We introduce Goedel-Architect, an agentic framework for formal theorem proving in Lean 4 centered on blueprint generation and refinement. A blueprint is a dependency graph of definitions and lemmas that builds up to the main theorem. First, Goedel-Architect generates a blueprint of formally stated definitions and lemmas, along with declared dependencies. This blueprint is optionally guided by a natural language proof. Then, a tool-equipped Lean prover component closes each open lemma node in parallel using relevant dependencies. Failed lemmas in turn drive refinement of the global blueprint. This strategy contrasts with other mainstream approaches which use recursive lemma decomposition, and can inefficiently loop on dead-end strategies. Using the open-weight DeepSeek-V4-Flash (284B-A13B) as the backbone, Goedel-Architect attains 99.2% pass@1 on MiniF2F-test and 75.6% pass@1 on PutnamBench. With an optional natural-language proof seeding the initial blueprint on the harder problems, we additionally close the remaining two MiniF2F-test problems (reaching 100%), lift PutnamBench to 88.8% (597/672), and solve 4/6 on IMO 2025, 11/12 on Putnam 2025, and 3/6 on USAMO 2026. This represents state-of-the-art performance for an open-source pipeline at a price point up to 500x less than comparable open-source pipelines.
Large Language Models (LLMs) exhibit strong informal mathematical reasoning but struggle to generate mechanically verifiable proofs in formal languages like Lean. We present LEAP, an agentic framework that enables general-purpose foundation models to achieve state-of-the-art performance on automated formal theorem proving. LEAP leverages foundation model capabilities, such as informal reasoning, instruction following, and iterative self-refinement. By decomposing complex problems into smaller units, the system bridges formal proof construction with informal blueprints through continuous interaction with the Lean compiler. To provide a rigorous evaluation beyond increasingly saturated benchmarks, we introduce Lean-IMO-Bench, a benchmark of IMO-style problems formalized in Lean, with short statements yet highly non-routine and multi-step proofs across a wide range of difficulty levels. Empirically, on the latest 2025 Putnam Competition, an annual mathematics competition for undergraduate students in North America, LEAP solves all 12 problems, matching recent breakthroughs by frontier formal mathematical models. On Lean-IMO-Bench, LEAP boosts the one-shot formal solve rate of general-purpose LLMs from below 10% to 70%, notably surpassing the 48% benchmark set by a specialized, gold-medal-caliber IMO system. Furthermore, we demonstrate LEAP's research-level utility by autonomously formalizing complex proofs for open combinatorial challenges, including a verified proof for a key subproblem in Knuth's Hamiltonian decomposition of even-order Cayley graphs.
We introduce DreamProver, an agentic framework that leverages a "wake-sleep" program induction paradigm to discover reusable lemmas for formal theorem proving. Existing approaches either rely on fixed lemma libraries, which limit adaptability, or synthesize highly specific intermediate lemmas tailored to individual theorems, thereby lacking generality. DreamProver addresses this gap through an iterative two-stage process. In the wake stage, DreamProver attempts to prove theorems from a training set using the current lemma library while proposing new candidate lemmas. In the "sleep" stage, it abstracts, refines, and consolidates these candidates to compress and optimize the library. Through this alternating cycle, DreamProver progressively evolves a compact set of high-level, transferable lemmas that can be effectively used to prove unseen theorems in related domains. Experimental results demonstrate that DreamProver substantially improves proof success rates across a diverse set of mathematical benchmarks, while also producing more concise proofs and reducing computational cost.