Title: Mind the Gap: Bridging Thought Leap for Improved Chain-of-Thought Tuning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Large Reasoning Models

LLM Reasoning
Abstract: Large language models (LLMs) have achieved remarkable progress on mathematical tasks through Chain-of-Thought (CoT) reasoning. However, existing mathematical CoT datasets often suffer from **Thought Leaps** due to experts omitting intermediate steps, which negatively impacts model learning and generalization. We propose the CoT Thought Leap Bridge Task, which aims to automatically detect leaps and generate missing intermediate reasoning steps to restore the completeness and coherence of CoT. To facilitate this, we constructed a specialized training dataset called **ScaleQM+**, based on the structured ScaleQuestMath dataset, and trained **CoT-Bridge** to bridge thought leaps. Through comprehensive experiments on mathematical reasoning benchmarks, we demonstrate that models fine-tuned on bridged datasets consistently outperform those trained on original datasets, with improvements of up to +5.87\% on NuminaMath. Our approach effectively enhances distilled data (+3.02\%) and provides better starting points for reinforcement learning (+3.1\%), functioning as a plug-and-play module compatible with existing optimization techniques. Furthermore, CoT-Bridge demonstrates improved generalization to out-of-domain logical reasoning tasks, confirming that enhancing reasoning completeness yields broadly applicable benefits.

Mind the Gap: Bridging Thought Leap for Improved Chain-of-Thought Tuning

Abstract

Large language models (LLMs) have achieved remarkable progress on mathematical tasks through Chain-of-Thought (CoT) reasoning. However, existing mathematical CoT datasets often suffer from Thought Leaps due to experts omitting intermediate steps, which negatively impacts model learning and generalization. We propose the CoT Thought Leap Bridge Task, which aims to automatically detect leaps and generate missing intermediate reasoning steps to restore the completeness and coherence of CoT. To facilitate this, we constructed a specialized training dataset called ScaleQM+ , based on the structured ScaleQuestMath dataset, and trained CoT-Bridge to bridge thought leaps. Through comprehensive experiments on mathematical reasoning benchmarks, we demonstrate that models fine-tuned on bridged datasets consistently outperform those trained on original datasets, with improvements of up to +5.87% on NuminaMath. Our approach effectively enhances distilled data (+3.02%) and provides better starting points for reinforcement learning (+3.1%), functioning as a plug-and-play module compatible with existing optimization techniques. Furthermore, CoT-Bridge demonstrates improved generalization to out-of-domain logical reasoning tasks, confirming that enhancing reasoning completeness yields broadly applicable benefits.

1 Introduction

Large Language Models (LLMs) [1-6] have demonstrated significant performance improvements through Chain-of-Thought (CoT) reasoning [7, 8], particularly on complex tasks [9] such as mathematics [10] and coding. CoT guides models to generate intermediate reasoning steps, simulating the human process of step-by-step problem solving, thereby enhancing both solution accuracy and interpretability [11]. The quality of these reasoning chains directly impacts model performance, serving as a critical foundation for advanced reasoning capabilities.

Despite substantial progress, we identify a prevalent but understudied phenomenon in CoT datasets [12-16]: Thought Leap . This refers to instances where one or more intermediate reasoning steps are omitted between adjacent steps, creating cognitive gaps in the reasoning chain. Unlike factual errors or answer inaccuracies that have been extensively studied, Thought Leap specifically concerns the completeness of reasoning structures. Figure 1 (a) illustrates this phenomenon through a mathematical problem where critical bridging steps (highlighted in gold) are missing in the

∗ Equal contribution.

† Corresponding Author.

0

What is the minimum number of times you must throw two fair

eight-sided dice to ensure that the same sum is rolled twice?

Question

To determine the minimum number of times you must throw two

fair eight-sided dice to ensure that the same sum is rolled twice, we

can use the pigeonhole principle. The pigeonhole principle states

that if  n items are put into m containers, with n > m, then at least

one container must contain more than one item.

Bridge 1

Consider two eight-sided dice, each numbered from 1 to 8. The

sums when rolling these dice range:

- Minimum sum: 1 + 1 = 2

- Maximum sum: 8 + 8 = 16

Step 1

Therefore, the possible sums when rolling two eight-sided dice are

the integers from 2 to 16, inclusive. This gives us a total of 16 - 2 +

1 = 15 possible sums.

Bridge 2

By the pigeonhole principle, to ensure at least one sum appears

twice, we need to roll the dice 15 + 1 = 16 times.

Step 2

Thus, to guarantee that the same sum appears twice when rolling

two eight-sided dice, the dice must be rolled $\boxed{16}$ times.

Step 3

Where did the 15 come from in Step 2?

What's the pigeonhole principle?

Question

Step 2

Step 1

Step 3

Great, I understand the solution!

Thought Leap

Thought Leap

Mind the Gap!

Bridge 1: To determine the minimum of times...

Bridge 2: Therefore, the possible sums...

Bridge Thought Leap

Question

Step 1

Step 2

Step 3

Bridge 1

Bridge 2

Bridge 1

Bridge 2

(b)

Meta-Llama3.1-8B

Qwen2.5-Math-1.5B

Base model 4-shot

SFT baseline

Bridge improvement

+ MetaMath

20.95

+12.14

+ 1.35

+ NuminaMath

20.95

+22.92

+ 5.87

+ NuminaMath

+10.82

42.05

+ 3.39

+  Distill

+13.81

42.05

+ 3.02

+  RFT

+18.39

42.05

+ 1.37

+ SFT + RL

+18.83

42.05

+ 3.1

+ MetaMath

42.05

+ 3.36

+0.5

Better Cold Start for RL

Bridge on Distill Data

Original Data

60

+

Mild Thought Leap

Moderate Thought Leap

Answer Only

Severe Thought Leap

40

30

20

10

10

30

50

70

90110130150

Training Step

170

190

O

60

》

(a)

(c)

original CoT. Without these bridges, models struggle to follow how the pigeonhole principle applies to dice probability and where the value "15" comes from-gaps that require implicit knowledge that may be obvious to experts but create significant barriers for models during learning.

Thought Leaps arise naturally in CoT datasets sourced from educational materials and expert demonstrations. Human experts often omit steps they consider trivial based on their extensive background knowledge, unconsciously creating gaps that can impede effective learning. This phenomenon is particularly problematic for both human learners and LLMs that lack the implicit knowledge necessary to bridge these gaps.

Our preliminary investigations reveal the detrimental effects of Thought Leap on model performance. By systematically introducing varying degrees of step omissions in the MetaMathQA [12] dataset, from mild to severe, we demonstrate that Thought Leaps significantly undermine training effectiveness. As shown in Figure 1 (b), models trained on datasets with Thought Leaps exhibit substantially lower performance ceilings (up to 27.83% reduction in accuracy for severe leaps, see details in Appendix C) and slower convergence rates compared to those trained on complete reasoning chains. These findings align with recent research suggesting that structural disruptions in reasoning chains can be more harmful than factual inaccuracies [17, 18].

To address this challenge, we propose the CoT Thought Leap Bridge Task , which aims to automatically detect Thought Leaps and generate the necessary intermediate reasoning steps to restore coherence and completeness in CoT demonstrations. As illustrated in Figure 1 (a), our approach effectively transforms incomplete reasoning chains into coherent step-by-step solutions by inserting appropriate bridging content, allowing learners to follow the complete reasoning process. Our approach involves creating a specialized dataset (ScaleQM+) for the Thought Leap Bridge Task by systematically removing intermediate steps from the structured ScaleQuestMath [19] dataset and pairing incomplete reasoning chains with their complete counterparts. We then develop CoT-Bridge, a fine-tuned model based on Qwen2.5-Math-7B [20] specifically designed to identify and bridge Thought Leaps in mathematical reasoning. Finally, we apply CoT-Bridge to existing mathematical reasoning datasets to enhance their completeness and coherence, thereby improving the quality of training data for downstream models.

