来自中国古老哲学阴阳五行相生相克拓扑理论表达式Penta-Cyclic Topological Theory (PCTT): A Rigorous Mathematical Formalization of Five-Element Dynamic Balance

PCTT五行环行拓朴理论:幼儿园大白话解读: 五行就是把五行相生相克简化到10对,其中5对相生(增益),5对相克(抵消或削弱)。金水木火土犹如五个幼儿园的小朋友面对面手拉手围成一圈,顺时针旋转(设计电路时禁止逆时针),前方相邻的小朋友必相生(比如金与水,水与木,木与火,火与土,土与金),前方隔一位的小朋友必相克(金克木,木克土等等)。。。游戏一直玩下去,旋转不止,直到天荒地老。 Abstract The canonical Five-Element (Wuxing) model—rooted in the unidirectional clockwise cycle of Metal-Water-Wood-Fire-Earth, with adjacent elements promoting (Sheng) and second-adjacent elements constraining (Ke) one another—captures fundamental dynamic balance in natural, philosophical, and engineered systems. Yet its mathematical essence has long remained a philosophical metaphor, lacking the rigor required for integration into modern systems science. This paper introduces the Penta-Cyclic Topological Theory (PCTT), an axiomatic framework grounded in graph theory and linear algebra that formalizes this intuitive model. At its core lies the PCTT identity A = sP - cP^2 , where P denotes a 5×5 cyclic permutation matrix encoding clockwise adjacency, and s, c > 0 are normalized strength coefficients for Sheng and Ke interactions. We prove three fundamental laws uniquely characterizing stable five-element balance: unidirectional clockwise cyclic topology, fixed Sheng-Ke polarity tied to topological distance, and irreversible evolution along the cycle. PCTT establishes a rigorous bridge between traditional Five-Element philosophy and contemporary system control theory, serving as a specialized instantiation of BSPP (Biao-Sha-Push-Pull) theory for five-element systems. Through spectral analysis, we derive explicit stability conditions ( 0 < s < 0.5 , 0 < c < 0.3 ) ensuring asymptotic convergence to dynamic balance. This work transforms a millennia-old intuition into a computable, verifiable, and deployable mathematical tool, with applications spanning ecological modeling, multi-agent coordination, and philosophical systems engineering. KEY PAPER (Full Theory & Proofs): A Flower's Smile, V-Functions Arise: Answering Lyapunov's Centennial Question DOI: 10.5281/zenodo.18927653 A Paradigm Shift in Control Theory CDEQ V1.2 The Countable Foundation of Stability DOI: 10.5281/zenodo.18939385 For the complete mathematical foundation, rigorous derivations, and general solution to Lyapunov's century-old problem — see the key paper above. A Paradigm Shift in Control TheoryCDEQ V1.2 The Countable Foundation of StabilityDOI: 10.5281/zenodo.18939385 CDEQ V1.3 Mathematical Proof of the PCTT-CW Penta-Cyclic Clockwise Topological TheoremDOI: 10.5281/zenodo.18988669 CDEQ V1.4: Mathematical Proof of the CENHE-LEX Nonary Central Holding Equilibrium TheoremDOI: 10.5281/zenodo. 19023440 CDEQ V1.5 GCVT-10 Global Convergence Validation TheoremDOI: 10.5281/zenodo. 19055969 Author: Jiang Wenjia (Shenzhen, China) ORCID: 0009-0000-3850-7286 Correspondence: jiangwenjiaszx@outlook.com