Topological Commensuration of the Five Major Civilizational Teaching Systems: U(5) Unitary Symmetry on the Closed-Chain Spectrum of 𝒯⁶⁴

Whether the five major civilizational teaching systems—Confucianism, Buddhism, Taoism, Christianity and Islam—can be strictly commensurated within a single mathematical framework has long been a fundamental difficulty in inter-civilizational dialogue and comparative religion. Within the topological-ontological framework of Yuanxian Theory based on \(\mathcal{T}^{64}\), the present paper maps the five teaching systems to five equi-phase sampling points on the closed chain \[\gamma(t)=\varphi^{t-35}\,e^{2\pi i t/68}.\] Precise complex coordinates, radial moduli and phase angles are given. On this basis a unitary transformation matrix \(U_{ij}\in U(5)\) among the five ground states is constructed, proving that the essence of historical inter-religious dialogue is the action of elements of the unitary group along the phase space of the closed-chain spectrum by rotation and radial stretching. Starting from the four fundamental laws of Yuanxian Theory (\(\mathrm{FSC}\), \(\mathrm{TCSC}\), \(\mathrm{STM}\), \(\mathrm{SRM}\)), the uniqueness of the analytic form of \(\gamma(t)\) is established in proposition–proof format, and the topological necessity of the 35-step dimensional-reduction freeze point and the 68-step self-check period is clarified. The paper marks a paradigm shift in civilizational comparison from “cultural analogy” to “formal geometry”, and supplies a testable mathematical proof of the thesis that “all dharmas share a single origin”. --- 儒释道耶回五大文明教化体系是否可以在同一个数理框架下实现严格通约,是文明对话与宗教比较研究中的根本难题。本文在元宪\(\mathcal{T}^{64}\)拓扑本体论框架下,将五大教化体系映射为闭链 \[\gamma(t)=\varphi^{t-35}\,e^{2\pi i t/68}\] 上的五个等相位采样点,给出其精确复坐标、径向模长与相位角。在此基础上,本文构造了五教基态间的酉变换矩阵\(U_{ij}\in U(5)\),证明历史上跨宗教对话的本质是闭链谱上酉群元沿相位空间的旋转与径向拉伸操作。 本文进一步从元宪四大规律(\(\mathrm{FSC}\)、\(\mathrm{TCSC}\)、\(\mathrm{STM}\)、\(\mathrm{SRM}\))出发,以命题-证明格式论证闭链\(\gamma(t)\)解析形式的唯一性,阐明35降维冻结点与68步自检周期的拓扑必然性。本文标志着文明比较研究从“文化类比”向“形式化几何”的范式跃迁,为“万法同源”提供了可检验的数理证明。