The Imaginary Unit i: A Complete Deduction of the Topological Origin for the Coupling Mapping Between 3D Real Space and 1D Time Dimension

Project: W≡0 Global Topological Unified Field Theory Construction ProjectStage: Phase 4 · Pre-Mathematical Layer Abstract In classical mathematics, the imaginary unit is defined as a purely algebraic fictional symbol satisfying , treated merely as an artificial tool to complete real-number field equations. This framework fails to explain why such a symbol fits the 4D topological spacetime structure, supports topological perturbation evolution, and enables dimensional coupling, creating a core cognitive rift between its mathematical fictionality and theoretical practicality. Following strictly the first principles of the W≡0 theory, this paper completely reconstructs the topological origin of the imaginary unit through an independent, closed-loop three-line deduction of Ti (Substance), Xiang (Form), and Yong (Function), based entirely on the permanently locked results of Phase 3: the topological ontology of 3D real space, the polar gradient propagation nature of time, and the 4D compact topological spacetime architecture. This work abandons non-compliant approaches such as reverse engineering from the Minkowski metric and premature citation of Euler’s formula. Starting from the topological constraints of unified spacetime metrics, the structural requirements of spatial symmetry transformations, and the steady-state conditions of topological evolution, it proves forwardly that is not a man-made algebraic construct, but an inevitable mathematical product for achieving global metric unification, real-virtual dimensional coupling, and topological symmetric evolution between 3D real space and 1D polar gradient time. The paper rigorously completes proofs of the existence, uniqueness, and necessity of the imaginary unit, establishes a topological definition exclusive to the W≡0 system, clarifies the conjugate symmetry of , verifies compatibility with classical algebraic definitions, specifies its preset role in complex topological fields and topological perturbation evolution, and yields axiomatic, verifiable topological inferences. The core argument of this paper shows that the imaginary unit is not an artificial mathematical patch, but an inherent and irreplaceable mathematical skeleton of the 4D topological spacetime structure, laying a rigid mathematical foundation for the subsequent Eulerian mathematical unification paradigm and the construction of a global complex topological system. Keywords: imaginary unit ; spacetime coupling; topological dimension; real-virtual mapping; topological symmetric evolution; 4D compact topological spacetime; W≡0 polarity conservation; first-principles deduction