PART 40: TOPOLOGICAL CLASSIFICATION OF DEFECTS IN ORIGIN GEOMETRY Hull Cohomology, Topological Charge, and State-Space Structure in Dual–H₄ Geometry

The previous stages of the Origin Geometry (OG) program established a geometric framework in which spacetime, matter precursors, and cosmological structure emerge from the dynamics of dual–H₄ geometric substrates embedded within the E₈ lattice. Parts 37–39 further developed an emergent cosmological sequence involving geometric renewal, phason-mediated reorganization, freeze-out dynamics, topological defect formation, primitive mass generation, and the emergence of matter precursors. A fundamental question naturally follows: How should these defects be classified? The present work addresses this question by constructing a topological classification framework for defects in Origin Geometry. Rather than introducing defects as external objects or phenomenological assumptions, we demonstrate that defect structures arise naturally from discontinuities generated during dynamic projection processes in the dual–H₄ substrate. To characterize these structures, we introduce the hull space associated with the family of admissible geometric configurations and employ algebraic-topological tools based on homology and cohomology. Within this framework, topological defects are classified by integer-valued invariants obtained through the pairing between cohomology classes and nontrivial cycles in the hull. This construction yields discrete topological sectors: Ω = ∪_w Ω_w where each sector is labeled by a conserved topological charge w. The framework does not introduce new dynamical laws and does not attempt to derive mass, dark matter, or cosmological observables. Instead, the objective is more fundamental: to establish the topological state-space structure underlying defect formation in Origin Geometry. The resulting classification framework provides a mathematical foundation upon which future dynamical theories may be constructed.