Lean Theorem Proving

Large language models are increasingly capable of mathematical reasoning, but the proofs they generate are often unreliable and hard to verify. Interactive theorem provers such as Lean 4 address this by accepting only kernel-checked proofs; however, their reach is bounded by the formalized knowledge available. While Mathlib, a repository of formalized Lean 4 theorems that covers diverse mathematical areas, certain specialized areas remain underrepresented; notably, the domain of Combinatorics on Words (CoW). CoW studies sequences, exploring their properties such as periodicity, borders, conjugacy, and morphisms. As a result, specialized provers, trained on Mathlib-centered data, lack the lemmas to operate in CoW. We present two contributions. First, we introduce a Lean 4 formalization of CoW containing eight modules and \textbf{93} declarations of core definitions and foundational lemmas. Second, we present LAMP, a multi-agent framework that synthesizes kernel-verified Lean 4 proofs by providing explicit, structured domain knowledge at inference time through an ontology, rather than by fine-tuning a prover. LAMP coordinates a Planner, Builder, and Verifier with Model Context Protocol based access to a domain-specific CoW ontology. In a suite of 90 CoW theorems that span all eight modules and three difficulty levels, LAMP synthesizes verified proofs for 96.7% of theorems, substantially exceeding both an unscaffolded baseline and existing specialized provers. An ablation shows that removing LAMP's tool-grounded architecture or its Planner/Builder separation each cost roughly 12 percentage points, even with the backbone model held fixed.

The axiom of choice has divided the foundations of mathematics for over a century, but the distinction between classical and constructive proofs has remained a philosophical and methodological one. We use Lean 4's kernel-level tracking of axiom dependence to show that the axiom of choice has a measurable geometric correlate in proof space that obeys a one-parameter mixture law and has operational consequences for neural theorem provers. To do this, we partition $471{,}260$ declarations of Mathlib by transitive dependence on the axiom of choice and represent a filtered population of $42{,}355$ traced theorems by their sequences of tactic invocations. We use the constructive proofs in this dataset to train a self-supervised proof encoder and show that when using it to measure classical proofs, three complementary measurements (anomaly score, reconstruction loss, and density-superlevel containment) exhibit a common decline with the proof's distance from the axiom in the dependency graph, from sharp separation at the shallow boundary (AUC $0.847$ at distance $2$) to indistinguishability at distance~$9{+}$. Robustness controls show that the signature survives length, file, author, and topic controls, and replicates under full-source encoders trained on normalised proof source. Operationally, we show that on an evaluation sample of $251$ Mathlib theorems, Lean's \texttt{aesop} tactic solves constructive theorems at $13\times$ the rate of classical ones, and a neural-guided hybrid using the ReProver tactic generator compresses the gap to $5\times$. The geometric anomaly score predicts \texttt{aesop} failure beyond proof length, providing an operational link between the geometric signature and prover performance.

Headline type-correctness (TC%) of LLM autoformalization has climbed from $\sim$53% to $\sim$76% in two years, yet this scalar conceals which errors each method resolves. We propose a signal-coverage matrix that crosses the Lean elaborator (pass/fail) with a semantic-equivalence judgment (equivalent/not), sorting every output into one of four cells: true success (TS), type-only (TO), semantic-only (SO), or both fail (BF). On ProofNet# and MiniF2F-test with DeepSeek V4-Pro across Vanilla, Lean-Retry, Sample-Filter, and Stratified Autoformalization (SAF): (1) the +34 to +36 TS gain across the three elab-feedback methods is $\sim$64% type-stratum recovery, with SO flat on net (87.5% of original semantic errors rescued, 8 newly created). (2) The TO-to-TS rate is 23/61 for each method (Wilson 95% CI [26.6%, 50.3%]), and this stratum-level recovery rate predicts $Δ$TS on held-out methods to within 2/186 and renders $Δ$TC linear in the Vanilla elab-fail rate across six (model, dataset) cells ($R^2=0.96$). (3) The two judges disagree by 26 to 37 pp on elab-feedback outputs (vs. 7 pp on Vanilla), with 30 to 56% of symbolic-judge false negatives traceable to elaborator-forced rewrites. The persistent residual reduces to two gold-formalization errors. TC% gains should be credited by which cell moved, not by the scalar alone.