Our experimental results demonstrate that models fine-tuned on bridged datasets achieve significant performance improvements compared to those trained on the original datasets with Thought Leaps. As illustrated in Figure 1 (c), addressing the Thought Leap phenomenon leads to consistent improvements across different model architectures and datasets, with performance gains of up to +5.87% on NuminaMath and +3.36% on MetaMathQA. These results highlight the broad applicability of our method and its ability to enhance reasoning capabilities across various mathematical and logical reasoning benchmarks.

Furthermore, our approach functions as a plug-and-play enhancement module that can be integrated with other advanced techniques. As shown in Figure 1 (c), it can improve the quality of distilled

data (+3.02%) or provide better cold start models for Reinforcement Learning (+3.1%), thereby amplifying the effectiveness of existing methods for enhancing reasoning capabilities. In summary, our contributions are:

To the best of our knowledge, we are the first to systematically identify and formalize the Thought Leap phenomenon in CoT reasoning. We introduce the CoT Thought Leap Bridge Task along with an evaluation framework for addressing this issue.

We develop a specialized dataset ( ScaleQM+ ) and a model ( CoT-Bridge ) for identifying and bridging Thought Leaps, demonstrating their effectiveness through comprehensive experiments.

We apply CoT-Bridge to existing mathematical reasoning datasets, achieving significant performance improvements and demonstrating good generalization capabilities on out-ofdomain logical reasoning benchmarks (+2.99%).

We validate that our approach can function as a plug-and-play enhancement module, compatible with methods such as knowledge distillation and reinforcement learning, to further amplify model performance.

2 Method

In this section, we provide a detailed discussion of our approach. Section 2.1 presents a rigorous formalization of the Thought Leap phenomenon and CoT Thought Leap Bridge task. Section 2.2 describes the construction of the Thought Leap dataset and the training of the bridge model, along with a brief introduction to a variant method. Section 2.3 outlines the process of applying the bridge model to existing step-by-step datasets.

2.1 Task Formalization

Let C ∗ = ( Q,s ∗ 1 , s ∗ 2 , . . . , s ∗ m ) represent an ideal complete CoT starting from question Q . In C ∗ , the transition from Q to s ∗ 1 , as well as transitions between any adjacent steps ( s ∗ i , s ∗ i +1 ) , are both explicit and coherent. Defining a completeness function V , for C ∗ we have V ( Q,s ∗ 1 ) = True and ∀ i ∈ [1 , m -1] , V ( s ∗ i , s ∗ i +1 ) = True. This function captures whether the reasoning transition between adjacent steps is sufficiently detailed and logically sound.

Now consider a CoT C = ( s 0 , s 1 , s 2 , . . . , s n ) that successfully derives an answer A (for convenience, we denote question Q as s 0 ). If there exists at least one pair of adjacent steps ( s k , s k +1 ) within C that fails the completeness criterion, i.e., V ( s k , s k +1 ) = False, then a Thought Leap exists between s k and s k +1 . Such incompleteness indicates that necessary intermediate reasoning steps S ′ miss = ( s ′ k. 1 , s ′ k. 2 , . . . , s ′ k.j ) , where j ≥ 1 , have been omitted between s k and s k +1 . These omitted steps must satisfy three conditions: (1) completeness from s k to the first missing step: V ( s k , s ′ k. 1 ) = True; (2) internal completeness within the missing sequence: ∀ i ∈ [1 , j -1] , V ( s ′ k.i , s ′ k.i +1 ) = True; and (3) completeness from the last missing step to s k +1 : V ( s ′ k.j , s k +1 ) = True.

Based on this formalization, we define the CoT Thought Leap Bridge Task as a two-stage process: first identifying all adjacent step pairs ( s k , s k +1 ) within C that satisfy V ( s k , s k +1 ) = False, and then for each identified leap, generating the missing intermediate step sequence S ′ miss to bridge the gap as s k ⊕ S ′ miss ⊕ s k +1 . This task directly addresses the completeness aspect of reasoning, which is complementary to the more commonly studied factual accuracy dimension.

2.2 Thought Leap Bridging Dataset Construction and Bridge Model Training

To implement the CoT Thought Leap Bridge task, we created a specialized training dataset named ScaleQM+ based on the structurally comprehensive ScaleQuestMath dataset. We chose ScaleQuestMath as an approximation of ideal CoT because its reasoning structure is relatively complete and well-formed, making it suitable for systematic step removal. As illustrated in Figure 2, we start with complete reasoning chains C ∗ = ( s 0 , s ∗ 1 , . . . , s ∗ m ) where s 0 = Q is the question, and strategically remove intermediate steps to produce incomplete chains C = ( s 0 , s 1 , . . . , s n ) containing Thought Leaps. The removed steps form the reference set of missing intermediate steps needed to bridge these leaps.

System:

You are a mathematics teacher reviewing a

solution that may be missing one or more steps.

User:

<incomplete_solution>

Step 0: Q

Step 1: S2

Step 2: S3

Step 3: S5

Step 4: S6

</incomplete_solution>

There are 6 blue, 7 red, and 4 black balls in the bowl. How many balls must be removed to make sure

you will remove at least 2 of the same color?

Q

S1

To determine the minimum number of balls that must be removed to ensure that at least 2 of the same

color are removed, we can use the Pigeonhole Principle. The Pigeonhole Principle states that if n items

are put into m containers, with n > m , then at least one container must contain more than one item.

S6

The final answer is \(\boxed{4}\).

S5

Therefore, the minimum number of balls that must be removed to ensure that at least 2 of the same

color are removed is \( 3 + 1 = 4 \).

After removing 3 balls (1 blue, 1 red, and 1 black), the next ball we remove must be one of the colors

we already have (blue, red, or black). This will ensure that we have at least 2 balls of the same color.

S3

There are 3 colors (blue, red, and black). In the worst-case scenario, we could remove 1 ball of each

color without having 2 balls of the same color. This means we could remove 1 blue, 1 red, and 1 black

ball, and still not have 2 balls of the same color. This totals to 3 balls.

S2

In this problem, the "containers" are the colors of the balls (blue, red, and black), and the "items" are the

balls. We want to ensure that at least one color has at least 2 balls.

Assitant:

1. The missing step is between

Step 0

and

Step 1

:

S1

2. The missing step is between

Step 2

and

Step 3

:

What is the minimum number of times you must

throw two fair eight-sided dice to ensure that the

same sum is rolled twice?

Consider two eight-sided dice, each numbered from

1 to 8. The sums when rolling these dice range:

- Minimum sum: 1 + 1 = 2

- Maximum sum: 8 + 8 = 16

Thus, to guarantee that the same sum appears twice

when rolling two eight-sided dice, the dice must be

rolled $\boxed{16}$ times.

By the pigeonhole principle, to ensure at least one

sum appears twice, we need to roll the dice 15 + 1 =

16 times.

Step 1

The missing step is between

Step 0

and

Step 1

:

To determine the minimum number of ...

The missing step is between

Step 1

and

Step 2

:

Therefore, the possible sums when  ...

.

