The Potential of CoT for Reasoning: A Closer Look at Trace Dynamics
Authors: Gregor Bachmann
Yichen Jiang
Seyed-Mohsen Moosavi-Dezfooli
Moin Nabi
Venue: ICLR 2026 Poster (2026)
Keywords: Reasoning
Chain-of-thought
Mathematical reasoning
Primary area: foundation or frontier models, including LLMs

Abstract: Chain-of-thought (CoT) prompting is a de-facto standard technique to elicit reasoning-like responses from large language models (LLMs), allowing them to spell out individual steps before giving a final answer. While the resemblance to human-like reasoning is undeniable, the driving forces underpinning the success of CoT reasoning still remain largely unclear. In this work, we perform an in-depth analysis of CoT traces originating from competition-level mathematics questions, with the aim of better understanding how, and which parts of CoT actually contribute to the final answer. To this end, we introduce the notion of a \textit{potential}, quantifying how much a given part of CoT increases the likelihood of a correct completion. Upon examination of reasoning traces through the lens of the potential, we identify surprising patterns including (1) its often strong non-monotonicity (due to reasoning tangents), (2) very sharp but sometimes tough to interpret spikes (reasoning insights and jumps) as well as (3) at times lucky guesses, where the model arrives at the correct answer without providing any relevant justifications before. While some of the behaviours of the potential are readily interpretable and align with human intuition (such as insights and tangents), others remain difficult to understand from a human perspective. To further quantify the reliance of LLMs on reasoning insights, we investigate the notion of CoT transferability, where we measure the potential of a weaker model under the partial CoT from another, stronger model. Indeed aligning with our previous results, we find that as little as 20% of partial CoT can ``unlock'' the performance of the weaker model on problems that were previously unsolvable for it, highlighting that a large part of the mechanics underpinning CoT are transferable.

Published as a conference paper at ICLR 2026
THE POTENTIAL OF COT FOR REASONING: A CLOSER LOOK AT TRACE DYNAMICS
Gregor Bachmann, Yichen Jiang, Seyed-Mohsen Moosavi-Dezfooli & Moin Nabi Apple
ABSTRACT
Chain-of-thought (CoT) prompting a de-facto standard technique elicit reasoning- like responses from large language models (LLMs), allowing them to spell out individual steps before giving final answer. While the resemblance human-like reasoning is undeniable, the driving forces underpinning the success CoT reasoning still remain largely unclear. In this work, we perform an in-depth analysis of CoT traces originating from competition-level mathematics questions, with the aim of better understanding how, and which parts of CoT actually con- tribute to the final answer. To this end, introduce the notion of potential, quantifying how much a given part of CoT increases the likelihood of a correct completion. Upon examination of reasoning traces through the lens of the poten- tial, we identify surprising patterns including (1) its often strong non-monotonicity (due reasoning langents), (2) very sharp but sometimes tough interpret spikes (reasoning insights and jumps) as well as (3) at times lucky guesses, where the model arrives at the correct answer without providing any relevant justifications before. While some of the behaviours of the potential are readily interpretable and align with human intuition (such as insights and tangents), others remain dif- ficult to understand from human perspective. To further quantify the reliance
LLMs on reasoning insights, we investigate the notion of CoT transferabilityy where we measure the potential of weaker model under the partial CoT from another, stronger model. Indeed aligning with our previous results, we find that little 20% of partial CoT can "unlock" the performance of the weaker model on problems that were previously unsolvable for it, highlighting that a large part of the mechanics underpinning CoT are transferable.
INTRODUCTION
Chain-of- thought (CoT) reasoning (Wei et al. 2023) has led to several breakthroughs in domains
ranging from mathematics (Cobbe et al. 2021; Gao et al. 2023 Luo et al. 2025; DeepSeek-A Al
Li al. 2022] Lozhkov et al. 2024] Rozière et al. 2024), enabling modern language models to now win gold medals at mathematical olympiads (Luong al. 2025). The underlying idea of CoT is very simple and intuitive: let the model reason through the given problem and explain its steps before giving final answer. This approach offers two main advantages:
et al. (225) to coding (Chen et al.) 2021 Austin al. 2021
(1) Generating additional tokens means more computation available to the model, allowing it to implement more complex routines. (2) CoT enables the model to decompose complex problems into more manageable sub-tasks, akin human reasoning.
The success of chain-of -hhough reasoning is undeniable, yet the precise mechanisms driving remain poorly understood. very tempting explanation, due to their (by design) strong resemblance to human reasoning, is that LLMs similarly benefit from spelling out bigger computations more slowly, using techniques such as backtracking and verification
explore several avenues before
finally arriving at the best answer (Zhou et al. 2023 [Shinn et al., 2023 Madaan et al.) 2023b Press) cet al. 2023). Other works however suggest that the content of CoTs might not always reflect the actual solving strategy of the model, for instance Lanham et al. 2023 ) Chen et al. (2025b) show that Lanham the model's s explanations to addition task do not line up with the underlying computation performed internally. This result seems to rather suggest that CoTs primarily act computational mechanisms, letting the model execute more complicated algorithms or heuristics "under the hood" while at the same time mimicking human reasoning.

