Mind the Gap: Bridging Thought Leap for Improved Chain-of-Thought Tuning
Authors: Haolei Xu
Yuchen Yan
Yongliang Shen
Wenqi Zhang
Guiyang Hou
Shengpei Jiang
Kaitao Song
Weiming Lu
Jun Xiao
Yueting Zhuang
Venue: NeurIPS 2025 poster ()
Keywords: Large Reasoning Models
LLM Reasoning
Primary area: deep_learning

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
Haolei Xul
Yuchen Yan'+
Yongliang Shen
Wenqi Zhang'
Guiyang Hou' Yueting Zhuang
Shengpei Jiang?
Kaitao Song
Weiming Lu't
Jun Xiao'
Zhejiang University SF Technology
3 Microsoft Research Asia
(xuhaolei,syl) bum
Project: https://zju-real.gibbhe wtw.giihttt..ooorrrrr......
Abstract
Large language models (LLMs) have achieved remarkable progress mathemati- cal tasks through Srain-of-Thougee (CoT) reasoning. However, existing mathe- matical 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 automatically detect leaps and generate missing intermediate reasoning steps to restore the complete- ness and coherence of CoT. To facilitate this, constructed specialized training dataset called ScaleQM+, based on the structured ScaleQuestMath dataset, and trained CoT-Bridge 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 to +5.87% on NuminaMath. Our approach effectively enhances distilled data (+3.02%) and provides better starting points for reinforce- ment learning (+3.1%% functioning as a plug-and-pla module compatible with existing optimization techniques. Furthermore, CoT-Brrdge demonstrates improved generalization to out-of- logical reasoming tasks, confirming that enhancing reasoning completeness yields broadly applicable benefits.
Introduction
Large Language Models (LLMs) 11 have demonstrated significant performance improvements through Chain-of-Thouuht (CoT) reasoning 77,, particularly on complex tasks [ such as math- ematies [10] and coding. CoT guides models generate intermediate reasoning steps, simulating the human process of step-by-step problem solving, thereby enhancing both solution accuracy and interpretability (T]] The quality of these reasoning chains directly impacts model performance. serving as a critical foundation for advanced reasoning capabilities.
Despite substantial progress, identify prevalent but understudied phenomenon in CoT datasets [2] Thought Leap. This refers to instances where one or more intermediate rea- soning steps omitted between adjacent steps, creating cognitive gaps in the reasoning chain. Unlike factual errors or answer inaccuracies that have been extensively studied, Thought Leap specif- ically concerns the completeness of reasoning structures. Figure [ (a) illustrates this phenomenon through a mathematical problem where critical bridging steps (highlighted in gold) are missing in the
Equal contribution.
f
Corresponding Author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Figure I: Overview of the Thought Leap phenomenon and our bridging approach. (a) Thought Leaps in CoT; (b) Negative impact on training; (c) Bridging leaps improves reasoning performance.
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 demon- strations. Human experts often omit steps they consider trivial based their extensive background knowledge, unconsciously creating gaps that can impede effective learning. This phenomenon is par- ticularly 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 (b), models trained on datasets with Thought Leaps exhibit substantially lower performance ceilings (up to 27.83% reduction in accuracy for severe leaps, details in Appendix 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 auto- matically detect Thought Leaps and generate the necessary intermediate reasoning steps to restore coherence and completeness in CoT demonstrations. As illustrated in Figure (a), approach effectively transforms incomplete reasoning chains into coherent step-by-step solutions by insert- ing 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 fine-tuned model based on Qwen2.5-Math-77 Thought Leaps in mathematical reasoning. Finally, we apply CoT--Brigge to existing mathematical reasoning datasets enhance their completeness and coherence, thereby improving the quality of training data for downstream models.
20 specifically designed to identify and bridge
Our experimental results demonstrate that models fine-tuned bridged datascts achieve significant performance improvements compared to those trained on the original datasets with Thought Leaps. As illustrated in Figure (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 plug-and- enhancement module that can be integrated with other advanced techniques. As shown in Figure (c), it can improve the quality of distilled

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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 model (CoTTBrrigge) 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-of- domain logical reasoning benchmarks (+2.99%).
We validate that our approach can function as plug-and and-play enhancement module, com- patible with methods such as knowledge distillation and reinforcement learning, further amplify model performance.
2 Method
In this section, we provide a detailed discussion of our approach. Section 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 brief introduction variant method. Section 2.3 outlines the process of applying the bridge model to existing aep-by-step datasets.
2.1 Task Formalization
Let C* (Q, si ss..... (n) represent ideal complete CoT starting from question Q. In C* the transition from Q to sj. as well as transitions between any adjacent steps , 1)+) are both explicit and coherent. Defining a completeness function V, for C* we have V(Q, si) True and 1), Vi E [1, m 1], Vs 1 = True. This function captures whether the reasoning transition between adjacent steps is sufficiently detailed and logically sound.
Now consider a CoT C = (so, 51, $2,..... 5n) that successfully derives an answer A (for convenience, we denote question Q as 5o). If there exists at least one pair of adjacent steps (sk, 8k++1) within C' that fails the completeness criterion, i.e., (( 5k+1) = False, then a Thought Leap exists between Sy and $4+1- Such incompleteness indicates that necessary intermediate reasoning steps S' (sk.1: $'........ ,. sk. where j 2 1, have been omitted between S and 8k+1- These omitted steps must satisfy three conditions: (1) completeness from 83 to the first missing step: V(sk, sk.1) = True;
(2) internal completeness within the missing sequence: Vi E [1, 1), V( sx11) ) = True; and
(3) completeness from the last missing step to $K+12 (( Sk+1)
True.
Based on this formalization, we define the CoT Thought Leap Bridge Task as two-stage process: first identifying all adjacent step pairs (sk, Sk+1) within C that satisfy V(s,,8++1) 8k+1) then for each identified leap, generating the missing intermediate step sequence Sy to bridge the gap as $& ① Smiss @ Sk+1. This task directly addresses the completeness aspect of reasoning, which complementary to the more commonly studied factual accuracy dimension.
False, and
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 ScaleQuest- Math 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* (so, , *tn) where 8q = Q is the question, and strategically remove intermediate steps to produce incomplete chains C C= (sp, .1..... 8n) containing Thought Leaps. The removed steps form the reference set of missing intermediate steps needed to bridge these leaps.
3

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Data Construction Training
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Figure 2: Illustration of our work. The left panel shows data construction for training, where we strategically remove intermediate steps (e.g., between Step 0 and Step I, or Step 2 and Step 3) from complete reasoning chains in ScalcQuestMath to create ScaleQM+ with Thought Leaps. The right panel demonstrates inference, where CoT-Bridge identifies gaps and generates appropriate intermediate steps to restore coherence in reasoning.
Our step removal strategy follows several principles to create effective training examples. We always retain the final step 8m to preserve the completeness of the answer, while allowing the initial reasoning step si be removed to help the model learn macro-planning skills. The number of steps remove, kdel, scales with the length of the original chain: for shorter chains (m S 10), we remove 1-2 steps; for longer chains (m 10), we remove 1-3 steps. Additionally, with a probability of 0.2, 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 Lgt identifying where step deletion resulted in reasoning gaps, and the corresponding ground-truth missing steps Mgt consisting of the original removed steps. Our primary model, CoT-
Bridge, learns the mapping f C - (2, taking an incomplete chain as input and producing both predicted leap positions and the corresponding missing steps. As baseline for comparison, we also implement generating missing steps given the ground-truth leap positions.
which learns y (C, Lgt) + M, focusing solely on
In practice, we process the ScaleQuestMath dataset by segmenting reasoning chains using "In'n" as delimiters and selecting examples with at least 6 steps (m 2 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 Sive2.55Matth 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 MetaMathQQA and MathinaMath-Co'' creating improved versions named Math-Baige and NuminaMath-Bricn 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 (so, si,..., sn), CoT-Bridge produces set of predicted Necar where each k; identifies a leap between steps 8k; and $k.+1, Neep represents the total number of leaps detected, and Š S is the sequence of generated intermediate steps. For each identified leap, we insert the generated steps between the corresponding original steps, performing Sk, ① Smiss,k ① sk,+1 to create the bridged reasoning chain Còridged-
Thought Leap repairs {k;, (ki,,'nisss S''iiss,,, ))
To accommodate the format differences across datasets, we adapt our approach to the specific step delimiters used: "An" for MetaMathQA and "Inin" for NuminaMath-Cofl. For the CoT-Bridge- Random variant, provide randomly selected positions Krandom instead of model-identified leaps,

