Complex Phase Cosmology: A Topological Attempt to Unify Cosmological Problems Using Euler's Identity and Juridical Structuring Methodology — Part 5: D(z)=f(dθ/dz) and γ≈0.15 — Verification Through DESI, JWST, and Gaia

Part 1 established the grammar: i as a 90-degree rotation operator and Euler's formula as the language of phase transitions. Part 2 extended this into dynamics, defining θ (z), γ (z), and cos² (πθ) /sin² (πθ) as energy partition ratios. Part 3 applied these tools to electromagnetic waves, defining the intrinsic phase θEM = 0. 25 as the perfect 50: 50 equilibrium. Part 4 extended the global phase function to a local phase field θ (x, z), interpreting black holes, voids, and filaments as topological defects on the phase manifold. Parts 1–4, however, did not yet confront these tools with observational data. This study (Part 5) concludes the 5-part series by designing a structural roadmap for observational verification. Six axes are proposed: (1) θ (z) vs DESI BAO, including an effective growth function DCPC (z) ∝ |dθ/dz| as a testable ansatz against fσ₈ (z) ; (2) the open question of whether θEM = 0. 25 and γ₀ ≈ 0. 15 share a constraint relationship; (3) convergence of Ωₘ, CPC, int (z) toward 0. 5 as a past achievement or future attractor (z ≈ −0. 83) ; (4) ∇θ (x, z) as a geometric predictor of filament rotation and galaxy spin alignment (Tudorache et al. 2025; Zee et al. 2025), with Gaia astrometry as an observational backbone; (5) CMB anisotropy as the initial distribution θ (x, z ≈ 1100), evolving into the present cosmic web θ (x, z = 0) ; and (6) JWST high-redshift massive structures reinterpreted as phase-transition boundaries — where "impossible" mass arises from unitary phase rotation (|e^ (iπθ) |² = 1), not gravitational accumulation, and the observable universe's boundary is redefined as a topological limit of imaginary-to-real energy conversion. Building on Part 1 (DOI: 10. 5281/zenodo. 19158235), Part 2 (DOI: 10. 5281/zenodo. 19332436), Part 3 (DOI: 10. 5281/zenodo. 19332612), and Part 4 (DOI: 10. 5281/zenodo. 19412396), Part 5 moves from theoretical construction to verification design, proposing that the observable universe is not bounded by the travel time of light, but by the topological boundary at which imaginary energy begins to convert into real energy. Core results established in Part 5: - DCPC (z) ∝ |dθ/dz| — effective growth function ansatz, testable against DESI fσ₈ (z) - ∇θ (x, z) as geometric alignment axis — filament rotation and galaxy spin alignment as phase-field consequences- CMB → cosmic web: θ (x, z ≈ 1100) initial variations evolving into present-day large-scale structure through phase-field dynamics- Observable universe boundary redefined: not a temporal limit (light travel time) but a topological limit (imaginary-to-real phase transition) - High-redshift "impossible" mass: unitary phase rotation bypasses ΛCDM's time-scale constraint- 68% of present energy partition is imaginary-direction: sin² (πθ) ≈ 0. 681 at θ (z=0) = 0. 309 This research aims to present a conceptual and structural roadmap for reconstructing the macroscopic skeleton of the universe through topological mathematics. Accordingly, the completion of specific numerical normalization and Lagrangian-based dynamical derivation are left as domains for subsequent research built upon this framework. Concrete numerical verification is expected to be developed through collaboration with researchers working with observational data. This research applies Juridical Structuring Methodology to cosmology, crossing traditional academic boundaries to propose a strictly falsifiable scientific framework.