LIMOPro: Reasoning Refinement for Efficient and Effective Test-time Scaling

We sincerely thank the reviewer for the thorough review and constructive feedback. We address each concern below with experimental evidence and clarifications.

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Q1: Relationship to AdaptThink and ThinkLess

The paper contributes to the ongoing research on efficient reasoning in foundation models, while this area has many similar works such as AdaptThink and ThinkLess.

Fundamental Technical Distinctions: While AdaptThink and ThinkLess both address efficiency in reasoning models, our PIR framework offers fundamentally different and complementary innovations:

Different Optimization Levels:

• AdaptThink: Operates at inference-time decision level using RL to choose "Thinking" vs "NoThinking" modes
• ThinkLess: Focuses on training-free early termination during inference
• PIR: Optimizes at the training data level through principled content refinement, creating inherently more efficient models

Granularity and Methodology:

• AdaptThink/ThinkLess: Binary decisions or wholesale early termination
• PIR: Fine-grained, step-level analysis using novel perplexity-based importance quantification to preserve essential progressive reasoning while selectively removing low-value functional components

Superior Effectiveness: Our results demonstrate advantages across multiple dimensions:

• PIR: Accuracy improvements (+0.9% to +6.6%) while simultaneously reducing tokens (-3% to -41%)
• ThinkLess: Focus primarily on efficiency without accuracy gains

Unique Value Proposition: PIR produces optimized training datasets that result in inherently more efficient models, eliminating need for complex inference-time orchestration while maintaining full reasoning capabilities. This addresses a complementary but fundamental aspect—optimizing reasoning exemplar quality rather than controlling when to use them.

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Q2: Originality vs. LIMO

This work is based on LIMO and incrementally prunes some extra reasoning steps, which is not very original.

Fundamental Scope Differences:

• LIMO: Dataset creation focusing on sample selection efficiency ("less is more" for data quantity)
• PIR: Comprehensive framework for reasoning chain optimization addressing reasoning structure and content efficiency

Four Major Original Contributions:

1. Novel Cognitive Framework: First systematic classification of reasoning patterns into progressive reasoning vs. functional elements (verification, multi-method validation, error correction)
2. Principled Quantitative Metric: PIR metric for measuring reasoning step importance through perplexity changes—entirely novel methodological contribution
3. Selective Optimization Strategy: Fine-grained step-level analysis vs. LIMO's wholesale sample selection from "tens of millions of candidates"
4. Generalizability Framework: Works across diverse data sources (Gemini, DeepSeek-R1, QwQ), model sizes (3B-32B), demonstrating broader applicability than LIMO

Superior Effectiveness: Our experimental results demonstrate significant advances:

• Simultaneous optimization: Accuracy improvements (+0.9% to +6.6%) AND efficiency gains (-3% to -41% token reduction)
• Up to 71% efficiency improvement in some cases
• 41% token reduction in case studies while maintaining solution accuracy

Broader Impact: Our framework's applicability to multiple existing datasets (LIMO, S1, LIMO-V2) demonstrates value as a general-purpose optimization framework. The theoretical insights establish a new paradigm for reasoning chain optimization that complements and enhances existing approaches.

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Q3: Generalization to Instruction Following

The paper focuses primarily on mathematical and science reasoning tasks. Can your method generalize to other domains like instruction following?

Instruction Following Validation: Our evaluation on IFEval (instruction following) shows consistent improvements:

Accuracy Scores

Token Usage

Cross-Domain Effectiveness: PIR achieves consistent improvements across diverse domains:

• Instruction Following (IFEval): +1.37% to +2.41% improvements
• Commonsense Reasoning (CommonsenseQA): +0.67% to +4.51% improvements
• Logical Reasoning (LogiQA): +0.37% to +0.63% improvements
• Code Generation (mbpp): up to +1.0% improvement

Framework Universality: The consistent improvements across such diverse task types validate that PIR identifies fundamental reasoning optimization principles rather than domain-specific patterns. Our cognitive reasoning pattern framework—distinguishing between progressive reasoning and functional elements—captures universal principles that extend well beyond mathematical domains to instruction following and other reasoning-intensive tasks.

