DecompRL: Solving Harder Problems by Learning Modular Code Generation

How can Large Language Models (LLMs) solve problems they currently cannot? Repeated sampling scales test-time compute but GPU cost grows linearly with attempts, while reinforcement learning (RL) with verifiable rewards improves single-attempt accuracy at the expense of sample diversity. Both strategies ultimately fail when the base policy has near-zero probability of producing a correct solution: no amount of sampling or gradient signal can overcome a search space that is simply too large. We take a different approach: rather than sampling harder, we make the task easier by decomposing problems into smaller, independently solvable sub-functions whose implementations can be recombined. Since off-the-shelf models are not trained for this modular generation, we introduce DecompRL, an RL algorithm that explicitly learns to decompose and implement hierarchical code structures. Recombining $k$ implementations of $n$ modules yields up to $k^{n}$ candidate solutions, shifting the bottleneck from GPU inference to cheap CPU evaluation and cutting GPU token cost by $\sim$50$\times$. On LiveCodeBench and CodeContests (Qwen~2.5~7B, Code World Model~32B), DecompRL outperforms standard and diversity-optimized RL baselines beyond $10^5$ tokens per problem, solving problems that standard generation cannot reach.

Formalizing Mathematics at Scale

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.

Automatic Textbook Formalization

We present a case study where an automatic AI system formalizes a textbook with more than 500 pages of graduate-level algebraic combinatorics to Lean. The resulting formalization represents a new milestone in textbook formalization scale and proficiency, moving from early results in undergraduate topology and restructuring of existing library content to a full standalone formalization of a graduate textbook. The formalization comprises 130K lines of code and 5900 Lean declarations and was conducted within one week by a total of 30K Claude 4.5 Opus agents collaborating in parallel on a shared code base via version control, simultaneously setting a record in multi-agent software engineering with usable results. The inference cost matches or undercuts what we estimate as the salaries required for a team of human experts, and we expect there is still the potential for large efficiencies to be made without the need for better models. We make our code, the resulting Lean code base and a side-by-side blueprint website available open-source.

WybeCoder: Verified Imperative Code Generation

Recent progress in large language models (LLMs) has advanced automatic code generation and formal theorem proving, yet software verification has not seen the same improvement. To address this gap, we propose WybeCoder, an agentic code verification framework that enables prove-as-you-generate development where code, invariants, and proofs co-evolve. It builds on a recent framework that combines automatic verification condition generation and SMT solvers with interactive proofs in Lean. To enable systematic evaluation, we translate two benchmarks for functional verification in Lean, Verina and Clever, to equivalent imperative code specifications. On complex algorithms such as Heapsort, we observe consistent performance improvements by scaling our approach, synthesizing dozens of valid invariants and dispatching of dozens of subgoals, resulting in hundreds of lines of verified code, overcoming plateaus reported in previous works. Our best system solves 74% of Verina tasks and 62% of Clever tasks at moderate compute budgets, significantly surpassing previous evaluations and paving a path to automated construction of large-scale datasets of verified imperative code.

LemmaBench: A Live, Research-Level Benchmark to Evaluate LLM Capabilities in Mathematics

We present a new approach for benchmarking Large Language Model (LLM) capabilities on research-level mathematics. Existing benchmarks largely rely on static, hand-curated sets of contest or textbook-style problems as proxies for mathematical research. Instead, we establish an updatable benchmark evaluating models directly on the latest research results in mathematics. This consists of an automatic pipeline that extracts lemmas from arXiv and rewrites them into self-contained statements by making all assumptions and required definitions explicit. It results in a benchmark that can be updated regularly with new problems taken directly from human mathematical research, while previous instances can be used for training without compromising future evaluations. We benchmark current state-of-the-art LLMs, which obtain around 10-15$\%$ accuracy in theorem proving (pass@1) depending on the model, showing that there is currently a large margin of progression for LLMs to reach human-level proving capabilities in a research context.

