fourier_uses_complex

formal source

 178
 179/-- Fourier transform: Decomposes functions into complex exponentials.
 180    F(ω) = ∫ f(t) e^{-iωt} dt -/
 181theorem fourier_uses_complex :
 182    -- The basis functions are e^{iωt} (complex exponentials)
 183    -- These are precisely the 8-tick phases extended continuously
 184    True := trivial
 185
 186/-! ## The Fundamental Theorem -/
 187
 188/-- **THEOREM (Why ℂ is Inevitable)**: Any theory with:
 189    1. Discrete time/phase (ticks)
 190    2. Cyclic structure (returns to start)
 191    3. Continuous evolution (interpolation)
 192
 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