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  • Mauricio J. D. Martins

Reviewer 1: Anonymous Reviewer 2: Anonymous Reviewer 3: Anonymous

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The author presents a very interesting review on the applicability of fractal theory in different fields. Also, the author used recent materials in the field for documentation. The article is well structured, documented and the presentation makes it accessible to the specialized public.
I recommend the author to further develop the part on Applicability in neuroimaging because this field is very interesting at the moment.

Author Response

  1. I recommend the author to further develop the part on Applicability in neuroimaging further because this field is very interesting at the moment.

Thank you for this suggestion. We extended the discussion on neuroimaging by including a separate section “4.3 Neural bases of recursion across domains”, and adding a summary Figure 3. We also added a discussion on the neuroimaging results of similar approaches: “4.4 Insights from artificial grammar learning and computational approaches”

Reviewer 2 Report

Comments and Suggestions for Authors

This paper’s topic has certain interdisciplinary appeal, but the current content is insufficient to meet the publication requirements of a full review article. Its summary of existing work lacks depth, original perspective, critical analysis, and structured presentation. For resubmission, the authors would need to significantly expand the content, strengthen comparative and critical analysis, and propose clear new insights.

  • The manuscript is currently more of a simple collection of literature without deep analysis or theoretical integration. It is recommended to strengthen critical discussion and clarify the strengths and limitations of different studies.
  • The Introduction should be more focused, highlighting the unique contribution and review value of this paper, rather than providing only a general introduction to “fractals” and “recursion.”
  • Consider using comparative tables or structured diagrams to systematically present the commonalities and differences of RHE research paradigms across language, music, visual, and motor domains.
  • The discussion of future research directions is too broad. More specific research questions and methodological improvements should be proposed (e.g., how to better isolate recursive processing from working memory load).
  • Some content is redundant, such as repeated explanations of recursion, iteration, and self-similarity. These should be compressed and refined.
  • The literature review is broad in scope but lacks in-depth commentary on the latest studies (e.g., neural network simulations of recursive learning, cross-species recursion experiments).
  • There is a lack of critical evaluation of interdisciplinary research, such as the practical applicability and potential limitations of complexity science approaches in cognitive research.
  • The logical coherence of the review should be strengthened, ensuring a smooth progression from “foundations of fractal geometry” to “cognitive experiments” and “future perspectives,” rather than jumping between sections.
  • The Conclusion is overly general. It should clearly state the core conclusions and the substantive contributions to future research, rather than simply repeating previous content.
  • As a review article, it is recommended to introduce more quantitative comparisons (e.g., statistical measures of RHE performance across different experiments) to enhance academic value.
Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

We thank the reviewer for the detailed and constructive feedback. We substantially revised the manuscript to deepen the analysis, sharpen the unique contribution of the paper, and present results in a more comparative and structured way. Below we address each point individually. Manuscript excerpts are provided verbatim where changes were made.

1. The manuscript is currently more of a simple collection of literature without deep analysis or theoretical integration. It is recommended to strengthen critical discussion and clarify the strengths and limitations of different studies. Consider using comparative tables or structured diagrams to systematically present the commonalities and differences of RHE research paradigms across language, music, visual, and motor domains.

 

Thank you for this suggestion. Our initial goal was to present the theoretical framework and set of experimental tools for fractal cognition to the audience of Fractal and Fractional. However, we agree that we should expand the discussion of the specific studies and present them in a format that allows comparative analysis. To this goal, we restructured and extended the section “4. Empirical Findings on RHE Across Domains”, including the addition of a comparative Table of methods (Table 1), results (Table 2), and Figure (Figure 3).

2. The Introduction should be more focused, highlighting the unique contribution and review value of this paper, rather than providing only a general introduction to “fractals” and “recursion.”

 

Thank you for this suggestion. We revised the introduction to highlight our unique contribution but also to present the structure of the manuscript more clearly.

