This paper establishes a multi-layer classification system for geometric structures based on fundamental geometric objects, transformation groups, local invariants, global topology, geometric principles, additional structures, and concrete geometric models. The system takes as successive refinement criteria the basic geometric type, transformation symmetry, dimension and metric signature, curvature and connection structure, geometric principles, extra geometric structures, and concrete geometric instances, forming an arbitrarily extensible classification tree. Each layer is equipped with corresponding axiom systems and fundamental theorems, so that any geometric structure can be uniquely placed into a specific node of the tree. Conversely, any parameter combination of a node can mechanically generate an axiom system and predict as-yet-unstudied geometric structures. The system possesses unity, completeness, and extensibility, analogous to the periodic table of chemical elements, and can be used to systematically discover and construct new geometric branches. This paper provides formal definitions, construction methods, fundamental theorems, and multiple examples, and shows how classical branches of geometry are embedded into the system, as well as how to build axiom systems and fundamental theorems for vacant parameter combinations. All theorems are given rigorous proofs (general theorems with at least 4 steps, important theorems with at least 8 steps), and all predictions are equipped with complete axiom systems and existence constructions, transforming the development of geometry from random discovery to fill-in-the-blank construction.
Liu S (Wed,) studied this question.