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 - cP², 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.
Wenjia Jiang (Fri,) studied this question.