solidAngle

formal source

 113/-! ## Part 3: Why 4π Specifically -/
 114
 115/-- The solid angle is defined as 4π in D=3. -/
 116noncomputable def solidAngle : ℝ := 4 * Real.pi
 117
 118/-- The solid angle equals the sphere surface area. -/
 119theorem solidAngle_is_sphere_area : solidAngle = unitSphereSurface 3 := by
 120  unfold solidAngle
 121  rw [unitSphereSurface_D3]
 122
 123/-! ## Part 4: Why No Other Factor Works -/
 124
 125/-- 2π is the circumference of a circle (D=2), not the surface of a sphere (D=3). -/
 126theorem two_pi_not_D3 : 2 * Real.pi ≠ 4 * Real.pi := by
 127  intro h
 128  have hpi_ne : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
 129  have : (2 : ℝ) = 4 := by
 130    have := mul_right_cancel₀ hpi_ne h
 131    linarith
 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 :