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. 真圆自洽律(TCSC)作为元宪理论的第一性原理,其逻辑严谨性直接决定了整个理论体系的全域自洽性与适用边界。本文依托 Lean 4 与 Rocq(原Coq)双交互式定理证明工具,完成了 TCSC 系统逻辑完备性、三大核心衍生规律(时空唯一性律、自指心场生成律、宇宙因子守恒律)以及64维时空拓扑性质的完整机器形式化核验。 研究首先构建了 TCSC 严格的公理化体系,依次定义了对合算子、自指迭代收敛规则、不动点判定规则、否定保持规则与可证性保持辅助约束。在此基础上,本文在 Lean 4 环境中完成了全域完备性主定理的无占位机器证明,严格阐明了 TCSC 体系内部不存在任何不可判定命题,从底层拓扑约束层面收缩并规避了哥德尔不完备定理的适用前提。继而,在统一的形式化模型下,逐条完成了三大基础衍生规律与64维紧致环面拓扑属性的推导验证。最后,借助模型论反例构造法,严格证明了 TCSC 四条核心规律彼此相互独立,确立了该体系作为描述全域自指闭环系统最小完备逻辑基底的地位。 本研究不仅为元宪理论搭建了机器可核验的数理逻辑根基,同时也为数理逻辑与理论物理基础领域的不可判定性困境提供了一套全新的形式化解决方案。