2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/147916One of the interesting features in unification models and supersymmetric unification models is that the chiral states of quarks and leptons in a family including a right-handed neutrino can be fitted neatly into a fundamental spinor representation (f.s) of dimension 16 for the SO(10) gauge group. However, it is shown in this paper that such a fundamental spinor representation of SO(10) for Weyl fermions will generate global (non-perturbative) gauge anomalies (of new type) when restricting to the $SU(2)\otimes SU(2)\otimes SU(2) \otimes SU(2)$ gauge subgroup. Such an example is the four SU(2) factors obtained through the reduction of the subgroup $SU(2)\otimes SO(7)$ of SO(10) with the SO(7) to the three SU(2) factors. The branching rule in this case is given by $(f.s)\rightarrow (2-1-2-2)\oplus (2-2-1-2)$ in terms of dimensions. A consistent gauge theory implies the gauge symmetry in a gauge subgroup, and then needs to be well-defined when restricting to the gauge subgroup. Consequently, a consistent SO(10) quantum theory needs to satisfy our selection rule $N_f+N_{mf}=even\geq 4$, namely the total number of generations with $N_f$ ordinary fermion families and $N_{mf}$ mirror fermion families is even and larger than three, and the three generations of chiral fermions in this content can not correspond to a consistent theory. Then we expect that there exist at least one additional fermion family including a right-handed neutrino or at least one family of mirror fermions including a left-handed mirror neutrino if SO(10) unification theory is relevant to our realistic world. The next possibility is to have total six generations that may subject to stringentLBL-35864,July,1994 LaTeX 20 pages Email:Zhang@Theorm.lbl.gov (current). Final version submittedHigh Energy Physics - Phenomenologytext