# Beyond Pass@k: Breadth-Depth Metrics for Reasoning Boundaries

arXiv: https://arxiv.org/abs/2510.08325 · HTML: https://mlsanity.com/paper/2510.08325/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 23 of its 25 references that are in the map.
- First posted on arXiv 9 Oct 2025.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 23 of 23)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [Deep Reinforcement Learning at the Edge of the Statistical Precipice](https://mlsanity.com/paper/2108.13264/) (2021)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Reinforcement Learning for Reasoning in Large Language Models with One Training Example](https://mlsanity.com/paper/2504.20571/) (2025)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [ProRL: Prolonged Reinforcement Learning Expands Reasoning Boundaries in Large Language Models](https://mlsanity.com/paper/2505.24864/) (2025)
- [Pass@k Training for Adaptively Balancing Exploration and Exploitation of Large Reasoning Models](https://mlsanity.com/paper/2508.10751/) (2025)
- [REASONING GYM: Reasoning Environments for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2505.24760/) (2025)
- [Reasoning or Memorization? Unreliable Results of Reinforcement Learning Due to Data Contamination](https://mlsanity.com/paper/2507.10532/) (2025)
- [Rewarding the Unlikely: Lifting GRPO Beyond Distribution Sharpening](https://mlsanity.com/paper/2506.02355/) (2025)
- [Revisiting Reinforcement Learning for LLM Reasoning from A Cross-Domain Perspective](https://mlsanity.com/paper/2506.14965/) (2025)
- [Are Your LLMs Capable of Stable Reasoning?](https://mlsanity.com/paper/2412.13147/) (2024)
- [OMEGA: Can LLMs Reason Outside the Box in Math? Evaluating Exploratory, Compositional, and Transformative Generalization](https://mlsanity.com/paper/2506.18880/) (2025)

## Built on by (most cited first, 9 of 9)

- [Smaller Models are Natural Explorers for Policy-Level Diversity in GRPO](https://mlsanity.com/paper/2605.30789/) (2026)
- [SFT Overtraining Predicts Rank Inversion via Entropy Collapse Under RLVR](https://mlsanity.com/paper/2606.18487/) (2026)
- [RLVR is a Kernel, Not a Function: Statistical Inference for pass@k Crossovers](https://mlsanity.com/paper/2609.22547/) (2026)
- [Thinking Before Thinking: Scaling Agentic Inference Through Meta-Reasoning](https://mlsanity.com/paper/2609.38147/) (2026)
- [Locked at the Entrance, Open Inside: Where RLVR Narrows the Solution Space](https://mlsanity.com/paper/2608.29188/) (2026)
- [Sampling via Decision-Flow: Training-Free Extraction of Improved Latent Reasoning Paths in Large Language Models](https://mlsanity.com/paper/2609.12317/) (2026)
- [Does CoT-Pass@k Really Check the CoT? A Multilingual Mathematical Audit](https://mlsanity.com/paper/2609.32622/) (2026)
- [Oracle Gaps in Reliability Coverage: Sampling Noise or Policy Specialization?](https://mlsanity.com/paper/2609.32996/) (2026)
- [Sharpening Tax in Post-Training](https://mlsanity.com/paper/2610.01509/) (2026)

## Abstract (arXiv)

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a powerful paradigm to improve Large Language Models on reasoning tasks such as coding, math or logic. To assess the reasoning boundary (the fraction of problems a model can solve) researchers often report Pass@k at large sampling budgets. Recent results reveal a crossover phenomenon: while RLVR models outperform the base model at small k values, the base model usually outperforms them when sampling a very large number of completions. This has been interpreted as evidence that base models have a larger reasoning boundary. We argue that on tasks with discrete answer spaces, such as math with numeric outputs, Pass@k at large k reflects the increasingly higher chance of success in the limit of the number of trials rather than genuine reasoning, and can therefore be misleading. We propose Cover@tau, which measures the fraction of problems that a model can solve for which at least a tau proportion of completions are correct. Unlike Pass@k, Cover@tau captures reasoning under an explicit reliability threshold: models that rely on random guessing degrade rapidly as tau increases. We evaluate several RLVR models using Cover@tau-based metrics and illustrate how the relative rankings of popular algorithms change compared to Pass@1, offering a different perspective on reasoning boundaries.