MOOSE-Chem2: Exploring LLM Limits in Fine-Grained Scientific Hypothesis Discovery via Hierarchical Search

We are grateful for the detailed comments. Here, we present our responses corresponding to the questions you've asked.

---

Q1: what this paper essentially proposes, and what is the core contribution of this paper.

A1: The core contribution of this paper is the formalization of the novel fine-grained hypothesis discovery task as a combinatorial optimization problem, along with the development of Hierarchical Heuristic Search (HHS) to optimize this task effectively for scientific discovery. To our knowledge, no prior LLM-based scientific discovery work has framed the task through an optimization lens. Furthermore, HHS adapts coarse-to-fine search methods to the challenging, language-based open-ended tasks like hypothesis generation, an approach that was previously unclear.

Specifically, we propose a novel framing of the task that enables the use of LLMs to provide the gradient for optimization. To enhance optimization, we introduce a hierarchical design that smooths the reward landscape, facilitating convergence to better local optima. Additionally, we adopt evolutionary search to interpolate among multiple local optima, further refining the optimization process.

Q2: The authors do not propose a new training method or architecture for LLMs but rather use existing models via APIs.

A2: The proposed method is essentially a test-time scaling approach, where computation is used as a trade-off for improved performance. This represents a cutting-edge research direction. For example, OpenAI’s Dan Roberts discussed in his talk “9 Years to AGI? OpenAI’s Dan Roberts Reasons About Emulating Einstein” that an ideal scientific discovery system might require 8 years of inference-time computation to propose a high-quality hypothesis, emulating Einstein's process in developing the theory of relativity. The key challenge is designing a scaling method that effectively leverages large amounts of compute (e.g., inference-time or API usage) to improve performance. While we currently have access to significant compute resources, the question remains how to best utilize them for generating high-quality research hypotheses. Our method makes a meaningful step in this direction.

Q3: Although the proposed method consistently outperforms baseline approaches, the baselines used for comparison are very basic—essentially just self-consistency. Better to compare with previous proposed methods.

A3: Thank you for the suggestion to compare with additional baselines. While previous works have addressed hypothesis generation, they do not specifically focus on fine-grained hypothesis discovery. We initially chose not to include them as baselines because they would not offer comparable performance in terms of providing detailed hypotheses. We have since added experiments with five strong baselines for comparison, with the newly added baselines highlighted in bold.

[1] ChemCrow: Augmenting large-language models with chemistry tools, Nature 2023

[2] Large Language Models for Automated Open-domain Scientific Hypotheses Discovery, ACL 2024

[3] Large language models are zero shot hypothesis proposers, COLM 2024

[4] MOOSE-Chem: Large Language Models for Rediscovering Unseen Chemistry Scientific Hypotheses, ICLR 2025

Q4: Regarding the experimental design, it is important to evaluate alignment with ground truth, but in this case, such evaluation was conducted using LLMs as judges. It was unclear what specific instructions were given to the LLM judges or how effective they were—for example, how well their evaluations aligned with human judgments.

A4: Thank you for your comment. Due to word limitations, we are unable to provide the full prompts here, but they are open-sourced and can be found in the evaluation_instruction_prompts() function in ./Method/utils.py.

We worked closely with chemistry experts during the development of the evaluation framework to ensure its quality. A significant portion of our research was dedicated to developing a robust evaluation process.

Additionally, we performed an analysis to assess the consistency between LLM judges and human expert evaluations. Specifically, 100 hypotheses were sampled and evaluated by two human experts (each reviewing 50). The experts were satisfied with 97 out of 100 evaluations.

Q5: Human evaluation results are only reported whether the proposed method outperformed existing methods or not. It was not clear how desirable the method’s properties were on their own, independent of its relative performance.

A5: Thank you for the suggestion. Table 2 in the paper presents the recall for each method, which reflects absolute performance rather than relative performance.

Additionally, we conducted further error analyses performed by human experts to gain a deeper understanding of the experimental results using very strict criteria to understand the absolute performance of HHS.

Analysis: Error analysis on the proposed method, HHS. Two chemistry PhD candidates conducted it, when one analyzed 30 hypotheses in terms of 30 research questions, the other analyzed another 21 hypotheses in terms of another 21 research questions. The results are as below:

Q6: While itemized evaluations are provided, only the overall human evaluation scores were reported. As a result, we cannot tell how each specific evaluation criterion was judged by human researchers.