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Published as a conference paper at ICLR 2026
Question:
Spawn muliiple responses
Whstlss
Vi car identity
Condition partel CoT CO
s/n 12/m 22
FS
withrs
Prompt
Partial CoT
Compute inide:
5-28 84, 764
Figure 1: Left:
Illustration of the calculation of the potential. Right: An example prompt and partial COT, which in this case should intuitively raise the probability of success (i.c. the potential) significantly once discovered or provided by another model.
These perspectives motivate a closer look at how CoT actually contributes in practice. We therefore closely examine reasoning traces produced by several models with focus on competition-level mathematics questions from AIME-2024, AIME-2025 (MAA 2025) and MATH-500 (Hendrycks) et al., (021), GPQA-Diamond (Rein et al. (023) for more general reasoning, as well as coding problems from HumanEval (Chen et al. 2021). p. In the main text we focus on AIME, as its difficult questions present an ideal arena study properties of reasoning chains, especially as modern models still produce highly variable CoTs for the same question, sometimes leading to the correct solution, but often failing to do In this paper, aim to understand what, which parts of CoT make successful or wrong? To gain insights into this question, we introduce the notion of the potential, defined the probability of success of the model when sampling conditioned chain of the CoT (see Eq. for precise definition). As the potential initially starts out low (models can only sometimes arrive the right answer), we can use it to monitor precisely which tokens (or collection thereof) increase or decrease it, equipping us with tool to understand what parts of CoT unlock a previously difficult problem. We observe that similarly to humans, LLMs often exhibit reasoning insights, i.e., strong increases of the potential due to the completion of a conceptually difficult step (see e.g., Fig. [ 5 9 12 13 or 15). Not all spikes the potential are easily interpretable however; we find that performance can significantly increase through seemingly trivial steps, coined reasoning far from monotonic, i.c. not every token contributes effectively towards the final answer but rather long durations of no progress or even sharp drops can occur. The latter are often due to reasoning tangents, i.e. approaches which initially look promising but ultimately lead to dead ends or wrong answers, (see e.g. Fig. 12 13 14
given partial
(see e.g. Fig. 5 or Fig. (6) Surprisingly, we observe that the potential is
To further study the usage of reasoning insights in language models, investigate the degree of transferability of CoT between different models. We focus on providing a weaker model with the (partial) CoT from stronger one, with the motivation that if models indeed struggle with conceptual understanding of the problem, their reasoning might be unblocked when being provided correct sub- steps. Indeed, difficult mathematical questions often involve solving several steps of non-uniform difficulty, with some problems even becoming mostly trivial for humans once a specific insight is obtained or provided. An illustrative example of such a question is shown in Fig. I taken from AIME-1988 1989 (MAA 2025). While the question might look intimidating to many math students at first sight, the problem becomes easily solvable when presented with the insight that n(n + 1) 1)(n + 2) + (n² 333+ 1)2. In other words, human reasoning is often able to transfer if the gap not too large. For LLMs, we find similar results; problems that were previously unsolved by the weaker model, gradually become solvable as more and more CoT is provided, even as little as 20% of CoT leads to a significant improvement in performance. We observe such transfcrability even between very different model classes, e.g. Qwen3-0.6B's accuracy significantly improves when provided with partial CoT of GPT-OSS-205 This suggests that language models can profit from insights provided by stronger models, suggesting that some CoT's work in model-agnostic way.
RELATED WORK
Chain-of- Chain-of-Thought reasoning has been very influential in recent years, with every modern language model now being trained to give reasoning-like responses. This characteristic has been strongly
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Published as a conference paper at ICLR 2026
Wrong
Correct
Correct
Wrong
1.0
0.5
Potential
0.6
Potential
0.6
0.4
0.
0.0
200
600
600
830
1000
CoT tokens
CoT tokens
Figure 2: Potential curves. Potential of correct and wrong CoTs for Qwen2.5-7B AIME-2024,
Question and 11. Strongly non-mon non-monotonic behaviour for both correct and incorrect CoTs.
exacerbated by the emergence of reasoning models such as ol (OpenAl al., 2024 and R1 [DeepSeek-Al et al. 2025), further encouraging longer responses by training with reinforcement learning with verifiable rewards. Such models now regularly require generating 128k tokens for difficult mathematics questions before returning final answer. The still human-lik nature of these reasoning chains has inspired a surge of works with the aim of interpreting and understanding how these long sequences of tokens actually contribute to the final answer. A line of work has investi- gated how models react when their CoT manipulated through insertion mistakes (Wang al.) 2023) or changes in symbols (Madaan et al. 2023a Madaan & Yazdanbakhsh 2022). finding them to be surprisingly robust. Other works have investigated several attribution strategies to identify important parts in CoT (Golovneva al. 2023 Berchansky al. [024 Wu et al. 2023). Opposite types of findings have also been made; Lyu et al. (2023): Lanham et al. (203)) ; Madsen et al. (2024) have observed that CoT does always reflect the underlying computation of the model, making thus difficult to pin-point helpful steps in the first place. Other works go step further and argue that CoT reasoning should not be compared human reasoning (Kambhampati et 2025 Stechly et al. 2025 |025 Bhambri et al. 2025) or that they outright imitate reasoning without actually performing any (Shojace al. 2025). Finally, the line of works most similar to ours also studies conditional generation from partial CoTs; Bigelow et al. (2025) investigate so-called "fork tokens" in the con- text of neural text generation, [Bogdan et al. (2025) also explore the notion conditional generation to find "thought anchors", parts of CoT that help the model arrive at correct answers. While their focus on more abstract reasoning concepts such as backtracking and self-verification, we focus on task-relevant insights and also explore the failure modes of CoT reasoning. Finally, Amani et al. (2025) also explore the notion of completing partial CoTs, incorporating the idea in reinforcement learning for better reward signal. Our latter definition of the potential shares strong resemblance to the value function in actor-critiuue models (Konda & Tsitsiklis) 1999; Sutton & Barto 2018), similarly measuring the quality of given state but in Monte-Carlo fashion.
POTENTIAL OF CoT
Setup. Let V denote the vocabulary. Assume we have a tokenized input prompt a E yD (e.g.. encoding a math question) and a ground truth answer y" E V (for simplicity represented by single token) encoding the expected response (e.g. "513"), Let LM, represent a language model with parameters 0, mapping a sequence of tokens to the logits of size [V). When answering to a prompt, models now generate T E N intermediate chain-of-thoughi tokens e VT auto- regressively before arriving at a final answer y. Le. given prompt æ. we generate & ~ LMa(+ x) auto-regressively and only then sample the answer, intermediate tokens have been observed to outperform models trained (or prompted) to directly provide answer in variety of settings (Wei et al. 2023). We will often abuse notation slightly by letting (y, C2.) ~ LMg (+|c2,, x) denote the (sequential) autoregressive generation, conditional on (C<t, 2. Typical decoding strategies in language models leverage this stochastic generation and
[Ma(+lc, æ)) Generations involving such
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Published as a conference paper at ICLR 2026
Potential
Optimal Potential
Potential
Optimal Potential
0.8
Accuracy
0.4
0.4
0.2
0.0
200
600
tokenss
takens
Figure 3: The potential of optimal and standard CoT for AIME-2025-1, question I and 5. While standard CoT eventually arrives at the right answer, optimal CoT does so in a more robust way.
hence interesting consider K N such generations by varying the random seeds, either unconditionally or starting from a partial CoT Can.
(k)
i.d. (M)(+Cc..))) for l =,,,,.. I,...,K
where we obtain most likely distinct CoT completions (k) and final answers (
Potential. Given prompt æ and an associated reasoning process e with final answer y. it natural ask which sub-steps in contributed most to the overall result. Let define the potential of chunk of CoT Cat on the prompt as the probability of correct generation conditioned on Cer.
pot(cer; x) = ((exxa))xXo(Heezee (y =y)
Intuitively, if chunk of CoT is useful or encompasses a step that the model tends struggle with, conditioning on it should subsequently lead to a higher potential. In mathematical terms, conditioning on a shorter prefix Ccs for pot(Cc.; z) < pot(c<e; x), this implies that the CoT chunk Caet "made progress" towards the final solution. On the other hand, the potential remains similar, pol((c.;; x) R pot(Cer), then chunk of CoT Cact did not solve step that is difficult to the model, as it can reliably reproduce it under sampling. This does not necessarily imply that such steps can skipped as they could en- tail necessary computations such as a long multiplication, which the model can reliably do but also needs to do. Finally, we can have situations where the potential decreases, with CoT's actively wors- ening the state of the model. On average however, improves monotonically over correct CoTs:
has a lower potential compared Cet i.e.
show mathematically that the potential
Proposition 1. Conditional on the event that the full CoT CuT yields the correct final answer y", it holds for every t < T that
E [pot(c)); S [po((1cc11;)))
We invite the reader to check the proof in Appendix A.2 Hence on average, every token e should push the potential higher, encouraging the model converge towards the correct solution, reflect- ing the intuition that chain-of-thought performs evidence accumulation. Calculating the potential exactly unfortunately intractable, so in practice use the following estimator instead,
pot (Ceri x)
Σ
(g(m)y'' where (1))),
(m)
LMa (+/c.r, x)
N
Sampling a higher number of trajectories N will provide a better approximation to the true potential. We observe that setting N 128 gives very reliable estimations of the potential and use it through- out this work. We provide more experimental details regarding the computational complexity in Appendix A.5
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Published as a conference paper at ICLR 2026
Pasagi Coreeced
Fassgk
20
AINE2025 Accuracy (%)
15
A1M72175
20-
Figure 4: Inflated k. We show pass@k accuracies and the corresponding corrected values for Qwen2.5-1.5B (left) and Qwen2.5-7B (right).
SHAPE OF POTENTIAL CURVES
We now empirically study the potential pot(Ce;; a)) function of the CoT chunk length f. When conditioning on CoT's e that lead to the correct answer yf, based on Prop. | we expect the potential be smooth and monotonic function in with every chunk of CoT Ca<s positively con- tributing to the overall solution. We will mainly focus on difficult competition-level mathematics questions, where the potential pot(c<u; x) corresponding tween O and 1, i.e. the model only sometimes produces the correct answer when prompted from scratch. the model is always correct, the potential does not offer any insight into the CoT; steps are equally easy to the model. In contrast, if performance starts significantly lower, we can precisely pinpoint where a successful CoT overcame hurdles that stopped most other attempts. We calculate the potential curves for variety of models, including both the non-thinking types of mod- els Qwen2.5 (sizes 1.5B and 7B), (Qwen al. 2025) and Llama-3. (sizes 8B and 70B) (Grattafiori et al., 2024). well the reasoning models Qwen3 (sizes 0.6B and 32B) YYng al. 2025). We display variety potential curves (both for correct and wrong trajectories) in Fig. 2 for two samples taken from AIMB-2024. Surprisingly, typical chain-of-thought exhibits quite erratic poten- tials, with certain sections of CoT actively worsening the probability of success, going against the theoretical result Prop. I We will examine the characteristics of potential curves qualitatively
the "empty" CoT Cęo is strictly be-
in close detail in Sec. 5 We quantify the following properties of potential curves often exhibited across AIME-2024:
small token window, we will later refer to these occurrences reasoning insights and jumps.
(1) Very sharp increases in the potential
(2) Very sharp drops in the potential, we coin this behaviour reasoning tangents or flaws. (3) Extremely late increases in the potential,
which previously remained flat and close O. We will show qualitatively in Sec. that such CoTs are very often associated with guessing, i.e. the model produces correct answer without relying on its previously generated reasoning and at times even admits to do so.
MODEL
REASONING
INSIGHTS
TANGENTS J
LATE SPIKE
MONOTONICITY
QWEN2.5-1.50
X
40%
5%
20%
45%
QWEN2.5-7B
X
62%
9.5%
14%
42%
LLAMA3. 1-88
X
46%
33%
6%
15%
LLAMA3. 1- 1-70B
X 37%
40%
5%
17%
QWEN3-0.6B
55%
41%
10%
15%
QWEN3-32B
36%
18%
0%
36%
Table 1: Behaviours of potential for several reasoning and non-reasoning models AIME-2024.
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Question ------VE--- Prediom 15)
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1.0
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6
Response
0.8
Fied mnmmnn ensmurnvaaaeee odterr loyenn
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0.6
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poly
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0.0
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1000
# CoT tokens
salve doweharddcccattnn stael Sika non 2n
4
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91 129 hrquent
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Figure Reasoning tangents and insights. Qwen2.5-7B's potential pot256(1;22) behaving strongly monooonot.... The reasoning tangent @ hurts the potential, while the reasoning insights 3 (observing the symmetry x = y of the problem) and ④ (finding the root of the cubic equation) push the potential back on track. Finally, the model performs reasoning jump (5) (for some non- obvious reason, this particular calculation is difficult for the model).
Quantifying the shape. In the following, we will derive some quantitative summary statistics corresponding to the observations we made based on the plots in Fig. 2 We calculate the potentials for 128 responses per sample on AIME-2024 (total of 30 x 128 samples) and filter out responses that led to wrong answers. We further only consider samples that are difficult enough for the given model to not reach perfect accuracy without any partial COT. We then derive four summary statistics that aim to describe the properties introduced above, we detail their precise definitions in Appendix A.7
We show analogous results on MATH-500 in Table in the Appendix.
We display the results in Table Our initial observations are substantiated; only half of the CoTs exhibit monotonicity, with reasoning models tending to produce even more erratic potentials. Non- reasoning models seem to exhibit more late spikes, which aligns with our qualitative observations later in Sec. [5] that such models tend to produce correct answers often through guessing on very difficult problems. Model size also seems to surpress this behaviour more, which expected since larger models generally tend perform better. Reasoning tangents occur the literature that such models have the tendency to
often for reasoning
models, aligning well with the observation overthink (Chen al., 2025a), i.e. they discard the discovered, correct answer and explore alterna- tive but flawed approaches. This also partially explains their less monotonic potential. All models exhibit high amount of reasoning insights, suggesting that most of the difficulty is concentrated in a few key steps instead of being uniformly spread out, more akin to human reasoning.
Amount of guessing. We now focus on the situations where the potential spikes very late in the reasoning process, which we connect almost uniquely with the model guessing answers correctly. This has strong effect the pass@k metric used to assess model capabilities, which show strongly impacted in the case of Qwen2.5-1.5B and 7B when used in conjunction with weak verification such as final answer verification in mathematical benchmarks. For a dataset consisting of P queries ) with corresponding answers {Mr He we sample responses y; per question from the model and measure if the correct answer is at least once among this set. i.e.
P
pass@k Is =
1 p
Σ
v Elv
.....)

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Published as a conference paper at ICLR 2026
Question - Probfem
speed
karmarrr per hast valk tncluding and mkmmidtee
the corfee shop. When
spont sperrss-pprr rumber minuss walk sakas incladingttin the munutte spenti cof
ding vunere
oaleelecoooo bupunse bye wat ata + filerdters per thoa, risd the
1.0
Response
We
I Most comp leated part human perspective, sclving o discarding negative soluton
Potential
0.6
eqpatlions,
2/4
[n.]
Relatively amail Increase In potential!
givecus atlone
-4.5
Sinee spoed carnet be regat we disuud
4.5 thendor, a = 2.5.
0.0
2
Seeriingly humane obsain ij but dffuuut for model
600
# CoT tokens
Thii the numbor of manatoo Sersttoioooo the with lakes he; incldding whc.rrnstooeerrre coee cofesshss 204
Figure 6: Unaligned difficulty. Qwen2.5-1.5B solves most difficult parts in 1 but only small increase in potential. Scemingly easier part @ of just obtaining given and adding the two turns out to be significantly more difficult.
The idea behind this metric is to measure whether the model does possess the ability sometimes achieve the right answer, albeit not reliably. Especially for large k, this metric could fall victim to lucky guesses (1) only takes one correct answer obtain the full score and (2) the reasoning process is usually not being assessed in the case of mathematics benchmarks. Indeed, in Fig. 4 we show that the pass@k scores can be very inflated by flagging samples with the late spike statistic, in this case on AIMB-2025.
Optimizing the potential. Given observation that CoT does not naturally follow monotonic curve, with many tokens even worsening performance, the following question emerges:
Can we search the space of CoT c such that every sub- CoT Cust contributes?
One way to try and maximize the potential of every chunk of CoTT is randomly explore candidate chunks, calculate their potential and keep the highest scoring chunk. In this manner,
a chunk size C N and
construct CoT that increases the potential at least gradually if the model admits such reasoning, ideally avoiding issues such as reasoning tangents. We summarize the recipe in Algorithm D and the incurred computational complexity in Appendix | more formally. We indeed find that models admit such optimized CoT, we display some associated potential curves in contrast with regular CoT Fig. 3 We can indeed see that the optimized CoT displays strong monotonicity with most tokens contributing to the potential. This is in stark contrast with the standard Co',, which either does not increase the potential for long token horizon (left side of the figure), or even actively worsens it (right side). We examine such CoTs more qualitatively in Appendix A.3
A CLOSER LOOK AT CHAIN-OF-
REASONING
We now perform a qualitative analysis of various chain-of-though reasonings on competition-level mathematics. Due to the verbosity of reasoning models such as Qwen3, we limit this section to the Qwen2.5 series, whose CoT is more concise and thus more amenable to direct interpretation. The only exception is Fig. 7 where we display parts of trace from Qwen3-0.6B. Our goal is to precisely align the potential curve with the underlying reasoning produced by the model, and as consequence obtain an understanding space reasons we defer from displaying the full CoT but instead show only the sections crucial to the potential. We refer the interested reader to Appendix A..3 for additional qualitative examples. We display the first sample obtained from Qwen2.5-7B strong non non-monotyy as we have often encountered (see Fig. 2). We dissect the reasoning into five segments according to the potential. Segment ① steadily makes progress towards the solution
the types of tokens that drive or hinder the progress. For
Fig. 5 potential curve that exhibits

