A Proof of the Zhuliang Prime Recursive Element Theorem

Within the axiomatic system of recursive elements, using arithmetic geometry, model theory, and analytic number theory, this paper presents a pure logical proof that the prime distribution $\mathcal{P} = \{p_1,p_2,\dots\}$ 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 arithmetic geometry to prove existence and generativity, model theory to prove encoding invariance, and analytic number theory to prove metabolic conservation. Combining these four parts yields the Zhu-Liang Prime Recursive Element Theorem, and we further prove at the meta-level the self-consistency of the recursively nested structure. The theorem reveals that primes are not only the atoms of arithmetic but also core recursive elements that recursively generate complex mathematical structures; their truth originates from the recursive self-consistency requirement of the formal system itself.