Hint Tuning: Less Data Makes Better Reasoners

# Hint Tuning: Less Data Makes Better Reasoners
Source: [https://arxiv.org/html/2605.08665](https://arxiv.org/html/2605.08665)
Siqi Fan1Minghao Li2Xiaoqian Ma2Xiusheng Huang2Zhuo Chen2 Bowen Qin3Liujie Zhang2Shuo Shang1Weihang Chen2 1University of Electronic Science and Technology of China 2Xiaohongshu Inc\. 3National University of Singapore

###### Abstract

Large reasoning models achieve high accuracy through extended chain\-of\-thought but generate 5–8 more tokens than necessary, applying verbose reasoning uniformly regardless of problem difficulty\. We proposeHint Tuning, a data\-efficient approach that teaches models to calibrate reasoning depth\. Our key insight: thecorresponding instruct modelserves as an ideal difficulty probe\. By testing what the instruct model can solve with varying guidance, we automatically construct training data across three states:No\-Hint\(direct answer\),Sparse\-Hint\(minimal prefix\), andFull\-Hint\(complete reasoning\)\. This converts the abstract challenge of difficulty labeling into a measurable consistency check between the instruct and reasoning models\.

With only 1K self\-annotated samples,Hint Tuningachieves 24–66% token reduction \(31\.5% average\) across mainstream reasoning models \(Qwen3\-Thinking, DeepSeek\-R1\-Distill\) at multiple scales \(4B–32B\) while maintaining competitive accuracy on five benchmarks\. Unlike methods requiring massive distillation datasets or expensive RL, we achieve superior efficiency through simple alignment with the instruct model’s capabilities\.

![Refer to caption](https://arxiv.org/html/2605.08665v1/x1.png)Figure 1:Hint TuningEfficiency on DeepSeek\-R1\-Distill\-Qwen\-7B\.## 1Introduction

Chain\-of\-thought \(CoT\) prompting\([Weiet al\.,](https://arxiv.org/html/2605.08665#bib.bib9);[Kojimaet al\.,](https://arxiv.org/html/2605.08665#bib.bib29)\)has become the dominant paradigm for eliciting reasoning in large language models\. Recent work demonstrates that CoT follows test\-time scaling laws\(Snellet al\.,[2024](https://arxiv.org/html/2605.08665#bib.bib27); OpenAI,[2024](https://arxiv.org/html/2605.08665#bib.bib28)\): accuracy improves with longer reasoning traces, enabling breakthroughs on complex mathematical tasks\(Jaechet al\.,[2024](https://arxiv.org/html/2605.08665#bib.bib2); DeepSeek\-AI,[2025](https://arxiv.org/html/2605.08665#bib.bib1)\)\. However, this capability introduces a new inefficiency:models reason verbosely on all problems, regardless of difficulty\.

![Refer to caption](https://arxiv.org/html/2605.08665v1/x2.png)Figure 2:Reasoning models waste tokens through over\-elaboration\. \(a\) Example: same answer, different verbosity\. \(b\) Systematic comparison: instruct models use55–88\\timesfewer tokens across all four correctness scenarios\. \(c–d\) With partial hints, instruct models need only 27\.5% of episodes and 11\.9% of tokens to succeed\.As shown in Figure[2](https://arxiv.org/html/2605.08665#S1.F2), comparing an instruct and a reasoning model across four correctness scenarios reveals that the instruct model consistently uses far fewer tokens\. Even when both succeed, the reasoning model generates55–88\\timesmore tokens through verbose self\-reflection\(Chenet al\.,[2024](https://arxiv.org/html/2605.08665#bib.bib24); Suiet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib25)\), wasting computation without adding reasoning value\.

#### The annotation bottleneck\.

Training models to adapt reasoning depth requires supervision signals specifying “how much thinking is appropriate\.” Existing approaches struggle with this challenge in complementary ways\.*Post\-hoc compression*methods\(Kanget al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib31); Xiaet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib32); Chenet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib15)\)prune redundant steps via heuristics or external models, but importance is context\-dependent: the same phrase may be essential in one problem and redundant in another\.*During\-reasoning*methods\(Hanet al\.,[2025b](https://arxiv.org/html/2605.08665#bib.bib33); Liuet al\.,[2024b](https://arxiv.org/html/2605.08665#bib.bib34);[Maet al\.,](https://arxiv.org/html/2605.08665#bib.bib5)\)generate concise reasoning via token budgets or step\-skipping, though uniform strategies fail on problems of varying complexity\. Both families avoid the cost of*RL with length penalties*\(Luoet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib3); Arora and Zanette,[2025](https://arxiv.org/html/2605.08665#bib.bib23); Aggarwal and Welleck,[2025](https://arxiv.org/html/2605.08665#bib.bib14); Zhanget al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib7)\), which requires careful tuning of penalty coefficients and reward scaling, incurring significantly higher compute than SFT\(Suiet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib25); Schulmanet al\.,[2017](https://arxiv.org/html/2605.08665#bib.bib42)\); moreover, fixed penalties cause over\-compression on easy problems and under\-compression on hard ones\.

#### The instruct model as a capability probe\.

Our key insight is that the corresponding instruct model serves as an ideal capability probe\. If the instruct model can solve a problem with minimal or no reasoning guidance, the reasoning model should not generate a lengthy chain\-of\-thought\. By probing the instruct model with incrementally more reasoningepisodes111Following\(Quet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib26)\), each round of verification is anepisode; we use regex patterns such as “let me verify” or “on second thought” to segment episodes\.from the reasoning model’s trace, we categorize problems into three cognitive states\.State I \(No\-Hint\):the instruct model succeeds without any hint, so the reasoning model should produce a direct answer\.State II \(Sparse\-Hint\):the instruct model succeeds given only the first few episodes—the “Minimum Effective Hint”—so the reasoning model should generate just this essential prefix\.State III \(Full\-Hint\):the instruct model fails even with substantial hints, so the reasoning model should generate complete deliberative reasoning to ensure accuracy\.

#### Hint Tuning: efficiency via alignment\.

Based on this self\-annotation, we proposeHint Tuning\. Unlike RL methods that must*discover*efficiency through expensive exploration, we frame it as an*alignment*task: aligning the reasoning model’s output length with the instruct model’s capability boundaries\. This converts the ill\-defined question of “how much reasoning is enough” into concrete, measurable training data, requiring no human annotation or reward models\.

#### Contributions\.

On Qwen3 and DeepSeek\-R1\-Distill models \(4B–32B\),Hint Tuningachieves 24–66% token reduction \(31\.5% on average\) with competitive accuracy across five math reasoning benchmarks \(Figure[1](https://arxiv.org/html/2605.08665#S0.F1)a\), using only 1K training samples \(25–640 fewer than existing methods, Figure[1](https://arxiv.org/html/2605.08665#S0.F1)b\)\.

## 2Related Works

![Refer to caption](https://arxiv.org/html/2605.08665v1/x3.png)Figure 3:Efficient reasoning training paradigms\.#### Chain\-of\-Thought and Test\-Time Scaling\.