Inference

ScaleQuestMath

Data Construction for Training

SFT

Step 0

Step 2

Step 3

CoT-Bridge

ScaleQM+

Original Chain of Thought

Our step removal strategy follows several principles to create effective training examples. We always retain the final step s ∗ m to preserve the completeness of the answer, while allowing the initial reasoning step s ∗ 1 to be removed to help the model learn macro-planning skills. The number of steps to remove, k del , scales with the length of the original chain: for shorter chains ( m ≤ 10 ), we remove 1-2 steps; for longer chains ( m> 10 ), we remove 1-3 steps. Additionally, with a probability of 0.2, we retain the complete chain unchanged ( C = C ∗ ) to train the model to recognize when no bridging is needed.

For each modified chain C , we generate two essential training components: the ground-truth Thought Leap positions L gt identifying where step deletion resulted in reasoning gaps, and the corresponding ground-truth missing steps M gt consisting of the original removed steps. Our primary model, CoT-

Bridge , learns the mapping f : C → ( ˆ L , ˆ M ) , taking an incomplete chain as input and producing both predicted leap positions and the corresponding missing steps. As a baseline for comparison, we also implement CoT-Bridge-Random , which learns g : ( C, L gt ) → ˆ M , focusing solely on generating missing steps given the ground-truth leap positions.

In practice, we process the ScaleQuestMath dataset by segmenting reasoning chains using "\n\n" as delimiters and selecting examples with at least 6 steps ( m ≥ 6 ) to ensure sufficient complexity. This process yields 588k training samples with 10k examples held out for testing. We fine-tune CoT-Bridge from the Qwen2.5-Math-7B base model using standard instruction tuning techniques for one epoch. We provide details such as instruction templates and training parameters in Appendix E.

2.3 Data Augmentation with Bridge Model

After training, we apply CoT-Bridge to enhance existing mathematical reasoning datasets, specifically MetaMathQA and NuminaMath-CoT , creating improved versions named MetaMath-Bridge and NuminaMath-Bridge respectively. As shown in Figure 2, CoT-Bridge processes reasoning chains to identify gaps and generate appropriate bridging content.

Given an input reasoning chain C = ( s 0 , s 1 , . . . , s n ) , CoT-Bridge produces a set of predicted Thought Leap repairs { ( k i , ˆ S ′ miss,k i ) } N leap i =1 , where each k i identifies a leap between steps s k i and s k i +1 , N leap represents the total number of leaps detected, and ˆ S ′ miss,k i is the sequence of generated intermediate steps. For each identified leap, we insert the generated steps between the corresponding original steps, performing s k i ⊕ ˆ S ′ miss,k i ⊕ s k i +1 to create the bridged reasoning chain C bridged .

To accommodate the format differences across datasets, we adapt our approach to the specific step delimiters used: "\n" for MetaMathQA and "\n\n" for NuminaMath-CoT. For the CoT-BridgeRandom variant, we provide randomly selected positions k random instead of model-identified leaps,

allowing us to evaluate the importance of accurate gap identification in improving reasoning quality. This approach enables us to assess both the leap detection and step generation aspects of the bridge task separately.

3 Experiments

3.1 Setup

To evaluate the generality and effectiveness of our proposed approach, we performed supervised finetuning (SFT) experiments using MetaMathQA, NuminaMath-CoT datasets and their bridged versions on representative base models. Specifically, we selected Meta-Llama3.1-8B [21] as a representative general-purpose model, and Qwen2.5-Math-1.5B [20] as a representative math-specialized model. To ensure comparability, we maintained a unified experimental configuration across all SFT experiments. The detailed training settings are provided in Appendix D.1.

3.2 Evaluation

We employed six benchmarks: GSM8K [15], MATH500 [22], and GaoKao2023EN [23] as basiclevel benchmarks, and MathOdyssey [24], OlympiadBenchEN [25], and AMC23 [26] as advanced competition-level benchmarks. The evaluation was carried out using the vLLM [27] inference library. To ensure consistency, all models adopted identical generation parameters, employing greedy decoding with zero-shot prompting, without external tools. To mitigate evaluation variance, we sampled four outputs for each problem and computed average accuracy as the final metric. The detailed evaluation settings are provided in Appendix D.2.

3.3 Baselines

We establish a series of baselines for comparison. The most important baseline is standard SFT performed directly on the original datasets. In addition, we use a zero-shot bridging which method leveraging general-purpose LLMs (in this paper, Qwen2.5-Instruct-7B/72B [28]) to explore Thought Leap bridging without direct training. This baseline aims to assess the effectiveness of pure prompt engineering in bridging Thought Leaps, determining whether specialized detection and repair mechanisms are necessary. Furthermore, we use CoT-Bridge-Random to generated intermediate steps at randomly chosen positions, to evaluate the importance of accurate Thought Leap localization for effective bridging. As a reference, we consider the performance of base models in a 4-shot setting. Additionally, we present the results of SFT using GSM8K+MATH and MathInstruct.

3.4 Main Results

Table 1 presents our main experimental results across all benchmarks, base models, and training datasets. We observe several consistent patterns that highlight the effectiveness of our CoT Thought Leap Bridge approach.

Bridging Thought Leaps consistently improves reasoning performance. When comparing CoTBridge with Direct SFT, we observe substantial improvements across almost all configurations. The most significant gains are seen with Meta-Llama3.1-8B tuned on NuminaMath, where CoT-Bridge achieves an average improvement of +5.87%, with particularly impressive gains on competition-level benchmarks (+15.63% on AMC23). Similarly, for Qwen2.5-Math-1.5B on MetaMathQA, CoTBridge yields a +3.36% average improvement, with a remarkable +7% on MATH500. These results indicate that addressing reasoning completeness through bridging enhances model performance, especially on more challenging problems that require rigorous step-by-step reasoning.

Accurate leap identification is crucial for effective bridging. Comparing CoT-Bridge with CoTBridge-Random reveals the importance of precise gap detection. While CoT-Bridge consistently improves performance, CoT-Bridge-Random shows highly variable results, degrading performance on certain benchmarks. For instance, with Qwen2.5-Math-1.5B on NuminaMath, CoT-Bridge-Random decreases accuracy on GSM8K (-0.56%), GaoKao2023EN (-1.56%), and MathOdyssey (-3.68%), with only marginal average improvement (+0.64%). In contrast, CoT-Bridge achieves a substantial

+3.39% average gain under the same setting. This stark difference highlights that merely inserting additional steps without strategic placement can disrupt reasoning coherence, whereas targeted bridging at identified leaps enhances logical flow and ultimately improves reasoning quality.

Zero-shot bridging shows promise but lacks consistency. Although zero-shot bridging demonstrates certain effectiveness on specific tasks, its overall performance still falls short compared to CoT-Bridge. Taking the LLaMA + NuminaMath setup as an example, the 72B bridge model improves the average accuracy by 4.54% over standard SFT, whereas CoT-Bridge further boosts it to 5.87%. Notably, the noise introduced by zero-shot bridge can degrade model performance in some scenarios. Under the LLaMA + MetaMathQA configuration, the 72B model shows decreased performance on GaoKao2023EN, MathOdyssey, and AMC23 by 1.43%, 0.20%, and 15.00%, respectively, leading to an overall average drop of 1.98%. This effect is more pronounced when using the weaker 7B model. In contrast, CoT-Bridge achieves more robust improvements under the same configuration, raising the average accuracy by 1.35%, highlighting its superior quality control and structural adaptability.

4 Analysis

4.1 Plug-and-Play Integration

CoT-Bridge can serve as a plug-and-play enhancement module that seamlessly integrates into existing training pipelines while delivering consistent performance improvements. To evaluate its adaptability and benefits across different training paradigms, we applied it to two representative scenarios: (1) improving the quality of generated data in knowledge distillation and rejection sampling, and (2)

assessing whether models fine-tuned on CoT-Bridge-enhanced data perform better in subsequent RL stages compared to models trained on original data.

