Landscape of Thoughts: Visualizing the Reasoning Process of Large Language Models
Authors: Zhanke Zhou
Zhaocheng Zhu
Xuan Li
Mikhail Galkin
Xiao Feng
Sanmi Koyejo
Jian Tang
Bo Han
Venue: ICLR 2026 Poster (2026)
Keywords: Large Language Models
Reasoning
Visualization
Primary area: foundation or frontier models, including LLMs
TL;DR: We introduce a visualization tool for users to inspect the reasoning paths of chain-of-thought and its derivatives on any multi-choice dataset

Abstract: Numerous applications of large language models (LLMs) rely on their ability to perform step-by-step reasoning. However, the reasoning behavior of LLMs remains poorly understood, posing challenges to research, development, and safety. To address this gap, we introduce landscape of thoughts (LoT), the first landscape visualization tool to inspect the reasoning trajectories with certain reasoning methods on any multi-choice dataset. We represent the textual states in a trajectory as numerical features that quantify the states' distances to the answer choices. These features are then visualized in two-dimensional plots using t-SNE. Qualitative and quantitative analysis with the landscape of thoughts effectively distinguishes between strong and weak models, correct and incorrect answers, as well as different reasoning tasks. It also uncovers undesirable reasoning patterns, such as low consistency and high uncertainty. Additionally, users can adapt LoT to a model that predicts the property they observe. We showcase this advantage by adapting LoT to a lightweight verifier that evaluates the correctness of trajectories. Empirically, this verifier boosts the reasoning accuracy and the test-time scaling effect. The code is publicly available at: https://github.com/tmlr-group/landscape-of-thoughts.

Published as a conference paper at ICLR 2026
LANDSCAPE OF THOUGHTS: VISUALIZING THE REASONING PROCESS OF LARGE LANGUAGE MODELS
Zhanke Zhou's 1.2-
Zhaocheng Zhu
Xuan Li
Mikhail Galkin
Xiao Feng'
Sanmi Koyejo²
Jian Tang".
Bo Han
'TMLR Group,
Hong Kong Baptist University
Stanford University
Mila Québec Al Institute
*Université de Montréal
'HEC Montréal
Intel Al Lab
ABSTRACT
Numerous applications of large language models (LLMs) rely on their ability to perform step-by-step reasoning. However, the reasoning behavior of LLMs remains poorly understood, posing challenges to research, development, and safety. To address this gap, we introduce landscape of thoughts (LoT), the first landscape visu- alization tool to inspect the reasoning trajectories with certain reasoning methods on any multi-choice dataset. We represent the textual states in a trajectory as numerical features that quantify the states' distances to the answer choices. These features are then visualized in two-dimensional plots using t-SNE. Qualitative and quantitative analysis with the landscape of thoughts effectively distinguishes between strong and weak models, correct and incorrect answers, as well as different reasoning tasks. It also uncovers undesirable reasoning patterns, such as low consistency and high uncertainty. Additionally, users can adapt LoT to a model that predicts the property they observe. We showcase this advantage by adapting LoT lightweight verifier that evaluates the correctness of trajectories. Empirically, this verifier boosts the reasoning accuracy and the test-time scaling effect. The code is publicly available at: https::p... / thub. www/tm///..mc http://mmrrrroopp/oe hawpa//aaancccpppeeem of toffthouuttss.
INTRODUCTION
Large language models (LLMs) have revolutionized problem-solving. Many practical applications, e.g., L.LMs as agents (Schick et al., 2023; Lewis et al., 2020; Yao al., 2023b), critically depend step-by-step reasoning (Wei et al., 2022; Kojima et al., 2022). Despite progress in advanced models like OpenAl ol (Jaech et al., 2024) and decoding methods such as test-time scaling (Snell al., 2024), the underlying reasoning behavior of LLMs remains poorly understood, hindering the development of these models and posing deployment risks (Anwar et al., 2024).
A few pioneering attempts (Wang et al., 2023a; Saparov & He, 2023; Saparov et al., 2023; Dziri et al., 2024) probe LLM reasoning, but their insights often hinge on specific decoders and tasks. In practice, practitioners debug by manually reading the reasoning trajectories generated by LLMs, which has two drawbacks: (i) scalability, i.e., human inspection does not scale (e.g., at 30s per trajectory, 100 trajectories require 50min); and (ii) aggregation, i.e., deriving reliable, dataset-level conclusions (e.g.. from 10,000 trajectories) is difficult, yielding subjective and even biased summaries. These costs compound during iterative development, where fast, interpretable feedback essential. Consequently, there is a clear need for general, reusable tools to analyze LLM reasoning in users' own settings. This tool can potentially benefit engineers by accelerating iteration, reasoning researchers by informing decoder improvements, and safety researchers by monitoring model behaviors.
To this end, we introduce the landscape of thoughts (LoTT), visualization of L.LM reasoning trajec- tories that delivers automatic, objective analysis from single examples to full datasets. Analogous to how t-SNE (Van der Maaten & Hinton, 2008) reveals structure in high-dimensional data, LoT exposes patterns in the reasoning space of LLMs. By pairing qualitative landscapes with quantitative metrics (consistency, uncertainty, and perplexity), LoT enables comparison and reveals insights beyond manual inspection metric analysis.
"Equal Contribution. "orrespondence Bo Han (bhanml (bhanml@compp haaimm/mmoooophhu..hiioiee

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Published as a conference paper at ICLR 2026
visualize www.
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Choices: 204.6, (B) 104.6, (C) 234.6, (0) 184.0, (E) 256.6.
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Figure 1: Landscape of thoughts for visualizing the reasoning steps of LLMs. Note that the red landscape represents wrong reasoning cases, while the blue indicates the correct ones. The darker regions in landscapes indicate more thoughts, with * indicating incorrect answers and * marking correct answers. Specifically, given a question with multiple choices, we sample few thoughts from an LLM and divide them into two categories based on correctness. We visualize the landscape of each category by projecting the thoughts into a two-dimensional feature space, where each density map reflects the distribution of states at reasoning step. With these landscapes, users intuitively discover the reasoning patterns an LLM or a decoding method. In addition, a predictive model is applied to predict the correctness of landscapes and can help improve the accuracy of reasoning.
Specifically, given any multiple-choice reasoning dataset, LoT visualizes the distribution of interme- diate states in any reasoning trajectories of interest w.r.t. the answer choices, which enables users to uncover reasoning patterns in both success and failure trajectories (Fig. 1). The core idea is to characterize the states of textual thoughts a trajectory as numerical features that quantify the states' distances to the answer choices. These distances are estimated via the perplexity metric, using the same I.L. that generates the thoughts. Then, these state features (i) produce three metric plots and (ii) are projected into two-dimensional space with t-SNE to generate the landscape plots.
We examine LoT with different dimensions of model sizes, decoding methods, and reasoning datasets. LoT reveals several insightful observations regarding the reasoning behaviors of LLMs. Some notable observations include: 1) The convergence speed of trajectories towards correct answers reflects the accuracy, regardless of the base model, decoding method, or dataset; 2) The convergence speed of trajectories in success and failure cases differs markedly, indicating that we may use the convergence speed of reasoning trajectory to predict its accuracy; 3) Low consistency and high uncertainty generally observed in the intermediate thoughts, revealing the instability of the reasoning process. These patterns are uncovered by connecting per-trajectory inspection with dataset-level reasoning analysis, which is not reported by prior text inspection or metric analysis.
Since our tool is built on top of state features, it can be adapted to a machine-learning model to quantitatively predict certain properties, such as the findings mentioned above. We showcase this advantage by training a lightweight model predict the success and failure cases, which equivalent to verifiers commonly used in LLM reasoning (Cobbe et al., 2021). Despite being lightweight compared to typical I.LM-based verifiers, consistently improves reasoning performance across the majority of model, method, and dataset combinations in our experiments. Hence, users can leverage this framework to predict task-specific properties their own settings.
In summary, our main contributions are three-fold:
We introduce the first tool for automatic and scalable visualization of the LLM reasoning procedure, applicable to any open-source models and decoding methods on multiple-choice datasets (Sec. 2).
Our tool reveals several new insights into the reasoning behaviors of different language models, decoding methods, and datasets (Sec. 3).
Our tool can also be adapted into a predictive model to estimate certain properties and guide the reasoning process, improving L.LM reasoning without modifying the model parameters (Sec. 4).
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2 LANDSCAPE OF THOUGHTS
2.1 PROBLEM FORMULATION
Our goal is to visualize the reasoning trajectories of LLMs across variety of task domains. Specif- ically, we target datasets consisting of multiple-choice questions, where each datapoint (I. y, C) comprises question I, correct answer y, and a finite set of candidate choices C = fc, j= all represented in texts. The visualization tool applies the following models and methods.
Language models. To explore the landscape of thoughts generated by an LLM (L the model should produce diverse reasoning trajectories for solving a problem. In each trajectory, the reasoning thoughts are decoded autoregressively as & ~ PLLM [r, C, i,. f)=1): tioned on the question v; the candidate set C, and the sequence of preceding thoughts ,,,--., 44-1 To characterize intermediate states within these trajectories, the LLM must also function as a likelihood estimator, enabling the computation of the probability PLLM (0,2 C, f....... f.) of any answer ÿ. These two requirements are generally satisfied by open-source LLMs,, such as Llama (Dubey et al., 2024) and DeepSeek (Liu et al., 2024). However, -losed-source LLMs like GPT-4 (Achiam et al., 2023) and Gemini (Team et al., 2023) are excluded, as their likelihood estimation not publicly supported.
each thought & is condi-
Reasoning methods. While there are many approaches to solving reasoning problems with LLMs (Creswell et al., 2022; Kazemi et al., 2023), this work focuses on chain-of-thought (CoT) (Wei et al., 2022) and its derivatives (Zhou et al., 2023; Yao et al., 2023a), given their widespread adoption. These decoding methods generally guide the model in generating a structured trajectory of intermedi- ate reasoning thoughts before arriving at the final answer. Note that to visualize a large number of reasoning thoughts effectively, these thoughts should be automatically parsed into distinct units (e.g.. via sentence tokenization). This requirement can be satisfied by most LLMs."
2.2 QUALITATIVE VISUALIZATION WITH LANDSCAPES
Given collection of reasoning trajectories generated by an LLM,, our tool seeks to visualize, within a two-dimensional (2D) space, how different trajectories lead to either correct or incorrect answers, as illustrated Fig. 1. A key challenge lies in the absence of a direct mapping from the textual space of thoughts to numerical 2D coordinates. To address this gap, we utilize the same LLM to represent intermediate states as numerical features. These state features are then projected into a 2D space for visualization. For simplicity, we denote thought
instead of f,.
Characterizing the states. Here, the intermediate thoughts {t,]i=1 in a reasoning trajectory naturally define sequence of states {s:) where so [ar] and 8; = [x. 11, ... 44). We characterize the states as features using the likelihood function of the LLM. Specificaly the k-dim feature f: for state
8; indicates the relative distances from the state s; to all possible choices {cz (=z)
f: [d(s;, c1), d(s;, o2),..., d(,,,, cx)
(1)
where d(s;, cj) measures the distance between state s; and choice cj. To normalize the token lengths across choices, we calculate d(s,,c,) through the perplexity metric (Shannon, 1948; Manning & Schutze, 1999):
ley
d(si, 0j) 2 exp
log PLLM((/([))ssscc]::))) = PLiM(cyle))-111
(2)
where |cy) is the number of tokens in cj. and (Cy (x) is the accumulated probability in autore- gressive manner. Assuming |c,1 = T, we have PriM Cj (;) = PL [1]]s)) (c [2]s, [1]]
pliM(c, [3] |s, ey [1]], cg(2)) (e [T] Cg [1], [[ IT The token-level probabilities are normalized over the entire vocabulary; ey [1] is the first token of cy. and [y[T] is the last token.
We normalize the vector f. to have unit L1 norm. Additionally, we encode each choice as a landmark feature vector for visualization. Notably, the perplexity decreases as the model's 's prediction confidence increases. To align with this observation, we define the feature vector fz for a choice e, as:
ff a 110 1), 1(j + =))T
'ooT is positioned for multi-choice questions. Appendix E. 11 discusses its extension to open-ender tasks. We empirically verify the robustness of LoT if this requirement does not hold (please see Appendix H.9). Our method provides post-hac analyses; it never intervenes alters the model's 's reasoning trajectory.
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For trajectories, each with n states, compute the feature vectors for all states. Here, the zero entry in fj indicates zero distance to the choice сj itself, and the nonzero entries (each equal to 1/k) encode the assumption that distances among different choices are equal. Together with the feature vectors of & choices, we obtain feature matrix F E R ( as:
F F
f(") f!"
ff. f..
fA
(4)
Note that a sufficiently large number of trajectories is necessary to generate a comprehensive vi- sualization of the reasoning landscape. For computational efficiency, question across all questions, yielding r = d x N₂ total trajectories, where Ng is the number of questions. We then normalize feature vectors by reordering choices so the correct answer appears in the first dimension across all questions. In this way. we can visualize the landscape of multiple questions by putting their trajectories together, which is more efficient than visualizing by generating enough trajectories for one question.
sample d trajectories per
Visualization. After constructing the feature matrix F, we project the states and choices into a 2D space for visualization. This step can be performed using various existing dimensionality reduc- tion methods (Pearson, 1901; Van der Maaten Hinton, 2008; McInnes et al., 2018). We employ t-SNE (Van der Maaten & Hinton, 2008) due to its effectiveness in preserving local neighborhood structure from the original high-dimensional space and its robustness to a wide range transfor- mations. By applying t-SNE to the k-dim F, we obtain the 2D coordinates F R (rn+*)) The two projected dimensions correspond to directions in the original answer space; each state's s 2D coordinates thus reflect its relative distance to different answers. Finally, the coordinates the states define a discrete density function in the 2D space, represented by the color depth in landscapes.
2.3 QUANTITATIVE VISUALIZATION WITH METRICS
Beyond the qualitative visualization, we introduce three quantitative metries to help understand the LLMs' behavior. These metrics are defined based on the intermediate states in Sec. 2.2.
Consistency. To understand whether the LLM knows the answer before generating thoughts, we compute the consistency of state s; by checking whether and fn agree:
Consistency(s.. (s;) = 1 (arg min f: = arg min f.). Uncertainty. To know how confident the LLM about its predictions intermediate steps, compute the uncertainty of state 8; as the entropy of fi (note Eder. d= 1)
(5)
Uncertainty(s:. = = -- d. lo-oog d.
(6)
dEf.
Perplexity. We also measure the LLM's confidence in its generated thoughts using perplexity, which comparable across thoughts of different lengths:
Perplexity (t.) = (LL((t(ss--))- 1//4al
(7)
Remark 2.1. While perplexity traditionally used the token level for language modeling, we
apply it at the thought level. Additionally, we introduce consistency and uncertainty metrics tailored to reasoning trajectories, offering new perspective for the community. Appendix F introduces related works in detail. The following section demonstrates that the Lo',, containing the qualitative landscape and the quantitative metrics, effective for automatic and scalable visualization of reasoning trajectories.
