Uncovering the Representation Geometry of Minimal Cores in Overcomplete Reasoning Traces

Reasoning-capable large language models solve hard problems by emitting long chains of thought, paying heavily in latency, GPU time, and energy. Casual inspection of their traces reveals extensive reformulation, verification, and circular self-reflection, yet how much of this deliberation is actually necessary has never been measured at scale or explained from first principles. This paper closes both gaps. We formalise reasoning redundancy directly in terms of the reasoning model itself: the redundancy of a correct trace is the largest fraction of its trailing segmented steps that can be truncated while $π$, forced to terminate thinking and emit a final answer, still produces the correct answer. A large-scale quantification across four frontier reasoning models and two mathematical benchmarks shows that step-level redundancy is consistently high -- between 61% and 93% across the 8 (model, benchmark) conditions we study, with the median critical prefix equal to a single segmented step in six of the eight conditions -- that the finding is robust to the choice of judge family, and that although $ρ$ decreases with problem difficulty on MATH-500, all four models remain substantially redundant ($ρ\in [46\%, 85\%]$) even on the hardest Level-5 problems. We then prove that this redundancy is a structural consequence of length-agnostic outcome rewards, not a model-specific artefact: under any such reward, no finite expected stopping time is optimal. The result holds regardless of RL algorithm, base model, data distribution, or whether the policy is obtained via RL or distillation; over-thinking is therefore not a bug to be patched in individual models but a structural property of how current reasoning models are trained. Code: https://github.com/zhiyuanZhai20/how-much-thinking-is-enough

Large Reasoning Models (LRMs) improve performance by generating explicit intermediate reasoning traces through increased test-time compute, yet the assumption that longer reasoning is consistently beneficial remains under-examined. While recent evidence shows that additional reasoning can lead models to overthink, we ask: "Once a model has reached the correct answer, does further reasoning refine the solution, or deviate from it?" To study the dynamics after correctness, we introduce a prefix-level trajectory evaluation protocol grounded in reasoning sufficiency, defining the minimum reasoning budget required for a model to first generate the correct answer. This allows us to disentangle verbose overthinking, where additional reasoning is redundant but harmless, from harmful overthinking, where continued reasoning destabilizes an already-correct trajectory. Starting from multimodal benchmarks, we find that many instances considered reasoning-intensive require surprisingly little reasoning. Moreover, stopping at the first correct prefix improves accuracy over standard reasoning up to 21%, revealing that current models are limited not only by their ability to reason, but also by their inability to stop at the right time. Furthermore, while common efficiency strategies like early stopping substantially reduce verbose overthinking (up to 50%), they fail to mitigate harmful overthinking. Failure analysis reveals that correctness deviations are mainly driven by logical drift and visual reinterpretation. Finally, we show that our findings generalize to language-only reasoning benchmarks, highlighting harmful overthinking as a broader reliability risk. Code available at https://simonecaldarella.github.io/thinking-past-the-answer.

Large reasoning models (LRMs) often exhibit overthinking, producing verbose Chain-of-Thought (CoT) traces that increase inference cost and obscure the underlying reasoning process. Existing CoT compression methods mainly rely on global length rewards, which conflate necessary intermediate reasoning with redundant text and may therefore compromise reasoning fidelity. This paper revisits overthinking from a semantic-efficiency perspective and decomposes CoT redundancy into two distinct forms: internal redundancy, defined as informational stagnation before the first correct answer, and external redundancy, defined as superfluous continuation after the first correct answer. Based on this decomposition, we propose a dual-penalty reinforcement learning framework that separately optimizes reasoning progress and termination behavior. Specifically, a sliding-window semantic similarity metric penalizes low-progress reasoning segments, while a normalized external-redundancy metric discourages post-answer continuation. Experiments on GSM8K, MATH500, and AIME24 across different model scales show that our method reduces average reasoning length by 41.3% on the 1.5B model and 40.1% on the 7B model, while preserving competitive accuracy and achieving the best overall accuracy-efficiency score among evaluated baselines. The learned compression behavior further transfers to out-of-domain reasoning tasks, including GPQA and LiveCodeBench. More importantly, our analysis reveals a clear asymmetry between the two redundancy types: external redundancy can be largely removed with little performance loss, whereas internal redundancy compression follows a sensitive accuracy-efficiency trade-off. These results suggest that effective CoT compression should optimize semantic efficiency rather than sequence length alone, offering a principled route toward more concise, efficient, and interpretable LRMs.