A6: Thank you for the suggestion. We have conducted an additional analysis by human researchers for each specific evaluation criterion. The results are provided below:

Q7: While the authors carefully explain that the process can be interpreted as combinatorial search, the actual proposal is just a particular way of prompting and combining outputs from an LLM—not a method involving any explicit combinatorial search technique.

A7: In line 122 we analyze that one of the challenges for the fine-grained hypothesis discovery task is that the full search space is unknown. We deal with this challenge by adopting heuristic search in each hierarchy instead of precise search over the full space, which can avoid the need to know the full search space. We use LLMs for this heuristic search, which is prompting looks from the outside.

Out of the classic combinatorial search method, MCTS and genetic algorithms require a clearly defined search space, making them not suitable for this task. Reinforcement learning, however, does not require a defined search space, and we have implemented it as an additional baseline shown in the Table of A3.

---

We greatly appreciate your comprehensive feedback. We hope that our responses have satisfactorily addressed all your queries. Should you have further questions or suggestions for enhancing our manuscript, we warmly welcome your input.

Thank you so much for the reply.

Q1: If I understand correctly, it seems that the method is not actually differentiating the loss landscape to compute gradients, but rather using the LLM to generate reward-like signals that help guide improvement. It’s not entirely clear to me how this framing has substantively contributed to devising new methods in a useful way.

Response: Thank you for this important question. There may be a misunderstanding about our approach that we'd like to clarify.

Our method does navigate a reward landscape relying on "gradients" for optimization, but within a hypothesis space rather than a continuous parameter space. Specifically:

Conceptual Framework: We conceptualize a hypothesis space where each point along the input dimensions (the x-axis, potentially multidimensional) represents a candidate hypothesis, and each point is assigned a reward value (on the y-axis) by the LLM based on its internal heuristics. This defines a reward landscape over the hypothesis space, with the highest peak corresponding to the hypothesis the LLM internally judges as most promising (as described in line 51~55).

Gradient Computation: At each step, we incorporate an addition/deletion/replacement of detail to update the hypothesis in the previous step (Equation 6). The pairwise comparison between the previous hypothesis and the modified version provides the optimization direction—essentially "computing" the "gradient" in hypothesis space. This pairwise evaluation serves both as assessment and gradient estimation.

Methodological Contributions: This framing enables us to:

1. Explain why our hierarchical method would lower the complexity (Sections 2.3, 2.6).
2. Explain why we choose to adopt heuristic search to further lower the complexity to make the task practical.
3. Identify that the essential task is to incorporate a set of details to the initial search point (a very coarse-grained research direction). We frame this incorporation sequentially (Equation 6 in the paper), with each step incorporating one or a few additions/modifications.

The key insight is that while we're not computing gradients in traditional parameter space, we are performing gradient-based optimization in the space of scientific hypotheses, where the LLM's comparative judgments provide directional information for improvement.

We hope this clarifies how our landscape differentiation approach substantively contributes to developing more efficient hypothesis optimization methods.

Q2: There is no issue with using APIs. My point was rather that, given the method doesn’t introduce a new model architecture, training procedure, or dataset, the design choices relying on the API are expected to offer a high degree of novelty or utility. I’m not fully convinced that the method reaches that level yet.

Response: Thank you for the thoughtful question, we believe that our addition of 5 baselines (including ChemCrow, MOOSE, and Reinforcement Learning) and the added expert analysis in terms of the four specific aspects (in A6) should be able to fully answer the question here. We are glad that you are satisfied with our rebuttal in A3, A6, and A7 correspondingly (e.g., the inclusion of baselines).

We additionally have conducted an error analysis by experts on how the baseline is weaker than the proposed method (HHS). Two experts conduct this experiment, one checks 30 pairs of hypotheses, the other checks another 21 pairs.

Q5: The table you’ve kindly shown here looks different from Table 2 in the paper. Could you clarify where this version appears?

Response: Yes, during the addition of baseline experiments, we discovered that the decimal precision in Table 2 was rounded to one decimal place. We have now updated the table to show two decimal places for greater precision.

Q3 & Q4 & Q6 & Q7: Thank you! Great to know that these questions are addressed!

We appreciate your insightful questions and helpful suggestions. For clarity, we have organized your queries along with our responses below:

---

Q1: The HHS method, with its iterative search within each level, multiple independent runs for recombination, and progression through several hierarchical levels, appears to be computationally expensive in terms of LLM API calls and tokens. Could you provide an analysis of the computational cost (e.g., approximate number of LLM calls or total tokens processed per problem instance) for HHS compared to the baselines?