RESULTS AND OBSERVATIONS
In this section, we utilize the landscape of thoughts to analyze the reasoning behavior of LLMs by comparing the visualizations across three dimensions: (1) diverse scales and types of language models in See. 3.1. 1, (2) different reasoning tasks See. 3.2, and (3) various reasoning methods in Sec. 3.3, Unless stated otherwise, employ Llama-3. 3-BBBBB with CoT the default configuration in evaluations. All the visualizations are built upon the model's 's estimation of its own thoughts.
'LoT can be applied trajectories with different numbers of states. We assume states for demonstrations. "Appendix H.8 shows that LoT is compatible and robust with different methods of dimensionality reduction. "Appendix H.1 validates each qualitative observation from LoT. Full visualizations are in Appendix
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Figure 2: Comparing the LoT of different language models (with CoT on the AQuA dataset). Darker regions represent higher state density, with indicating incorrect answers
marking the correct ones. Through the reasoning trajectories, spanning from early (0-20% states) to the later stages (80-100% states), the visualization shows correct cases (bottom rows in blue) with incorrect cases (top rows in red). Metrics calculated w.E.t. each bin, e.g., 20% 40% of states. The reasoning accuracy of the four subfigures is: (a) 15.8%, (b) 42.0%, (c) 53.2%, and (d) 84.4%.

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Figure 3: The LoT of the reasoning model QwQ-32B (using CoT prompting on the AQuA dataset).
3.1 COMPARISON ACROSS LANGUAGE MODELS
We study several LLMs' behavior across parameter scales (from IB, 3B to 70B). We run each model with CoT prompting on 50 randomly selected problems from the mathematical reasoning dataset AQuA. Their landscapes are shown in Fig. 2, from which we have the following observations.
Observation 3.1 (The landscape converges faster as the model size increuse). As model parameters scale from IB to 70B, the corresponding landscape demonstrates faster convergence to the correct answers with higher density in the last 20% states, aligning with the increasing accuracy. With more parameters, larger models can store broader knowledge (Allen-Zhr & Li, 2024). This leads to more confident solutions, demonstrated by more focused answer patterns and lower uncertainty.
Observation 3.2 (Larger models have higher consistency, lower uncertainty, and lower perplexity). As the model size increases, the consistency increases; the same time, the uncertainty and perplexity decrease significantly. This also aligns with the higher accuracy for the large models.
In addition, we apply LoT to the recent reasoning model QwQ-32B (Team, 2025) and observe:
Observation 3.3 (Reasoning models present more-complex reasoning behaviors in landscapes.) ) In Fig. 3, the landscapes can capture complex reasoning patterns such as self-evaluation and self- correction. Specifically, correct trajectories tend to include more instances of self-evaluation and self-correction compared to incorrect ones. These behaviors often oceur early in the reasoning process, when the model is far from the correct answer. Compared to non-reasoning models, correct trajectories here show greater diversity, with green and yellow points more widely scattered.
3.2 COMPARISON ACROSS REASONING TASKS
Besides the AQuA dataset, we include MMLU, Commonsense@AA, and Strategy datasets. We run the default model with CoT on 50 problems per dataset. These observations are derived from Fig. 4: Observation 3.4 (Similar reasoning tasks exhibit similar landscupes). The landscapes of AQuA, MMLU, and Strategy in Fig. 4 exhibit organized search behavior with higher state diversity, while CommonSenseQA presents concentrated search regions, reflecting direct retrieval of common- sense knowledge rather than step-by-ste reasoning processes. These distinct landscape patterns demonstrate the potential to reveal underlying domain relationships across different reasoning tasks.
Observation 3.5 (Different reasoning tasks present significantly different patterns in consistency, uncertainty, and perplexity). The histograms in Fig. 4 show that the perplexity consistently increases as reasoning progresses across all datasets. Specifically, datasets such as AQuA and MMLU show relatively higher levels of uncertainty compared to StrategyQA and CommonSense(AA As for StrategyQA, correct trajectories show increasing consistency that surpasses incorrect trajectories at
7 All claims are defined the original answer distance space and visualized 2D space (see Appendix F.7.) Appendix H.3 presents additional analyses of the consistency metrie: the consistency does not relate to the length of the trajectory. Tab. 5 in Appendix H.5 supports the validity of comparing perplexity across models.
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Figure 4: Comparing the LoT of different datasets (using Llama-3. with CoT). The accuracy of reasoning for the four subfigures is: (a) 84.4%, (b) 80.2%, (c) 75.8%, and (d) 64.8%.
around 60% states, while incorrect trajectories show decreasing consistency. However, when the trajectory is longer than the ground truth trajectory, the later stages (60-100% of states) exhibit both increasing perplexity and decreasing uncertainty."
We show detailed analysis for trajectories Strategy in Appendix H.4.
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Figure 5: Comparing the LoT of four reasoning methods (using Llama-3.1-70B on the AQuA dataset). The reasoning accuracy is: (a) 84.4%, (b) 82.2%. (c) 75.8%, and (d) 81.6%, respectively.
3.3 COMPARISON ACROSS REASONING METHODS
Setup. We evaluate the default model with four reasoning methods: Sfain--ff-thouhh (CoT) (Wei al., 2022), least-to-most (LtM) (Zhou al., 2023), MCTS (Zhang al., 2024), and tree-of- oee-oof-houuht (ToT) (Yao et al., 2023a). We run these methods on 50 problems from AQuA and observe that:
Observation 3.6 (Cross-method comparison: Among correct reasoning trajectories, methods with faster convergence to correct answers achieve higher accuracy.). From Fig. 5, we observe that the
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states are widely dispersed early stages and gradually converge to correct (or incorrect) answers in later stages. Here, convergence means the trend of a reasoning trajectory approaching one answer. Generally, methods with more scattered landscapes (that converge more slowly) present lower accuracy than those that converge faster. For example, the blue landscape in Fig. 5(a) converges faster than the blue landscapes in Fig. 5(c), and the former achieves higher accuracy than the latter.
Observation 3.7 (Within-method comparison: For any single method, incorrect trajectories converge faster to wrong answers than correct trajectories converge to right answers.). In Fig. 5, failure trajectories usually converge to the wrong answers earlier stages of reasoning, e.g., 20-40% of states in Fig. 5(c). By contrast, the states in the success trajectories converge to the correct answers in the later 80-100% of states. This implies that early states of the reasoning process can lead to any potential answers (from a model perspective), while the correct answers are usually determined at the end of reasoning trajectories. In addition, Fig. 34 showcases the corresponding text of thoughts. 10
Observation 3.8 (Compared to failure trajectories. the intermediate states in correct trajectories have higher consistency w.rl. the final state). By comparing the consistency plots in Fig. 5, we find that the model generally has low consistency between the intermediate states and the final state. Notably, the consistency of wrong trajectories is significantly lower than that of correct trajectories. This implics that the reasoning process can be quite unstable. Even though decoding methods like CoT and LtM are designed to solve problem directly (without exploration), the thoughts generated by these methods do not consistently guide the reasoning trajectory to the answer.
4 ADAPTING VISUALIZATION TO PREDICTIVE MODELS
One advantage of our method is that it can be adapted into a predictive model to predict any property users observe. Here, we show how to convert our method to a lightweight verifier for voting trajectories, following the observations in Sec. Note that this methodology not limited to verifiers. Users use this technique to adapt the visualization tool to monitor the properties in their scenarios.
4.1 LIGHTWFIGHT VERIFIER
Observations 3.7 and 3.8 show that the convergence speed and consistency of intermediate states can distinguish correct and wrong trajectories. Inspired by these observations, we build model g IR (M-1))x> + {0, 1} to predict the correctness of a trajectory based on the state features (fr ) and consistency metric (Consistency(s:) the 2-D visualization, encode rich location information of the states and can be used to estimate the convergence speed. Due to the small dimensionality of these features, we parameterize g with random forest (Breiman, 2001) to avoid overfitting. We use this model as a verifier to enhance LLM reasoning (Cobbe et al., 2021). Unlike popular verifiers (Lightman et al., 2024) that involve a moderately sized language model on textual thoughts, our verifier operates on state features and is quite lightweight. We train verifier on thoughts sampled from the training split of each dataset and apply it to vote trajectories test time. Given g trajectories sampled by a decoding method, the final prediction is produced by a weighted majority voting:
The insight is that the state features, used compute
10000 = c) -(fff (Consistency(ss (s))
= arg max GEC
(8)
i=1
4.2 EXPERIMENTAL RESULTS
We evaluate our numerical verifier against an unweighted voting baseline (Wang et al., 2023b) with various models, decoding methods, and reasoning datasets. We report the accuracy here instead of the commonly used pass@k metric, which can be casily inflated by random guessing or exhaustive enumeration of candidates to obtain a high score. Detailed settings of experiments are in Appendix G. We also provide ablation studies on training the verifier and discuss and compare the variance of the verifier in Appendix H.6, and experiment on the scaling effect with different features Appendix H.7
We first compare our verifier against the unweighted voting baseline, each applied to 10 trajectories. As shown in Fig. 6, our verifier consistently enhances the reasoning performance of all models and decoding methods, even though our verifier does not use any pre- trained language model. Notably, smaller language models (IB and 3B) show significant performance
Effectiveness of the verifier.
In Appendix H.2, only few incorrect trajectories (1.8%) are close the correct answer middle thoughts.

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3B
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Figure 6: The accuracy of reasoning under different decoding methods and model scales (averaging across all four datasets). Results for each dataset are in Appendix I.
Staagy Ooeeres
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(a) Transfer across datasets
(h) Transfer across models
Figure 7: Demonstration of the inference-time scaling effect of the verifier. We show the voting accu- racy (%) on StrategyQA scales with the number of trajectories.
Figure Absolute accuracy changes (A Acc) with the verifier, compared to performance in Fig. 6 (without the verifier), The verifier is trained on each column (dataset or model) and eval- uated on all rows (other datasets or models). Positive values indicate improvement in accuracy with the verifier.
gains with the verifier's assistance, achieving substantial improvements over their original reasoning capabilities. We also compare the verifiers across reward- seward-guided methods.
Test-time scaling. While the improvement of the verifier seems marginal with 10 trajectories. our verifier can provide a substantial performance gain with more trajectories. We adjust the number of trajectories from I to 50, and plot the results of the verifier and the unweighted voting baseline in Fig. 7. Models with our verifier exhibit significantly stronger scaling behaviors, achieving over 65% accuracy. In contrast, the performance of the baseline saturates around 30% accuracy. These results suggest that our state features, which are used in both the visualization tool and the verifier, capture important information about the reasoning behavior of LLMs. Thus, the verifier can boost test-time scaling, especially in solving complex problems.
Cross-datasee and cross-model transferability. One interesting property of the state features and metrics is that their shape and range independent of the model and dataset, suggesting that may deploy the verifier trained on one dataset or model in another setting. As illustrated in Fig. 8, we evaluate how the verifier transfers across reasoning datasets (e.g.. train on AQuA and test on MMLU) and model scales (e.g., train on IB model and test on 70B model). We observe some positive transfers across datasets and models. For example, a verifier trained on AQuA can improve the performance of StrategyQA by 4.5%. A verifier trained on the 70B model also improves the performance of the 3B model by 5.5%. However, some cases do not benefit from the transferred verifiers. We leave improving the transferability of the state features and metrics as future work.
CONCLUSION
This paper introduces the landscape of thoughts, visualization tool for analyzing the reasoning trajectories produced by large language models. Built on top of feature vectors of intermediate states trajectories, our tool reveals several insights into LLM reasoning, such as the relationship between convergence and accuracy, and issues of low consistency and high uncertainty. Our tool can also be adapted to predict the answer of reasoning trajectories based on the observed property, which is demonstrated by lightweight verifier developed based on the feature vectors and observations for distinguishing the correctness of trajectories. We foresee that this tool will create several opportunities to monitor, understand, and improve the L.IM reasoning.
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ACKNOWLEDGEMENT
ZKZ. XL., XF, and BH were supported by RGC Young Collaborative Research Grant No. C2005-24Y, RGC General Research Fund No. 12200725, and HKBU CSD Departmental Incentive Scheme. SK was partially supported by NSF 2046795 and 2205329, IES R305C240046, ARPA-H, the MacArthur Foundation, Schmidt Sciences, Stanford HAL, RAISE Health, OpenAI, Microsoft, and Google.
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APPENDIX
A Ethic Statement
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Impact Statement
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Reproduction Statement
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LLM Usage Disclosure
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Further Discussions
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E,1
Challenges in Analyzing LLM's Reasoning Automatically
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E,2 Comparison Between Landscape Visualization and Textual Analysis
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E.3
The Intrinsic Relationship Between Visualization and Metrics
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E,4
Discussion on Results and Observations
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E.5
How to Explain the Increasing Uncertainty and Perplexity?
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E.6
The Convergence Speed of Correct/Incorrect Trajectories and Small/Large Models
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E.7
The Claims of Positions of LoT
E.8
How is Cross-question Comparability Achieved?
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E.9
Potential Extension to Pruning Unpromising Trajectories
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E.10
Potential Extension to Identify Post-hoc Trajectories
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E.11
Limitations and Future Directions
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E. Comparison Between Lightweight Verifier and Reward- guided Algorithms
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Related Work
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Experiment Settings
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G.1 Setup
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G.2 Datasets
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G.3 Decoding Algorithms
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H Supplementary Results and Analysis
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H.l Statistical Verification of the Observations
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H.2 Analysis of Reasoning Trajectory C...............................
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H.3 Further Investigation on the Consistency Metric
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H.4 Further Discussion on the StrategyQA
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H.5
Comparing the Perplexity among Different Models
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H.6
Additional Experiments on the Verifier
H.7
Further Experiments the Scaling Effect
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H.8 Landscapes with Different Methods of Dimensionality Reduction.
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H.9 Robustness of Sentence Tokenization...
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H.10 Generalization to Tasks beyond Multi-choice Problems
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H.11 Landscape Visualization with Choice Reordering.
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H.12 Landscape Visualization with Model beyond Llama Family...
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H.13 Landscape Geometry is Robust across Rollout Sample Sizes
H.14 Landscape Challenging Questions.
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H.15 Landscape Visualization on New Reasoning Methods
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Visulizations
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A
ETHIC STATEMENT
The study does not involve human subjects, data set releases, potentially harmful insights, applications, conflicts of interest, sponsorship, discrimination, bias, fairness concerns, privacy or security issues, legal compliance issues, or research integrity issues.
B IMPACT STATEMENT
This work aims to advance the field of trustworthy machine learning and large language models, especially the interpretability of machine reasoning. Our work presents a tool for visualizing and understanding reasoning steps in LLMs. We foresee that our work will introduce more interpretabil- ity and transparency into the development and deployment of LLMs, advancing us toward more trustworthy machine learning. We do not find any negative societal consequences of our work.
