Semantic Search over 9 Million Mathematical Theorems

TL;DR

This paper introduces a large-scale semantic search system for mathematical theorems, demonstrating its effectiveness in retrieving specific theorems from a vast corpus of research-level mathematical statements.

Contribution

It presents the largest publicly available corpus of research-level theorems and systematically analyzes factors affecting semantic theorem retrieval quality.

Findings

Semantic search significantly improves theorem retrieval accuracy.

Representation choices impact retrieval performance.

The system is effective at web-scale theorem search.

Abstract

Searching for mathematical results remains difficult: most existing tools retrieve entire papers, while mathematicians and theorem-proving agents often seek a specific theorem, lemma, or proposition that answers a query. While semantic search has seen rapid progress, its behavior on large, highly technical corpora such as research-level mathematical theorems remains poorly understood. In this work, we introduce and study semantic theorem retrieval at scale over a unified corpus of $9.2$ million theorem statements extracted from arXiv and seven other sources, representing the largest publicly available corpus of human-authored, research-level theorems. We represent each theorem with a short natural-language description as a retrieval representation and systematically analyze how representation context, language model choice, embedding model, and prompting strategy affect retrieval quality. On a curated evaluation set of theorem-search queries written by professional mathematicians, our approach substantially improves both theorem-level and paper-level retrieval compared to existing baselines, demonstrating that semantic theorem search is feasible and effective at web scale. The project page, search tool, dataset, REST API, and MCP server are available at theoremsearch.com.