Demystifying Reasoning Dynamics with Mutual Information: Thinking Tokens are Information Peaks in LLM Reasoning

Thank you for your great efforts in reviewing this paper. We will try our best to answer all your questions. Please let us know if you still have further concerns, or if you are not satisfied with the current responses, so that we can further update the response ASAP.

---

Q1: "It is intuitively not very clear to me what exactly MI is measuring. If MI is to be taken as how informative the hidden representation at each step is about the correct answer, should we not expect MI to increase throughout the reasoning process as the model arrives closer to the answer? (Perhaps this can be quantified instead as the MI between the answer and the concatenated representations up to time .) Why does MI increase sharply at thinking tokens but fall back down afterwards? The paper does a great job at showing the importance of thinking tokens but could have more discussion on what this signifies or why this happens."

A1: Thank you for your insightful comment!

• Intuitively, your understanding of "what exactly MI is measuring" is completely correct, i.e., "MI is to be taken as how informative the hidden representation at each step is about the correct answer." However, what we compute is the MI between the representation of each generated token and the answer. Since the representation of a single token may not contain all information up to the current step, the MI of the whole sequence is not necessarily monotonically increasing.

• Following your suggestion, we conduct new experiments to quantify the MI between the answer and the concatenated representations up to each time step. Specifically, we follow the same experimental setting as in the paper and use the DeepSeek-R1-Distill-Qwen-7B model on the training split of the MATH dataset. We compute Spearman’s ρ and Kendall’s τ to measure the monotonicity of the MI sequence. Both metrics take values in [−1,1], with values closer to +1 indicating stronger monotonic increasing trends.

  Spearman’s ρ		Kendall’s τ
  value	p-value	value	p-value
  0.9204	0.0051	0.8150	$4.2e^{-11}$

  The results show that when computing MI between the concatenated representations and the answer, the MI sequence indeed increases as reasoning progresses, consistent with your insight.

• Regarding "Why does MI increase sharply at thinking tokens but fall back down afterwards", we propose a possible explanation from the perspective of the definition of MI: $I(X;Y) = H(X) - H(X|Y)$, where $H(X)$ measures the uncertainty of $X$. During reasoning, when some tokens represent transitions or explore new reasoning paths (e.g., “Wait”, “But”), they may have higher uncertainty. In contrast, most other tokens simply continue along an already established reasoning path in a more deterministic manner (e.g., “1+1=2”, “and”).

• Finally, thank you for your recognition of our paper. We fully agree with your point that it is interesting and important to further investigate why the MI peaking phenomenon occurs (which we also discussed in the limitations section). We hope our work can inspire future explorations in this direction.

---

Q2: "It should be mentioned that Theorem 1 basically follows from an application of Fano's inequality (Eq.6). Also, the result is stated with the conditional MI , so the precise connection with the experiments is unclear. Can we also compute the conditional MI in the experiments?"

A2: Thank you for your comment.

• We will make the connection to Fano’s inequality explicit in the revised manuscript.
• Regarding the conditional MI $I(y; h_j | h_{<j})$ in Theorem 1, we follow your suggestion to improve clarity. By the chain rule and the definition of mutual information, we can rewrite: $I(y; h_j | h_{<j}) = I(y; h_j) + I(h_j; h_{<j} | y) - I(h_j; h_{<j})$. Since in the setting of this paper, we assume that y is independent to $h_j$, $h_{<j}$, respectively (only the input X affects $h_j$, $h_{<j}$. Therefore, $I(h_j; h_{<j} | y) = I(h_j; h_{<j})$. As a result, the conditional MI in Theorem 1 $I(y; h_j | h_{<j})$ can be directly replaced by $I(y; h_j)$。

---

Q3: "Some results may not be statistically significant, in particular the gaps between the original and proposed models in Figure 7 are not very large and would benefit from adding deviations/error bars over a couple of trials."

A3: Thank you for your suggestion. We have followed your instructions to conduct new experiments, running both the origin and TTTS settings in Figure 7 three times and reporting the mean and standard deviation. The results are as follows:

MATH500:

GSM8K：

AIME24：

The results show that across three runs, TTTS consistently outperforms the original model, demonstrating stable performance improvements.

We are grateful for your engagement in the discussion process. We are pleased that our response has adequately addressed your concerns. We will remain available to address any further questions or clarifications until the end of the discussion phase.

