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

Dear Reviewer Ayux:

Thank you for reviewing our paper and providing valuable feedback. Below, we present our responses to the weaknesses you have mentioned:

Question: 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? In that case, is CoT-bridge utilizing one additional dataset ScaleQuestMath? Does the baseline (direct SFT) use MetaMathQA / NuminaMath + ScaleQuestMath, or just the former only?

Response to Question:

Thank you for your question. We would like to clarify the CoT-Bridge and baseline setup.

1. CoT-Bridge Setup:

   • We first train the CoT-Bridge model on ScaleQM+ (derived from a subset of ScaleQuestMath, ~588k samples from the original 1M, as mentioned in Section 2.2, lines 128-132)
   • Then apply CoT-Bridge to augment MetaMathQA/NuminaMath
   • Finally perform SFT on the augmented datasets

2. Baseline Setup: The baseline model (Direct SFT) is trained only on MetaMathQA or NuminaMath.

We understand your concern regarding the fairness of this comparison. We will address this issue through additional experiments in our response to Weakness 1.

Weakness 1: For the results in Table 1, do CoT-Bridge use an additional dataset ScaleQuestMath where the baselines don't?

Response to Weakness 1:

Thank you for your insightful observation regarding the fairness of comparison.

• Indeed, CoT-Bridge utilizes ScaleQM+ (derived from ScaleQuestMath) during the augmentation process of MetaMathQA and NuminaMath. However, this utilization differs from direct SFT using ScaleQuestMath. CoT-Bridge is designed to learn structural patterns for identifying and bridging thought gaps rather than directly acquiring mathematical knowledge from the dataset.

• Simply incorporating the 588k ScaleQuestMath subset (used to construct ScaleQM+, as mentioned in Section 2.2, lines 128-132) into the baseline's SFT training would also create an imbalance by significantly increasing the baseline's training samples.

To ensure equitable comparison, we conducted additional experiments where both the baseline and CoT-Bridge-augmented setup incorporated this 588k subset. Due to the limited time and computational resources available during the rebuttal period, we could not conduct comprehensive experiments across all model sizes and datasets as in our main study. We focused our testing on Meta-Llama3.1-8B + NuminaMath as a representative configuration. As shown in the table below, the performance improvement is still significant and consistent.

Results on Meta-Llama3.1-8B + NuminaMath:

We sincerely appreciate your observation. We plan to include these additional results in our paper to enhance its rigor.

Weakness 2: 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?

Response to Weakness 2:

Thank you for your kind words of our motivation. We would like to clarify how our approach works in practice. CoT-Bridge does not require any external guidance about where thought leaps occur during inference. When processing any reasoning chain (e.g., from MetaMathQA or NuminaMath) that potentially contains thought leaps, CoT-Bridge simultaneously:

1. Identifies WHERE thought leaps occur (position detection)
2. Generates WHAT content to bridge the gaps (content generation)

• As formally defined in Section 2.1 (lines 123-127), CoT-Bridge learns the mapping $f: C → (L̂, M̂)$, taking a reasoning chain $C$ as input and autonomously producing both the leap positions $L̂$ and corresponding missing steps $M̂$.

• Actually, Figure 1(a) in the paper demonstrate this capability with a real-world example. In Figure 1(a), CoT-Bridge independently identifies the thought leaps (between Question - Step 1 and between Step 1 - Step 2) and generates appropriate bridging content (Bridge 1 and Bridge 2) without external guidance.

• The prompt used in inference is consistent with the one in Section E.1, which shows that CoT-Bridge does not require users to provide the bridge positions.

We hope this clarification addresses your concern regarding the practical application of CoT-Bridge to datasets with unknown thought leap positions. If you have any further questions, we would be happy to answer them.

Sincerely,
CoT-Bridge Authors

Dear Reviewer Wqes:

Thank you for the time and effort you have dedicated to reviewing our paper. Below, we present our responses to the weaknesses and questions you have mentioned:

Weakness 1: Model scalability concerns

Response to Weakness 1:

Thank you for pointing out the potential model scalability concern. We would like to respectfully clarify that our core data augmentation tool CoT-Bridge is a 7B-scale model fine-tuned from Qwen2.5-Math-7B as described in Section 2.2 (lines 130-132), rather than "10x larger language models (72B)". Actually, the 72B models are used to highlight the efficiency and practicality of CoT-Bridge-7B.