Auto-formalization is critical for scalable formal verification, but existing progress largely focuses on isolated statements, while theory-scale auto-formalization, which coherently translates hundreds of interdependent definitions, lemmas, and theorems, remains open due to challenges in consistency, faithfulness, scalability, and correctness. In this paper, we introduce LCS-Bench, a stand-alone, theory-scale benchmark based on Logics for Computer Science. LCS-Bench is built through a novel semi-automated agentic pipeline that leverages concept graphs, formal signature planning, issue tracking, sorry-filling with counter-example search, complemented by faithfulness review from human experts. The resulting artifact covers 327 textbook items, over 4,076 Lean declarations, and more than 85K lines of Lean code. The dataset supports broad evaluation through a data engine that automatically derives five tracks of evaluation benchmarks, measuring different aspects of auto-formalization and theorem-proving capabilities. We also introduce a novel evaluation protocol featuring definitional equivalence checkers, enabling more fine-grained and faithful assessment. Through extensive evaluation on 14 models, we demonstrate that (1) LCS-Bench is of high quality, consistent, and faithful; (2) the benchmark is challenging, with state-of-the-art models achieving only 20.1% on auto-formalization tasks; and (3) our analysis reveals key findings regarding theory-scale auto-formalization and suggests promising directions for future work.

We present AXLE (Axiom Lean Engine), a cloud service for Lean 4 proof manipulation, extraction, and verification. Recent progress in AI for mathematics -- reinforcement learning pipelines, agentic proving workflows, dataset curation -- demands Lean 4 tooling that scales to millions of requests while remaining correct and robust; existing infrastructure offers parallel compilation but not scalable proof verification, higher-level proof manipulation, multi-version support, or per-request isolation at the throughput modern AI workflows require. AXLE provides 14 Lean 4 metaprogramming tools spanning strict proof verification, declaration metadata extraction, semantic source manipulation, deterministic proof repair and simplification, and lemma extraction. The service runs as a multi-tenant cloud deployment with per-request isolation and concurrent support for multiple Lean 4 and Mathlib versions, accessible via a Python SDK, command-line interface, web UI, MCP server, and raw HTTP API. AXLE is publicly available and free to use at https://axle.axiommath.ai and via the axiom-axle PyPI package, with no local Lean 4 installation required. It has served over 500 million requests to date and is the underlying infrastructure for Axiom Math's proving efforts, including its 12/12 score on the 2025 Putnam competition.

While reinforcement learning from verifiable rewards (RLVR) typically has relied on a single binary verification signal, symbolic proof assistants in formal reasoning offer rich, fine-grained structured feedback. This gap between structured processes and unstructured rewards highlights the importance of feedback that is both dense and sound. In this work, we demonstrate that the Lean proof assistant itself can serve as a symbolic process oracle, supplying both outcome-level and fine-grained tactic-level verified feedback during training. Proof attempts are parsed into tactic sequences, and Lean's elaboration marks both locally sound steps and the earliest failing step, yielding dense, verifier-grounded credit signals rooted in type theory. We incorporate these structured rewards into a GRPO-style reinforcement learning objective with first-error propagation and first-token credit methods that balances outcome- and process-level advantages. Experiments with STP-Lean and DeepSeek-Prover-V1.5 show that tactic-level supervision outperforms outcome-only baselines in most settings, delivering improvements on benchmarks such as MiniF2F and ProofNet. Beyond empirical gains, our study highlights a broader perspective: symbolic proof assistants are not only verifiers at evaluation time, but can also act as process-level reward oracles during training. This opens a path toward reinforcement learning frameworks that combine the scalability of language models with the reliability of symbolic verification for formal reasoning.

