two_pi_not_D3

formal source

 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 :
 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 :