Complex Phase Cosmology: A Topological Attempt to Unify Cosmological Problems Using Euler's Identity and Juridical Structuring Methodology — Part 1: The Imaginary Unit i and the Complex Plane

In linear number systems, the imaginary unit i has long been treated as a "non-existent" or "fictitious" number. However, from the perspective of the complex plane, i is defined as a geometric operator that performs a 90-degree phase rotation. This study (Part 1) reinterprets the algebraic definition i² = −1 as a geometric rotation and argues that Euler's formula e^ (iθ) = cosθ + i·sinθ is not merely a convenient identity, but the fundamental language for describing phase transitions in the universe. This document establishes the mathematical foundation for the 5-part series Complex Phase Cosmology. While the earlier V1 overview (DOI: 10. 5281/zenodo. 19007198) provided a broad sketch, this V2 (Part 1) lays down the rigorous geometric and conceptual framework upon which Parts 2–5 are built. Core Equations established in Part 1: - i² = −1 (algebraic definition of phase rotation) - e^ (iθ) = cosθ + i·sinθ (Euler's formula: the language of phase transition) - |e^ (iπθ) |² = 1 (mathematical necessity of energy conservation) This research applies Juridical Structuring Methodology to cosmology, crossing traditional academic boundaries to propose a strictly falsifiable scientific framework.