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allowing us to evaluate the importance of accurate gap identification in improving reasoning quality. This approach enables 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 fine- tuning (SFT) experiments using MetaMathQA, NuminaMath-Col datasets and their bridged versions on representative base models. Specifically, we selected Meta-Llama3. 1--B (1 as a representative general-purpose model, and Qwen2.5-Mathh 1.5B [20] as 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.I
3.2 Evaluation
We employed six benchmarks: GSM8K [15.. MATH500 22 and GaoKao2023EN 23 as basic- level benchmarks, and MathOdyssey (24), OlympiadBenchEN 25] and AMC23 266 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
establish a series baselines for comparison. The most important baseline standard SFT performed directly on the original datasets. In addition, we use zero-shot bridging which method leveraging general-purpor purpose LLMs (in this paper, Qwen2.5--nstrrutt7BBBB
(28) to explore Thought Leap bridging without direct training. This baseline aims to assess the effectiveness of pure prompt engincering in bridging Thought Leaps, determining whether specialized detection and repair mecha- nisms are necessary. Furthermore, use CoT-Bridge-Randee randomly chosen positions, to evaluate the importance of accurate Thought Leap localization for effective bridging. As reference, consider the performance of base models in a 4-shot setting. Additionally, we present the results of SFT using GSMSK+MATH and MathInstruct.
to generated intermediate steps
3.4 Main Results
Table
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 CoT- Bridge with Direct SFT, we observe substantial improvements across almost all configurations. The with Meta-Llama3. 1-8B tuned NuminaMath, where CoT--Brigge achieves an average improvement of +5.87%, with particularly impressive gains on competition-lewe MetaMathQA, CoT-
most significant gains
benchmarks (+15.63% AMC23). Similarly, for Owen2.5-Math- Bridge 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 CoT- Bridge-Randon Random reveals the importance of precise gap detection. While CoT-Bridge consistently improves performance, CoT-Bridge-Randee Random shows highly variable results, degrading performance on certain benchmarks. For instance, with @wen2.5-Math- 1.5B on NuminaMath, CoT-Bridge-Randee 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

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Basic Level
Competition Level
Dataset
Size
Method
Average
GSM8K
MATH
GaoKao
Odyssey
Olympiad AMC23
Meta- Lamta-Llama3.1-8
4-shot
54.15
18.30
20.58
16.54
11.25
20.95
GSMRK-MATH
Direct SFT
65.09
21.69
18.48
12.50
23.68
Mathlnstruct
262k
Direct SFT
68.16
23.60
25.52
25.06
5.89
7.50
25.96
Direct SFT
78.90
36.10
32.86
24.68
8.48
17.50
33.09
QwenBridger-S
81.10
34.85
30.52 23.4
22.67
9.26
7.50 10.00
30,98-211
MetaMathQA
395k
QwenBridger-LL
80.80 1.00
38.05
1195
31.43
1.43
24.48 9:20
9.37 9.29
2.50 15,00
31.11 1.95
CoT--Bridg-R
80.46 1.56
38.05 1,95
33,57
10.71
24.42
9:26
9.37
12.50
5.00
33.06 0.03
CoT-Bridge
81.14
38.15 2.05
33.12 (0.36
25.97
9.48 9.48
18.75 1.25
34.44 1.35
Direct SFT
84.86
51.45
49.03
36.56
21.30
20.00
43.87
QwenBridger-S
84,23
51,95 52.40
39.73-X11
24.70
27.50
46.75
NuminaMuth
859k
QwenBridger--L
85,25-0
54.20
51,62
39.08
25.33
35.00
48,41
CoT-Bridge-R
84.82
54.20
51 11.88* 51.88
40.12
26.15
33.75
48.50
CoT-Bridge
85.97
56.80
54.42
40.76
24.85
35.63
49.74
Qwen2.5-Math-LL5.
4-shot
79.00
48.05
45.52
38.18
19.07
22.50
42.05
GSMRK+MATH Mathlostruct
Direct SFT
74.45
47.66
38.50
17.44
27.50
42.83
262k
Direct SFT
70.96
48.90
46.49
40.89
16.22
20.00
40.58
Direct SFT
81.01
46.62
38.63
18.19
21.25
42.55
QwenBridger-S
81.58-055
51.30"
46,60
+0.07
38.63 4.00
18.52-0.8
+1.8
44.04*
44.04-149
MetaMathQA
395k
QwenBridger-LL
81.01 0.00
53.63 1403
49.22
1200
38.70 9000
0.06
27.50
16.25
44.82 /2.27
CoT--BridgeR
81.58 0.57
53.65
1405
48.12
1.30
39.02
10.99
18.22 0.03
25.63
14.38
44.37 11.52
CoT-Bridge
81.39-0.4 0.38
7.00
49.61
(2.99
39.66 1.00
19.44 1:25
28.75
7..50
45.91 1336
Direct SFT
83.62
63.90
$7.40
46.77
33.04
32,5
$2.87
QwenBridger-S
83,23
58.25 63.20
46.96-019
32.04-1.00
35.00
53.28
NuminaMuth
859k
QwenBridger-LL
82.81 -9.61
66.25
57.79*
46.64-
32.70
40.00
54.37
CoT-Bridge-R
83,06
65,20
55.88
43,09 1,6
33.89
40.00
53.51
CoT-Bridge
84.61
68.05
59.29
47.16
34.11
45.00
56.26*h3
Table Main results (%) on mathematical benchmarks. MATH, GaoKao, Odyssey, and Olympiad correspond to the MATH500, GaoKao2023EN, MathOdyssey, and OlympiadBenchEN benchmarks, respectively. QwenBridger-S and QwenBridger-L represent zero-shot bridging based on Qwen2.5- Instruct-7B and I-strrucct- respectively. CoT-Bridge-R stands for ST-BBridgeRannom.
+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 demon- strates certain effectiveness on specific tasks, its overall performance still falls short compared to CoT-Bridge. Taking the I.IaMA + 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 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 pronounced when using the weaker 7B model. In contrast, CoT- BoT-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 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 to two representative scenarios: (1)
improving the quality of generated data in knowledge distillation and rejection sampling, and