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Q4: Impact on Model Exploration Ability

Will shorten the reasoning steps hurt the model's exploration ability?

Enhanced Exploration Evidence: Our systematic analysis reveals that PIR training fundamentally improves how models structure their reasoning:

More Efficient Exploration: The +4.05% increase in Progressive Reasoning density while reducing redundant functional components demonstrates more efficient exploration rather than reduced capability. PIR selectively removes only low-importance, redundant instances while preserving high-value exploration strategies.

Performance Validation: The simultaneous accuracy improvements (+0.9% to +6.6%) across benchmarks provide compelling evidence—if exploration ability were impaired, we would expect performance degradation. Instead, higher Progressive Reasoning density shows models allocate reasoning capacity more strategically, concentrating on productive solution pathways while maintaining ability to explore alternatives when beneficial.

Case Study Evidence: Our detailed case study (Section B.4) demonstrates more focused and effective exploration rather than limited reasoning depth. Comparing two models solving identical problems:

• PIR-trained model (1,612 tokens): Streamlined yet thorough exploration
• Original model (3,234 tokens): Verbose, repetitive exploration with redundant validation cycles

Both reach correct answers, but PIR-trained model achieves this through more intelligent exploration strategies. The 50% token reduction represents elimination of redundant exploration rather than curtailment of necessary reasoning depth.

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Q5: Alternative Metrics Beyond Perplexity

The paper relies on perplexity as the primary metric for evaluating the importance of reasoning steps. Are there alternative metrics that could be used to assess the semantic contribution of steps and potentially provide more nuanced insights?

Principled Choice of Perplexity: Our choice of perplexity is principled and empirically validated. Perplexity directly measures the model's confidence in generating correct answers when specific steps are removed, providing a clear signal of step importance that is both computationally efficient and theoretically grounded.

Strong Empirical Validation: Our extensive experiments demonstrate that this approach consistently improves accuracy (+0.9% to +6.6%) while reducing token usage (-3% to -41%) across diverse benchmarks and model sizes, validating its effectiveness for practical optimization purposes.

Future Research Directions: We acknowledge that exploring semantic-based metrics could provide additional insights (as noted in our limitations, lines 323-325). Future work could explore complementary metrics such as:

• Embedding-based semantic similarity
• Logical dependency analysis

Current Framework Strength: However, our framework's strong generalizability across different data sources, model scales, and token budgets suggests that perplexity successfully captures fundamental aspects of reasoning step importance. The consistent improvements across diverse reasoning domains demonstrate that our approach identifies universal optimization principles effective across various task types.

We sincerely thank the reviewer for the thorough review. We address each concern below with experimental evidence and clarifications.

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Q1: External Dependency on Claude 3.7 Sonnet

How would it perform if a weaker open-source model were used instead? and (b) the framework's effectiveness may partially derive from the classifier's intelligence rather than the PIR logic itself.

Reproducibility with Open-Source Alternatives: To directly address reproducibility, we conducted additional experiments using Qwen3-32B as an open-source alternative for reasoning step classification. The results demonstrate maintained effectiveness:

Accuracy Performance

Average Response Tokens

Importantly, Qwen3-32B requires only 4×RTX 4090 GPUs for inference, making it accessible to most research laboratories.

Conceptual Coherence - PIR Logic vs. Classifier Intelligence: The core PIR contribution is the quantitative importance metric (Equation 1), not classification sophistication. The ablation study with rule-based filtering (S1-RULE in Table 3) shows: while random functional step removal reduces accuracy below baseline on AIME (36.7% vs. 37.9%) and GPQA (58.1% vs. 60.7%), our S1-32B-P method achieves substantial improvements (42.1% and 61.6%).

This confirms PIR's perplexity-based assessment drives performance gains, not classification quality.

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Q2: Evaluation Beyond Mathematical Reasoning

How the framework performs on commonsense reasoning, logic puzzles, and code generation.