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Published as a conference paper at ICLR 2026
Question
1.0
intagerss Then Bob fsis nenerpply B inttaees with the property that the masimum element /2 telongsto A. fubis list has 2004 wats, Find the sum
08-
6.6
Response
[-.]
e Long ressoning tangents and flaws
Given that cannot resolve the contradictio, but the problem givee,, and assuming that there typo, and intanded answerts that |4] 11, then
Altematively irt''s
Since
0.2
2024 is 2025 maybe answeris 55, Therefore, think answer 55
2 Model takes guess and admils so even!
0.0
Answer
0
1000
2000
3000
4000
9000
6000
CoT tokens
Figure 7: Reasoning tangents and guessing. Qwen3-0.6B goes on a long reasoning tangent in ! that does not increase the potential over long token horizon. Finally it outputs final answer in 2, itself admitting that the guess is not backed by the reasoning prior but seems likely to the model.
by correctly expressing the radius as a function of the sides of the box and formulating the optimiza- tion problem. In segment ( the model goes on a reasoning tangent, a step that is not necessarily wrong but happens to not work out for the particular problem (AM-GM inequality gives a non-tight lower bound for the minimum). The model manages ignore this step this particular trajectory, but on average suffers from this distraction, leading to sharp drop in the potential. In segment ( and ( the model correctly recognizes the symmetry of the problem as well as discovers the root of the cubic equation, with both insights consequently boosting the accuracy akin human reasoning. Finally, segment (5) we observe the final spike the potential, stemming from a simple arith- metics step that the model tends to get wrong. While the previous spikes were readily interpretable, the last one seems more unintuitive, given that the model manages to very reliably perform the ar- guably harder arithmetics steps just before. We coin this reasoning jump, very sharp increase in the potential that largely seems due to a very model-specifie issue.
Such misalignment in perceived difficulty of sub-steps is often present in CoT, in Fig, 6 we dis- play another reasoning trace of Qwen2.5-78 7- along with the associated potential which exhibits this surprising characteristic. Segment ! here does the conceptual heavy-lifting: it correctly deduces the associated system of equations in two variables, simplifies and obtains the solution for the first variable The completion of these seemingly involved steps is only rewarded with small increase in potential, as opposed to humans, the model does not struggle here. Instead, the more difficult steps contained in segment ( consist of now obtaining the second variable which only involves plugging the value for s into the previously derived equation. Compared to the previous segment, finishing the problem starting from the end of ! would be significantly simpler task for humans.
Another surprising insight we obtained is that models can be very capable of guessing solutions to such problems. In Fig. (and Fig. TOp we display the reasoning of Qwen3-0.6B, While the content of segment ! at first sight looks relevant, closer inspection reveals that the final answer "80" is not all deduced from the reasoning performed before. The answer seems be lucky guess, most likely informed by the fact that answers to such competition-level questions usually take the form of an integer value. This guessing is elegantly reflected in the potential curve; the reasoning segment I (which essentially encompasses the entire CoTT) does not make any progress at all towards the final answer, precisely because the model is most likely making a guess in the end, which more often than not ends up being wrong.
TRANSFERABILITY OF COT THROUGH THE LENS OF POTENTIAL
Motivated by the insights from Sec. 4 and 5 we now investigate if reasoning insights transfer be- tween different families of models, which would further underscore that the mechanisms underlying CoT reasoning share parallels with human reasoning. We hypothesize that if the sub-steps present information gain through reasoning insights, (similar to e.g. Fig. [.. weaker models could able to solve problems that were previously too difficult. We study this scenario for both reasoning and non- reasoning models. In the first setup we consider Qwen3-0.66 as the weak model, which is provided with partial CoT from its bigger version Qwen3-32B. We also explore traces from GPT-OSS-20R to
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Ower2 5.79
Owen2 7-72
007-055522 COT
Owenn 328 CST
62:2
2x-37
AIME-2025
AMM: 201-
10
04 Praction CoT
Fraction of CoT
Figure 8: Transferability of COT. Left: Accuracy on AIME-2025 weaker reasoning model Qwen3-0.6B when provided with partial CoT from Qwen3-32B (red) and from GPT-OSS-200 (or- ange), leading to very quick improvements. Right: Accuracy of non-reasoning models Qwen2.5-7B and Qwen2.5-72B when provided with partial CoT based the final summary output of reasoning model Qwen3-235B.
further assess how robust transferability is with respect to out- out-of-distribution scenarios. For the non- reasoning models we instead create dataset of gold CoT, using one of the strongest public models Qwen3-235B to produce answers on AIME-2024 in thinking mode. We then extract the CoT after thinking, which presents clean summary of the long thinking traces and use these as partial traces. We then test the weaker Qwen2.5-7B and Qwen2.5-72B models, letting them complete the partial responses for various percentages. We display the resulting test accuracies as a function the frac- tion of partial CoT in Fig. 8 We observe that surprisingly, in both reasoning and non-reasoning scenarios, the models manage to not only maintain their original accuracies but quickly improve (with little 20% CoT), answering previously unsolved questions. While the CoT does seem to transfer better within the same family, Qwen3-0.61 can still leverage the significantly different traces from GPT-OSS-20R suggesting that the mechanisms driving the performance universally shared between models to strong degree.
CONCLUSION
this work we have investigated chain-of-thought reasoning large language models through notion of the associated potential. We have performed in-depth analysis of parts of CoT that strongly move the potential upwards (reasoning insights and jumps), as well as tokens that actively worsen the performance due reasoning tangents. We further investigate reasoning insights by introducing the notion of CoT transferability, which measures to what degree a weaker model can profit from the partial CoT of stronger one. We show that the insights of the stronger model indeed help push the performance of the weaker one beyond what it can typically solve on its own, highlighting that CoT indeed relies on such interpretable mechanisms.
We believe that the potential can be used in future work to further understand how CoT contributes to successful completions of L.Mss, helping pin-point its important parts within the very long chains of today's is reasoning models. The potential could also serve as a partial reward in RI training to perform more fine- -reigra credit assignment, indeed concurrent work has already started exploring this angle (Guo et al. 22225] Hou et al. 2025). Finally, we also hope that the transferability of partial CoTs can be leveraged reinforcement learning to reduce sparsity of rewards with Amani et al. 2025 already obtaining positive initial results.
REFERENCES
Mohammad Hossein Amani, Aryo Lotfi, Nicolas Mario Baldwin, Samy Bengio, Mehrdad Fara- jtabar, Emmanuel Abbe, and Robert West. RI for reasoning by adaptively revealing rationales,

---

Published as a conference paper at ICLR 2026
2025. UKL/Etpps//axxxxv.oo//////////
Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, and Charles Sutton. Program synthesis with large language models, 2021. URL http:::: httppt://axxiii..ooooeee
/ 280.0001010 07732
Moshe Berchansky, Daniel Fleischer, Moshe Wasserblat, and Peter Izsak. CoTAR: Chain-of-thought attribution reasoning with multi-level granularity. In Yaser Al-Onaizan, Mohit Bansal, and Yun- Nung Chen (eds.), Findings of the Association for Computational Linguistics: EMNLP 2024, pp. 236-246, Miami, Florida, USA, November 2024, Association for Computational Linguistics. doi: 10. 101138866110000 1ine2044.inningg-enn....... URL htttps://cllantholoog.ooo wrg.:/looooor//22220 Eindings- www.gs--emn..... 13/
Siddhant Bhambri, Upasana Biswas, and Subbarao Kambhampati. Do cognitively interpretable
reasoning traces improve Ilm performance?, 2025. URK//ttps::/axxi.oo///////
16695
Eric Bigelow, Ari Holtzman, Hidenori Tanaka, and Tomer Ullman. Forking paths in neural text generation. In The Thirteenth International Conference on Learning Representations, 2025. URL
https::/openreview net oot.:e//ooormm22dddeeeeeeeeeee
Paul C. Bogdan, Uzay Macar, Neel Nanda, and Arthur Conmy. Thought anchors: Which I1m rea- soning steps matter?, 2025. URL https::://// /ttps:///rxivv org/ abs 2506 19143
Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Broekman, Alex Ray, Raul Puri, Gretchen Krueger, Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke Chan, Scott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet, Felipe Petroski Such, Dave Cummings, Matthias Plappert, Fo- tios Chantzis, Elizabeth Barnes, Ariel Herbert-Voss William Hebgen Guss, Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin, Suchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, Andrew N. Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder, Bob Mc- Grew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojeiech Zaremba. Evaluating large language models trained on code, 2021. URL BoLhhtss:: /ttp///axvvvv/// org/ abs 03374
Xingyu Chen, Jiahao Xu, Tian Liang, Zhiwei He, Jianhui Pang, Dian Yu, Linfeng Song, Qiuzhi Liu, Mengfei Zhou, Zhuosheng Zhang, Rui Wang, Zhaopeng Tu, Haitao Mi, and Dong Yu. Do NOT think that much for 2+3=? on the overthinking of long reasoning models. In Forty-second International Conference on Machine Learning, 2025a. URL
MSbU3
Yanda Chen, Joe Benton, Ansh Radhakrishnan, Jonathan Uesato, Carson Demison, John Schulman, Arushi Somani, Peter Hase, Misha Wagner, Fabien Roger, Vlad Mikulik, Samuel R. Bowman, Jan Leike, Jared Kaplan, and Ethan Perez. Reasoning models don't 't always say what they think, 2025b. URL lhttps://axxii org/ /abs/2555. 05410
Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Rejichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems, 2021. URL atttps://arxx.
org/abs/ /2100. 14168
DeepSeck-AL, Daya Guo, Dejian Yang, Haowei Zhang. Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang. Xiao Bi, Xiaokang Zhang, Xingkai Yu, Yu Wu, Z. F. Wu, Zhibin Gou, Zhihong Shao, Zhuoshu Li, Ziyi Gao, Aixin Liu, Bing Xue, Bingxuan Wang, Bochao Wu, Bei Feng, Chengda Lu, Chenggang Zhao, Chengqi Deng, Chenyu Zhang, Chong Ruan, Damai Dai, Deli Chen, Dongjie Ji, Erhang Li, Fangyun Lin, Fucong Dai, Fuli Luo, Guangbo Hao, Guanting Chen, Guowei Li, H. Zhang, Han Bao, Hanwei Xu, Haocheng Wang, Honghui Ding, Huajian Xin, Huazuo Gao, Hui Qu, Hui Li, Jianzhong Guo, Jiashi Li, Jiawei Wang, Jingchang Chen, Jingyang Yuan, Junjie Qiu, Junlong Li, J. L. Cai, Jiaqi Ni, Jian Liang, Jin Chen, Kai Dong, Kai Hu, Kaige Gao, Kang Guan, Kexin Huang, Kuai Yu. Lean Wang, Lecong Zhang, Liang Zhao, I Litong Wang, Liyue Zhang, Lei Xu, Leyi Xia, Mingchuan Zhang, Minghua Zhang,
10