• Zero-Shot Baseline: In Tables 1 and 3, we use the 72B model QwenBridger-L (Qwen2.5-Instruct-72B without fine-tuning) for zero-shot bridging as a strong baseline. The experiment results clearly demonstrate that our CoT-Bridge-7B model significantly outperforms 10x larger model both in terms of enhanced dataset quality and bridging task performance.

• Enhancing Data from Larger Models: In Section 4.1, Table 2, we use the 72B model to generate distilled data. The purpose of this experiment is to demonstrate that even data generated by 10x larger models still exists thought leaps. Our 7B CoT-Bridge model can effectively enhance the high-quality but still imperfect data, leading to further performance improvements (+3.02%).

Furthermore, we strongly agree that the bridge model size is indeed worth exploring. We believe that the thought gap bridge task requires a task-specialized model rather than necessarily a larger-size one with strong capabilities. Since the bridge task has access to the original solution CoT, it doesn't require an especially large model compared to directly solving the problem. To test this hypothesis, we fine-tuned Qwen2.5-Math-1.5B on ScaleQM+ to create CoT-Bridge-1.5B. We then used this smaller model to enhance NuminaMath and performed SFT on Meta-Llama3.1-8B with the enhanced dataset. As shown in the table below, the performance using CoT-Bridge-1.5B for bridging is not significantly different from our main results in the paper.

Results on Meta-Llama3.1-8B + NuminaMath:

Weakness 2.1: Limited effects on RL

Response to Weakness 2.1

Thank you for raising this important point about the relationship between our method and RL. We would like to clarify that the main purpose of our RL analysis experiments in Section 4.1 was to verify whether better SFT initialization can lead to a higher performance ceiling for subsequent RL training.

The results in Table 9 of our paper (Appendix G.2) demonstrate that models initialized with CoT-Bridge-enhanced data achieved 63.98% accuracy after RL, compared to 60.88% for models without bridging (+3.1% improvement). Furthermore, when compared to directly applying RL to Qwen2.5-Math-1.5B (59.33%), our approach shows a more substantial +4.65% improvement.

These findings align with DeepSeek-R1 [1], which demonstrates that models with SFT initialization (DeepSeek-R1) achieve superior reasoning performance compared to directly trained with RL from scratch (DeepSeek-R1-Zero).

[1] DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning

Weakness 2.2: Narrow evaluation

Response to Weakness 2.2

We understand your concern about the comprehensiveness of our experiments. Due to computational resource constraints and the limited time available during the rebuttal period, we conducted additional RL experiments using another smaller dataset (MATH training set, difficulty levels 3-5) with the Llama3.1-8B fine-tuned on NuminaMath and NuminaMath-Bridged. Additionally, we included results from SimpleRL-Zoo [2] on direct RL training of Llama3.1-8B for your reference.

As shown in the table below, the results continue to show that using datasets enhanced with thought bridges for RL leads to a higher performance ceiling across all benchmarks.

Results:

Experimental setup:

• Hardware: 16 × A800 80G GPUs
• Algorithm: GRPO with same hyperparameters as original experiments

[2] SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild

Weakness 3: Information-theoretical inefficiency

Response to Weakness 3:

Thank you for offering this novel analytical perspective. We understand the theoretical perspective of Data Processing Inequality (DPI) that you mentioned, and respectfully offer a different view. DPI is based on the premise that information in a transmission process can only be preserved or reduced, not increased. However, we consider this conclusion mainly applies to scenarios of transmitting static information (such as mathematical theorems). In our setting, CoT-Bridge is not focusing on transmitting the mathematical knowledge from ScaleQuestMath itself, but learning structural patterns of reasoning and universal thinking strategies in the reasoning process. This structural information has high generalizability and transferability, and can be transferred to millions of CoT data containing thought gaps. More importantly, CoT-Bridge can directly access the original CoT chains when performing thought gap bridge tasks, a process that resembles pattern induction rather than information transmission. Therefore, this is not completely consistent with the input-output channel model defined by DPI, and the applicability of DPI in this context is limited.