Enhancing the formal math reasoning capabilities of Large Language Models (LLMs) has become a key focus in both mathematical and computer science communities in recent years. While significant progress has been made in using state-of-the-art Auto-Regressive (AR) LLMs for formal theorem proving, these models suffer from inherent limitations. Their next-token prediction generation methods may yield suboptimal performance due to the challenges of long-range coherence and the compounding of errors over long sequences. Recent advancements in diffusion LLMs (dLLMs), which generate text through iterative denoising of a multi-token block, offer a promising alternative. However, the application of dLLMs to formal mathematics, where maintaining long-range coherence is critical, remains largely understudied. To address the challenges above, we propose Diffusion-Proof, to the best of our knowledge, the first framework to train and apply dLLMs for formal theorem proving. Our frameworks contain training and inference methods for two models. The first one is dLLM-Prover-7B, which performs whole-proof writing with long-range coherent tactic usage. The second one is dLLM-Corrector-7B, which is a novel large block diffusion-based correction model. It leverages the in-filling capabilities of dLLMs to perform local proof correction using bi-directional information. Extensive experiments demonstrate that Diffusion-Proof relatively significantly outperforms the AR LLM baseline trained under the same dataset. Diffusion-Proof achieves an absolute improvement of 1.61% on ProofNet-Test and 6.14% on MiniF2F-Test benchmarks compare to the baseline. Notably, Diffusion-Proof successfully resolves one IMO problem that more advanced thinking model DeepSeek-Prover-V2-7B could not solve, showcasing the unique advantage of dLLMs in formal theorem proving.

We present a dependent-type-based prover designed around the way LLMs (and humans) tend to write mathematics, complementing existing systems such as Lean and Rocq. Its core design choices are a surface that imitates mathematical natural language and a rule-driven automation layer that closes the routine steps a textbook would omit, so that an accepted proof can be re-emitted as a checked Lean file. Early experiments suggest that, even without any prover-specific training data, LLMs can learn to use it effectively on the miniF2F benchmark. Lean output excerpts: https://github.com/xiyuzhai-husky-lang/visored/

Autoformalization, translating natural-language mathematics into formal proof assistants, is bottlenecked not by translation fluency but by \emph{faithfulness}: a formal statement can typecheck and be provable, yet still encode a different theorem than the source intended. We introduce \emph{Bidirectional Provability Fingerprinting} (\bpf{}), a framework that certifies faithfulness by characterizing each candidate through its forward and backward consequence neighborhoods in the ambient theory and matching these against probes derived from the natural-language statement. We further introduce four novel components: (i) \emph{Counterfactual Probe Generation} (\cpg{}), a contrastive procedure that synthesizes probes targeting specific drift directions; (ii) the \emph{Equivalence Spectrum}, a continuous faithfulness score that replaces brittle binary verdicts; (iii) \emph{Adaptive Probe Budget Allocation} (\apba{}), an information-theoretic budget router; and (iv) \emph{Faithfulness-Guided Decoding} (\fgd{}), which uses \bpf{} signals as a reward during autoformalization. We prove a \emph{drift detection theorem} and a \emph{PAC-faithfulness} result establishing that the equivalence class of a natural language statement is learnable from $\mathcal{O}(\log(1/δ)/\varepsilon)$ probes under mild assumptions. We release \driftbench{}, a benchmark of $2{,}183$ NL/Lean~4 pairs with controlled drift labels across six subfields of mathlib4. \bpf{},+,\cpg{} detects $89.6\%$ of drifted formalizations at a $3.0\%$ false-positive rate-against $41.2\%$ for typecheck and $63.3\%$ for LLM-judge baselines, and \fgd{} reduces the rate at which a state-of-the-art autoformalizer emits drifted statements by $47\%$. https://pmlrbd.github.io/BPF/