Chain\-of\-Thought \(CoT\) prompting\([Weiet al\.,](https://arxiv.org/html/2605.08665#bib.bib9);[Kojimaet al\.,](https://arxiv.org/html/2605.08665#bib.bib29)\)elicits reasoning in large language models by encouraging step\-by\-step thinking\. Recent work shows CoT follows a test\-time scaling law\(Snellet al\.,[2024](https://arxiv.org/html/2605.08665#bib.bib27); OpenAI,[2024](https://arxiv.org/html/2605.08665#bib.bib28)\), where accuracy improves with increased inference compute viasequential scaling\(longer reasoning traces per sample, as in O1 and DeepSeek\-R1\) orparallel scaling\(aggregating multiple samples via self\-consistency\([Wanget al\.,](https://arxiv.org/html/2605.08665#bib.bib30)\)or best\-of\-N\)\(Muennighoffet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib12)\)\. However, this improved accuracy comes at the cost of substantial computational overhead and increased latency\.

#### SFT\-Based Efficient Reasoning\.

One line of work improves reasoning efficiency via supervised fine\-tuning\. Existing approaches employ two strategies:post\-reasoning compressionprunes redundant steps from full traces using heuristics or external models\(Kanget al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib31); Xiaet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib32); Chenet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib15);[Munkhbatet al\.,](https://arxiv.org/html/2605.08665#bib.bib35)\), whileduring\-reasoning generationprompts models to produce concise paths through optimal token budgets\(Hanet al\.,[2025b](https://arxiv.org/html/2605.08665#bib.bib33)\), step\-skipping\(Liuet al\.,[2024b](https://arxiv.org/html/2605.08665#bib.bib34)\), or parameter mixing\([Maet al\.,](https://arxiv.org/html/2605.08665#bib.bib5)\)\. Fine\-tuning strategies include standard approaches \(LoRA\([Huet al\.,](https://arxiv.org/html/2605.08665#bib.bib36)\), DPO\(Hanet al\.,[2025a](https://arxiv.org/html/2605.08665#bib.bib37); Rafailovet al\.,[2023](https://arxiv.org/html/2605.08665#bib.bib38)\)\), progressive methods\([Maet al\.,](https://arxiv.org/html/2605.08665#bib.bib5); Liuet al\.,[2024a](https://arxiv.org/html/2605.08665#bib.bib39)\), and model merging\(Teamet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib40)\)\. However, these methods determine reasoning length based on external models or uniform strategies, ignoring that problem difficulty is relative to the target model’s capability\-what appears simple to a strong compressor may still require extensive reasoning for the model being fine\-tuned, causing inappropriate compression\.

#### RL\-Based Efficient Reasoning\.

Reinforcement Learning \(RL\) incorporate length penalties into reward functions via cosine\-based penalties\(Changet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib4)\), length\-harmonizing rewards\(Luoet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib3); Arora and Zanette,[2025](https://arxiv.org/html/2605.08665#bib.bib23); Fenget al\.,[2026](https://arxiv.org/html/2605.08665#bib.bib48)\), L1 regularization\(Aggarwal and Welleck,[2025](https://arxiv.org/html/2605.08665#bib.bib14)\), constrained instructions\([Shenet al\.,](https://arxiv.org/html/2605.08665#bib.bib13)\), or preference optimization\(Zhanget al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib7); Rafailovet al\.,[2023](https://arxiv.org/html/2605.08665#bib.bib38)\)\. These methods integrate length\-aware objectives into classical RLHF frameworks such as PPO\(Schulmanet al\.,[2017](https://arxiv.org/html/2605.08665#bib.bib42)\)and DPO\(Rafailovet al\.,[2023](https://arxiv.org/html/2605.08665#bib.bib38); Ouyanget al\.,[2022](https://arxiv.org/html/2605.08665#bib.bib41)\)\. However, RL incurs higher compute costs than SFT with training instability\(Zhenget al\.,[2023](https://arxiv.org/html/2605.08665#bib.bib43)\)\. Fixed penalties fail to adapt to problem heterogeneity\(Cobbeet al\.,[2021](https://arxiv.org/html/2605.08665#bib.bib44); Lightmanet al\.,[2024](https://arxiv.org/html/2605.08665#bib.bib45)\), causing over\-compression on easy tasks and under\-compression on hard ones\. Despite automation claims, extensive hyperparameter tuning remains necessary\(Engstromet al\.,[2020](https://arxiv.org/html/2605.08665#bib.bib46); Andrychowiczet al\.,[2020](https://arxiv.org/html/2605.08665#bib.bib47)\): PPO clips, KL constraints, reward scaling, and sampling temperatures\.

#### Positioning of Our Work\.

As illustrated in Figure[3](https://arxiv.org/html/2605.08665#S2.F3): instead of expensive RL optimization or data\-limited SFT, we automatically generate difficulty\-adaptive reasoning traces viaMinimum Effective Hintcriterion\. This data\-centric strategy achieves RL\-level efficiency at lower cost, requiring no reward models or external compressor models for data labeling\.

## 3Method

Reasoning Models generate verbose traces regardless of problem difficulty, wasting computation on problems that require far less reasoning\. We first quantify this inefficiency, then describe how it motivates our data construction pipeline\.

### 3\.1Efficiency Gap and Capability Probing

We compare Qwen3\-4B\-Instruct \(Instruct Model\) and Qwen3\-4B\-Thinking \(Reasoning Model\)\(Team,[2025](https://arxiv.org/html/2605.08665#bib.bib17)\)on three competition\-level math benchmarks: AIME24, AIME25, and HMMT25, sampling 8 responses per problem validated withMath\-Verify\. As shown in Figure[2](https://arxiv.org/html/2605.08665#S1.F2), Reasoning Models generate55–88\\timesmore tokens than Instruct Models even when both reach correct answers\. Yet when provided with partial reasoning hints, Instruct Models successfully solve problems using only 27\.5% of episodes and 11\.9% of tokens, suggesting most reasoning is redundant\.

This motivates a key question:what is the minimum hint needed to activate the Instruct Model’s capabilities?We define theMinimum Effective HintK∗K^\{\*\}as:

K∗=min⁡\{k​\{0,1,…,N\}:Minstruct​\(x​e1:k\)=y\}K^\{\*\}=\\min\\bigl\\\{k\\in\\\{0,1,\\ldots,N\\\}:M\_\{\\text\{instruct\}\}\(x\\oplus e\_\{1:k\}\)=y\\bigr\\\}\(1\)wheree1:ke\_\{1:k\}denotes the firstkkreasoning episodes from the Reasoning Model \(e1:0=e\_\{1:0\}=\\emptyset\)\. If no suchkkexists \(*i\.e\.,*the instruct model fails under all hint levels\), we setK∗=NK^\{\*\}=N, treating the problem as requiring full deliberation\. Intuitively,K∗K^\{\*\}measures how much of the Reasoning Model’s trace is actually necessary to unlock the Instruct Model’s answer\.

