A Pure Logical Proof of the Zhu-Liang Logical Equivalence Relation Recursive Element Theorem

Within the axiomatic system of recursive elements, using model theory, proof theory, and categorical logic, this paper presents a pure logical proof that the logical equivalence relation \(\equiv\) (defined by \(\phi\equiv\psi\) iff \(\models\phi\leftrightarrow\psi\)) is a fundamental recursive element 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 model theory and proof theory to prove existence and generativity, model theory to prove encoding invariance, and proof theory with algebraic logic to prove metabolic conservation. Combining these four parts yields the Zhu-Liang Logical Equivalence Relation Recursive Element Theorem, and we further prove at the meta-level the self-consistency of the recursively nested structure. The theorem reveals that the logical equivalence relation is not only the cornerstone of reasoning but also a core recursive element that recursively generates Lindenbaum–Tarski algebras, model-theoretic structures, and proof-theoretic systems; its truth originates from the recursive self-consistency requirement of the formal system itself.