With large language models (LLMs) increasingly applied to mathematical reasoning, formal proof assistants such as Lean can be leveraged to verify reasoning outputs with machine-checkable rigor, enabling use cases such as answer selection in test-time scaling with K sampled candidate answers. However, employing Lean requires that LLM outputs, originally in natural language, first be formalized. Existing Lean-based answer-selection work uses an autoformalization model to generate a formal statement in Lean for each candidate answer independently, incurring a significant computational cost. We propose BASE, a base-and-edit pipeline that formalizes a single base candidate per problem and derives the remaining K-1 statements by editing the answer expression in place. To facilitate this, we train a rewriter model LEANSCRIBE to localize the answer in the base formalization and generate a reusable edit function for the other K-1 candidates. BASE simultaneously improves selection accuracy and reduces formalization cost - a Pareto improvement that holds on all 12 (dataset, solver) configurations across four benchmarks and three solvers, cutting autoformalizer calls by about 5x at K=8, with the reduction expected to become larger as K grows. Code is available at https://github.com/ucr-rai/base-and-edit.

Proof autoformalization aims to translate a mathematical informal proof written in natural language into a formal proof in a formal language such as Lean~4. Several works have developed LLM-based models for proof autoformalization. However, existing evaluations have typically focused on translating well-formed informal proofs from curated datasets. We argue that a robust proof autoformalizer must remain faithful even for informal proofs that diverge from these idealized ones, and we present the first study on the robustness of proof autoformalization models. We formulate two categories of perturbations and evaluate robustness under each: a global perturbation paraphrases the informal proof in a different style, under which the formalization should remain consistent; a local perturbation alters a value, symbol, or proof step, possibly in a counterfactual way, and a robust formalization should faithfully reflect the perturbation rather than reverting to the original one or inferring a different one on its own. We build a benchmark with both perturbations on miniF2F and MATH-500, and automatically measure how stable a proof autoformalization's correctness is under global perturbations and how faithfully its output reflects local perturbations. We evaluate seven recent models, all of which are sensitive to global perturbations and mostly fail to remain faithful under local perturbations. Code and data are available via https://github.com/ucr-rai/robust-proof-autoformalization.

Large language models can often close proof gaps in interactive theorem provers, but a verified theorem is not the same thing as a reusable library contribution. We study this distinction through a detailed case study: a semi-autonomous formalization of Grothendieck's vanishing theorem. The initial version compiles with no sorries, but an expert review found serious problems in definitions, theorem generality, file organization, and the API. We then ran a review-driven refactor and compression process and obtained a second expert review. The before-and-after comparison shows a sharp split: agents adapted well to local, mechanically checkable feedback, but remained weak at choosing definitions and designing APIs. We argue that autoformalization should be evaluated not only by closed sorries, but by whether the resulting formalization survives expert review.

Large Language Models (LLMs) have made notable progress in automated theorem proving, yet existing formal benchmarks remain limited in both mathematical coverage and difficulty. Most are concentrated in areas that are easier to formalize, such as algebra and elementary number theory, and provide limited coverage of subfields that require deeper reasoning, including mathematical analysis. To address this gap, we introduce MA-ProofBench, to the best of our knowledge, the first formal theorem-proving benchmark dedicated to Mathematical Analysis. The benchmark contains 200 formalized theorems covering 6 core topics and 27 subcategories, including measure and integration theory, complex analysis, and functional analysis. The problems are divided into two difficulty levels, an undergraduate level (Level I, 100 problems) and a Ph.D. qualifying level (Level II, 100 problems), to evaluate how well LLMs perform formal reasoning at different mathematical depths. Each problem is constructed through a human-led, LLM-assisted formalization pipeline followed by independent expert review, ensuring that the formal statements remain faithful to the original mathematics. We evaluate a range of recent general-purpose reasoning models and formal theorem provers on MA-ProofBench. However, most models perform poorly: even the best-performing model, GPT-5.5, achieves only 16% Pass@8 on Level I and 5% on Level II, while most models stay close to 0% on Level II. Further analysis identifies Mathlib hallucinations and incomplete proofs as the two dominant failure modes, while an evaluation on the natural-language version of the benchmark exposes a clear gap between informal and formal reasoning. MA-ProofBench is intended to serve as a reliable reference for tracking progress in formal mathematical reasoning in advanced domains.