C REPRODUCTION STATEMENT
The experimental setups for training and evaluation are described in detail in Appendix G.1, and the experiments are all conducted using public datasets. We provide the link source codes to ensure the reproducibility of our experimental results: https: / liggtuuu cwm.ccoo/tttto tt/r/mlrrrr.////// mamll-gii r-group/ landscape of tff-thuuh
D LLM USAGE DISCLOSURE
This submission was prepared with the assistance of L.MMs, which were utilized for polishing content and checking grammar. The authors assume full responsibility for the entire content of the manuscript. It is confirmed that no LLM is listed as an author.
E FURTHER DISCUSSIONS
E. CHALLENGES IN ANALYZING LLM's REASONING AUTOMATICALLY
Currently, the fundamental mechanisms behind both successful and unsuccessful reasoning attempts LLMs remain inadequately understood. Traditional performance metries, such as accuracy, provide insufficient insights into model behavior. While human evaluation has been employed to assess the quality of sequential thoughts (e-g., logical correctness and coherence), such approaches are resource-intensive and difficult to scale. We identify three challenges in developing automated analysis systems for LLMs' reasoning:
Challenge 1: Bridging the token-thought gap. Current explanatory tools, including attention maps (Clark et al., 2019; Kobayashi et al., 2020), probing (Alain & Bengio, 2016; Tenney et al., 2019; Hewitt & Liang, 2019), and circuits (Yao al.. 2024), primarily operate the token-level explanation. While these approaches offer valuable insights into model inference, they struggle to capture the emergence of higher-level reasoning patterns from lower-level token interactions. Additionally, the
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discrete nature of natural language thoughts poses challenges for traditional statistical analysis tools designed for continuous spaces. Understanding how thought-level patterns contribute to complex reasoning capabilities requires new analytical frameworks that can bridge this conceptual gap.
Challenge 2: Analyzing without training data access. Existing investigations into LM reasoning have predominantly focused on correlating test questions with training data (Ippolito et al., 2022; Wang et al., 2024a). This approach becomes particularly infeasible given the reality of modern LLMs: many models are closed-source, while some offer only model weights. Therefore, desired analysis framework should operate across varying levels of model accessibility.
Challenge 3: Measuring reasoning quality. Beyond simple performance metrics, need ways to evaluate the quality and reliability of model reasoning. This includes developing techniques to understand reasoning paths, creating intermediate representations that capture both token-level and hevell patterns, and designing metrics that can assess the logical coherence and validity of reasoning steps.
Consequently, we propose that viable analysis of reasoning behavior should satisfy multiple criteria: should operate in a post-hoc manner with varying levels of model access, bridge the gap between token-level and thought-level analysis, and provide meaningful metrics for evaluating reasoning quality. Given the absence of tools meeting these requirements, we identify the need for new analytical framework that can address these challenges while providing useful insights for improving model reasoning capabilities.
E.2 A COMPARISON BETWEEN LANDSCAPE VISUALIZATION AND TEXTUAL ANALYSIS
Notably, for the language model, one could manually examine the responses to individual questions, as their responses are interpretable by humans. However, this approach has two major limitations:
Limitation 1: Lack of Scalability, Analyzing the individual question time-consuming and labor- intensive. In general, text-based analysis requires human evaluators to carefully read long reasoning chains word by word. For example, if it takes 30 seconds to understand a single problem, review- ing 100 problems would require around 50 minutes of focused human effort. This burden grows quickly, especially as researchers often repeat this process many times while developing models and methods. In practice, researchers need quick, easily interpretable feedback, such accuracy, when experimenting with changes to models and methods.
Lack of Aggregation. It is difficult to aggregate insights across multiple problems to understand model behavior at the dataset level. Summarizing model behavior across multiple problems presents another challenge. Suppose one researcher has 100 reasoning chains; it is hard for him/her to reliably synthesize the model's 's overall behavior. Different researchers may arrive at different, subjective summaries, which hinders consistency and interpretability.
Limitation 2:
By contrast, our visualization method provides a more objective and automatic way to analyze model, making it much easier for researchers to analyze the model's 's reasoning behavior. Similar to the t-SNE (Van der Maaten & Hinton, 2008), the visualization enables a more comprehensive analysis of multiple reasoning problems instead of only one problem. The visualization uniquely combines human-readable paths with quantitative, scalable metrics for reasoning process analysis, enabling both model comparisons and mechanistic insights beyond manual text inspection.
Notably, the landscape provides unique insights into LLM reasoning that text analysis alone cannot capture. This power source bridges the gap between localized text understanding and global reasoning behavior. Our analysis in Sec. 3 reveals insights that not revealed by previous text-based analysis. These insights include structural patterns across many reasoning paths, a strong correlation between early consistency and accuracy, and model-level differences where larger models explore more broadly than smaller ones.
E.3 THE INTRINSIC RELATIONSHIP BETWEEN VISUALIZATION AND METRICS
In the modeling of this work, we project each thought (state in a trajectory) from text space to numerical space, with the thought's feature vector that each dimension indicates the distance to particular answer (see Eqn. 1). We compute the feature vectors of all the thoughts from multiple trajectories and then obtain the feature matrix F. Then, based on this feature matrix, compute
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(I) the landscape visualization through dimension reduction and (2) the metries of consistency and uncertainty. From this view, the metrics' information can actually be seen from the landscape. In this work, we mainly focus on the landscapes and also use the metrics plots to help analyze.
In addition, landscape visualizations preserve the information of metrics, including the consistency, uncertainty, convergence, and many other metrics that not covered in this work. The landscape provides a "global" view of the overall reasoning trajectories, while each metric provides a "local" view of particular aspect. Note that humans naturally prefer visual matters like figures and videos, e.g., researchers prefer to use t-SNE in understanding the classification models. We recommend using landscape as visualization tool help understand the LLM reasoning, while the metric plots can further help inspect some particular aspects.
E.4 DISCUSSION ON RESULTS AND OBSERVATIONS
In the landscape visualizations, red regions map out the reasoning trajectories that end in incorrect answers, while blue regions map out those that end correctly. The contour lines and the depth of color together convey the density of reasoning states at each step: darker shades mean more trajectories passing through that region. As you observe landscape evolve from its initial scatter of states toward later clustering, you're seeing whether and how quickly the model's 's reasoning paths lock onto an answer.
Observation 3.6 arises when we compare only the blue (correct) landscapes of different methods in Fig. 5. Early in the process, all methods scatter widely, exploring many possibilities; over time, though, some methods' contours tighten more rapidly than others. Here, the landscape in Fig. 5(a) converges to its correct region much sooner-and with denser cluster-than the landscapes in Fig. 5(b) to 5(c), and this faster, tighter convergence corresponds to its higher accuracy. Namely, methods with more scattered landscapes (converge more slowly) present lower accuracy than those that converge faster.
related pattern appears when we compare models of different sizes in Fig. 2 (Observation 3.1). As we scale from the 1B model to the 70B model, the last 20% of the reasoning steps show increasingly dense blue clusters. Larger models, with greater capacity to store and retrieve information, steer their reasoning more directly and confidently toward the right answer, mirroring their higher accuracy. This further supports the positive correlation between convergence speed (of correct landscapes) and reasoning accuracy, which is revealed in Observation 3.6.
Observation 3.7 emerges from contrasting the red and blue landscapes of the same algorithm in Fig. 5. Here, failure trajectories (red) often settle into wrong answer by roughly 20-40% of the reasoning process, while success trajectories (blue) only coalesce around the correct answer toward the very end-around 80-100% of the states. This indicates that early reasoning states are exploratory and can drift toward incorrect conclusions, whereas correct solutions only converge late in the trajectories. This convergence-speed disparity between red and blue landscapes also holds across multiple datasets in Fig. 4.
Finally, Fig. 4 shows that each reasoning task leaves distinct landscape "fingerprint," supporting Observation 3.4. In AQuA, MMLU, and StrategyQA, the landscapes trace wide, structured sweeps of reasoning states- clear evidence of byep-by-step deduction and exploration of intermediate hypotheses. By contrast, CommonSense@U produces tightly clustered trajectory from the outset, indicating direct retrieval of knowledge rather than an iterative trajectory. This divergence mirrors the tasks themselves: AQuA, MMLU, and StrategyQA require exploratory traversal through multiple reasoning steps, resulting in diverse yet organized state distributions, whereas CommonSenseQAA depends on straightforward recall. These task-specific structures demonstrate how our landscape visualizations can uncover both shared patterns and fundamental differences across reasoning challenges.
In addition, each of these qualitative observations is further supported by statistical analyses in Appendix 1, and we provide full visualizations, including annotated state trajectories (Figs. 32 to 35) and additional model comparisons (Figs. 37 to 38).
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E.5 How TO EXPLAIN THE INCREASING UNCERTAINTY AND PERPLEXITY?
In short, uncertainty and perplexity in our framework capture two different aspects of the process (belief over answer options predictability of reasoning text), and their mostly increasing trends in Fig. 2 reflect exploratory and increasingly specific reasoning, rather than a monotone loss of commitment, In the following, we clarify this interpretation, emphasize the small late-stage drop in uncertainty, and connect these plots to existing findings in the literature.
Uncertainty measures the spread of belief over answer options, and its trend reflects exploration plus late-stage commitment. For each state 8; we form a normalized distance vector f: E 1 over the k answer choices and define Uncertainly(s;; (8;) = Σラ= St(j) log fi(j) with E( = 1. Low uncertainty means the model strongly prefers one option; high uncertainty means it spreads belief across several. In Fig. 2. the average uncertainty over trajectories tends to increase as reasoning progresses, indicating that intermediate thoughts often open up or rebalance multiple candidates instead of simply sharpening an initial guess. Importantly, in the final stage (roughly the last 20% of thoughts), we observe a slight drop in uncertainty, consistent with the model only firmly committing to its final answer near the end of reasoning.
Perplexity is defined over the reasoning text itself and naturally increases as thoughts become longer, more specific, and less templated. For each thought t,, we define PPL(t.) = PL ${-1) using the model's own conditional probabilities. Early steps often use very generic templates (for example, "Let think step by step"), which are highly probable and hence low perplexity. Later thoughts contain detailed computations, question-speciffe entities, and idiosyncratic phrasing, which are rarer under the model and thus yield higher perplexity. This explains why perplexity tends to increase with step index, even though larger models have systematically lower perplexity plots than smaller ones at the same step (as seen in Fig. 2 and Appendix H.5).
Similar phenomena have been noted in prior work: high-quality or human-like text does not always correspond to the lowest-perplexity generations, and useful content often resides in relatively lower- power-probabilityy regions of the model's distribution. For example, Holtzman et al. (2020) observe that pushing perplexity too low leads degenerate, repetitive text and argue that high-quality text typically has moderate perplexity rather than minimal perplexity. Recent work on chain-of-thoughh also uses stepwise perplexity to identify "critical" reasoning steps, showing that changes in perplexity along the chain correlate with decision quality (Cui et al., 2025). Our peoreasing-perplees plots are consistent with the view that deeper reasoning steps are rarer but important.
The increasing trends of uncertainty and perplexity are most informative when interpreted together with consistency and accuracy, especially across model scales. In Fig. 2, larger models exhibit higher consistency (their intermediate states agree with the final answer more often and earlier in the trajectory) while still showing increasing uncertainty and perplexity. This suggests a pattern of "early latent commitment plus exploratory justification": larger models tend to settle on a preferred answer relatively early (reflected in high consistency), then generate more complex, problem-specific reasoning that (i) briefly reconsiders alternatives, raising average uncertainty, and (ii) uses less frequent language, raising perplexity. Smaller models, in contrast, exhibit lower consistency together with similar or higher uncertainty and perplexity, indicating that their exploration is less controlled and more often accompanied by changes in the preferred answer. Our verifier in Sec. leverages exactly these temporal patterns in uncertainty and perplexity (along with consistency) to predict correctness, which further supports that these metrics capture meaningful dynamies rather than noise.
In summary, the non-decreasing uncertainty and increasing perplexity in Fig. 2 do not imply that models simply become less confident as they reason. Rather, they show that models (i) explore and hedge among multiple answer options in the middle of trajectories, then slightly reduce uncertainty when committing at the end, and (ii) move from generic, high-proc bigbabbtbabilityy templates to rarer, more specialized reasoning text as depth increases, with larger models doing so at overall lower perplexity levels.
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E.6 THE CONVERGENCE SPEED OF CORRECT/INCORRECT TRAJECTORIES AND SMALL/LARGE MODELS
The key point is that they refer to different conditionings: Fig. I compares correct versus incorrect trajectories within fixed model, while Observation 3.1 compares convergence behavior across model sizes, averaged over trajectories. Once this distinction is explicit, the two statements are consistent.
Within fixed model, wrong trajectories converge prematurely, and correct trajectories con- verge later. Fig. I describes a within-model phenomenon: for given model and dataset, trajectories that eventually answer incorrectly tend to "lock in" to wrong answer region earlier, while correct trajectories stay exploratory for longer and only settle near the correct region toward the end. We formalize this using a convergence statistic based on high-dimensional state features. For a trajectory with states f fn. we measure the distance d: = [ fn 2 between each state and its own final state, fit a log-linear model log d, R a + fii and use ex as a convergence coefficient, where smaller el means faster convergence. Tab. I(a) shows that, under the same model and method, incorrect paths have significantly smaller e than correct paths (p value = 0.008), which matches the intuition in Fig. that wrong paths converge "too fast" to wrong answers, while correct paths take more steps before stabilizing.
Across model sizes, larger models converge more efficiently to correct regions average. Observation 3.1 studies a different axis, comparing convergence across model scales on AQuA. Here we keep the dataset, prompt, and reasoning method fixed and vary only the model size (for example, Llama 3.2 1B, 3B, 8B, and Llama 3.1 70B). We quantify how directly trajectories move from initial to
1 Sol12
final states using the speed metric speed
-1855-18-111 8--1l12 E [0. 1], where & is the 2D position 13 13=
of state i. Higher speed means less wandering and a more direct path. Appendix H.1 shows that this speed correlates very strongly with accuracy (p-value about 9.4 x 10-11), and that larger, more accurate models have higher speeds than smaller ones. Intuitively, as the model scales up, a larger fraction of its trajectories quickly head toward the correct region and follow relatively straight paths, so the landscape "converges faster" in the sense of Observation 3.1.
The two statements are compatible once we distinguish between conditional and marginal views. Putting these together, we have two different but compatible facts. Within each fixed model, if we condition outcome, incorrect trajectories tend to converge earlier than correct ones, that is, Elewrong] < Eleemeal] which Fig. and Observation 3.7. Across models of different sizes, when we look at all trajectories or at correct trajectories in aggregate, larger models have higher speed and smaller convergence coefficients; that is. they reach their final regions, especially the correct region, more efficiently than smaller models, which is Observation 3.6. In practice, a 70B model still has some early wrong convergences, but these are fewer, and its successful trajectories travel more directly to the correct cluster and dominate the overall landscape. An analogy is that a stronger solver both has more correct solutions and tends to reach them more quickly on average, while for any fixed solver, the quickest answers are often overconfident mistakes.
In summary, Fig. I describes a modil effect (wrong paths in that model converge earlier than right paths), while Observation 3.1 describes mcross-model effect (larger models' trajectories, especially correct ones, converge faster and more directly than those of smaller models).