Alternative 1 - Manual filling: Manual annotation lacks scalability. CoT-Bridge learns transferable structural patterns, requiring only 5 minutes for ScaleQM+ construction and 10 hours for fine-tuning, then efficiently processes 1 million CoTs in ~5 hours. Manual annotation of datasets like MATH500 requires ~100 expert hours, making our automated approach orders of magnitude more efficient while maintaining broad task transferability.

Alternative 2 - Direct training on ScaleQuestMath: As mentioned in Section 2.2, lines 128-132, ScaleQM+ is constructed from a 588k subset of ScaleQuestMath. Therefore, we fine-tuned Qwen2.5-Math-1.5B on ScaleQuestMath-588k. Combining the results from Table 1 and Table 2, we can observe from the table below:

1. Direct training on ScaleQuestMath 588k underperforms NuminaMath-bridge,
2. 72B-distill SFT outperforms the ScaleQuestMath subset, suggesting that CoT-Bridge is capable of further enhancing data of even higher quality.

Results on Qwen2.5-Math-1.5B:

Question 1

Response to Question 1:

Thank you for the insightful question. We agree that augmenting data with intermediate steps increases the number of training tokens, and this could potentially benefit performance due to scaling laws. However, we believe the gains from CoT-Bridge are not simply due to more tokens.

In fact, the Qwen2.5-Instruct-72B zero-shot bridge (QwenBridger-L) can be viewed as adding extra tokens to the basic baseline (Direct SFT) without structural optimization. It serves as a good control to examine whether token count alone explains the improvement.

As shown in the table below, the number of additional tokens introduced by Qwen2.5-Instruct-72B is comparable to CoT-Bridge (even greater on NuminaMath):

Despite having more tokens, QwenBridger-L consistently underperforms compared to CoT-Bridge (see Table 1). This suggests that it’s not just the quantity of tokens that matters.

Question 2

Response to Question 2:

Thank you for the thoughtful question. To clarify, we do not assume that ScaleQM+ is entirely free of Thought Leaps. Rather, our assumption is that ScaleQuestMath offers relatively complete and structurally coherent CoTs compared to many existing datasets. We plan to clarify this in a future version of the paper.

When constructing ScaleQM+, we deliberately introduce Thought Leaps by systematically removing intermediate steps. In doing so, our goal is not to create a dataset without Thought Leaps, but rather to ensure that any Thought Leaps we simulate exhibit finer granularity, making the detection of leaps and the transfer of relatively complete reasoning patterns more effective.

Regarding the alternative approaches you mentioned, we have addressed them in our response to Weakness 3 above.

Thank you again for your insightful review comments, especially the new perspective from information theory. If you have any further questions, we are very willing to discuss them with you.

Sincerely,
CoT-Bridge Authors

Dear Reviewer Wqes:

Thank you very much for taking the time to read our response. We are glad that our additional clarification and experiments have addressed most of your concerns. We appreciate your thoughtful follow-up question about the performance comparison between ScaleQuestMath 588k subset and NuminaMath. This seemingly counter-intuitive phenomenon you identified is precisely what motivates our work. Considering the characteristics of the two datasets, we can derive the following insights:

• While the overall average performance of ScaleQuestMath is slightly lower than NuminaMath (52.66% vs. 52.87%), this gap mainly comes from three competition-level benchmarks: Odyssey, Olympiad, and AMC23 (46.77%, 33.04%, 32.50% vs. 45.48%, 32.44%, 30.00%). As NuminaMath contains a large number of high-difficulty competition problems, its stronger performance on these tasks is expected.

• ScaleQuestMath performs better on basic datasets such as GSM8K and MATH (85.67%, 65.20% vs. 83.62%, 63.90%). This indicates that for simpler problems, its more structurally complete reasoning chains (indirectly reflected in the average token length below) offer a clear advantage. However, ScaleQuestMath hits a performance bottleneck on harder tasks due to limited coverage of advanced competition-level knowledge.