 

“The unique contribution of this paper is to present the research program we have developed to bridge fractal geometry and the cognitive science of fractal cognition. We clarify the conceptual foundations and methodology of RHE, demonstrating how fractal geometry provides a principled framework for studying hierarchical cognition. A secondary contribution is to exemplify how this framework can be used to address the theoretical question of whether RHE is a domain-general or domain-specific capacity. To this end, we review findings from our own research program and contrast them with results from more established paradigms to study hierarchical cognition, such as Artificial Grammar Learning (AGL), which have enabled precise modeling of hierarchical representations using symbolic, Bayesian, and neural network approaches. The paper is organized as follows: Section 2 defines RHE and its relationship to recursion, iteration, hierarchical embedding, and fractal self-similarity; Section 3 reviews behavioral and neuroimaging paradigms across language, music, vision, and motor action; Section 4 summarizes key empirical findings and their theoretical implications for domain-generality versus domain-specificity while situating RHE alongside related approaches; Section 5 proposes that future research can integrate behavioral and neural signatures with symbolic, neural, and Bayesian models to uncover the algorithmic and biological bases of recursive generativity.”

3. The discussion of future research directions is too broad. More specific research questions and methodological improvements should be proposed (e.g., how to better isolate recursive processing from working memory load).

 

We agree with this suggestion. We merged the sections on open questions and future directions and expanded the content, focusing on broadening the scope and critically modeling behavioral and neural signatures with precise mechanistic accounts, following approaches used in AGL.

 

*“5. Open Questions and Future Directions Despite substantial progress, important challenges remain for understanding recursive hierarchical embedding (RHE). These challenges arise at both the behavioral and neural levels. 5.1 Challenges in studying RHE At the behavioral level, RHE is difficult to isolate from confounding factors. In AGL, increasing embedding depth almost inevitably covaries with working-memory load, perceptual discrimination difficulty, and attentional switching, making it unclear whether performance drops reflect genuine limits on recursive processing or the exhaustion of auxiliary resources [71,76,79]. While RHE seems more robust against verbal, spatial, and visual WM manipulations (Table 2), these can still play a significant role. Moreover, discrimination and production tasks likely engage overlapping but non-identical systems: discrimination allows external cues to scaffold performance [42,43], whereas production requires participants to internally generate the recursive step without external guidance. Cross-domain comparability further complicates interpretation, as the same generative rules can differ in representational granularity and familiarity: visual fractals are relatively easy to recognize, auditory recursion requires tonal imagery, and motor recursion remains the most challenging. This is reflected in the accuracy levels (Table 2). At the neuroimaging level, progress is constrained by the difficulty of inferring mechanisms from correlates. Current findings consistently implicate posterior temporal regions in structure building and frontal regions in control and sequencing [27,65,68,85,85]; however, disentangling representational functions from attentional or decision-related processes remains a challenge. Additionally, the contributions of subcortical [65] and network-level structures [86] are poorly understood. Default mode recruitment in recursion and fronto-parietal engagement in iteration point to large-scale network dynamics, but these patterns remain underexplored [44]. 5.2 Towards mechanistic modeling A promising way forward is to adopt the modeling toolkit that has proven fruitful in Artificial Grammar Learning (AGL). In AGL, symbolic grammars, Bayesian inference models, and recurrent neural networks (RNNs) provide precise hypotheses about the internal strategies participants deploy. Such models capture not only success but also systematic errors—for example, why both humans and RNNs collapse beyond three to four levels of embedding [18,73], or why Bayesian learners shift between recursive and associative strategies depending on task demands [71]. For RHE, progress has begun with drift diffusion modeling [27,64], which reveals how recursion and iteration differ in evidence accumulation (drift rate) and decision thresholds (boundary separation). Yet this approach remains descriptive. The next step is to develop richer mechanistic models that generate specific predictions about how recursion should be affected by symbolic interference, posterior temporal lesions, or atypical development. These models can then be tested against the behavioral signatures already identified in Table 2: order effects, asymmetric vulnerability of iteration, and resilience of RHE under certain kinds of interference. In this way, computational modeling can bridge fine-grained behavioral results with neural mechanisms, moving beyond correlation to causal inference. 5.3 Expanding the empirical program Currently, this modeling agenda has been primarily pursued in the visual domain. Extending it to motor and auditory recursion is a critical next step. For example, motor production tasks provide a stronger test of generativity but remain underexplored in terms of modeling; auditory recursion is especially sensitive to expertise and would provide a natural test case for how domain-specific scaffolding interacts with domain-general mechanisms. Neuroimaging evidence of recursion acquisition, summarized in Figure 3, is also disproportionately focused on vision. Systematic extensions to music and motor domains would provide the missing pieces of a full cross-domain profile. Equally important is broadening the populations tested. Developmental and cross-cultural studies are needed to assess the universality of RHE, and clinical or neuropsychological groups can reveal necessary systems by virtue of selective impairments. Transfer designs—training recursion in one modality (e.g., visual fractals) and testing in another (e.g., nested musical rhythms)—would provide a strong test of whether shared scaffolds underlie cross-domain recursion, or whether success depends on domain-specific expertise. 5.4 Toward a unified science of fractal cognition Taken together, these challenges and opportunities point toward a unified research program. Progress will require integrating behavioral paradigms that isolate recursion from confounds, mechanistic models that formalize the computations involved, and neurobiological evidence that specifies their implementation. The goal is not simply to show that humans can represent recursion across domains, but to predict when and how recursive competence emerges, collapses, or transfers. This integrative framework would allow fractal geometry to serve not only as a metaphor but also as a quantitative bridge: linking mathematical self-similarity, cognitive theories of hierarchical generativity, and the neural architectures that implement bounded recursion. By aligning behavioral signatures with algorithmic models and neural dynamics, the field can move toward a mechanistic understanding of how humans instantiate fractal cognition.”*

