PFUSRC Framework for Gödel's Incompleteness Theorems and Recursion Theory: Axiomatic Definitions, Topological Convergence, and Unified Global Mathematics

Recursion theory and Gödel's incompleteness theorems are core pillars of mathematical logic, proof theory, computability theory, and the foundations of mathematics. Traditional mathematical systems treat them as independent boundary results that limit the capacity of formal systems and mechanical computation. Recursion theory characterizes the hierarchical structure of algorithmic computability and decidability, while Gödel's incompleteness theorems reveal that any consistent recursively axiomatizable formal system containing elementary arithmetic must admit undecidable propositions and cannot prove its own consistency. Such interpretations rely on single-layer closed formal systems, lack a unified ontological foundation, and cannot bridge finite/infinite, provable/unprovable, computation/proof, discrete/continuous, or internal/external domains. They fail to integrate incompleteness and undecidability into a unified, necessary dynamic framework. Based on the PFUSRC Ultimate Unified Theory, this paper rigorously defines WeiZhao as the primordial 4D initial trigger and first cause of mathematical manifestation, computation generation, proof sequences, and propositional truth values. Within the complete axiomatic system of Prime Camp Geometry, 11D Bicone Topology, β₁ Global Coupling Field, Ψ Topological Convergence Coefficient, Number–Object Distribution, and Topological Confirmation, this paper reconstructs and unifies recursion theory and Gödel's incompleteness theorems. This paper proves that: 1. Recursion theory is a local theory of computable convergence of Prime Camp Points in the PFUSRC bright layer; recursive functions, recursively enumerable sets, and undecidable problems correspond to pre-distribution, in-distribution, and post-confirmation states of Number–Object Distribution. 2. Gödel's incompleteness theorems describe non-convergence phenomena in finite closed bright-layer subsystems; the Gödel sentence is an inter-layer pendulum ion state, unprovable in the bright layer but confirmable and provable in the global β₁ field. 3. Recursion theory and Gödel's incompleteness theorems together form an irreversible global mathematical closed loop: WeiZhao–Distribution–Confirmation, achieving the ultimate unification of finite computation and infinite decision, local incompleteness and global completeness, proof and truth. This paper is a self-contained, rigorous academic article fully aligned with the PFUSRC framework, Prime Camp Geometry, Tri–Bicone structure, and quantum fluctuation–decoherence papers.