Uniqueness of SU(5) and SO(10) grand unified theories

We will prove that SU(5) and SO(10) grand unified theories are almost uniquely chosen under a few reasonable assumptions. The main ansatz is that there exist only left-handed particles in an SU(2) doublet and their antiparticles in an SU(2) singlet. For the case of a single-multiplet unification, SO(10) is unique, using the facts that the charge of the $d$ quark is not neutral and the theory should be anomaly free. For the case of two-multiplet unification, SU(5) is unique among $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ ($N\le 500$). The color group is effectively determined as the standard SU(3) with triplets of colored quarks. Quark charges must be the standard $\frac{2}{3}$ and $-\frac{1}{3}$. No unifying simple group exists for $G\left(\mathrm{horizontal}\right) \times G\left(\mathrm{GUT}\right)$ under the assumptions above. For the case of SU(5), the cancellation of the triangle anomaly emerges as a consequence rather than a hypothesis.