---

Published as a conference paper at ICLR 2026
Minghui Tang, Meng Li, Miaojun Wang, Mingming Li, Ning Tian, Panpan Huang, Peng Zhang, Qiancheng Wang. Qinyu Chen, Qiushi Du, Ruiqi Ge, Ruisong Zhang, Ruizhe Pan, Runji Wang, R. J. Chen, R. L. Jin, Ruyi Chen, Shanghao Lu, Shangyan Zhou, Shanhuang Chen, Shengfeng Ye, Shiyu Wang, Shuiping Yu, Shunfeng Zhou, Shuting Pan, S. S. Li, Shuang Zhou, Shaoqing Wu, Shengfeng Ye, Tao Yun, Tian Pei, Tianyu Sun, T. Wang, Wangding Zeng, Wanjia Zhao, Wen Liu, Wenfeng Liang, Wenjun Gao, Wenqin Yu, Wentao Zhang, W. L. Xiao, Wei An, Xiaodong Liu, Xiaohan Wang, Xiaokang Chen, Xiaotao Nie, Xin Cheng, Xin Liu, Xin Xie, Xingchao Liu, Xinyu Yang, Xinyuan Li, Xuecheng Su, Xuheng Lin, X. Q. Li, Xiangyue Jin, Xiaojin Shen, Xi- aosha Chen, Xiaowen Sun, Xiaoxiang Wang. Xinnan Song, Xinyi Zhou, Xianzu Wang, Xinxia Shan, Y. K. Li, Y. Q Wang. Y. X. Wei, Yang Zhang. Yanhong Xu, Yao Li, Yao Zhao, Yaofeng Sun, Yaohui Wang, Yi Yu, Yichao Zhang, Yifan Shi, Yiliang Xiong, Ying He, Yishi Piao, Yisong Wang, Yixuan Tan, Yiyang Ma, Yiyuan Liu, Yongqiang Guo, Yuan Ou, Yuduan Wang, Yue Gong, Yuheng Zou, Yujia He, Yunfan Xiong. Yuxiang Luo, Yuxiang You, Yuxuan Liu, Yuyang Zhou, Y. X. Zhu, Yanhong Xu, Yanping Huang, Yaohui Li, Yi Zheng, Yuchen Zhu, Yunxian Ma, Ying Tang, Yukun Zha, Yuting Yan, Z. Z. Ren, Zehui Ren, Zhangli Sha, Zhe Fu, Zhean Xu, Zhenda Xie, Zhengyan Zhang. Zhewen Hao, Zhicheng Ma, Zhigang Yan, Zhiyu Wu, Zihui Gu, Zijia Zhu, Zijun Liu, Zilin Li, Ziwei Xie, Ziyang Song, Zizheng Pan, Zhen Huang, Zhipeng Xu, Zhongyu Zhang, and Zhen Zhang. Deepseek-rl: Incentivizing reasoning capability in Ilms via reinforce- ment learning, 2025. URL htLthhtt::/aart
org/ abs/ 250. 11550.1220 12948
Luyu Gao, Aman Madaan, Shuyan Zhou, Uri Alon, Pengfei Liu, Yiming Yang. Jamie Callan, and Graham Neubig. Pal: Program-aided language models, 2023, URL htttps://arxiv...
abs/2211 0435
Olga Golovneva, Moya Peng Chen, Spencer Poff, Martin Corredor, Luke Zettlemoyer, Maryam Fazel-Zarandi, and Asli Celikyilmaz. ROSCOE: A suite of metrics for scoring step-by-step rea- soning. In The Eleventh International Conference on Learning Representations, 2023. URL net http:/orrmmmmdxxy1P-----
Sttpss//oppereevin
Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, Amy Yang, Angela Fan, Anirudh Goyal, Anthony Hartshom, Aobo Yang, Archi Mitra, Archie Sravankumar, Artem Ko- renev, Arthur Hinsvark. Arun Rao, Aston Zhang, Aurelien Rodriguez, Austen Gregerson, Ava Spataru. Baptiste Roziere, Bethany Biron, Binh Tang, Bobbie Chern, Charlotte Caucheteux, Chaya Nayak, Chloe Bi, Chris Marra, Chris McConnell, Christian Keller, Christophe Touret, Chunyang Wu, Corinne Wong, Cristian Canton Ferrer, Cyrus Nikolaidis, Damien Allonsius, Daniel Song, Danielle Pintz, Danny Livshits, Danny Wyatt, David Esiobu, Dhruv Choudhary, Dhruv Mahajan, Diego Garcia-Olano, Diego Perino, Dieuwke Hupkes, Egor Lakomkin, Ehab AlBadawy, Elina Lobanova, Emily Dinan, Eric Michael Smith, Filip Radenovic, Francisco Guzmán, Frank Zhang, Gabriel Synnaeve, Gabrielle Lee, Georgia Lewis Anderson, Govind That- tai, Graeme Nail, Gregoire Mialon, Guan Pang, Guillem Cucurell, Hailey Nguyen, Hannah Kore- vaar, Hu Xu, Hugo Touvron, Iliyan Zarov, Imanol Arrieta Ibarra, Isabel Kloumann, Ishan Misra, Ivan Eytimov, Jack Zhang, Jade Copet, Jaewon Lee, Jan Geffert, Jana Vranes, Jason Park, Jay Ma- hadeokar, Ject Shah, Jelmer van der Linde, Jennifer Billock, Jenny Hong, Jenya Lec, Jeremy Fu, Jianfeng Chi, Jianyu Huang, Jiawen Liu, Jie Wang, Jiecao Yu, Joanna Bitton, Joe Spisak, Jong- soo Park, Joseph Rocca, Joshua Johnstun, Joshua Saxe, Junteng Jia Kalyan Vasuden Alwala, Karthik Prasad, Kartikeya Upasani, Kate Plawiak, Ke Li, Kenneth Heafield, Kevin Stone, Khalid El-Arini, Krithika Iyer, Kshitiz Malik, Kuenley Chiu, Kunal Bhalla, Kushal Lakhotia, Lauren Rantala-Yeary, Laurens van der Maaten, Lawrence Chen, Liang Tan, Liz Jenkins, Louis Martin, Lovish Madaan, Lubo Malo, Lukas Blecher, Lukas Landzaat, Luke de Oliveira, Madeline Muzzi, Mahesh Pasupuleti, Mannat Singh, Manohar Paluri, Marcin Kardas, Maria Tsimpoukelli, Mathew Oldham, Mathicu Rita. Maya Pavlova, Melanie Kambadur, Mike Lewis, Min Si, Mitesh Ku- mar Singh, Mona Hassan, Naman Goyal, Narjes Torabi, Nikolay Bashlykov, Nikolay Bogoy- chev, Niladri Chatterji, Ning Zhang, Olivier Duchenne, Onur Çelebi, Patrick Alrassy, Pengchuan Zhang, Pengwei Li, Petar Vasic, Peter Weng, Prajjwal Bhargava, Pratik Dubal, Praveen Krishnan, Punit Singh Koura, Puxin Xu, Qing He, Qingxiao Dong, Ragavan Srinivasan, Raj Ganapathy, Ra- mon Calderer, Ricardo Silveira Cabral, Robert Stojnic, Roberta Raileanu, Rohan Maheswari, Ro- hit Girdhar, Rohit Patel, Romain Sauvestre, Ronnie Polidoro, Roshan Sumbaly, Ross Taylor, Ruan Silva, Rui Hou, Rui Wang, Saghar Hosseini, Sahana Chennabasappa, Sanjay Singh, Sean Bell, Seohyun Sonia Kim, Sergey Edunov, Shaoliang Nie, Sharan Narang, Sharath Raparthy, Sheng
11

---

Published as a conference paper at ICLR 2026
Shen, Shengye Wan, Shruti Bhosale, Shun Zhang, Simon Vandenhende, Soumya Batra, Spencer Whitman, Sten Sootla, Stephane Collot, Suchin Gururangan, Sydney Borodinsky, Tamar Herman, Tara Fowler, Tarek Sheasha, Thomas Georgiou, Thomas Scialom, Tobias Speckbacher. Todor Mi- haylov, Tong Xiao, Ujjwal Karn, Vedanuj Goswami, Vibhor Gupta, Vignesh Ramanathan, Viktor Kerkez, Vincent Gonguet, Virginie Do, Vish Vogeti, Vítor Albiero, Vladan Petrovic, Weiwei Chu, Wenhan Xiong, Wenyin Fu, Whitney Meers, Xavier Martinet, Xiaodong Wang, Xiaofang Wang, Xiaoqing Ellen Tan, Xide Xia, Xinfeng Xie, Xuchao Jia, Xuewei Wang, Yaelle Gold- schlag, Yashesh Gaur, Yasmine Babaei, Yi Wen, Yiwen Song, Yuchen Zhang, Yue Li, Yuning Mao, Zacharie Delpierre Coudert, Zheng Yan, Zhengxing Chen, Zoe Papakipos, Aaditya Singh, Aayushi Srivastava, Abha Jain, Adam Kelsey, Adam Shajnfeld, Adithya Gangidi, Adolfo Victoria, Ahuva Goldstand, Ajay Menon, Ajay Sharma, Alex Boesenberg, Alexei Baevski, Allie Feinstein, Amanda Kallet, Amit Sangani, Amos Teo, Anam Yunus, Andrei Lupu, Andres Alvarado, An- drew Caples, Andrew Gu, Andrew Ho, Andrew Poulton, Andrew Ryan, Ankit Ramchandani, An- nie Dong, Annie Franco, Anuj Goyal, Aparajita Saraf, Arkabandhu Chowdhury, Ashley Gabriel, Ashwin Bharambe, Assaf Eisenman, Azadeh Yazdan, Beau James, Ben Maurer, Benjamin Leon- hardi, Bernie Huang. Beth Loyd, Beto De Paola, Bhargavi Paranjape, Bing Liu, Bo Wu, Boyu Ni, Braden Hancock, Bram Wasti, Brandon Spence, Brani Stojkovic, Brian Gamido, Britt Mon- talvo, Carl Parker, Carly Burton, Catalina Mejia, Ce Liu, Changhan Wang. Changkyu Kim, Chao Zhou, Chester Hu, Ching-Hsiang Chu, Chris Cai, Chris Tindal, Christoph Feichtenhofer, Cynthia Gao, Damon Civin. Dana Beaty, Daniel Kreymer, Daniel Li, David Adkins, David Xu, Davide Testuggine, Delia David, Devi Parikh, Diana Liskovich, Didem Foss, Dingkang Wang, Duc Le, Dustin Holland, Edward Dowling, Eissa Jamil, Elaine Montgomery, Eleonora Presani, Emily Hahn, Emily Wood, Eric-Tuan Le. Erik Brinkman, Esteban Arcaute, Evan Dunbar, Evan Smoth- ers, Fei Sun, Felix Kreuk, Feng Tian, Filippos Kokkinos, Firat Ozgenel. Francesco Caggioni, Frank Kanayet, Frank Seide, Gabriela Medina Florez, Gabriella Schwarz, Gada Badeer, Georgia Swee, Gil Halpern, Grant Herman, Grigory Sizov, Guangyi, Zhang, Guna Lakshminarayanan, Hakan Inan, Hamid Shojanazeri, Han Zou, Hannah Wang. Hanwen Zha, Haroun Habceb, Harri- son Rudolph, Helen Suk, Henry Aspegren, Hunter Goldman, Hongyuan Zhan, Ibrahim Damlaj, Igor Molybog, Igor Tufanov, Ilias Leontiadis, Irina-Flena Veliche, Itai Gat, Jake Weissman, James Gehoski, James Kohli, Janice Lam, Japhet Asher, Jean-Baptiste Gaya, Jeff Marcus, Jeff Tang, Jen- nifer Chan, Jenny Zhen, Jeremy Reizenstein, Jeremy Teboul, Jessica Zhong, Jian Jin, Jingyi Yang, Joe Cummings, Jon Carvill. Jon Shepard, Jonathan McPhie, Jonathan Torres, Josh Ginsburg, Jun- jie Wang. Kai Wu, Kam Hou U, Karan Saxena, Kartikay Khandelwal, Katayoun Zand, Kathy Matosich, Kaushik Veeraraghavan, Kelly Michelena, Keqian Li, Kiran Jagadeesh, Kun Huang, Kunal Chawla, Kyle Huang, Lailin Chen, Lakshya Garg, Lavender A, Leandro Silva, Lee Bell, Lei Zhang, Liangpeng Guo, Licheng Yu, Liron Moshkovich, Luca Wehrstedt, Madian Khabsa, Manav Avalani, Manish Bhatt, Martynas Mankus, Matan Hasson, Matthew Lennie, Matthias Reso, Maxim Groshev, Maxim Naumov, Maya Lathi, Meghan Keneally, Miao Liu, Michael L. Seltzer, Michal Valko, Michelle Restrepo, Mihir Patel, Mik Vyatskov, Mikayel Samvelyan, Mike Clark, Mike Macey, Mike Wang, Miquel Jubert Hermoso, Mo Metanat, Mohammad Rastegari, Munish Bansal, Nandhini Santhanam, Natascha Parks, Natasha White, Navyata Bawa, Nayan Singhal, Nick Egebo, Nicolas Usunier, Nikhil Mehta, Nikolay Pavlovich Laptev, Ning Dong. Norman Cheng, Oleg Chernoguz, Olivia Hart, Omkar Salpekar, Ozlem Kalinli, Parkin Kent, Parth Parekh, Paul Saab, Pavan Balaji. Pedro Rittner, Philip Bontrager, Pierre Roux, Piotr Dollar, Polina Zvyagina, Prashant Ratanchandani, Pritish Yuvraj. Qian Liang, Rachad Alao, Rachel Ro- driguez, Rafi Ayub, Raghotham Murthy, Raghu Nayani, Rahul Mitra, Rangaprabhu Parthasarathy, Raymond Li, Rebekkah Hogan, Robin Battey, Rocky Wang, Russ Howes, Ruty Rinott, Sachin Mehta, Sachin Siby, Sai Jayesh Bondu, Samyak Datta, Sara Chugh, Sara Hunt, Sargun Dhillon, Sasha Sidorov, Satadru Pan, Saurabh Mahajan. Saurabh Verma, Seiji Yamamoto. Sharadh Ra- maswamy, Shaun Lindsay. Shaun Lindsay, Sheng Feng, Shenghao Lin, Shengxin Cindy Zha, Shishir Patil, Shiva Shankar, Shuqiang Zhang, Shuqiang Zhang, Sinong Wang, Sneha Agarwal, Soji Sajuyigbe, Soumith Chintala, Stephanie Max, Stephen Chen, Steve Kehoe, Steve Satter- field, Sudarshan Govindaprasad, Sumit Gupta, Summer Deng, Sungmin Cho, Sunny Virk, Suraj Subramanian, Sy Choudhury, Sydney Goldman, Tal Remez, Tamar Glaser, Tamara Best, Thilo Koehler, Thomas Robinson, Tianhe Li, Tianjun Zhang, Tim Matthews, Timothy Chou, Tzook Shaked, Varun Vontimitta, Victoria Ajayi, Victoria Montanez, Vijai Mohan, Vinay Satish Ku- mar, Vishal Mangla, Vlad Ionescu, Vlad Poenaru, Vlad Tiberiu Mihailescu, Vladimir Ivanov, Wei Li, Wenchen Wang, Wenwen Jiang. Wes Bouaziz, Will Constable, Xiaocheng Tang, Xiao- jian WU, Xiaolan Wang, Xilun Wu, Xinbo Gao, Yaniv Kleinman, Yanjun Chen, Ye Hu, Ye Jia,
12

