# CircuitLens: Reasoning Circuits as Data Selection Signals for Reinforcement Learning with Verifiable Rewards

arXiv: https://arxiv.org/abs/2609.07183 · HTML: https://mlsanity.com/paper/2609.07183/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 38 references that are in the map.
- First posted on arXiv 7 Sep 2026.

## Builds on (most cited first, 30 of 31)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [LESS: Selecting Influential Data for Targeted Instruction Tuning](https://mlsanity.com/paper/2402.04333/) (2024)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [Improve Mathematical Reasoning in Language Models by Automated Process Supervision](https://mlsanity.com/paper/2406.06592/) (2024)
- [The Lessons of Developing Process Reward Models in Mathematical Reasoning](https://mlsanity.com/paper/2501.07301/) (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)
- [Rewarding Progress: Scaling Automated Process Verifiers for LLM Reasoning](https://mlsanity.com/paper/2410.08146/) (2024)
- [How to use and interpret activation patching](https://mlsanity.com/paper/2404.15255/) (2024)
- [Free Process Rewards without Process Labels](https://mlsanity.com/paper/2412.01981/) (2024)
- [LIMR: Less is More for RL Scaling](https://mlsanity.com/paper/2502.11886/) (2025)
- [Superfiltering: Weak-to-Strong Data Filtering for Fast Instruction-Tuning](https://mlsanity.com/paper/2402.00530/) (2024)
- [REASONING GYM: Reasoning Environments for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2505.24760/) (2025)
- [DsDm: Model-Aware Dataset Selection with Datamodels](https://mlsanity.com/paper/2401.12926/) (2024)
- [Language Models Use Trigonometry to Do Addition](https://mlsanity.com/paper/2502.00873/) (2025)
- [RLVE: Scaling Up Reinforcement Learning for Language Models with Adaptive Verifiable Environments](https://mlsanity.com/paper/2511.07317/) (2025)
- [Depth-Breadth Synergy in RLVR: Unlocking LLM Reasoning Gains with Adaptive Exploration](https://mlsanity.com/paper/2508.13755/) (2025)
- [Data Selection via Optimal Control for Language Models](https://mlsanity.com/paper/2410.07064/) (2024)
- [VCRL: Variance-based Curriculum Reinforcement Learning for Large Language Models](https://mlsanity.com/paper/2509.19803/) (2025)
- [Smaller Language Models are capable of selecting Instruction-Tuning Training Data for Larger Language Models](https://mlsanity.com/paper/2402.10430/) (2024)
- [Reasoning Circuits in Language Models: A Mechanistic Interpretation of Syllogistic Inference](https://mlsanity.com/paper/2408.08590/) (2024)
- [Efficient Automated Circuit Discovery in Transformers using Contextual Decomposition](https://mlsanity.com/paper/2407.00886/) (2024)
- [Causal Head Gating: A Framework for Interpreting Roles of Attention Heads in Transformers](https://mlsanity.com/paper/2505.13737/) (2025)

## Built on by (most cited first, 1 of 1)

- [What Will Post-Training Fix? Per-Problem Gains Are Shared Across Independent RL Runs, and Existing Checkpoints Predict Them Better Than A Priori Signals](https://mlsanity.com/paper/2610.04978/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) is sensitive to which problems a model trains on, yet existing selection criteria--difficulty filtering, hand-curation, reward-trajectory scoring--assess data value as an intrinsic property of problems, independent of the model that will learn from them. We introduce Circuit Reasoning Score (CRS), a selection signal derived from 46 reasoning-sensitive attention heads identified via contrastive ablation, computed in a single forward pass on the frozen base model without reward labels or rollouts. CRS runs against the intuitive hypothesis that stronger reasoning-circuit engagement produces better training data: on Qwen2.5-Math-7B, the lowest-engagement decile improves over random selection on three medium-difficulty benchmarks (GSM8K +2.0 pp, OlympiadBench +1.6 pp, Minerva +2.9 pp), while the highest-engagement decile gains less and is indistinguishable from the middle decile. The advantage has boundary conditions: on a domain-curated pool no selection method separates from the others; at 1.5B scale the useful direction differs; and the lowest-reward training condition produces the strongest downstream generalization. Within the Qwen2.5-Math settings tested, RLVR data selection appears regime-dependent rather than reducible to a static ranking of problem quality.