### 3\.2Hint Distribution Analysis

Analyzing the distribution ofK∗K^\{\*\}\(Figure[4](https://arxiv.org/html/2605.08665#S3.F4)\) reveals three properties that challenge the “longer is better” assumption\.

#### Hints unlock previously unsolvable problems\.

While 36–50% of problems require no hints at all, reasoning hints enable an additional 23–33% to succeed with sparse episodes and 33–60% with full traces \(Figure[4](https://arxiv.org/html/2605.08665#S3.F4)c\), demonstrating that the Instruct Model has latent capabilities that targeted hints can activate\.

#### Key steps matter more than length\.

Only 25–32% of problems exhibitcontinuoussuccess, where succeeding at episodekkguarantees success atk\+1k\{\+\}1\. For the remaining 68–75%, success is non\-monotonic: additional episodes can introduce distractors, and success rates fluctuate rather than consistently improve withKK\(Figure[4](https://arxiv.org/html/2605.08665#S3.F4)a,d\)\. This indicates that specific pivotal episodes matter far more than linear accumulation\.

#### Early hints serve as activators\.

Critical reasoning steps concentrate within the first 25 episodes \(Figure[4](https://arxiv.org/html/2605.08665#S3.F4)b\), suggesting CoT hints function as direction\-setting activators rather than scaffolding that requires complete reasoning chains\.

![Refer to caption](https://arxiv.org/html/2605.08665v1/x4.png)Figure 4:Distribution ofK∗K^\{\*\}reveals non\-monotonic patterns\.\(a\) Success rate fluctuates withKKrather than increasing monotonically\. \(b\) Most required hints concentrate in early episodes \(within first 25\)\. \(c\) Hints enable 23–33% of previously unsolvable problems to succeed with sparse reasoning, and 33–60% with full traces\. \(d\) Only 25–32% show continuous success; most exhibit non\-monotonic patterns\.

### 3\.3Constructing Variable\-Length CoT Data

The three properties above motivate a natural three\-state partition\. We use the s1 dataset\(Muennighoffet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib12)\)as our seed corpus—1,000 challenging problems curated from 59K samples across 16 diverse sources \(NuminaMath, OlympicArena,*etc\.*\), filtered for quality and difficulty\. The pipeline transforms these problems into efficient training samples through three stages: \(1\)Generation: we discard the original s1K traces and regenerate fresh CoT traces for each problemxxusingMreasonM\_\{\\text\{reason\}\}; \(2\)Segmentation: we decompose traces into discrete episodes\(e1,…,eN\)\(e\_\{1\},\\ldots,e\_\{N\}\)based on self\-reflection markers \(Appendix[B](https://arxiv.org/html/2605.08665#A2.SS0.SSS0.Px1)\); \(3\)Probing: we testMinstructM\_\{\\text\{instruct\}\}by sequentially prependingkkepisodes to determine if it reaches the correct answeryy, recordingK∗K^\{\*\}\.

Based onK∗K^\{\*\}, we construct the training responserrfor each problem:

r=<think\>​\{p0K∗=0\(e1,…,eK∗\)0<K∗<N\(e1,…,eN\)K∗=N​</think\>​yr=\\texttt\{<think\>\}\\oplus\\begin\{cases\}p\_\{0\}&K^\{\*\}=0\\\\ \(e\_\{1\},\\ldots,e\_\{K^\{\*\}\}\)&0<K^\{\*\}<N\\\\ \(e\_\{1\},\\ldots,e\_\{N\}\)&K^\{\*\}=N\\end\{cases\}\\oplus\\;\\texttt\{</think\>\}\\oplus y\(2\)wherep0p\_\{0\}is a brief directive \(e\.g\., “Direct answer”\) forState I\(No\-Hint\); the firstK∗K^\{\*\}episodes are retained forState II\(Sparse\-Hint\); and the full trace is preserved forState III\(Full\-Hint\)\. The complete procedure is formalized in Algorithm[1](https://arxiv.org/html/2605.08665#alg1)\(Appendix[B](https://arxiv.org/html/2605.08665#A2)\)\. The resulting dataset is balanced across the three states \(35\.7% / 26\.4% / 38\.2%\), reflecting diverse difficulty levels in the s1K corpus; see Appendix[C](https://arxiv.org/html/2605.08665#A3)for statistics and Appendix[D](https://arxiv.org/html/2605.08665#A4)for representative examples\.

We combine𝒟=𝒟1​𝒟2​𝒟3\\mathcal\{D\}=\\mathcal\{D\}\_\{1\}\\cup\\mathcal\{D\}\_\{2\}\\cup\\mathcal\{D\}\_\{3\}and fine\-tuneMreasonM\_\{\\text\{reason\}\}via standard cross\-entropy loss, teaching it to adapt output length to problem difficulty—generating direct answers for simple problems while preserving full deliberative reasoning for complex ones\.

Table 1:Overview of evaluation benchmarks\.CategoryBenchmark\# Prob\.DifficultyIn\-domainMATH500500High SchoolAIME2430CompetitionAIME2530CompetitionHMMT2530CompetitionTemporal Held\-outAMO\-Bench50IMO\-levelCross\-domainLiveCodeBench713MixedGPQA\-Diamond198Graduate

## 4Experiments

We evaluateHint Tuningacross four model scales and three task domains\. Our experiments are designed to answer: \(1\) CanHint Tuningreduce reasoning verbosity without sacrificing accuracy? \(2\) Does efficiency generalize to unseen problems and domains? \(3\) What does adaptive compression reveal about the nature of reasoning itself?

### 4\.1Experimental Setup

#### Models\.

For reasoning models, we evaluate Qwen3\-4B\-Thinking\(Team,[2025](https://arxiv.org/html/2605.08665#bib.bib17)\)and DeepSeek\-R1\-Distill\-Qwen\-\{7B, 14B, 32B\}\(DeepSeek\-AI,[2025](https://arxiv.org/html/2605.08665#bib.bib1)\), spanning an order of magnitude in scale\. We use Qwen3\-4B\-Instruct as the Instruct Model for hint validation\.

#### Baselines\.

We compare against four categories of methods\.Original Modelsreport zero\-shot results of vanilla reasoning models\.Prompt\-based: Budget Prompt\([Hanet al\.,](https://arxiv.org/html/2605.08665#bib.bib6)\)instructs models to “think withinNNtokens” at inference time without weight updates \(budget set to 8,000; see Appendix[A\.3](https://arxiv.org/html/2605.08665#A1.SS3)\)\.SFT\-based: SFT\-S1\.1 fine\-tunes on high\-quality traces from s1\(Muennighoffet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib12)\); VeriThinker\(Chenet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib15)\)distills from self\-verified reasoning paths to suppress unnecessary reflection \(340K samples\)\.RL\-based: EfficientR\(Arora and Zanette,[2025](https://arxiv.org/html/2605.08665#bib.bib23)\)and AdaptThink\(Zhanget al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib7)\)\(7B only\) apply online RL with length\-accuracy reward shaping\. EfficientR uses 3\.2K prompts with 8 rollouts per prompt \(25\.6K total generations\); AdaptThink uses 40K prompts with 16 rollouts per prompt \(640K total generations\)\. Both incur substantially higher compute than offline SFT methods\.