---

Published as a conference paper at ICLR 2026
Ye Qi, Yenda Li, Yilin Zhang, Ying Zhang, Yossi Adi, Youngjin Nam, Yu, Wang. Yu Zhao, Yuchen Hao, Yundi Qian, Yunlu Li, Yuzi He, Zach Rait, Zachary DeVito, Zef Rosnbrick. Zhao- duo Wen, Zhenyu Yang, Zhiwei Zhao, and Zhiyu Ma. The llama 3 herd of models, 2024. URL
pittps:/eardt
org/ abs /20//2407-22 21783
Yiran Guo, Lijie Xu, Jie Liu, Ye Dan, and Shuang Qiu. Segment policy optimization: Effective segment-level credit assignment in RL for large language models. In The Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025. URL net/ 2d= 1odiiddvooovbbbb
Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. Measuring mathematical problem solving with the math dataset, 2021.
URL httpps://aaxii..../:// org/ abs/ 2103. 03874
Zhenyu Hou, Ziniu Hu, Yujiang Li, Rui Lu, Jie Tang, and Yuxiao Dong. TreeRL: LLM reinforce- ment learning with on-policy tree search. In Wanxiang Che, Joyce Nabende, Ekaterina Shutova, and Mohammad Taher Pilehvar (eds.), Proceedings of the 63 Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 12355-12365 Vienna, Austria, July 2025. Association for Computational Linguistics. ISBN 979-8-89176-251-40 doi: 10.18653/v1/ 604/
wal...cc--ong.6000 URLhttts::/ httptttsss///aaaaaaae/// www.aannooooooooooooooooooooc UBL.attees//acc..nttoooooo.oo//p
Subbarao Kambhampati, Kaya Stechly, Karthik Valmeekam, Lucas Saldyt, Siddhant Bhambri, Vardhan Palod, Atharva Gundawar, Soumya Rani Samineni, Durgesh Kalwar, and Upasana Biswas. Stop anthropomorphizing intermediate tokens as reasoning/thinking traces!, 2025. URL
attps://arxivv org hbpp///b///// 2504 09762
Vijay Konda and John Tsitsiklis.
Actor-critic algorithms. In S. Solla, T. Leen, and K. Müller (eds.), Advances in Neural Information Processing Systems, volume 12. MIT Press, 1999. URL httt:::://pooeeiiinnnnrrrrr.oo.s
http:rppppccppppp....ppppe saappe_ffiees/ee files//aper/1999 ppff]
Woosuk Kwon, Zhuohan Li, Siyuan Zhuang. Ying Sheng, Lianmin Zheng, Cody Hao Yu, Joseph E. Gonzalez, Hao Zhang, and lon Stoica. Efficient memory management for large language model serving with pagedattention. In Proceedings of the ACM SIGOPS 29th Symposium on Operating Systems Principles, 2023.
Tamera Lanham, Anna Chen, Ansh Radhakrishnan, Benoit Steiner, Carson Denison, Danny Her- nandez, Dustin Li, Esin Durmus, Evan Hubinger, Jackson Kernion, Kamilé Lukošiūté, Karina Nguyen, Newton Cheng, Nicholas Joseph, Nicholas Schiefer, Oliver Rausch, Robin Larson, Sam McCandlish, Sandipan Kundu, Sauray Kadavath, Shannon Yang, Thomas Henighan, Tim- othy Maxwell, Timothy Telleen-Lawton, Tristan Hume, Zac Hattield-Dodds, Jared Kaplan, Jan Brauner, Samuel R. Bowman, and Ethan Perez. Mcasuring faithfulness in chain-of-thoug rea- soning. 2023. httt:ttt:////attt
org/ abs 2307 13702
Yujia Li, David Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser, Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal Lago, Thomas Hubert, Peter Choy, Cyprien de Masson d' Autumme Igor Babuschkin, Xinyun Chen, Po-Sen Huang, Johannes Welbl, Sven Gowal, Alexey Cherepanov, James Molloy, Danicl J. Mankowitz, Esme Sutherland Robson, Push- meet Kohli, Nando de Freitas, Koray Kavukcuoglu, and Oriol Vinyals. Competition-level code generation with alphacode. Science, (866 1097, 2022. doi: 10.1126/science..be 158.
URL https::///// www. sc www.eeee.....o...cc org/ doi abs 10. /226 wwweeceeeeeeccccceeeeoo.eeeoeeee www.eecee.bllllg.. abg1158/
Anton Lozhkov, Raymond Li, Loubna Ben Allal, Federico Cassano, Joel Lamy-Poirier, Noua- mane Tazi, Ao Tang, Dmytro Pykhtar, Jiawei Liu, Yuxiang Wei, Tianyang Liu. Max Tian, De- nis Kocetkov, Arthur Zucker, Younes Belkada, Zijian Wang, Qian Liu. Dmitry Abulkhanov, Indraneil Paul, Zhuang Li, Wen-Ding Li, Megan Risdal, Jia Li, Jian Zhu, Terry Yue Zhuo Evgenii Zheltonozhskii, Nii Osae Osae Dade, Wenhao Yu, Lucas KrauB, Naman Jain, Yix- uan Su, Xuanli He. Manan Dey, Edoardo Abati, Yekun Chai, Niklas Muennighoff, Xian- gru Tang, Muhtasham Oblokulov, Christopher Akiki, Marc Marone, Chenghao Mou, Mayank Mishra, Alex Gu, Binyuan Hui, Tri Dao, Armel Zebaze. Olivier Dehaene, Nicolas Patry, Can- wen Xu. Julian McAuley, Han Hu, Torsten Scholak, Sebastien Paquet, Jennifer Robinson, Car- olyn Jane Anderson, Nicolas Chapados, Mostofa Patwary, Nima Tajbakhsh, Yacine Jernite, Car- los Muñoz Ferrandis, Lingming Zhang, Sean Hughes, Thomas Wolf, Arjun Guha, Leandro von
13