Modern Lean theorem provers achieve strong performance only with substantial training and inference compute, driven in part by scarce verified proof data and the long reasoning traces of formal proof search, making both supervised fine-tuning (SFT) and sampling expensive. We introduce Pythagoras-Prover, a compute-efficient open-source family of Lean theorem provers built for practical compute budgets. The family spans two generation paradigms: autoregressive models at 4B and 32B parameters, and a first proof-of-concept diffusion-based prover (4B) that iteratively refines Lean proofs at inference time. For training efficiency, we build a Lean-verified corpus stratified into easy, medium, and hard problems for curriculum SFT, so models acquire proof skills progressively from shorter, simpler proofs to longer, harder ones. During SFT, a dynamic proof-reasoning filtering scheme preserves informative proof traces while keeping each instance within an 8k-token context budget. We also introduce Augmented Lean Formalisation (ALF), which expands scarce verified corpora into variants of formal statements, populated via self-distillation for extra training signal without formally verifying every mutated instance. By perturbing known problems while preserving their formal character, ALF reduces reliance on any statement's surface form. Empirically, Pythagoras-Prover-4B surpasses DeepSeek-Prover-V2-671B at pass@32 on MiniF2F-Test (86.1% vs 82.4%) with ~167x fewer parameters, while Pythagoras-Prover-32B sets the open-source state of the art at 93.0% on MiniF2F-Test and solves 93 of 672 PutnamBench problems. We release MiniF2F-ALF, an ALF-mutated contamination-sensitive benchmark on which every evaluated model loses accuracy; here our 32B remains strongest and our 4B matches the prior state of the art, Goedel-Prover-V2-32B.

We present Trellis: an autoformalization system that leverages LLM agents in a deterministically constrained workflow to enforce incremental progress in Lean autoformalization tasks through iterative refinement of natural language proofs. Our approach is motivated by the common mathematician's notion of what it means to have a rigorous proof in the first place: namely, that it would be routine to elaborate any part of the proof in further detail. The result is a system which aims to achieve reliable autoformalization on a modest budget and with generalist agents, with specialization to autoformalization coming not from any task-specific agent training but instead from a meaning-of-rigor inspired workflow enforced by process semantics. We link to an end-to-end Lean formalization of a recent Ramsey theory breakthrough produced by the process.

LLMs have recently achieved strong results on formal proving benchmarks. However, existing evaluations remain heavily concentrated on competition-style problems and often fail to capture how models behave on longer, more dependency-rich mathematical developments. We introduce TheoremBench, a Lean4 benchmark designed to evaluate theorem provers beyond contest settings. The benchmark is built from nearly one hundred classical theorems and is released in two complementary forms: a plain main version containing one target theorem per instance, and a premised version that expands each theorem into a structured family of related proving tasks consisting of the main theorem together with automatically extracted supporting subtheorems. This design enables evaluation of not only whether the final theorem was proved from scratch, but also of partial progress through the internal proof structure of a theorem. Our experiments show that explicit premises substantially improve performance for Lean4-capable prover models. To provide a comprehensive evaluation, we introduce theorem-level coverage and token-efficiency metrics that expose qualitative differences in proof behavior. The results show that current provers remain strongly biased toward easy subtheorems and often solve theorems through long and inefficient tactic traces rather than compact proof plans. TheoremBench therefore provides a more fine-grained view of formal reasoning ability and highlights the importance of structural benchmark design for evaluating Lean4 theorem provers.

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.