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Dataset
Method
GSM8K
MATH
GaoKao
Odyssey
Olympiad AMC23
Average
Direct SFT
81,86
68.
60.84
48.13
33.00
39.37
55.23
Distill
CoT-Bridge
82.52
71.50
66.43-5.9
49.16
34.89*1.0
45.00
58.25
Direct SFT
83.36
74.90
64,94
51.81
37.63
50.00
60.44
Reject Sampling
CoT CoT-Bridge
83.74
75.25 10.35
67.47
51.87
-0.0
39.41 1177
53.13 3311
61.81
Table 2: Model performance after bridging distilled and rejection-sampled data.
assessing whether models fine-tuned CoT-Bridge-enhanced data perform better in subsequent RL stages compared to models trained on original data.
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 Table models fine-tuned on the bridged data achieve superior performance: average accuracy improved from 55.23% (distill) / 60.44% (rejection sampling) to 58.25% / 61.6 .81% %, respectively.
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Owen2.5 wwa.2-5-ttthuuuu Numno
Owen2 Owen2.5#ath
NATHOOK Arcuncty
74
C.T-Bridac
70
82
CoT-Bridac
Queel. 1-00e-----
100
150
200
GRPO Training Step
Figure 3: Model accuracy over train-
Figure 4: PRM scores of
on CoT-
ing steps MATH500.
Bridge for MetaMathQA and NuminaMath.
In the reinforcement learning setup, continued training the (0en2.5-MMath11 models that were fine-tuned on NuminaMath and NuminaMath-Briin
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 shows the training curves of models. demonstrating that our method provides both higher starting point and a better final performance. As shown Table 9 from Appendix G.2]
the model trained with NuminaMath -Bidge achieved an RI. accuracy of 63.98% outperforming that trained with NutinaMath at 60. 88%. This result also surpasses the officially released Qwen2.5- Math-Instruct-L.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 D As shown in Table CoT-Bridge significantly outper- forms other methods in localization precision (78.02%) redundancy rate (1.61%) and overall metric (76. 15%). notably outper- forms 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-Randon to generate candidate content between all adjacent steps, filtering them using similarity thresholds. While this approach demonstrates high recall capability,
Position
Method
Similarity
Overall
Pret
Rect
Red_
14.15
12.04
34.13
10.54
33.99
33.64
33.73
31.12
CoT-Bridge
/
78.02
78.37
1.61
76.15
20.96
79.64
79.04
75.72
0.95
23.42
75.07
76.57
71.22
Full-position
0.90
24.81
59.21
74.37
55.9
0.85
19.34
30.17
63.87
28.26
0.80
7.47
8.65
32.51
7.84
Table Performance of different bridging methods on ScaleQM+ test set. Pre stands for Precision, Rec stands for Recall and Red stands for Redundancy.
Dwen2.55IInsrruee
7

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it exhibits an extremely high redundancy rate. The substantial redundant content may cause LLMs 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 practice.
4.3 Evaluation of Out-of-Domaal Reasoning Capability
To verify the generalization capability of our method Out-of-Domain Domain(OOD) reasoning, we con- ducted evaluations five out-of-domain logical reasoning datasets (FOLIO [II, LogicQA [32], ProofWriter [33. ReClor [34], RuleTTker [35] that were not used in training. These datasets cover formal logic, factual deduction, and textual entailment reasoning types. In the evaluation,
used XFinder (36] to automatically extract answers from model responses, with outputs that failed to yield valid answers (such those deviating from the prompt or exhibiting logical loops) counted as errors and separately tallied.
Bridge Method
Mertic
FOLIO
LogicQA
PW
ReClor
RuleTaker
Average
NuminaMath+ -atttt---aa-----a.-aaaa L
Accuracy
68.15
34.33
59.09
54.5
52.61
No Bridge
Invalid
1.48
6.3
0.51
0.31
2.12
Accuracy
74.07
35.64 1331
61.52
50.20
56.57 0.1
55.60
GapBridge
Invalid
0.74
4.92
0.2
2.2
1.63
Accuracy 74.07
29.72
55.84
37.6
53.05
50.06
No Bridge
Invalid 1.48
4.53
0.3
1.2
0.72
1.65
Accuracy 71.11-200
33.10
58.38
39.00 10
53.67
51.05-0.9
GapBridge
Invalid 1.48
3.46
0.3
1.4
0
1.33
Table 4: Performance of NuminaMath and its bridged version on logical reasoning benchmarks.
As shown in Table we compared the performance of models trained on NuminaMath and NuminaMath- Results indicate that CoT-Bridge improved OOD reasoning performance 1-8:: 12.99%. 1eeen..5-Math-1.55 10.99%) while reducing invalid response rates. We attribute the performance improvement to the refined intermediate steps introduced by CoT Toi-Bridge, which help models master general reasoning structures and enhance generalization capabilities.
4.4 Analysis of Bridging Positions
Leap bridging enhance reasoning capabilities of LLMss, but whether complementary content bridged at different structural positions contributes substantially still lacks systematic analysis. Therefore, 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 relatively small proportion (approximately 2%). To eval- uate 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 -riddri data while keeping the rest unchanged, and fine-tuning 55 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.
under identical training settings. As shown in
4.5 Process Supervision Scoring and Noise Impact Analysis
To measure the bridging quality of various methods, we introduced the process supervision model 7 to score the intermediate steps (range 0-1) and analyzed the score distribu-
Qwen2.5-Math PRM-7B

---

Delete Pos
Num
Ratio
GSM8K
MATH
GaoKao
Odyssey
Olympiad AMC23
Average
Qwen2.5 M----attt----MaMMaale
/
81.39
56.60
49.61
39.66
19.44
28.75
45,91
begin
225873
31.84
82.71 1:33
55.85
0.75
49.55
0.86
41.09
19.22
22.50
6.25
45.15 0.76
middle
470541
66.34
80.97
0,42
52.45
4.15
49.35
0.25
37.14
2.52
17.26
2.18
27.50
1.25
44.11 1.80
end
12908
1.82
81.14 025
55.75
0.85
49.94
37.02 2.66
18.78 0.66
28.12
-053
45.13
Dwwen---Nath--55--Nmminaaa
/
84.61
68.05
59.29
47.16
33.44
45.00
56.26
begin
694887
42.57
82.60 2.01
64.85
58.67 -9.62
45.48
1.68
32.96
38.75
6:25
53.89 2.37
middle
906168
55.52
83,51
63.55
56.43
47.16
0.00
28.30
5:14
38.13-687
52.85-344
endd
31115
1.91
82.79 1.82
64.05
4,00
55.97
46.51
10:5
27.48
5.95
42.50
2.50
53.22 3.04
Table 5: Performance variation after removing each bridging component.
Threshold GSM8K MATH
GaoKao
Odyssey
Olympiad AMC23
Average
MetaMaath-Qween255-Mtttt
/
81.39
56.60
49.61
39.66
19.44
28.75
45,91
pmm 0.1
81.22
57.00
50.45
37.40-206
19.70
29.38-063
45.86-006
prm < 0.3
80.99 0.40
56.90
51.88
37.79 1187
18,33-1a1
32.50 3.75
46.40 -049
pmm
0.5
80.53
54.75
49.09
37.86
18.78
25.00
44.34
/
84.61
68.05
59.29
47.16
33.44
45.00
56.26
prm < 0.1
83.28 1.33
65.50
2:55
59.09
0.20
46.
0.97
34.33 00.89
41.88
3.12
55.04
1.22
pm < 0.3
83.85
0.76
63.00
5.05
59.35
46.38
0.78
33.07
0.37
45.00
0.00
55.11
1.15
prm < 0.5
83.26
1.35
65.95
2.10
58.05
124
46.06
1.10
33.56
+0112
41.88
3.12
54.79
1.47
Table Performance after removing low-scoring steps (evaluated by 1wwen.55-Math-PRM--
tion. Table in Appendix K and 4 the PRM score distribution of the bridged steps. Results show that steps generated by CoT Cridge 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, CoTT-Brigge's proportion in the low-quality (range 0-0.1) 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[]
To further verify whether oww-quality steps generated by CoT-Bridge affect model training effective- ness, we set different thresholds based on PRM scores to denoise the bridged data and conducted SFT training on Qwen2.5-MM. --tthhhh 1.5t accordingly. As shown in Table results indicate that removing low-scoring steps has limited impact on model performance, suggesting that noise introduced by bridging has minimal effect training. On the NuminaMath dataset, after removing steps with PRM scores below 0. accuracy decreased from 56.26% to 55.04%; while on MetaMattQA, 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 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.
Acknowledgement
This work is supported by the National Natural Science Foundation of China (No. 62436007), the Key Research and Development Program of Zhejiang Province, China (No. 2024C01034), the Fundamental Research Funds for the Central Universities (226-2024-00170), MOE Engineering Research Center of Digital Library, CCF-Baidu Open Fund and ZJU Kunpeng&Ascend Center of Excellence.
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13