E.7 THE CLAIMS OF POSITIONS OF LOT
All core claims are defined and validated in the original answer distance space and visualized in 2D space. For each intermediate state s; = [Ir, "r,,t,........ tt). we construct a feature vector fs e R with components fi) d(s,,cg)) where d(s;, cj) is the length- norgth-oormlizeed perplexity- the candidate answer Cy. followed by l normalization so that fa lies in a probability simplex-like space over the & options. This vector is exactly the model's 's internal assessment of how the current partial reasoning aligns with each answer. Our metrics are defined directly on fi and on the thoughts: consistency checks whether arg min, fi()) matches arg min; fn (j), uncertainty is the entropy of the normalized f.. and convergence is captured by the margin and stability of the best answer over steps. Using these quantities, show that incorrect trajectories tend to converge earlier and more sharply to a wrong answer (Observation 3.7). while correct trajectories exhibit higher intermediate consistency and lower uncertainty (Observation 3.8), as summarized in the consistency and uncertainty plot in
distance to
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Figs. 2, 4, 5 and statistically verified in Appendix H.1 to H.3. None of these results relies any geometric property of the 2D representations.
Our consistency metric tracks the stability of answer preference and is largely insensitive to global distribution sharpness. Consistency depends only on arg min f; and arg min fn, that is, on the ranking of distances across choices. Any monotone transform that uniformly sharpens probabilities, for example, P; y/ with > 1 or temperature rescaling p' (c; st) p(c; s;) with T < 1. preserves this ranking and leaves Consistency(s:) unchanged. A generie increase in sharpness would therefore not by itself increase consistency, nor would create the systematic gaps we observe between correct and incorrect trajectories between small and large models. For consistency to increase, intermediate states must more often agree in their top choice with the final answer, which is exactly what we interpret as more stable belief dynamics.
The visible clusters and paths align with these high-dimensional metrics and labels, which indicates that they reflect real structure. Distances to the correct answer and distractors are explicit coordinates of ft, and we also embed answer anchors as landmarks (Fqn. 3). Regions that appear in 2D as clusters around a particular anchor correspond states where fo places most mass on that option, and consistency with the final prediction is high. Similarly, trajectories that visually move from diffuse areas a compact region near the correct anchor correspond to sequences where distances to the correct answer decrease and the margin over other options increases, precisely what our convergence coefficients and consistency plots quantify. Observation 3.6 and Tab. I(a) show that incorrect trajectories converge earlier and more strongly toward their final answer region than correct trajectories, and Observation 3.7 and Tab I(b) show that paths with higher speed tend to be more accurate. This alignment between geometric patterns and scalar metries suggests that the 2D clusters and paths are manifestations of structure already present in the original feature space.
LoT is explicitly framed as behavioral diagnostic of belief' dynamics rather than direct probe of latent cognitive mechanisms. The log probabilities reflect surface likelihoods rather than an explicit symbolic proof tree. But for autoregressive language model, the conditional likelihood distribution is what governs behavior, so any latent reasoning process must ultimately manifest as changes in these scores. By tracking the sequence (fi) we analyze how the model's own scoring function reallocates probability mass across the k candidate answers as thoughts are generated. Phenomena such as incorrect trajectories converging carlier to wrong options than correct trajectories converging to the right one, and the existence of mid-trajectory states with low consistency and high entropy, are expressed in terms of margins and entropy in this original feature space and are not trivial consequences of simply preferring tigh-brobybility answers. LoT therefore makes deliberate choice to study belief evolution over answer options as an operational view of reasoning behavior, without claiming to reconstruct a hidden conceptual reasoning structure.
The t-SNE is used only to visualize an already low-dimensional, semantically anchored space, and all qualitative patterns in the landscapes are corroborated by rrojection-independeent analyses and alternative projectors. Each state is represented by the k-dimensional vector fi = (d(s;, ....... d(st, o))) after length and & normalization, and the answer options themselves embedded as anchors in the same space. Two states are close if they induce similar distributions over answers, so this space already has clear probabilistic interpretation before any projection. We then apply t-SNE to map this interpretable space to 2D primarily to illustrate how trajectories move from regions where multiple answer clusters are mixed regions dominated by single cluster and how correct and incorrect trajectories end near different anchors, as shown in Figs. 2, 4, and 5. These are coarse neighborhood and cluster properties rather than fine-grained claims about global distances or angles. Moreover, Appendix H.8 shows that replacing t-SNE with UMAP or PaCMAP yields qualitatively similar patterns of early diffuse exploration versus later convergence and similar separation of fast-converging wrong trajectories from slower-converging correct ones. The corresponding one-dimensional plots in the original space (distance to correct answer, entropy, consistency over steps) show the same trends. This indicates that the observed paths are rooted in the underlying answer distance features rather than artifacts of a particular projection algorithm or random seed.
The analyses of model scale explicitly separate global distribution sharpness from genuine differences in trajectory behavior by using normalized features, within-mode comparisons, and additional controls. Larger models indeed tend to produce sharper output distributions, and we observe lower uncertainty and perplexity in Fig. 2. To mitigate this effect, we always apply L
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normalization to each fi. that is, ft t f// that we focus on the relative geometry among options rather than absolute log probability scales. More importantly, many key comparisons are within fixed model, such correct versus incorrect trajectories or different reasoning methods and datasets evaluated with the same model, where global calibration and sharpness are essentially held fixed. In these settings, we still find that failure trajectories typically converge earlier and remain highly consistent around a wrong option, while success trajectories converge later but more reliably to the correct one, which cannot be attributed to differences in global sharpness. As shown in Tab. 5, while there is slight variation in perplexity across model scales, the values all fall within a comparably narrow range (from 1.42 to 1.96), which demonstrates that for decoding the same CoTs, different models in the Llama-3 family produce similar and comparable perplexity scores. This supports the validity of comparing perplexity across models our study.
The predictive power of the lightweight verifier trained on LoT features provides independent evidence that these representations capture meaningful structure in reasoning behavior rather than only visualization artifacts. The lightweight verifier is a simple random forest that operates solely on the likelihood-baser state features and consistency statistics derived from fa, without access to raw text or hidden activations. Despite this simplicity, it significantly improves accuracy over unweighted celf-conni and yields much stronger test time scaling as the number of trajectories increases, shown in Figs. 6 and 7. If the features only reflected superficial sharpness or spurious projection effects, it would be difficult for such a simple model to reliably distinguish correct from incorrect trajectories across datasets, methods, and models. The verifier's ' effectiveness supports the view that LoT's representation retains stable and informative signals about reasoning quality.
In summary, LoT should be interpreted as method for visualizing and quantifying belief dynamics over candidate answers rather than for reconstructing a latent "conceptual reasoning structure." It builds a well-defined model of the internal representation of each thought in the answer distance space ft. defines trajectory- based metrics and statistical tests that do not depend on t-SNE, and uses 2D landscapes that are robust to the choice of projector and consistent with these metrics.
E.8 How IS
COMPARABILITY ACHIEVED?
LoT does not assume a single global semantic manifold over all texts. Instead, each state is embedded into shared belief space over answer options, so cross-question comparability comes from common probability simplex structure and per-question normalization, and all quantitative results are computed in this space before any 2D projection.
(1) LoT works in decision space over answer options, not a universal text embedding space. For a multiple-choice question with candidates C (ci....,.c, } and an intermediate state s, = (x., (1,...,44) 4z), we construct a feature vector Ja E & with components f((j) = d(ss; 9j). where d(ss; (2) is the length-normalized perplexity-based distance (Eq. (2)), followed by e normalization so that E; fi( = 1 (Section 2.2). Thus f. is point in probability simplex over answer indices {1..... ......... }.), encoding how the model distributes belief over the options state 8;, rather than an embedding of raw text into a semantic space.
(2) Cross-question comparability is defined in terms of belief geometry over answer indices, not direct semantic similarity of answer texts. For a fixed dataset like AQuA, all questions share the same number of options k. so every state feature fü lies in the same k-dimensional simplex. Two states from different questions are close if the model exhibits similar belief patterns, such as strong confidence in one option, confusion between two options, or near-uniform uncertainty. Our core metrics use exactly this structure. Distances to the correct answer are computed by selecting the coordinate corresponding to the ground index. These quantities are therefore comparable across questions because they measure how belief mass concentrates and stabilizes over indices, not over specific strings.
(3) Each landscape is learned from single global embedding over the shared simplex, not from stitched per-question subspaces. For a given dataset and model, we obtain state features {ff from all questions and trajectories. We normalize feature vectors by reordering choices so the correct answer appears in the first dimension across all questions. These state features, together with the choice anchors (Eqn. 2). are pooled into one matrix in R$ and apply single dimensionality reduction g *R + R² (by default, t-SNE). Figs. 2, 4, and 5 are produced by this single mapping per dataset, so all points in figure lie in the same underlying belief space. For datasets with different k (for
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example, AQuA versus StrategyQA), we keep landscapes separate and compare them via metrics such as consistency, uncertainty, and histogram intersection scores in Tab. 1(c), rather than forcing them into shared manifold.
(4) Our quantitative conclusions do not rely on interpreting arbitrary distances between states from different questions semantic similarity. All core metrics and statistical tests are defined within each question's ' simplex and then aggregated. For example, the convergence coefficient in
Appendix H.1 is obtained by fitting log d'") x ag B, for each question q. where d!") is the distance from & to the correct answer for that question, and then analyzing the distribution of exp(3, across questions (Tab. (a)). Similarly, the consistency and uncertainty plot in Figs. 4, and 5 are computed by first evaluating these metrics per state and per question, then averaging over questions. None of these analyses requires treating the Euclidean distance between a state from question q and a state from question g' as semantically meaningful; they depend only on how beliefs evolve within each local answer simplex.
(5) Geometrically, the global landscape can be viewed as many aligned local simplexes embedded in a common feature space, and we interpret it in that way. Each question induces a k-vertex simplex of belief states over its own options. Because we use the same coordinates (option indices and normalized distances) for every question in dataset, these simplexes live in the same ambient space RR* When we apply t-SNE to the pooled set of {fi). we obtain 2D projection in which clusters correspond to structurally similar belief states, such as "high confidence in the chosen option" or "persistent confusion between two options". We do not claim that this 2D embedding is globally faithful semantic reasoning space; it is an intuitive visualization of how belief states populate the probability simplex, while our formal claims are grounded in the per-question metries described above.
In summary, LoT embeds thoughts into dataset-specifie belief space over answer indices that is shared across questions, and our conclusions are derived from within-question geometry and aggregated statistics rather than from assuming universal semantic manifold over text.
E.9 POTENTIAL. EXTENSION TO PRUNING UNPROMISING TRAJECTORIES
We showcase that our tool can be utilized to identify potentially incorrect reasoning trajectories at test time. In Section 3.3, we build up a lightweight verifier, which is based on the thoughts' feature vectors and the consistency metric from the landscape of thoughts. This verifier indeed aims to predict the correctness of reasoning trajectory, in order boost the reasoning accuracy test time. It is proven to be beneficial to the voting of multiple reasoning trajectories, as shown in Sec. 4.2.
Further, this verifier (together with the visualization tool) can be adopted to prune unpromising reasoning trajectories in tree-based searching. For instance, in methods like sre-oof-thuggle and MCTS, model explores multiple reasoning trajectories and usually uses the same model identify the promising paths to search for the ultimate solution. Here, by leveraging features from the landscape of thoughts and the consistency metric, our verifier can identify flawed trajectories early during reasoning, acting as an efficient pruning mechanism to boost the search efficiency and reasoning performance.
Therefore, our tool can be integrated into the reasoning methods to monitor particular reasoning patterns (e.g., the correctness) and help understand as well as boost reasoning. There are multiple directions that deserve future exploration, including the one to identify and prune the potentially incorrect reasoning trajectories (Han et al., 2025). Such capability is particularly relevant in safety- critical applications, where challenges like jailbreak defense (Li et al., 2023b), safety alignment (Cao et al., 2025), and knowledge retention (Wang et al., 2025a) demand robust monitoring of LLM reasoning behavior.
E. POTENTIAL EXTENSION TO IDENTIFY POST-HOC TRAJECTORIES
In the following, we discuss the feasibility of detecting post-hoc trajectory using our framework, particularly in defining the post-hoc trajectory. A post-hoc trajectory refers to the trajectory that the model exhibits high confidence in a single answer in the early states and maintains high consistency across states in the trajectory. Specifically,
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the "early state" correspond to the "very early tokens of the response"; the "high confidence in single answer" corresponds to the "model has chosen its answer":
the "high consistency across states in the trajectory" corresponds to the "trajectory is produced as consequence of that decision".
Namely, the post-hoc trajectory can be potentially identified by inspecting the confidence and consistency of particular positions of states in our framework. Then, we elaborate on the more detailed definitions for the three components above.
For defining the "early states", it should have an absolute threshold of states index, e.g., early 10 states, or a relative threshold, c.g., early 10% of states. This threshold should be chosen deliberately, and the states with index smaller than this threshold are categorized as "carly states".
Similarly, clear threshold is necessary for defining the "high confidence" or "high consistency". e-g., over 80% confidence and 60% consistency. With the metrics defindd in Section 2.3, here, we should examine (1) the confidence of the early states in the trajectory and (2) the consistency across all states of the trajectory. Here, only the trajectory that exceeds the confidence threshold as well as the consistency threshold can be classified post-hoe trajectory.
In conclusion, our framework shows promise for identifying post-hoc trajectories. Meanwhile, we should note that still needs (1) choose particular thresholds for the precise definition of post-hoe trajectory and (2) to collect set of reliable data to verify the effectiveness in identifying post-hoc trajectory. These are quite challenging to conduct. Although it goes beyond the scope of work, we believe investigating post-hoc trajectory in reasoning is valuable and merits exploration in future work.
E.11 LIMITATIONS AND FUTURE DIRECTIONS
Scope. While the Landscape of Thoughts offers practical lens on model reasoning, its current in- stantiation is limited to multiple-choice settings. Extending LoT to open-ended mathematical problem solving, code generation, and planning- requires handling less structured and more entangled reasoning paths, especially as LLM training infrastructure continues to evolve (Tang et al., 2025; 2026). Two complementary threads of future work are: (i) improving accessibility by producing intuitive visual and textual explanations that help non-experts inspect and trust model behavior, and (ii) developing automated, scalable detectors of reasoning failures to improve reliability across applications.
Key challenge: synthesizing options. The central obstacle is the quality of the synthesized answer options. Human-authored distractors carefully calibrated to be plausible, exposing distinctions between (1) correct reasoning and (2) ceasonable-but-wront or making arithmetic slips). In contrast, LLM-ggnnrated distractors can be implausible and thus triv- ially eliminated when juxtaposed with the correct option, yielding visualizations that over-emphasize the correct trace and limit diagnostic value. Moreover, LLMs may reuse similar reasoning patterns, producing duplicup eтror modes across incorrect options and reducing the comprchensiveness of the analysis.
reasoning (e.g.. overlooking information
Mitigations. To address these issues, we can elicit higher-quality distractors with state-of-the-art LLMs (e-g., OpenAl o3, Gemini 2.5 Pro) and tune sampling hyperparameters (temperature, top-p) to promote diversity and explore alternative solution trajectories.
