eight_pi_not_unit

formal source

 132  norm_num at this
 133
 134/-- 8π is the surface of a sphere with radius √2, not the unit sphere. -/
 135theorem eight_pi_not_unit : 8 * Real.pi ≠ 4 * Real.pi := by
 136  intro h
 137  have hpi_ne : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
 138  have : (8 : ℝ) = 4 := by
 139    have := mul_right_cancel₀ hpi_ne h
 140    linarith
 141  norm_num at this
 142
 143/-! ## Part 5: The Complete Exclusivity Proof -/
 144
 145/-- **Main Theorem**: 4π is the unique geometric factor in the α seed for D=3. -/
 146theorem four_pi_unique_for_D3 :
 147    ∀ (factor : ℝ),
 148      (∀ (D : ℕ), D = 3 → factor = unitSphereSurface D) →
 149      factor = 4 * Real.pi := by
 150  intro factor h
 151  have := h 3 rfl
 152  rw [unitSphereSurface_D3] at this
 153  exact this
 154
 155/-- The geometric seed is uniquely determined by the solid angle. -/
 156theorem geometric_seed_eq_solidAngle_times_11 :
 157    geometric_seed = solidAngle * 11 := by
 158  unfold geometric_seed solidAngle
 159  -- geometric_seed = 4 * π * geometric_seed_factor
 160  -- We need: geometric_seed_factor = 11
 161  have h : geometric_seed_factor = 11 := by
 162    unfold geometric_seed_factor passive_field_edges cube_edges D active_edges_per_tick
 163    rfl
 164  simp only [h]
 165  ring