Therefore, the training performance of the ScaleQuestMath subset is not necessarily better than that of the bridged datasets. This is where CoT-Bridge demonstrates its unique value:

• Rather than simply training on ScaleQuestMath directly (which would inherit its knowledge limitations), CoT-Bridge learns to identify and bridge thought gaps—the universal thinking patterns of how reasoning steps should connect.

• CoT-Bridge is not just transferring static knowledge—it's learning generalizable patterns of reasoning coherence that enhance any dataset it's applied to. NuminaMath-Bridge (56.26%) outperforms both the original NuminaMath (52.87%) and ScaleQuestMath subset (52.66%), with particularly strong gains on competition benchmarks where ScaleQuestMath alone struggles.

Thank you for this insightful question, which has allowed us to clarify the fundamental value proposition of our work. If you have any further questions, we are very willing to discuss them with you.

Sincerely,
CoT-Bridge Authors

Dear Reviewer 9f3z:

Thank you for your time and effort in reviewing our paper, as well as for the valuable feedback you provided. Below, we present our responses to the mentioned weaknesses.

Weakness 1: 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.

Response to Weakness 1:

We thank the reviewer for this insightful observation about position-aware generation improvement. Your comment prompted us to conduct a deeper analysis. As shown in Table 5, Section 4.4, removing middle bridging consistently causes the largest average performance drops across both datasets:

• MetaMath: -1.80% (compared to -0.76% for begin and -0.78% for end positions)
• NuminaMath: -3.41% (compared to -2.37% for begin and -3.04% for end positions)

This indicates that middle position bridges contribute most significantly to the model's effectiveness.

When conducting further position-based analysis on the results in Table 3, Section 4.2 (Thought Leap Bridge Task), we found that CoT-Bridge achieves higher recall rates for identifying thought gaps at begin and end positions (92.1%). This implies that the accuracy for identifying thought gaps in middle positions is relatively lower. This performance difference is expected since begin and end positions typically have more distinct structural characteristics, while middle positions require more nuanced understanding of logical flow.

Based on these insights, we recognize that to improve position-aware generation, we need to enhance the CoTBridge's ability to precisely identify thought gaps in middle positions. To address this, we implemented a position-aware enhancement specifically targeting middle positions by incorporating a middle-position identification loss on top of the original SFT loss when training CoT-Bridge. This additional position loss is formulated as a binary cross-entropy loss focused on middle positions:

$L_{middle-position} = -\sum_{i=1}^{n-2} [y_i \log(p_i) + (1-y_i) \log(1-p_i)]$ $L_{new} = L_{SFT} + \lambda \cdot L_{middle-position}$

where $y_i$ indicates whether a gap exists between steps $i$ and $i+1$, $p_i$ represents the probability of gap position predicted (obtained from the model's output distribution when generating the token $i$), $L_{SFT}$ is the original SFT loss used for CoT-Bridge training, and $\lambda = 0.2$ serves as the weighting hyperparameter.

Due to the limited time available during the rebuttal period, we could not conduct the same comprehensive experiments as in our main study. However, we performed preliminary experiment following Section 4.2 on the Thought Leap Bridge Task. As shown in the table below, the enhanced approach demonstrates moderate but consistent improvements across all metrics. This provides the community with a potential direction for Position-Aware improvement.

Weakness 2: 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.

Response to Weakness 2:

Thank you for this insightful observation. As discussed in Section 4.5 (lines 296-299), we hypothesized that: (1) the proportion of noise introduced by CoT-Bridge is already very low (Table 10 & Figure 4), and (2) the PRM model may misjudge certain steps in complex reasoning scenarios.

