Tetrahedron of flavors: One Three to rule them all

Under the conception that the total number three of fermion families must have the one and the same gauge theoretical origin as all other threes which accompany the single family grand unifiable group structure, we trade the trinification $SU(3)$ symmetry building block for its semi-simple Vertical-Horizontal symmetric $SU(3{)}_V\otimes SU(3{)}_H$ extension. The anomaly free flavor-chiral fermion representation is then constructed, solely out of threes and without any superfluous replication, by treating each standard $V$-triplet (and anti triplet) as an $H$-singlet, and conversely, by letting each standard $V$-singlet transform as an $H$-triplet (or anti triplet). The model can be schematically described by a tetrahedron of flavors. In its bi-trinification phase, the model exhibits two coupled trinification cycles, sharing the same color group, and is furthermore accompanied by a built-in dark sector. In its isomorphic quartification phase, the model presents a novel non-Pati-Salam version of quark/lepton correspondence, where quarks are paired with anti-leptons to cancel horizontal anomalies, and all fermion masses stem from one and the same Yukawa source.