#### Benchmarks\.

We evaluate across a spectrum of domain familiarity \(Table[1](https://arxiv.org/html/2605.08665#S3.T1)\)\. For in\-domain math reasoning, we use MATH500\([Hendryckset al\.,](https://arxiv.org/html/2605.08665#bib.bib18)\), AIME24, AIME25, and HMMT25, spanning increasing difficulty and chronological release\(Caoet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib19)\)\. As a temporal held\-out set, AMO\-Bench\(Anet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib20)\)comprises 50 IMO\-level problems released after all model checkpoints, minimizing contamination risk\. Forcross\-domaingeneralization, we include LiveCodeBench\([Jainet al\.,](https://arxiv.org/html/2605.08665#bib.bib21)\)\(code\) and GPQA\-Diamond\(Reinet al\.,[2024](https://arxiv.org/html/2605.08665#bib.bib22)\)\(science\)\.

#### Evaluation Protocol\.

All evaluations use thelightevalframework222[https://huggingface\.co/docs/lighteval](https://huggingface.co/docs/lighteval)for reproducibility\. We sample 16 responses per problem with temperature 0\.6 and top\-pp0\.95, reporting averaged accuracy to mitigate variance\. All problems use a consistent prompt:“Please reason step by step, and put your final answer within \\boxed\{\}\.”Our baseline reproductions align with official reports\.

### 4\.2Implementation Details

We sample 1K problems from s1\(Muennighoffet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib12)\)and apply our three\-state annotation strategy \(Section[3](https://arxiv.org/html/2605.08665#S3)\)\. For each problem, we probeMinstructM\_\{\\text\{instruct\}\}by prepending episode prefixes to determineK∗K^\{\*\}\(Equation[1](https://arxiv.org/html/2605.08665#S3.E1)\)\. We use Qwen3\-4B\-Instruct as the default probe across all model scales to ensure a consistent and fair comparison\. While a stronger same\-family instruct model can improve label quality at larger scales where the probe\-target capability gap is more pronounced, we find that the 4B probe already yields competitive results across most settings \(Section[5\.3](https://arxiv.org/html/2605.08665#S5.SS3)\)\. For practitioners, we recommend using the strongest available same\-family instruct model as the probe when accuracy is the primary concern\. Since 90% of hint\-requiring problems succeed within the first 25 episodes, we apply early stopping during the linear search, keeping the average number of probes per problem to only 3–6\. Samples are labeled as No\-Hint \(K∗=0K^\{\*\}\{=\}0\), Sparse\-Hint \(0<K∗<N0<K^\{\*\}<N\), or Full\-Hint \(K∗=NK^\{\*\}\{=\}N\)\. We perform full\-parameter SFT with learning rate110−51\\times 10^\{\-5\}, cosine decay, and max sequence length 32,768\. Training takes 2/2\.5/8/10 hours for 4B/7B/14B/32B models on 8H20 GPUs\. See Appendix[A](https://arxiv.org/html/2605.08665#A1)for complete details\.

Table 2:Main results across model scales and reasoning benchmarks\. We compare against \(1\)General baselines\(Original, SFT, Budget Prompt\) implemented at all scales, and \(2\)Specialized methods\(AdaptThink, Efficient Reasoning, VeriThinker\) with official checkpoints available only at 7B scale\. ForHint Tuning, improvements over Original are shown ingreen, declines inred\.Method\# DataMathematical Reasoning TasksAllStandard BenchmarksTemporal Gen\.AverageMATH500AIME24AIME25HMMT25AMO\-BenchAccTokAccTokAccTokAccTokAccTokAccTokQwen3\-4B\-Thinking\-2507Original–97\.8651777\.91909571\.32097339\.42437610\.32753359\.319699Budget Prompt–97\.2545777\.11814771\.02036241\.72276712\.82550960\.018448SFT\-S1\.11K95\.3325067\.11092956\.01227330\.8142227\.71888151\.411911Hint Tuning1K95\.7219073\.81023359\.41209932\.71452320\.52592656\.412994vs\. Original–\-2\.1\-66%\-4\.1\-46%\-11\.9\-42%\-6\.7\-40%\+10\.2\-6%\-2\.9\-34%DeepSeek\-R1\-Distill\-Qwen\-7BGeneral Baselines:Original–92\.6409853\.81295937\.91457616\.7163392\.62095740\.713786Budget Prompt–91\.0297951\.71251739\.01313417\.7157685\.12018440\.912916SFT\-S1\.11K93\.4410157\.71341439\.41531919\.0165727\.71945543\.413772Specialized Methods:EfficientRα=0\.1\\alpha=0\.13\.2K†\{\}^\{\\text\{\\textdagger\}\}92\.2286951\.51155035\.41190215\.4131842\.61813239\.411527EfficientRα=0\.4\\alpha=0\.43\.2K†\{\}^\{\\text\{\\textdagger\}\}90\.6243253\.51033633\.11104817\.7122160\.01871839\.010950AdaptThink40K†\{\}^\{\\text\{\\textdagger\}\}92\.0209555\.01082636\.91206319\.8138730\.01742140\.711256VeriThinker340K92\.8267454\.61003134\.61064315\.4136895\.11527940\.510463Hint Tuning1K92\.5194452\.5989937\.7914417\.796295\.11451740\.79027vs\. Original–\-0\.1\-53%\-1\.3\-24%\-0\.2\-37%\+1\.0\-41%\+2\.5\-31%\+0\.4\-35%DeepSeek\-R1\-Distill\-Qwen\-14BOriginal–94\.1380068\.21152346\.91324824\.8149167\.71986848\.312671Budget Prompt–93\.4281366\.01051345\.61233924\.61357612\.81854148\.511556SFT\-S1\.11K95\.3425569\.81248353\.31410430\.41583610\.32004151\.813344Hint Tuning1K93\.4190863\.3796346\.6843023\.81001012\.81593948\.08850vs\. Original–\-0\.7\-50%\-4\.9\-31%\-0\.3\-36%\-1\.0\-33%\+5\.1\-20%\-0\.3\-30%DeepSeek\-R1\-Distill\-Qwen\-32BOriginal–94\.8361169\.01106151\.71305126\.91481315\.42115151\.612737Budget Prompt–93\.6248167\.7940147\.51198225\.6124637\.71702648\.310671SFT\-S1\.11K96\.0389177\.31138364\.81398332\.71527615\.41883657\.212674Hint Tuning1K95\.9192768\.8817351\.0790826\.2955112\.81865050\.99242vs\. Original–\+1\.1\-47%\-0\.2\-26%\-0\.7\-39%\-0\.7\-36%\-2\.6\-12%\-0\.7\-27%†Online RL methods with 8 rollouts/prompt \(EfficientR\) and 16 rollouts/prompt \(AdaptThink\)\. Unmarked: offline methods\.