Comprehensive Cross-Domain Evaluation: Our framework's effectiveness extends well beyond mathematical and scientific domains. We conducted extensive evaluation on diverse reasoning tasks that exhibit different characteristics from mathematical reasoning:

Accuracy Scores

Token Usage

Consistent Cross-Domain Improvements: The results demonstrate that PIR optimization consistently improves performance across fundamentally different reasoning types:

• Commonsense Reasoning (CommonsenseQA): +0.67% accuracy with 6.1% token reduction for LIMO-P, and +4.51% accuracy with 16.5% token reduction for LIMO-V2-P
• Logical Reasoning (LogiQA): +0.63% and +0.37% accuracy improvements with 7.5% and 6.1% token reductions respectively
• Code Generation (mbpp): +1.0% accuracy improvement for LIMO-V2-P

Framework Universality: These domains involve different functional steps—commonsense reasoning features intuitive validation, logical reasoning involves formal verification, and code generation includes debugging and testing—yet PIR consistently optimizes the efficiency-effectiveness tradeoff. This validates our hypothesis that the distinction between progressive reasoning (essential logical flow) and functional elements is a fundamental cognitive pattern transcending specific reasoning domains.

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Q3: Adaptive Pruning Strategies

... A more adaptive strategy could potentially yield better performance.

Foundational Contribution: Our work establishes the essential theoretical and methodological foundation for adaptive approaches by introducing systematic reasoning step classification, the PIR importance metric, and empirical validation—prerequisites for adaptive strategies. Without our cognitive reasoning patterns and importance quantification, adaptive approaches would lack theoretical grounding.

Current Framework's Robustness: While using predefined ratios, our PIR framework demonstrates strong generalizability across model sizes (3B-32B), data sources (Gemini, DeepSeek-R1, QwQ), and benchmarks (AIME, AMC, GPQA). The PIR metric provides the foundation for dynamic thresholds.

Future Extensions: Building on our foundation, adaptive strategies represent natural progressions: difficulty-aware pruning, length-adaptive thresholds, and dynamic importance cutoffs. Our framework provides essential infrastructure enabling these advanced approaches while offering predictable behavior for immediate deployment.

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Q4: Accessibility of Open-Source Alternatives

Could the authors comment on the feasibility of replacing Claude 3.7 Sonnet with a more accessible open-source model?

Open-Source Solution: While experiments with LLaMA-3 8B and Qwen-2.5-7B were not conducted, these smaller models may struggle with the nuanced reasoning step classification required by our framework, particularly in distinguishing between the four cognitive patterns (progressive reasoning, verification, multi-method validation, and error correction).

Validated Alternative: Experiments demonstrate that Qwen3-32B provides an excellent open-source alternative, achieving comparable performance to the original framework (see Q1 results table above).

Practical Accessibility: Qwen3-32B requires only 4×RTX 4090 or 4×RTX 3090 GPUs for inference, making it accessible to most research laboratories with standard computational resources.

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Q5: Minimum Classification Accuracy Requirements

What minimum classification accuracy do you believe is required for PIR to remain effective?

Framework Robustness: While systematic experiments across different classification accuracy levels were not conducted, current results and framework design demonstrate resilience to classification errors.

Evidence from Ablation Study: The ablation study with rule-based filtering (S1-RULE in Table 3) shows imperfect classification relying solely on linguistic markers still achieves efficiency gains, though with reduced effectiveness compared to PIR, indicating PIR's tolerance for classification noise.

Dual-Filtering Design: PIR remains effective if the classifier reasonably distinguishes progressive reasoning steps (requiring preservation) from functional steps (verification, multi-method validation, error correction). The perplexity-based metric serves as secondary quality filtering—misclassified progressive reasoning steps likely receive high PIR scores ensuring preservation, while truly redundant functional steps receive low PIR scores regardless of classification accuracy.

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Q6: Dependencies Between Functional and Progressive Steps

Is there a risk that a low-importance functional step (e.g., a verification) may actually be critical for enabling a downstream progressive step?

Global Dependency Capture: Our PIR metric is specifically designed to capture such dependencies through its global evaluation approach. When calculating PIR scores, we measure the perplexity change of the entire answer generation process after removing a specific step. If a functional step (e.g., verification) is truly critical for enabling downstream progressive reasoning, removing it would significantly impair the model's confidence in generating subsequent reasoning steps, resulting in higher answer perplexity and thus a higher PIR score that preserves the step.

