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.