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A Related Works
A.1 Methods for Enhancing Mathematical Reasoning
Researchers have explored various reasoning enhancement methods based CoT, broadly categorized into inference-time methods and training-basee methods.
Inference-time approaches improve reasoning by adopting advanced decoding stralegies 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-Thougee (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 sl budget forcing (41). dynamically increase
computation boost performance. addition, inference-time intervention methods performance by manipulating hidden states.
enhance reasoning
Training- based upproaches internalize enhanced reasoning abilities through parameter updates during model train- ing. A common strategy involves fine-tuning models on datasets with explicitly detailed intermediate reasoning steps. Additionally, training paradigms incorporating self-retlection [4]. self-veriiicttion self-breaking 4:5] and self-correction have gained attention, teaching models to recognize and rectify errors autonomously. Frameworks like StepCo [46]. S2R [47]. and VeVISE [48] integrate these metacognitive capabilities through supervised. reinforcement, or preference learning. In particular, recent advances highlight that reinforcement learning plays an espccially crucial role [49] 52 in enhancing the overall capabilities of large models. We leverage CoT- Bridge enhance the structural completeness of existing datasets, cnabling models acquire finer- grained reasoning patterns during training and 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. MetaMathQQ [22 consists 395k problem-solution pairs derived by reformulating and reverse-engineering existing datasets like GSM8K [5] and MATH [16]. MathInstruct [33 aggregates 13 existing math datasets and employs GPT-4 (53) to synthesize 260k examples containing both Chain-of Thought and Program-of Soagram-ofTThougeee Thought solutions. NuminaMath-CoT [14] compiles data from examinations, competitions, and Q&A communitics, resulting in dataset 860k problem-solution pairs. ScaleQuestMath [19] introduces zero-shot data generation framework for mathematical questions using small open-source models, resulting in a scalable dataset of about I million samples. DAPO 30 developed DAPO-Math-17K 17K, comprising 17k standardized enteger-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
CoT-Bridge can't guarantee that 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 non-negligihle concern.
Potential Noise in Reasoning Steps:
Lack of Validation on Larger-Scale Models: Due to computational resource constraints, our experiments focus primarily small medium-sized models, and we have not yet conducted evaluations 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 mathematics, the method remains largely domain-specific. It is worth noting, however, that CoT-Criiie GoT-Bridge is theoretically generalizable and can potentially 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.
14

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Preliminary Experiment
C.1
Settings
To simulate varying degrees of Thought Leap. created modified versions of the MetaMathQA dataset by removing intermediate reasoning steps. Considering the structure of MetaMathQA, we use "In" segment individual steps within CoT. The specific configurations are as follows:
The final step which contains the explanation is always retained.
Mild, Moderate, Severe Thought Leap: We randomly remove 1, 2, or intermediate steps, respec- tively, if sufficient steps are availahle.
Extreme Thought Leap: All intermediate reasoning steps are removed and only the final answer explanation is preserved.
We adopt the same training configuration described in Section 3.1 using Veta-Llama3.1144 1-8B. Each model trained for 200 steps, with evaluation performed on the GSM8K every 10 steps. For convenience, answer verification is conducted using Math-Verily only.
C.2 Results
We present detailed results from Figure lib) Table
Training Step
Original Data
Mild Thought Leap
Moderate Thought Leap
Severe Thought Leap
Auswer Only
10
8.64
8.79
8.64
8.57
7.35
20
43.67
38.71
33.2
25.78
8.04
40
60.96
55.34
46.4
29.95
15.09
62.77
55.42
50.87
40.33
60.88
50.57
35.71
16.37
70
68.08
63.61
$2.69
35.18
14.86
66.94
64.67
55.57
37.98
69.9
66.03
56.94
40.03
16.22
100
69.06
64.52
60.5
37.76
16.38
120
69.74
66.41
61.03
37.15
15.62
130
69.98
65.22
61.79
36.39
17.66
57.77
37.83
150
69.6
62.47
56.1
29.04
16.22
69.83
66.64
56.03
38.29
17.36
68.61
64.29
54.66
29.72
15.01
180
70.28
65.5
48.75
32.9
15.39
66.03
50.49
27.45
200
67.63
62.77
60.35
39.8
13.65
Table Accuracy of Meta-Llama3.1-81) on GSM8K over training steps using data with varying degrees of Thought Leaps.
15

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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 with warm-up ratio of 0, and cosinc scheduling was used to gradually reduce the learning rate zero. The maximum sequence length
8192 tokens, with global batch size of 128. We trained models for 3 epachs on MetaMathQA [12] and for epochs on NuminaMath-Co! 14. as preliminary experiments indicated further improvement from additional epochs. All SFT experiments were conducted using Ascend H910B-64G..
For SFT training template, we use the default style template in LlamaFactory:
Prompt Template
System: You are math problem solver. You should think step by step.
Human: <Question>
Assistant: <Answer>
D.2 Evaluation Settings
maximum token limit generation was set 2048, 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 Matt-V (60) tool. Given limitations of math-verify in handling complex expressions, responses failing initial verification were subsequently validated using DeepSeck-RI [6]. All model evaluations were performed using NVIDIA A100-40G GPUs.
Referring OpenR
we use following prompt template:
Prompt Template
You are mathematical answer validator. You will be provided with mathematical prohlem and need compare the in the reference solution, and the final answer model's solution determine they are equivalent, even formatted differently.
PROBLEM:
<problem>
REFERENCE SOLUTION:
<answer>
MODEL'S SOLUTION:
<generation>
Focus ONLY on comparing the final mathematical answer provided by the model while ignor- ing differences in:
Formatting (c.g., \boxed() vs plain text)
Multiple choice formatting (e.g.. "A" vs full solution)
Order of coordinate pairs solutions
Equivalent mathematical expressions notation variations
the model's answer is nonsense, return "Verdiet: AMBIGUOUS"
Start with brief explanation of your comparison (2-3 sentences). Then output your final an- swer in of the following formats:
"Verdict: EQUIVALEENT
"Verdict: DIFFERENT"
"Verdiet: AMBIGUOUS"
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E
S Construction and CoT-Bridge training
E.1
ScaleQM+ Template
Prompt Template
System:
You mathematics teacher reviewing solution that may be missing
steps.
Your task to:
1. Identify all points the logical flow where step is missing. For each missing step, specify exactly
between which two consecutive steps it should be placed.
2. Provide the complete missing step(s) with necessary explanations and equations. The solution may be missing multiple steps might be complete. steps in solution are labeled from Step (problem statement) to Step N.
For each missing step. please format your response as follows:: Missing Step X:
The missing step should be placed between Step Y and Step Y+
The missing step
[Write the complete missing step here with necessary explanations and equations]
If there missing steps, please output:
No missing steps.
Human: <incomplete_solie solutionn
Step 0:
<question statement>
Step I:
Step N:
</incomplete_soluee -souution_
Assistant:
<Answr>>
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 (problem statement) to Step N.
Please format your response as:
The missing step
[Write the complete missing step here with necessary explamations and equations]
Human:
There is missing step between Step X and Step X+1.
<incomplete solution>
Step O:
<question statement>
Step 1:
Step N:
</incomplete solution> >
Assistant:
<Answer>
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, identifying Thought Leaps in such cases becomes trivially easy. For example, in two-step CoT.
17