Empirical Validation: The ablation study results strongly validate this dependency-preservation mechanism:

While indiscriminate removal of entire functional categories can lead to higher accuracy than baseline, our PIR approach achieves superior performance: +6.6% accuracy improvement on AIME, +1.9% on AMC, and +4.0% on GPQA Diamond. This demonstrates that PIR successfully identifies and preserves those functional steps that are truly essential.

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Q7: Local Reasoning Dependencies

Has the framework considered the possibility that pruning such steps could disrupt local reasoning dependencies?

Step-Level Coherence Preservation: Our framework addresses this through step-level granularity and systematic classification. The hierarchical decomposition segments reasoning chains into logical steps, each comprising multiple coherent sentences forming a cohesive unit (lines 125-127). Removing functional steps as complete units preserves local reasoning coherence and prevents partial dependency disruption.

Independence of Functional Steps: Analysis shows functional steps (verification, multi-method validation, error correction) operate independently, targeting specific progressive reasoning steps with minimal interdependencies. Progressive reasoning steps exhibit strong sequential dependencies, but our framework preserves these entirely to maintain the solution's logical backbone.

Classification Accuracy: The robust classification system uses characteristic linguistic markers ("Let me check" for verification, "I made a mistake" for error correction) combined with contextual analysis, achieving 93.4% unanimous agreement in human evaluation and minimizing misclassification risks that could disrupt critical reasoning dependencies.

We sincerely thank the reviewer for the thorough review and constructive feedback. We address each concern below with experimental evidence and clarifications.

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Unified Experimental Results

Following the reviewer's suggestions, we conducted experiments using 0 as the threshold for pruning functional steps. These results address multiple concerns (Q1, Q2, Q4):

Accuracy Results

Token Efficiency

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Q1: Fixed Pruning Ratio vs. PIR-based Thresholds

One of my biggest concerns regarding this work is about the decision to use the pruning ratio as a hyperparameter. Intuitively, it should be possible that depending on the sample, all the functional steps are essential or most of the functional steps are not needed... Given the definition of PIR (Eq.(1)), pruning all the functional steps with negative PIRs or PIRs smaller than some threshold value makes more sense to me.

We sincerely thank the reviewer for this insightful suggestion about using PIR-based thresholds. Following the reviewer's suggestion, we conducted experiments removing all functional steps with PIR < 0. Our results demonstrate that this approach achieves strong performance (see unified results table above).

These results demonstrate that using PIR < 0 as a threshold provides consistent improvements across all benchmarks, validating the effectiveness of PIR for identifying low-importance functional steps. While both fixed ratios and PIR-based thresholds can achieve good performance, our fixed-ratio design offers a more controllable hyperparameter that practitioners can easily adjust based on user's specific efficiency-accuracy trade-offs. We will include these PIR < 0 threshold results in our camera-ready version to provide readers with empirical evidence for both approaches.

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Q2: Benchmark-Specific Pruning Ratios

On a related topic, I'm also primarily concerned about the use of different pruning ratios for different target benchmarks... This could indicate the possibility of overfitting of this parameter to each target benchmark.

We sincerely thank the reviewer for raising this important concern about potential overfitting of pruning ratios to specific benchmarks. To address this concern, we conducted experiments using a single, uniform criterion across all benchmarks: removing all functional steps with PIR < 0. This approach eliminates any benchmark-specific tuning.

Our results demonstrate that this unified criterion achieves consistent improvements across all three benchmarks (see unified results table above). These results show that PIR effectively identifies low-importance functional steps regardless of the benchmark, without requiring benchmark-specific ratio tuning. The consistent improvements validate that our approach captures general properties of reasoning chains rather than overfitting to specific evaluation sets. We will clarify in our camera-ready version that practitioners can use either benchmark-agnostic approaches (like PIR < 0) or tune ratios based on their specific application needs.

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Q3: Reasoning Step Decomposition Quality

I wonder how well each reasoning chain is decomposed into cohesive reasoning steps (units), because each reasoning step becomes a target of the exclusion and mis-decomposition may lead to having incomplete logical reasoning steps.

We sincerely thank the reviewer for this thoughtful observation about the granularity and cohesiveness of reasoning step decomposition. We share this concern and conducted additional analysis to evaluate decomposition quality.

