The LLMs show significant improvements in the mathematical reasoning when they are extended through natural language. This results in improved performance on benchmarks like MATH and AIME. Reinforcement learning (RL), which is used to train such models, faces a problem: it’s very hard to verify the accuracy of natural-language proofs, and each step of reasoning must be checked manually. The application of RL to train mathematical theorem proving models is limited by this. Despite the fact that formal languages, such as Lean, offer automated correctness checks, LLM’s formal provers are limited. The step-level compilers generate incremental code, but they require scaffolding. They also lack the high-level reasoning capability.
ByteDance Seed Team has introduced Seed-Prover – a whole-proof reasoning system based on lemmas. The model refines proofs using Lean Feedback, lemmas that have been established, and self summarization. The Seed-Prover uses three specialized inference methods that are tested at the test time to allow for deep and wide reasoning. Lemma-style proofs are its primary innovation, as they place lemmas front and center in the reasoning process. Moreover, this paper introduces Seed-Geometry, a complementary geometry reasoning engine that overcomes Lean’s limitations in handling geometric support.
Multi-task RL, based on the VAPO, is used to interact between Seed-Prover, Lean and other RLs. In the training dataset, open-source datasets are combined with formal problems created in-house. A proposer is used to simplify difficult tasks. This excludes simple problems whose proof rate is above 25%. Seed-Geometry’s backend allows for large-scale problems generation. It identifies over 230,000,000 unique problems in seven days, with an 8-fold increase in search efficiency. Although a separate value model and policy are created, extensive testing indicates that value models could reduce performance as a result of estimation error. The result is that step-bystep generation using beam search has been adopted for distributed setups.
The results of Seed-Prover are state-of-the art in terms of multiple mathematical benchmarks. Seed Prover is able to solve 5 of the 6 problems in IMO-2025. Seed Geometry can instantly solve Problem 2 and Seed Prover will derive proofs from various settings for remaining problem. It has successfully solved 121 tasks out of 155 in the past IMO, achieving 78.1% across all difficulties. It shows good results in all problem categories. 47 of the 55 easiest problems were solved, followed by 47 of 56 medium and hard problems.
MiniF2F researchers have achieved a 99.6% rate of proof for both test and validation sets in medium settings. This is despite the fact that they are solving difficult questions such as IMO P3 1990. The PutnamBench shows a performance improvement of 201 out of 657 solved problems when switching from the light inference setting to the medium one. This is compared to previous undergraduate level mathematical reasoning systems. Seed-Prover, on CombiBench solves 30 of 100 combinatorics questions, outperforming other methods, but still revealing the challenges associated with combinatorial reasoning. Researchers achieved 81.8% on MiniCTX v2, demonstrating strong generalization outside of competition problems. They also outperformed the baseline o4 mini’s 44.3% for Pass@8.
ByteDance Seed concludes by presenting Seed-Geometry, and Seed-Prover. These two formal reasoning techniques integrate the abilities of LLMs. Seed Geometry uses accelerated search, a more powerful verification mechanism and a faster way to verify. Seed Prover is based on iterative refining and sophisticated test-time strategies. These methods are effective in tackling mathematical elite competitions, as shown by their ability to solve 5 of the 6 problems at the IMO 2025. Lean is a form of formal language which allows rapid verification, more accurate than LLM-based judgements and cheaper than experts. The future research focus will be on the combination of formal systems and LLMs in order to tackle open conjectures.
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Sajjad A. Ansari, a student in the final year at IIT Kharagpur. Tech-enthusiast, Sajjad Ansari focuses on the real world implications of AI and its practical applications. He strives to make complex AI ideas clear and understandable.
