arXiv2026
We investigate the minimal Yukawa sector of $\mathrm{SO}(10)$ grand unified theories with Higgs representations $\mathbf{10}_{\mathbb{R}}\oplus\mathbf{120}_{\mathbb{R}}\oplus\mathbf{126}$. Taking $\mathbf{10}_{\mathbb{R}}$ and $\mathbf{120}_{\mathbb{R}}$ to be real scalars, we derive the corresponding reality conditions for their weak-doublet components and revisit previously reported fermion mass relations. We find a relative sign difference between the reality constraints on the two weak doublets in $\mathbf{120}_{\mathbb{R}}$, introducing a new magnitude parameter in the mass relations. We establish this result through four complementary approaches: an explicit $\mathrm{SO}(10)$ tensorial calculation of invariants, a Pati-Salam embedding map, an algorithm transporting reality structures from parent to daughter irreps, and an $\mathrm{SU}(5)$ calculation using the $\mathrm{SO}(10)$ oscillator method, all yielding consistent results. The methods for determining reality conditions can be applied to any parent-daughter representation pair of $\mathrm{SO}(10)$ and its Pati-Salam subgroup, while the transport algorithm generalizes to arbitrary groups $H\subset G$. Incorporating the correct mass relations, we perform an extensive numerical scan and find that the model successfully reproduces SM fermion masses and mixings, including recent precision measurements of solar oscillation parameters by JUNO. It accommodates both octants of $θ_{23}$ while mildly disfavoring $δ_\mathrm{PMNS} \sim (140^\circ - 220^\circ)$. The model predicts a strongly hierarchical right-handed neutrino spectrum $(10^{5},10^{12},10^{15})$ GeV and a neutrinoless double beta decay parameter $m_{ββ}\sim 3$-$4$ meV, just below future experimental sensitivity. Proton decay is dominated by $p\toπ^+\overlineν$ and $p\toπ^0 e^+$, making these channels testable in upcoming experiments.