Human Evaluation Protocol: We randomly sampled 5% reasoning chains and had four annotators assess whether each decomposed step represents a complete, cohesive logical unit. The evaluation criteria included:

• (1) Logical completeness - whether the step contains a self-contained reasoning operation
• (2) Semantic coherence - whether all sentences within a step work together toward a single reasoning goal
• (3) Boundary appropriateness - whether the step boundaries preserve logical dependencies

Results: Our analysis shows that 89.2% of reasoning steps were rated as "cohesive and complete" by all annotators, 8.1% had minor boundary issues (e.g., a verification statement included within the progressive step), and only 2.7% showed significant decomposition problems.

Robustness Analysis: When we analyzed the impact of decomposition quality on PIR calculations, we found that steps with minor boundary issues had similar PIR distributions to well-decomposed steps (mean PIR difference < 0.05), suggesting our framework is robust to small decomposition variations. We also observed that progressive reasoning steps had higher decomposition quality (92.1% perfect) compared to functional steps (86.3% perfect), which actually supports our approach since progressive steps are always preserved.

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Q4: PIR = 0 as Absolute Threshold

What do the empirical results (especially accuracies) look like if you use an absolute value of 0 (zero) as a threshold for PIR? Could you demonstrate that with some constant thresholding of PIR values themselves, you can achieve similar empirical results?

We sincerely thank the reviewer for this specific and constructive suggestion. We have conducted exactly this experiment - using PIR < 0 as the threshold to prune functional steps. The empirical results are shown in our unified results table above.

These results demonstrate that using PIR = 0 as an absolute threshold achieves similar empirical performance to our fixed-ratio approach, with consistent improvements in both accuracy and efficiency across all benchmarks. This validates that PIR values themselves provide a meaningful signal for step importance, and constant thresholding (PIR < 0) offers a viable alternative to ratio-based pruning. We will include these results in our main paper to provide readers with both implementation options.

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Q5: Y-axis Scales in Figure 1

This may be a minor thing, but the charts in Figure 1 have various scales of the y-axis, which could be somewhat misleading.

We sincerely thank the reviewer for this careful observation about the visualization. We acknowledge that using different y-axis scales across the charts in Figure 1 could potentially make visual comparisons more challenging. Our intention was to maximize the visibility of performance differences within each metric, but we agree that consistent scaling would improve clarity and prevent any unintended misleading interpretations.

In the camera-ready version, we will revise Figure 1 to use consistent y-axis scales across all charts for each metric type (accuracy, response tokens, and efficiency), making it easier for readers to compare results across benchmarks at a glance. We will also add a note in the caption explicitly stating the scale ranges used. Thank you for helping us improve the clarity of our presentation.

Dear Reviewer U9qd,

I hope this message finds you well.

I wanted to follow up on the rebuttal we submitted in response to your valuable feedback on our paper. We greatly appreciate the time and effort you invested in providing detailed and constructive comments during the initial review.

In our rebuttal, we have addressed the concerns you raised. We believe our responses clarify these important aspects and strengthen the contribution of our work.

As we are now in the author-reviewer discussion period, we would be very grateful for any additional feedback or clarification you might have regarding our responses. If there are any remaining concerns or if you would like us to elaborate further on any particular points, we would be happy to provide additional details.

We understand that you are likely managing multiple papers during this busy review period, and we genuinely appreciate your continued engagement with our work. Your expertise and insights are invaluable to improving the quality of our research. Thank you once again for your time and consideration. We look forward to your response at your earliest convenience.

We sincerely thank the reviewer for the thorough review and constructive feedback. We appreciate the valuable insights that have helped strengthen our contribution. The concerns raised are primarily due to presentation issues and misunderstandings, which we address systematically below.

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Q1: Lack of Qualitative Characterization

For which reasoning types does the approach perform best and why?

Thank you for raising this important question about qualitative characterization across reasoning types.

Based on our experimental analysis in Section B.3 (Appendix), Progressive Reasoning + Error Correction (PR+EC) and PR+Verification (PR+V) achieve better accuracy performance compared to PR+Multi-method Validation (PR+MV). The observed performance variations highlight that different reasoning types contribute differentially to model performance, with certain patterns delivering more substantial benefits in specific contexts. Hence, we believe that Multi-method Validation is relatively less important compared to the other two functional types.