A1: Thank you for the question. Below is an analysis of the average reasoning steps for each method. The HHS method, while more computationally intensive, uses computation as a trade-off for improved performance. This trade-off is reasonable, as evidenced by OpenAI’s Dan Roberts, who discussed in his talk titled “9 Years to AGI? OpenAI’s Dan Roberts Reasons About Emulating Einstein” that an ideal scientific discovery system might require 8 years of inference-time computation to propose a high-quality hypothesis, emulating Einstein's process.

Q2: The paper title aims at scientific discovery, but the paper does not discuss how this approach would generalize to other scientific domains (e.g., biology, materials science, physics) beyond chemistry. Could you elaborate on the process of designing the 5-level chemistry hierarchy in Figure 1? How much expert effort was involved? More importantly, how sensitive is the performance of HHS to the specific design of this hierarchy? Can you easily extend the design to other scientific domain? For instance, what would happen if you collapsed levels 2 and 3, or used a different number of levels? This would help clarify the robustness and practical requirements of the method.

A2: Thank you for the suggestion. The method is largely domain-agnostic in its design, with the only domain-specific aspect being the hierarchy design. The chemistry hierarchy was designed by a chemistry PhD candidate in approximately 2 hours, based on their understanding of the chemistry research process. Similar hierarchies for other scientific disciplines can be designed by domain experts. We will discuss this further in the paper. To evaluate the robustness of the hierarchy design, we conducted an analysis by collapsing levels 2 and 3. The comparison results are shown below:

The results are very similar, particularly in comparison to the baselines, demonstrating the robustness of the hierarchy design.

Q3: The "smoothing of the reward landscape" is the central theoretical claim motivating HHS. However, this claim is supported only by illustrative schematics (Figures 3 & 4) and the downstream performance of the system. While the results are strong, they are indirect evidence. The paper would be significantly strengthened by a more direct, quantitative analysis attempting to measure this smoothing effect.

A3: We appreciate the reviewer's feedback on the theoretical support for our "hierarchy smooths reward" claim. We connect fine-grained hypothesis discovery to established theories in image and signal processing, where smoothing convolution (averaging pixels in an image patch) acts as a low-pass filter. The sufficient condition for this claim is our assumption that "LLM’s evaluation for a hypothesis with a general concept can be viewed as an aggregated estimate--approximating an average or soft maximum--of LLM’s evaluation for hypotheses with specific concepts under that general concept" (lines 197–199). While this assumption is commonsensically true but challenging to prove formally, we conducted additional experiments to verify it: We prompted LLMs to score hypotheses with general concepts and their corresponding specific variants on a 0–100 scale, finding an average difference of only 3.47 between the general score and the averaged specific scores (in total 204 pairs compared).

Q4: The results show HHS consistently outperforming baselines. Were there any specific types of problems or hypotheses where HHS struggled or failed to produce a superior result? An error analysis could provide valuable insights into the limitations of the current framework.

A4: Appendix B has four case studies including detailed expert's analysis (we will mention it in the main body of the paper). In addition to it, we conducted two more error analyses to better understand the experiment results.

Analysis 1: Error analysis on the proposed method, HHS. Two chemistry PhD candidates conducted it, when one analyzed 30 hypotheses in terms of 30 research questions, the other analyzed another 21 hypotheses in terms of another 21 research questions. The results are as below:

Analysis 2: Error analysis on why the baselines are weaker than HHS. Two chemistry PhD candidates conducted it: one analyzed 30 hypotheses for 30 research questions, the other analyzed 21 hypotheses for 21 research questions. The results are as follows:

Q5: Table captions should be placed above the table.

A5: Thank you for the advice! We will adjust it in the next version.

---

We greatly appreciate your comprehensive feedback. We hope that our responses have satisfactorily addressed all your queries. Should you have further questions or suggestions for enhancing our manuscript, we warmly welcome your input.

Thank you for your insightful inquiries. In the following sections, we've structured our responses to each of your points raised.

---

Q1: The definition of the fine-grained hypothesis relies on the annotation by domain experts, which may have subjective bias, and the benchmark scale is relatively small (based on 51 chemistry papers), so the robustness needs to be verified with larger-scale data.

A1: We appreciate the reviewer’s concern. The hypotheses were annotated by chemistry PhD students, who are among the highest-quality annotators in the field.

