The Golden Ratio Constant φ: A Complete Derivation of the Topological Origin for the Optimal Steady-State Ratio in Spacetime Spontaneous Organization

Project: Construction of the W≡0 Global Topological Unified Field TheoryStage: Phase 4 · Mathematical Preliminaries Layer Abstract In classical mathematics, the definition of the golden ratio constant relies on empirical geometric construction and proportional presupposition, which can only summarize phenomenal laws without explaining the existential necessity and unique steady-state properties of the constant from first-principles topological conservation constraints. Based on the W≡0 global polarity conservation axiom and locked topological iron laws of Phase 1, within the framework of three-dimensional compact closed-domain topology, this paper completes a purely mathematical origin derivation of via the three-line paradigm (ontology–phase–application) with topological self-similarity consistency as the core constraint. Starting from the self-similar steady-state division condition of a unit topological line segment, this paper derives the core constraint equation of the golden ratio in full, step-by-step detail, solves for the unique valid positive root, and proves the topological ontological existence of . Through the domain uniqueness of solutions to algebraic equations, it rigorously proves that is the unique proportion satisfying self-similar steady-state constraints and inherently possesses topological optimality. Based on a discrete recursive mapping, a perturbation iteration model is constructed; combining the monotone convergence theorem and the contraction mapping principle, is verified as the globally stable fixed attractor of the topological division system. All invalid functional arguments and out-of-bounds content have been thoroughly removed. The text fully aligns with the archival specifications of the 4.0 series, completes bidirectional compatibility verification with classical mathematical definitions, clarifies the exclusive role of as a discrete recursive steady state in the innate constant system, and perfects the full-dimensional closed loop of “static–dynamic–steady state” in the topological mathematical system. Keywords: golden ratio constant ; topological self-similarity; steady-state division; recursive iteration; topological invariant; W≡0 conservation; innate mathematical constant