Theorem Evolution Graphs: A Structured Representation of How Mathematical Theorems Build on One Another

Much of mathematics research can now be automated: proofs can be automatically generated, formalized, verified, and written up for dissemination. Coming up with a promising conjecture is harder. Automated methods can generate conjectures, but figuring out whether one is sound, plausible, and interesting to work on still takes considerable human oversight, since a conjecture cannot be verified the way a proof can. Mathematicians might typically do this by surveying prior results and identifying which ideas can be meaningfully extended. Inspired by this evolutionary process, in which new results build on existing ones, we introduce the theorem evolution graph (TEG), a graph where nodes are theorems and edges are typed relations naming how one theorem builds on another (e.g. generalizing it, weakening a hypothesis). We construct TEGs over 1,213 arXiv papers, seeded both from open mathematical conjecture datasets and from iterative citation search from a seminal paper. We prompt frontier LLMs to extend a given theorem augmented with different contexts, and judge the resulting conjectures against the work that actually cited it. TEG-augmented prompting outperforms both a citation-graph baseline and zero-shot prompting (13.8\% vs. 9.5\% and 8.0\%). These results demonstrate the effectiveness of encoding the evolutionary structure of mathematical progress for automated conjecture generation.