A Pure Logical Proof of the Zhu-Liang Group Structure Recursive Element Theorem

Within the axiomatic system of recursive elements, using higher category theory, homotopy type theory, and algebraic topology, this paper presents a pure logical proof that group structures (in particular, the symmetric groups $S_n$ and abstract groups $G$) are fundamental recursive elements of the mathematical universe. We first define the four axioms that a recursive element must satisfy: Existence (A1), Encoding Invariance (A2), Metabolic Conservation (A3), and Generativity (A4). Subsequently, we employ category theory to prove existence and generativity, homotopy type theory to prove encoding invariance, and algebraic topology to prove metabolic conservation. Combining these four parts yields the Zhu-Liang Group Structure Recursive Element Theorem, and we further prove at the meta-level the self-consistency of the recursively nested structure. The theorem reveals that symmetry is not only a core concept in mathematics but also a core recursive element that recursively generates complex algebraic and geometric structures; its truth originates from the recursive self-consistency requirement of the formal system itself.