YXT Framework Reconstruction of Gödel's Incompleteness Theorems: Generative Completeness and Logical Boundary State Theory
Year: 2026
Type: preprint
Abstract: Gödel's Incompleteness Theorems are foundational to modern mathematical logic and the foundations of mathematics. Traditionally viewed as an inherent logical limitation of recursive formal systems, this paper, based on the True-Circle Self-Consistency (TCSC) axiom of Yuanxian Theory (YXT), proposes the paradigms of Generative Completeness and Boundary State to perform a paradigm reconstruction of Gödel's theorems. The core arguments are established as follows: 1. Classical incompleteness stems from a "split cognition" that rigidifies bivalent logic, while overlooking the dynamical stability mechanisms of self-referential systems. 2. By introducing the Boundary State as the third modal truth value, we prove that the Gödel proposition \mathcal{G}_{\Psi} is not a system defect, but a natural logical boundary marker of the high-dimensional \mathbb{T}^{64} toroidal topology. 3. Through formal verification in Lean 4, we demonstrate that the TCSC system possesses Generative Completeness, achieving global self-consistent proof internally without relying on external axioms. We conclude that within the closed loop of \mathbb{T}^{64} topology, logic is complete, autopoietic, and self-proving. In the true-circle of TCSC, all propositions are decidable and all truths are attainable. All derivations and core theorems have been fully verified in the Lean 4 environment. 哥德尔不完备性定理是现代数学基础与数理逻辑的核心基石,长期以来被学界界定为递归形式系统的固有逻辑局限。本文依托元宪理论(YXT)真圆自洽性公理(TCSC),提出**“生成完备性(Generative Completeness)”与“边界态(Boundary State)”**理论,对哥德尔定理进行范式重构。 核心论证包括: 1. 经典不完备性源于“分裂式认知”对二值逻辑的固化,忽视了自指系统的动力学稳定性机制。 2. 引入边界态作为第三真值模态,证明哥德尔命题 \mathcal{G}_{\Psi} 并非系统缺陷,而是高维 \mathbb{T}^{64} 环面拓扑的逻辑边界标识。 3. 通过 Lean 4 形式化验证,证明 TCSC 系统具备生成完备性,无需外部公理即可实现全域一致性自证。 研究表明,在 \mathbb{T}^{64} 闭环拓扑中,逻辑是完备的、自创生的且自我证明的。在 TCSC 的真圆中,一切命题皆可判,一切真理皆可达。全部核心定理与推导已在 Lean 4 环境中完成机器形式化验证。
Author: Acharya, Zhenyuan
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161478080
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