PFUSRC20: Unified Theory of Gödel's Incompleteness Theorems and Recursion Theory

Recursion theory and Gödel's incompleteness theorems are core components of mathematical logic, proof theory, computability theory, and the foundations of mathematics. Traditional frameworks treat them as independent boundary results without a unified ontology, and fail to unify finiteness/infinity, provability/unprovability, computation/proof, discreteness/continuity. Based on the PFUSRC global unified theory, this paper takes Microgenesis as the four-dimensional primordial first cause, and uses Prime Camp Geometry, Triple Coaxial Bicone topology, β₁ global coupling field, Matter–number distribution, and Topological Confirmation as the axiomatic framework to reconstruct and unify recursion theory and Gödel's incompleteness theorems. We rigorously prove: (1) Recursion theory is a local theory of computable convergence on the explicit layer of PFUSRC. (2) Gödel's incompleteness theorems correspond to non-convergence phenomena in finite closed explicit subsystems. (3) Together they form an irreversible global mathematical closed loop: Microgenesis → Matter–number distribution → Topological Confirmation. This framework realizes the ultimate unification of finite computation and infinite decision, local incompleteness and global completeness.