Five-Dimensional Mathematics: A New Mathematical Framework for Systems Science

Traditional mathematics is constrained by the linear superposition paradigm, making it difficult to precisely characterize the nonlinear synergistic mechanisms and holistic emergent properties of heterogeneous systems. Based on five-dimensional ontology, this paper constructs a new five-dimensional mathematical system, proves the five-dimensional minimal completeness theorem, and defines the synergy coefficient k through multiplicative coupling. The total intensity satisfies Ibase · (1+k), achieving the unification of physical conservation and systemic emergence. It is further demonstrated that traditional mathematics is a degenerate special case when k=0. Moreover, this paper reveals that the synergy coefficient exhibits reference-frame relativity—the same combination may present different k values under different functional intentions or observational scales. Based on differences in subjective initiative, systems are categorized into three types (none, weak, strong) with corresponding five-dimensional operation types, revealing the core reason why complex system problems such as entropy increase are difficult to resolve using traditional mathematics. This framework achieves a unified description of the full spectrum of systems and provides a new quantitative foundation for general system theory.