# When RLVR Shrinks the Reasoning Boundary: Diagnosing Pass@k Inversion

arXiv: https://arxiv.org/abs/2607.20543 · HTML: https://mlsanity.com/paper/2607.20543/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 26 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Builds on (most cited first, 18 of 18)

- [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)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [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)
- [Conservative Q-Learning for Offline Reinforcement Learning](https://mlsanity.com/paper/2006.04779/) (2020)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Omni-MATH: A Universal Olympiad Level Mathematic Benchmark For Large Language Models](https://mlsanity.com/paper/2410.07985/) (2024)
- [ReMax: A Simple, Effective, and Efficient Reinforcement Learning Method for Aligning Large Language Models](https://mlsanity.com/paper/2310.10505/) (2023)
- [Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining](https://mlsanity.com/paper/2504.07912/) (2025)
- [Understanding Diversity Collapse in RLVR via the Lens of Overtraining](https://mlsanity.com/paper/2606.15455/) (2026)

## Built on by (most cited first, 3 of 3)

- [RLVR is a Kernel, Not a Function: Statistical Inference for pass@k Crossovers](https://mlsanity.com/paper/2609.22547/) (2026)
- [Does Learning to Predict the World Help Agents Act? Auditing World-Model Post-Training](https://mlsanity.com/paper/2609.33335/) (2026)
- [Sharpening Tax in Post-Training](https://mlsanity.com/paper/2610.01509/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) can improve one-sample accuracy while making a model worse under repeated sampling. We study this pass@k inversion: after training, the policy may solve fewer distinct problems than its base model at large k. The failure concentrates on boundary prompts, where the base model contains rare correct trajectories that are recoverable by sampling but too sparse to reliably appear in finite RLVR rollout groups. We argue that a two-mode account explains this as an absence-of-evidence failure: rare correct trajectories may disappear before RLVR samples and reinforces them often enough. The main contribution is this diagnostic and mechanistic framing. Per-Problem Base Anchoring (PBA) is a deliberately simple proof-of-concept: sharpen prompts with sufficient frozen-base correct evidence, and anchor risky prompts to the base distribution. Across three training seeds on Omni-MATH-Test, with MATH500 as a secondary high-coverage validation benchmark, PBA improves both PassK1 and high-budget coverage over matched GRPO. A 3000-prompt regime-controlled diagnostic study is consistent across seeds with the expected signature: ordinary GRPO loses base-solvable boundary prompts, while PBA preserves rare verifier-positive trajectories. We use mathematical verifiers as a controlled testbed for verifier-guided optimization; the same pass@k inversion risk applies to ECCV-relevant vision-language agents when repeated visual, spatial, or chart-reasoning attempts are checked by external tools or verifiers. Reasoning post-training should decide not only how strongly to optimize, but which prompts are safe to optimize.