PMNSElement

formal source

 165
 166/-- Neutrinos also come in 3 generations (flavors).
 167    The PMNS matrix is the leptonic analog of CKM. -/
 168structure PMNSElement where
 169  /-- From flavor (e, μ, τ). -/
 170  fromFlavor : Fin 3
 171  /-- To mass eigenstate (1, 2, 3). -/
 172  toMass : Fin 3
 173  /-- Mixing amplitude. -/
 174  amplitude : ℂ
 175
 176/-- **THEOREM (Neutrino Generations = Charged Lepton Generations)**:
 177    Both are 3 because both arise from the same 8-tick × 3D structure. -/
 178theorem neutrino_generations_match :
 179    List.length [Generation.first, Generation.second, Generation.third] = 3 := rfl
 180
 181/-! ## Experimental Tests -/
 182
 183/-- Ways to test the 3-generation prediction:
 184    1. Look for 4th generation at colliders (ruled out)
 185    2. Precision measurement of Z → νν̄ (gives N_ν ≈ 3)
 186    3. Check if mass ratios follow φ^n pattern
 187    4. Test CKM/PMNS structure against RS predictions -/
 188def experimentalTests : List String := [
 189  "LEP Z-width: N_ν = 2.984 ± 0.008",
 190  "LHC: No 4th generation quarks up to TeV scale",
 191  "Mass ratios approximately follow φ^n",
 192  "CKM unitarity verified to 10⁻⁴"
 193]
 194
 195/-! ## Falsification Criteria -/
 196
 197/-- The 3-generation derivation would be falsified by:
 198    1. Discovery of a 4th generation