Extended Trinification-Based E₆ Grand Unification: Non-Supersymmetric and Supersymmetric Variants with Full Standard Model Embedding

This preprint extends a previous non-supersymmetric trinification-based E₆ grand unified theory (GUT) by incorporating the complete three-generation Standard Model fermion spectrum, detailed two-loop renormalization group equations with threshold corrections, and supersymmetric generalizations. Gauge couplings unify at M_GUT ≈ 10¹⁶ GeV, the Higgs vacuum remains metastable up to the Planck scale with long lifetime, and proton decay (p → e⁺ π⁰) is predicted in the range 10³⁴–10³⁶ years, testable by next-generation experiments such as Hyper-Kamiokande. The SUSY extension embeds the model into E₈ for family unification and is compatible with heterotic string theory compactifications that include gravity. The framework successfully reproduces the full SM below the intermediate scale while offering predictive beyond-SM phenomenology, including seesaw neutrino masses and potential TeV-scale exotics.