Scientific discovery is different from the traditional reasoning task that one discovered high-quality hypothesis could lead to significant effect. Also the data is harder to annotate. Previous efforts all have only a few dozen annotation data (purely as test data). For example, [1] [2] annotates ~50 data, [3] done by OpenAI only annotates only 20 data.

[1] Large Language Models for Automated Open-domain Scientific Hypotheses Discovery, ACL 2024

[2] MOOSE-Chem: Large Language Models for Rediscovering Unseen Chemistry Scientific Hypotheses, ICLR 2025

[3] PaperBench: Evaluating AI's Ability to Replicate AI Research, OpenAI

Q2: The HHS framework cannot guarantee finding the global optimal solution (as admitted in the manuscript), and the hierarchical division relies on domain knowledge (such as the five-level structure in chemistry), which may carry the risk of over-reliance on prior assumptions. If the domain structure is complex or difficult to stratify, the method's effectiveness may decline.

A2: HHS can be seen as an optimization method. Currently none of the optimization methods is guaranteed to find the global optimum on non-convex problems.

The chemistry hierarchy is designed by a chemistry phd candidate in 2 hours, and it works quite well. The hierarchy is about very general descriptions and therefore should be relatively quite robust.

Q3: The paper mentions using the internal heuristics of LLM as a reward signal to guide hypothesis search. Then, what is the specific form of the reward function? Is it defined solely through pairwise comparisons by LLM? During the optimization process, how can the stability and consistency of the reward function be ensured? Especially in different levels of search, will there be conflicts or deviations in the reward signals? For instance, the rewards at a higher level may favor the innovativeness of concepts, while those at a lower level focus more on the feasibility of experiments. How can the differences in rewards between these levels be balanced?

A3: The reward function is solely defined through pairwise comparisons by LLM. The pairwise comparison prompt is consistent in terms of its main body, where in different hierarchies, different specific subprompt is adopted, stressing what that hierarchy focuses on. Each hierarchy is developed with one specific subprompt.

Q4: The paper defines the discovery of fine-grained hypotheses as a combinatorial optimization problem, with the complexity of the search space being C{^m}_n. When m and n are large, the computational complexity will increase exponentially. Although the hierarchical search framework alleviates this problem by restricting the search space at each level, in practical applications, how to determine the optimal search range at each level? When facing very complex hypotheses (such as those involving a large number of components and parameters), can HHS effectively avoid getting stuck in local optima? Are there theoretical analyses or experimental data to support the efficiency and effectiveness of HHS in high-dimensional search spaces?

A4: In line 122 we analyze that one of the challenges for the fine-grained hypothesis discovery task is that m and n are all unknown. We deal with this challenge by adopting heuristic search in each hierarchy instead of precise search over the full space, which can avoid the need to know the full search space.

The fine-grained hypothesis discovery task itself has high-dimensional search spaces, and our experiments are all conducted in it. We show that theoretically, HHS can find better local optima than the baselines, and empirically HHS does outperforms baselines. This work introduces the fine-grained hypothesis discovery task, and we believe that an ultimate solution for this challenge has yet to be established. We encourage the research community to explore these intriguing questions further.

---

We greatly appreciate your comprehensive feedback. We hope that our responses have satisfactorily addressed all your queries. Should you have further questions or suggestions for enhancing our manuscript, we warmly welcome your input.

We appreciate your insightful questions and helpful suggestions. For clarity, we have organized your queries along with our responses below:

---

Q1: Only ~15 references are listed, Key recent work on agent-based scientific discovery published in NeurIPS/ICLR/ACL 2024-25 is absent.

A1: We thank the reviewer for raising concerns about the references. As this work introduces the novel task of fine-grained scientific hypothesis discovery—for which no prior research exists—there are inherently fewer directly relevant papers to cite.

We did cite key recent works on agent-based scientific discovery published in NeurIPS/ICLR/ACL 2024-25, including:

[1] Scimon: Scientific inspiration machines optimized for novelty, ACL 2024.
[2] Large Language Models for Automated Open-domain Scientific Hypotheses Discovery, ACL 2024.
[3] Large language models are zero shot hypothesis proposers, COLM 2024.
[4] MOOSE-Chem: Large Language Models for Rediscovering Unseen Chemistry Scientific Hypotheses, ICLR 2025.
[5] Can llms generate novel research ideas? A large-scale human study with 100+ NLP researchers, ICLR 2025.
[6] NOVA: An Iterative Planning Framework for Enhancing Scientific Innovation with Large Language Models, ACL 2025.

