Determinant Closure and Uniqueness in Grand Unified Theories: A Structural Derivation of SO(10)

We introduce a determinant-based closure criterion for Grand Unified Theories and apply it to the classification of viable unification groups.

Working entirely within standard Lie-group and representation-theoretic frameworks, we show that imposing determinant closure as a consistency condition on fermion representations uniquely selects SO(10) as the admissible grand unified gauge group. Competing candidates, including SU(5), flipped SU(5), and E₆, are excluded by explicit failure of determinant closure.

Unlike traditional approaches based on coupling unification, aesthetic minimality, or renormalization-group coincidences, the criterion developed here is purely structural and representation-theoretic. The analysis requires no assumptions about supersymmetry, quantum gravity, or ultraviolet completion, and does not rely on phenomenological tuning.

The results establish determinant closure as a rigorous internal selection principle within conventional GUT model-building and provide a novel explanation for the privileged role of SO(10) in grand unification.