Continual Graph Memory for Mathematical Research Agents

Junyi Zhang; Jinxi Yu; Eric Hanchen Jiang; Jiachen Lu; Zhi Zhang; Xinjie He; Hyunsik Chae; Ethan Ji

cs.AIOct 2, 2026

Continual Graph Memory for Mathematical Research Agents

Authors: Junyi Zhang, Jinxi Yu, Eric Hanchen Jiang, Jiachen Lu, Zhi Zhang, Xinjie He, Hyunsik Chae, Ethan Ji,

Organizations: University of California, Los Angeles

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.

Figures & tables

Figure 1: Continual Graph Memory. (a) The harness consists of workers, strategy consultation, and verifiers. The graph memory system coordinates the exploration process of all agents. Agents update and access the memory through tool calls. (b) Dependency-aware retrieval leverages the dependency information in the graph to provide multi-hop information about the target theorem. (c) Curator organizes the whole graph, update node states based on edge-based information propagation, revisits stale annotations and record structured lessons at orderly project close. (d) Scoped recall retrieves fact statements, negative findings, and methodological lessons together with provenance. The twenty most recent eligible lessons are included in the context of a later curator. Cross-project statements must be re-admitted locally under the agent protocol, while temporal isolation requires sequential orchestration.

	Official Participating Team				Our Reported results
Task	ETH	UCLA	OpenAI	Princeton	Plain	Codex	Danus	Ansatz
LLM	GPT-5.5 Pro ‡	GPT-5.5 Pro	GPT-5.5 Pro	Gemini 3.1 Pro Preview	GPT-5.6 Sol	GPT-5.6 Sol	GPT-5.6 Sol	GPT-5.6 Sol
01 Automorphism	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
02 Möbius	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
03 Bernoulli	$\checkmark$	$\times$	$\times$	$\times$	$\times$	$\times$	$\times$	$\checkmark$
04 Dilation	$\times$	$\times$	$\times$	$\times$	$\times$	$\times$	$\times$	$\checkmark$

Table 1: Experimental results for First Proof Second Batch.

\checkmark

: Pass (essentially flawless or minor revisions);

\triangle

: major revisions;

\times

: reject; —: no submission. ‡ ETH additionally calls GPT-5.5, Gemini 3.1 Pro Preview, Claude Opus 4.7.

Figure 3: (a) Memory retrieval activity across First Proof Second Batch tasks. The bars report the number of memory retrieval calls made by Ansatz for each task. Blue denotes calls to the embedding-based memory_recall tool, while green denotes calls to the graph-based multi-hop fact_neighbors tool. (b) Independent verifier review of selected records approved by workers through self-review. Bar labels indicate rejected/reviewed counts. We use a census for Tasks 01 and 06 and random sampling for Tasks 03 and 10.

Component study	01	02	03	05	06	07	08	09	10	Closed
A. Flat fact notebook	$\checkmark$	$\checkmark$	—	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	—	7/7
B. No exploration graph or curator	$\checkmark$	$\checkmark$	—	$\times$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	—	6/7
C. Worker self-review	$\checkmark$	—	$\times$	—	$\checkmark$	—	—	—	$\times$	2/4
D. No strategy consult	$\checkmark$	—	$\times$	—	$\times$	—	—	—	$\times$	1/4
Ansatz	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	10/10

Table 2: Ablation study of the components in Ansatz. Since conducting ablation studies of Ansatz on research-level tasks is very costly, we selected seven problems for ablation studies A and B, and four problems for ablation studies C and D.

Problem	Current best	Ansatz	Target	Solved
Jamison caterpillar conjecture [ Jamison, 1983 ; Jamison, 1984 ]	$n\leq 24$ [ Mol and Oellermann, 2019 ]	all $n$	all $n$	Solved
#289 Erdős–Graham interval problem [ Erdős and Graham, 1980 ]	Open [ Bloom, 2026b ]	all large $k$	all large $k$	Solved
#488 density of multiples [ Bloom, 2026e ]	Open [ Bloom, 2026e ]	counterexample; ratio $>256/67$	decide factor- $2$ bound	Solved
#348 complete-sequence deletion problem [ Erdős and Graham, 1980 ]	$m=0,1$ known [ Bloom, 2026d ]	exactly $m=0,1$	classify all $m<n$	Solved
#811 balanced-rainbow problem [ Erdős and Tuza, 1993 ]	most $q$ [ Axenovich and Clemen, 2024 ] ; $K_{4}$ [ Clemen and Wagner, 2023 ]	cliques $q\geq 4$	all graphs	Partial
#644 pair-transversal conjecture [ Erdős et al., 1992 ]	$7/8$ [ Fon-Der-Flaass et al., 1999 ]	$6/7$	$3/4$	Partial

Table 3: Ansatz outcomes on open problems. Solved indicates that the target is met, while Partial indicates meaningful progress toward the target without fully meeting it.

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Recent LLM-based mathematical reasoning agents have begun to tackle research-level problems and, in several cases, have contributed to the resolution of open problems. However, scaling and orchestrating such agents effectively remains challenging, due to the difficulty of coordinating parallel proof search while keeping intermediate claims organized and reliable. In this paper, we propose Danus, an orchestration system for research-level mathematical reasoning centered on a shared fact graph as a global memory-management mechanism. Danus consists of a main agent that performs planning and coordination, multiple worker agents that carry out proof search in parallel, and a stateless verifier that checks proposed mathematical claims before they are admitted into the fact graph. Each verified fact is stored together with its proof and logical dependencies, allowing the system to build long arguments incrementally while keeping the shared proof state organized. The main agent periodically summarizes the evolving proof state, redirects workers across promising directions, and supports interaction with human mathematicians through progress reports. We evaluate Danus through six research-level case studies in algebraic geometry, singularity theory, and combinatorics, illustrating how the fact-graph memory mechanism enables Danus to construct long, detailed mathematical proofs. Our results suggest that fact-graph-based orchestration provides an effective route toward scaling mathematical reasoning agents for long-horizon research problems. Danus is open source at https://github.com/frenzymath/Danus.

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