To investigate hypothesis (2) more deeply, we manually labeled 100 randomly sampled low-PRM bridged steps (PRM<0.5) and categorized them into four types:

• Steps with genuine errors (19%)
• Completely correct steps (18%)
• Reflective/inspirational steps that provide valuable reasoning guidance (11%)
• Steps that are mathematically correct but have suboptimal linguistic coherence (52%)

This distribution reveals that 81% of low-PRM steps still contain useful information, explaining why removing them may not improve performance. In particular, We found that CoT-Bridge sometimes generates valuable reasoning patterns (Reflective/inspirational steps that provide valuable reasoning guidance, 11%) that receive low PRM scores. As shown in the case study below from MetaMathQA, the original CoT contains a logical error claiming "the only possible value of n that satisfies $1<n<10$ is 61". Bridge3 added by CoT-Bridge attempts to reconcile this inconsistency by reconsidering the constraints. However, Bridge3 received a low PRM score (0.3086). We plan to include these statistical results and case study in the paper to provide more insights, and we sincerely thank you for your careful observation that led to this investigation.

Case Study: (from MetaMathQA)

Question: Natasha has more than $\$1$ but less than $\$10$ worth of dimes. When she puts her dimes in stacks of 3, she has 1 left over. When she puts them in stacks of 4, she has 1 left over. When she puts them in stacks of 5, she also has 1 left over. How many dimes does Natasha have?

Solution: 
(Bridge1) To determine how many dimes Natasha has, we can use modular arithmetic and the Chinese Remainder Theorem. [PRM score=1.0000]

Let $n$ be the number of dimes Natasha has.
We know that $1<n<10$.
From the given information, we can write the following equations:
$n\equiv 1\pmod{3}$,
$n\equiv 1\pmod{4}$,
$n\equiv 1\pmod{5}$.

We can solve this system of congruences using the Chinese Remainder Theorem.
Since the modulus values 3, 4, and 5 are pairwise relatively prime,
there is a unique solution modulo $3\cdot 4\cdot 5=60$.

By inspection, we find that $n\equiv 1\pmod{60}$ satisfies the congruences.

(Bridge2) We can write this as:

$
 n-1 = 60k 
$

for some integer $ k $. This implies:

$
 n = 60k + 1. 
$
 [PRM score=0.9609]

Therefore, the possible values of $n$ are $1, 1+60=61, 1+2(60)=121, 1+3(60)=181,\dots$.
Since the only possible value of $n$ that satisfies $1<n<10$ is $\boxed{61}$,

(Bridge3) However, we need to find the values of ( n ) that lie in the range ( 1 < n < 10 ). The only value that satisfies this condition is ( n = 1 ), but since ( n ) must be greater than 1, we need to check the next possible value, which is ( n = 61 ). Since each dime is worth $0.10, so actually we have ( 10 < n < 100 ), [PRM score=0.3086]

we conclude that Natasha has 61 dimes.
The answer is: 61

If you have any further questions, we would be happy to answer them.

Sincerely,
CoT-Bridge Authors

Dear Reviewer 9f3z:

We hope this message finds you well. As the discussion period is gradually approaching its conclusion with only a few days remaining, we would like to kindly follow up to ensure that all of your concerns have been fully addressed.

We have carefully responded to your thoughtful review and constructive feedback in the detailed rebuttal above, including:

• Position-aware generation improvements, with a dedicated middle-position identification loss function showing consistent metric improvements (+1.68% overall)

• Comprehensive analysis of low-PRM steps, revealing that 81% still contain valuable information despite low scores

• A detailed case study illustrating how steps with low PRM scores contribute to the reasoning process

We would greatly appreciate it if you could take a moment to review our responses, particularly the additional experimental results and analyses that directly address your concerns regarding position-aware generation and the counter-intuitive behavior of low-PRM steps.

Your insights have been instrumental in improving our work, and we remain fully committed to addressing any remaining concerns you might have.

Sincerely,
CoT-Bridge Authors

Dear Reviewer H9iA:

Thank you for the effort and time you dedicated to reviewing our paper. We have greatly benefited from your valuable feedback. Below are our responses to the weaknesses you pointed out.

Weakness 1: issue of step segmentation

Response to Weakness 1:

Thank you very much for pointing out the issue of step segmentation. We acknowledge that step segmentation remains a challenging open problem. Actually, our use of "\n\n" as delimiters is a practical approach, and many studies[1,2,3] have adopted this method as well.

