Continual Graph Memory for Mathematical Research Agents
Abstract Overview
The paper presents Ansatz, a mathematical research agent built around Continual Graph Memory, a structured memory system for long-horizon proof search. The memory explicitly stores accepted facts, exploratory artifacts such as plans and counterexamples, and cross-project lessons in a unified graph, while retrieval and reuse are controlled through dependency-aware context selection, curator annotations, and scoped recall. The system is designed to help agents sustain, resume, and transfer mathematical exploration across projects without treating prior results as automatically valid in a new problem. The evaluation includes the ten First Proof Second Batch tasks, component ablations, and a set of open mathematical problems.
Novelty
The distinctive contribution is a cross-problem, graph-based memory architecture for mathematical research agents that separates verified facts from exploratory state and reusable lessons. Its combination of typed exploration graphs, evidence-sensitive curation, dependency-aware retrieval, and scoped recall with mandatory local re-proving appears to be the main new system design element.
Results
In the reported benchmark evaluation, Ansatz achieves closure on all ten First Proof Second Batch problems, exceeding the other systems listed in the paper, including GPT-5.6-based baselines. The component studies indicate that exploration-and-curation, independent verification, and strategy consultation contribute materially to performance, whereas the explicit fact-graph structure alone does not show a measurable gain on the tested subset. Beyond the benchmark, the authors report complete solutions to the Jamison caterpillar conjecture and Erdős Problems 289, 348, and 488, along with partial progress on six additional open problems.
Key Points
- Ansatz organizes mathematical proof search with a unified memory that links accepted facts, plans, counterexamples, and other intermediate findings under explicit dependency and provenance structure.
- The benchmark evidence reports 10/10 task closure on First Proof Second Batch, with graph-based retrieval used heavily and variably across problems rather than at a fixed rate.
- Ablations and audits suggest that curation, external verification, and strategy consultation are practically important, while cross-project reuse is constrained by local re-proving instead of unconditional transfer.