power_of_2_forces_D3

formal source

 136theorem eight_tick_is_2_cubed : eight_tick = 2^3 := rfl
 137
 138/-- If 2^D = 8, then D = 3. -/
 139theorem power_of_2_forces_D3 (D : Dimension) (h : 2^D = 8) : D = 3 := by
 140  match D with
 141  | 0 => norm_num at h
 142  | 1 => norm_num at h
 143  | 2 => norm_num at h
 144  | 3 => rfl
 145  | n + 4 =>
 146    have h16 : 2^(n+4) ≥ 16 := by
 147      have : 2^n ≥ 1 := Nat.one_le_pow n 2 (by norm_num)
 148      calc 2^(n+4) = 2^n * 2^4 := by ring
 149        _ ≥ 1 * 16 := by nlinarith
 150        _ = 16 := by ring
 151    rw [h] at h16
 152    norm_num at h16
 153
 154/-- The eight-tick cycle forces D = 3. -/
 155theorem eight_tick_forces_D3 (D : Dimension) :
 156    EightTickFromDimension D = eight_tick → D = 3 := by
 157  intro h
 158  unfold EightTickFromDimension eight_tick at h
 159  exact power_of_2_forces_D3 D h
 160
 161/-! ## The Clifford Algebra / Spinor Argument
 162
 163The spinor argument for D=3 is grounded in Clifford algebra theory:
 164
 1651. **Clifford algebras Cl_D**: The algebra generated by {e₁, ..., e_D} with
 166   eᵢ² = -1 and eᵢeⱼ = -eⱼeᵢ for i ≠ j.
 167
 1682. **Dimension dependence**:
 169   - Cl₁ ≅ ℂ (complex numbers)