To validate the use of "\n\n" for step segmentation, we conducted both human and GPT-4o-based step segmentation on 100 randomly sampled examples from NuminaMath. We calculated the Alignment Rate as $|S_{nn} ∩ S_{ref}| / |S_{ref}|$, where $S_{nn}$ represents step boundaries identified by "\n\n" delimiter and $S_{ref}$ represents step boundaries identified by human or GPT-4o. The results below indicate that the segmentation method using "\n\n" shows a high level of alignment with both human judgment and advanced LLMs in identifying logical steps.

[1] The Lessons of Developing Process Reward Models in Mathematical Reasoning
[2] LLMs Can Easily Learn to Reason from Demonstrations Structure, not content, is what matters!
[3] SEAL: Steerable Reasoning Calibration of Large Language Models for Free

Weakness 2: 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.

Response to Weakness 2:

Thank you very much for your kind words about the clarity of our writing.

Regarding domain generalizability:
While we primarily focus on mathematics, as it is one of the most common forms of structured reasoning tasks, our approach is domain-agnostic. Thought leaps could occur across any domains that require structured reasoning, such as science, law, and medicine.

Our experiments in Section 4.3 support this claim: models trained on mathematical datasets show a +2.99% improvement on OOD logical reasoning tasks, confirming the potential of our approach to transfer across domains. Future work could involve training domain-specific models for detecting and briding thought leaps in specialized fields such as science, law, and medicine.

Regarding strong teacher model:
We would like to respectfully clarify that CoT-Bridge is designed as a specialized model rather than a more powerful one. On the benchmark MATH, Qwen2.5-Instruct-72B achieves 83.1[4], while CoT-Bridge's predecessor (Qwen2.5-Math-7B) achieves only 55.4[4]. However, despite this mathematical capability gap, our 7B CoT-Bridge model significantly outperforms the 72B model on the Thought Leap Bridge Task (overall score: 76.15 vs. 31.12). Additionally, as shown in Table 1 of our paper, datasets bridged by CoT-Bridge yield better SFT results than those bridged by zero-shot Qwen2.5-Instruct-72B. This striking contrast occurs because the Thought Leap Bridge Task differs from generating reasoning steps from scratch - it requires specialized capabilities to identify and fill logical gaps within existing solutions rather than raw mathematical power.

[4] Qwen2.5 Technical Report

Weakness 3: 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.

Response to Weakness 3:

Thank you very much for pointing out the potential issue. Due to the limited time during the rebuttal phase, it was challenging for us to conduct experiments on larger models as thoroughly as our main experiments. However, we did perform SFT experiments on Llama3.1-70B using MetaMathQA and MetaMathQA-Bridged. As shown in the table below, we acknowledge that the relative performance gains do narrow when applied to larger, more capable models (Llama3.1-8B +1.35%, Llama3.1-70B +1.02%). However, bridged datasets still consistently outperform the original ones across all benchmarks.

Question L196 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.

Response to Question:

Thank you for your insightful question. Due to limited experimental budget and time for rebuttal, we were unable to use GPT-4o or Gemini 2.5 Pro to bridge entire datasets. However, following Section 4.2, we randomly sampled 100 instances in ScaleQM+ test set and evaluated them on the Thought Leap Bridge Task.

The results below show that zero-shot bridging using models with stronger instruction-following capabilities (GPT-4o and Gemini 2.5 Pro) achieve significantly better bridging performance than Qwen2.5-Instruct-7B/72B. However, this can lead to a relatively high API cost.

Case Study:
Thank you for your suggestion. This will enable readers to better understand zero-shot bridging. We plan to incorporate a case study in a future revision of the paper. 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.

As can be seen in the case study below:

• Although a planning-style bridge between the question and step 1 was missing, GPT-4o, Gemini-2.5-PRO, 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.

# Original Problem (Step 0) and Solution
Step 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?
Step 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 2:
By the pigeonhole principle, to ensure at least one sum appears twice, we need to roll the dice 15 + 1 = 16 times.
Step 3:
Thus, to guarantee that the same sum appears twice when rolling two eight-sided dice, the dice must be rolled $\boxed{16}$ times.

# 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:

$$
16 - 2 + 1 = 15 \text{ distinct possible sums}.
$$

If you have any further questions, we would be happy to answer them.

Sincerely,
CoT-Bridge Authors