TopoMath: A Concept Graph Topology-Guided Approach for LLM Mathematical Reasoning

Although large language models (LLMs) have shown immense potential for mathematical problem solving, their intermediate derivations frequently suffer from premise drift, computational errors, and exploratory "dead ends". A fundamental limitation is that models navigate an entirely implicit conceptual space without structural guidance on mathematical concepts and their interrelations. In this paper, we propose TopoMath, an inference-time framework that establishes a relationship graph of mathematical concepts to guide and verify LLM reasoning. Specifically, TopoMath builds a directed acyclic graph (DAG) from Wikidata's mathematical taxonomy and projects each problem onto a compact scaffold containing grounded anchors, taxonomic paths, shared ancestors, and local neighbors. At inference time, the scaffold can serve as a global navigational prior for solution generation, while a dual-verification mechanism combining SymPy's symbolic execution with topological concept-membership checks detects computational flaws and conceptual drifts to trigger targeted revision. Evaluated on six benchmarks spanning standard to Olympiad-level mathematics across three backbone models, TopoMath consistently outperforms direct inference and state-of-the-art inference-time baselines, exhibiting significant improvements on PASS@1 and perfect reasoning rates (PRRs). Process-level graph analysis further reveals that TopoMath eliminates 81.2% of irrelevant reasoning nodes and shortens proof depth by 34.5%, indicating that topological concept priors effectively foster concise, robust, and structurally closed mathematical reasoning.