Leveraging frontier agents to tackle mathematical research problems has become an effective way to advance mathematics. Solving such frontier problems often requires a massive number of agents working in parallel for extended periods to construct proofs, which generates an enormous volume of intermediate proof results. Organizing these intermediate results throughout a long‑horizon proof‑search process and reusing knowledge from prior explorations remain major challenges. We introduce Ansatz, a mathematical research agent built around Continual Graph Memory. Continual Graph Memory is a graph‑based, evolvable, cross‑problem memory system that explicitly organizes the entire proof search and reuses information from exploration trajectories of previous problems. Specifically, Ansatz creates a unified graph memory where nodes represent facts, plans, and counterexamples, and edges explicitly encode their relationships; dependency‑aware retrieval supplies precisely targeted local context; an evidence‑sensitive curator updates the research frontier and distills lessons from prior attempts; 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 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 Erd\H{o}s Problems 289, 348, and 488 without human intervention, and makes partial progress on several open problems, illustrating its strong capability to solve open mathematical research challenges.
Review