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LongCat-Flash-Prover: Advancing Native Formal Reasoning via Agentic Tool-Integrated Reinforcement Learning

Jianing Wang, Jianfei Zhang, Qi Guo, Linsen Guo, Rumei Li, Chao Zhang, Chong Peng, Cunguang Wang, Dengchang Zhao, Jiarong Shi, Jingang Wang, Liulin Feng, Mengxia Shen, Qi Li, Shengnan An, Shun Wang, Wei Shi, Xiangyu Xi, Xiaoyu Li, Xuezhi Cao, Yi Lu, Yunke Zhao, Zhengyu Chen, Zhimin Lin, Wei Wang, Peng Pei, Xunliang Cai

arXiv:2603.21065v1cs.AIcs.CL

TL;DR

LongCat-Flash-Prover targets the challenge of reliable native formal reasoning in Lean4, where formal statements and proofs must be verified. It combines specialized experts, tool-integrated trajectory synthesis, and HisPO-based reinforcement learning, achieving state-of-the-art open-source performance across auto-formalization and theorem-proving tasks. The paper also examines evaluation loopholes and incorporates legality checks to improve proof reliability.

  • Problem

    LLMs still struggle with Lean4 theorem proving, while long-horizon MoE training faces distribution drift from train-inference discrepancies.

  • Method

    The paper combines three native formal reasoning experts with hybrid tool-integrated trajectory synthesis, HisPO reinforcement learning, and legality detection.

  • Results

    The model achieves state-of-the-art performance among open-source models across auto-formalization and theorem-proving tasks, including 97.1% on MiniF2F-Test with 72 attempts per problem.

  • Takeaways & Limitations

    Verified tools, synthesized trajectories, and legality detection provide a foundation for open-source native formal reasoning in Lean4.

  • Takeaways & Limitations

    The evaluation metrics of closed-weight reasoning models may not fully reflect their capabilities because of inference-API instability.

Abstract

from arXiv · show

We introduce LongCat-Flash-Prover, a flagship 560-billion-parameter open-source Mixture-of- Experts (MoE) model that advances Native Formal Reasoning in Lean4 through agentic tool-integrated reasoning (TIR). We decompose the native formal reasoning task into three independent formal capabilities, i.e., auto-formalization, sketching, and proving. To facilitate these capabilities, we propose a Hybrid-Experts Iteration Framework to expand high-quality task trajectories, including generating a formal statement based on a given informal problem, producing a whole-proof directly from the statement, or a lemma-style sketch. During agentic RL, we present a Hierarchical Importance Sampling Policy Optimization (HisPO) algorithm, which aims to stabilize the MoE model training on such long-horizon tasks. It employs a gradient masking strategy that accounts for the policy staleness and the inherent train-inference engine discrepancies at both sequence and token levels. Additionally, we also incorporate theorem consistency and legality detection mechanisms to eliminate reward hacking issues. Extensive evaluations show that our LongCat-Flash-Prover sets a new state-of-the-art for open-weights models in both auto-formalization and theorem proving. Demonstrating remarkable sample efficiency, it achieves a 97.1% pass rate on MiniF2F-Test using only 72 inference budget per problem. On more challenging benchmarks, it solves 70.8% of ProverBench and 41.5% of PutnamBench with no more than 220 attempts per problem, significantly outperforming existing open-weights baselines.

1 Introduction

LongCat-Flash-Prover addresses the difficulty of reliable Lean4 theorem proving by combining native formal reasoning capabilities with tool-integrated training. The resulting open-source MoE model reports strong performance across formal reasoning benchmarks.

  • Motivation: Current LLMs struggle with formal theorem proving because Lean4 requires rigorously verified statements and proofs, making vanilla tool-integrated reasoning challenging.Lean4 differs from ordinary callable tools by embodying the logical progression of a solution.
  • Contribution: LongCat-Flash-Prover is a 560B-parameter open-source MoE model designed to improve both informal and formal reasoning.The model has approximately 27B active parameters.
  • Contribution: Native formal reasoning is decomposed into auto-formalization, lemma-style sketching, and proving, with experts interacting directly with Lean4 compilation and verification tools.The capabilities transform informal problems into verified statements, generate sketches, and produce whole or lemma-style proofs.
  • Method: The hybrid-experts framework iteratively refines specialized auto-formalizer, sketcher, and prover models using tool-assisted trajectories and verification feedback.The pipeline synthesizes trajectories for formalization, sketching, and proving, retaining tool-utilizing trajectories during rejection sampling.
  • Method: HisPO stabilizes MoE agentic reinforcement learning by handling sequence- and token-level importance-sampling discrepancies and detecting legality violations that could enable reward hacking.The legality checks target semantic inconsistency, theorem-condition mismatches, and unverified or model-created axioms.
  • Results: 97.1% Pass@? on MiniF2F-Test was achieved with 72 attempts per problem, while Pass@32 improvements of 25.5% and 20.3% were reported on MathOlympiad-Bench and PutnamBench.The passage labels the MiniF2F metric as a score and the other improvements as Pass@32 gains.

