The Geometry of Power Through Omission: Engagement Topology and Adaptive Viability — GTSET Part III.

This paper completes Girum's Theory of Selective Engagement and Transformation (GTSET), a trilogy establishing universal principles of adaptive viability across biological, organizational, economic, and artificial systems. Part I established why selective engagement emerges as optimal strategy through the Capability Concentration Threshold and superlinear opportunity costs. Part II introduced how selectivity accumulates consequences over time through Transformation Debt and Selectivity Momentum. This work—Part III—reveals where selectivity reshapes reality through geometric topology. While integrated within the trilogy, this paper stands independently with complete theoretical development, comprehensive empirical validation, and practical computational tools. The core principle is counterintuitive: systems gain power not through universal engagement but through systematic refusal. This structured omission warps possibility spaces into curved Riemannian manifolds where bypassed regions become "voids"—not passive absences, but active generators of diversity and hierarchy. The framework presents seven contributions: (1) The Engagement Manifold ℳ with selectivity-induced curvature κ(R,t) encoding refusal intensity; (2) Structured voids quantified through persistent homology as topological generators; (3) Void inheritance mechanisms where high-capability refusals create inevitable niches for lower systems; (4) The Meta-Convertibility Constraint MSC ≥ κ·SIR—a universal viability law; (5) The Topological Evolvability Index (TEI) validated at 92% accuracy for collapse prediction; (6) Geometric proof that cooperation emerges as mathematical necessity from negative curvature; (7) Cross-scale coupling operators explaining systemic cascades. Empirical validation spans: 100 organizations (2015-2024) with collapse predicted 18 months in advance at 92% accuracy; 15 AI models showing hallucination rates correlated with curvature tears (R²=0.76, p