A Self-Contained Axiomatic Algebraic System for Geometric Growth and Structural Closure

This paper constructs a self-contained axiomatic algebraic system for geometric and structural growth, starting from a primitive unit element. It defines a set of core operations and axioms, and establishes a closed, logically consistent formal mathematical framework without reliance on external number systems. The system introduces original operators to describe structural coupling and evolution, and examines the internal consistency and closure properties of the resulting structure. It provides a formal foundation for analyzing hierarchical geometric and algebraic systems.