We used Qwen2.5-Instruct-72B to generate data via distillation and rejection sampling on GSM8K and MATH training sets. For rejection sampling, we sampled 4 responses per prompt (temperature = 1.0, top-p = 1.0) and retained the correct answers, and then bridged them using CoT-Bridge. As shown in Table 2, models fine-tuned on the bridged data achieve superior performance: average accuracy improved from 55.23% (distill) / 60.44% (rejection sampling) to 58.25% / 61.81%, respectively.

78

+

Qwen2.5-Math-Numina

76

+1

Qwen2.5-Math-NuminaBridge

74

72

70

68

66

64

62

0

50

100

150

200

250

GRPOTrainingStep

MetaMath

CoT-Bridge

Qwen2.5-Instruct-72B

Qwen2.5-Instruct-7B

CoT-Bridge

Qwen2.5-Instruct-72B

Qwen2.5-Instruct-7B

0k

100k

200k

300k

400k

500k

600k

700k

800k

NuminaMath

0k

100k

200k

300k

400k

500k

600k

700k

800k

900k 1000k1100k1200k1300k1400k1500k1660k1700k

0-0.1

0.1-0.2

0.2-0.3

0.3-0.4

0.4-0.5

0.5-0.6

0.6-0.7

0.7-0.8

0.8-0.9

0.9-1

In the reinforcement learning setup, we continued training the Qwen2.5-Math-1.5B models that were fine-tuned on NuminaMath and NuminaMath-Bridge in the main experiments, using the GRPO [29] algorithm and the DAPO-Math-17K [30] dataset. Additional training details are provided in the Appendix G. Figure 3 shows the training curves of models, demonstrating that our method provides both a higher starting point and a better final performance. As shown in Table 9 from Appendix G.2, the model trained with NuminaMath-Bridge achieved an RL accuracy of 63.98%, outperforming that trained with NuminaMath at 60.88%. This result also surpasses the officially released Qwen2.5Math-Instruct-1.5B [20] model, which was also trained with GRPO algorithm.

4.2 Evaluation of Thought Leap Bridge Task

To evaluate the capability of various methods in CoT Thought Leap Bridge Task, we constructed a standardized evaluation framework on ScaleQM+ test set, covering leap identification and generation quality. Calculation methods for these metrics are detailed in Appendix H. Case study is detailed in Appendix I. As shown in Table 3, CoT-Bridge significantly outperforms other methods in localization precision (78.02%), redundancy rate (1.61%) and overall metric (76.15%). Qwen2.5-Instruct-72B notably outperforms the 7B model, indicating that

stronger generative models possess certain zero-shot localization and restoration capabilities. We also evaluated an exhaustive approach based on full-position generation and similarity filtering . This method uses CoT-Bridge-Random to generate candidate content between all adjacent steps, filtering them using similarity thresholds. While this approach demonstrates high recall capability,

it exhibits an extremely high redundancy rate. The substantial redundant content may cause LLMs to learn low-quality repetitive patterns during fine-tuning, thereby compromising reasoning ability. Additionally, this strategy is highly sensitive to similarity thresholds, incurs significant computational costs, and is difficult to scale in practice.

4.3 Evaluation of Out-of-Domain Reasoning Capability

To verify the generalization capability of our method in Out-of-Domain(OOD) reasoning, we conducted evaluations on five out-of-domain logical reasoning datasets (FOLIO [31], LogicQA [32], ProofWriter [33], ReClor [34], RuleTaker [35]) that were not used in training. These datasets cover formal logic, factual deduction, and textual entailment reasoning types. In the evaluation, we used XFinder [36] to automatically extract answers from model responses, with outputs that failed to yield valid answers (such as those deviating from the prompt or exhibiting logical loops) counted as errors and separately tallied.

As shown in Table 4, we compared the performance of models trained on NuminaMath and NuminaMath-Bridge. Results indicate that CoT-Bridge improved OOD reasoning performance (Meta-Llama3.1-8B: ↑ 2.99%, Qwen2.5-Math-1.5B: ↑ 0.99%) while reducing invalid response rates. Weattribute the performance improvement to the refined intermediate steps introduced by CoT-Bridge, which help models master general reasoning structures and enhance generalization capabilities.

4.4 Analysis of Bridging Positions

Leap bridging can enhance reasoning capabilities of LLMs, but whether complementary content bridged at different structural positions contributes substantially still lacks systematic analysis. Therefore, we categorized the bridging content in CoT-Bridge by its position in the reasoning chain as begin (reasoning starting point, planning strategy), middle (key calculations and logical progression), and end (result verification and summary).

Statistics show that middle type bridgings dominate (MetaMathQA: 66.34%; NuminaMath: 55.52%), followed by begin , with end accounting for a relatively small proportion (approximately 2%). To evaluate the specific role of each type of content, we designed position component ablation experiments: removing one type of bridging ( begin / middle / end ) from the CoT-Bridge data while keeping the rest unchanged, and fine-tuning Qwen2.5-Math-1.5B under identical training settings. As shown in Table 5, experimental results indicate that removal leads to performance degradation. This suggests that all three types of completion have a positive effect on model performance. We believe that begin bridgings help clarify problem objectives and establish solution frameworks; middle bridgings support critical links in reasoning paths; and end bridgings, although proportionally smaller, help strengthen logical closure and result reasonability in certain tasks.

4.5 Process Supervision Scoring and Noise Impact Analysis

To measure the bridging quality of various methods, we introduced the process supervision model Qwen2.5-Math-PRM-7B to score the intermediate steps (range 0-1) and analyzed the score distribu-

tion. Table 11 in Appendix K and Figure 4 present the PRM score distribution of the bridged steps. Results show that steps generated by CoT-Bridge have higher quality compared to other methods. On the NuminaMath dataset, CoT-Bridge's proportion in the high-quality (range 0.9-1) reaches 83.51%, significantly higher than 72B Fill (53.45%) and 7B Fill (56.8%). Furthermore, CoT-Bridge's proportion in the low-quality (range 0-0.1) is only 2.08%, compared to 14.81% and 13.12% for 72B and 7B Fill, respectively. We also used DeepSeek-R1 [6] to score the CoT comprehensively, CoT bridged by CoT-Bridge scoring higher than the original chains. The scoring prompt, results and examples are provided in the Appendix J.

To further verify whether low-quality steps generated by CoT-Bridge affect model training effectiveness, we set different thresholds based on PRM scores to denoise the bridged data and conducted SFT training on Qwen2.5-Math-1.5B accordingly. As shown in Table 6, results indicate that removing low-scoring steps has limited impact on model performance, suggesting that noise introduced by bridging has minimal effect on training. On the NuminaMath dataset, after removing steps with PRM scores below 0.1, accuracy decreased from 56.26% to 55.04%; while on MetaMathQA, removing steps with scores below 0.3 slightly improved performance from 45.91% to 46.40%. We attribute this phenomenon to two factors: first, the proportion of noise introduced by CoT-Bridge itself is already very low; second, the PRM model may misjudge in complex reasoning scenarios, with certain steps marked as low-scoring still possessing heuristic and training value.

