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.
Zhenmin Wang (Sun,) studied this question.