---

Published as a conference paper at ICLR 2026
Werra, and Harm de Vries. Starcoder 2 and the stack v2: The next generation, 2024. URL htttp:://axxi..xx//bbbaass http://bbs/2422.99..
Haipeng Luo, Qingfeng Sun, Can Xu, Pu Zhao, Jianguang Lou, Chongyang Tao, Xiubo Geng, Qingwei Lin, Shifeng Chen, Yansong Tang, and Dongmei Zhang. Wizardmath: Empower- ing mathematical reasoning for large language models via reinforced evol-instruct, 2025. URL
https://arxix: org/ 2308 09583
Thang Luong, Dawsen Hwang, Hoang H. Nguyen, Golnaz Ghiasi, Yuri Chervonyi, Insuk Seo, Junsu Kim, Garrett Bingham, Jonathan Lee, Swaroop Mishra, Alex Zhai, Clara Huiyi Hu, Hen- ryk Michalewski, Jimin Kim, Jeonghyun Ahn, Junhwi Bae, Xingyou Song, Trieu H. Trinh, Quoe V. Le, and Junehyuk Jung. Towards robust mathematical reasoning. 2025. https:::
www.aa/......a..o 01846
Qing Lyu, Shreya Havaldar, Adam Stein, Li Zhang, Delip Rao, Eric Wong, Marianna Apidianaki, and Chris Gallison-Bureh Faithful chain-of oft--huttthhhtttt reasoning. In Jong C. Park, Yuki Arase, Bao- tian Hu, Wei Lu, Derry Wijaya, Ayu Purwarianti, and Adila Alfa Krisnadhi (eds.), Proceedings of the 13th International Joint Conference on Natural Language Processing and the 3rd Conference of the Asia- Sai--Pacific Chapter of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 305 329, Nusa Dua, Bali, November 2023, Association for Computational Linguis- tics. doi: 10.116533v1 1/20:/.1222222/mnp/pp main.20. 20. URL httpps:/aclaantaclogy.oook org.://ooooo.g/22//// 1jcnlp-ma 207]
MAA. American invitational mathematics examination, 2025. URL hattps:/mmat
maa invi tat aontiiinnllooooo i it ioss Accessed: 2025-08-08.
Aman Madaan and Amir Yazdanbakhsh. Text and patterns: For effective chain of thought, it takes two tango, 2022. URL http:/ttt//pxiiiiiee wwwpriiwv..rg//b..oeooeee http://b//22000 07686
Aman Madaan, Katherine Hermann, and Amir Yazdanbakhsh. What makes chain-of-thought prompting effective? a counterfactual study. In The 2023 Conference on Empirical Methods in Natural Language Processing, 2023a. URL hatpss://penreviev..eet net forum?id- va 7ZRRsbA4
Aman Madaan, Niket Tandon, Prakhar Gupta, Skyler Hallinan, Luyu Gao, Sarah Wiegreffe, Uri Alon, Nouha Dziri, Shrimai Prabhumoye, Yiming Yang, Shashank Gupta, Bodhisattwa Prasad Majumder, Katherine Hermann, Sean Welleck, Amir Yazdanbakhsh, and Peter Clark. Self-refine: Iterative refinement with self-feedback, 2023b. URL
17651
Andreas Madsen, Sarath Chandar, and Siva Reddy. Are self-explanations from large language mod- els faithful? In Annual Meeting of the Association for Computationall Linguistics, 2024. URL https://api www.nt
wwtp/orpggID: :269997762669977
OpenAl, Aaron Jaech, Adam Kalai, Adam Lerer, Adam Richardson, Ahmed KI-Kishky, Aiden Low, Alee Helyar, Aleksander Madry, Alex Beutel, Alex Carney, Alex Iftimic, Alex Karpenko, Alex Tachard Passos, Alexander Neitz, Alexander Prokofiev, Alexander Wei, Allison Tam, Ally Bennett, Ananya Kumar, Andre Saraiva, Andrea Vallone, Andrew Duberstein, Andrew Kondrich, Andrey Mishchenko, Andy Applebaum, Angela Jiang, Ashvin Nair, Barret Zoph, Behrooz Ghor- bani, Ben Rossen, Benjamin Sokolowsky, Boaz Barak, Bob McGrew, Borys Minaiev, Botao Hao, Bowen Baker, Brandon Houghton, Brandon McKinzie, Brydon Eastman, Camillo Lugaresi, Cary Bassin, Cary Hudson, Chak Ming Li, Charles de Bourcy, Chelsea Voss, Chen Shen, Chong Zhang, Chris Koch, Chris Orsinger, Christopher Hesse. Fische, Clive Chan, Dan Roberts, Daniel Kappler, Daniel Levy, Daniel Selsam, David Dohan, David Farhi, David Mely, David Robinson, Dimitris Tsipras, Doug Li, Dragos Oprica, Eben Freeman, Eddie Zhang, Edmund Wong. Eliz- abeth Proehl, Enoch Cheung, Eric Mitchell, Eric Wallace, Erik Ritter, Evan Mays, Fan Wang, Felipe Petroski Such, Filippo Raso, Florencia Leoni, Foivos Tsimpourlas. Francis Song, Fred von Lohmann, Freddie Sulit, Geoff Salmon, Giambattista Parascandolo, Gildas Chabot, Grace Zhao, Greg Brockman, Guillaume Leclere, Hadi Salman, Haiming Bao, Hao Sheng, Hart An- drin, Hessam Bagherinezhad, Hongyu Ren, Hunter Lightman. Hyung Won Chung, Ian Kivlichan, Ian O' Connell, Ian Osband, Ignasi Clavera Gilaberte, Ilge Akkaya, Ilya Kostrikov, Ilya Sutskever, Irina Kofman, Jakub Pachocki, James Lennon, Jason Wei, Jean Harb, Jerry Twore, Jiacheng Feng.
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Published as a conference paper at ICLR 2026
Jiahui Yu, Jiayi Weng, Jie Tang, Jieqi Yu, Joaquin Quiñonero Candela, Joe Palermo, Joel Parish, Johannes Heidecke, John Hallman, John Rizzo, Jonathan Gordon, Jonathan Uesato, Jonathan Ward, Joost Huizinga, Julie Wang, Kai Chen, Kai Xiao, Karan Singhal, Karina Nguyen, Karl Cobbe, Katy Shi, Kayla Wood, Kendra Rimbach, Keren Gu-Lemberg, Kevin Liu, Kevin Lu, Kevin Stone, Kevin Yu, Lama Ahmad, Lauren Yang, Leo Liu, Leon Maksin, Leyton Ho, Liam Fedus, Lilian Weng, Linden Li, Lindsay McCallum, Lindsey Held, Lorenz Kuhn, Lukas Kon- draciuk, Lukasz Kaiser, Luke Metz, Madelaine Boyd, Maja Trebacz, Manas Joglekar, Mark Chen, Marko Tintor, Mason Meyer, Matt Jones, Matt Kaufer, Max Schwarzer, Meghan Shah, Mehmet Yatbaz, Melody Y. Guan, Mengyuan Xu, Mengyuan Yan, Mia Glaese, Mianna Chen, Michael Lampe, Michael Malek, Michele Wang, Michelle Fradin, Mike McClay, Mikhail Pavlov, Miles Wang, Mingxuan Wang, Mira Murati, Mo Bavarian, Mostafa Rohaninejad, Nat McAleese, Neil Chowdhury, Neil Chowdhury, Nick Ryder, Nikolas Tezak, Noam Brown, Ofir Nachum, Oleg Boiko, Oleg Murk, Olivia Watkins, Patrick Chao, Paul Ashbourne, Pavel Izmailov, Peter Zhokhov, Rachel Dias, Rahul Arora, Randall Lin, Rapha Gontijo Lopes, Raz Gaon, Reah Miyara, Reimar Leike, Renny Hwang, Rhythm Garg, Robin Brown, Roshan James, Rui Shu, Ryan Cheu, Ryan Greene, Saachi Jain, Sam Altman, Sam Toizer, Sam Toyer, Samuel Miserendino, Sandhini Agar- wal, Santiago Hernandez, Sasha Baker, Scott McKinney, Scottie Yan, Shengjia Zhao, Shengli Hu, Shibani Santurkar, Shraman Ray Chaudhuri, Shuyuan Zhang, Siyuan Fu, Spencer Papay, Steph Lin, Suchir Balaji. Suvansh Sanjeev, Szymon Sidor, Tal Broda. Aidan Clark, Tao Wang. Tay- lor Gordon, Ted Sanders. Tejal Patwardhan, Thibault Sottiaux. Thomas Degry. Thomas Dimson, Tianhao Zheng, Timur Garipov, Tom Stasi, Trapit Bansal, Trevor Creech, Troy Peterson, Tyna Eloundou, Valerie Qi, Vineet Kosaraju, Vinnie Monaco, Vitchyr Pong, Vlad Fomenko, Weiyi Zheng, Wenda Zhou, Wes McCabe, Wojeiech Zaremba, Yann Dubois, Yinghai Lu, Yining Chen, Young Cha, Yu Bai, Yuchen He, Yuchen Zhang, Yunyun Wang, Zheng Shao, and Zhuohan Li. Openai ol system card, 2024. URL httth::////aaaee
org/ abs 221224122.. 16720
Ofir Press, Muru Zhang, Sewon Min, Ludwig Schmidt, Noah A. Smith, and Mike Lewis. Measuring and narrowing the compositionality gap in language models, 2023. URL brg/abs/2210) 03350
Qwen, :, An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, Huan Lin, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianxin Yang. Jiaxi Yang. Jingren Zhou, Junyang Lin, Kai Dang, Keming Lu, Keqin Bao, Kexin Yang, Le Yu, Mei Li, Mingfeng Xue, Pei Zhang, Qin Zhu, Rui Men, Runji Lin, Tianhao Li, Tianyi Tang, Tingyu Xia, Xingzhang Ren, Xuancheng Ren, Yang Fan, Yang Su, Yichang Zhang, Yu Wan, Yugiong Liu, Zeyu Cui, Zhenru Zhang, and Zihan Qiu. Qwen2.5 technical report, 2025. URL / arx
/ /4llaa/s//2...2 15115
David Rein, Betty Li Hou, Asa Cooper Stickland, Jackson Petty, Richard Yuanzhe Pang, Julien Dirani, Julian Michael, and Samuel R. Bowman. Gpqa: A graduate-level google-proof q benchmark, 2023. URL https: http::://wxvvvv....../
2311 12022
Baptiste Rozière, Jonas Gehring, Fabian Gloeckle, Sten Sootla, Itai Gat, Xiaoqing Ellen Tan, Yossi Adi, Jingyu Liu, Romain Sauvestre, Tal Remez, Jérémy Rapin, Artyom Kozhevnikov, Ivan Ev- timov, Joanna Bitton, Manish Bhatt, Cristian Canton Ferrer, Aaron Grattafiori, Wenhan Xiong. Alexandre Défossez, Jade Copet, Faisal Azhar, Hugo Touvron, Louis Martin, Nicolas Usunier, Thomas Scialom, and Gabriel Synnaeve. Code Ilama: Open foundation models for code, 2024.
URL
org/ abs/ 220//220...0 12950
Noah Shinn, Federico Cassano. Edward Berman, Ashwin Gopinath, Karthik Narasimhan, and Shunyu Yao. Reflexion: Language agents with verbal reinforcement learning, 2023. URL
fttps://arza wwwpiii.rrg/abb/o// wttp:////aba//....111. 11366
Parshin Shojace, Iman Mirzadeh, Keivan Alizadeh, Maxwell Horton, Samy Bengio, and Mehrdad Farajtabar. The illusion of thinking: Understanding the strengths and limitations of reasoning models via the lens of problem complexity, 2025. URL hRtpstt:s///axtt wtwpll..rrgabb//555000
06941
Kaya Stechly, Karthik Valmeekam, Atharva Gundawar, Vardhan Palod, and Subbarao Kambham- pati. Beyond semantics: The unreasonable effectiveness of reasonless intermediate tokens, 2025.
URL htRlsttts:// poittss://ariie www.r///b//aae 2505. 2
15