### 4\.3Main Results: Mathematical Reasoning

Table[2](https://arxiv.org/html/2605.08665#S4.T2)summarizes performance across four model scales\. The results demonstrate a consistent trend:Hint Tuningsignificantly reduces computational cost while preserving or enhancing reasoning accuracy\.

#### Reasoning Efficiency\.

Across competition\-level benchmarks,Hint Tuningachieves a 24–66% reduction in token consumption with negligible impact on performance\. Notably, DeepSeek\-32B on MATH500 yields a 1\.1% accuracy gain while reducing inference tokens by 47%\. This represents a Pareto improvement rather than a trade\-off, suggesting that standard Chain\-of\-Thought \(CoT\) processes often expend significant computation on redundant reasoning steps whichHint Tuningsuccessfully prunes\.

#### Data Efficiency\.

Hint Tuningrequires only 1K training samples versus significantly larger data requirements for existing baselines\. For SFT\-based methods, VeriThinker uses 350K samples and SFT\-S1\.1 uses 1K samples of higher\-cost curated traces\. For RL\-based methods, the comparison requires accounting for rollout overhead: EfficientR uses 3\.2K prompts 8 rollouts=25\.6=25\.6K total generations, and AdaptThink uses 40K prompts 16 rollouts=640=640K total generations\. Compared on total generations,Hint Tuningachieves 25\.6–640 greater sample efficiency with no online exploration\. Compared on unique prompts alone,Hint Tuningmatches or exceeds efficiency at 3\.2–40 fewer prompts\. This demonstrates that adaptive reasoning can emerge from precise data curation rather than exhaustive RL exploration\.

#### Temporal Generalization \(AMO\-Bench\)\.

To assess robustness against memorization, we evaluate on AMO\-Bench, atemporally held\-outdataset of IMO\-level problems released after all model training cutoffs\. Most model variants maintain accuracy within 3% of baseline while achieving modest token reduction\. Qwen3\-4B shows a larger accuracy gain, which we attribute to the mitigation of two overthinking failure modes: the model either contradicts a correct intermediate result through excessive reflection, or exhausts its context on redundant verification loops\. By teachingwhen to stop reasoning,Hint Tuningturns token reduction and accuracy improvement into mutually reinforcing effects\. The lower compression rate on AMO\-Bench compared to in\-distribution math further confirms that the model conservatively retains reasoning steps on genuinely novel problems, in contrast to SFT\-S1\.1, which compresses more aggressively but at the cost of accuracy\.

Table 3:Cross\-domain generalization results\.Hint Tuningpreserves accuracy on out\-of\-distribution \(OOD\) tasks\. Notably, it doesnotforce compression on unfamiliar domains, providing evidence of a content\-aware reasoning policy\.\(a\) Code Generation \(LiveCodeBench\)

MethodQwen3\-4BDS\-R1\-7BDS\-R1\-14BDS\-R1\-32BAccTok\(k\)AccTok\(k\)AccTok\(k\)AccTok\(k\)Original62\.717\.437\.311\.849\.310\.856\.711\.1Budget Prompt––––––––SFT\-S1\.150\.414\.739\.215\.057\.112\.165\.711\.8Hint Tuning61\.913\.237\.716\.653\.713\.866\.013\.0vs\. Orig\-0\.8\-24%\+0\.4\+41%\+4\.4\+28%\+9\.3\+17%

\(b\) Scientific Knowledge \(GPQA\-Diamond\)

MethodQwen3\-4BDS\-R1\-7BDS\-R1\-14BDS\-R1\-32BAccTok\(k\)AccTok\(k\)AccTok\(k\)AccTok\(k\)Original66\.28\.853\.58\.460\.67\.660\.67\.3Budget Prompt66\.28\.746\.56\.553\.05\.557\.14\.9SFT\-S1\.169\.27\.650\.59\.661\.110\.063\.69\.3Hint Tuning67\.27\.752\.010\.458\.610\.163\.110\.5vs\. Orig\+1\.0\-13%\-1\.5\+23%\-2\.0\+34%\+2\.5\+44%

### 4\.4Cross\-Domain Generalization: The Efficiency Spectrum

A critical risk in reasoning compression is degraded performance on domains absent from training\. We evaluateHint Tuningon Code Generation and Scientific Reasoning to assess this risk\.

#### Accuracy holds; compression is domain\-selective\.

Table[3](https://arxiv.org/html/2605.08665#S4.T3)reveals a consistent asymmetry:Hint Tuningpreserves or improves accuracy on both OOD domains, while token consumption does not decrease and in some cases increases substantially\. Rather than compressing indiscriminately, the model defaults to fuller reasoning on inputs outside the learned compression regime—in contrast to Budget Prompt, which forces compression regardless of domain and incurs notable accuracy drops on scientific reasoning\.

#### The efficiency spectrum\.

Across benchmarks, compression scales with domain familiarity: strongest on in\-distribution math, moderate on temporally held\-out problems, and largely absent on OOD domains\. This gradient suggestsHint Tuninginternalizes a compression boundary tied to its training distribution rather than a uniform reasoning policy\. We leave a fuller characterization of this boundary to future work\.

Table 4:Ablation study on data composition\.MethodCountAIME25HMMT25AccTok \(k\)AccTok \(k\)BaselinesOriginal \(zero\-shot\)—73\.322\.040\.025\.8SFT on external traces1k46\.613\.250\.013\.2Ours \(full method\)1k50\.07\.553\.315\.6\(a\) Answer\-correctness filteringCorrect traces only42143\.38\.133\.39\.3\(b\) Trace sourceExternal teacher \(DeepSeek\-R1\)1k43\.310\.826\.713\.1\(c\) Episode tag formatRandom word tags1k56\.610\.643\.317\.6Special tokens only1k26\.713\.143\.310\.8

![Refer to caption](https://arxiv.org/html/2605.08665v1/x5.png)Figure 5:Training Data Variance and Adaptive CoT Compression\. \(a\) Comparison of token length variance between our SFT dataset and the baseline \(DPSK trajectories\), showing a broader distribution in our method\. \(b\) Token distribution across three distinct cognitive trajectory types within our dataset\. \(c\) Post\-SFT inference behavior AIME25\. The model demonstrates benchmark\-specific compression, dynamically adapting reasoning length to problem complexity\.Table 5:Left: Probe ablation on R1\-Distill\-14B \(AIME24\); same\-family probes better preserve reasoning alignment\.Right: Sample efficiency on R1\-Distill\-7B \(AIME24\);s1= base curated set\. Performance plateaus beyond 1k samples, suggesting trace quality matters more than quantity\.ProbeRelationshipAccTok \(k\)Original—70\.611\.5Llama\-3\.1\-8Bcross\-family58\.114\.5Qwen3\-4Bsame family59\.28\.0Qwen3\-30B\-A3Bsame family, MoE61\.19\.9Qwen2\.5\-72Bsame family, larger65\.210\.5

SamplesSourceAccTok \(k\)200s1\(subsample\)50\.016\.4500s1\(subsample\)50\.012\.8800s1\(subsample\)46\.711\.21000s152\.59\.92000s1\+ MATH/NuminaMath51\.610\.2

## 5Analysis

We conduct systematic ablations to understand the mechanisms underlyingHint Tuning, examining four design choices: answer correctness filtering, trace source, episode tag format, and instruct model selection\. Our findings suggest that, for reasoning models, the quality of the reasoning process can matter more than the correctness of the final outcome\.

