Complex Phase Cosmology: A Topological Attempt to Unify Cosmological Problems Using Euler's Identity and Juridical Structuring Methodology — Part 4: Topological Defects — A Phase-Topological Interpretation of Black Holes, Voids, and the Cosmic Large-Scale Structure

Part 1 established the grammar: ii 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 θ\_EM = 0.25 as the perfect 50 :50 equilibrium of energy partition. Parts 1–3, however, addressed only the equilibrium state and the macroscopic average. This study (Part 4) addresses the opposite side of that equilibrium — the regions where symmetry collapses to its extremes. By extending Part 2's global phase function θ(z) to a local phase field θ(x,z), it becomes possible to read the inhomogeneous large-scale structure of the universe as a single phase manifold, reinterpreting Kibble's (1976) topological-defect formation mechanism within the CPC framework. Building on Part 1 (DOI: 10.5281/zenodo.19158235), Part 2 (DOI: 10.5281/zenodo.19332436), and Part 3 (DOI: 10.5281/zenodo.19332612), Part 4 moves from the equilibrium analysis of light to the extreme structures of the universe — black holes, voids, and filaments — as topological defects on the phase manifold. Core results established in Part 4: - θ(z) → θ(x,z) — extension from global phase to local phase field, enabling description of cosmic inhomogeneity - Black hole = phase fixed point at θ → 0 (cos² → 1): real saturation, loss of dynamical fluidity (dθ/dt → 0), structurally consistent with SMBH staticity - Void = phase fixed point at θ → 0.5 (sin² → 1): imaginary saturation, symmetric to black holes by construction (cos² + sin² = 1) - Filament = steep phase gradient (∇θ ≫ 0): cosmic web read as iso-phase surface pattern - Three-point symmetry: Black Hole (θ→0) ↔ Light (θ=0.25) ↔ Void (θ→0.5) As an exploratory note, the reversed-phase identity cos²(π(θ+1)) = cos²(πθ) is proposed as a topological candidate for antimatter, and (i·sin(πθ))² = −sin²(πθ) as a candidate for structural consistency with negative energy density of dark energy. Quantitative verification — including connections with w(z), DESI BAO, JWST, and Gaia data — is reserved for future research. This research applies Juridical Structuring Methodology to cosmology, crossing traditional academic boundaries to propose a strictly falsifiable scientific framework.