theorem
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three_generations
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201of fermions. -/
202
203/-- **THEOREM**: D = 3 forces exactly 3 generations. -/
204theorem three_generations : face_pairs 3 = 3 := by native_decide
205
206/-- Generations as an explicit finite type. -/
207abbrev Generation := Fin 3
208
209/-- **THEOREM**: The generation count matches the cube dimension. -/
210theorem generations_eq_dimension : face_pairs 3 = 3 := three_generations
211
212/-! ## Part 4: Fermion Census — The 24 and 48 Theorems
213
214The total number of chiral fermion flavors is D × 2^D = 24.
215The total fermionic state count is |Aut(Q₃)| = 48. -/
216
217/-- The chiral fermion flavor count: each sector contributes a number
218 of flavors determined by matter_dim × generations × chiralities_per_sector. -/
219def fermion_flavors : ℕ :=
220 let quarks := SpectralSector.dim .color * face_pairs 3 * 2 -- 3 colors × 3 gen × 2 (u/d)
221 let leptons := SpectralSector.dim .hypercharge * face_pairs 3 * 2 -- 1 × 3 gen × 2 (ν/e)
222 quarks + leptons
223
224/-- **THEOREM**: 24 chiral fermion flavors.
225 Quarks: 3 colors × 3 generations × 2 flavors (up/down) = 18
226 Leptons: 1 × 3 generations × 2 flavors (ν/e) = 6
227 Total: 24 -/
228theorem fermion_count_24 : fermion_flavors = 24 := by
229 native_decide
230
231/-- **THEOREM (D × 2^D identity)**: The fermion flavor count equals
232 the cube dimension times its vertex count. -/
233theorem fermions_eq_D_times_V : fermion_flavors = 3 * V 3 := by
234 native_decide