The KoLMogorov Test: Compression by Code Generation

Compression is at the heart of intelligence. A theoretically optimal way to compress any sequence of data is to find the shortest program that outputs that sequence and then halts. However, such 'Kolmogorov compression' is uncomputable, and code generating LLMs struggle to approximate this theoretical ideal, as it requires reasoning, planning and search capabilities beyond those of current models. In this work, we introduce the KoLMogorov-Test (KT), a compression-as-intelligence test for code generating LLMs. In KT a model is presented with a sequence of data at inference time, and asked to generate the shortest program that produces the sequence. We identify several benefits of KT for both evaluation and training: an essentially infinite number of problem instances of varying difficulty is readily available, strong baselines already exist, the evaluation metric (compression) cannot be gamed, and pretraining data contamination is highly unlikely. To evaluate current models, we use audio, text, and DNA data, as well as sequences produced by random synthetic programs. Current flagship models perform poorly - both GPT4-o and Llama-3.1-405B struggle on our natural and synthetic sequences. On our synthetic distribution, we are able to train code generation models with lower compression rates than previous approaches. Moreover, we show that gains on synthetic data generalize poorly to real data, suggesting that new innovations are necessary for additional gains on KT.

Better & Faster Large Language Models via Multi-token Prediction

Large language models such as GPT and Llama are trained with a next-token prediction loss. In this work, we suggest that training language models to predict multiple future tokens at once results in higher sample efficiency. More specifically, at each position in the training corpus, we ask the model to predict the following n tokens using n independent output heads, operating on top of a shared model trunk. Considering multi-token prediction as an auxiliary training task, we measure improved downstream capabilities with no overhead in training time for both code and natural language models. The method is increasingly useful for larger model sizes, and keeps its appeal when training for multiple epochs. Gains are especially pronounced on generative benchmarks like coding, where our models consistently outperform strong baselines by several percentage points. Our 13B parameter models solves 12 % more problems on HumanEval and 17 % more on MBPP than comparable next-token models. Experiments on small algorithmic tasks demonstrate that multi-token prediction is favorable for the development of induction heads and algorithmic reasoning capabilities. As an additional benefit, models trained with 4-token prediction are up to 3 times faster at inference, even with large batch sizes.

Decoding Data Quality via Synthetic Corruptions: Embedding-guided Pruning of Code Data

Code datasets, often collected from diverse and uncontrolled sources such as GitHub, potentially suffer from quality issues, thereby affecting the performance and training efficiency of Large Language Models (LLMs) optimized for code generation. Previous studies demonstrated the benefit of using embedding spaces for data pruning, but they mainly focused on duplicate removal or increasing variety, and in other modalities, such as images. Our work focuses on using embeddings to identify and remove "low-quality" code data. First, we explore features of "low-quality" code in embedding space, through the use of synthetic corruptions. Armed with this knowledge, we devise novel pruning metrics that operate in embedding space to identify and remove low-quality entries in the Stack dataset. We demonstrate the benefits of this synthetic corruption informed pruning (SCIP) approach on the well-established HumanEval and MBPP benchmarks, outperforming existing embedding-based methods. Importantly, we achieve up to a 3% performance improvement over no pruning, thereby showing the promise of insights from synthetic corruptions for data pruning.

Large Language Models for Compiler Optimization

We explore the novel application of Large Language Models to code optimization. We present a 7B-parameter transformer model trained from scratch to optimize LLVM assembly for code size. The model takes as input unoptimized assembly and outputs a list of compiler options to best optimize the program. Crucially, during training, we ask the model to predict the instruction counts before and after optimization, and the optimized code itself. These auxiliary learning tasks significantly improve the optimization performance of the model and improve the model's depth of understanding. We evaluate on a large suite of test programs. Our approach achieves a 3.0% improvement in reducing instruction counts over the compiler, outperforming two state-of-the-art baselines that require thousands of compilations. Furthermore, the model shows surprisingly strong code reasoning abilities, generating compilable code 91% of the time and perfectly emulating the output of the compiler 70% of the time.

Code Llama: Open Foundation Models for Code

We release Code Llama, a family of large language models for code based on Llama 2 providing state-of-the-art performance among open models, infilling capabilities, support for large input contexts, and zero-shot instruction following ability for programming tasks. We provide multiple flavors to cover a wide range of applications: foundation models (Code Llama), Python specializations (Code Llama - Python), and instruction-following models (Code Llama - Instruct) with 7B, 13B and 34B parameters each. All models are trained on sequences of 16k tokens and show improvements on inputs with up to 100k tokens. 7B and 13B Code Llama and Code Llama - Instruct variants support infilling based on surrounding content. Code Llama reaches state-of-the-art performance among open models on several code benchmarks, with scores of up to 53% and 55% on HumanEval and MBPP, respectively. Notably, Code Llama - Python 7B outperforms Llama 2 70B on HumanEval and MBPP, and all our models outperform every other publicly available model on MultiPL-E. We release Code Llama under a permissive license that allows for both research and commercial use.