# From Correctness to Utility: Gain-Based Prefix Evaluation for LLM Reasoning

arXiv: https://arxiv.org/abs/2606.07190 · HTML: https://mlsanity.com/paper/2606.07190/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 10 of its 21 references that are in the map.
- First posted on arXiv 5 Jun 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- LLM alignment / RLHF / preference optimisation / [Process reward models for reasoning](https://mlsanity.com/alignment/process-reward-models-for-reasoning/)

## Builds on (most cited first, 10 of 10)

- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [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)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Lost at the Beginning of Reasoning](https://mlsanity.com/paper/2506.22058/) (2025)
- [Linking Process to Outcome: Conditional Reward Modeling for LLM Reasoning](https://mlsanity.com/paper/2509.26578/) (2025)
- [Bounds of Chain-of-Thought Robustness: Reasoning Steps, Embed Norms, and Beyond](https://mlsanity.com/paper/2509.21284/) (2025)

## Built on by (most cited first, 2 of 2)

- [Before the Rollout Ends: Early Terminal Reward Prediction for Long-horizon Coding Agents](https://mlsanity.com/paper/2609.31995/) (2026)
- [Balancing Reference Guidance and Free Generation in Trajectory Rollouts for Reasoning RL](https://mlsanity.com/paper/2610.11128/) (2026)

## Abstract (arXiv)

Reasoning prefixes shape the future trajectory of LLM problem solving, yet existing process reward models usually evaluate them through local step correctness. We argue that correctness is a useful but indirect proxy for the effect we ultimately care about: whether a prefix increases the probability of successful completion. We define this effect as prefix gain, the solve-rate improvement induced by conditioning lightweight student model group on a prefix, and use it to train a Prefix Utility Model (PUM) with a simple pairwise ranking objective. PUM learns outcome-grounded prefix utility and can score both complete trajectories and partial reasoning prefixes. Across Best-of-N selection, beam search, and reinforcement learning on mathematical reasoning, PUM provides a strong prefix-level supervision signal, especially when candidate pools are large, search budgets increase, or rule-based rewards are sparse. We release all data, models, and code at https://zhiqix.github.io/pum-project-page.