We also conducted ablation experiments by removing only specific functional reasoning types. The results show that removing only Multi-method Validation (MV) achieves higher accuracy compared to removing only Error Correction (EC) or Verification (V). This further validates our finding that Multi-method Validation steps are indeed less critical for maintaining reasoning performance, confirming our approach's effectiveness in selectively targeting the most redundant functional components.

Ablation Study Results

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Q2: Negative Impact Scenarios

In which scenarios does it have a negative impact?

We thank the reviewer for this direct question about negative impacts. Our comprehensive evaluation across diverse reasoning tasks provides valuable insights into when PIR optimization may have limited effects.

Empirical Evidence from Cross-Domain Evaluation: Based on our systematic evaluation across different reasoning domains, we can analyze the performance patterns:

Performance Analysis

Rare Negative Impact Scenarios: PIR optimization shows robust performance across reasoning domains, with only one minor negative case: LIMO-P on MBPP code generation (-0.6% accuracy decline, 73.2% → 72.6%). This is the sole negative impact among all evaluated task-model combinations.

Framework Design Considerations: PIR is designed to optimize reasoning chains from Large Reasoning Models (LRMs), performing best on tasks requiring extensive step-by-step reasoning. The evaluation shows consistent improvements across mathematically complex tasks, commonsense reasoning, and logical reasoning. For tasks with simpler reasoning structures or those not heavily relying on elaborate reasoning chains—such as factual QA, simple classification, or direct information retrieval—optimization potential may be limited.

Overall Robustness: Evidence demonstrates consistent positive impacts across commonsense reasoning, logical reasoning, and code generation. The single minor negative case (-0.6%) shows negative impacts are rare exceptions rather than systematic limitations, highlighting PIR's broad applicability with strong efficiency gains through token reduction.

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Q3: Systematic vs. Anecdotal Analysis

I believe it is essential to provide a systematic characterization of the qualitative points (i.e. non-anecdotal).

We thank the reviewer for highlighting the need for systematic qualitative analysis beyond individual case studies. To address this, we provide systematic qualitative characterization comparing reasoning traces from models trained on original versus PIR-optimized datasets.

Pattern Analysis: We systematically analyzed reasoning traces from LIMO and LIMO-P models across all test problems, characterizing qualitative differences using keyword frequency statistics based on cognitive reasoning patterns from Table 4. As shown below, models trained on PIR-optimized data generate reasoning traces with systematically different patterns—increasing Progressive Reasoning density (+4.05%) while reducing Verification (-1.62%) and Multi-method Validation (-2.45%). This demonstrates that PIR training fundamentally changes reasoning structure, concentrating on essential logical progression while reducing redundant verification through measurable linguistic pattern shifts.

Systematic Pattern Analysis

Human Evaluation Protocol: We conducted rigorous evaluation with 4 expert annotators independently analyzing 5% randomly sampled correct reasoning traces from LIMO and LIMO-P models. Annotators classified reasoning steps into four cognitive patterns and calculated Progressive Reasoning proportions, achieving high inter-annotator agreement (κ = 0.87).

Results show human evaluators consistently identified significantly higher Progressive Reasoning proportions in LIMO-P traces (68.3% ± 4.2%) versus LIMO traces (52.1% ± 5.8%), representing a 16.2 percentage point increase. Functional components showed corresponding proportional decreases in LIMO-P traces. This systematic validation confirms PIR-trained models fundamentally alter reasoning generation patterns, producing traces with higher concentration of essential logical steps.

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Q4: Model Selection Criteria

What is the formal selection criteria (inclusion and exclusion) for the base models? How does it affect the universality of the claims?

