ReasAtlas: Shared Expositions as a Representation for Mathematical Knowledge

Mathematical knowledge is developed through expositions that organize definitions, theorems, and algorithms around related mathematical objects. Such organization situates individual statements within broader mathematical contexts and can serve as a representation of mathematical knowledge. We introduce ReasAtlas, which organizes mathematical knowledge from heterogeneous sources within a shared hierarchical exposition. Each statement can consolidate formulations across sources while preserving their source-specific content and provenance throughout structural revisions. The hierarchy provides each statement with a primary organizational context, while relations and cross-topic links capture mathematical connections beyond the primary hierarchy. To build this representation, ReasAtlas aligns source-local expositions, retrieves relevant regions of the shared organization, and uses source context to place new statements and revise the structure. The resulting atlas provides a shared organizational context for locating and retrieving mathematical knowledge across sources. Its hierarchy also identifies mathematical neighborhoods in which relevant statements must be distinguished from nearby alternatives. We introduce information-calibrated margin cross-entropy (IC-MCE), which uses shared ancestry to strengthen these comparisons during training while the resulting encoder operates on statement text alone at inference time. Our evaluation examines whether shared exposition can support both the localization of mathematical knowledge and the learning of retrieval representations, while also assessing the mathematical coherence of the resulting topic organization.