The origin of three generations of fermions in the Standard Model remains one of the most profound unanswered questions in fundamental physics. While the empirical fact of three generations is well-established, no first-principles derivation of this number exists within the Standard Model itself — it is simply an input parameter. This paper presents a rigorous mathematical theorem — the Representation Splitting Theorem — which derives the existence of exactly three generations from the geometry of the quantum flag manifold Oq (SU (3) /T). The theorem is built upon the Branching Theorem of Aldenhoven–Koelink–Román (2017): the restriction of the fundamental representation of Uq (su (3) ) to the right coideal subalgebra B (the function algebra of the quantum flag manifold) decomposes multiplicity-freely into exactly three irreducible one-dimensional representations: 3 ₁ = V₁ V₂ V₃ The theorem demonstrates that three generations are not an empirical coincidence but a necessary consequence of the noncommutative geometry of the quantum flag manifold. The classical limit q 1 recovers the familiar three-dimensional fundamental representation, explaining why the three generations become indistinguishable at low energies.
Zheng Xinyu (Fri,) studied this question.