We introduce Trinity Cluster Theory, a novel mathematical framework based on ternary dependency relations among three elements. In this theory, any two elements uniquely determine the third, forming a cluster that captures a fundamental three-way relationship. Unlike classical algebraic systems built on binary operations, this framework emphasizes three-way interdependencies, enabling the construction of cluster chains, loops, and networks. The theory originates from geometric intuition, particularly the sum of angles in a triangle, and extends naturally to generalized forms including weighted, nonlinear, and higher-dimensional clusters. Trinity Cluster Theory offers a new perspective on algebraic structures derived from multi-variable dependency rather than conventional composition, with potential applications in geometry, dynamical systems, and network analysis.
Ren Matsuoka (Sun,) studied this question.