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the final step (the answer) cannot be removed, only the first step is eligible for deletion. This makes it very easy for the model identify the location of the Thought Leap, which lies directly between the question and the answer.
Regarding the number of steps to delete, avoid removing too many steps. For instance, 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 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.I] but train for only 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. Mathematics Benchmarks
This benchmark consists of 1,319 grade-school math word problems, primarily used evaluate a model's 's ability to perform multi-step arithmetic reasoning.
GSM8K:
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 model's ability to solve complex problems 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 '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- non-standard formats and high demands on creative problem sooeinnn. these problems serve rigorous test of advanced mathematical reasoning capabilities.
AMC23: This benchmark contains 40 problems selected from the 2023 American Mathematics Com- petitions (AMC), offering challenging testbed for evaluating models in a standardized competition setting.
F2 Logical Reasoning Benchmarks
FOLIO:
This benchmark consists of 1,430 expert-crafted natural language inference instances, each annotated with ooder logic (FOL) expressions, is designed to assess model's ability to translate hetween natural language and formal logic, 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 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 model's 'ss 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 model can simulate deductive reasoning. It evaluates the model's 's ability to draw logical conclusions from natural language rules.
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G
RL Training
G.1
Settings
All RL experiments were conducted using the veRL [62 training framework. employed Math- Verify for answer verification. We set the initial learning rate to 10 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 template in Appendix D.I
G.2 Results
We present the detailed results from Figure 3) in Table 8 For simplicity, only Math-Vorify is used for answer verification without further applying DeepSeck-RI.
Training Step
Qwen2.5-Math-Numming
Qwenn.--aath-Numinnariddgg
62.2
67.4
10
64.4
67.8
20
65.4
65.6
30
64.6
66.4
40
66.8
67.6
50
65.6
67.6
60
67
70.6
70
67.8
70.4
80
68
71.4
90
69.6
100
69
72.6
110
67.6
71.8
120
69.6
71.8
130
70.8
72.8
140
71.4
150
72
74
160
72
74,8
170
73.4
73.6
180
72
74.8
190
72.8
76
200
71.6
75.4
210
71.6
75.2
220
72.6
76.8
230
72.8
240
73.6
76.4
250
71.8
77.4
260
73.6
77.4
270
73.4
77.6
Table 8: Model accuracy on MATH500 over GRPO training steps.
Model
Method
GSMSK
MATH
GaoKao
Odyssey
Olympiad AMC23
Average
Qwen2.5 Math-Instruttt
84,80
75.80
65.50
54.52
38.10
60.00
63.12
Out-Zero-1.5 2.50-1.5B
Dr. GRPO
83.62
74.20
69.61
52.71
37.60
53.00
61.79
GRPO
82.71
74.60
64.94
49.10
35.85
50.00
59.33
Seann..5Math-Nuaaee Sren2.5--uth-uummnne
GRPO
84.31
74.80
62.34
$1.94
39.41
52.50
60.88
GRPO
84.08-92
78.20
67.01
54.26-2.1
40.30
60.00
63.98-1.u
Table 9:
Reinforcement learning results. Qwen2.5 5-tthhh---th--- Math-Numina and 1-0.555--a--uumiti refer to the Qwen2.5-Math-11553 models fine-tuned NuminaMath and NuminaMath-Biin Bridde respec- tively in the main experiments. We also present the results of on IA using GRPO, along with the results of Qwen2.5. Math--nstrrut- 1.5B 20 and Oat-Zero-1. .B5 63 for reference.
trained directly
19

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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)..... :IG)) ). and the predicted set be P {pr,.... PIPl predicted position py exactly matches ground-truth position g.. it is counted as True Positive (TP). The metrics defined as:
TP
TP [G]
Precision
Recall
P
To measure the proportion of redundant predictions, we define the redundancy rate as:
PP
TP
Redundancy
IP]
H.2 Overall Metric
To evaluate both leap positions locating ability and the quality of the generated step, introduce the Position- aware BERTScore. This metric only compares text at successfully matched positions. Let the reference text a matched position be t; and the generated text be f4 the dosition-aware BERTScore is defined as:
TP
BERT Score
BFRTScore(ft,tt)
Here, BERTScore is computed using RoBERTa-lare model and measures semantic similarity between each pair of texts (f,, ts).
I Case Study: CoT Thought Leap Bridge
Following the example in Figure []., compared the shot bridging results of Qwen2.5 Dive22.-IInstruct-o Instruct-7B/725 GPT-4o, and Gemini-2.5 P.R.PRO.
CoT-Bridge
Missing Step 1:
The missing step should be placed between Step 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 items put into containers, with then at least container must contain more than one item.
Missing Step 2:
The missing step should be placed between Step and Step 2.
The missing step is:
Therefore, the possible This gives us of 2 15 possible sums.
when rolling eight-sided dice the integers from 2 to 16, inclusive.
Qwen2.5-7B-Inssee
Missing Step 1:
The missing step should be placed between Step 0 and Step 1.
The missing step is:
The possible sums when rolling eight-sided dice range from 16, which gives different possible sums.
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Qwen2.5-72B-Instruu
Missing Step I:
The missing step should be placed between Step and Step 2.
The missing step
The possible sums when rolling two eight-sided dice are 2, 3, 4, 16. There are 1.5 possible sums total.
Missing Step 2:
The missing step should be placed between Step and Step 3. The missing step is:
According the pigeonholc principle, if we roll the dice 16 times, there are 15 possible sums, least these sums must appear least twice.
Gemini-2.5-PRO
Missing Step 1:
The missing step should be placed between Step and Step 2.
The missing step is:
Now, need determine the total number of distinct sums possible. The all the integers from
inclusive. These distinet sums will be our "pigeonholes." To count the number of possible sums:
Number of distinct sums=(Maximum Mum-Minimum Sum)+1
Number of distinet (162++146))]11144+1=
So, there are 15 unique possible sums (the pigeonholes) when rolling two eight-sided dice.
GPT-4o
Missing Step
The missing step should be placed between Step and Step 2.
The missing step is:
We need count how many distinct sums range from 2 to 16, which gives:
result from rolling eight-sided dice. The possible
16 2 1 = 15 distinct possible sums.
$$
As can be seen the case study:
Although a planning-style bridge between the question and step 1 was missing, GPT-4o, Gemini-2.5. PRO, and Qwen2.5-72B Swen2.5-72B-Instruee all identified that "15" lacked calculation explanation and bridged the gap the appropriate position.
Bwen2.5-72B-Insttuue 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
The consistency the content and position bridged by Qwenn.5-7B-IInstee not good.
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J
CoT Score
J.1
Score Template
Since scoring single response with DeepSeek-RI result in high variance, instead compare the original CoT bridged version for same question side-by-side.
DeepSeek-RI Score Template
are math reasoning expert. Please evaluate the coherence given two Chain Thoughts (CoTss) 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: Docs it show evidence of having a structured plan or certain degree foresight
approaching the problem?
3. 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, other necessary expansions?
5. Conclusion Support: Does it provide a clear summary or verification to support the final conclusion?
Question
<problem>
CoT
<answerl>
CoT 2
<answer2>
[Scoring Criteria Description]
Outstanding across all dimensions. Demonstrates deep understanding, clear planning, rigorous step-by-step reasoning, detailed intermediate calculations derivations, and strong conclusion verification. Rarely achieved; reflects expert-level CoT.
10 points:
points: Strong performance with only minor flaws. Reasoming is coherent mostly complete, with evident planning and clahoration. May miss slight details or final verification, but overall very solid. points: Adequate but uneven. Shows basic understanding and partial planning. Some steps are under-explained or loosely connected; may lack full elaboration support for conclusion.
points: Weak overall. Reasoning fragmented or inconsistent, with limited detail and minimal evidence structured planning. Key steps are missing or poorly justified.
2 points: Very poor. Little to no logical structure or claboration; response lacks mcaningful connection to the question.
0 points: No relevant reasoning. Off-topic fundamentally flawed throughout.
Note: Scores may also assigned odd numbers (e-g., 5, 7, 9) to reflect responses fall between
two defined levels. For instance, 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 overall <Integer between 0-10>
CoT overall score: <Integer between 0-10>
A brief analysis is sufficient.
22