5 Conclusion

In this paper, we addressed the critical issue of Thought Leaps in CoT reasoning by introducing the CoT Thought Leap Bridge Task and developing the CoT-Bridge model trained on ScaleQM+ dataset. Our approach automatically detects and fills these reasoning gaps, demonstrably enhancing the completeness and coherence of reasoning chains. Comprehensive experiments revealed that models tuned on these bridged datasets lead to significant performance improvements and better generalization to out-of-domain reasoning tasks. Furthermore, CoT-Bridge acts as an effective plug-

and-play module, improving outcomes in knowledge distillation and reinforcement learning, thereby highlighting the substantial benefits of rectifying thought leaps for more robust and capable large language models.

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A Related Works

A.1 Methods for Enhancing Mathematical Reasoning

Researchers have explored various reasoning enhancement methods based on CoT, broadly categorized into inference-time methods and training-based methods.

Inference-time approaches improve reasoning by adopting advanced decoding strategies or utilizing extra computational resources. Self-Consistency (SC) [37] enhances robustness by sampling multiple reasoning paths and applying majority voting. Search-based methods, including Tree-of-Thought (ToT) [38] and Monte Carlo Tree Search (MCTS) [39], systematically explore reasoning spaces for optimal solutions. Recent test-time expansion techniques, such as multi-round thinking [40] and s1 budget forcing [41], dynamically increase computation to boost performance. In addition, inference-time intervention methods [42, 43] enhance reasoning performance by manipulating hidden states.

Training-based approaches internalize enhanced reasoning abilities through parameter updates during model training. A common strategy involves fine-tuning models on datasets with explicitly detailed intermediate reasoning steps. Additionally, training paradigms incorporating self-reflection [44], self-verification, self-breaking [45] and self-correction have gained attention, teaching models to recognize and rectify errors autonomously. Frameworks like StepCo [46], S2R [47], and ReVISE [48] integrate these metacognitive capabilities through supervised, reinforcement, or preference learning. In particular, recent advances highlight that reinforcement learning plays an especially crucial role [49-52] in enhancing the overall capabilities of large models. We leverage CoT-Bridge to enhance the structural completeness of existing datasets, enabling models to acquire finer-grained reasoning patterns during training and to be seamlessly integrated into existing training pipelines.

A.2 High-Quality Mathematical Datasets for Training

Developing high-quality datasets is essential for enhancing mathematical reasoning capabilities in LLMs. MetaMathQA [12] consists of 395k problem-solution pairs derived by reformulating and reverse-engineering existing datasets like GSM8K [15] and MATH [16]. MathInstruct [13] aggregates 13 existing math datasets and employs GPT-4 [53] to synthesize 260k examples containing both Chain-of-Thought and Program-of-Thought solutions. NuminaMath-CoT [14] compiles data from examinations, competitions, and Q&A communities, resulting in a dataset of 860k problem-solution pairs. ScaleQuestMath [19] introduces a zero-shot data generation framework for mathematical questions using small open-source models, resulting in a scalable dataset of about 1 million samples. DAPO [30] developed DAPO-Math-17K, comprising 17k standardized integer-formatted math problems sourced primarily from platforms such as AoPS for effective reward model evaluation.

Beyond creating new datasets, extensive research has focused on data augmentation (e.g., MathFusion [54], PersonaMathQA [55], MathFimer [56]) and selective strategies (e.g., QaDS [57], DELIFT [58]) to enhance data efficiency and training effectiveness. CoT-Bridge contributes by bridging Thought Leap issues in existing mathematical reasoning datasets, thus improving their overall data quality.

B Limitations

Potential Noise in Reasoning Steps: CoT-Bridge can't guarantee that all bridged reasoning steps are entirely correct, which may inevitably introduce a certain level of noise during automated data construction. Although our analysis in Section 4.5 suggests that this noise has a limited impact on overall model performance, its presence remains a non-negligible concern.

Lack of Validation on Larger-Scale Models: Due to computational resource constraints, our experiments focus primarily on small to medium-sized models, and we have not yet conducted evaluations on larger models such as 32B or 72B-scale LLMs. This may limit the verification of our method's generalization capability on very large-scale pretrained models. Future work could explore its performance and adaptability on larger architectures.

Training Limited to Mathematical Data: CoT-Bridge is trained exclusively on ScaleQuestMath in mathematical domain. While we demonstrate promising OOD generalization to logical reasoning tasks that are relatively close to mathematics, the method remains largely domain-specific. It is worth noting, however, that CoT-Bridge is theoretically generalizable and can potentially be applied to other multi-step reasoning tasks in domains such as law, medicine, and scientific QA. Future research could further investigate its adaptability across diverse task types.

C Preliminary Experiment

C.1 Settings

To simulate varying degrees of Thought Leap, we created modified versions of the MetaMathQA dataset by removing intermediate reasoning steps. Considering the structure of MetaMathQA, we use "\n" to segment individual steps within CoT. The specific configurations are as follows:

The final step which contains the answer explanation is always retained.

Mild, Moderate, Severe Thought Leap: We randomly remove 1, 2, or 3 intermediate steps, respectively, if sufficient steps are available.

Extreme Thought Leap: All intermediate reasoning steps are removed and only the final answer explanation is preserved.

We adopt the same training configuration as described in Section 3.1 using Meta-Llama3.1-8B. Each model is trained for 200 steps, with evaluation performed on the GSM8K every 10 steps. For convenience, answer verification is conducted using Math-Verify only.

C.2 Results

We present the detailed results from Figure 1(b) in Table 7.

D Experimental Details

D.1 Model Training Settings

We utilized LlamaFactory [59] as SFT training framework. The initial learning rate was set to 1 × 10 -5 with a warm-up ratio of 0.1, and cosine scheduling was used to gradually reduce the learning rate to zero. The maximum sequence length was set to 8192 tokens, with a global batch size of 128. We trained models for 3 epochs on MetaMathQA [12] and for 2 epochs on NuminaMath-CoT [14], as preliminary experiments indicated no further improvement from additional epochs. All SFT experiments were conducted using 8 Ascend H910B-64G.

For SFT training template, we use the default style template in LlamaFactory:

D.2 Evaluation Settings

The maximum token limit for generation was set to 2048, and prompt templates remained consistent with training ones. It is noteworthy that, although we set the decoding temperature parameter to zero, vLLM outputs still exhibited some randomness. For answer extraction and comparison, we employed the Math-Verify [60] tool. Given limitations of math-verify in handling complex expressions, responses failing initial verification were subsequently validated using DeepSeek-R1 [6]. All model evaluations were performed using 4 NVIDIA A100-40G GPUs.

Referring to OpenR1 [61], we use the following prompt template:

Prompt Template

You are a mathematical answer validator. You will be provided with a mathematical problem and you need to compare the answer in the reference solution, and the final answer in a model's solution to determine if they are equivalent, even if formatted differently.

PROBLEM:

<problem>

REFERENCE SOLUTION:

<answer>

MODEL'S SOLUTION:

<generation>

Focus ONLY on comparing the final mathematical answer provided by the model while ignoring differences in:

Formatting (e.g., \boxed{} vs plain text)

Multiple choice formatting (e.g., "A" vs full solution)

Order of coordinate pairs or solutions

Equivalent mathematical expressions or notation variations

If the model's answer is nonsense, return "Verdict: AMBIGUOUS"

Start with a brief explanation of your comparison (2-3 sentences). Then output your final answer in one of the following formats:

"Verdict: EQUIVALENT"

"Verdict: DIFFERENT"

"Verdict: AMBIGUOUS"

E ScaleQM+ Construction and CoT-Bridge training

E.1 ScaleQM+ Template

E.2 Variant Template

Prompt Template

System:

You are a mathematics teacher reviewing a solution that appears to be missing one step. Given the position of the missing step, your task is to fill in the missing step.

