pith. sign in

arxiv: 2606.03303 · v2 · pith:7Q35S6WKnew · submitted 2026-06-02 · 💻 cs.AI

LEAP: Supercharging LLMs for Formal Mathematics with Agentic Frameworks

Po-Nien Kung , Linfeng Song , Dawsen Hwang , Jinsung Yoon , Chun-Liang Li , Simone Severini , Mirek Ol\v{s}\'ak , Edward Lockhart
show 5 more authors
Quoc V Le Burak Gokturk Thang Luong Tomas Pfister Nanyun Peng
This is my paper

Pith reviewed 2026-06-28 09:48 UTC · model grok-4.3

classification 💻 cs.AI
keywords formal theorem provingLeanlarge language modelsagentic frameworksPutnam competitionIMO problemsformal verificationcombinatorial proofs
0
0 comments X

The pith

LEAP agentic framework lets general LLMs reach 70 percent formal solve rate on IMO-style problems by decomposing them and iterating with the Lean compiler.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents LEAP as a way to convert the informal reasoning strengths of ordinary large language models into mechanically correct formal proofs in Lean. It does this by splitting hard problems into smaller pieces, generating informal blueprints, and then using repeated rounds of Lean compiler feedback to correct and complete the formal code. On the 2025 Putnam Competition the system solves every problem, and on the new Lean-IMO-Bench it lifts one-shot success from below 10 percent to 70 percent, beating the 48 percent mark of a specialized IMO solver. The same setup also produces a verified formal proof for a subproblem in an open combinatorial question about Cayley graphs. Readers would care because the method removes the need for hand-written formal proofs or model retraining while still producing checkable results.

Core claim

LEAP solves all twelve problems on the 2025 Putnam Competition and raises the one-shot formal solve rate of general-purpose LLMs from below 10 percent to 70 percent on Lean-IMO-Bench, surpassing the 48 percent benchmark of a specialized IMO system, by decomposing complex problems into smaller units, bridging informal reasoning with formal construction, and performing iterative self-refinement through continuous interaction with the Lean compiler; the same process autonomously produces a verified proof for a key subproblem in Knuth's Hamiltonian decomposition of even-order Cayley graphs.

What carries the argument

The LEAP agentic framework that decomposes complex problems into smaller units and bridges formal proof construction with informal blueprints through continuous interaction with the Lean compiler.

If this is right

  • General-purpose LLMs can match or exceed the formal proving performance of models built specifically for IMO problems.
  • The same decomposition-plus-compiler loop can be applied to open research questions in combinatorics without human-written formal sketches.
  • One-shot formal proving becomes feasible for problems whose proofs require many interdependent steps.
  • Formal verification of competition-level mathematics no longer requires specialized training beyond the base LLM.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same interaction pattern could be tested on formal systems other than Lean to check whether the gains depend on Lean's particular error messages.
  • Success on Putnam and IMO problems suggests the method might accelerate formalization of textbook mathematics where proofs are already known but not yet coded.
  • If the decomposition step can be automated further, the framework might reduce the human effort needed to verify claims in active research areas.

Load-bearing premise

Continuous interaction between the LLM's informal reasoning and the Lean compiler, combined with decomposition and self-refinement, is sufficient to produce mechanically correct formal proofs for non-routine multi-step problems without additional human guidance or domain-specific training.

What would settle it

A new collection of IMO-style Lean problems with short statements but non-routine multi-step proofs on which LEAP achieves a one-shot solve rate below 40 percent would show the framework does not generalize as described.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper presents LEAP, an agentic framework that combines LLM informal reasoning, problem decomposition, continuous Lean compiler interaction, and self-refinement to generate formal proofs. It claims to solve all 12 problems from the 2025 Putnam Competition, raise the one-shot formal solve rate of general-purpose LLMs from below 10% to 70% on the newly introduced Lean-IMO-Bench (surpassing the 48% rate of a specialized IMO system), and autonomously produce a verified formal proof for a key subproblem in Knuth's Hamiltonian decomposition of even-order Cayley graphs.

