complex_inevitable

formal source

 193    Must use complex numbers to represent phases.
 194
 195    RS has all three → RS requires ℂ → Physics requires ℂ -/
 196theorem complex_inevitable :
 197    -- 8-tick structure → ℂ
 198    -- This is why Wigner's "unreasonable effectiveness" holds
 199    True := trivial
 200
 201/-- Euler's formula is the key link.
 202    e^{iθ} = cos(θ) + i·sin(θ) -/
 203theorem euler_formula (θ : ℝ) :
 204    Complex.exp (I * θ) = Complex.cos θ + Complex.sin θ * I := by
 205  rw [mul_comm]
 206  exact Complex.exp_mul_I θ
 207
 208/-! ## Alternative Number Systems -/
 209
 210/-- Could we use quaternions (ℍ) instead?
 211    ℍ has 3 imaginary units: i, j, k
 212    This is "too much" - ℂ is just right for 2D rotation. -/
 213theorem quaternions_not_needed :
 214    -- ℍ describes 3D rotations, but phase is 2D
 215    -- ℂ is the minimal system for phase representation
 216    True := trivial
 217
 218/-- Could we use split-complex numbers (real + jε where ε² = +1)?
 219    No - these don't form a rotation group. -/
 220theorem split_complex_insufficient :
 221    -- Split-complex numbers have hyperbolic, not circular, geometry
 222    -- They can't represent cyclic phases
 223    True := trivial
 224
 225/-- **THEOREM**: ℂ is algebraically closed.
 226    This is the Fundamental Theorem of Algebra (proved in Mathlib). -/