The steps in the solution are labeled from Step 0 (problem statement) to Step N.

Please format your response as:

The missing step is:

[Write the complete missing step here with necessary explanations and equations]

Human:

There is a missing step between Step X and Step X+1.

E.3 Discussion of Hyperparameter Settings

We set the minimum CoT length m to 6, based on the following considerations. CoTs with too few steps should be avoided, as identifying Thought Leaps in such cases becomes trivially easy. For example, in a two-step CoT,

the final step (the answer) cannot be removed, so only the first step is eligible for deletion. This makes it very easy for the model to identify the location of the Thought Leap, which lies directly between the question and the answer.

Regarding the number of steps to delete, we avoid removing too many steps. For instance, in a 6-step CoT, deleting 4 steps would leave too little context, making the task resemble generating a new solution based on the final answer rather than bridging a partial reasoning chain.

As for the probability of retaining the original CoT without modification, this parameter is introduced to avoid introducing redundancy into already complete reasoning chains.

E.4 CoT-Bridge Training Settings

We broadly follow the setup in D.1, but train for only 1 epoch with a global batch size of 1024, considering the large amount of data and the primary goal of learning the structural integrity of CoT rather than introducing new knowledge.

F Evaluation Benchmarks

F.1 Mathematics Benchmarks

GSM8K : This benchmark consists of 1,319 grade-school math word problems, primarily used to evaluate a model's ability to perform multi-step arithmetic reasoning.

MATH500 : A curated subset of 500 representative problems from the larger MATH dataset, designed to reflect the original dataset's coverage across diverse mathematical topics and difficulty levels.

Gaokao2023EN : This benchmark includes 385 English-translated questions from the 2023 Chinese Gaokao (college entrance exam) mathematics section. It tests a model's ability to solve complex problems at the advanced high school level.

MathOdyssey : A benchmark of 387 problems spanning difficulty levels from high school to early undergraduate mathematics. It emphasizes evaluating a model's performance on problems requiring deeper understanding and more complex reasoning steps.

OlympiadBenchEN : A collection of 675 problems at the difficulty level of international mathematical olympiads. Known for their non-standard formats and high demands on creative problem-solving, these problems serve as a rigorous test of advanced mathematical reasoning capabilities.

AMC23 : This benchmark contains 40 problems selected from the 2023 American Mathematics Competitions (AMC), offering a challenging testbed for evaluating models in a standardized competition setting.

F.2 Logical Reasoning Benchmarks

FOLIO : This benchmark consists of 1,430 expert-crafted natural language inference instances, each annotated with first-order logic (FOL) expressions. It is designed to assess a model's ability to translate between natural language and formal logic, as well as perform multi-step reasoning over complex logical structures.

LogicQA : Collected from the Chinese National Civil Servant Examination, this benchmark targets logical reasoning in reading comprehension tasks.

ProofWriter : Composed of multiple small rulebases, each containing natural language descriptions of facts and rules, along with questions labeled as true, false, or unknown. This benchmark supports evaluating a model's ability to generate reasoning chains and perform multi-step logical inference.

ReClor : Derived from standardized graduate-level entrance exams such as the LSAT and GMAT, ReClor evaluates a model's performance on complex logical reasoning questions.

RuleTaker : Built from synthetically generated rules and facts, this benchmark tests whether a model can simulate deductive reasoning. It evaluates the model's ability to draw logical conclusions from natural language rules.

G RL Training

G.1 Settings

All RL experiments were conducted using the veRL [62] training framework. We employed Math-Verify for answer verification. We set the initial learning rate to 1 × 10 -6 and used a global batch size of 512. The maximum response length was limited to 4096 tokens. During rollout, 4 samples were generated per input. KL regularization was disabled, and evaluation was performed with zero temperature every 10 epochs on MATH500. We use the same template in Appendix D.1.

G.2 Results

We present the detailed results from Figure 3(b) in Table 8. For simplicity, only Math-Verify is used for answer verification without further applying DeepSeek-R1.

H Metrics for CoT Thought Leap Bridge Task

H.1 Leap Position Identification Metrics

Let the ground-truth set of leap positions be G = { g 1 , . . . , g | G | } , and the predicted set be P = { p 1 , . . . , p | P | } . If a predicted position p j exactly matches a ground-truth position g i , it is counted as a True Positive (TP). The metrics are defined as:

To measure the proportion of redundant predictions, we define the redundancy rate as:

H.2 Overall Metric

To evaluate both leap positions locating ability and the quality of the generated step, we introduce the Positionaware BERTScore . This metric only compares text at successfully matched positions. Let the reference text at a matched position be t i and the generated text be ˆ t i , the Position-aware BERTScore is defined as:

Here, BERTScore is computed using the RoBERTa-large model and measures the semantic similarity between each pair of texts ( ˆ t i , t i ) .

I Case Study: CoT Thought Leap Bridge

Following the example in Figure 1(a), we compared the zero-shot bridging results of Qwen2.5-Instruct-7B/72B, GPT-4o, and Gemini-2.5-PRO.

CoT-Bridge

Missing Step 1:

The missing step should be placed between Step 0 and Step 1.

The missing step is:

To determine the minimum number of times you must throw two fair eight-sided dice to ensure that the same sum is rolled twice, we can use the pigeonhole principle. The pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.

Missing Step 2:

The missing step should be placed between Step 1 and Step 2.

The missing step is:

Therefore, the possible sums when rolling two eight-sided dice are the integers from 2 to 16, inclusive. This gives us a total of 16 - 2 + 1 = 15 possible sums.

Qwen2.5-7B-Instruct

Missing Step 1:

The missing step should be placed between Step 0 and Step 1.

The missing step is:

The possible sums when rolling two eight-sided dice range from 2 to 16, which gives us 15 different possible sums.

Qwen2.5-72B-Instruct

Missing Step 1:

The missing step should be placed between Step 1 and Step 2.

The missing step is:

The possible sums when rolling two eight-sided dice are 2, 3, 4, ..., 16. There are 15 possible sums in total.

Missing Step 2:

The missing step should be placed between Step 2 and Step 3.

The missing step is:

According to the pigeonhole principle, if we roll the dice 16 times, there are 15 possible sums, so at least one of these sums must appear at least twice.

Gemini-2.5-PRO

Missing Step 1:

The missing step should be placed between Step 1 and Step 2.

The missing step is:

Now, we need to determine the total number of distinct sums possible. The sums are all the integers from 2 to 16, inclusive. These distinct sums will be our "pigeonholes."

To count the number of possible sums:

Number of distinct sums=(Maximum Sum-Minimum Sum)+1

Number of distinct sums=(16-2)+1=14+1=15

So, there are 15 unique possible sums (the pigeonholes) when rolling two eight-sided dice.

GPT-4o

Missing Step 1:

The missing step should be placed between Step 1 and Step 2.

The missing step is:

We need to count how many distinct sums can result from rolling two eight-sided dice. The possible sums range from 2 to 16, which gives:

$$

16 - 2 + 1 = 15 distinct possible sums.

$$

As can be seen in the case study:

Although a planning-style bridge between the question and step 1 was missing, GPT-4o, Gemini-2.5PRO, and Qwen2.5-72B-Instruct all identified that "15" lacked a calculation explanation and bridged the gap in the appropriate position.

Qwen2.5-72B-Instruct generated a redundant step (almost identical to step 2). This phenomenon is quite common and aligns with its 33.73% redundancy rate shown in Table 3.

The consistency of the content and position bridged by Qwen2.5-7B-Instruct is not good.

