# CORE: Concept-Oriented Reinforcement for Bridging the Definition-Application Gap in Mathematical Reasoning

arXiv: https://arxiv.org/abs/2512.18857 · HTML: https://mlsanity.com/paper/2512.18857/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 30 of its 46 references that are in the map.
- First posted on arXiv 21 Dec 2025.

## Placed in

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

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

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [DeepSeek-V3 Technical Report](https://mlsanity.com/paper/2412.19437/) (2024)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Llemma: An Open Language Model For Mathematics](https://mlsanity.com/paper/2310.10631/) (2023)
- [A Diverse Corpus for Evaluating and Developing English Math Word Problem Solvers](https://mlsanity.com/paper/2106.15772/) (2021)
- [Seed-Prover: Deep and Broad Reasoning for Automated Theorem Proving](https://mlsanity.com/paper/2507.23726/) (2025)
- [CMATH: Can Your Language Model Pass Chinese Elementary School Math Test?](https://mlsanity.com/paper/2306.16636/) (2023)
- [Reasoning or Memorization? Unreliable Results of Reinforcement Learning Due to Data Contamination](https://mlsanity.com/paper/2507.10532/) (2025)
- [Winning Gold at IMO 2025 with a Model-Agnostic Verification-and-Refinement Pipeline](https://mlsanity.com/paper/2507.15855/) (2025)
- [BREAD: Branched Rollouts from Expert Anchors Bridge SFT & RL for Reasoning](https://mlsanity.com/paper/2506.17211/) (2025)
- [Mathematical Proof as a Litmus Test: Revealing Failure Modes of Advanced Large Reasoning Models](https://mlsanity.com/paper/2506.17114/) (2025)
- [Conceptual and Unbiased Reasoning in Language Models](https://mlsanity.com/paper/2404.00205/) (2024)
- [BIRD: A Trustworthy Bayesian Inference Framework for Large Language Models](https://mlsanity.com/paper/2404.12494/) (2024)
- [Decomposing Elements of Problem Solving: What "Math" Does RL Teach?](https://mlsanity.com/paper/2505.22756/) (2025)
- [Right Is Not Enough: The Pitfalls of Outcome Supervision in Training LLMs for Math Reasoning](https://mlsanity.com/paper/2506.06877/) (2025)
- [ReasonAgain: Using Extractable Symbolic Programs to Evaluate Mathematical Reasoning](https://mlsanity.com/paper/2410.19056/) (2024)
- [Self-supervised Analogical Learning using Language Models](https://mlsanity.com/paper/2502.00996/) (2025)

## Abstract (arXiv)

Large language models (LLMs) often solve challenging math exercises yet fail to apply the concept right when the problem requires genuine understanding. Popular Reinforcement Learning with Verifiable Rewards (RLVR) pipelines reinforce final answers but provide little fine-grained conceptual signal, so models improve at pattern reuse rather than conceptual applications. We introduce CORE (Concept-Oriented REinforcement), an RL training framework that turns explicit concepts into a controllable supervision signal. Starting from a high-quality, low-contamination textbook resource that links verifiable exercises to concise concept descriptions, we run a sanity probe showing LLMs can restate definitions but fail concept-linked quizzes, quantifying the conceptual reasoning gap. CORE then (i) synthesizes concept-aligned quizzes, (ii) injects brief concept snippets during rollouts to elicit concept-primed trajectories, and (iii) reinforces conceptual reasoning via trajectory replacement after group failures, a lightweight forward-KL constraint that aligns unguided with concept-primed policies, or standard GRPO directly on concept-aligned quizzes. Across several models, CORE delivers consistent gains over vanilla and SFT baselines on both in-domain concept-exercise suites and diverse out-of-domain math benchmarks. CORE unifies direct training on concept-aligned quizzes and concept-injected rollouts under outcome regularization. It provides fine-grained conceptual supervision that bridges problem-solving competence and genuine conceptual reasoning, while remaining algorithm- and verifier-agnostic.