 6 := by exact_mod_cast faces_at_D3
 148  rw [per_face_solid_angle_eq, h6]; ring
 149
 150/-! ## Part 4: Geometric Seed Assembly -/
 151
 152/-- The geometric seed factor is the passive edge count.
 153    This is WHERE THE 11 COMES FROM. -/
 154def geometric_seed_factor : ℕ := passive_field_edges D
 155
 156/-- Verify: geometric_seed_factor = 11 -/
 157theorem geometric_seed_factor_eq_11 : geometric_seed_factor = 11 := by native_decide
 158
 159/-- The full geometric seed: Ω(∂Q₃) × E_passive.
 160    Both factors derive from Q₃ geometry:
 161    - 4π = Gauss-Bonnet total curvature of ∂Q₃ (Part 3)
 162    - 11 = cube edges − 1 active edge (Part 2) -/
 163noncomputable def geometric_seed : ℝ := solid_angle_Q3 * geometric_seed_factor
 164
 165/-- The geometric seed equals 4π·11. -/
 166theorem geometric_seed_eq : geometric_seed = 4 * Real.pi * 11 := by
 167  unfold geometric_seed
 168  rw [solid_angle_Q3_eq]
 169  simp only [geometric_seed_factor_eq_11, Nat.cast_ofNat]
 170
 171/-- The alpha seed factorizes into solid angle × passive channels,
 172    both derived from Q₃ cube geometry with zero imported constants. -/
 173theorem alpha_seed_structural :
 174    geometric_seed = solid_angle_Q3 * (passive_field_edges D : ℝ) := rfl
 175