SHAPE: Stage-aware Hierarchical Advantage via Potential Estimation for LLM Reasoning

Reasoning in Large Language Models incurs significant inference-time compute, yet the token-level information structure of reasoning traces remains underexplored. We observe that reasoning tokens split into two functional types: low-entropy structural tokens (recurring phrases that scaffold the reasoning process) and higher-entropy organic tokens (problem-specific content that drives toward a solution). This asymmetry motivates a simple, model-agnostic compression pipeline: apply cross-word BPE merges on a model's own reasoning traces to derive supertokens that capture frequent structural patterns, then teach the model to adopt them via supervised fine-tuning. Across three model families and five mathematical reasoning benchmarks, our approach shortens reasoning traces by 8.1% on average with no statistically significant accuracy loss on any model--benchmark pair. Beyond compression, supertokens act as interpretable reasoning-move annotations (backtracking, verification, strategy shifts), exposing the model's high-level strategy at a glance. Analyzing transitions between structural categories reveals systematic differences between correct and incorrect traces: correct traces show productive recovery (backtracking followed by strategy shifts and verification), while incorrect traces are dominated by confusion cycles (repeated hedging and unresolved contradictions). These diagnostic signals suggest applications in reward shaping and early stopping for RL-based reasoning training.

Large Reasoning Models (LRMs) have shown remarkable reasoning capabilities, yet they still suffer from overthinking, generating redundant reasoning steps which incur substantial token consumption. Existing methods, such as suppressing reflective keywords or forcing shorter reasoning lengths, attempt to mitigate this issue but inevitably truncate necessary steps and induce underthinking, thereby compromising performance. To address this dilemma, we investigate the latent representations and observe that efficient reasoning steps naturally cluster into a concentrated region in latent space, while those deviating from this region tend to produce verbose sequences. To leverage this, we keep reasoning focused within this region via a quadratic program which projects deviating hidden states back into the region. Then we propose a novel training-free framework to achieve efficient reasoning that reduces token generation costs without sacrificing performance. Extensive experiments conducted on four models ranging from 1.5B to 14B, and across six benchmarks in math reasoning, coding, and scientific QA, validate the effectiveness of our method, up to a 12.1% improvement in accuracy while reducing generated tokens by 11.8% to 52.8%. Codes are available at https://github.com/hzn18/Opt4Reasoning.

Large language models (LLMs) achieve strong performance on mathematical reasoning benchmarks, yet final-answer accuracy reveals little about the strategies they use. We introduce SHAPE (Structured Heuristic Analysis for Problem-solving Evolution), a framework grounded in mathematics-education research that characterizes local mathematical actions in reasoning trajectories using a problem-independent heuristic taxonomy. Using SHAPE, we examine two previously hypothesized phenomena in LLM reasoning. First, we revisit the hypothesis that LLMs reuse learned solution techniques under problem perturbations. Across six models on MATH-Perturb, perturbations requiring a different solution approach yield lower accuracy but greater heuristic shifts than those preserving it. Second, we characterize mode-seeking in RLVR through mathematical heuristic profiles. Across four base--RL pairs, successful RL solutions overlap substantially with base-model profiles. We further introduce Heuristic-Augmented Plan-GRPO (HA-Plan-GRPO), which guides RLVR planning through heuristic selection and applicability reasoning. Across three models and five evaluation sets, it improves mean Avg@16 by up to 7.0% absolute and mean Pass@16 by up to 2.4% absolute over planning-only RLVR. Our code is available at https://github.com/holi-lab/shape-of-CoT.