Geometric Structure and Symmetry Groups as Extensions of a Unified Physical Framework

This paper develops a structural framework for introducing symmetry groups, their representations, and their geometric realizations as explicit extensions of a unified physical theory. By treating symmetry as structural data rather than a foundational assumption, it clarifies the distinction between physical symmetries and gauge redundancy, establishes criteria for consistent geometric realization, and formulates anomaly control as a fundamental consistency requirement. The framework provides a disciplined pathway from abstract invariance principles to concrete symmetry-based model building within a unified theoretical setting.