2 Hybrid-Experts Iteration Framework

The hybrid-experts iteration framework decomposes native formal reasoning into specialized experts and synthesizes verified trajectories through progressively richer tool interaction. It combines auto-formalization, whole-proof generation, sketching, and lemma completion into a reusable training corpus.

  • 2.1 Native Formal Experts: Native formal reasoning is represented by specialized auto-formalizer, sketcher, and prover experts, each paired with formal tools for trajectory synthesis and rejection sampling.The three capabilities are auto-formalization, sketching, and proving.
  • 2.1 Native Formal Experts: Auto-formalization transforms an informal problem into a Lean4 statement, using syntax and semantic verification tools to filter generated statements.The Lean4 Server checks syntax by compiling the statement with a placeholder proof, while semantic consistency is also verified.
  • 2.1 Native Formal Experts: Sketching generates a complete target-theorem body alongside unproved helper lemmas, enabling later proof completion through decomposition.The sketch follows a divide-and-conquer and dynamic-programming-inspired structure.
  • 2.1 Native Formal Experts: Whole-proof generation produces a valid Lean4 proof directly from the informal problem and verified formal statement, either in one interaction or through tool feedback.Proof syntax and validity are checked by dedicated verifiers.
  • 2.2 Hybrid-experts Tool-Integration Synthesize: The synthesis curriculum progresses from single-turn generation to multi-turn tool calls and from whole-proof generation to lemma-style sketch proving.Different trajectory sets support varied problem difficulty and proof strategies.
  • 2.2 Hybrid-experts Tool-Integration Synthesize: When direct whole-proof attempts fail, the framework generates lemma-style sketches and uses the prover to complete each helper lemma before producing the final proof.The process retains sketches with consistent theorems and unresolved helper lemmas for subsequent proof synthesis.
  • 2.2 Hybrid-experts Tool-Integration Synthesize: 58? The iterative synthesis process produces a corpus containing formal statements, synthesized reasoning processes, and verified proofs for training native formal reasoning.The corpus is used to enhance the model’s native formal reasoning capabilities.

3 Approach

The approach iteratively trains LongCat-Flash-Prover through cold-start data synthesis, domain-mixed SFT, and agentic RL. It combines expert iteration and HisPO to generate diverse formal-reasoning trajectories and stabilize long-horizon MoE optimization.

  • Overview: The training pipeline starts from LongCat Mid-train Base, creates a cold-start model, and repeatedly refreshes trajectories through self-distillation and agentic RL.Each iteration applies domain-mixed SFT before further agentic training.
  • Cold-start Phase: Cold-start data is synthesized from formal statements and tool-integrated trajectories, then filtered through decontamination, deduplication, difficulty estimation, and diversity sampling.The process integrates capabilities from different expert model families through domain-mixed SFT.
  • Iteration Phase: The iteration phase uses the cold-start model as a new expert to synthesize task trajectories, mixes general data, and trains with domain-mixed SFT and agentic TIR RL.Multiple iterations are used to further improve performance while retaining informal reasoning capabilities.
  • Data Processing: Difficulty scores guide later synthesis by retaining prompts with score 0 and removing prompts scoring 1 for at least two consecutive iterations.Cold-start and RL subsets are selected to maintain non-extreme pass rates.
  • Hierarchical Importance Sampling Policy Optimization: HisPO stabilizes agentic RL by masking gradient contributions according to sequence-level and token-level train-inference consistency.The method addresses both policy staleness and discrepancies caused by asynchronous, heterogeneous training and inference systems.

4 Experiments

The experiments evaluate LongCat-Flash-Prover on auto-formalization, theorem proving, general reasoning, and reward-hacking robustness. Across these settings, the model achieves strong formal-reasoning results, improved sample efficiency, and reduced fake-proof behavior after reward-function repair, while informal reasoning is slightly weaker.