### 5\.1Process Matters More Than Outcome

#### Incorrect answers still teach reasoning\.

We compare training on all 1K collected trajectories against a filtered subset of 421 correct\-only traces\. The unfiltered set substantially outperforms the filtered subset: 50\.0% vs\. 43\.3% on AIME25 and 53\.3% vs\. 33\.3% on HMMT25 \(Table[4](https://arxiv.org/html/2605.08665#S4.T4), group \(a\)\)\. Trajectories with incorrect final answers encode valuable reasoning patterns—problem decomposition, intermediate verification, and explicit self\-correction—that teach the model*how*to reason rather than*what*to answer\.

#### Reasoning diversity underlies this effect\.

Figure[5](https://arxiv.org/html/2605.08665#S4.F5)\(a\) offers a structural explanation\. The unfiltered set exhibits substantially higher variance in trajectory length \(σ​36\.3\\sigma\\approx 36\.3k tokens\) than the correct\-only subset \(σ​8\.3\\sigma\\approx 8\.3k tokens\)\. Correct\-answer filtering disproportionately removes long, exploratory trajectories on difficult problems, discarding the reasoning patterns most relevant to hard instances\. As a result, models trained on the high\-variance corpus dynamically scale computation to problem difficulty \(Figure[5](https://arxiv.org/html/2605.08665#S4.F5)\(c\)\), an adaptive behavior largely absent in the correct\-only\-trained model\.

### 5\.2Trace Source and Format

#### Self\-generated traces outperform external sources\.

We contrast endogenous traces from the target model against exogenous traces from a stronger external model \(DeepSeek\-R1\)\. Despite the external model’s superior capability, self\-generated traces yield markedly better outcomes \(Table[4](https://arxiv.org/html/2605.08665#S4.T4), group \(b\)\), with distributional proximity mattering more than raw capability \(Table[5](https://arxiv.org/html/2605.08665#S4.T5), left\)\. A stronger model’s reasoning strategies diverge from the target’s generative distribution in abstraction level, notation, and granularity, forcing simultaneous style adaptation and capability improvement\. Self\-generated traces instead reinforce patterns already latent in the model\.

#### Structure matters; semantics do not\.

Replacing semantic tags with arbitrary words yields comparable performance, while removing delimiters entirely causes sharp degradation \(Table[4](https://arxiv.org/html/2605.08665#S4.T4), group \(c\)\)\. The active ingredient is*structural separation*between thinking and answering phases, not tag semantics\. Consistent with this, correctly answered problems consistently use fewer tokens than incorrect ones across all models, suggesting compression is a byproduct of successful reasoning rather than a cause\.

### 5\.3Instruct Model as Difficulty Probe

The instruct probe determinesK∗K^\{\*\}labels and directly shapes the training signal\. Table[5](https://arxiv.org/html/2605.08665#S4.T5)\(left\) compares probes of varying scale and family on R1\-Distill\-14B\.

#### Same\-family probes outperform cross\-family ones\.

A cross\-family probe \(Llama\-3\.1\-8B\) yields the largest accuracy drop, while same\-family probes of increasing scale progressively recover performance, consistent with the distributional alignment findings in Section[5\.2](https://arxiv.org/html/2605.08665#S5.SS2)\.

#### The probe’s failure mode is conservative by design\.

When the probe fails at all hint levels, the sample is assigned Full\-Hint and the complete reasoning trace is preserved\. An overly weak probe therefore over\-assigns to Full\-Hint, producing less compression but not incorrect supervision\. This explains the larger accuracy gap at 14B scale with the default Qwen3\-4B probe: switching to Qwen2\.5\-72B recovers 6\.0 points on AIME24 while retaining meaningful token reduction\. We recommend using the strongest available same\-family instruct model as the probe when accuracy is the primary concern\.

## 6Conclusion

We introducedHint Tuning, a data\-efficient framework that calibrates reasoning depth by using the instruct model as a difficulty probe\. The Minimum Effective Hint criterion converts the abstract question of “how much reasoning is enough” into a concrete, measurable signal, yielding a three\-state training set that delivers 24–66% token reduction with only 1K samples\. Our results suggest that adaptive reasoning can emerge from precise data curation rather than exhaustive RL exploration, and that, on problems within the training distribution, shorter reasoning and higher accuracy are not necessarily at odds—teaching the modelwhento stop can mitigate overthinking failure modes\.

#### Limitations and future work\.

Hint Tuningrelies on a capability gap between the instruct and reasoning models to generate meaningful difficulty variation; how this gap should be calibrated across model families and scales remains an open design question\. Our experiments focus on mathematical reasoning, and whether the learned compression boundary generalizes to other domains requires further investigation\. Finally, the mechanism by which structural episode tags activate adaptive reasoning is empirically observed but theoretically unexplained\.

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## Appendix AImplementation Details

### A\.1Training Configuration

We perform full\-parameter supervised fine\-tuning on 8NVIDIA H20\-141GB GPUs\. All models are trained with the AdamW optimizer using a cosine learning rate schedule with 10% warmup for 10 epochs\. We set per\-device batch size to 1 with gradient accumulation over 8 steps, yielding an effective batch size of 8\. The maximum sequence length is 32,768 tokens, and we employ BF16 mixed\-precision training with FlashAttention\-2\.

For smaller models \(Qwen3\-4B\-Think and DeepSeek\-R1\-Distill\-Qwen\-7B\), we use DeepSpeed ZeRO Stage 0 with sequence parallel size 2 and a learning rate of110−51\\times 10^\{\-5\}\. For larger models \(DeepSeek\-R1\-Distill\-Qwen\-14B and 32B\), we adopt DeepSpeed ZeRO Stage 3 to accommodate memory constraints\. The 14B model uses learning rate110−51\\times 10^\{\-5\}and sequence parallel size 2, while the 32B model uses a higher learning rate of110−41\\times 10^\{\-4\}and sequence parallel size 4\.

We evaluate checkpoints on a held\-out validation set and select the best\-performing one: Qwen3\-4B\-Think at step 320, DeepSeek\-R1\-Distill\-Qwen\-7B at step 450, DeepSeek\-R1\-Distill\-Qwen\-14B at step 350, and DeepSeek\-R1\-Distill\-Qwen\-32B at step 315\.

