Continual Graph Memory for Mathematical Research Agents

2026-10-02

AI Summary

This study addresses the challenges of organizing intermediate results and reusing knowledge during extended proof searches by mathematical research agents. To this end, we propose the Ansatz agent, centered on an evolvable unified graph memory architecture. This architecture employs graph neural networks to explicitly model dependencies among facts, plans, and counterexamples, while integrating dependency-aware retrieval, evidence-sensitive curation, and scoped recall mechanisms to optimize long-horizon reasoning. Furthermore, the system incorporates multi-agent parallel search and dynamic context management techniques. Empirically, Ansatz successfully closes ten frontier benchmark tasks and independently resolves several well-known conjectures alongside open problems posed by Erdős, demonstrating robust capabilities in autonomous mathematical discovery.

Abstract

Using frontier agent harnesses to tackle mathematical research problems has emerged as an effective means of advancing mathematics. However, solving frontier problems in mathematics may require a massive number of agents working in parallel for extended periods to construct proofs, thereby generating an enormous volume of intermediate proof results. Organizing these intermediate results throughout a long-horizon proof-search process and reusing knowledge gained from prior explorations remain major challenges. We present Ansatz, a mathematical research agent built around Continual Graph Memory, a graph-based, evolvable, cross-problem mathematical research memory system that explicitly organizes the entire proof search process and reuses information from exploration trajectories of previous problems. Specifically, we develop a unified graph memory that represents all intermediate exploration results, including facts, plans, and counterexamples, together with edges that explicitly represent the relationships among them; dependency-aware retrieval supplies precisely targeted local context; an evidence-sensitive curator updates the research frontier and distills lessons from prior attempts; and scoped recall surfaces earlier statements and negative findings for local re-proving rather than uncritical reuse. Experiments cover runs across all ten First Proof Second Batch problems, together with four component studies. Ansatz reports closure on all ten research tasks, demonstrating its ability to sustain and resume long-horizon mathematical search. Beyond these problems, Ansatz also produces solutions to the Jamison caterpillar conjecture and Erdős Problems 289, 348, and 488 without human intervention, and makes partial progress on several open problems, illustrating its strong ability to solve open mathematical research problems.

Problem

• mathematical research agents
• continual graph memory
• long-horizon proof search
• intermediate proof results
• knowledge reuse

Innovation

• Continual Graph Memory
• Mathematical Research Agent
• Dependency-aware Retrieval
• Graph-based Memory
• Proof Search