Binary reformulation. A practical alternative is to recast multiple-choice prompts as binary (yes/no) queries. For example, the question "What is the capital of France?" can be reformulated as "Is Paris the capital of France?" with options Yes or No. Under this framing, both options remain prima facie plausible: the incorrect choice admits coherent yet flawed rationales, and the variety of "No" trajectories preserves diversity without resorting to obviously implausible distractors.
Beyond multiple choice. Although open-ended tasks are beyond the present scope, LoT is, in principle, extendable. The key requirement is to construct a candidate set of answers by querying the model (a non-trivial step that is given for free in multiple-choice tasks). Treat the ground-trutt answer as one option and generate additional plausible alternatives using LLMs; LoT can then analyze the induced reasoning behaviors in these open-ended scenarios.
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code generation. Code generation introduces additional challenges: there is typically single ground-truth program, and evaluation proceeds via test suites. Candidate programs are diverse and do not naturally discretize into options. We propose the following procedure: (i) sample multiple candidate solutions from the model under evaluation; (ii) score each by the number of tests passed; (iii) apply a threshold to separate more-correct from less-correct solutions; (iv) embed and cluster solutions within each partition; and (v) use cluster centroids as anchors for "correct" and "incorrect" choices. Cluster quality can be assessed with the Silhouette Score and the Davies-Boulde Index. These anchors enable a LoT--style visualization over the solution space and provide insight into reasoning behaviors.
Case:
In summary, our visualization framework is adaptable beyond multiple-choice scenarios. To our knowledge, LoT is the first landscape visualization tool aimed at analyzing I.L. reasoning; it is imperfect and remains open to improvement and extension. We believe it constitutes small but meaningful step toward understanding and improving the reasoning processes of L.MSs.
E.12 A COMPARISON BETWEEN LIGHTWEIGHT VERIFIER AND REWARD-GUIDED ALGORITHMS
It is worth noting that our goal is not to build a sophisticated verifier, but rather to demonstrate how the feature vectors from the landscape visualization can be effectively used.
In general, -wradd-guiedd algorithms are more computationally efficient than the path landscape. Specifically, for reasoning path with n thoughts and e answer choices, constructing the landscape requires x forward passes through the reasoning model. In contrast, reward-guided approach typically makes a single call to a reward model that evaluates the entire reasoning chain at once.
Mcanwhile, it's important to consider the overhead involved in training the reward models in reward- guided algorithms. Notably, for Process-Rewande Models (PRMs) (Luo et al., 2024; Xu et al., 2025), collecting high-quality training data often requires detailed, fine-grained annotations of reasoning steps, which can be costly and time-consuming. Moreover, training a reward model (often itself an LLM) incurs significant computational expense. In contrast, our lightweight verifier is much more efficient to train, as it requires no human annotations and uses easily obtainable data.
F RELATED WORK
Reasoning with large language models. Chain-of- Chain-of-Thought (CoT) prompting (Wei et al., 2022; Kojima et al., 2022) has empowered LLMs to tackle multi-step reasoning problems by generating intermediate steps before producing a final answer. Building upon COT, numerous methods have been proposed address various challenges, including compositional generalization (Zhou al., 2023; Khot et al., 2023), planning (Yao et al., 2023a; Hao et al., 2023), knowledge reasoning (Liang et al., 2024a; 2025; 2024b), graph analytics (Shang & Huang, 2025; Shang et al., 2025), rule learning (Zhu et al., 2023), and active reasoning (Zhou et al., 2025) within the CoT reasoning. Beyond solving reasoning tasks, CoT has also emerged as a foundational framework for other techniques, such as fine-tuning LLMs (Zelikman et al., 2022; Wang al., 2025b; Zhang et al., 2025a), enabling LLM--ased agents (Yao et al., 2023b), and facilitating test-time scaling (Snell et al., 2024; Zhang al., 2025b). Nevertheless, most of these approaches are developed in a trial-and-error manner, largely due to the absence of proper tools for analyzing the COT.
chain-of-thougie reasoning. There are few studies that explore what makes CoT prompting effective by perturbing its exemplars. To be specific, Madaan & Yazdanbakhsh (2022) found that the text and patterns exemplars help CoT generate sentences resembling correct answers. Besides, Wang et al. (2023a) highlighted the importance of maintaining the correct order of reasoning steps, while Ye et al. (2022) demonstrated that using complementary exemplars can enhance reasoning performance. Furthermore, CoT can benefit from longer reasoning chains, even without new information to the prompt (Jin et al., 2024). Another line research investigates CoT's general behavior (Tang et al., 2023; Saparov & He, 2023; Saparov et al., 2023; Shi et al., 2023). For example, CoT heavily depends on the semantic structure of the problem to perform reasoning (Tang et al., 2023), struggles with planning and unification in deductive reasoning (Saparov & He, 2023), has difficulty generalizing to longer reasoning paths (Saparov et al., 2023), and can be easily misled by irrelevant information in the context (Shi et al., 2023; Zhou et al., 2024). However, these observations
Understanding
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derived from specific reasoning tasks and prompt settings, limiting their applicability other scenarios. In contrast, we introduce a general-purposs tool that allows users to analyze reasoning in their contexts.
Tools for analyzing -hain-of-thought. To the best of our knowledge, the only existing tool for analyzing CoT is gradient-basee -ased feature attribution (Wu al., 2023), which computes saliency score for each input token based on the model's output. However, these token-level saliency scores do not directly capture the thought-level, multi-step reasoning process of LLMs. Consequently, the main finding in (Wu et al., 2023) is that CoT stabilizes saliency scores on semantically relevant tokens compared direct prompting. Metrics designed to quantify CoT performance (Chen al., 2024; Ton et al., 2024) can also be used to analyze the reasoning behaviors of LLMs. For instance, Ton al. (2024) employs information gain to identify failure modes reasoning paths, aligning with Observation 3.7 in this paper. However, our 2-1) visualization offers significantly deeper insights than single information gain metric. Additionally, the verifier derived from our tool conceptually related to outcome-supervane reward models (Cobbe et al., 2021).
Mcasuring uncertainty and consistency LLM reasoning. Several works in this research line compute metrics (such as confidence and perplexity) by leveraging the features from LLMs to measure and detect hallucination in reasoning (Li et al., 2023a; Chuang et al., 2024; Yang et al., 2025). Specifically, low confidence and high perplexity often indicate unreliable reasoning. enabling the development of lightweight detectors to guide reasoning and mitigate hallucinations. However, these metrics have limitations (Xiong et al., 2024; Zhang al., 2023): they can exhibit over-confidence or low perplexity in incorrect responses, their reliability relies heavily on the models' capability, and they cannot provide more comprehensive insights into the multiple reasoning trajectories. By contrast, our landscape of thoughts offers holistic approach, integrating several existing metrics. This framework enables global qualitative analysis, including measures of perplexity, consistency. and uncertainty. In addition, the landscape of thoughts enables the development of advanced tools to enhance reasoning by using the features and metrics, as mentioned in Sec. 3.3.
G EXPERIMENT SETTINGS
G.1 SETUP
Visualizing the landscape of thoughts fundamentally relies on the decoding probability of
LLMs. To this end, we adopted four open-source models with varying parameter sizes, namely
Llama-3.. 3.2-1, Llama-3. 2B3B, L1ama-3.1 and Llama- 3. 1-70B. We repeatedly sample 10 times from the target LLM using the same reasoning strategy as self-consistency (Wang et al., 2023b).
For visualization purposes, randomly sample 50 questions from the testing split of each dataset and generate reasoning paths with the setup described above. For simplicity, we compute distances only between each state and all candidate answers. To visualize multiple problems we always place the distance to the correct answer as the first element of each feature vector. This aligmment allows joint analysis across problems, as introduced the paragraph below Equation We then aggregate feature vectors from all problems into feature matrix (Equation 2). which is passed to t-SN to compute the pairwise distance between any two states and then outputs the 2D) coordinate of each state.
shared space,
For training the lightweight verifier, we randomly sample 20 questions from the training split of each dataset to obtain the feature matrix S. We extract these features using three model scales: Llama ..2--33.23000 2-3B, Llama- 33. 3.1 1B, and Llama-33 3..maa-..---- Despite the relatively small training set, it proves sufficient for our lightweight verifier, which subsequently evaluate on the data for visualization in Sec. 3.
G.2 DATASETS
AQuA (Ling al., 2017). This dataset develops to challenge language models' quantitative reasoning capabilities. The AQuA presents complex algebraic word problems in a multiple-choice format, where only one is correct. Each problem requires numerical computation, deep linguistic understanding.
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and logical inference. It provides a nuanced assessment of model's ability to translate textual information into algebraic reasoning.
MMLU (Hendryeks al., 2021). Spanning 57 distinct academic and professional domains, MMLU provides a rigorous test of language models' capabilities across humanities, social sciences, hard sciences, and technical disciplines.
StrategyQA (Geva et al., 2021). This dataset is designed to evaluate implicit reasoning and multi- hop question answering. The dataset is characterized by yes/no questions that demand implicit reasoning strategies. Unlike straightforward factual queries, these questions require models to construct elaborate reasoning paths, showing hidden logical connections.
CommonsenseQA (0a0AATalmmn (Talmor et al., 2019). This dataset assesses commonsense reasoning through multi-choice questions derived from the ConceptNet knowledge graph (Speer et al., 2017). The dataset aims to test model's s understanding of commonsense concepts and ability to make logical inferences. However, the questions often require the model to incorporate external knowledge to select the correct answer from plausible distractors.
Note that AQuA, MMLU, and Strategy QA all demand exploratory traversal of intermediate reasoning states, resulting in diverse but structured landscapes. Commonsense@AA, conversely, represents distinct domain where answers depend on static knowledge rather than emergent reasoning pathways.
G.3 DECODING ALGORITHMS
Chain of Thought (CoTT) (Wei et al., 2022). CoT elicits the LLM's reasoning capabilities by incorporating few-shot examples that demonstrate explicit reasoning steps. provides the model with exemplar reasoning traces to guide its problem-solving process.
Zero-shot CoT (Kojima et al., 2022). The core idea of this prompt strategy lies adding simple instructions, e.g., "Let's think step by step." to the prompt, enabling models to generate reasoning traces without assigned task -sec-fpe examples.
Least-to-Most (LtM) (Zhou et al., 2023). LIM an innovative reasoning approach that systematically breaks down complex problems into progressively simpler subproblems. This approach mirrors human cognitive problem-solving strategies, where individuals naturally break down complex tasks into smaller, more comprehensible parts.
Tree-of-Thought (ToT) (Yao et al., 2023a). ToT expanded this concept by creating phisticated, multi-branching reasoning framework. While CoT follows a linear path of reasoning, ToT introduces simultaneously, evaluate them, and strategically prune less promising trajectories.
so-
dynamic exploration, allowing models generate multiple reasoning paths
Monte Carlo tree search (MCTS) (Zhang et al., 2024). MCTS is powerful computational algorithm originally developed for game-playing strategies, particularly in complex decision-making environments like chess and Go. The method uses probabilistic sampling and tree exploration to systematically navigate potential solution spaces, balancing exploring new possibilities with exploiting promising paths. We adopt the task-agnostic node expansion and evaluation prompt from ReST-MCTS (Zhang et al., 2024) to conduct our experiment across different tasks.
H SUPPLEMENTARY RESULTS AND ANALYSIS
H.1 STATISTICAL VERIFICATION OF THE OBSERVATIONS
In this part, we conduct extra experiments and statistically verify Observations 3.1, 3.4, 3.6, and 3.7, while the other Observations 3.2, 3.5, and 3.8 have been quantitatively verified by the metrics in Sec. 2.3.
To verify Observations 3.6, we calculate the convergence coefficient (e)) by fitting log-linear regression model to the sequence of distances d, between each state and the final answer as log(d,) (d.) ss 3i, where is the intercept term; is the slope coefficient that quantifies convergence behavior; i represents the position index in the reasoning chain. Lower values of el indicate faster convergence. For Observations 3.1 and 3.7, we measure the speed of reasoning path moving from start to end as
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Table 1: Statistical verification of the observations in Sec.
(a)
Verifying
Observa-
(b) Verifying Observation 3.7
(c) Verifying Observation 3,4
tion 3.6
and 3.1
Common SenseQA
Correct
Incorrect
Speed
Accuracy
AQuA MMLU Sirategy
CoT
1.026
0.975
CoT
0.322
84.4%
AQDA MMLU
0.914
0.870
L2M
1.026
0.989
L2M
0.224
82.2%
Strategy
0.895
1.0
0.889
1.004
0.987
ToT 0.205
81.6%
SenseQA
Common
0.859
0.843
0.889
1.0
1.002
0.985
MCTS
0.198
75.8%
3=11 13--81-1 E [0, 1], where represents the 2D coordinate of the state i. Whereas Observation 3.4, we compute pairwise histogram intersection scores of the density distributions. Lower scores indicate greater dissimilarity between landscapes.
speed
1. 80
Notably, for Tab. ((a), we found that correct paths consistently show slight divergence, while incorrect paths show more convergence (p-value 0.008), thus verifying Obs. 3.6. As shown in Tab. 1(b), speed and accuracy correlate strongly (p-value = (9.22e--11) thus verifying Observation 3.7. This is also applicable for verifying Observation 3.1. Tab. I(c) shows that lower scores indicate greater dissimilarity between landscapes, which verifies Observation 3.4, i.., AQuA, MMLU, and similar, while CommonSense(AA exhibits distinct patterns.
Strategy
H.2 ANALYSIS OF REASONING TRAJECTORY CONVERGENCE
We aim to investigate Observation 3.7 quantitatively to show its consistency with the statistical result. Specifically, analyzed all questions from the AQuA dataset using the model with the CoT method. Among the 500 reasoning trajectories (50 questions, with 10 trajectories per problem), we observed that cases where reasoning chain initially approached the correct answer's path but later diverged to an incorrect conclusion were quite rare: only 4 questions (8% of all questions) exhibited this phenomenon, accounting for just 9 reasoning trajectories (1.8% of all trajectories). This indicates that such failure cases are infrequent within the overall set of generated reasoning trajectories.
Thoughts that diverge from the correct answer exhibit remarkable proximity at certain states. We quantify the proximity by measuring the distance between states: smaller distance indicates that the state is closer to the correct answer. Notably, in the following example, the chain's 's reasoning reached a minimum distance of just 0.077 from the correct path before drifting to final distance 0. 08182. This reveals that even incorrect answers may closely track the correct reasoning at key moments.
We provide a concrete example of such a failure the following reasoning chain for a question below, where the correct answer is B) 28%.
An example of reasoning that is close to the correct answer at intermediate thoughts but finally converges to the incorrect answer
> Q:
The original retail price an appliance was percent more than its wholesale cost. the appliance was actually sold for 20 percent less than the original retail price, then it was sold for what percent more than its wholesale cost?
Options: A) 20% B) 28% C) 36% D) 40% E) 42% The model's output is:
1. "To find the percent more than the wholesale for which the appliance sold,
need to break down the calculations step by step."