Significance. If the empirical results are reproducible and the generated proofs are mechanically verified, the work would demonstrate that general-purpose LLMs can reach competitive performance on non-routine formal mathematics via agentic scaffolding, with potential implications for scaling automated theorem proving beyond specialized models.

major comments (3)
  1. [Abstract and evaluation sections] Abstract and § on evaluation: the central performance claims (all 12 Putnam problems solved; 70% one-shot rate on Lean-IMO-Bench) are stated without any experimental protocol, prompt templates, interaction logs, ablation studies, or explicit verification that the output Lean scripts type-check and are accepted by the compiler. This renders the quantitative results unverifiable from the manuscript.
  2. [Benchmark description] Lean-IMO-Bench introduction: the benchmark is presented as containing short-statement yet multi-step IMO-style problems, yet no problem list, difficulty distribution, formalization source, or comparison methodology against the 48% specialized baseline is supplied, making the 70% claim impossible to assess or replicate.
  3. [Applications to open problems] Knuth subproblem demonstration: the claim of autonomous formalization of a verified proof is made at a high level with no artifact, interaction trace, or statement of the precise subproblem formalized, so the research-level utility assertion cannot be evaluated.
minor comments (2)
  1. [Framework description] Notation for the agentic loop (decomposition, refinement, compiler feedback) should be formalized with a diagram or pseudocode to clarify the continuous interaction process.
  2. The manuscript should include at least one full example of a generated Lean proof script together with the corresponding informal blueprint to illustrate the bridging mechanism.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback highlighting the need for greater transparency and reproducibility. We agree that the current manuscript would benefit from expanded details on protocols, benchmarks, and artifacts to allow independent verification of the claims. We outline our responses to each major comment below and commit to incorporating the requested information in a revised version.

read point-by-point responses

  1. Referee: [Abstract and evaluation sections] Abstract and § on evaluation: the central performance claims (all 12 Putnam problems solved; 70% one-shot rate on Lean-IMO-Bench) are stated without any experimental protocol, prompt templates, interaction logs, ablation studies, or explicit verification that the output Lean scripts type-check and are accepted by the compiler. This renders the quantitative results unverifiable from the manuscript.

    Authors: We acknowledge this limitation in the current draft. The manuscript presents high-level results but omits the detailed experimental setup. In the revision we will add a dedicated subsection describing the full interaction protocol, all prompt templates used, sample interaction logs for representative problems, ablation studies on key components (e.g., decomposition and self-refinement), and explicit compiler verification steps confirming that generated Lean scripts type-check. This will enable readers to reproduce the reported solve rates. revision: yes

  2. Referee: [Benchmark description] Lean-IMO-Bench introduction: the benchmark is presented as containing short-statement yet multi-step IMO-style problems, yet no problem list, difficulty distribution, formalization source, or comparison methodology against the 48% specialized baseline is supplied, making the 70% claim impossible to assess or replicate.

    Authors: We agree that the benchmark description is currently insufficient for replication. The revision will include the complete list of problems (or a representative subset with links to the full set), the difficulty distribution, the source of the formalizations, and a clear methodology section explaining how the 48% baseline was obtained and how our 70% one-shot rate was measured under identical conditions. revision: yes

  3. Referee: [Applications to open problems] Knuth subproblem demonstration: the claim of autonomous formalization of a verified proof is made at a high level with no artifact, interaction trace, or statement of the precise subproblem formalized, so the research-level utility assertion cannot be evaluated.

    Authors: We recognize that the open-problem demonstration lacks the necessary supporting artifacts. In the revised manuscript we will state the precise subproblem from Knuth's Hamiltonian decomposition, provide the full interaction trace, and include (or link to) the verified Lean artifact demonstrating that the proof was autonomously generated and mechanically checked. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical results on external benchmarks

full rationale

The paper reports direct empirical measurements of solve rates on the 2025 Putnam Competition and the introduced Lean-IMO-Bench, along with an autonomous formalization example. These outcomes are presented as experimental results from applying the LEAP framework, not as quantities derived from fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the central claims to tautologies. No equations, uniqueness theorems, or ansatzes are invoked in a manner that collapses the reported performance into the inputs by construction. The derivation chain consists of framework description followed by benchmark evaluation, which remains self-contained against external verification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that foundation models already possess usable informal reasoning and self-refinement abilities that can be channeled into formal proofs via agentic interaction; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Foundation models exhibit strong informal mathematical reasoning, instruction following, and iterative self-refinement capabilities
    Explicitly invoked in the abstract as the capabilities leveraged by LEAP.