Within the past few years, the ability of Large Language Models (LLMs) to generate formal mathematical proofs has improved drastically. We provide a comparison of various LLMs' effectiveness in producing formal proofs in Lean 4 with the goal of assisting those seeking to use LLMs to support their own projects. We utilize both pass@$k$ and refine@$k$ metrics as the benchmark for our comparison and evaluate on subsets of both miniF2F and miniCTX datasets. Our testing shows that overall, Gemini 3.1 Pro and Claude Opus 4.7 perform best. Gemini 3.1 Pro achieved a 92% success rate on miniF2F via refine@32 whereas Opus 4.7 achieved a 86% success rate on miniCTX via refine@32. When taking cost into account, NVIDIA Nemotron 3 Super and GPT-OSS 120B were the most efficient, with competitive accuracies and average costs of <\0.01$ per correct proof.

Long-horizon autoformalization of research mathematics fails not only at hard lemmas, but at scale: statements drift, dependencies tangle, context decays, and local repairs corrupt distant work. We present LeanMarathon, a multi-agent harness for reliable research-level Lean autoformalization. Its core abstraction is an evolving blueprint: a Lean file that serves simultaneously as formal proof skeleton, natural-language proof graph, and shared system of record. Four contract-scoped agents construct, audit, prove, and repair this blueprint. These agents are coordinated by a two-stage orchestrator that first stabilizes target fidelity through adversarial review and then discharges the proof directed acyclic graph (DAG) from its dynamic leaves upward in parallel CI-gated rounds. LeanMarathon turns one brittle multi-hour run into many local, recoverable, parallel transactions. We evaluate LeanMarathon on two recent research papers spanning four Erdős problems (#1051, #1196, #164, #1217). Across three autonomous runs, it formalizes all seven target theorems with no sorry, proving 258 lemmas and theorems. These results show that reliable AI co-mathematics requires not only stronger provers, but durable harnesses that preserve target fidelity across long mathematical developments. The code can be found at https://github.com/YuanheZ/LeanMarathon.

Asymptotic statistical theory is a challenging domain for AI-assisted formalization: its central results mix convergence statements, asymptotic expansions, functional analysis, and regularity conditions that have a large gap from existing infrastructure in Lean 4 formalization. To address these challenges, we propose a hypothesis-disciplined Lean 4 formalization pipeline built from multiple agents: a manager that coordinates seven specialist roles for proof planning, skeleton scaffolding, Mathlib reconnaissance, proof construction, integration, independent review, and audit. The main methodological discipline is the hypothesis-disciplined audit, implemented by the Auditor agent: every main-theorem hypothesis and concept-layer field must be anchored in the source mathematical prose, justified as a Lean encoding adapter, marked as source-implied, or rejected as an unsupported strengthening. Using this workflow, we build a systematic formalization of asymptotic statistical theory, especially the parametric and semi-parametric models' asymptotic distribution and efficiency results. The resulting Lean development is axiom-clean and source-faithful, with Lean-checked and human-audited proofs of core parametric and semi-parametric theorems organized so that theorem-agnostic infrastructure and statistical concept definitions are separated from theorem-specific assembly. The formalization results are available at https://github.com/junwei-lu/Lean-Asymptotic-Statistical-Theory.

Large language models (LLMs) are increasingly used in workflows for generating formal proofs in Lean. These workflows often decompose problems into smaller lemmas, sample many proof attempts, and use compiler feedback to guide search. However, they can be prohibitively expensive, often spending substantial compute on attempts that ultimately fail. In this work, we address this problem with an action routing agent that consists of a data plane and a control plane. The data plane generates natural-language lemma decompositions, formalizes them in Lean, and samples proof attempts for the resulting theorem and lemma targets. The control plane observes previous failed Lean attempts, estimates both the likelihood of success and the cost of another attempt, and decides whether to continue proving the current target or restart from a new breakdown. On a subset of PutnamBench, our agent decreases the cost by 28.9% over a fixed-step baseline on average, preserving performance while using substantially less compute. These results suggest that failed Lean trajectories provide actionable signals for cost-aware resource allocation in agentic theorem proving.