---

Published as a conference paper at ICLR 2026
Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction. A Bradford Book, Cambridge, MA, USA, 2018. ISBN 0262039249.
Boshi Wang, Sewon Min, Xiang Deng, Jiaming Shen, You Wu, Luke Zettlemoyer, and Huan Sun. Towards understanding chain-of-thought prompting: An empirical study of what matters. In Anna Rogers, Jordan Boyd-Graber, and Naoaki Okazaki (eds.), Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 2717- 2739, Toronto, Canada, July 2023, Association for Computational Linguistics. doi: 10. 18653/v1/ a023.acl-long. httpa///a..//////aaaaaaaaaoooooa acl- 153/
Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed Chi, Quoc Le. and Denny Zhou, Chain-of-thought prompting elicits reasoning in large language models, 2023. URL httptttts///arttee .rg. abs wttp///aa//...11111111 11903
Skyler Wu, Eric Meng Shen, Charumathi Badrinath, Jiaqi Ma, and Himabindu Lakkaraju. Analyz- ing chain-of of-thought prompting large language models via gradient-based feature attributions, 2023. URL https: //wxxvva.vv.///////v//.a./aaia org/ abs 12.30711333
An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, Chujie Zheng. Dayiheng Liu, Fan Zhou, Fei Huang, Feng Hu, Hao Ge, Haoran Wei, Huan Lin, Jialong Tang, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianxin Yang. Jiaxi Yang, Jing Zhou, Jingren Zhou, Junyang Lin, Kai Dang, Keqin Bao, Kexin Yang, Le Yu, Lianghao Deng, Mei Li, Mingfeng Xue, Mingze Li, Pei Zhang, Peng Wang. Qin Zhu, Rui Men, Ruize Gao, Shixuan Liu, Shuang Luo, Tianhao Li, Tianyi Tang, Wenbiao Yin. Xingzhang Ren, Xinyu Wang, Xinyu Zhang, Xuancheng Ren, Yang Fan, Yang Su, Yichang Zhang, Yinger Zhang, Yu Wan, Yugiong Liu, Zekun Wang, Zeyu Cui, Zhennu Zhang, Zhipeng Zhou, and Zihan Qiu. Qwen3 technical report, 2025. URL https: /ttp:i//aaivv.vvaa org/ abs/ 2bs/2500 2505 09388
Denny Zhou, Nathanael Schärli, Le Hou, Jason Wei, Nathan Scales, Xuezhi Wang, Dale Schu- urmans, Claire Cui, Olivier Bousquet, Quoe V Le, and Ed H. Chi. Least-to-most prompting enables complex reasoning in large language models. In The Eleventh International Confer- ence on Learning Representations, 2023. URL https: /ttp::///ppeeeeeeeeeeee net forum?id= W2H7099tqfM
A APPENDIX
A.1 EXPERIMENTAL DETAILS
We use vllm (won 2023) for all of our experiments. For potential calculation we set N 128 and use temperature of 0.6 and 0.95 as sampling parameters. For all models and datasets we generate T 32k tokens excluding the prompt. To ensure that the potential does not increase due to higher generation length, we always subtract the length of the partial CoT from 32k and use this number as T.
A.2 PROOF OF PROPOSITION
Here we present the previously omitted proof of Proposition By Bayes' rule, for any token c++1 we have
J SEPI(Ce41)
f41 = Ply Cra,, ((1)) fo p(((++)) + (1 ft) po(ce+))
Taking expectation with respect 6+11 drawn from Pr, i.e. conditioned on the event that the rest of the run is correct, gives
pi(c++)
E[fe+1] fr Σ pi(cet))
fipi(ce1) (1 f()
c++1
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Equivalently,
p1(c++122 fipi (ct-1) (1 f()po(cc+1)
E f
Σ
Now apply the Cauchy-Schward Schwarz inequality with weights g(c(+1) =
2
Σ pi(cet1) g(cem))
Σ a(can))
Σ pilcet1) Co+1
Ce+1
Since Ean (((+1) = and Ean, Pi(c(+1) = 1, it follows that
(c(+1)2
2 1.
Gc+1 (((+11)
Therefore,
B[f++] f.
Finally, taking expectation over prefixes Cı.t distributed as correct yields
B[f++1] 2 Bift).
which is the desired result.
A.3 MORE COT EXAMPLES
We display more quantitative results in Table for the MATH-500 dataset, in Table we show results for the coding benchmark HumanEval (Chen et al. (2021) and in Table[4]te ones for GPQA- Diamond (Rein et 2023). We find that the same trends observed for AIME--2024 hold also this case. Interestingly. models seem more stable coding benchmarks, with their CoT displaying more monotonic behaviour and general less tangents. For GPQA we find trends more consistent with AIME, most likely because more questions very difficult compared to HumanEval. As expected due to GPQA being multiple choice benchmark with only four choices, we observe a significantly higher guessing rate. In this case we adjusted the threshold for guessing to 25% (as opposed to the previously used 5%) as a random guessing baseline would course always achieve 25%.
We also show
annotated CoT in Fig. Fig. 9 and for coding in Fig, 14 and Fig. 15 In Fig.
we have again have the model performing a reasoning insight, correctly realizing that the exponents can be deduced from the binary representation of the number. We then finally have reasoning jump, where the model experiences a strong boost in potential from the word "correspond". While first sight not clearly interpretable, hypothesize that this word forces the model output concrete values for a; 's otherwise a common failure model as the model tries to further refine their computation.
MODEL
REASONING
INSIGHTS
TANGENTS J
LATE SPIKE
MONOTONICITY
QWEN2.5-1.50
X
23%
11%
1%
40%
QWEN2.5-7B
X
24%
12%
1%
41%
LLAMA3. 1-8B
X
19%
21%
1.8%
31%
LLAMA3. 1--0B 1-70B
X 22%
23%
1.5%
30%
QWEN3-0.6B
30%
40%
1.4%
30%
QWEN3-32B
28%
35%
0.9%
45%
Table 2: Behaviours of potential for several reasoning and non-reasoning models MATH-500.
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Quedtio 12004 12004-AMtE-11 AINE-11 Problem
chooses snt coatioa
Brtte pe
properythul madirue df belongete thrratedte
Rhoponse so
0.8
Samming io of oeti mbtrr ofusts
ge
2004
Potential
0.6
8
fidtth
efoose rtatton numberr
form.
aaanas clatuct of
Indight thai benser
roprocooill01 findl eepooeent
- 221 ------PP
0.2
That the teceepppadnng Naaw it, andte Thardoww, the muatt
N. 10,11) Thn wwrtte A
Word "corneporrdngg
4
undiar
# CoT tokens
Figure 9: Unintuitive reasoning jumps. Qwen2.5-7B's potential (5 x) remains flat in ! although crucial insights are obtained. The potential then increases due to a reasoning insight in 2 (realizing that the binary representation determines the exponents). In ( obtain the final spike at the word "corresponding", a reasoning jump, which seems strange from a human perspective. We hypothesize that it might force the model to output values for a; 's, which indeed is the next logical step. We indeed observe that without this word, the model continues to perform unnecessary calculations, subsequently leading wrong values for a;.
MODEL
REASONING
INSIGHTS f TANGENTS |
LATE SPIKE
MONOTONICITY
QWEN2.5-1.5B
39%
7%
0%
55%
QWEN2.5-7B
X
30%
4.8%
0%
73%
L1AMA-3.1-8B
X
41%
16.5%
0.4%
42.1%
Table 3: Behaviours of potential for several reasoning and non-reasoning models on the coding benchmark HumanEval.
MODEL
REASONING
INSIGHTS
TANGENTS J
LATE SPIKE
MONOTONICITY
QWEN2.5-1.5B
X
17%
6.8%
19%
20.9%
LLAMA-3.1-8B
X 29.1%
20.0%
17.6%
25.8%
Table 4: Behaviours of potential for several reasoning and non-reasoning models on the coding benchmark GPQA- Diiamodd.
In Fig. 10 we again observe reasoning guess from Qwen2.5-1.5B, where the CoT in segment ①. while seemingly making sense first sight, actually does not contribute to the final answer at all. In fact the number 80 does not relate at all to the computations made before. This is reflected in the potential, that shows a spike only towards the very end, highlighting that the CoT indeed did not contribute.
Finally, we show an instance of optimized CoT introduced in Sec We observe that the potential is now strongly monotonic, with almost every partial CoT leading to some improvement in the potential. This is also reflected qualitatively, we can see that the CoT is more concise in language, in fact display all of it here. In segment ! the model makes slower progress those are steps can reliably do. Finally, the model undergoes a reasoning insight ② with the model discovering that d needs to divide 56.
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AIMEI Problem
bas sncerp littrrl houggn appodte oidee paralal entenga esdensions regreett CJ, and linghts 2HD, aea
1.0
longth euggee.
Response
To find the side linghh ofth haragon ABC DDE rtart netintl-4
drrighttonwwe proachh recogrton
Potential
Thes hove
Height of the triangle Av.
200
The model goea
0.4
puin
Thinghee
Litrural play
0.2
Solingfers::
200./3
0.0
100
200
300
400
500
600
Thanfor, lengeh bauage
CoT tokens
Figure 10: Reasoning tangents and guessing. Qwen2.5-1.5B goes on a long reasoning tangent in ① that does not increase the potential over a long token horizon. Finally it outputs a final answer in unrelated the previous reasoning that happens be correct.
Queston (2025-AMME-1 Problem
beanch titer El oeberre MT)
T dhe Zund the snd (il wi to 20- 7. uo intaee > mukina Telices
0.8
Potential
Blow but sterdy
Indight foond divdes to
0.2
aniak Foleroo sccivaldd
20 Stred sstitieh Siriooent
enl 90-17 teed di untdtech aett
0.0
colit servide
100
200
300
400
500
CoT tokens
Figure 1I: Optimized COT. We show trajectory based on the optimized CoT from Qwen2.5- show it here full length. The potential
1.5B. The CoT is more concise, actually allowing is monotonic anticipated and all tokens contribute to it. In segment ! the model makes slower progress as those are steps it can reliably do. Finally, the model undergoes reasoning insight with the model discovering that d needs to divide 56.
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Question MATH |Problem
wins nert becomee dutss tars On boot's furm, be fiip coin, gets talk, fe wio...ffrcc Finetit becomes Allor's sum. What the probucp fiy that Alla wine parnal
Alcce andi plyling gane dicee stans finst Alcc's fipe cain. heeds,
1.0
0.9
Response
0.8
ore the grim continue brd ja but gane essen by sequemoce indepe sdont csen
dimplity problers by thohingtattt
Rotenbal
Let's defice
? The model goos on reasening tangeems, the recurntvv doesn't work out
Phey 11211 this ai rand vazz
Stensiiiping stteerguaattett equatiin, see ger
Gog Lpp -a2--alaitt
dind probarility than wine by finctin the clent gree genarsting funetion.
0.2
indeper zok ritps.
Howeret
preiiemi furthes
cose of
100
200
300
400
500
600
700
Lette delire probuiiiity thai-All
tollowingg exuartion:
model Rigines out protots dry sucoess
sincciljinng eqgattin,,
So. pnodubiil by thet virt thegamme
Figure 12: Reasoning tangents and insights for I.lama.1-88 on Math-500. We show an example for another dataset using Llama3.1-4 Similar observations hold; the models tend to reason- ing tangents as shown here with the model exploring a more difficult recursive approach, which ultimately abandons by deriving the key insight, expressing the probability simple linear equa- tion.
Question MATH-600
Find the unique odid integart such thae ot <c 25 and 2 th
inthei tnvema of module 23.
1.0
Response
0.8
To End the urique irteger that sumtehes the given cond toons, on followr thene stepe: S
I-]
0.6
bothaldec
(mod 25),
Poserball
square oo modulo can the 2 (mod. 23) and (mod
We need fixd such that 21 (mod 23), By inspection, sse than Siraa 18.
0.4
I.11
The meate corner
incegers
0.2
Since we need + 2o oddd, we chossed 17 (mod 23).
0.0
100
200
300
400
500
600
700
The arower
TIES 8 a1 SUCDer
CaT tokans
Figure 13: Reasoning tangents and insights for Llama3.1-8 3-BB on Math-500. We show an example for another dataset using Llama3.1-8B. Again the model displays key insight, realizing directly that 5 and 18 are the square roots. A small tangent is also encountered, with the model missing that 17 an odd number, but subsequently corrected later in the reasoning.
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(HumanEval,
Thercoop kyst and comettt handion conepprring to rolowing doctiing spanes the plen tent
1.0
that
nuebera reprativ
deeilec et 3, 7,,
2trferencel: 4, (2, 03)
1
Potembal
If ligus Ltat enaty, retura
0.4
Response
0.2
ceutte, the #ffffrrncc latt:
integer dlnsart thes postiiie Entegerca 2
eddud rogalie numbers wiich plouied
wove ecculirragttt wow.eeeeeeee....................
eittione
A
soreddy song aatraanne
Figure 14: Potential for coding tasks. We analyze trace of Qwen2.5-1.5-Insttee on HumanEval and identify occurence of tangents and insights. Here, model gets distracted initially by outlining that it will convert all numbers to positive integers, and indeed using commands abs() and int(), dropping the potential significantly as consequence. The model then recovers by fixing the for-loop using if statements, which actually render abs() and int() effectless as the loop anyways only goes over positive integers.
Question (HumarEris Erai 125)
1.0
def pred, wwe..sigg..on band):
You are given on anrey ane troegers and you reed to retur san magir toces ietegers miltiaal by product of slow namler array, represented by Sota:: return Hooe anaty arm.
0.8
Potental
0.6
Ca 2a
0.4
11
Response
3
0.2
Ixport wasy
eut ared, wwt.attttenntn tene)) net arei
Number of chunis
10
34
return
Ealeulate predict eiorents errty wwebettaffrleeees prodlaar)
intettion products
which
load wrong cuppus
Colicllte asponitudes auterplted decenstned
resutt abaiamm)) sign
Model gate the sign comedd special
return readts
Figure 15: Potential for coding tasks. We analyze a trace of Q-wn2.5-11.5-nnee on HumanEval and identify the occurence of tangents and insights.
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Wrong Correct
Wrong
1.0
0.8
Potential
Potential
200
400
500
800
1000
1200
500
1000
1500
2000
2500
tokens
CoT tokens
Figure 16: Stability profiles. Stability profiles for Qwen2.5-1.5B and Qwen2.5-77 on AIME 7 and 26 respectively. Correct and wrong answers exhibit similar profiles across models and questions.
A.4 STABILITY OF CoT
We can also consider slight variation of the potential, called the stability of CoT. Given prompt z, CoT reasoning and answer pair (c. y) we define the stability of a sub-chain Cea as
1 N
Σ (1())=y) where (y(1), Ozz
(n)
IMO (M(et)))))
with the slight variation that instead of considering the ground truth y", final answer of the chain c as the target. L.c. the potential is a special of stability, when = g". Stability measures how determined the final answer is throughout the reasoning process of the model. Somewhat surprisingly, we observe that correct answers do not necessarily always display higher stability, indicating that models can become convinced very early on in their reasoning about wrong answers. We display various stability curves in Fig. 16
consider the reached
A.5 COMPUTATIONAL COMPLEXITY POTENTIAL CALCULATION
The computational load to obtain the potential curve given prompt scales with the number of evaluations N produced to estimate the potential and the number of points Netunks calculate the potential for. More precisely, say for a generation from scratch we produce tokens, the total amount of tokens produced estimate the potential curve will be given by
Nannks
N Netunkss
Tuo N
Nehunks
2
Nehonks hence also gives us way to control the amount of compute and we set it to moderate values of Nehunks 15 for the quantitative evaluations, while we use higher values of around Nehunks 50 for most figures, to obtain a more fine-graines understanding. Advanced inference techniques such as prefix-caching and continuous batching in vLLM further help speed up the potential calculation, as all of the prefill is shared among the prompts, and every prompt can in theory be generated in parallel, thus allowing for continuous batching. This is different for the optimal CoT, where inference has to be done sequentially as things depend eachother, making it significantly slower.
A.6 CALCULATION OF OPTIMAL CoT
We detail the recipe calculate the optimal CoT in Algorithm
below:
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Algorithm 1 Generating potential-oppimizze CoTs
1: Initialize the CoT Cct <- 0
2: while the chosen candidate does not contain the answer do
3: Sample M candidate CoT chunks C(((+T)
[m] 1.1.d.
LMa(- Cet, ()) of length C, for m
1,...,M
MM
4:
Compute potentials Pm < x p (Cet+713)
(c(m)
5: Select mn t arg m&Xm Pm
6:
Update Cc++ < [c<:: (e((Tt))
(m)
7:
end while
The computational complexity of the optimal CoT caleulation is quite high with s0 MNNamt.T to- kens needed, while also not enjoying the same benefits from prefix caching and continuous batching, the potential curve calculation did. We thus do not recommend its usage in practical setting but view it more as proof of concept.
A consequence of greedy strategy is the following: The optimal CoT could degenerate "lucky guessing" answer when M goes to infinity, given that the model has non-zero support across all tokens, this would effectively minimize the objective. Since we restricted using finite M however in practice, we are implicitly regularizing the objective to sequences of sufficient proba- bility mass, naturally circumventing this degeneracy. To make this regularization more explicit, one could potentially enhance the objective with regularization term to require a given chain to meet certain probability budget, eliminating this possibility. We observe however that M being finite is enough to avoid such degenerate cases.
We do however want stress that the optimal CoT does depend on M.
A.7 MORE DETAILS ON SUMMARY STATISTICS
Here provide the definitions for the statistics used in Sec. In all experiments we divide the CoT into 20 chunks, getting thus potential curves consisting of 20 points.
Insight: We say that given potential contains an insight if the difference between two consecutive chunks of CoT exceeds 40%, i.e. if one step of CoT raised the potential by at least 40%. We exclude the last two steps to make sure we don't count the late reasoning spikes as insights.
Tangent: We define a potential to exhibit tangent if the potential drops by at least 30%, not necessarily consecutively.
Guess:
We define late reasoning spikes guesses the case when the potential the second to last step is smaller than 5%.
Monotonicity: We call a potential monotone if its consecutive steps do not decrease by more than 10%.
23


