Incompatible Closure Criteria: Projection Structure and the Necessity of Duality

When closure criteria conflict, projection and duality become unavoidable. This paper derives projection structure and duality from purely pre-physical constraints on describability. Building on prior results establishing irreversible transition, ordered asymmetry, forced quantitative measure, and thresholded evaluation, we introduce plural closure criteria as jointly necessary but not globally compatible. We show that no single global evaluation (even if multi-valued) can preserve all required distinctions once repair requirements conflict. The minimal structural resolution is criterion-indexed stabilization: projection operators Pi characterized by viability, idempotence, and fixing of already-viable states. Duality arises as the existence of two necessary stabilized descriptions of one underlying state, and complementarity appears as non-commutation of projections under mutual disruption. The paper remains prior to probability, Hilbert space structure, and physical interpretation, and closes by identifying the next forced step: a minimal theory of selection and interaction (interference-like effects) between incompatible projections.