TheoremGraph: Bridging Formal and Informal Mathematics

2026-06-23

AI Summary

This work addresses the lack of fine-grained alignment between formal and informal mathematical knowledge. The authors propose TheoremGraph, the first unified dependency graph linking 11.7 million informal theorems from arXiv with 388,000 formal statements in Lean 4. Their approach combines theorem environment parsing, elaborator-level extraction, semantic embeddings, and an LLM-based discriminator to achieve statement-level cross-domain alignment. The method enables tunable precision–coverage trade-offs in dependency extraction and achieves a Recall@10 of 0.775 for formal concept retrieval—approaching state-of-the-art performance. It yields 47,952 high-confidence matches (cosine similarity ≥ 0.8). The dataset and an accompanying API are publicly released.

Abstract

Mathematical knowledge is organized around statements and their dependencies, but this structure is exposed unevenly: informal papers cite mostly at the document level, while formal libraries record fine-grained dependencies over a much smaller body of mathematics. We introduce TheoremGraph, a unified statement-level dependency graph spanning both informal and formal mathematics. On the informal side, we parse 11.7M theorem-like environments from mathematics arXiv and recover 18.3M candidate directed dependencies, each labeled by the extractor that proposed it so downstream users can trade coverage for precision. On the formal side, we release LeanGraph, a Lean 4 elaborator-level extractor producing 388,105 declaration nodes and 11.3M typed edges across 25 Lean projects. We bridge the two graphs by embedding generated natural-language slogans into a shared semantic space, linking related statements across papers and across the informal/formal divide; an LLM judge affirms 47,952 such matches above a 0.8 cosine floor, with the judge-acceptance rate rising from 48% across the floor to 87% in the >=0.9 tier. On formal concept retrieval, our name-and-signature representation with graph expansion comes within 0.5pp of LeanSearch v2's reranked Recall@10 (0.775 vs. 0.780) without an LM reranker. We release the dataset, extractors, HTTP API, and MCP interface as infrastructure for mathematical search, attribution, and retrieval-augmented reasoning, available at theoremsearch.com and huggingface.co/datasets/uw-math-ai/theorem-matching.

Problem

• mathematical knowledge representation
• formal-informal mathematics bridging
• theorem dependency graph
• semantic alignment
• mathematical retrieval

Innovation

• TheoremGraph
• mathematical dependency graph
• formal-informal alignment
• semantic embedding
• Lean 4 elaboration