On the Structural Limits of Ranking Under Non-Separable Valuation

This paper studies the structural limits of ranking systems under non-separable valuation. Many real-world selection environments operate under capacity constraints: search engines return finite result sets, platforms recommend limited inventories, and hiring systems filter candidate pools. In such settings, the value of an item may depend on which other items are simultaneously selected. The paper formalizes a class of allocation problems where valuations are non-separable and demonstrates that ranking cannot, in general, guarantee optimal allocation under such complementarities. It introduces a separability sufficiency theorem establishing that ranking guarantees optimal allocation if and only if valuations are separable. The work further distinguishes retrieval from allocation as formally distinct computational problem classes. An information-theoretic characterization is developed showing that ranking necessarily compresses the valuation space from exponential interaction structures to linear ordering structures, discarding interaction information. The paper also analyzes inferential infrastructure and shows how inferential costs can affect inclusion probabilities in constrained allocation systems. The contribution is theoretical and implementation-independent. It does not propose a commercial platform or application-specific architecture, but instead provides a formal framework for understanding the computational and allocative limits of ranking-based selection systems.