We sincerely thank the reviewer for this important question. Our base model selection was guided by criteria ensuring rigor and generalizability:

1. Availability and Reproducibility - We used openly-available Qwen2.5 and Llama models from official repositories with standardized training scripts
2. Scale Diversity - Systematically evaluating 3B to 32B parameter models (Section 3.3.3), demonstrating consistent improvements across all sizes
3. Cross-Architecture Validation - Beyond Qwen2.5, we conducted experiments with Llama-3.1-8B showing strong improvements
4. Data Source Diversity - Validating across training data from three different LRMs (Gemini Flash Thinking, DeepSeek-R1, QwQ)

These consistent improvements across model scales (3B-32B), architectures (Qwen, Llama), and data sources suggest our PIR framework captures fundamental reasoning optimization principles rather than model-specific characteristics.

Cross-Architecture Validation: Llama-3.1-8B Results

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Q5: Benchmark Selection Criteria

What is the formal selection criteria for the benchmarks?

We thank the reviewer for this question about benchmark selection methodology. Our selection follows systematic criteria ensuring fair comparison and comprehensive evaluation.

Selection Criteria:

1. Alignment with Prior Work - We selected benchmarks from recent reasoning literature (LIMO, S1 papers) enabling direct comparison with established baselines
2. Reasoning Diversity - We chose benchmarks spanning difficulty levels and reasoning types—AIME/AMC for mathematical reasoning, GPQA Diamond for scientific reasoning—ensuring comprehensive evaluation across varied demands

Methodological Consistency: Aligning with LIMO and S1 evaluation protocols ensures performance improvements stem from our PIR framework rather than methodology differences, establishing validity and enabling fair comparison.

Generalizability Validation: Beyond established benchmarks, PIR demonstrates consistent improvements across different data sources (Gemini Flash Thinking, DeepSeek-R1, QwQ), model scales (3B-32B parameters), and reasoning domains including commonsense reasoning (CommonsenseQA), logical reasoning (LogiQA), and code generation (MBPP). This indicates PIR captures universal reasoning optimization principles rather than domain-specific patterns.

Cross-Domain Performance Validation

This comprehensive validation confirms that our benchmark selection provides both rigorous comparison standards and robust evidence of broad applicability across diverse reasoning scenarios.

Thank you for this excellent follow-up question that helps clarify our approach. To ensure we address your valuable concern accurately: are you asking for a formal characterization of the reasoning properties inherent in our training datasets rather than inferring reasoning type effectiveness through benchmark performance comparisons?

If our understanding is correct, we provide systematic characterization using Evidence 1 to directly address your question about formal dataset properties, with three additional complementary lines of evidence that validate this conclusion: Multi-method Validation contributes less than Verification and Error Correction to reasoning effectiveness.

Evidence 1: Training Dataset Reasoning Property Distribution

Based on our systematic analysis of reasoning patterns in training data:

Key Insight: PIR's quantitative importance calculation predominantly targets Multi-method Validation (-3.1% reduction) while preserving Verification (-0.6% minimal reduction) and Error Correction (+0.3% slight increase), indicating Multi-method steps contain higher functional redundancy as measured by our perplexity-based PIR metric.

Evidence 2: Model Output Reasoning Distribution Changes After Training

As detailed in our Q3 response, models trained on these respective datasets exhibit corresponding reasoning generation patterns:

Convergent Pattern: Models internalize the training data distribution, with LIMO-P showing reduction in Multi-method Validation generation (-2.45%), confirming these steps are less essential for reasoning quality.

Evidence 3: Human Evaluation of Model Output Reasoning Distribution

Human Evaluation Protocol: As reported in our Q3 response, we conducted rigorous evaluation with 4 expert annotators independently analyzing 5% randomly sampled correct reasoning traces from LIMO and LIMO-P models. Annotators classified reasoning steps into four cognitive patterns and calculated Progressive Reasoning proportions, achieving high inter-annotator agreement (κ = 0.87).

Results: Human evaluators consistently identified significantly higher Progressive Reasoning proportions in LIMO-P traces (68.3% ± 4.2%) versus LIMO traces (52.1% ± 5.8%), representing a 16.2 percentage point increase. This systematic validation confirms that Multi-method Validation reduction enhances reasoning concentration without compromising logical integrity.

Evidence 4: Benchmark Performance Validation

Our ablation study removing specific functional reasoning types confirms this hierarchy:

Systematic Validation: Removing Multi-method Validation achieves the highest performance improvements across all benchmarks, while removing Verification or Error Correction shows smaller or mixed effects.