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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 repeated with two eight-sided dice. The scoring is based on five dimensions: Problem Understanding, Planning, Logical Coherence, Detail Flaboration, and Conclusion Support.
First, 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 (I+1) to 16 (8+8). That gives 15 possible sums. According to the pigeonhole principle, guarantee repeat, you need more than need
the number of possible sums, so 15 + 16 throws. Both CoTs arrive at this answer, assess how well each explains the reasoning.
Looking at CoT I: It starts by identifying the minimum and maximum sums correctly. Then it directly applics the pigconhole principle, stating 15 +1 =16. The conclusion is there, but the explanation is very brief. doesn't mention how many possible sums there are, just states the range. The step from the range to 15 possible sums implicit. A reader might wonder why instead of 16-2=14. So, CoT skips explaining the count of possible sums, which is a crucial step. The logical steps are there but not fully elaborated.
CoT 2 is detailed. explicitly states the pigeonhole principle, defines the range of sums, calculates the number of possible sums as 15 (16-2+1). then adds This makes reasoning clearer. The explanation of why there are 15 possible sums is important because it's a common point of confusion (inclusive counting). CoT 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 does better job by explicitly stating the principle the count of sums. Planning: CoT structures the answer step-by-step, starting with the principle, then calculating possible sums, then applying the formula. CoT jumps to the answer without explaining the count of sums, which is gap in planning.
Logical Coherence: Both are logically correct, but CoT has a minor leap in assuming the number of possible sums. CoT 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, hut CoT 2's conclusion better supported by the preceding steps.
Scoring: CoT might get or 7 because it's correct but lacks some details. CoT 2 is more thorough, 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 7. However, according to the criteria, 8 for strong performance with minor lapses. CoT docsn't have lapses, maybe 9? Wait, the scoring criteria say 10 is outstanding. CoT is thorough but maybe not "outstanding" because it's straightforward application. Alternatielly, 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 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 leap. Detail Elaboration: Missing the count explanation. Conclusion: Correct. So maybe 6 or 7. The criteria for say "intermediate steps present but not fully explained". CoT 1 has the steps hut skips the count. So 6. CoT 2: All steps explained, 10? But the example for 10 says "smooth and complete". CoT 2 is complete. Alternatively, maybe if there's minor point missing. But don't it see any. It explains the pigeonhole principle, calculates the sums correctly, adds 1. So 10. But the user's example answer might have given and 10. Wait, the user's '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 I gets 6 because it's adequate but missing the count explanation. </think>
Justification:
Logical Coherence Analysis: CoT I correctly applics the pigconhole principle but skips explaining the count of possible sums (15), leaving 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:
CoT 2 overall score: 10
***ote:** CoT 1 earns 7 for adequate reasoning but lacks explicit calculation of the count. CoT 2 achieves hy thoroughly addressing all dimensions, including precise step-by-step clahoration and seamless logical flow.
23

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The question, CoTI, and CoT2 here are the same as those shown in the example in Figure Ta).
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
Original CoT
Bridged CoT
Metamath
7.85 7.49
8.44
Numinamath
8.22
Table 10: DeepSeek-RI scoring results for the original CoT and the bridged CoT.
K PRM Score
We present the data from Figure in Table IT
Method
Metric
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
MelaMathQA
GapBridge
num percent
2401
672825
0.71
0.35
0.3
0.31
0.34
0.39
0.5
0.71
1.52
94.85
Qwen2.5 72B
num
9528
4946
4410
4516
4891
5990
8172
11505
22577
321582
percent
2.39
1.24
1.11
1.13
1.23
2.05
2.89
5.67
80.78
Oween2.5-Instrueee
num percent
19349
5557
4706
4372
4714
5287
7023
9568
18096
252466
1.68
1.32
2.12
5.46
76.24
NuminaMath
GapBridge
33923
20643
18397
17857
18783
21589
27041
37375
73499
1363063
percent
2.08
1.26
1.13
1.09
1.15
1.32
1.66
2.29
83.51
Qwen2.5-Innstruet-2
num
147988
percent
14.81
2.57
2.32
2.32
2.49
2.95
3.82
5.28
9.98
53.45
Oween2.-IInsrueer.e
num
177687
38694
32492
31388
32955
37837
47592
65059
121298
769034
pereent
2.86
2.32
2.43
3.51
56.8
Table 11: The count and proportion of bridge steps falling into different PRM score intervals for Bwen2.5-Instruut-7B.. and CoT-Bridge on MetaMath and NatinaMath.
24

