unique_gauge_factorization

formal source

 335    where a = D!, b = 2^(D-1), c = 2 (and D = 3) is 6 × 4 × 2.
 336
 337    This means the gauge group decomposition is forced by the cube geometry. -/
 338theorem unique_gauge_factorization :
 339    ∀ a b c : ℕ,
 340      a * b * c = 48 →
 341      a = Nat.factorial 3 →
 342      (∃ k, b = 2 ^ k ∧ k + 1 = 3) →
 343      c = 2 →
 344      a = 6 ∧ b = 4 ∧ c = 2 := by
 345  intro a b c habc ha hb hc
 346  subst ha; subst hc
 347  obtain ⟨k, hbk, hk3⟩ := hb
 348  have hk : k = 2 := by omega
 349  subst hk
 350  simp at hbk
 351  subst hbk
 352  norm_num at habc ⊢
 353
 354/-- **THEOREM (No Alternative Gauge Groups)**:
 355    The cube Q₃ does not support gauge groups with fundamental
 356    representation dimensions other than (3, 2, 1).
 357
 358    In particular:
 359    - (4, 2, 1) would require D = 4, but D = 3 is forced
 360    - (3, 3, 1) would require |even sign flips| = 6, but it's 4
 361    - (3, 2, 2) would require |parity| = 4, but it's 2 -/
 362theorem no_alternative_321 :
 363    ¬(color_layer.fund_rep_dim = 4) ∧
 364    ¬(weak_layer.fund_rep_dim = 3) ∧
 365    ¬(hypercharge_layer.fund_rep_dim = 2) := by
 366  simp [color_layer, weak_layer, hypercharge_layer]
 367
 368/-! ## Part 8: The Complete Gauge Structure -/