three_layer_factorization

formal source

 213    48 = 6 × 4 × 2
 214
 215    Each factor corresponds to one layer of the gauge group. -/
 216theorem three_layer_factorization :
 217    Fintype.card (SignedPerm 3) =
 218    axis_perm_count 3 * even_sign_flip_count 3 * parity_quotient_order := by
 219  rw [cube_aut_order]
 220  native_decide
 221
 222/-- The factorization in terms of the Standard Model structure. -/
 223theorem sm_factorization :
 224    (48 : ℕ) = 6 * 4 * 2 := by norm_num
 225
 226/-! ## Part 5: Gauge Rank Correspondence -/
 227
 228/-- The **gauge rank** of a layer: the number of independent generators.
 229
 230    For a Lie group of rank r:
 231    - SU(n) has rank n - 1, acts on ℂⁿ (fundamental rep dimension n)
 232    - U(1) has rank 1
 233
 234    The cube layers provide the FUNDAMENTAL REPRESENTATION DIMENSIONS:
 235    - S₃ acts on 3 axes → fundamental rep dimension 3 → SU(3)
 236    - (ℤ/2ℤ)² acts on 2-element subsets → fundamental rep dimension 2 → SU(2)
 237    - ℤ/2ℤ acts on parity → fundamental rep dimension 1 → U(1) -/
 238structure GaugeLayer where
 239  name : String
 240  fund_rep_dim : ℕ
 241  discrete_order : ℕ
 242
 243/-- The three gauge layers extracted from B₃. -/
 244def color_layer : GaugeLayer :=
 245  { name := "SU(3) color"
 246    fund_rep_dim := 3