quaternions_not_needed

formal source

 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). -/
 227theorem complex_is_unique :
 228    -- ℂ is algebraically closed: every polynomial over ℂ has a root in ℂ
 229    IsAlgClosed ℂ := Complex.isAlgClosed
 230
 231/-! ## The RS Interpretation -/
 232
 233/-- In RS, complex numbers arise because:
 234
 235    1. The ledger has 8 ticks (discrete)
 236    2. Ticks are phases (cyclic)
 237    3. Phase differences matter (interference)
 238    4. Phase is additive under composition
 239    5. The unique structure satisfying these is ℂ
 240
 241    Complex numbers aren't a human invention - they're forced by nature! -/
 242theorem complex_from_ledger :
 243    -- 8-tick ledger → cyclic phases → ℂ