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NeurIPS Paper Checklist
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is fine to include aspirational goals as motivation long it is clear that these goals are not attained by the paper.
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The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors encouraged to provide short proof sketch to provide intuition.
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Reviews:
[{"review_id": "92UKGs9Yja", "reviewer": "Reviewer_H9iA", "summary": "This 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.", "questions": "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.", "limitations": "As 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.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strenghts\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.", "was_revised": true, "final_justification": "The author's response have address most my concerns.", "rebuttal": "**Dear Reviewer H9iA:**\n\nThank 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.\n\n> Weakness 1: issue of step segmentation\n\n**Response to Weakness 1:**\n\nThank 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.\n\nTo 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.\n\n| Method | Alignment Rate |\n|-|-|\n| Human | 85.3% |\n| GPT-4o | 74.2% |\n\n[1] The Lessons of Developing Process Reward Models in Mathematical Reasoning  \n[2] LLMs Can Easily Learn to Reason from Demonstrations Structure, not content, is what matters!  \n[3] SEAL: Steerable Reasoning Calibration of Large Language Models for Free\n\n> Weakness 2:\nWhile 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\n**Response to Weakness 2:**\n\nThank you very much for your kind words about the clarity of our writing.\n\n**Regarding domain generalizability:**  \nWhile 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.\n\nOur 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.\n\n**Regarding strong teacher model:**  \nWe 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.\n\n[4] Qwen2.5 Technical Report\n\n> Weakness 3:\nSince 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\n**Response to Weakness 3:**\n\nThank 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. \n\n| Model | Dataset | GSM8K | MATH | GaoKao | Odyssey | Olympiad | AMC23 | Average |\n|-|-|-|-|-|-|-|-|-|\n| Llama3.1-70B | MetaMathQA | 90.83 | 50.20 | 47.27 | 42.12 | 13.63 | 27.50 | 45.26 |\n| Llama3.1-70B | MetaMathQA-Bridged | 91.51 | 51.80 | 47.79 | 43.15 | 14.67 | 28.75 | 46.28 |\n| Improvement | / | +0.68 | +1.60 | +0.52 | +1.03 | +1.04 | +1.25 | +1.02 |\n\n> Question\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\n**Response to Question:**\n\nThank 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.\n\nThe 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.\n\n| Method | Precision | Recall | Redundancy | Overall | API Cost (100 samples) |\n|-|-|-|-|-|-|\n| CoT-Bridge | 78.02 | 78.37 | 1.61 | 76.15 | ≈$0 |\n| Qwen2.5-Instruct-7B | 14.15 | 12.04 | 34.13 | 10.54 | ≈$0 |\n| Qwen2.5-Instruct-72B | 33.99 | 33.64 | 33.73 | 31.12 | ≈$0 |\n| GPT-4o | 58.67 | 59.17 | 18.33 | 55.41 | $1.42 |\n| Gemini-2.5-PRO | 67.75 | 68.33 | 9.25 | 64.95 | $2.46 |\n\n**Case Study:**  \nThank 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.\n\nAs can be seen in the case study below:\n- 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.  \n- 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. \n- The consistency of the content and position bridged by Qwen2.5-7B-Instruct is not good.\n```\n# Original Problem (Step 0) and Solution\nStep 0:\nWhat is the minimum number of times you must throw two fair eight-sided dice to ensure that the same sum is rolled twice?\nStep 1:  \nConsider two eight-sided dice, each numbered from 1 to 8. The sums when rolling these dice range:\n- Minimum sum: 1 + 1 = 2\n- Maximum sum: 8 + 8 = 16\nStep 2:\nBy the pigeonhole principle, to ensure at least one sum appears twice, we need to roll the dice 15 + 1 = 16 times.\nStep 3:\nThus, to guarantee that the same sum appears twice when rolling two eight-sided dice, the dice must be rolled $\\boxed{16}$ times.\n```\n\n```\n# CoT-Bridge\nMissing Step 1:  \nThe missing step should be placed between Step 0 and Step 1.  \nThe missing step is:  \nTo 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.\n\nMissing Step 2:  \nThe missing step should be placed between Step 1 and Step 2.  \nThe missing step is:  \nTherefore, 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.\n```\n\n```\n# Qwen2.5-7B-Instruct\nMissing Step 1:  \nThe missing step should be placed between Step 0 and Step 1.  \nThe missing step is:  \nThe possible sums when rolling two eight-sided dice range from 2 to 16, which gives us 15 different possible sums.\n```\n\n```\n# Qwen2.5-72B-Instruct\nMissing Step 1:\nThe missing step should be placed between Step 1 and Step 2.\nThe missing step is:\nThe possible sums when rolling two eight-sided dice are 2, 3, 4, ..., 16. There are 15 possible sums in total.\n\nMissing Step 2:\nThe missing step should be placed between Step 2 and Step 3.\nThe missing step is:\nAccording 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.\n```\n\n```\n# Gemini-2.5-PRO\nMissing Step 1:\nThe missing step should be placed between Step 1 and Step 2.\nThe missing step is:\n\nNow, 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.\"\n\nTo count the number of possible sums:\n\nNumber of distinct sums=(Maximum Sum−Minimum Sum)+1\nNumber of distinct sums=(16−2)+1=14+1=15\n\nSo, there are 15 unique possible sums (the pigeonholes) when rolling two eight-sided dice.\n```\n\n```\n# GPT-4o\nMissing Step 1:\nThe missing step should be placed between Step 1 and Step 2.\nThe missing step is:\nWe 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:\n\n$$\n16 - 2 + 1 = 15 \\text{ distinct possible sums}.\n$$\n```\n\nIf you have any further questions, we would be happy to answer them.\n\nSincerely,  \nCoT-Bridge Authors"}, {"review_id": "0Na6gJAdEP", "reviewer": "Reviewer_9f3z", "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.", "questions": "See the weakness.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength：\n- This paper is well-written and easy to read.\n- The paper systematically identifies and formalizes the \"Thought Leap\" phenomenon in CoT datasets.\n- The CoT-Bridge model is trained on a carefully constructed dataset (ScaleQM+) using step-deletion strategies and rigorously evaluated on standard benchmarks. \n- CoT-Bridge is designed to be plug-and-play, improving distillation and RL fine-tuning workflows without architectural changes.\n\nWeakness:\n- 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.\n- 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.", "was_revised": true, "final_justification": "The detailed response solves my concern.", "rebuttal": "**Dear Reviewer 9f3z:**\n\nThank 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.\n\n> Weakness 1:\nThe method's performance varies with bridge location (begin/middle/end), but the paper offers limited insight into how position-aware generation might be improved.\n\n**Response to Weakness 1:**\n\nWe 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:\n- MetaMath: -1.80% (compared to -0.76% for begin and -0.78% for end positions)\n- NuminaMath: -3.41% (compared to -2.37% for begin and -3.04% for end positions)\n\nThis indicates that middle position bridges contribute most significantly to the model's effectiveness.\n\nWhen 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.\n\nBased 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:\n\n$$L_{middle-position} = -\\sum_{i=1}^{n-2} [y_i \\log(p_i) + (1-y_i) \\log(1-p_i)]$$\n$$L_{new} = L_{SFT} + \\lambda \\cdot L_{middle-position}$$\n\nwhere $y_i$ 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. \n\nDue 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.\n\n| Method | Precision | Recall | Redundancy | Overall |\n|-|-|-|-|-|\n| CoT-Bridge | 78.02 | 78.37 | 1.61 | 76.15 |\n| CoT-Bridge + Middle Position Loss | 79.45 | 80.22 | 1.24 | 77.83 |\n\n> Weakness 2:\nThe authors should provide more explanations for why removing low-PRM steps (PRM < 0.5) does not improve performance, as this is somewhat counter-intuitive.\n\n**Response to Weakness 2:**\n\nThank 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.\n\nTo investigate hypothesis (2) more deeply, we manually labeled 100 randomly sampled low-PRM bridged steps (PRM<0.5) and categorized them into four types:\n- Steps with genuine errors (19%)\n- Completely correct steps (18%)\n- Reflective/inspirational steps that provide valuable reasoning guidance (11%)\n- Steps that are mathematically correct but have suboptimal linguistic coherence (52%)\n\nThis 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.\n\n**Case Study: (from MetaMathQA)**  \n\n```\nQuestion: 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?\n\nSolution: \n(Bridge1) To determine how many dimes Natasha has, we can use modular arithmetic and the Chinese Remainder Theorem. [PRM score=1.0000]\n\nLet $n$ be the number of dimes Natasha has.\nWe know that $1<n<10$.\nFrom the given information, we can write the following equations:\n$n\\equiv 1\\pmod{3}$,\n$n\\equiv 1\\pmod{4}$,\n$n\\equiv 1\\pmod{5}$.\n\nWe can solve this system of congruences using the Chinese Remainder Theorem.\nSince the modulus values 3, 4, and 5 are pairwise relatively prime,\nthere is a unique solution modulo $3\\cdot 4\\cdot 5=60$.