While Large Language Models (LLMs) have shown strong performance in generating formal proofs, their outputs often remain less readable, modular, maintainable, and reusable than proofs in mature formal mathematics libraries. We argue that this gap stems in part from the compile-first objective implicit in most proof-generation pipelines, which encourages monolithic or ad hoc proof scripts rather than library-quality artifacts. Existing approaches to proof-quality improvement often rely on explicit, computable optimization objectives. In practice, however, the most tractable and experimentally validated objectives are largely length-based, while higher-level qualities such as readability, modularity, maintainability, and reusability are difficult to reduce to reliable automatic metrics. Instead of optimizing proof improvement against a single proxy metric, we take a process-guided approach inspired by human proof-refactoring workflows. We propose an agentic framework $\textbf{Proof-Refactor}$ that decomposes proof refactoring into four phases: extracting candidate proof fragments, designing helper declarations, formally proving the extracted and designed components, and repairing the original proof using the verified components. On generated Lean proofs from PutnamBench and Putnam2025, Proof-Refactor improves rubric-based refactoring scores over a strong Claude Code refactoring baseline, with the largest gains in signature quality and human readability. These results suggest that process-guided refactoring can improve proof structure without treating proof length as the primary objective.

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 present AutoformBot, a multi-agent system for building an Autoformalized Textbook Library At Scale (Atlas) in Lean 4. AutoformBot orchestrates thousands of LLM agents, equipped with formal verification tools, dependency-aware task scheduling, and collaborative version control, to translate informal textbook prose into machine-checked definitions and proofs. We apply our methods to a corpus of 26 open-access textbooks spanning analysis, algebra, topology, combinatorics, and probability, producing Atlas: a verified library of over 45,000 Lean 4 declarations and 500 thousand lines of code. We release two artifacts: (i) AutoformBot, the open-source multi-agent framework; and (ii) Atlas, the resulting formal library. Our results suggest that autoformalizing the core content of graduate-level mathematics at scale is now economically and technically feasible. This opens the door to the automated verification of both human- and machine-generated mathematics at a research level.

Self-evolving scientific agents capable of conquering the hard tail of formal mathematics require Compositional Learning Behaviours (CLBs) -- the capacity to ground and recombine novel symbolic structures in context, beyond mere recombination of prelearned atoms. We propose S2B-LM, an adaptation of the CLB-evaluating Symbolic Behaviour Benchmark that removes numerical processing as a confound and adds chain-of-thought scaffolding to elicit rather than merely probe latent CLB competency. Cross-evaluating ten Lean~4 theorem provers on CLB competency in S2B-LM and miniF2F whole-proof performance, we find correlational and causal evidence of our claim: First, a necessary-condition analysis via quadrant test yields $p=0.004$, with model scale being ruled out as a confound. Second, extracting a CLB-encoding activation direction from DeepSeek-Prover-V2-7B using S2B-LM traces via Contrastive Activation Addition and applying it during miniF2F whole-proof generation on the AIME subset, CLB suppression collapses solve rate from $32.3\%$ to $2.9\%$, without loss of coherence, while suppressing a random activation direction of equal magnitude leaves it at $31.9\%$. Together, these results show that CLB competency is necessary but not sufficient for the hard tail of formal mathematical verification.

Lean is increasingly used to judge natural-language mathematical answers, but its signal is partial: many answers never formalize, and a failed proof may reflect an ill-typed statement or a missing library fact, not a wrong answer. On MATH-500 we show this signal is (i) sharply coverage-dependent, that is the proof-winning answer is correct 96% of the time at high proved coverage but 20% at low, and (ii) sparse and often unfaithful: a 7B autoformalizer proves a class for only 28% of problems, and a manual audit finds only approximately 43% of those proofs faithful. We propose COVCAL, a selector over Lean-trace diagnostics that certifies a finite-sample selective-risk bound on accepted answers or abstains, under two regimes (a conservative Bonferroni bound and a tighter dev-then-cal rule). Feasibility depends on autoformalization coverage: with the 7B formalizer the signal is too sparse and Bonferroni abstains on all 20 bootstrap partitions, whereas a prover-specialized formalizer reaches 79% coverage and flips it to feasible on 17 of 20, accepting approximately 48% of problems at 0.98 accepted accuracy. Since self-consistency alone is already 91% accurate, our contribution is a precise account of when, and with which formalizer, a partial formal signal can be trusted under risk control.

