Relational Singularities: Bitopological Rings and Obstruction Geometry

Many singular phenomena in algebra are relational rather than pointwise. They persist under localization, completion, and quotienting, and cannot be resolved by refining neighborhoods or passing to local data. This book develops a geometric framework for such behavior through bitopological rings, where relations between elements carry intrinsic geometric meaning. Regularity and failure are expressed through relational structure rather than pointwise conditions, leading to a new notion of singularity based on interaction rather than locality. Failures of relational regularity are organized using radicals and obstruction spectra, which replace classical prime spectra as geometric carriers of singular data. Completion distinguishes intrinsic obstructions from removable artifacts, giving rise to stable notions of depth, dimension, and stratification. These ideas culminate in obstruction geometry, a self-contained geometric theory independent of sheaf-theoretic or analytic foundations. The book further develops homology, spectral sequences, and deformation theory directly from relational geometry, introducing invariants that measure interaction between singularities rather than their isolated presence. This monograph establishes the first comprehensive theory of bitopological rings and introduces obstruction geometry as a new language for persistent singular behavior. It is written for researchers and advanced students in algebra, topology, and geometry seeking alternatives to classical point--based frameworks.