Pi (π): Complete Derivation on Topological Origin of Symmetric Structure of Three-Dimensional Compact Topological Space

Project: W≡0 Global Topological Unified Field Theory System Construction ProjectStage: Stage 4 · Mathematical Precondition Layer (Locked Final Version) Abstract Within the framework of classical mathematics, pi (π) is defined empirically based on two-dimensional Euclidean plane geometry, which can only describe superficial geometric phenomena, and fails to interpret its high-dimensional topological origin as well as the inevitable logic of its inherent existence. To make up for the dimensional limitations and ontological deficiencies of traditional definitions, this paper adopts the three-dimensional compact, simply-connected and isotropic 3-sphere with positive constant curvature as the sole standard research model based on the locked axiom system of W≡0 Theory in Stage 3 and Stage 4, and completes the original construction of the topological origin definition of π relying on the first principles of geometric symmetry. This paper thoroughly avoids the circular reasoning existing in traditional deduction, determines the universal scale attribute of π based on the constant scale rule of great geodesics, clarifies the basic position of π in spatial angle calibration combined with the parameter periodic rule of rotation symmetry group, completes the rigorous demonstration of geometric invariance of π with the help of conformal deformation rule under positive constant curvature, and verifies the mathematical compatibility between topological definition and classical Euclidean geometric definition through the limit degeneration deduction when the curvature radius tends to infinity. The research results show that π is not a geometric derived constant artificially defined, but a geometric invariant and symmetric scale constant inherent to the innate structure of three-dimensional closed symmetric space. Together with the benchmark constant 0, it can form a static double-layer benchmark system within the theoretical system, undertaking the functions of zero-point equilibrium benchmark and spatial scale symmetric scale benchmark respectively. The full text has a complete closed-loop mathematical deduction without conceptual ambiguity and logical faults, which fully consolidates the basic mathematical status of this constant in the global topological unified field theory system. Keywords: Pi (π); 3-sphere ; compact manifold with positive constant curvature; rotation symmetry group; geometric invariant; topological symmetric scale; Euclidean limit degeneration