MerLean-Prover is an end-to-end Lean4 theorem prover that replaces sorry declarations with kernel-checkable proofs. It is built from three agent types (Planning, Check, and Lean) composed by a recursive outer loop whose unit of revision is the proof plan itself, and uses no fine-tuning, no custom RL objective, and no theorem-specific scaffolding. On FormalQualBench, a benchmark of 23 PhD-qualifying-exam theorems, MerLean-Prover solves 10/23, surpassing the strongest published open-source baseline (OpenGauss, 8/23). On Putnam2025, the same harness closes 12/12 with substantially lower total wall-clock than the next-best system that closes the full set. The harness also transfers to smaller models: Sonnet closes all four tested FormalQualBench problems, and Haiku closes the two short ones. These results suggest that harness design is a central factor in end-to-end Lean4 theorem proving, alongside raw model capability, and that a relatively simple harness can already be effective.

Automated theorem proving systems built on Lean 4 increasingly rely on parallel tactic search over partially specified proofs, such as those generated by Draft-Sketch-Prove (DSP) pipelines. In current systems, each search branch reconstructs a proof state by re-running elaboration, leading to substantial per-branch overhead. In Lean 4 with Mathlib, this cost has two components: (1) import loading, which deserializes pre-compiled libraries (~60 s per branch); and (2) theorem-body elaboration, which re-checks the theorem context up to the target goal (estimated 18-735 s depending on proof complexity). Together, these account for >99% of per-branch wall time, making portfolio-based search impractical at scale. We observe that this overhead arises from a mismatch between the structure of proof search and its execution model: branching is implemented via repeated reconstruction of proof states rather than direct reuse. To address this, we introduce proof-state snapshotting, which captures the elaborated proof state once and reuses it across branches via a small extension to the Lean 4 language server. Across 48 miniF2F-v2 problems (45 prove-phase benchmarks and 3 full end-to-end runs), our approach achieves a 5.6-50x wall-time speedup over the standard fallback (average 14x, median 9.7x). Speedup increases with the number of proof branches. Our method is orthogonal to import-level caching (e.g., Kimina Lean Server), which avoids import loading but not theorem-body elaboration. The patched Lean binary and the Snapshot-DSP pipeline will be released as open source upon publication.

Agentic systems have recently emerged as state-of-the-art approaches for automated theorem proving in formal mathematics. To assess how far these capabilities extend to program verification, we evaluate Claude Code in an agentic proving framework on CLEVER, a Lean 4 benchmark for verifiable code generation. Our results show that Claude generates arguably valid specifications for 98.8% of problems (with 81.3% also accepted by CLEVER's isomorphism-based scoring on the correct portion of the benchmark), certifies implementations against correct ground-truth specifications for 87.5% of problems, and reaches a 98.1% success rate on the end-to-end program generation and verification pipeline over entries with self-consistent premises. Across all stages, Claude further provides high-quality feedback on its own attempts (as confirmed under manual review), identifying underlying causes of failure and lingering bugs in the dataset. These findings highlight a growing mismatch between the difficulty of existing program verification benchmarks and the capabilities of modern agentic provers, and point to the need for more rigorous, bug-resilient evaluation methodologies, and in particular for alternatives to isomorphism-based scoring of generated specifications. More broadly, our results provide empirical evidence that tight compiler-in-the-loop agentic paradigms are currently the most effective approach for foundational program verification.

Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.