Q2: Previous work like Chemcrow already works on hypothesis generation.

A2: We thank the reviewer for highlighting ChemCrow. While previous works like ChemCrow address hypothesis generation, it does not focus on fine-grained hypothesis discovery. We did not adopt it as a baseline initially because they won’t have a comparable performance on providing the hypothesis details. We have added 5 additional baselines including ChemCrow [7] as well as four other strong baselines (three published in top conferences in the recent two years, and one combinatorial optimization optimizer as required in Q3), with newly added baselines boldened. We will refer to them in the paper.

[7] ChemCrow: Augmenting large-language models with chemistry tools, Nature

Q3: Baselines omit standard combinatorial optimisers such as reinforcement learning, MCTS, genetic algorithms.

A3: We thank the reviewer for suggesting additional combinatorial optimizers as baselines. As fine-grained scientific hypothesis discovery is a new task introduced in this paper, and while we formalize it as a combinatorial optimization problem, the search space is inherently unknown (line 122: "n is unknown"), unlike traditional problems.

MCTS and genetic algorithms require a clearly defined search space, precluding direct implementation here. We are aware that prior works like [8][9] use MCTS, but in tasks with explicit search spaces.

Reinforcement learning, however, does not require a defined search space, and we have implemented it as shown in the Table of A2.

[8] Monte Carlo Thought Search: Large Language Model Querying for Complex Scientific Reasoning in Catalyst Design, EMNLP 2023.
[9] CHEMREASONER: Heuristic Search over a Large Language Model's Knowledge Space using Quantum-Chemical Feedback, ICML 2024.

Q4: HHS is essentially a coarse-to-fine or evolutionary search variant.

A4: We appreciate the reviewer's observation that HHS incorporates coarse-to-fine and evolutionary elements. However, our core contributions lie in formalizing the novel fine-grained hypothesis discovery task as a combinatorial optimization problem and developing HHS to optimize it effectively for scientific discovery. No prior LLM-based scientific discovery work has framed the task through an optimization lens. Moreover, HHS advances coarse-to-fine search by adapting it to challenging, language-based open-ended tasks like hypothesis generation, where such implementations were previously unclear.

Q5: The “hierarchy smooths reward” claim is supported only by schematic plots, lacking formal metrics or convergence theory.

A5: We appreciate the reviewer's feedback on the theoretical support for our "hierarchy smooths reward" claim. We connect fine-grained hypothesis discovery to established theories in image and signal processing, where smoothing convolution (averaging pixels in an image patch) acts as a low-pass filter.

The sufficient condition for this claim is our assumption that "LLM’s evaluation for a hypothesis with a general concept can be viewed as an aggregated estimate--approximating an average or soft maximum--of LLM’s evaluation for hypotheses with specific concepts under that general concept" (lines 197–199).

While this assumption is commonsensically true but challenging to prove formally, we conducted additional experiments to verify it: We prompted LLMs to score hypotheses with general concepts and their corresponding specific variants on a 0–100 scale, finding an average difference of only 3.47 between the general score and the averaged specific scores (in total 204 pairs compared).

---

We greatly appreciate your comprehensive feedback. We hope that our responses have satisfactorily addressed all your queries. Should you have further questions or suggestions for enhancing our manuscript, we warmly welcome your input.

Thank you for your insightful inquiries. In the following sections, we've structured our responses to each of your points raised.

---

Q1: Evaluation metrics & methodology can benefit from more details.
A1: We thank the reviewer for this suggestion. We will add more details on evaluation metrics and methodology in the next version, as the submitted paper's dense content limited space in the main text. All relevant details, including prompts, will be updated. (Given the rebuttal space constraints, we cannot include them here.)

Q2: The conclusions drawn from the experiments for each research question would benefit from deeper analysis.
A2: Appendix B has four case studies including detailed expert's analysis (we will mention it in the main body of the paper). In addition to it, we conduct two more analyses to better understand the experiment results.

Analysis 1:

Error analysis on the proposed method, HHS. Two chemistry PhD candidates conducted it, when one analyzed 30 hypotheses in terms of 30 research questions, the other analyzed another 21 hypotheses in terms of another 21 research questions. The results are as below:

Analysis 2:

Error analysis on why the baselines are weaker than HHS. Two chemistry PhD candidates conducted it: one analyzed 30 hypotheses for 30 research questions, the other analyzed 21 hypotheses for 21 research questions. The results are as follows:

Q3: What are “redundant or tangential elements”.

