Part 40: TOPOLOGICAL CLASSIFICATION OF DEFECTS IN ORIGIN GEOMETRY Hull Cohomology, Topological Charges, Projection-Sector Classes, and the State-Space Closure of Phase-I OG

Overview Parts 37–39 of Origin Geometry developed the transition-and-renewal branch of the program [5–7]. In that sequence, large-scale geometric saturation leads to phason-mediated projection restructuring, renewed accessibility, finite-rate phason ordering, Kibble–Zurek type freeze-out, and the formation of topological defects [25–28]. Part 39 established that such defects may act as primitive geometric precursors to later mass localization, projection-sector imbalance, hidden-sector populations, and primordial geometric perturbations. The present Part addresses the next mathematical question: once defects are generated, how should they be classified? Defect Classification and Topological Charge This work constructs a kinematical topological classification framework for defects in Origin Geometry. It does not introduce new equations of motion, does not derive particle masses, does not identify dark matter, and does not claim a completed theory of matter. Instead, it defines the global state-space architecture required before such dynamical questions can be meaningfully formulated [35, 36]. The key object is the hull Ω of admissible projected configurations [11, 18–20]. Rather than classifying defects only by local appearance, we classify them by their relation to nontrivial cycles, cohomology classes, and projection-sector structure within Ω. A persistent defect is interpreted as a topological obstruction: a local or regional incompatibility that cannot be removed by smooth deformation within the admissible configuration space [21, 22]. Given a defect-associated cycle [γ_D] ∈ H₁(Ω, ℤ) and a cohomology class [ω] ∈ H¹(Ω, ℤ), one obtains an integer-valued topological charge by the pairing: w_ω(D) = ⟨[ω], [γ_D]⟩ ∈ ℤ More generally, if multiple independent cohomology classes are relevant, a defect may carry a vector charge: w(D) = (w₁, ..., w_r) ∈ ℤʳ Sector Decomposition This leads to a sector decomposition of the admissible state space: Ω = ∪_w Ω_w where Ω_w denotes the set of configurations with the same topological charge vector. Continuous evolution within the hull preserves w, while transitions between sectors require singular events, boundary crossings, annihilation processes, or projection restructuring. Scope and Limitations The central result of Part 40 is therefore a classification principle rather than a dynamical theorem: topological defects in OG are classified by persistent obstruction classes and by their associated homology–cohomology pairings within the hull [18, 20]. This closes the first phase of OG by organizing the defect structures generated in Parts 37–39 into a mathematically defined topological state space.