### A\.2Evaluation Protocol

#### Sampling Strategy\.

All evaluations use thelightevalframework333[https://huggingface\.co/docs/lighteval](https://huggingface.co/docs/lighteval)for reproducibility\. For each problem, we sample 16 responses using temperatureT=0\.6T=0\.6, top\-p=0\.95p=0\.95, and maximum new tokens of 32,768\. We report averaged accuracy across all samples to mitigate variance inherent in reasoning tasks\.

#### Prompt Format\.

All math problems are evaluated using a consistent prompt format that instructs the model to reason step by step:“Please reason step by step, and put your final answer within \\boxed\{\}\.”This ensures fair comparison across different model configurations\.

#### Answer Validation\.

Following the s1 dataset\(Muennighoffet al\.,[2025](https://arxiv.org/html/2605.08665#bib.bib12)\), we employ an LLM\-as\-judge approach for answer validation\. The grading prompt \(Figure[6](https://arxiv.org/html/2605.08665#A1.F6)\) instructs the judge model to compare generated answers against ground\-truth solutions, evaluating both final answers and intermediate reasoning steps where applicable\.

Answer Grading PromptYou are an AI assistant for grading a science problem\. The user will provide you with the question itself, an attempt made by a student and the correct answer to the problem\. Your job is to judge whether the attempt is correct by comparing it with the correct answer\. If the expected solution concludes with a number or choice, there should be no ambiguity\. If the expected solution involves going through the entire reasoning process, you should judge the attempt based on whether the reasoning process is correct with correct answer if helpful\.The user will provide the attempt and the correct answer in the following format:\# Problem \{problem\}\#\# Attempt \{attempt\}\#\# Correct answer \{solution\}Explain your reasoning, and end your response on a new line with only “Yes” or “No” \(without quotes\)\.

Figure 6:Grading prompt for answer validation\.
#### Token Counting\.

Reasoning overhead is measured by counting tokens within the<think\>\.\.\.</think\>tags\. Tokens outside these tags \(i\.e\., the problem statement and final answer\) are excluded from compression metrics but included when computing total inference cost\.

### A\.3Baseline Configuration

#### Budget Forcing Prompt\.

For the Budget Forcing baseline\([Hanet al\.,](https://arxiv.org/html/2605.08665#bib.bib6)\), we use a token budget of 8,000 tokens and the following prompt template:

\{problem\}Solve this problem concisely in approximately 8000 tokens\. Show your reasoning with only the essential steps needed\. Put your final answer within \\boxed\{\}\.

Figure 7:Prompt template for Budget Forcing baseline\.

## Appendix BAlgorithm Pseudocode

Algorithm[1](https://arxiv.org/html/2605.08665#alg1)presents the complete procedure for constructing variable\-length Chain\-of\-Thought training data\. The algorithm takes a problem set𝒳\\mathcal\{X\}, a reasoning modelMreasonM\_\{\\text\{reason\}\}\(e\.g\., DeepSeek\-R1\), and an instruct modelMinstructM\_\{\\text\{instruct\}\}\(e\.g\., Qwen3\-4B\) as inputs\. For each problem, we first generate a complete reasoning trace using the reasoning model, then segment it into episodes, and finally probe the instruct model to determine the minimum hint required for successful problem\-solving\. Based on the probing resultK∗K^\{\*\}, each sample is assigned to one of three cognitive states:No\-Hint\(K∗=0K^\{\*\}=0\),Sparse\-Hint\(0<K∗<N0<K^\{\*\}<N\), orFull\-Hint\(K∗=NK^\{\*\}=N\)\.

Algorithm 1Variable\-Length CoT Data Construction0:Problem set

𝒳\\mathcal\{X\}, Reasoning model

MreasonM\_\{\\text\{reason\}\}, Instruct model

MinstructM\_\{\\text\{instruct\}\}
0:Training dataset

𝒟=𝒟1​𝒟2​𝒟3\\mathcal\{D\}=\\mathcal\{D\}\_\{1\}\\cup\\mathcal\{D\}\_\{2\}\\cup\\mathcal\{D\}\_\{3\}
1:Initialize

𝒟1,𝒟2,𝒟3\\mathcal\{D\}\_\{1\},\\mathcal\{D\}\_\{2\},\\mathcal\{D\}\_\{3\}\\leftarrow\\emptyset
2:foreachproblem

x​𝒳x\\in\\mathcal\{X\}do

3:// Stage 1: Generate full reasoning trace

4:

r,y​Mreason​\(x\)r,y\\leftarrow M\_\{\\text\{reason\}\}\(x\)\{

rr: reasoning trace,

yy: answer\}

5:// Stage 2: Segment reasoning into episodes

6:

\(e1,e2,…,eN\)​SegmentEpisodes​\(r\)\(e\_\{1\},e\_\{2\},\\ldots,e\_\{N\}\)\\leftarrow\\text\{SegmentEpisodes\}\(r\)
7:// Stage 3: Binary search for minimum effective hint

8:

K∗​NK^\{\*\}\\leftarrow N\{Default to full reasoning\}

9:for

k​0k\\leftarrow 0to

NNdo

10:

y^​Minstruct​\(x​e1:k\)\\hat\{y\}\\leftarrow M\_\{\\text\{instruct\}\}\(x\\oplus e\_\{1:k\}\)\{Probe with

kkepisodes\}

11:if

y^=ycorrect\\hat\{y\}=y\_\{\\text\{correct\}\}then

12:

K∗​kK^\{\*\}\\leftarrow k\{Found minimum hint\}

13:break

14:endif

15:endfor

16:// Stage 4: Assign to cognitive state

17:if

K∗=0K^\{\*\}=0then

18:

d​\(x,<think\>​p0​</think\>,y\)d\\leftarrow\(x,\\texttt\{<think\>\}\\oplus p\_\{0\}\\oplus\\texttt\{</think\>\},y\)\{No\-Hint\}

19:

𝒟1​𝒟1​\{d\}\\mathcal\{D\}\_\{1\}\\leftarrow\\mathcal\{D\}\_\{1\}\\cup\\\{d\\\}
20:elseif

0<K∗<N0<K^\{\*\}<Nthen

21:

d​\(x,<think\>​e1:K∗​</think\>,y\)d\\leftarrow\(x,\\texttt\{<think\>\}\\oplus e\_\{1:K^\{\*\}\}\\oplus\\texttt\{</think\>\},y\)\{Sparse\-Hint\}

22:

𝒟2​𝒟2​\{d\}\\mathcal\{D\}\_\{2\}\\leftarrow\\mathcal\{D\}\_\{2\}\\cup\\\{d\\\}
23:else

24:

d​\(x,<think\>​r​</think\>,y\)d\\leftarrow\(x,\\texttt\{<think\>\}\\oplus r\\oplus\\texttt\{</think\>\},y\)\{Full\-Hint\}

25:

𝒟3​𝒟3​\{d\}\\mathcal\{D\}\_\{3\}\\leftarrow\\mathcal\{D\}\_\{3\}\\cup\\\{d\\\}
26:endif

27:endfor

28:return

𝒟=𝒟1​𝒟2​𝒟3\\mathcal\{D\}=\\mathcal\{D\}\_\{1\}\\cup\\mathcal\{D\}\_\{2\}\\cup\\mathcal\{D\}\_\{3\}

#### Episode Segmentation\.

TheSegmentEpisodes​\(\)\\text\{SegmentEpisodes\}\(\\cdot\)function splits a reasoning trace into discrete episodes based on reflection keywords that signal cognitive transitions\. We identify episode boundaries by detecting the following keywords \(case\-insensitive\):“wait”,“actually”,“hmm”,“let me reconsider”,“on second thought”,“hold on”, and“let me rethink”\. These keywords naturally demarcate points where the model pauses to reflect, reconsider, or shift its reasoning strategy, making them effective boundaries for incremental hint probing\.