Thank you for all the valuable comments and questions. Your comments indeed help us further improve the paper, and we will incorporate these discussions into the manuscript.

Sincerely,

The authors

Thank you for your great efforts in reviewing this paper. We will try our best to answer all your questions. Please let us know if you still have further concerns, or if you are not satisfied with the current responses, so that we can further update the response ASAP.

---

Q1: "Although this work discovered and demonstrated the positive correlation between the MI peaks and reasoning performance from both the empirical and theoretical perspectives, the investigation and evaluation were mainly across different math problem-solving benchmarks, involving no other domains that demand reasoning, such as GPQA and MedQA, undermining the effectiveness for all reasoning tasks."

A1: Thank you for your constructive comment. We have followed your instructions to conduct new experiments on GPQA and MedQA to examine whether the MI peaks phenomenon still exists. We used DeepSeek-R1-Distill-Llama-8B, DeepSeek-R1-Distill-Qwen-14B, and their corresponding base models.

1. On both GPQA and MedQA, the MI peaks phenomenon persists. Since figures cannot be included here, we instead report the AOM metric (as defined in Lines 160-162, larger values indicate stronger outlier peaks) for reference.

   		GPQA	MedQA
   Llama-3.1-8B	Origin	3.5193	3.6337
   	Reasoning	4.0809	3.8436
   Qwen2.5-14B	Origin	2.7671	2.4355
   	Reasoning	2.9444	2.6281

2. Regarding the tokens at MI peaks:

   • For GPQA, the token set largely overlaps with the one obtained from the MATH dataset in the manuscript.

   • For MedQA, while tokens like "Let," "So," and "But" still show high overlap, we also observe some new tokens such as "Admin," "Perform," "She," and "He," which may be related to the medical domain.

In summary, these experiments further validate that the MI peaks phenomenon generalizes to other reasoning domains.

---

Q2: "The token budgets of DeepSeek-R1-Distill-Qwen models were usually set the maximum as 32768, and the accuracy in Figure 7 seemed not converged, the authors might consider for more max_length to show the overall trends."

A2: Thank you. We follow your suggestions to further increase the token budgets on AIME24 in Figure 7. The results show that after accuracy saturates, TTTS consistently outperforms the baseline.

---

Q3: "Generally, the evaluation of reasoning models should consider the pass@k metric with more sampling times, as the LRMs perform unstably, merely the one-time inference is not that faithful."

A3: Thank you for your suggestion. We have followed your suggestions to conduct new experiments by adding pass@3 as an evaluation metric for RR and TTTS, following settings in Lines 248-260 and 277-280. The results on AIME24 with DeepSeek-R1-Distill-Llama-8B are as follows:

The results show that when evaluated with pass@3, both TTTS and RR consistently outperform the baselines.

---

Q4: "I notice that during the investigation, the authors conduct experiments from 7B to 70B, which is very comprehensive, while the application experiments were conducted only on 7B. I understand that there may be computation resources limited, why did not complement a 14B experiments to make the validation more convincible?"

A4: Thank you for your constructive suggestions. We have followed your suggestions to conduct new experiments to validate the effectiveness of RR and TTTS using DeepSeek-R1-Distill-Qwen-14B on AIME24 following the settings in Lines 248-260, 277-280. The following experimental results indicate that 1) TTTS performs worse than the origin model at early stages of the token budget but continues to grow steadily later (similar to the phenomenon described in Lines 184–190); 2) RR shows a clear improvement over the origin model.

---

Q5: "Intervening at the test-time has demonstrated the performance, have the authors considered for post-training phase methods? Just for personal curiosity."

A5: Thank you for your valuable suggestions. For simplicity and ease of use, we only applied thinking tokens during the test-time phase. Meanwhile, it can also be developed into post-training phase methods, and here are two potential integration directions:

1. During RL (Reinforcement Learning) and SFT (Supervised Fine-Tuning), higher weights can be assigned to these "thinking tokens."
2. During RL, multiple rollouts could be performed for thinking tokens to allow more exploration.

Some more recent works are exploring similar ideas, and we are also happy to explore and verify these post-training applications in future work.

---

Q6: "Except for the MI, are there other alternative approaches that can support the analysis results?"