pith-pipeline@v0.9.1-grok · 5838 in / 1350 out tokens · 23994 ms · 2026-06-28T09:48:42.272880+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

18 extracted references · 3 canonical work pages · 2 internal anchors

  1. [1]

    URLhttps://arxiv.org/abs/2603.12744. G. Tsoukalas, J. Lee, J. Jennings, J. Xin, M. Ding, M. Jennings, A. Thakur, and S. Chaudhuri. Put- nambench: Evaluating neural theorem-provers on the putnam mathematical competition. In 14 LEAP: Supercharging LLMs for Formal Mathematics with Agentic Frameworks A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J....

    work page arXiv

  2. [2]

    Advancing Mathematics Research with AI-Driven Formal Proof Search

    doi: 10.52202/079017-0368. URLhttps://proceedings.neurips.cc/paper_files /paper/2024/file/1582eaf9e0cf349e1e5a6ee453100aa1-Paper-Datasets_and_Be nchmarks_Track.pdf. G. Tsoukalas, A. Kovsharov, S. Shirobokov, A. Surina, M. Firsching, G. Bérczi, F. J. Ruiz, A. Suggala, A. Z. Wagner, E. Wieser, et al. Advancing mathematics research with ai-driven formal proo...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.52202/079017-0368 2024

  3. [3]

    Kimina-Prover Preview: Towards Large Formal Reasoning Models with Reinforcement Learning

    URLhttps://openreview.net/forum?id=ljAHonPrs1. H. Wang, M. Unsal, X. Lin, M. Baksys, J. Liu, M. D. Santos, F. Sung, M. Vinyes, Z. Ying, Z. Zhu, J. Lu, H. de Saxcé, B. Bailey, C. Song, C. Xiao, D. Zhang, E. Zhang, F. Pu, H. Zhu, J. Liu, J. Bayer, J. Michel, L. Yu, L. Dreyfus-Schmidt, L. Tunstall, L. Pagani, M. Machado, P. Bourigault, R. Wang, S. Polu, T. B...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.18653/v1/2025.acl-long.1565 2025

  4. [4]

    Polynomial Roots and Splitting: Since𝑃 has natural degree𝑘 and 𝑘 distinct real roots, it splits completely overℝ, and the multiset of its roots𝑆has cardinality𝑘

  5. [5]

    Vieta’s Formulas: Vieta’s formulas express the coefficients𝑐0, 𝑐1, 𝑐2 in terms of the elementary symmetric polynomials of𝑆: 𝑐0 =𝑐 𝑘 (−1) 𝑘 𝐸𝑘 𝑐1 =𝑐 𝑘 (−1) 𝑘−1 𝐸𝑘−1 𝑐2 =𝑐 𝑘 (−1) 𝑘−2 𝐸𝑘−2 where𝐸 𝑖 =esymm 𝑖 (𝑆)

  6. [6]

    Purely multiset inductive identities show:Í 𝑌=𝐸 𝑘−1 esymm2 (𝑌)=𝐸 𝑘 𝐸𝑘−2Î 𝑌=(𝐸 𝑘)𝑘−1 4.Sum of Squares: For the multiset𝑍=𝑐 𝑘𝑌, we evaluate the sum of its squares𝑊={𝑧2 |𝑧∈𝑍}

    Multiset Identity: We construct a new multiset𝑌 by mapping each root𝑥∈𝑆 to the product of all other roots,(𝑆\ {𝑥}).prod. Purely multiset inductive identities show:Í 𝑌=𝐸 𝑘−1 esymm2 (𝑌)=𝐸 𝑘 𝐸𝑘−2Î 𝑌=(𝐸 𝑘)𝑘−1 4.Sum of Squares: For the multiset𝑍=𝑐 𝑘𝑌, we evaluate the sum of its squares𝑊={𝑧2 |𝑧∈𝑍}. By the relation(Í 𝑍) 2 = Í(𝑍 2) +2esymm 2 (𝑍), we can algebraic...