J CoT Score

J.1 Score Template

Since scoring a single response with DeepSeek-R1 can result in high variance, we instead compare the original CoT and its bridged version for the same question side-by-side.

DeepSeek-R1 Score Template

You are a math reasoning expert. Please evaluate the coherence of the given two Chain of Thoughts (CoTs) based on the provided question.

Perform an overall assessment of the CoTs based on the following five dimensions:

1. Problem Understanding: Does the CoT accurately identify the core of the question and interpret its requirements appropriately?

2. Planning: Does it show evidence of having a structured plan or certain degree of foresight in approaching the problem?

Logical Coherence: Is the reasoning step-by-step, consistent, and free of major logical leaps?

4. Detail Elaboration: Does it include detailed intermediate calculations, formula derivations, or other necessary expansions?

5. Conclusion Support: Does it provide a clear summary or verification to support the final conclusion?

Question

<problem>

CoT 1

<answer1>

- CoT 2

<answer2>

[Scoring Criteria Description]

- 10 points: Outstanding across all dimensions. Demonstrates deep understanding, clear planning, rigorous step-by-step reasoning, detailed intermediate calculations or derivations, and strong conclusion verification. Rarely achieved; reflects expert-level CoT.

- 8 points: Strong performance with only minor flaws. Reasoning is coherent and mostly complete, with evident planning and elaboration. May miss slight details or final verification, but overall very solid.

- 6 points: Adequate but uneven. Shows basic understanding and partial planning. Some steps are under-explained or loosely connected; may lack full elaboration or support for conclusion.

- 4 points: Weak overall. Reasoning is fragmented or inconsistent, with limited detail and minimal evidence of structured planning. Key steps are missing or poorly justified.

- 2 points: Very poor. Little to no logical structure or elaboration; response lacks meaningful connection to the question.

- 0 points: No relevant reasoning. Off-topic or fundamentally flawed throughout.

Note: Scores may also be assigned in odd numbers (e.g., 5, 7, 9) to reflect responses that fall between two defined levels. For instance, a score of 7 indicates performance stronger than the 6-point criteria but not fully reaching the 8-point standard.

[Output Format]

- Justification:

Logical Coherence Analysis: <Your brief analysis>

- Score:

CoT 1 overall score: <Integer between 0-10>

CoT 2 overall score: <Integer between 0-10>

A brief analysis is sufficient.

J.2 Score Example

Example

Okay, let's tackle this problem. The user wants me to evaluate two Chain of Thoughts (CoTs) for the question about the minimum number of throws needed to ensure a repeated sum with two eight-sided dice. The scoring is based on five dimensions: Problem Understanding, Planning, Logical Coherence, Detail Elaboration, and Conclusion Support.

First, I need to understand the question thoroughly. The key here is the application of the pigeonhole principle. The possible sums when rolling two 8-sided dice range from 2 (1+1) to 16 (8+8). That gives 15 possible sums. According to the pigeonhole principle, to guarantee a repeat, you need one more than the number of possible sums, so 15 + 1 = 16 throws. Both CoTs arrive at this answer, but I need to assess how well each explains the reasoning.

Looking at CoT 1: It starts by identifying the minimum and maximum sums correctly. Then it directly applies the pigeonhole principle, stating 15 +1 =16. The conclusion is there, but the explanation is very brief. It doesn't mention how many possible sums there are, just states the range. The step from the range to 15 possible sums is implicit. A reader might wonder why 15 instead of 16-2=14. So, CoT 1 skips explaining the count of possible sums, which is a crucial step. The logical steps are there but not fully elaborated.

CoT 2 is more detailed. It explicitly states the pigeonhole principle, defines the range of sums, calculates the number of possible sums as 15 (16-2+1), then adds 1. This makes the reasoning clearer. The explanation of why there are 15 possible sums is important because it's a common point of confusion (inclusive counting). CoT 2 addresses that, making it more thorough.

Now, evaluating each dimension for both CoTs:

For Problem Understanding: Both correctly identify the core of the question, which is applying the pigeonhole principle. CoT 2 does a better job by explicitly stating the principle and the count of sums. Planning: CoT 2 structures the answer step-by-step, starting with the principle, then calculating possible sums, then applying the formula. CoT 1 jumps to the answer without explaining the count of sums, which is a gap in planning.

Logical Coherence: Both are logically correct, but CoT 1 has a minor leap in assuming the number of possible sums. CoT 2 connects each step more clearly.

Detail Elaboration: CoT 2 elaborates on the number of sums, which CoT 1 omits. This makes CoT 2's explanation more complete.

Conclusion Support: Both reach the correct conclusion, but CoT 2's conclusion is better supported by the preceding steps.

Scoring: CoT 1 might get a 6 or 7 because it's correct but lacks some details. CoT 2 is more thorough, so probably an 8 or 9. Since CoT 2 explains each step clearly without missing key points, it's closer to the 8-9 range. CoT 1 is adequate but not as detailed, so maybe a 7. However, according to the criteria, 8 is for strong performance with minor lapses. CoT 2 doesn't have lapses, so maybe 9? Wait, the scoring criteria say 10 is outstanding. CoT 2 is thorough but maybe not "outstanding" because it's a straightforward application. Alternatively, since it's complete, maybe 10. But the example given in the criteria for 10 is "precise, step-by-step reasoning" with "careful attention to clarity". CoT 2 does that. CoT 1 is missing the explanation of the number of sums, so maybe a 6 or 7. Let me check again.

CoT 1: Problem Understanding: Correct. Planning: Minimal, skips the count of sums. Logical Coherence: Correct but with a leap. Detail Elaboration: Missing the count explanation. Conclusion: Correct. So maybe 6 or 7. The criteria for 6 say "intermediate steps present but not fully explained". CoT 1 has the steps but skips the count. So 6. CoT 2: All steps explained, so 10? But the example for 10 says "smooth and complete". CoT 2 is complete. Alternatively, maybe 9 if there's a minor point missing. But I don't see any. It explains the pigeonhole principle, calculates the sums correctly, adds 1. So 10. But the user's example answer might have given 8 and 10. Wait, the user's example in the criteria for 10 is "outstanding across all dimensions". CoT 2 is thorough, but perhaps not outstanding. Hmm. Alternatively, maybe 9. But given that it's correct and explains all steps, 10. CoT 1 gets 6 because it's adequate but missing the count explanation. </think>

- Justification:

Logical Coherence Analysis: CoT 1 correctly applies the pigeonhole principle but skips explaining the count of possible sums (15), leaving a minor gap. CoT 2 explicitly calculates the number of sums (16 2 + 1 = 15) and links each step clearly, ensuring full logical continuity.

- Score:

CoT 1 overall score: 7

CoT 2 overall score: 10

**Note:** CoT 1 earns a 7 for adequate reasoning but lacks explicit calculation of the sum count. CoT 2 achieves a 10 by thoroughly addressing all dimensions, including precise step-by-step elaboration and seamless logical flow.

The question, CoT1, and CoT2 here are the same as those shown in the example in Figure 1(a).

J.3 Score Result

We randomly sampled 1,000 CoT pairs from both MetaMath and NuminaMath for scoring, and the results are shown in Table 10.

