QED-Nano: Teaching a Tiny Model to Prove Hard Theorems

We study how model post-training and test-time inference design affect natural-language proof generation for hard olympiad mathematics. Starting from Nemotron 3 Ultra, we train two specialist checkpoints using supervised fine-tuning and reinforcement learning, and evaluate checkpoint choice, verification, and refinement. Based on these findings, we present an open-model test-time-compute pipeline. The system operates entirely in natural language, with no formal prover, external tools, or internet access. Three Nemotron 3 Ultra checkpoints - the general-availability model and two post-trained specialists - power an iterative search that generates, verifies, and refines candidate proofs; a separate high-compute stage then selects each final submission. The system scored 30 out of 42 points at IMO 2026, reaching the gold-medal threshold. We release the two post-trained checkpoints as well as the training data, the training and inference code, the submitted solutions, and Nemotron-IMO-Bench, a new benchmark of 200 novel olympiad-level problems.

The frontier of mathematics is defined by problems whose solutions are not yet known. However, whether language models can meaningfully engage with such problems without human intervention remains unclear. A major obstacle is the lack of large-scale research-level math datasets. To this end, we introduce ResearchMath-14k, a set of $14{,}056$ problems curated from academic sources via a multi-agent pipeline. ResearchMath-14k spans 11 mathematical domains and ranks above existing math datasets on knowledge, novelty, and procedural difficulty. To our knowledge, it is the largest research-level mathematical problem set available for training. We additionally generate $220$K teacher trajectories through targeted prompting, followed by behavioral filtering. Notably, however, generating correct trajectories is nontrivial at this level, and two LLM judges label only $3.7\%$ and $4.3\%$ of sampled ResearchMath training trajectories as correct. Nevertheless, across three model families, full-parameter training on ResearchMath improves performance on graduate- and research-level mathematics benchmarks by $2.1$ points over the starting checkpoints. In comparison, training on existing datasets such as DASD and Nemotron-SFT-Math-v4 changes performance by $0.0$ and $-0.5$ points, respectively. Notably, mixing DASD with ResearchMath yields higher scores than token-matched DASD alone on benchmarks covering olympiad short-form ($+2.0$), graduate- and research-level short-form ($+0.8$), graduate- and research-level symbolic ($+2.6$), and proof evaluation ($+5.7$). Further analysis suggests that research-level mathematical content and greater reasoning diversity may help explain why ResearchMath provides complementary supervision to contemporary datasets. We make ResearchMath-14k publicly available for future works on research-level mathematical reasoning.

We present QED, an open-source multi-agent system that turns human-provided research questions into complete mathematical proofs without further human guidance. Its pipeline is designed to overcome common failures of single-query proof generation by separating planning, proving, and verification: a decomposition agent structures the proof search, prover agents generate candidate arguments, and verifier agents check correctness. In collaboration with domain experts, we evaluated QED on 18 research-level projects of varying difficulty. QED produced five original works across algebraic geometry, fluid PDEs, probability, and inverse problems. Expert assessments regard these works as solid specialized research contributions, with three comparable in difficulty and scope to work commonly published in established specialist mathematics venues. QED is released at https://github.com/proofQED/QED.