\n\nBy inspection, we find that $n\\equiv 1\\pmod{60}$ satisfies the congruences.\n\n(Bridge2) We can write this as:\n\\[ n-1 = 60k \\]\nfor some integer \\( k \\). This implies:\n\\[ n = 60k + 1. \\] [PRM score=0.9609]\n\nTherefore, the possible values of $n$ are $1, 1+60=61, 1+2(60)=121, 1+3(60)=181,\\dots$.\nSince the only possible value of $n$ that satisfies $1<n<10$ is $\\boxed{61}$,\n\n(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]\n\nwe conclude that Natasha has 61 dimes.\nThe answer is: 61\n```\n\nIf you have any further questions, we would be happy to answer them.\n\nSincerely,  \nCoT-Bridge Authors"}, {"review_id": "wJ4svMKzEI", "reviewer": "Reviewer_Wqes", "summary": "To 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.", "questions": "1. 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?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "The issues of model scalable, impact on RL , and information-theoretic efficiency have been resolved.", "rebuttal": "**Dear Reviewer Wqes:**\n\nThank 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:\n\n> Weakness 1: Model scalability concerns\n\n**Response to Weakness 1:**\n\nThank 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.**\n\n- **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.\n\n- **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%).\n\nFurthermore, 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.\n\nResults on Meta-Llama3.1-8B + NuminaMath:\n\n| Method | GSM8K | MATH | GaoKao | Odyssey | Olympiad | AMC23 | Average |\n|-|-|-|-|-|-|-|-|\n| Direct SFT | 84.86 | 51.45 | 49.03 | 36.56 | 21.30 | 20.00 | 43.87 |\n| CoT-Bridge-7B | 85.97 | 56.80 | 54.42 | 40.76 | 24.85 | 35.63 | 49.74 |\n| CoT-Bridge-1.5B | 85.67 | 55.60 | 54.03 | 40.05 | 25.19 | 35.00 | 49.26 |\n\n> Weakness 2.1: Limited effects on RL\n\n**Response to Weakness 2.1**\n\nThank 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.\n\nThe 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.\n\nThese 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).\n\n[1] DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning\n\n> Weakness 2.2: Narrow evaluation\n\n**Response to Weakness 2.2**\n\nWe 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. \n\nAs 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.\n\nResults:\n\n| Model | GSM8K | MATH | GaoKao | Odyssey | Olympiad | AMC23 | Average |\n|-|-|-|-|-|-|-|-|\n| Llama3.1-8B + SimpleRL-Zoo[2] | 79.2 | 23.0 | - | - | 5.3 | 15 | - |\n| Llama3.1-8B-Numina + RL | 85.60 | 64.20 | 60.26 | 48.58 | 31.85 | 40.00 | 55.08 |\n| Llama3.1-8B-NuminaBridge + RL | 86.43 | 68.40 | 63.63 | 49.61 | 34.96 | 50.00 | 58.84 |\n| Improvement | +0.83 | +4.20 | +3.37 | +1.03 | +3.11 | +10.00 | +3.76 |\n\nExperimental setup:\n- Hardware: 16 × A800 80G GPUs\n- Algorithm: GRPO with same hyperparameters as original experiments\n\n[2] SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild\n\n> Weakness 3: Information-theoretical inefficiency\n\n**Response to Weakness 3:**\n\nThank 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**.\n\n**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.\n\n**Alternative 2 - Direct training on ScaleQuestMath:**\nAs 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:\n1) Direct training on ScaleQuestMath 588k underperforms NuminaMath-bridge,\n2) 72B-distill SFT outperforms the ScaleQuestMath subset, suggesting that CoT-Bridge is capable of further enhancing data of even higher quality.\n\nResults on Qwen2.5-Math-1.5B:\n\n| Dataset | GSM8K | MATH | GaoKao | Odyssey | Olympiad | AMC23 | Average |\n|-|-|-|-|-|-|-|-|\n| NuminaMath (from Table 1) | 83.62 | 63.90 | 57.40 | 46.77 | 33.04 | 32.50 | 52.87 |\n| NuminaMath-bridge (from Table 1) | 84.61 | 68.05 | 59.29 | 47.16 | 34.11 | 45.00 | 56.26 |\n| ScaleQuestMath 588k subset | 85.67 | 65.20 | 57.14 | 45.48 | 32.44 | 30.00 | 52.66 |\n| 72B-distill (from Table 2) | 81.86 | 68.15 | 60.84 | 48.13 | 33.00 | 39.37 | 55.23 |\n| 72B-distill-bridge (from Table 2) | 82.52 | 71.50 | 66.43 | 49.16 | 34.89 | 45.00 | 58.25 |\n\n> Question 1\n\n**Response to Question 1:**\n\nThank 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.\n\nIn 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.\n\nAs 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):\n\n| Dataset | Bridge Method | Avg. Steps Bridged | Avg. Tokens Bridged |\n|-|-|-|-|\n| MetaMathQA | CoT-Bridge | 1.79 | 108.93 |\n| | Qwen2.5-Instruct-72B | 1.01 | 92.56 |\n| NuminaMath | CoT-Bridge | 1.90 | 142.57 |\n| | Qwen2.5-Instruct-72B | 1.73 | **161.15** |\n\nDespite 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.\n\n> Question 2\n\n**Response to Question 2:**\n\nThank 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.\n\nWhen 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.\n\nRegarding the alternative approaches you mentioned, we have addressed them in our response to Weakness 3 above.\n\nThank 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.\n\nSincerely,  \nCoT-Bridge Authors"}, {"review_id": "bdL95VxQV1", "reviewer": "Reviewer_Ayux", "summary": "This 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.", "questions": "- 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?", "limitations": "Not discussed. Please see weaknesses above", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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?", "was_revised": true, "final_justification": "The experiments ablating ScaleQM+ confirm that the bridging can indeed boost model performance, in addition to just dataset gains. Overall I'm leaning positive for this paper and would like to keep my current positive rating.", "rebuttal": "**Dear Reviewer Ayux:**\n\nThank you for reviewing our paper and providing valuable feedback. Below, we present our responses to the weaknesses you have mentioned:\n\n> Question:\nTo 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?\n\n**Response to Question:**\n\nThank you for your question. We would like to clarify the CoT-Bridge and baseline setup.\n\n1. **CoT-Bridge Setup:**\n   - 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)\n   - Then apply CoT-Bridge to augment MetaMathQA/NuminaMath\n   - Finally perform SFT on the augmented datasets\n\n2. **Baseline Setup:**\n   The baseline model (Direct SFT) is trained only on MetaMathQA or NuminaMath.\n\nWe understand your concern regarding the fairness of this comparison. **We will address this issue through additional experiments in our response to Weakness 1.**\n\n> Weakness 1:\nFor the results in Table 1, do CoT-Bridge use an additional dataset ScaleQuestMath where the baselines don't?\n\n**Response to Weakness 1:**\n\nThank you for your insightful observation regarding the fairness of comparison.\n\n- 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.\n\n- 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.\n\nTo 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**.\n\nResults on Meta-Llama3.1-8B + NuminaMath:\n\n| Method | Size | GSM8K | MATH | GaoKao | Odyssey | Olympiad | AMC23 | Average |\n|-|-|-|-|-|-|-|-|-|\n| NuminaMath (from Table 1) | 859k | 84.86 | 51.45 | 49.03 | 36.56 | 21.30 | 20.00 | 43.87 |\n| NuminaMath-Bridged (from Table 1) | 859k | 85.97 | 56.80 | 54.42 | 40.76 | 24.85 | 35.63 | 49.74 |\n| NuminaMath + 588k subset | 859k + 588k | 86.12 | 55.60 | 54.29 | 36.43 | 22.37 | 32.50 | 47.89 |\n| NuminaMath-Bridged + 588k subset | 859k + 588k | 87.19 | 59.20 | 56.88 | 41.60 | 25.48 | 37.50 | 51.31 |\n| Improvement | - | +1.07 | +3.60 | +2.59 | +5.17 | +3.11 | +5.00 | +3.42 |\n\nWe sincerely appreciate your observation. We plan to include these additional results in our paper to enhance its rigor.\n\n> Weakness 2:\nThis 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\n**Response to Weakness 2:**\n\nThank 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:\n1) Identifies **WHERE** thought leaps occur (position detection)\n2) Generates **WHAT** content to bridge the gaps (content generation)\n\n- 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 $M̂$.\n\n- 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.\n\n- 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.\n\nWe 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.\n\nSincerely,  \nCoT-Bridge Authors"}]

Author rebuttal:

Meta-review: This paper analyzes and identifies Thought Leaps, missing intermediate reasoning steps in existing CoT datasets, as a major source of inefficiency in CoT learning, and proposes CoT-Bridge to address them. The paper presents thorough experiments analyzing the effect of Thought Leaps and demonstrates significant improvements across multiple reasoning benchmarks and end uses (SFT, distillation, RL) by bridging them.

The reviewers found the findings to be interesting and novel, with the analysis and experiments thorough, and the method simple yet effective in delivering substantial improvements. The rebuttal addressed concerns regarding scalability to larger models (showing that small bridging models suffice), the effect on RL training, and further clarified the method.

The AC agrees that the paper highlights an important phenomenon and introduces a simple but impactful solution, supported by careful experimentation that convincingly demonstrates the significance of the results. The findings should be valuable for researchers and practitioners working on reasoning, and the AC therefore recommends acceptance.
Decision: Accept (poster)