A6: A good question. Some similarity and distance measuring metrics and tools, such as Center Kernel Alignment (CKA) and Kullback–Leibler (KL) divergence, can also be used to support the analysis results. We are happy to use different metrics in future work.

---

Q7: "This paper listed some frequent tokens such as: “Wait”, “Hmm” and “So”, have the authors statistical the “Alternatively” and how did it perform? As during my investigation, this word is also very common."

A7: A good question. Yes, we did record the token “Alternatively”, which appears on the left of Figure 1(b) in the main text. Regarding its frequency, it ranks roughly within the 20th and 40th across DeepSeek-R1-Distill-Llama-8B, DeepSeek-R1-Distill-Qwen-14B, and QwQ-32B, while Figure 4 only demonstrates top-15 tokens.

Thank you for your great efforts in reviewing this paper. We will try our best to answer all your questions. Please let us know if you still have further concerns, or if you are not satisfied with the current responses, so that we can further update the response ASAP.

---

Q1: "One of the major weaknesses of this work is regarding the core theorems presented in this work. The core theorems presented (Theorems 1 and 2) essentially restate or slightly extend well-known results related to Fano’s inequality in mutual information theory. However, the paper neither cites this foundational work nor sufficiently elaborates on how its results substantively advance understanding of MI in the context of LRMs. Furthermore, Theorem 2’s bound scales with the number of tokens T, becoming tighter as T grows, a dependence that appears counterintuitive and is not adequately justified, raising concerns about the practical significance and interpretation of the theoretical analysis. A more detailed justification for increasing the context length and a suitable relation to Fano’s inequality would be good to add in the updated version of the paper."

A1: Thank you for your valuable comments. We will follow your suggestions to add more connections with Fano's inequality and cite relevant works, so as to make the positioning of the theoretical part clearer.

• In Lines 138-141 and 144-151, we discuss how the theoretical part helps us understand the MI peaks phenomenon. Specifically, the higher the accumulated MI during the reasoning process, the lower the probability of the LLMs making mistakes. The emergence of MI peaks will potentially increase the final accumulated MI. Experimental results in Figure 3 and Table 2 verify that LRMs have emerged with the phenomenon of MI peaks, and improve the overall MI during the reasoning process (Lines 171-179) compared with their base models. At the same time, the reasoning performance of LRMs is also significantly better than that of base models.
• Explaining the number of tokens T in Theorem 2. Intuitively, as T increases, allowing more MI to be accumulated enables the LLM to explore more during the reasoning stage to approach the correct answer, thus reducing the lower bound of the model's error probability. In addition, the experimental results in Figure 7 in the main text verify the theoretical results: as the number of tokens T increases, the model's performance improves (i.e., the lower and upper bounds of errors in Theorems 2 & 3 are getting tighter, respectively).

---

Q2: "The Test-Time Thinking Token Scaling approach strongly resembles the “Simple Test-Time Scaling” technique from existing literature (s1: Simple test-time scaling). The paper could contrasts with this prior work (as mentioned in one of the inspirations [line 270]), highlighting the originality and potential impact of the TTTS method."

A2: Thank you. The prior work "s1: Simple test-time scaling" has already been cited and discussed in Lines 267-271. The main innovation and contribution of this paper is to analyze the underlying dynamic reasoning mechanisms of LRMs, discovering the MI Peaks phenomenon, and further analyzing the token distribution corresponding to MI Peaks based on information-theoretic tools. Furthermore, to demonstrate the potential real-world application of MI Peaks' theoretical insights, we show two case studies (RR and TTTS) for improving reasoning performance in a training-free manner. In this way, this paper aims to demystify and explain the reasoning mechanisms of LRMs rather than merely focusing on Test-time scaling.

---

Q3: "There are some ambiguities in the MI calculation setup, particularly it is not clearly specified whether the “golden answer” representation used for computing MI includes the pre-question context or only the answer itself. This lack of clarity affects reproducibility and the interpretation of the MI peak dynamics, making it hard to understand in one go and keeping things ambiguous in general." "... Was the pre-question context included in the target representation for MI computation? ... "

A3: Thank you. As introduced in Lines 90-91, we describe that "extract the representation of the gold answer by feeding y into the LLM." In this way, the representation used for computing MI only includes the answer itself.