A3: In this context, "redundant elements" refer to additional chemistry concepts that, while potentially relevant in some specific cases, are unnecessary for the hypothesis. For example, if the groundtruth hypothesis requires concepts [A, B, C, D, E], and the hypothesis includes [A, B, C, D, E, F, G, H, I], the concepts [F, G, H, I] would be considered redundant. "Tangential elements" are a subset of redundant elements, specifically referring to those that are irrelevant to the research question.

Q4: Quantitative analysis of the edits (e.g., number of insertions, deletions, or rewrites).

A4: We performed an additional analysis, revealing that the average number of optimization steps for HHS is 282. Of these, 113 steps involve insertions, 82 steps involve deletions, and 152 steps involve rewrites. Overall the three edits are evenly distributed across the hierarchies.

Q5: How are the hierarchical levels (p) and the number of hierarchical levels defined across different research topics or domains? Moreover, how are these levels conveyed to the LLM-driven agentic process?

A5: The chemistry hierarchy is designed by a chemistry phd candidate in 2 hours, mostly by their understanding of chemistry research. Different discipline’s hierarchy can be designed by their domain experts.

These levels are explicitly defined and explained in the prompt. The code is open-sourced. Due to the space limit, please understand that we can’t list the prompt here.

Q6: The paper mentions LLM-based dimension-specific evaluations. a) What scoring system is used for these assessments? b) How are these individual scores aggregated into an “overall” (LLM) evaluation?

A6: The evaluation is based on pairwise comparisons conducted by LLMs. The "overall" evaluation has a separate prompt that includes all four dimensions, and the results for the overall evaluation are obtained independently from the individual dimension scores.

Q7: The authors claim to observe two trade-offs. How were these conclusions drawn?

A7: The conclusions are primarily based on the definitions of the four dimensions in the prompt. Effectiveness is supported by similar principles or methods, but this often sacrifices novelty. Feasibility (lines 255–258) is inversely related to implementation complexity, where adding details to enhance thoroughness inevitably increases complexity by introducing more elements and concepts into the hypothesis.

Q8: It is unclear whether the LLM-based evaluation of hypothesis alignment is simply matching keywords or phrases from the ground truth. If so, how do the authors ensure these matches are not out of context or semantically misleading (e.g., terms that appear similar but differ in meaning)?

A8: The full hypotheses are provided during the comparison to ensure complete context, and the evaluation prompt explicitly emphasizes this requirement. Each generated hypothesis is approximately 600 words and requires domain-specific chemistry knowledge for proper understanding. While providing specific examples here may not be feasible, we extensively consulted with chemistry experts during the development of the evaluation framework to ensure its quality. A significant portion of our research efforts was dedicated to developing a robust evaluation framework.

Q9: Table 3 - There appears to be a high % of ties. It would be helpful to explicitly discuss the implications of these ties on the strength of the conclusions drawn. I would imagine that the high tie rate might suggest that differences between committees are less pronounced than the win/loss counts alone indicate.

A9: Thank you for your insightful observation. The conclusion focuses on determining which committee is better, and the tie rate does not affect the qualitative outcome. The conclusion is supported by multiple pairwise committee comparisons, many of which show clear win/loss discrepancies. We include the tie results to provide a more comprehensive understanding of the comparison process.

Q10: Table 3 - It has a confusing structure, as the comparison order is reversed between the left and right columns. Reformatting the table for consistent comparison direction and row order would improve clarity.

A10: Thank you for your careful review and helpful suggestion. We will reformat Table 3 for improved clarity and consistency.

Q11: Could the authors elaborate on whether Table 2 serves this purpose, i.e., does the recall of ground-truth components by HHS, greedy search, and self-consistent search reflect their respective distances from the global optimum? If so, it would be helpful to explicitly frame Table 2 as a measure of this “distance to optimum” and discuss how often HHS actually recovers (or nearly recovers) the full ground-truth hypothesis.

A11: Thank you for the insightful suggestion. Yes, the recall of ground-truth components reflects the respective distances from the global optimum. We will explicitly frame Table 2 as a measure of "distance to optimum" and discuss how often HHS recovers (or nearly recovers) the full ground-truth hypothesis in the paper.

---

We greatly appreciate your comprehensive feedback. We hope that our responses have satisfactorily addressed all your queries. Should you have further questions or suggestions for enhancing our manuscript, we warmly welcome your input.