euler_from_8tick

formal source

 153    - Complex numbers (rotation)
 154
 155    All unified by the 8-tick phase structure! -/
 156theorem euler_from_8tick :
 157    ∀ θ : ℝ, cexp (I * θ) = Complex.cos θ + I * Complex.sin θ := by
 158  intro θ
 159  rw [Complex.exp_mul_I]
 160  ring
 161
 162/-- e^{iπ} + 1 = 0 (Euler's identity)
 163
 164    Often called "the most beautiful equation."
 165    In RS, it says: 4 ticks around the circle returns to -1. -/
 166theorem euler_identity : cexp (I * Real.pi) + 1 = 0 := by
 167  rw [Complex.exp_mul_I, Complex.cos_pi, Complex.sin_pi]
 168  simp
 169
 170/-! ## Deep Implications -/
 171
 172/-- The 8-tick structure explains:
 173
 174    1. **Why i² = -1**: Half-way around the phase circle
 175    2. **Why waves oscillate**: Phase accumulation
 176    3. **Why QM is unitary**: Phase-preserving evolution
 177    4. **Why fermions get sign flip**: π rotation (4 ticks) = -1 -/
 178def implications : List String := [
 179  "i² = -1 from 4 ticks = π rotation",
 180  "Waves from continuous phase accumulation",
 181  "Unitarity from phase conservation",
 182  "Fermion sign from 2π rotation = 8 ticks = 1"
 183]
 184
 185/-! ## The 8th Roots of Unity -/
 186