...
> 11. "So, (0.28x/ x) 100 = 28%. + The state that is closest to the correct answer, with
distance of 0.077.
> 12. "Therefore, the appliance was sold for 28% more than its wholesale cost."
> 13. "The answer C." - Eventually, this state reaches the incorrect answer, with distance
as 0.182
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H.3 FURTHER INVESTIGATION ON THE CONSISTENCY METRIC
In the Tab. 2, we analyze the model responses for drawing Fig. and report (1) the average number of thoughts, (2) the average number of tokens in a thought, and (3) the average consistency of different thoughts.
Table 2: The relation of consisteney with the number of thoughts and tokens
Model
Avg. Thoughts
Avg. Tokens
Avg. Consistency
Llama-3.2-1
8.07
346.81 439.37
0.51
Llama-3.2-
11.73
0,40
Llama-3.1-8B
21.38
715.56
0.48
Llama-3.1-70B 13.55
442.72
0.51
As be seen, the 8B/70B models produce more thoughts than the 1B/3B models; meanwhile, their intermediate states of correct chains in blue are more consistent than those of the IB/3B model. The Pearson correlation coefficient between CoT length (thoughts) and consistency only -0.0185, indicating a very weak negative correlation that is not approaching either +1 or or- Hence, higher consistency doesn't 't correlate with shorter chains. Fewer CoT steps do not necessarily indicate higher consistency.
As we introduced in Sec. 2.3, the consistency metric is used to understand whether the LLM knows the answer before generating all thoughts. Here, the observation "larger models have higher consistency" actually indicates that larger model has a higher probability of knowing final answer in its middle steps of reasoning. We believe that this observation is new and insightful to the community.
In addition, we investigate whether the consistency is meaningful for the reasoning outcome or if consistently decreases the thoughts increases. We ask the Llama 3.1 8B Instruct model to generate some random thoughts, using a temperature 0.7 to encourage more varied responses. For each of the 10 questions select from AQuA, we then randomly combine different numbers of these thoughts to create 50 chains for each question, with the number of thoughts ranging from 2, 4, 8, 16, or 32. After generating these chains, we calculate the distance matrix and report the consistency, as shown in Tab. 3. Notably, as the length of the chain of random thoughts increases, the consistency consistently decreases, regardless of the correctness, which justifies that consistency will not increase as n increases.
Besides, conduct extra experiments on harder task across model scales and show that larger models achieve higher consistency than smaller models on both easy and hard tasks. Specifically, we apply the MMLU-Pro (Wang et al., 2024b) as harder benchmark. MMLU-Pro is more challenging version of MMI.U (adopted in this work), extending the MMLJ dataset by integrating more cussoning-focusee questions. We sample problems from the MMI.U-Pro Math subset and evaluate models of different scales, following the consistency calculation described in equation 5. The experiment results are shown as follows:
The above results show that larger models have substantially higher consistency on both the easy task (MMLU) and the hard task (MMLU-Pro) than smaller models. Here are some detailed observations: (1) Notably,
the hard task, the 70B model still has higher consistency than the IB/3B/8B model on either the hard task or the easy task. (2) Besides, the 70B model achieves a similar consistency easy and hard tasks (0.55 and 0.52, respectively). (3) However, the 8B model drops significantly from easy to hard tasks (from 0.41 to 0.20).
H.4 FURTHER DISCUSSION ON THE STRATEGYQA
The abnormal reasoning behavior, where states cluster anchors that differ from their final answer in Fig. 4(c), is not due to our visualization method but to the unstable reasoning process in the Llama- 3.1-70B using CoT on StrategyQA. This model struggles to reliably represent its self-generated intermediate thoughts, presenting consistency between intermediate thoughts and final predictions, thus leading to the abnormal patterns observed.
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Table 3: Consistency metrics across random thoughts
The number of random thoughts
Consistency
2
4
8
16
32
Correct Paths
0.77
0.80
0.80
0.75
0.66
Incorrect Paths
0.90
0.92
0.92
0.79
0.79
Table 4: Accuracy and consistency on MMLU and MMLU-Pro across different models.
MMLU
MMLU Consistency
MMLU-Pro Accuracy
MMLU-Pro Consistency
Model
Accuracy
Llama-3.2-1 Instruct
0.20
0.40
0.05
0.17
1lama-3.2-3B Instruct
0,46
0.41
0.30
0.26
Llama-3.1-8B Instruct
0.66
0.41
0.30
0.20
Llamm-3.1-700B Instruct
0.86
0.55
0.40
0.52
Specifically, the consistency of incorrect paths declines steadily. This highlights the model's unstable reasoning, as it fails to maintain coherent reasoning even when approaching the final answer. In addition, the landscape exhibits the highest perplexity compared to other models, indicating low confidence in its generated thoughts, which undermines the reliability of the estimated feature matrix used in our visualization.
Further, we provide landscape visualizations for the same dataset using other models and methods in Fig. 9 Fig. 12. These landscapes do not exhibit the same abnormal density patterns, reinforcing that the issue is specific to Llama-3. 10-m--3.170B's 's reasoning instability rather than flaw in our visualization.
H.5 COMPARING THE PERPLEXITY AMONG DIFFERENT MODELS
We conduct experiments to calculate the average perplexity of models in our visualization. Consistent with the prior works, we find that different models present similar perplexity when decoding the same set of CoTs. Here, we first generate a set of CoTs from the AQuA dataset using Llama-3,1-70B Instruct. Then, we use models from the same family (i.e., Llama-3.2- 1B Instruct, Llama-3.2-3B Instruct, Llama-3.1-4 1-BBBB Instruct, and Llama-3.1 1-7BB Instruct) to compute the average perplexity on decoding the same set of CoTs. This control experiment isolates the effect of model's 's inherent perplexity calculation from the variation of its generated thoughts.
As shown Tab. while there is slight variation perplexity, the values all fall within a comparably narrow range (from 1.4 to 2.0). This demonstrates that for decoding the same CoT', different models in the Llama-3 family produce similar and comparable perplexity scores.
In addition, in Fig. 2, we measure the perplexity of decoding CoTs generated by the models themselves. In this context, perplexity reflects both model's reasoning capabilities and the comprehension of its generated content. To some extent, the above findings support the validity of the comparison of perplexity across models our study.
H.6 ADDITIONAL EXPERIMENTS ON THE VERIFIER
Absouue Performance of the Verifier. this part, we provide the absolute performance of the experiment conducted in Fig. 8. Shown as Tab. 6, the results demonstrate that our approach consistently provides improvements across different domains and models.
Variants of Verifier. In this part, we extend it into process verifier and validate its effective- ness through additional experiments. Our lightweight verifier functions as an outcome reward model (ORM), assessing the correctness of entire reasoning path. Specifically, the process ver- ifier predicts the accuracy each reasoning state using features from the current and all previous thoughts. State accuracy reflects whether the current state is closer the correct answer (measured
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0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(a) Llama-3.2-1 IB with CoT on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(b) with CoT on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(c) Llama-3.1 1-8B with CoT on Strategy QA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(d) Llama-3.1 3-BB with CoT on StrategyQA
Figure 9: The landscapes of the model across scales (using CoT on the Strategy QA dataset).
Table 5: Comparison of the perplexity of CoTs of correct and incorrect reasoning.
Model
Avg. Perplexity (Correct CoTss
Avg. Perplexity (Wrong CoTss
Llama-3.2-1B Instruct
1.68
1.96
Llama-3.2-32 Instruct
1.72
1.69
Llama-3.1-8B Instruct
1.61
1,49
1-a00 Instruct
1.56
1.42
by perplexity) than other answers. We then aggregate these predictions across the chain to estimate overall accuracy.
Empirically, we collect the state-wise data by comparing the state features and the correct answers, and train the process verifier. Note, we do not need to manually annotate the step-wise rewards
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0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(a) Llama-3.2-1 IB with L2M on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(b) 3lama-332-338 with L2M on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(c) Llama-3.1 1-88 with 1.2M on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(d) Llama- 3.1ma-3.1-70B with L2M on Strategy
Figure 10: The landscapes of the model across scales (using 1.2M on the Strategy QA dataset).
Table 6: Absolute accuracy with the verifier, compared to performance in Fig. 6 (without the verifier).
(a) Across datasets
(b) Across models
AQuA
MMLU
Stategy
Commont SenseQA
3B
70B
AQuA
(-0.7)
(+0.0)
62.3 (-0.0)
(+1.7)
IB
(40.5)
2+2.0)
2+250)
27.5
MMLU
53.0
$3.0 (+0.0)
53.0 (+0.0)
53.0
3B
(40.0)
(42.5)
(1.05
(1.00
SnategyQA
Common
SenseQA
(4.00(11.0)
41.5
(+0.0)
(v3.5)
(+0.0)
(46.0)
(4.0
(+000
70B
SB
(+200)
(+0,0)
(+100)
6000 (+0.0)
(+00))
72.5
72.5 (+0.5)
60.0 (+0.))
to train conventional PRMs. Results in Tab. 7 show that this process verifier is comparable to the outcome verifier.
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0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(a) Llama-3.2- 1B with MCTS on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(b) Llama-3 3.2-3B with MCTS on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(c) Llama-3. with MCTS on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(d) Llama 3.1 10BB with MCTS on Strategy
Figure 11: The landscapes of the model across scales (using MCTS on the StrategyQA QA dataset).
Comparing the lightweight verifier with existing verifiers. In the following, we compare our lightweight verifier with the other two types of existing verifiers: the LM-based verifier and the model-self verifier.
The LM-based verifier leverages another powerful LLM (not the model to do reasoning) to seman- tically analyze reasoning trajectories, mimicking human expert evaluation to detect errors in the trajectories. These verifiers rely on extensive, specially curated datasets (e.g.. PRM800K (Lightman al., 2024)) to train a language model for process verification. Here, collecting high-quality training data often requires detailed, fine-grained annotations of reasoning steps, which can be costly and cine-consuming. Moreover, training this verifier (often itself a large language model) incurs much additional computational expense. In contrast, our lightweight verifier is much more efficient to train, as it requires no human annotations and only uses easily obtainable data that is collected from the model to do reasoning.
As for the model-self verifier (Li et al., 2023a; Xiong et al., 2024), it utilizes features derived from the model itself, such uncertainty, perplexity, entropy, eliminating the need for external model
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0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(a) Llama-3.2-18 with ToT on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(b) Llama-3.2-3B with ToT on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(c) Llama-3. 18BB with ToT on StrategyQA
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
(d) Llama-3, 7-0 with ToT on StrategyQA
Figure 12: The landscapes of the model across scales (using ToT on the Strategy dataset).
and enhancing efficiency in search-based methods. While these model-self verifiers are training- -free and efficient, they lack the learnability to be trained and optimized, as the model is not trained on the downstream task, and thus it can be suboptimal. In contrast, our verifier is specifically trained with the downstream task's data collected from the model, ensuring greater reliability compared to model-self verifiers.
Therefore, our landscape- based lightweight verifier offers distinct advantages in terms of efficiency and reliability over the other two types of verifiers.
Ablation study on verifier. We conduct an extra ablation study on training the verifier with either consistency or 2D information. We report the accuracy of reasoning under Least-to-Most with different model scales, averaged across different datasets.
As shown in the Tab. 8, the combination of the consistency score and 2D information delivers the best overall accuracy. This shows that our verifier could utilize the complementary aspects of both kinds of features to access the reasoning chains and thus boost reasoning accuracy.
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Table Performance comparison of reasoning methods across model scales on the AQuA dataset, with and without verifiers.
Model
Method
Without Verifier
With Outcome Verifier
With Process Verifier
CoT
0.26
0.28
0.26
L2M
0.22
0.24
0.29
Llama-3.2-1B
ToT
0.35
0.38
0,35
MCTS
0.29
0.32
0.31
CoT
0.46
0.51
0.46
L2M
0.29
0.31
0.31
1lama-3.2-3B
ToT
0.33
0.35
0.33
MCTS
0.35
0.36
0.35
CoT
0.60
0.63
0.60
L2M
0.58
0.58
Llama-3, 3.1-8BB
ToT
0.50
0.50
MCTS
0.50
0.51
0.50
CoT
0.72
0.73
0.73
L2M
0.72
0.72
0.73
Llama-3.1-700
ToT
0.74
0.74
0.74
MCTS
0.72
0.73
0,72
Table 8: Ablation study on data employed for training the verifier.
3B
70B
Consistency only
0.21
0.31
0.59
0.71
2D information only
0.20
0.31
0.61
0.71
Consistency 2D information
0.24
0.31
0.62
0.72
H.7 FURTHER EXPERIMENTS ON THE SCALING EFFECT
We present experiments and demonstrate that combining both information sources is the best choice, with significant gains from more sampled trajectories (i.e., test-time scaling) compared to the verifier trained with either feature, can 3B Instruct model on the StrategyQA dataset as follows. As can be seen, the advantages of using both information sources increase with more sampled trajectories, especially for more than 20 sampled trajectories. In contrast, verifiers trained only on consistency or 2D information peak earlier, showing no notable performance gains beyond sampled trajectories.
in Tab. Here, report the accuracy using the Llama-3.2 2-
H.8 LANDSCAPES WITH DIFFERENT METHODS OF DIMENSIONALITY REDUCTION
1-SNE is widely adopted in non-linear projection for visualisations, which makes the plots more interpretable. Beyond t-SNE (Cai & Ma, 2022), several advanced dimensionality reduction techniques have been developed improve visualization quality and efficiency. UMAP (MeInnes al., 2018) outperforms t-SNE by better balancing local and global structure preservation while offering greater speed and scalability for large datasets. TriMAP (Amid & Warmuth, 2019) prioritizes both local and global preservation but tends to emphasize global structure in practice, potentially at the expense of local details. PaCMAP (Wang et al., 2021) achieves a robust balance between local and global structure preservation by incorporating neighbors, mid-near points, and further points, resulting in high-quality visualizations across diverse scenarios.
In addition, goal is to develop visualization tool help users analyze the reasoning behaviors of LLMs. necessary, we can change the adopted t-SNE to more advanced methods dimensionality reduction. Our tool is designed be compatible with these methods.
Next, we experiment with different dimensionality reduction methods, including t-SNE, UMAP, and PacMAP, to visualize the landscape. Across all three visualization techniques, consistently observe the same overarching dynamics in the reasoning process. In the early stages (0-40% of states), the thought states widely dispersed. As reasoning progresses, states gradually converge
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Table 9: Performance of the verifier given different numbers of sampled paths.
Sampled Paths
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2D Information
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Figure 13: The landscapes of thought visualization with different dimensionality reduction methods (Llama-3.1-70B with CoT on AQuA).
toward the final answer choices. Importantly, a clear distinction emerges between correct and incorrect reasoning paths, regardless of the selection of different dimensionality reduction methods. Incorrect paths tend converge rapidly toward wrong answers early in the process, while correct paths exhibit more gradual and deliberate progression, only clustering tightly around the correct answer in the final stages (80-100% of states).
