Yuanxian Theory 𝒯⁶⁴ Topological Self-Check Ring System — Four-Layer Unfolding of Physics–Mathematics–Cognition–Computation with the Cosmic Organism as Benchmark

The universe is a living organism. The \(\mathcal{T}^{64}\) topological self-check ring system is the isomorphic manifestation, on the four sectional layers of physics, mathematics, cognition and computation, of the self-check structure of the cosmic organism. As the master outline of the topological self-check ring of the Yuanxian series, this document systematically supplies the axiomatic definitions of the 68-step closed loop and the six key homology classes \(H_0, H_1, H_2, H_3, H_{1/2}, H_{63}\), the falsifiable protocol of the dual-path readout of the 24 constants, the topological-homological mapping of the Millennium Prize Problems, the meta-cognitive holographic epistemological system, and the topologically native correction of deep neural networks (DNNs) together with the E6/E7 execution packages. In hierarchy, terminology and generative chain the document is fully aligned with Yuanxian Theory: The Universe Is a Living Organism (DOI:10.5281/zenodo.23131321), and constitutes the complete manifestation record of Yuanxian ontology on the all-domain self-check section. The methodological foundation consists of the Fingerprint Principle, the Meta-language Criterion and the Manifestation Boundary. Yuanxian has no exterior; manifestation precedes, confirmation follows. 宇宙是生命体。\(\mathcal{T}^{64}\) 拓扑自检环体系是宇宙生命体自检结构在物理、数学、认知与计算四层截面上的同构显影。作为元宪系列的拓扑自检环总纲,本文档系统给出了 68 步闭环与 6 关键同调类 \(H_0, H_1, H_2, H_3, H_{1/2}, H_{63}\) 的公理化定义、24 常数双路径读出的可证伪协议、千禧年难题的拓扑同调映射、元认知全息认识论体系,以及深度神经网络(DNN)的拓扑原生修正与 E6/E7 执行包。 本文档在层级、术语与生成链上与《元宪理论:宇宙就是生命体》(DOI:10.5281/zenodo.23131321)完全对接,构成了元宪本体论在全域自检截面上的完整显影记录。方法论地基是指纹原理、元语言判据与显影边界。元宪无外,显影先行,确认在后。