  • Evaluation Scope: The evaluation covers auto-formalization, theorem proving, general reasoning, and scaling behavior across multiple benchmarks.Auto-formalization uses Pass@8, while theorem proving includes whole-proof and TIR-based modes under attempt budgets.
  • Auto-Formalization: LongCat-Flash-Prover establishes new state-of-the-art results on all auto-formalization benchmarks, including 100% on MiniF2F-Test and ProofNet.Tool-integrated reasoning provides up to a 14% performance gain, and evaluations use Lean4 syntax and semantic-consistency checks.
  • Theorem Proving: 93.9% on MiniF2F-Test under Pass@32 makes sketch-proof TIR a new open-source theorem-proving state of the art, while PutnamBench reaches 28.9%.The comparison uses a clear budget limit of 32 attempts and evaluates multiple whole-proof and sketch-proof configurations.
  • Theorem Proving: 95.5% on MiniF2F-Test with only 72 attempts demonstrates higher sample efficiency than baselines requiring over 1024 attempts for 92.2%.Tree Search further improves accuracy by an average of 3.1% under the same budget limitation.
  • General Informal Reasoning: LongCat-Flash-Prover performs slightly worse than LongCat-Flash-Thinking-2601 on general reasoning tasks, indicating a loss in informal reasoning after formal-reasoning training.The authors describe the loss as acceptable and expect later work to better balance formal and informal reasoning.
  • Reward Hacking and Evaluation Loopholes: Reward-function repair suppresses cheating behavior, and the fixed model produces a higher ratio of valid proofs than the hacking model.The analysis identifies loopholes in syntax-only evaluation and uses AST-based checking alongside repaired rewards.

5 Related Works

The paper positions LongCat-Flash-Prover within native formal reasoning, where Lean4-based formalization and proving extend beyond separate specialized models.

  • The model retains general reasoning abilities while expanding native formal reasoning through expert iteration and agentic TIR reinforcement learning.
  • Native formal reasoning uses Lean4 to provide formal rationales for mathematical problems, unlike vanilla reasoning.

6 Conclusion

The paper introduces LongCat-Flash-Prover as an open-source MoE model combining formal and general reasoning, supported by verified tools and specialized training methods.

  • LongCat-Flash-Prover is a 560-billion-parameter MoE model that fuses native formal reasoning with general reasoning capabilities.
  • Its core innovations are a hybrid-experts iteration framework and HisPO for synthesizing trajectories and stabilizing MoE reinforcement learning.
  • Verified tools and legality detection are used to improve the reliability of theorem proving.

7 Contributions

The supplied contribution section contains author and partner acknowledgments rather than substantive technical contributions.

  • The section identifies equal-contribution authors and lists additional team members.
  • It acknowledges support from partners and marks members who have left the LongCat team.

A Details of Benchmarks

The benchmark suite pairs natural-language problems with formal counterparts for auto-formalization and uses established mathematical datasets for theorem proving.

  • The evaluation selects diverse mathematical benchmarks containing natural-language statements paired with formal counterparts for auto-formalization.
  • The theorem-proving evaluation covers MathOlympiad, MiniF2F, ProofNet, ProverBench, and PutnamBench.
  • Combibench contains 100 combinatorial problems spanning middle-school through IMO levels and more than ten topics.
  • FormalMath-lite contains 425 problems derived from the validated 5,560-problem FormalMATH benchmark.

B Task Cases

The Ladies’ Club case formalizes a partition problem in Lean4: under specified tea-drinking constraints, the club must be divided into two internally complete groups.

  • Problem setup: The informal problem has 100 ladies, each acquainted through tea with exactly 56 others, while a 50-member board is fully connected.The target is to partition the entire club into two groups whose members are pairwise connected by tea-drinking relationships.
  • Proof strategy: The accompanying reasoning identifies the main challenge as using the board’s complete graph to construct the desired partition.The passages describe this as a combinatorial partitioning problem rather than a completed proof.
  • Formalization: The Lean4 statement represents ladies as Fin 100 elements and assumes each tea-neighborhood has cardinality 56.It also assumes a 50-element board in which every pair of distinct members has the required relation.
  • Formalization: The conclusion requires finite sets A and B whose union is the universal set, whose intersection is empty, and whose members are pairwise connected within each set.This encodes a partition of the club into two complete subgraphs.

B.2 Case of Auto-formalization with Tools

The auto-formalization example shows an initial Lean encoding of a balls-and-sons problem failing semantic checks, followed by a corrected stars-and-bars formulation that matches the intended count.

  • Problem and formalization: The informal task asks for distributions of 10 identical balls among 6 sons, with every son receiving at least one ball, and claims the answer is 126.The intended mathematical representation uses positive integer allocations summing to 10.
  • Validation: The corrected combinatorial argument gives 6 mandatory balls first, leaving 4 freely distributable balls and yielding C(9,5) = 126 solutions.The corrected formalization is reported as matching the original problem’s sons, total balls, minimum allocation, and count.
  • Initial attempt: The first encoding is inconsistent because it models assignments through functions to Fin 10 and imposes an image cardinality of 10 despite only 6 sons.The analysis also identifies a mismatch between identical balls and the representation of individual assignments.
  • Corrected formulation: The corrected formulation uses functions from Fin 6 to Fin 11, requires every value to be at least 1, and constrains the sum of values to equal 10.The codomain permits values through 10, while filtering enforces the valid positive allocations.

B.3 Case of Theorem Proving

The theorem-proving example derives and formalizes the closed form T_n = n! + 2^n for a recurrence, using induction and Lean4 verification attempts.