Reviews:
[{"review_id": "uSIy5ZNMqS", "reviewer": "Reviewer_1SnR", "summary": "The authors endeavor the investigate which parts of the CoT are most \"important\" in the model's reasoning and eventual production of the answer. To do so, the authors define the notion of potential, which, given some partial CoT, is defined as the probability that the model will output the correct answer after generating the completion of the CoT. Thus, the authors can ask which parts of the CoT have the largest potential or the largest change in potential.", "questions": "The notion of potential is very closely related to the value function in actor-critic models in RL, but the paper doesn't really draw this connection. A vague connection is made at the end of Section 2, but the similarity between the two notions warrants further discussion and elaboration.\n\nAlgorithm 1 describes a \"greedy\" strategy for finding the optimal CoT. However, there may exist more \"globally optimal\" CoTs that find the correct answer faster than the CoT produced by Algorithm 1, while at the same time, do not increase the potential as quickly. But I suppose the probability of the CoT is an important consideration, as there exists the possibility of a \"degenerate\" CoT where the model guesses the correct answer immediately, which would result in an immediate jump in the potential at the very beginning of the CoT (due to lucky guessing). As such, the result of Algorithm 1 is sensitive to the choice of M, where in the limit as M goes to infinity, the algorithm would eventually recover this degenerate (lucky guess) CoT.\n\nSome of the key observations were made in the qualitative analysis, such as the one in Figure 6, where the portion of the CoT that is more difficult for humans actually corresponds to a small increase in potential, whereas other portions that seem quite simple to humans are associated with larger jumps in potential. It is unclear to what extent these observations generalize to other examples without any accompanying quantitative evidence. How often does the behavior observed in Figures 7 and 8 occur?\n\n\nMore detailed comments and questions follow. While the paper is largely well-written, there are a small number of grammatical and stylistic errors. I include some of them from the first three sections in the list below. This list is not comprehensive, so I encourage the authors to carefully read through the paper and correct all such errors.\n\nLine 28: A weaker what? Is this a typo?\n\nLine 31: Grammatical error: there should be a complete clause following \"that\", such as \"...highlighting that the grass is green.\" Here, note that \"grass is green\" can stand on its own as a complete sentence. However, \"a large part of the mechanics underpinning reasoning transfer\" is not a complete sentence/clause.\n\nLine 36: \"lead to\" -> \"led to\"\n\nThere are a number of claims in the introduction that are stated without supporting citations. For example \"[CoT] reasoning (Wei et al. 2023) has [led] to several breakthroughs in domains spanning mathematics\"; and similarly for \"...coding.\" There are several others.\n\nLine 298: \"Model size also seems to surpress [guessing behaviour] more, which is expected since larger models generally tend to perform better\"  Wouldn't this only be expected if the model's final answers are faithful with respect to their CoT? For example, it is possible that larger models are better at finding the correct answer *latently* (in a fashion that is not verbalized in their CoT), in which case, they may be able to more accurately find the correct answer via the guessing tactic.\n\nLine 306: Why is the pass@k metric prone to suffer from guessing behavior?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The authors characterize a new way to study which parts of the CoT are most impactful/useful in the model's reasoning via a new metric they call \"potential\".\n - Some interesting observations are made that connect the changes in the potential with different parts/strategies in the CoT.\n - The paper is largely well-written.", "weaknesses": "- The implications and future applications of the potential metric are unclear (or they seem rather limited). Additional quantitative analysis may enable us to draw more general observations about the reasoning of LLMs. The conclusion is missing a discussion on future work, which could have alleviated this concern.\n - Experiments were focused on math competition examples, and it's unclear if similar observations would be drawn in other domains.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ranosL4dJZ", "reviewer": "Reviewer_v32n", "summary": "The paper studies chain-of-thought (CoT) reasoning in large language models using a new measure called “potential,” which quantifies how each reasoning step contributes to getting the correct answer. The authors analyze competition-level math problems to reveal how CoT progress fluctuates — sometimes increasing sharply with key insights, other times dropping due to reasoning tangents or random guesses. They further explore how partial CoT from stronger models can boost weaker ones, suggesting reasoning transfer across architectures.", "questions": "see above", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper addresses an important gap in understanding how CoT actually helps models reason, not just appear to. Introducing “potential” as a metric is novel and provides a quantitative way to evaluate reasoning progress. The experiments on AIME problems are well-motivated and the qualitative analysis of “reasoning tangents” and “insights” is compelling. The study also goes beyond introspection by testing CoT transferability, showing interesting empirical results that small parts of reasoning from stronger models can unlock weaker ones", "weaknesses": "The proposed concept of potential is not rigorously justified beyond empirical correlation, and its interpretation remains vague. The assumption that higher potential implies genuine reasoning progress is too strong, as sampling-based estimation might simply capture distributional artifacts. The analysis often reads descriptive rather than explanatory; the paper reports patterns (insights, jumps, tangents) without providing mechanisms or theoretical grounding. The heavy reliance on visual examples and qualitative claims weakens the argument, since the method’s reliability is unclear. The “transferability” results are interesting but confounded — providing partial CoT is effectively giving a hint, so improved performance does not prove reasoning generalization. There is also limited diversity in tasks and model families; focusing only on AIME math and Qwen variants narrows generality. The paper overstates human-analogy comparisons despite limited evidence. Overall, while the work is original and thought-provoking, it lacks methodological rigor and over-interprets its empirical patterns.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "V3uqee39Fk", "reviewer": "Reviewer_m7iu", "summary": "This paper introduces a novel metric called \"potential\" to analyze the reasoning process in large language models using Chain-of-Thought (CoT) prompting. By examining how the potential evolves over the course of a CoT on competition-level math problems, the authors identify several distinct patterns: \"reasoning insights\" where the potential sharply increases, \"reasoning tangents\" where it drops, and \"lucky guesses\" where the potential remains low until the very end .", "questions": "Please refer to weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "(1) Novel and Intuitive Methodology: The core contribution, the \"potential\" metric (Eq. 1), offers a principled and intuitive method for quantifying progress within a CoT. This provides a valuable tool for moving beyond simple final-answer accuracy to analyze the intermediate reasoning steps.\n\n(2) Strong Qualitative Analysis: The paper excels in its qualitative analysis, providing clear and well-annotated examples that connect the behavior of the potential curve to specific segments of the model's generated text.\n\n(3) Interesting Empirical Findings: The study uncovers compelling and non-obvious dynamics in CoT reasoning.", "weaknesses": "(1) Estimating the potential metric requires sampling a large number of completions ($N=128$) at every intermediate CoT step to obtain a stable estimate. The manuscript does not address the resulting computational overhead, which may pose a serious limitation for scalability and broader adoption.\n\n(2) The quantitative analysis in Table 1 depends on fixed, manually chosen thresholds to categorize behaviors such as “insights” (potential increase > 40%), “tangents” (potential drop > 30%), and “guesses” (potential < 5% at the penultimate step). The absence of justification for these cutoffs makes the reported statistics appear somewhat subjective.\n\n(3) The empirical study centers almost entirely on competition-level mathematics problems (AIME) and on the Qwen family of models. It remains unclear whether the observed potential trajectories and reasoning patterns would extend to other tasks—such as commonsense reasoning or code generation—or to other model architectures.\n\n(4) Algorithm 1 employs a brute-force approach that randomly samples and evaluates candidate CoT segments to maximize potential. This process is computationally inefficient and appears intended as a proof of concept. However, the manuscript does not discuss its scalability or the resources required to produce the results in Figure 3.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)