= Solution
The <SU(5) grand unified theory> embeds the strong and electroweak groups in a single simple gauge group. Its 24 gauge generators contain the eight <gluons>, three weak generators and one <hypercharge> generator, plus twelve additional generators connecting color and weak directions. A common <gauge coupling> applies at the unification scale with the properly normalized <hypercharge> generator; low-energy couplings differ through their <renormalization group> evolution.
Using left-chiral fields throughout, one <Standard Model> generation fits into
$$
\overline{\mathbf5}=(d_R^c,L_L),\qquad
\mathbf{10}=(Q_L,u_R^c,e_R^c).
$$
The dimensions are $3+2=5$ and $6+3+1=10$, so no <right-handed neutrino> is required in these multiplets. A singlet neutrino can be added. A suitable adjoint scalar vacuum breaks SU(5) to the <Standard Model> subgroup; a fundamental scalar contains an electroweak doublet and a colored triplet. The extra heavy vector <bosons> can mediate transitions between <quarks> and <leptons>, hence <proton decay> in the minimal model. This is a proposed unification, not an experimentally established identification of new particles.
The <hierarchy problem> concerns keeping an elementary weak-scale scalar light in the presence of much heavier physical scales. A heavy state coupled to it generically induces a threshold correction of order $\delta m_H^2\sim cM^2/(16\pi^2)$, with $c$ a product of couplings. Without protection, a weak-scale result then needs cancellation against a much larger parameter. In SU(5), this is sharpened by <doublet-triplet splitting>: the electroweak doublet must stay light while its colored partner must be sufficiently heavy to avoid rapid <proton decay>. Choosing a small tree-level doublet <mass> is not by itself a mechanism protecting it from subsequent heavy-particle corrections. <Supersymmetry> and other proposals change this protection problem, but the minimal SU(5) embedding alone does not solve it.
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