Five-Dimensional Mathematics: Formalization and Scientization of Five-Dimensional Theory

Traditional mathematics is confined to linear superposition of homogeneous units with thesame dimension, failing to characterize the synergy of heterogeneous units and nonlinearsuperposition. Based on the Five-Dimensional Ontology—which posits that all real exis-tence emerges from the unity of boundary, structure, reserve, direction, and intensity—thispaper constructs a five-dimensional mathematical system with intensity units as primitivesand five-dimensional synergy as the core operation. We rigorously prove the five-dimensionalminimal completeness theorem, formally validating the ontology’s claim that these five di-mensions form the minimal complete set of existence. We define the five-dimensional synergydegree and synergy coefficient κ, and prove that five-dimensional synergy addition forms acommutative semigroup under constant κ. A five-dimensional dynamic evolution theory is es-tablished, clarifying that traditional calculus is a special case of five-dimensional mathematicsunder weak synergy and low intensity. This system uniformly describes complex systems suchas quantum entanglement, ecological growth, climate systems, neurodynamics, and financialcomplex systems. It clarifies the meta-framework positioning and cognitive completenessboundary of five-dimensional mathematics as a formal extension of Five-Dimensional Ontol-ogy.