---

Q4: "Regarding the analysis of “thinking tokens”, the paper does not provide a clear, explicit identification method of the tokens termed as “thinking tokens.” (seems more like a heuristic). Moreover, many of these tokens seem like the sentence-initial markers, which could confound the interpretation that MI peaks correspond to reasoning steps rather than simply sentence boundaries. A systematic analysis of the overlap between sentence-start tokens and thinking tokens is missing, making it more difficult to come to a proper conclusion/finding about the thinking tokens." "... how was temporal alignment handled between question and answer during MI estimation across timesteps? ..." "... Another question is regarding the thinking tokens; it seems many of the identified thinking tokens (e.g., “So,” “Therefore,” “Wait”) are commonly used to begin new sentences (i.e., the sentence always begins with these words/tokens). Did the authors analyze whether these tokens are simply serving as sentence-onset (SOS) markers, rather than uniquely contributing to reasoning via high-MI representations? ..."

A4: Thank you for your suggestions. As introduced in Lines 51-53 and 212-216, we define the tokens at MI peaks as thinking tokens.

We have followed your suggestion and extracted the words at the beginning of all sentences, then sorted them by their occurrence frequency to get the top ten: "So, Wait, Let, But, Therefore, The, Alternatively, Then, Hmm, First". Interestingly, we found that there is a significant overlap between these sentence-onset (SOS) tokens and the thinking tokens at MI peaks in Figure 4. Here, we would like to make the following two statements:

1. The large overlap between the set of tokens at MI peaks and the set of SOS tokens is not surprising. The beginning of a sentence usually affects the overall direction of the subsequent content, so it may contain more information, which precisely corresponds to those MI peaks. From another perspective, $I(X;Y) = H(X) - H(X|Y)$, where $H(X)$ measures the uncertainty of $X$. The tokens at the beginning of a sentence usually affect the subsequent trend and require transitions, choices, etc., thus having higher uncertainty; while for tokens in the middle of a sentence, since the semantics of the previous part of the sentence have gradually been determined, it is only necessary to continue generating along the established logic/semantics, thus having higher certainty (this point has also been mentioned in some concurrent works [c1]).
2. Considering the frequency of token occurrence further, the distribution of tokens at MI peaks and the distribution of SOS tokens are quite different. This also indicates that not all SOS tokens have high mutual information.

[c1] Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning

---

Q5: Minor Comments

A5: Thank you for your constructive suggestions. We will follow your suggestions to make the following revisions:

1. Cite and discuss the "logit lens" technique.
2. Add more notations in Figure 2 to highlight the MI peaks phenomenon, and enrich the captions with descriptions and brief conclusions, so that each figure is more self-contained.
3. Move the limitation section into the main paper.

Thank you for your great efforts in reviewing this paper. We will try our best to answer all your questions. Please let us know if you still have further concerns, or if you are not satisfied with the current responses, so that we can further update the response ASAP.

---

Q1: "While the paper identifies MI peaks as important phenomena, it does not address why MI values across different models (even of the same family) are not directly comparable. Additionally, although model sizes vary (e.g., 7B, 8B, 14B, etc.), no clear relationship between model size and MI behavior is established. Given that larger models generally exhibit better reasoning capabilities, the lack of correlation between MI and model scale raises questions about the universality of the MI peak phenomenon."

A1: This is a good question.

• There is no clear relationship between model size and MI behaviors according to experimental results in Tables 1 and 2 of the main text. Specifically, quantitative metrics including #MI Peaks, #All Steps, Ratio of MI Peaks, and Max Interval do not show obvious patterns as the model size scales. Different families and sizes of LLMs have different architectures and dimensions of latent representations, and are trained with different strategies and data. These factors are all closely related to the reasoning capabilities, which need to be further investigated in the community [c1-4].
• Nevertheless, MI peak is a universal phenomenon across LLMs' families (Llama and Qwen) and sizes (7B，8B，14B，32B, and 70B), as shown in Figures 2-3 in the main text, Section 2, and Figures 2-13 in Appendix.

[c1] Demystifying Long Chain-of-Thought Reasoning in LLMs

[c2] SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild

[c3] LIMO: Less is More for Reasoning

[c4] Dr.GRPO Understanding R1-Zero-Like Training: A Critical Perspective

---

Q2: "The experiment comparing the suppression of thinking tokens versus randomly selected non-thinking tokens may not be entirely fair. Since non-thinking tokens include many function words or punctuation marks that naturally carry less semantic content, randomly suppressing them might disproportionately affect meaningless tokens. A more rigorous and informative evaluation necessitates the careful selection of token pools that are both semantically meaningful and sufficiently challenging for comparative analysis."

