MechMath: Sorrifier-Driven Formal Decomposition Workflow for Automated Theorem Proving

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.

AI reasoning has become a central focus in contemporary artificial intelligence, largely driven by the success of large language models. However, mathematical research, which is characterized by non-linear derivation paths, rigorous logical requirements, and protracted exploration cycles, poses severe challenges for existing reasoning systems. To overcome these limitations, we present the MechMath Agent Team (MMAT), which is a large language model driven agent designed to serve as a co-pilot throughout the full cycle of mathematical research. We design a tripartite Harness Architecture that decouples system responsibilities into Control, Execution, and Augmentation planes, thereby reconciling rigorous logical control with the agility demanded by open-ended research. Building upon this framework, we instantiate three specialized agents: a Knowledge Base Manager, a Natural Language Prover, and a Formal Language Prover, all operating in a closed loop to produce formally certified mathematical proofs. We evaluate MMAT on open problems in Number Theory, Algebraic Complexity Theory, Differential Algebra, Operator Algebra, and Inequalities. Across a two-month deployment, 11 problems have been solved, demonstrating its capacity to act as a co-pilot throughout the entire research cycle. The contributions are threefold: a general decoupled Harness Architecture for multi-agent mathematical reasoning, its concrete instantiation in the MMAT system, and empirical validation on a diverse suite of open problems.

Lean is increasingly used to verify mathematical proofs and software produced by LLM agents, but writing formal proofs remains hard. Existing prover systems take a divide-and-conquer approach that recursively decomposes hard proof goals into easier lemmas and splits again any lemma that resists proof. However, a lemma can resist proof for different reasons, including insufficient guidance, a false statement, or a flaw in the decomposition that produced it. Existing systems cannot tell these apart as a failed attempt reports little beyond its failure, and responses are confined to the failing node without considering the rest of the proof or revising the plan above it. The system may thus keep decomposing obligations along a flawed proof direction until the budget runs out, causing the whole proof attempt to fail. We introduce Maestro, a Lean prover harness that handles failures through global coordination. Provers report why they failed, not just that they failed, and a coordinator that sees the whole proof chooses an appropriate response, such as guiding a stuck prover, correcting a false lemma, or retracting a flawed subtree and replanning it. The same harness extends to verified code synthesis, where a checked counterexample sends the code back for repair. We evaluate Maestro with four frontier LLMs on seven mathematics and three software verification benchmarks. It proves more targets than each model's native coding agent (Claude Code or Codex) on every benchmark the agent has not already saturated, with the largest gains on research-level mathematics. With Claude Opus 5, it discovers its own proofs of 34 of the 48 theorems in the June 2026 release of arXivLean, against 19 for the native agent. It also outperforms all published prover systems on every benchmark they share, solving 669/672 PutnamBench, 59/60 Lean-IMO-Bench, and 6/6 IMO 2026.