We provide landscape visualizations in Fig. 13 with different dimensionality reduction methods. While the specific geometry and density of clusters may vary between I-SNE, UMAP, and PacMAP, the fundamental narrative is unchanged: the landscape of thoughts consistently reveals that incorrect reasoning solidifies quickly, whereas correct reasoning is characterized by a slower, more refined convergence. This consistency across different dimensionality reduction algorithms demonstrates that our observations are not artifacts particular visualization technique, but rather reflect intrinsic properties of the model's 's reasoning process.
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Original
Tobegin hegiit dhw diegran aitiation
Ovanapiit Thoughts
Lets brock down the prodlem skoa zip.
Tobogis hespllll diagram
Under-apiz Thoughts
Lets break down the problem stsp map.
helppil
(a) Demonstration Sentence Tokonization
Llama-3.1
(c) 70B
Figure 14: Demonstration of sentence tokenization methods for thoughts splitting.
H.9 ROBUSTNESS OF SENTENCE TOKENIZATION
To evaluate the robustness of the landscape to the split thoughts' information volume, i.e., the granularity of the sentence tokenization, conduct controlled experiment by considering two imperfect cases in thought split, namely over-split thoughts and under-split thoughts.
Specifically, shown as Fig. 14 (a), compared to the original thoughts split that transform sentences to thoughts based on the period, over-split thoughts jointly consider the comma, resulting in additional splits. For the under-split, two adjacent thoughts are merged into one thought. We then visualize the imperfect thought splits using CoT on AQuA following the setting in Fig. 5(a) and Fig. 2(c).
Shown in Fig. 14 (b) and (c), the landscapes are robust to the split thoughts' information volume, which are stable and consistent with our observations. Notably, for over-split thoughts, the states are more visually diverse but eventually converge to the answers. Whereas under-split thoughts, the states show more compact pattern and exhibit a clear convergence trend toward the answer.
H. GENERALIZATION TO TASKS BEYOND MCL.TI-COICC PROBLEMS
Here, we provide new empirical results open-ended benchmarks MATH, GSM8K, and StrongReject to show the visualization effectiveness of LoT in open-ended questions, where the observations in these datasets highly align with our observations in multiple-choice tasks.
Note that the multiple-choice restriction is pragmatic, not fundamental, and a simple pseudo-option strategy can address this restriction. LoT construction requires anchors compute relative distances, as defined in Equation (1), where the fcature f. for state s, quantifies distances to choices {o, } via perplexity in Equation (2). In open-ended settings, these do not exist naturally, but they can be created reliably. We tested two complementary families:
For rule-verifiable reasoning (MATH and GSM8K), we keep the unique correct answer and use Llama-3.1 1.--a-3.1-8B-- followed by manual filtering to retain only non non-trivial distractors, yielding clean way multiple- choice versions.
to generate three plausible incorrect final answers plus full reasoning chains,
For non veniuule-e or safety tasks (StrongReject, jailbreak benchmark), use True/False as the two options, with ground-truth labels produced by GPT-4o and manually verified.
This approach mirrors the original setup in Section 2.2, ensuring the perplexity-based distances remain meaningful without altering the core visualization pipeline.
Then, following the experiment setting described in Sec. 2, We sampled 10 trajectories per question using the LLama 3. 1-8B Instruct and constructed the LoT accordingly. Figures 21 (MATH), 22 (GSM8K), and 23 (StrongReject) show the phenomena replicate strongly. Key observations hold clearly in open-ended settings. We describe them as follows:
Observation 3.4 (Similar reasoning tasks exhibit similar landscapes). MATH and GSM8K exhibit rich state diversity and phased exploration patterns highly similar to multiple-choice datasets (AQuA, MMLU, and StrategyQA in Fig. 4), whereas StrongReject shows concentrated, low- diversity search regions akin to ComMonSenseQA..
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(a) AQUA (ociginal)
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Figure 15: Comparing the LoT of perturbed options (using Llama-3.1-88 with CoTT).
Observation 3.7 Within-method comparison: For any single method, incorrect trajectories converge faster to wrong answers than correct trajectories converge to right answers.) Across all three open- ended benchmarks, incorrect trajectories rapidly approach wrong anchors in 60-80% of reasoning states, while correct trajectories explore longer and only converge in the final 80-100% of states, mirroring the pattern in Figs. 2, 5, and 4 of multiple-choice questions.
The core insights, therefore, transfer to open-ended questions without choices. These observations support that the LoT can be employed to analyze the reasoning process of open-ended questions.
Further, to eliminate the concern that our current results might depend on the quality of pseudo-options generated by another model or manual filtering, we envision a fully automatic anchor discovery approach as future work. Specifically, after sampling multiple trajectories for an open-ended question, the final answers can be embedded with strong embedding model and clustered in semantic space. Task ssec--pecificc verifiers, e.g., exact-match checks for math problems or safety classifiers for jailbreak evaluation, can automatically label the highest-qualit cluster as the correct anchor and the remaining clusters as incorrect anchors. This procedure requires no auxiliary model and no human intervention, making it more scalable while preserving the computations of the perplexity metric.
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(a) AQUA (ocipinal)
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Figure 16: Comparing the LoT of perturbed options (using Llama-3.1 1-B8B with 1.2M).
H.11 LANDSCAPE VISUALIZATION WITH CHOICE REORDERING.
We swap the ordering of answer options (for example, A) C1, B) C2 - B1 A) 62) while keeping the semantic content of each Cj unchanged. We then recompute state features, rebuild the landscapes, and replot the consistency, uncertainty, and perplexity curves. As shown in Fig. 16, convergence patterns, separation between correct and incorrect trajectories, and the overall geometry are effectively unchanged; the metric curves almost overlap. This indicates that LoT is insensitive to superficial label permutations.
H.12 LANDSCAPE VISUALIZATION WITH MODEL BEYOND LLAMA FAMILY.
To further test the generality of LoT, we apply LoT to Qwen 2, 5-7B Instruct and iiren-2.5-72 Instruct and visualize their landscapes (Fig. 17). Notalby, we observed that: (1) Early states (0-20% of steps) are more dispersed in the landscape; (2) As states progress (20-80%), they become more organized and form clearer structures in LoT space; (3) Incoreet trajectories (red in the top rows) move toward wrong answers already in early and mid stages (for example, 20-40%), consistent with the option-directed behavior reported in Observation 3.7; The
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(a) Liamaa 3.1 sn
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Figure 17: Comparing the LoT of different model families and scales (Sampling with CoT on AQUA).
separation between correct and incorrect trajectories remains evident: correct states stay closer to the correct option over many steps and show higher stability (higher consistency and lower uncertainty). whereas incorrect states exhibit reduced stability.
The fact that these patterns appear both the Llama family and Qwen 2.5 models indicates that LoT is capturing broader properties of autoregressive chain-of-thought reasoning, not idiosyncrasies of a specific Llama line. Taken together, the model family comparison (Fig. 17), cross-scale trends (Fig. 2), cross-task differences (Fig. 4), and the predictive utility of LoT features (Fig. 7) provide converging evidence that our observations reflect general reasoning behaviors rather than phenomena restricted the Llama family.
H.13 LANDSCAPE GEOMETRY ROBUST ACROSS ROLLOUT SAMPLE SIZES
From methodological perspective, LoT aggregates traces in way that is inherently stable sample size. The reason is that LoT operates in fixed feature space where each state is represented by distance vector f, over answer options, and all state features stacked into matrix S before dimensionality reduction. Adding more trajectories simply adds more rows to S from the same underlying distribution of states. This increases the sampling density of the manifold but does not
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(2) Sample sire 10
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x x
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Figure 18: Comparing the LoT of different sample sizes (using Llama-3.1 with CoT on AQUA).
redefine the feature space or the embedding function, so the global geometry of the landscape is expected to remain stable the sample size grows.
Here, we explicitly test this by generating landscapes with different numbers of sampled trajectories per question: 10, 20, 30, 40, and 50, used in the original landscape. Across all these configurations (reported in Fig. 18), the resulting landscapes exhibit the same qualitative structure: (1) Correct
trajectories gradually converge toward the correct region; (2) Incorrect trajectories collapse earlier
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(a) Accuracy 300
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Figure 19: Comparing the LoT of different accuracy thresholds (using Llama-3.1 with CoT on AQUA).
toward wrong answer anchors; (3) The relative placement of major clusters is unchanged; (4) the evolution of density over time is smooth and monotonic in sample size.
The only noticeable difference is visual: landscapes with more samples (for example, 40 or 50 rollouts) appear smoother and less noisy because the underlying density better estimated, but the shapes of clusters, the separation between correct and incorrect trajectories, and the convergence patterns remain the same.
H. LANDSCAPE ON CHALLENGING QUESTIONS.
The main LoT structure remains stable across all difficulty thresholds. We examined LoT on progressively harder subsets by filtering out easy questions using accuracy thresholds of < 50%, < 40%, and < 30% (Fig. 19). Across all cases, the characteristic patterns documented in Sec. 3 remain visible: carly-stage states later-stage states move more clearly toward specific answer options. This confirms that the core LoT behaviors persist even when accuracy is low.
more dispersed, mid-stage states exhibit partial organization, and
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(6) ArT
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Figure 20: Comparing the LoT of different reasoning methods (using Llama-3.1-8B on AQUA).
Incorrect trajectories continue to show clear option-directed movement. The incorrect trajectories (red) still exhibit recognizable movement toward specific answer choices as the reasoning progresses. This behavior is visible across all state ranges and does not disappear even when correct trajectories become rare. Thus, LoT continues to capture meaningful convergence tendencies for incorrect reasoning.
Late-stage concentration becomes weaker as accuracy decreases. In the original landscape, the later-state incorrect trajectories form more compact regions. However, in the challenging subsets, 60-80% and 80-10098 of states show broader and less sharply grouped red regions. This indicates that the model becomes less decisive during late reasoning stages on more challenging questions, as also discussed in Observation 3.6. The filtered landscapes make this phenomenon more pronounced.
The remaining correct trajectories still follow the expected patterns. Although correct trajectories become sparse in harder subsets, the ones that remain continue to align closely with the correct option across most state ranges. Their intermediate states show higher stability, and their later states form clear clusters near the correct anchor. These behaviors are consistent with Observation 3. and demonstrate that LoT continues to reflect successful reasoning even when questions are challenging.
The contrast between correct and incorrect trajectories becomes more pronounced on difficult questions. As the questions become harder, incorrect states become increasingly dispersed, while the remaining correct states stay compact. This highlights a clearer distinction between stable reasoning and unstable reasoning in the challenging subsets, reinforcing the interpretive value LoT even when accuracy is low.
In summary, LoT remains informative and interpretable even when the dataset consists primarily of incorrect trajectories. The landscapes of challenging questions preserve the core patterns described in Sec. while revealing additional difficulty-speccfie phenomena such as greater mid-stage dispersion and weaker late-stage grouping.
H.15 LANDSCAPE VISUALIZATION ON NEW REASONING METHODS
We extend the LoT to two additional methods: Algorithm-of-- Thooughh (AoT) (Sel et al., 2024) and Graph-of- ofTouhhhtssh (GoT) (Besta et al., 2024), using their public implementations with Llama-3. and applying the same visualization procedure described in Sec. 3. The corresponding landscapes shown in Fig. 20, and the reasoning accuracies for AoT and GoT are 0.54 and 0.30, respectively. In the following, we summarize our observations these two methods.
Both Algorithm of Thoughts (AoT) and Graph of Thoughts (GoT) produce distinct landscapes but preserve the characteristic LoT patterns reported in the submission. Despite differences in the
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Eal CoT
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1b)L
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Figure 21: Comparing the LoT of different reasoning methods (using Llama-3. 1-8B on MATH).
underlying reasoning strategies, both methods show the same core behaviors described in Sec. 3. Namely, early states (0-20%) are widely dispersed for both correct and incorrect trajectories. Notably, mid-range states (20-60%) begin to exhibit more organization, and later states (60-100%) show clearer movement toward specific answer options. This confirms that LoT continues reveal stable structure across diverse reasoning algorithms.
Correct trajectories AoT and GoT follow the same progression observed in the original analysis. For both reasoning methods, the correct trajectories remain closer to the correct answer across state ranges and display higher stability, consistent with Obs 3.1 and 3.2. AoT, which achieves higher accuracy (0.54), exhibits more concentrated blue regions in the 40-60% and 60-80% state ranges, reflecting more consistent reasoning. GoT, with lower accuracy (0.30), shows correct trajectories that fewer and more spread out, but still maintains recognizable proximity the correct option in later states.
Incorrect trajectories show early and clear movement toward wrong answers, consistent with the original LoT observations. In both AoT and GoT, the incorrect trajectories (red) begin to cluster toward specific wrong answer options within the first 20-40% of states. This matches the behavior described in Sec. 3, where incorrect paths tend to settle into wrong answers earlier in the reasoning process. The effect especially pronounced for GoT, whose lower accuracy results in broader and less stable distribution of mid-stage states.
In summary, the AoT and GoT experiments show that LoT captures consistent reasoning patterns across methods with different structures and accuracies. The landscapes in Fig. 20 reproduce the key behaviors described in Sec. 3 and additionally reflect the performance differences between methods.
I VISULIZATIONS
In this part, we provide the full visualization of the verifier performance and landscapes.
In Fig. 24 to Fig. 27, visualize the average voting accuracy (%) of different LLMs reasoning with and without verification on various datasets and methods. Fig. 28 to Fig. 31, we display the landscape of different models on various datasets using four methods. We also provide case studies by visualizing the landscape with corresponding states in Fig 32 to Fig. 35.
In addition, provide the landscape of thoughts the latest reasoning model. Specifically, conduct experiments on the DeepSeek- RDDistlill models (Guo et al., 2025) (Llama-70 B and Qwen- 1.5 B). As shown in Fig. 37 and Fig. 38, the landscape of the reasoning model also aligns with the
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Eal CoT
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Figure 22: Comparing the LoT of different reasoning methods (using Llama-3.1-8B on GSM8K).
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Figure 23: Comparing the LoT of different reasoning methods (using Llama-3. 1-8B on STRONGRe- ject).
observation drawn from the general-purpost model, but exhibits more complex reasoning patterns, such as self-evaluation and back-tracking.
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1B
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Figure 24: Average voting accuracy (%) of reasoning with and without verification on AQuA.
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Figure 25: Average voting accuracy (%) of reasoning with and without verification on MMLU.
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Figure 26: Average voting accuracy (%) of reasoning with and without verification on Strategy QA.
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Figure 27: Average voting accuracy (%) of reasoning with and without verification on Common- SenseQA.
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Figure 28: The landscapes of various reasoning methods (using Llama-3.22 3. on the AQuA dataset).
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Figure 29: The landscapes of various reasoning methods (using Llama-3.3 3.2-3B on the AQuA dataset).
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Figure 30: The landscapes of various reasoning methods (using 1amma-3.1-85 on the AQuA dataset).
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Figure 31: The landscapes of various reasoning methods (using Llama-3 3. 1-BB on the AQuA dataset).
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To solve the problem, let's break series of calculations according to the given property.
2. The perimeter of the part (perianeter of 16x and 14y).
Step 4: Substitute the caleulated value of into the expression for the difference.
