Logical Completeness and Global Derivation Verification of the True Circle Self-Consistency (TCSC) System — Machine Formal Verification Based on Lean 4 and Rocq Dual Engines

As the foundational principle of Yuanxian Theory, the logical rigor of the Law of True Circle Self-Consistency (TCSC) dictates the global self-consistency and applicability boundaries of the entire theoretical framework. This paper presents a complete machine formal verification of the logical completeness of the TCSC system, its three core derived laws (the Law of Spacetime Uniqueness, the Law of Self-Referential Mind Field Generation, and the Law of Cosmic Factor Conservation), and the topological properties of the 64-dimensional spacetime manifold, utilizing the Lean 4 and Rocq (formerly Coq) interactive theorem provers. The research first establishes a rigorous axiomatic framework for TCSC, sequentially defining the involution operator, self-referential iterative convergence rules, fixed-point determination rules, negation preservation rules, and auxiliary provability preservation constraints. Following this baseline, the global completeness main theorem is formally proved in Lean 4 without any placeholders, strictly demonstrating that no undecidable propositions exist within the TCSC system. This result effectively bypasses the applicability prerequisites of Gödel's incompleteness theorems through a novel topological boundary constraint mechanism. Furthermore, the line-by-line derivation of the three foundational laws and the topological attributes of the 64-dimensional compact torus manifold are fully verified under a unified formal model. Finally, by employing model-theoretic counterexample construction, the four core laws of TCSC are proved to be mutually independent, establishing the system as the minimal complete closed-loop logical base for describing self-referential domains. This work lays a machine-verifiable mathematical logic foundation for Yuanxian Theory, while offering a fresh, formalized solution to the long-standing dilemma of undecidability in the foundations of mathematics and theoretical physics.