Topo-Mathematical Interpretation of the Euler Mathematical Unified Paradigm

Project: Construction of the W≡0 Global Topological Unified Field Theory SystemStage: Stage 4 · Mathematical Precursor Layer Abstract The Euler identity serves as the core fundamental identity integrating five basic constants in classical mathematical systems. For a long time, it has only been regarded as a perfect algebraic formal equivalence, lacking topological spacetime origin, dimensional coupling mechanism, and verifiable mathematical boundaries. Remaining at the level of instrumental application, it cannot explain its endogenous necessity and objective spacetime correspondence. Based on the locked topological mathematical achievements of the W≡0 Global Topological Unified Field Theory System (Stage 4), and strictly abiding by the research boundaries of the pure mathematical precursor layer, this paper adopts the "structure-phase-application" three-line independent mutual verification framework to conduct a full-dimensional traceability and closed-loop interpretation of the Euler unified paradigm. From three core levels including the symmetric structure of three-dimensional compact space, orthogonal coupling of virtual and real dimensions, and steady-state limit of spacetime evolution, this paper clarifies the precise division of labor, coupling logic, and hierarchical correlation of the five fundamental constants , eliminating the instrumental cognition of classical pure mathematics. Without breaking the core boundaries of this paper, it paves the way for the gradient field homologous dual relationship between the natural constant and the golden section constant, reserving expansion interfaces for advanced theories. This paper ultimately upgrades the Euler formula from an artificial algebraic identity to an endogenous top-level mathematical unified paradigm of four-dimensional compact topological spacetime, and constructs concrete and quantifiable topological invariant verification criteria. It realizes the dual closed loop of theoretical self-consistency and mathematical verifiability, providing rigid underlying paradigm support for global fundamental constant linkage, complex topological field construction, and unified spacetime evolution laws of the W≡0 Global Topological Unified Field Theory System. Keywords Euler identity; topological mathematics; structure-phase-application framework; spacetime dimensional coupling; polar gradient field; verifiability; W≡0 conservation