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

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 $\textit{CORE}$ (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. $\textit{CORE}$ 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, $\textit{CORE}$ delivers consistent gains over vanilla and SFT baselines on both in-domain concept--exercise suites and diverse out-of-domain math benchmarks. $\textit{CORE}$ 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.