K PRMScore

We present the data from Figure 4 in Table 11.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes to augment existing SFT in LLMs by thought bridge data: i.e., CoT data that's specifically constructed to be missing intermediate reasoning steps, to enhance the model's ability in reasoning.\n\nThe authors did comprehensive experiments and show that the constructed ScaleQM+ data can help improve model's performance compared to regular SFT.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper is overall well-written.\n- The overall idea about model should learn to bridge thought gaps is quite interesting, and the authors managed to construct a dataset to quantify this behaviour, and improve the model's reasoning performance significantly.\n- Experiments are performed over two models, LLama and Qwen, and over several math datasets, to show the effectiveness of CoT-Bridge.\n\nWeaknesses:\n- For the results in Table 1, do CoT-Bridge use an additional dataset ScaleQuestMath where the baselines don't?\n- This paper is well-motivated and to support its claims, I would expect seeing data with real thought leaps being bridged in training. The current way of processing the data is kind of artificial, where you know where the missing step is and the model is asked to predict that exact step. In practice, I wonder given any dataset (possibly with thought leaps but you don't know where that is), how can you enable the model to bridge such thought leaps?\n\nQUESTIONS:\n- To clarify, for the results in Table 1, how exactly is the ScaleQM+ dataset used? Is it used to create the CoT-Bridge model, then the model was used to augment MetaMathQA and NuminaMath, and then SFT on those augmented datasets were done?\nIn that case, is CoT-bridge utilizing one additional dataset ScaleQuestMath? Does the baseline (direct SFT) use MetaMathQA / NuminaMath + ScaleQuestMath, or just the former only?\n\nLIMITATIONS:\nNot discussed. Please see weaknesses above", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nTo address the problem of thought leap in language model's Chain-of-Thought (CoT) reasoning where model skips intermediate steps, this paper proposes to finetune a thought-bridge model on a curated thought leap bridging dataset in order to augment existing mathematical reasoning datasets with sufficient intermediate steps. The authors experiment on six math reasoning datasets models and demonstrate that models tuned on the bridged datasets improves performance.\n\nSTRENGTHS AND WEAKNESSES:\n## Strength:\n\n### Identification of novel task\n\nThis works identifies the task of CoT Thought Leap Bridge that presents an automatic approach to detect reasoning leaps and generate missing steps. This task is useful to enhance the dataset construction of LLM training and has rich applicability to various training pipelines such as knowledge distillation and RLHF.\n\n## Weakness:\n\n### Model scalability concerns\n\nIn order to demonstrate improvement on Meta-Llama3.1-8B and Qwen2.5-Math-1.5B, this work uses 10x larger language models  (72B) for data augmentation. This raises a model scalability concern whether the advertised improvement is scalable to larger model sizes. If improvements are limited to smaller models, the contribution is less significant, especially given the abundance of existing methods that generate synthetic data using larger language models.\n\n### Limited effects on RL and narrow evaluation\n\nSince CoT-Bridge operates on augmenting thought processes within offline data, its benefits are mostly realized during supervised fine-tuning (SFT), rather than in reinforcement learning (particularly outcome-based RL), which is more prevalent and impactful in finetuning reasoning models. Although the work shows that CoT-Bridge can provide a better SFT initialization for RL (Appendix G), the evaluation is limited: it is demonstrated on only one dataset and with a single small model, restricting the generalizability and strength of the claims.\n\n### Information-theoretical inefficiency\n\nThe performance gains from SFT-trained models stem from the bridge model’s ability to provide thought-bridging supervision, which in turn relies on training with a large, customized dataset (ScaleQM+, derived from the structurally rich ScaleQuestMath dataset). However, this process is information-theoretically inefficient, due to the data processing inequality. There are at least two alternative and potentially more efficient strategies: (1) redirecting human annotation efforts towards filling in missing reasoning steps directly within the training datasets, or (2) training models directly on the comprehensive customized dataset, rather than relying on an intermediate bridge model.\n\nQUESTIONS:\n1. When you augment the dataset by filling the missing intermediate skips, you are also increasing the number of tokens used for training which also improves the model performance due to the scaling law with respect to the number of tokens. Is it a fair comparison to compare augmented datasets with more tokens to original datasets?\n\n2. While you create a specialized training dataset ScaleQM+ and remove intermediate steps to produce incomplete chains, are you assuming that ScaleQM+ is then free of thought leap issue? If so, why not directly train the models on this dataset? In other words, the (human) efforts you used to construct the thought-leap-free dataset in order to train the bridging model can be otherwise used to complete existing math reasoning datasets such as MATH500, then why would we prefer to do the former?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-29"}

comments: SUMMARY:
This paper addresses the issue of Thought Leaps in Chain-of-Thought (CoT) reasoning, where intermediate steps are often omitted in mathematical CoT datasets, hindering model learning and generalization. The authors propose the CoT Thought Leap Bridge Task, aimed at automatically detecting and bridging these reasoning gaps. To facilitate this, they construct the ScaleQM+ dataset and develop a dedicated bridging model, CoT-Bridge, trained on it. Their approach demonstrates significant improvements in performance across multiple reasoning benchmarks. The method also works as a plug-and-play module, enhancing knowledge distillation and cold starts in reinforcement learning pipelines.

STRENGTHS AND WEAKNESSES:
Strength：
- This paper is well-written and easy to read.
- The paper systematically identifies and formalizes the "Thought Leap" phenomenon in CoT datasets.
- The CoT-Bridge model is trained on a carefully constructed dataset (ScaleQM+) using step-deletion strategies and rigorously evaluated on standard benchmarks. 
- CoT-Bridge is designed to be plug-and-play, improving distillation and RL fine-tuning workflows without architectural changes.

Weakness:
- The method's performance varies with bridge location (begin/middle/end), but the paper offers limited insight into how position-aware generation might be improved.
- The authors should provide more explanations for why removing low-PRM steps (PRM < 0.5) does not improve performance, as this is somewhat counter-intuitive.

QUESTIONS:
See the weakness.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work describes the phenomenon of Thought Leaps in LLMs' chain of thought process, such as logical gaps between reasoning steps, making the chain of thought reasoning data incoherent. The authors also propose a method to construct training data for Thought Leaps by applying rule-based interventions to existing data with a complete reasoning process, and this training data contains the annotation of which step is missing and what content ihas been omitted. Based on this constructed training dataset, the authors fine-tune a LLM to perform chain of thoughts data synthesis. This work also attempts the prompt engineering methods for guiding LLMs to complete chain of thoughts.\n\nSTRENGTHS AND WEAKNESSES:\nStrenghts\n1. I strongly agree with the motivation of this work. The completeness and rigor of chain of thoughts data are crucial for the reasoning capabilities of LLMs.\n\nWeakness\n1. Despite the solid motivation, the design of this work is not solid. For example, the reasoning steps are segmented by simple heuristics such as splitting \"\\n\\n\", which may not align with the actual logical steps. Defining the minimal unit of general chain of thoughts data remains a challenging research question.\n\n2. While Section 2.2 is clearly written, the construction of training data by modifying existing complete chain of thoughts data in mathematics domain raises concerns about generalizability and scalability. My feeling is that the difficult of finding thought leaps is not easier than solve the problem, rasing the dependency of having a strong teacher model. \n\n3. Since missing steps in different chain of thoughs data may exhibit \"complementary\" and considering the LLM's generalization ability, I doubt the performance gap between CoT-Bridge and fine-tuning would narrow when applied to larger, more capable models.\n\nQUESTIONS:\nL196 discussed the \"Zero-shot bridging shows promise but lacks consistency\". What if we choose LLMs with stronger instruction following ability, such as GPT-4o or Gemini. It's also better to have a case study comparsion for zero-shot and fine-tuned CoT bridger.\n\nLIMITATIONS:\nAs mentioned in weakness, I think the difficult of finding thought leaps is not easier than solve the problem. So, the proposed method may not be helpful for chasing a higher reasoning ability for the SOTA LLMs.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}