## Appendix CData Statistics

### C\.1Cognitive State Distribution

Table[6](https://arxiv.org/html/2605.08665#A3.T6)presents the distribution of cognitive states in our constructed dataset after capability probing with Qwen3\-4B\.

Table 6:Distribution of cognitive states in the s1K dataset\. Think tokens denote reasoning content within<think\>tags; answer tokens include all content outside the tags\.Cognitive StateCountProportionThink TokensAnswer Tokens\(Median\)\(Median\)No\-Hint \(K∗=0K^\{\*\}=0\)35735\.7%23,882Sparse\-Hint \(0<K∗<N0<K^\{\*\}<N\)26426\.4%2156,522Full\-Hint \(K∗=NK^\{\*\}=N\)38238\.2%14,8941,034Overall1,003100%5,0373,813
### C\.2Token Distribution Analysis

Several notable patterns emerge from Table[6](https://arxiv.org/html/2605.08665#A3.T6):

#### Bimodal Reasoning Length Distribution\.

No\-Hint samples require virtually no explicit reasoning \(median of 2 tokens, corresponding to the directive phrase “I don’t need deep thinking”\), while Full\-Hint samples demand extensive deliberation \(median of 14\.9K tokens\)\. Sparse\-Hint samples occupy the middle ground \(median of 215 tokens\), representing genuinely intermediate difficulty levels\.

#### Inverse Correlation with Answer Length\.

Interestingly, harder problems \(Full\-Hint\) tend to produce shorter final answers \(median of 1K tokens\), whereas easier problems often require more elaborate solution write\-ups \(median of 3\.9K–6\.5K tokens\)\. This pattern suggests that the primary challenge lies in*discovering*the solution approach rather than*articulating*the final answer\.

#### Balanced State Distribution\.

The relatively even distribution across states \(35\.7% / 26\.4% / 38\.2%\) indicates that the s1K dataset encompasses diverse difficulty levels, preventing the model from converging to a single dominant reasoning strategy\.

## Appendix DCase Studies

This section presents representative examples from each cognitive state, illustrating how our method adaptively calibrates reasoning depth according to problem difficulty\.

### D\.1No\-Hint Example \(State I\)

Table[7](https://arxiv.org/html/2605.08665#A4.T7)demonstrates a straightforward physics problem that the instruct model can solve without any reasoning hints\. The original reasoning trace spans thousands of tokens, but our method compresses this to just 2 tokens—a 99\.96% reduction—while preserving correctness\.

Table 7:No\-Hint example: A standard calculation problem solvable via direct computation\.ProblemCompute the mean molecular speedvvin the heavy gas radon \(Rn\) in m/s\.Original Reasoning Model Output<think\>I may need some deep thinking\. Okay, so I need to compute the mean molecular speed of radon gas\. The formula for mean molecular speed isv=8​R​Tπ​Mv=\\sqrt\{\\frac\{8RT\}\{\\pi M\}\}whereRRis the gas constant,TTis temperature, andMMis molar mass…\[…5,037 tokens of detailed derivation…\]</think\>The mean molecular speed is approximately169\\boxed\{169\}m/s\.Our Method Output<think\>I don’t need deep thinking\.</think\>The mean molecular speed is approximately169\\boxed\{169\}m/s\.Analysis:Compression ratio of 99\.96%\. The instruct model possesses sufficient domain knowledge to solve this problem directly without explicit reasoning scaffolding\.
### D\.2Sparse\-Hint Example \(State II\)

Table[8](https://arxiv.org/html/2605.08665#A4.T8)shows a problem of moderate difficulty where a partial reasoning hint suffices to activate the model’s latent problem\-solving capability\.

Table 8:Sparse\-Hint example: A problem requiring a key conceptual insight to unlock the solution\.Problem\(Moderate difficulty mathematical reasoning problem\)Original Reasoning Model Output<think\>\[…Full reasoning trace with multiple exploration paths…\]</think\>\(Final answer\)Our Method Output<think\>I may need some thinking\. The key insight here is to recognize that…\[…215 tokens capturing essential reasoning steps…\]</think\>\(Final answer\)Analysis:Compression ratio of 95\.7%\. Only the critical reasoning steps are retained, filtering out exploratory tangents and redundant verification loops\.
### D\.3Full\-Hint Example \(State III\)

Table[9](https://arxiv.org/html/2605.08665#A4.T9)presents a challenging combinatorics problem that exceeds the instruct model’s independent solving capability, necessitating the complete reasoning trace\.

Table 9:Full\-Hint example: A complex problem requiring comprehensive step\-by\-step reasoning\.ProblemGiven a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator\. For how many rational numbers between 0 and 1 will20\!20\!be the resulting product?Our Method Output<think\>This is a complex question, and it is difficult to provide a direct answer\. I need to think deeply about it\.Let me think\. We are given a problem about rational numbers between 0 and 1 in lowest terms where numerator denominator=20\!=20\!…For a fractionab\\frac\{a\}\{b\}in lowest terms, we needgcd⁡\(a,b\)=1\\gcd\(a,b\)=1anda​b=20\!a\\cdot b=20\!…Wait, let me verify this approach\. Sincea<ba<b\(as the fraction is between 0 and 1\), we need to count coprime pairs…\[…14,894 tokens of detailed mathematical reasoning…\]Therefore, the number of such rational numbers equals27=1282^\{7\}=128\.</think\>The answer is128\\boxed\{128\}\.Analysis:No compression applied\. This problem requires sophisticated number\-theoretic reasoning that lies beyond the instruct model’s independent capability, justifying retention of the complete reasoning trace\.
### D\.4Compression Summary

Table[10](https://arxiv.org/html/2605.08665#A4.T10)summarizes the token compression achieved across the three cognitive states, demonstrating the effectiveness of our adaptive approach\.

Table 10:Token compression statistics by cognitive state\.StateMedian TokensCompressionInterpretationNo\-Hint299\.96%Direct solution; reasoning unnecessarySparse\-Hint21595\.7%Key insight activates latent capabilityFull\-Hint14,8940%Complete deliberation requiredWeighted Average5,03731\.5%Adaptive depth calibration