  7. [7]

    Since 𝑃 has degree𝑘 and 𝑐0 ≠ 0, both 𝑐0 and𝑐 𝑘 are non-zero integers

    Integer Product Bound: The product of𝑍 evaluates to𝑐𝑘 ((− 1)𝑘𝑐0)𝑘−1. Since 𝑃 has degree𝑘 and 𝑐0 ≠ 0, both 𝑐0 and𝑐 𝑘 are non-zero integers. Thus, the product of𝑍is a non-zero integer, implyingÎ 𝑊=( Î 𝑍) 2 ≥1

  8. [8]

    Required Global Definitions, Variables, or Structures No new definitions, axioms, or structures are needed

    AM-GM Inequality: Applying the AM-GM inequality to the multiset𝑊 (which consists of𝑘 non-negative real numbers whose product is≥1), we obtainÍ 𝑊≥𝑘=⇒𝐾(𝑐) ≥𝑘. Required Global Definitions, Variables, or Structures No new definitions, axioms, or structures are needed. We use purely standard Mathlib components (likeMultiset, Polynomial, andesymm). Smaller Lemm...

  9. [9]

    Define𝑃as the sum Í 𝑖∈Finset.Icc0𝑘 monomial𝑖(𝑐 𝑖)and𝑃 𝑅 as𝑃.map(algebraMapℤ ℝ)

  10. [10]

    Applycoeff_of_sum_Iccto assert𝑃 𝑅.coeff𝑖=(𝑐 𝑖 :ℝ)for𝑖∈ {0,1,2, 𝑘}

  11. [11]

    Establish 𝑃𝑅.roots.card=𝑘 using card_roots_eq_of_ncard_rootSet and the natural degree injectivity

  12. [12]

    Establish that𝑃 𝑅.splits(RingHom.idℝ)follows fromPolynomial.splits_iff_card_roots

  13. [13]

    InvokeVieta’sformulas(Polynomial.coeff_eq_esymm_roots_of_splits)toexpress 𝑐0,𝑐 1, and𝑐 2 in terms of𝑠.esymm𝑖

    Let 𝑠=𝑃 𝑅.roots. InvokeVieta’sformulas(Polynomial.coeff_eq_esymm_roots_of_splits)toexpress 𝑐0,𝑐 1, and𝑐 2 in terms of𝑠.esymm𝑖

  14. [14]

    Define multisets 𝑌 and 𝑍 matching the theoretical blueprint. Use multiset_sum_sq_eq, multiset_map_erase_prod_sum, andmultiset_map_erase_prod_esymm_twoto show that the sum of the squared elements of𝑍expands algebraically to exactly(𝑐2 1 −2𝑐 0𝑐2 :ℝ)=(𝐾(𝑐):ℝ)

  15. [15]

    Usemultiset_map_erase_prod_prodto find𝑍.prod=𝑐 𝑘 ((−1) 𝑘𝑐0)𝑘−1

  16. [16]

    Observe that since𝑐0 and 𝑐𝑘 are non-zero integers, their algebraic combination𝑍.prod represents a non-zero integer, so its square (the product of𝑊=𝑍2) is≥1

  17. [17]

    Feed𝑊tomultiset_sum_ge_card_of_prod_ge_oneto deduce that𝑊.sum≥𝑊.card

  18. [18]

    Use norm_cast to translate this back to(𝑘:ℤ) ≤𝐾(𝑐)

    Using 𝑊.card=𝑘 and 𝑊.sum=(𝐾(𝑐) : ℝ), deduce (𝑘 : ℝ) ≤ (𝐾(𝑐) : ℝ). Use norm_cast to translate this back to(𝑘:ℤ) ≤𝐾(𝑐). 24