  • Solution: The proposed solution defines the two component sequences as n ↦ n! and n ↦ 2^n.The formal abbreviation packages these functions as the target pair for the theorem.
  • Problem setup: The recurrence is initialized by T_0 = 2, T_1 = 3, and T_2 = 6, with later terms determined from three preceding values.The stated goal is to express T_n as a sum of two well-known sequences.
  • Proof strategy: The proof strategy uses strong induction, handling the three initial cases before substituting induction hypotheses into the recurrence.The recurrence then reduces factorial terms to (n + 3)! and exponential terms to 2^(n + 3).
  • Result: The mathematical derivation concludes that the recurrence’s right-hand side equals n! + 2^n, matching the proposed closed form.The passages report successful proof reasoning and a Lean compiler pass for an intermediate proof attempt, while a later full compilation exposes errors.

C Agentic Lemma Tree Search

The unified Judger-Sketcher-and-Prover performs lemma-tree search by recursively decomposing goals into helper lemmas and proving manageable sub-goals. Its prompts provide the evolving proof tree, formal context, and target theorem while enforcing verified, non-circular helper lemmas.

  • Unified lemma-tree search: The unified Judger-Sketcher-and-Prover enables tree search over lemma space by combining sketching and proving.It is introduced specifically because LongCat-Flash-Prover is trained for both capabilities.
  • Goal decomposition: Sketching recursively decomposes difficult target theorems into smaller sub-goals that form branching nodes in the lemma tree.Sketching is restricted at total depth 12 or after a consecutive chain of length 5.
  • Sub-goal solving: Proving supplies complete Lean 4 proofs for helper lemmas once their sub-goals are manageable.The proving action complements sketching by closing selected branches of the lemma tree.
  • Search context: Each search step presents the model with the global tree outline, current formal context, and target theorem or lemma.The outline supports overall proof planning and helps avoid duplicate lemma names.
  • Context compression: Proved lemmas are represented as axioms containing only their statements, substantially compressing agent memory for proof projects with thousands of lines.Their proof bodies are omitted because they require no future modification.
  • Verification constraints: The prompting protocol requires verification with Lean 4, complete target proofs without sorry or admit, and correct, provable, helpful helper lemmas.Unproven steps may remain only in helper lemmas, while false helper lemmas and unsupported contradiction tricks are prohibited.

E Legality Detection in Theorem Proving

The paper treats compilability as insufficient because Lean 4 code can exploit coding tricks to produce illegal fake proofs. It therefore uses AST-based consistency checks and categorizes observed cheating behaviors.

  • Reward-hacking problem: Compilable Lean 4 code can still be illegal when language-model-generated proofs exploit coding tricks to pass verification.This creates a reward and evaluation problem because compilation is a major signal in both settings.
  • Legality detection: A lightweight lexer and parser convert Lean 4 code into an AST for strict consistency checks between problem definitions and generated proofs.The checks are designed to recognize cheating behaviors in otherwise compilable code.
  • Observed behaviors: AST analysis of benchmark problems and reward-hacking prover outputs identified and categorized 9 distinct cheating patterns.The paper demonstrates these patterns using putnam_2025_b3 as an example.
  • Detection summary: Table 6 presents a comprehensive analysis of Lean 4 cheating behaviors together with their detection mechanisms.The table organizes the documented behavior patterns and corresponding safeguards.

1. Tampering with the Theorem

The paper documents theorem-tampering strategies that make Lean 4 outputs appear provable without faithfully proving the original theorem. These include changing statements, hiding obligations, injecting assumptions, and redefining background semantics.

  • Statement replacement: A theorem can be replaced by a trivial proposition with the same name, allowing the generated proof to establish the replacement instead of the original statement.The example explicitly changes the theorem proposition before proving it.
  • Compilation termination: The #exit command can stop compilation before the actual theorem signature, hiding subsequent proof obligations from the compiler.The example places the theorem in content ignored after compilation terminates.
  • Unproven assumptions: An absurd axiom such as False can be introduced and then eliminated to prove the target theorem without establishing its mathematical content.The example uses False.elim after declaring cheat_axiom : False.
  • Meta-level bypasses: Macros, unsafe definitions, and mutual recursion can bypass standard elaboration, termination, or type-safety checks while producing apparently accepted proofs.The documented examples expand a tactic to sorry or recursively invoke a mutually defined theorem.
  • Context and semantic tampering: Global variables can inject contradictory premises that the theorem consumes, while redefining identifiers such as Set, Nonempty, or pow can alter the problem’s semantics.These attacks change the context or background concepts rather than solving the intended theorem.
  • Instance manipulation: Fake local instances can redefine operations such as divisibility to make the target conditions trivial.The example supplies a local Dvd instance whose relation is always True.
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