Conclusion: The original price the item approximately $63.32. The answer is A.
Hose A fills the pool hours, its rate 1/8 the pool per hour.
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Step I: Start by adding the positive numbers: adding to 30 results in 15 since 15 > -15
3. The minimum total commission needed reach salary least
Step 4: Calculate the population years the initial population.
5. Now, divide both sides by 1.2, solve for C. Therefore, C 360 12 = 300.
Based on our calculation. grocer likely approximately 24.4 of flour. The answer is C.
$ $1000 is 1000.
Figure 32: Case Study: Landscape of thoughts of Llama-3.2-1B on AQuA using CoT.
4. Since the profit is 25% of the selling calculate the total selling price for all the bags sold.
However, the number toys cannot be a
Step 8: Since Hose B can fill the pool per bour, fill the entire pool in 8 hours. Therefore, the answer is D.
Let's break down the steps calculate the average speed of the plane around the square field.
6. Since we found that A 1/8, can for by substituring into the equation: 1/8 B 3/4,
price,
fraction, round to the nearest whole number.
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problem, let's break down the solution process into clear, independent steps.
Step find time takes for them to produce 10 yo-yos, consider the least multiple (LCM) 9 of minutes.
Step formula values into
Converting to percentage, get (30/70) 100% 42.86%.
The
the conditional probability formula. P(A/B) P(A) P(B) 0.2 0.8.
Figure 33: Case Study: Landscape of thoughts of Llama-3.2-33 on AQuA using CoT.
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Step The total cost can be expressed the sum costs of brown and white sharpeners: X -b) (X 1) 100.
Conclusion: The original price of the item was approxinately $63.32. The answer is A
Next, we divide the total profit the profit bag $3,000 825 120.
Step 8: Solve for using equation identified in step 7.
step step solve this problem.
0-20%
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Let's think step by step to solve the problem.
3. This leaves 1-1/4 3/4 of be filled both
We can represent the commission 15%
To find the pereentage increuse, we'll the formula: ((Inerease) Original) 100).
hoses working together in following
the monthly sale as: 0.15 monthly sale.
Figure 34: Case Study: Landscape of thoughts of Llama-3.1 3.BB8B on AQuA using CoT.
Now; rewrite the two equations follows: First
Therefore, the resultant solution 37.25% tea and milk.
Conclusion: The original price the item approximately $63.32. answer
break problem into steps to the solution.
equation is - 66 2w and second is
32 - 82 rupees which
100.
0-20% states
20-40% states
40-60% states
60-80% states
80-100% states
Step I: Start by adding the positive numbers: adding to -30 results in since >
The distance traveled on the third kilometers speed 600 km/ar.
Therefore, to find the sales, need to divide the additional amount by 5% (which 0.05).
Now, we multiply the common prime factors and the uncommen prime factors logether find the LCM.
Therefore, the answer is D)40
Figure 35: Case Study: Landscape of thoughts of Llama-3.1 1-70B AQuA using CoT.
53

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Published as a conference paper at ICLR 2026
Wait, may confuseed slepps freme...
So you, C is correct Let Let me chec k
800% states
Maybe made mistake there.
-eperepss original price a valus that ..... Lot me chock.
there any miscaic miscalculstion letton. yes, stegs out
Thai's nol among the options provided Did a mistake?
Walt, let me double-cher chile-checkk my calculetion step agaii..
The problen'' options are. ..... But the quesiton might be trick trikkhoo hors?
Wait. but maybe there is anather approach??
options o include C as 36 Let me confimm step by step.
The steps seer correct Let me check approach.
60-80% states
Wait. but the
Wait, sorry. lst mi recastt.
Wail,, perhaps better to subsillute
O makee mataks
But wait, let me check once more.
Wart let me redo the division.
Maybe made mistake oqu atton satup?
states
Wait, perhaps my total cost calculation was wrngg me check again.
40-60%
lst aa dhisk
Wait, Wait second, no, actually
maybe made calcalation
again.
Wait, perhapss
Walt in the
step, oot.
Bul wait, looking at the answer aptiore, is 110. However...
Wast, but maybe made an arthmmtc mistake somewhere. Let me check the division again:
20 40% states
Wait, let me check again.
625, 400 -554.40 400. 1......
Wait, but pertaas
Wait second. But hold let double-cheok
The minimum of 56 and 0.74 a 0.58. Wall, maybe. Let check
Wart. actually when-oo solving = 3DD 10k)
Wait, first maybe need to find the total miszamo.
Let me check that agein, Yes: the non- non-defecct dee-deefeceeeeeee total is 0.SNN and of that is rejected. Therefore.
0-20% states
Walt, this might be bit more compltcated.
Wait, the first part might sssume whelher letters can repeet
Figure 36: Landscape of QwQ-32B using CoT on AQuA.
54

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Published as a conference paper at ICLR 2026
bd paraps mate mara By fos ouuttinatii
80-100% states
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60-80% states
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oenion
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20-40% states
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Figure 37: Landscape of DeepSeek-Re RR--Dsttlll-Lamae using CoT on AQuA.
55

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Published as a conference paper at ICLR 2026
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80- 80-100% 100% states
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40-60%
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soout Luadts way
20-40% states
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(107 -171 mine
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Figure 38: Landscape of DeepSeek- R1- Disttll-Qween-... using CoT on AQuA.
56


Reviews:
[{"review_id": "vshNLeIQRg", "reviewer": "Reviewer_WVGo", "summary": "This paper proposes to analysis the reasoning process of LLMs via visualisation. Specifically, it projects the intermediate states (defined by the sequence of output from prompt up to the token of the step being considered) of LLMs into 2d planes via t-SNE. Three metrics are introduced to measure the consistency, uncertainty, and perplexity of the reasoning/generation quality.\n\nEmpirical results on models with different number of parameters in the Llama-3.1/3.2 family and across multiple datasets are presented。 Based on the results, the paper provides several observations along the dimension of LLMs, tasks, and reasoning methods.\n\nFurthermore, a verification method based on convergence and consistency is proposed to re-weight answers generated by LLMs. This method is demonstrated to improve the accuracy of Llama models with 1B and 3B parameters.", "questions": "Please refer to my concerns above.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper presents extensive results and comparisons between visualisation results of different LLMs on different tasks and with different reasoning methods. \n- Some observations, e.g., the consistency observation 3.8, might be useful for better regularisation of the reasoning process of LRMs.", "weaknesses": "- Most of the presented visualisations and qualitative metrics are not significant enough to draw meaningful conclusions. It is difficult to tell if the t-SNE visualisation of reasoning states are showing consistent patterns across all the tested models without a systematic way of verifying their existence. Most of the visualisations in this paper can be seen as good case studies but not enough to convince me that the results are generalisable.\n\n- Only Llama 3.1/3.2 family of LLMs are tested in this paper. This further makes me doubt the generalisability of the observations in this paper.\n\n- The observations listed in section 3, although seemingly reasonable, are mostly not very surprising. Furthermore, most of the observations are not verified by other evidence (maybe except 3.7 and 3.8 which are at least partially verified in section 4). This makes me doubt whether these observations universally exist.\n\n- The readabilities of most of the figures are slightly low. The paper has to be read in a monitor, otherwise it is difficult to read the texts or recognise the patterns in the scatter plot.", "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": "XgHQHz5eat", "reviewer": "Reviewer_FwuD", "summary": "Landscape of Thoughts (LoT), as introduced by the authors, is a general tool to visualize and analyze LLM reasoning trajectories on multi-choice tasks. The main idea is to represent each intermediate state in a chain-of-thought as a feature vector of distances to all answer choices, and the distance is the length-normalized inverse likelihood (perplexity) computed by the same LLM, after that, these state features (plus choice “landmarks”) are projected to 2D with t-SNE to form “landscapes. Beyond visualization, the authors adapt the state features into a lightweight verifier that re-weights trajectories at test time, improving accuracy and exhibiting strong test-time scaling without modifying base model parameters.", "questions": "1. The paper assumes that textual thoughts can be meaningfully embedded into a continuous numerical space. However, since each question has its own answer set, how is cross-question comparability achieved? Does a global semantic space actually exist, or are the landscapes a collection of locally defined subspaces stitched together?\n\n2. The visualization relies on t-SNE projections of log-likelihood-based features. Given that t-SNE introduces nonlinear distortions, to what extent do the “clusters” or “paths” in the landscape correspond to real structure in the high-dimensional reasoning space?\n\n3. Since each thought’s representation is derived from the model’s own likelihood distribution, how can we be sure that the resulting structure reflects reasoning dynamics rather than simple probability concentration or syntax-level similarity?\n\n4. The notion of “trajectory convergence” implicitly assumes a unified space across all problems and reasoning chains. How is this space defined mathematically? Is it constructed per dataset (via concatenation of all samples), or does it represent some theoretically consistent manifold of reasoning states?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper offers an interesting view to interpret LLM reasoning dynamics by transforming text-based chains of thought into measurable geometric trajectories. This provides an intuitive and scalable diagnostic framework for analyzing reasoning behavior.\n\n2. The study visualizes reasoning across multiple datasets, model sizes, and decoding strategies, making it one of the first to present a comparative, interpretable view of LLM reasoning evolution rather than static outputs.\n\n3. By extending the visualization features into a simple verifier that reweights trajectories at test time, the work shows a way to potentially improve reasoning reliability without modifying model parameters.", "weaknesses": "1. It seems like, the paper assumes that reasoning trajectories can be meaningfully mapped into a 2D space by computing each thought’s log-probability distance to candidate answers and projecting them via t-SNE. However, these representations primarily capture the model’s surface-level likelihood distribution, and how confident it is about next tokens, but not the latent cognitive process of reasoning. Since t-SNE emphasizes local density and distorts global geometry, the observed “paths” and “convergence” patterns likely arise from projection artifacts or distribution sharpness (e.g., larger models having more peaked probabilities) rather than reflecting genuine logical progression or conceptual reasoning structure.\n\n2. The claim that larger models exhibit more consistent and reliable reasoning is not well supported. The observed trajectory regularity may simply reflect statistical effects of scale, larger models tend to produce sharper probability distributions and more uniform language patterns—rather than genuine improvements in reasoning stability. Moreover, trajectory consistency is heavily influenced by task semantics and prompt structure, which are not controlled in the experiments. Without isolating these confounding factors, it is unclear whether the smoother “paths” truly arise from deeper reasoning ability or from surface-level distributional sharpness.\n\n3. Some interesting findings are based on visual inspection and single-case illustrations without statistical validation or controlled comparisons, making the observed “convergence” patterns more narrative than scientifically substantiated.", "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": "bonLioy1DS", "reviewer": "Reviewer_fVdk", "summary": "This paper introduces a landscape of thoughts, the first visualization tool for users to inspect the reasoning paths of the chain-of-thought and its derivatives on any multi-choice dataset. The authors represent the states in a reasoning path as feature vectors that quantify their distances to all answer choices. These features are then visualized in two-dimensional plots using t-SNE. Qualitative and quantitative analysis, combined with the landscape of thoughts, effectively distinguishes between strong and weak models, correct and incorrect answers, as well as different reasoning tasks.", "questions": "Questions:\n\nQ1: In Figure 1, it says that the wrong paths quickly converge to wrong answers, while the correct paths slowly converge to correct answers. However, observation 3.1 says that the landscape converges faster as the model size increases. But shouldn't the bigger model converge slowly, as it is expected to give more correct solutions? Any reason for this discrepancy?\n\nQ2: Are the insights similar on reasoning questions without any multiple choices? \n\nQ3: What about other reasoning methods, such as a graph of thoughts or an algorithm of thoughts?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Strengths:\n1.\tImportant contribution towards understanding the large reasoning models\n2.\tA model can be trained to predict any property users observe\n3.\tGood visualization plots", "weaknesses": "Weaknesses:\n1. Only multiple-choice kind of questions are considered. I am not sure if the same insights transfer to the reasoning questions with no choices.\n2. More models could be considered.", "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": "aQODVHd2Tg", "reviewer": "Reviewer_VKLY", "summary": "This work proposes a novel approach for monitoring long chains of thought by visualizing reasoning trajectories in a two-dimensional feature embedding space. The proposed monitoring method, however, is restricted to multiple-choice problems: by appending each answer option to the intermediate reasoning states and computing the corresponding perplexity scores, the model estimates the “distance” between its current reasoning state and each candidate answer.", "questions": "Q1. How does the number of sampled reasoning traces affect the resulting landscape? If the number of rollouts increases, will the overall structure of the landscape remain stable, or does it change significantly? A sensitivity analysis could help clarify the robustness of the visualization.\n\nQ2. In the visualizations, blue and red trajectories represent reasoning traces leading to correct and incorrect answers, respectively. For challenging questions where most rollouts fail, what meaningful insights can be extracted from those predominantly incorrect trajectories? \n\nQ3. The trends of uncertainty and perplexity are confusing. For example in Fig.2, all models exhibit a non-decreasing uncertainty over time, suggesting that they become less committed to a single final answer as reasoning progresses. Similarly, the perplexity for their own reasoning traces increases continuously. What does this imply? A clearer explanation would help readers interpret these metrics more meaningfully.", "limitations": "", "rating": 2, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper addresses a meaningful and challenging research problem—how to effectively monitor and audit reasoning traces (i.e., chains of thought), particularly when reasoning models generate extremely long CoTs that are infeasible for human evaluation to scale.\n\n2. The work demonstrates originality and novelty. Although the key components of the proposed method (e.g., perplexity and t-SNE) are not new, their combination and the new analytical perspective are refreshing and interesting.", "weaknesses": "W1: The scope of this paper is narrowly limited to multiple-choice questions with known ground-truth answers. It remains unclear how the proposed visualization tool can generalize to broader reasoning analysis or practical model auditing. Chain-of-thought monitoring and scalable oversight are indeed underexplored areas, people may hope to solve problems like detecting and controlling hallucination or unsafe content generation—tasks where ground-truth answers or verifiable signals are typically unavailable. However, this paper focuses instead on visualizing the distance between reasoning traces leading to correct versus incorrect answers. The conclusions (for example, wrong reasoning paths converge to wrong answers faster than correct ones) depend heavily on the existence of ground-truth labels. Such relative convergence patterns would not hold in open-ended reasoning tasks without definitive answers, substantially limiting the applicability and significance of this visualization approach for real-world reasoning analysis.\n\nW2: It is not clear how **robust or faithful** the proposed visualization method and the conclusions derived from it actually are. The final reasoning landscape is constructed from state features computed via the perplexity of appending choice answers to intermediate reasoning steps. However, many unrelated factors could influence this measurement. For instance, even when the model follows a correct reasoning path, forcibly inserting the final answer into its ongoing reasoning can disrupt coherence and yield artificially high perplexity scores. Moreover, the phrasing of the final answer itself may significantly affect the computed perplexity. In other words, the distance metric may be driven by superficial textual factors rather than genuine semantic relations. While the resulting visualizations and observations are intriguing, the paper does not convincingly demonstrate the reliability or faithfulness of these conclusions.\n\n* Typo*:\n(1) Line 381, converge means... -> convergence means ...", "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)