Complex Phase Cosmology: A Topological Attempt to Unify Cosmological Problems Using Euler's Identity and Juridical Structuring Methodology — Part 2: Structural Isomorphism Between e's Self-Referential Property and Fractal Self-Similarity

In Part 1 of this series, Euler's formula e^ (iπθ) and the identity |e^ (iπθ) |² = 1 were established as the mathematical language for describing phase transitions. Part 1, however, treated the phase angle θ as a static constant, while the observed universe is not static. This study (Part 2) extends that framework by introducing a dynamical function θ (z) that describes how the phase angle evolves with cosmological redshift z, and by defining the core dynamical parameter γ (z) — "Phase Tension" — as the spacetime resistance and elasticity arising when imaginary energy converts into real matter. The structural isomorphism between e's self-referential property d/dx (eˣ) = eˣ and fractal self-similarity provides the geometric basis for this extension. Building on Part 1 (DOI: 10. 5281/zenodo. 19158235), Part 2 moves from pure mathematical foundations to cosmological dynamics, introducing boundary conditions derived from Planck 2018 data and comparing integral-averaged predictions with DESI 2024 BAO observations. Core equations established in Part 2: - d/dx (eˣ) = eˣ — self-referential structure of e- θ (z) = 0. 5 − 0. 191· (1+z) ^ (−γ (z) ) — phase evolution function- Ωm, CPC, int (z) = (1/z) ·∫₀ᶻ cos² (πθ (z′) ) dz′ — integral average of matter density Using the geometric reference value γ₀ ≈ 0. 15 for the present universe, this integral average Ωm, CPC, int (z) corresponds to Ωm = 0. 295 ± 0. 015 reported by DESI over the range z = 0–0. 9. This research applies Juridical Structuring Methodology to cosmology, crossing traditional academic boundaries to propose a strictly falsifiable scientific framework.