A2: Thank you for your comment.

• We would like to first clarify that in the original experiments, when selecting random tokens, we filtered out punctuation and other meaningless symbols, keeping only English words.

• In addition, we have followed your instructions to select more meaningful tokens for comparison. Specifically, using the training set of the MATH dataset, we extract tokens from model answers by (i) removing the thinking tokens, (ii) requiring English tokens with length > 2, and (iii) selecting the top-frequency tokens. These tokens include answer, final, can, which, from, have, etc. Based on this, we follow the experimental setting of Figure 5 and set the number of suppression tokens to 15.

  • Dataset	GSM8K	MATH500	AIME24
    Random	82.9	79.2	33.3
    Thinking tokens	69.0	43.8	20.0
    Top-frequency tokens	81.3	69.8	26.7

  • The results show that suppressing top-frequency tokens hurts model performance more than suppressing random tokens. However, suppressing thinking tokens still leads to the most significant performance drop.

---

Q3: "The RR method, introduced in Lines 240–248, lacks sufficient detail and hard to understand. Specifically, it is unclear whether this mechanism is equivalent to prompting the model to “think again” using the same input—a strategy akin to asking the model to generate multiple responses and choosing the best one. To validate RR’s novelty and effectiveness, the authors should compare it against a baseline where the model generates multiple responses without recycling and evaluate whether the improvement stems from the recycling mechanism itself or simply from ensemble-like effects."

A3: Thank you for your valuable suggestions.

• Intuitively, the core idea of RR design is that when the LLM aims to generate "thinking tokens", we do not let LLM directly generate subsequent tokens. Instead, we feed the representation of the current token back into the same layer once more, as introduced in Lines 244-245, hoping that the LLM can better utilize the information in the representation of the "thinking token". For example, suppose the sequence currently generated by the LLM is "... Step 4 is finished. " At this point, we detect that the next token going to be generated is "So" (one of "thinking tokens" with high MI). Instead of letting the LLM generate "So" directly, we send the corresponding representation of this token back to the current layer once again. Then, the LLM may realize that it needs to check the previous results, thus generating "Wait". The subsequent generation process then returns to normal, unless a "thinking token" is encountered again, in which case the above process is repeated. Therefore, RR is not equivalent to prompting the LLM to "think again" using the same input.

• Prompting the LLM to "think again" means asking the LLM to re-answer the question, while RR only reuses the representations of some key steps to "temporarily" intervene in its thinking process. In this way, the former requires far more computing resources than RR, so a direct comparison is unfair. In fact, prompting the LLMs to "think again" and RR are two orthogonal methods that can be used in combination. To verify this, we conducted new experiments using DeepSeek-R1-Distill-Llama-8B on AIME24.

  • Method	Result
    Original	33.0
    "think again"	50.0
    RR	40.0
    "think again" + RR	56.66

• The experimental results show that both "think again" and RR can improve the performance, but their working principles are different ("think again" requires significantly more computing resources). Moreover, "think again" can be used in combination with RR to further improve the model's performance.

---

Q4: "In evaluating TTTS, the paper assumes that appending thinking tokens improves reasoning by encouraging deeper reflection. However, the improvement could also stem from merely extending the output length. A more rigorous evaluation would involve testing whether appending other non-thinking but semantically neutral continuation tokens (e.g., “continue,” “more”) yields similar benefits. Without such a control, it is hard to determine whether the improvement is specifically due to the semantic role of thinking tokens or just the increased token budget."

A4: Thanks for your comments.

• We agree with your point that "merely extending the output length can improve model performance." We already mentioned this in Lines 267–271, and the performance of the "Origin" line in Figure 7 (where only the token budget is increased) also confirms this observation.

• In addition, we have followed your suggestion to conduct new continuous generation experiments using neutral tokens such as "continue" and "more."

  • Budget	Origin	TTTS	Continue&More
    512	57.2	59.83	59.9
    1024	63.2	65.65	65.2
    2048	70.4	72.02	70.5
    3072	73.2	75.35	72.9
    4096	75	77.27	75.4

  • The results show that continuation with neutral tokens like "continue" and "more" is less effective than continuation with thinking tokens.