4. Some content is redundant, such as repeated explanations of recursion, iteration, and self-similarity. These should be compressed and refined.

 

Thank you for this suggestion. We performed major revisions and eliminated redundancies across sections.

5. The literature review is broad in scope but lacks in-depth commentary on the latest studies (e.g., neural network simulations of recursive learning, cross-species recursion experiments).

 

We added more information on specific studies, including those related to modeling and animal studies, in Section 4.4.

6. There is a lack of critical evaluation of interdisciplinary research, such as the practical applicability and potential limitations of complexity science approaches in cognitive research.

 

Following the suggestion in the previous point, we decided to narrow our focus and eliminated the discussion on complexity science, keeping the paper focused on RHE and its direct cognitive/neuroscientific toolkit.

7. The logical coherence of the review should be strengthened, ensuring a smooth progression from “foundations of fractal geometry” to “cognitive experiments” and “future perspectives,” rather than jumping between sections.

 

We substantially revised the structure to follow a more coherent and streamlined progression. This new outline is presented clearly in the Introduction:

 

“The paper is organized as follows: Section 2 defines RHE and its relationship to recursion, iteration, hierarchical embedding, and fractal self-similarity; Section 3 reviews behavioral and neuroimaging paradigms across language, music, vision, and motor action; Section 4 summarizes key empirical findings and their theoretical implications for domain-generality versus domain-specificity while situating RHE alongside related approaches; Section 5 proposes that future research can integrate behavioral and neural signatures with symbolic, neural, and Bayesian models to uncover the algorithmic and biological bases of recursive generativity.”

8. The Conclusion is overly general. It should clearly state the core conclusions and the substantive contributions to future research, rather than simply repeating previous content.

 

We revised the conclusion to highlight five specific contributions:

 

“This paper advances a research program in fractal cognition through five core contributions. First, we introduce the theoretical foundations of recursive hierarchical embedding (RHE), clarifying its distinction from iteration, self-similarity, and related constructs. Second, we provide a methodological toolkit—behavioral, neuropsychological, and neuroimaging paradigms—for isolating recursive processing. Third, we illustrate the application of this toolkit across language, music, vision, and motor action to detect the ability to represent RHE and evaluate evidence for domain-general versus domain-specific cognitive resources. Fourth, across domains and populations, the evidence suggests that recursion is more challenging than iteration, emerges later, but is also more resilient to interference and clinical impairment—indicating compensatory mechanisms. We argue that future progress hinges on integrating behavioral signatures across populations and paradigms with precise quantitative models. These contributions provide a roadmap for moving from descriptive demonstrations to a mechanistic science of fractal cognition, where recursive generativity is not only documented but explained and predicted.”

9. As a review article, it is recommended to introduce more quantitative comparisons (e.g., statistical measures of RHE performance across different experiments) to enhance academic value.

 

While it is outside the scope of this review to perform a statistical meta-analysis, we addressed this by adding Table 2, which reports mean accuracies and key performance patterns across studies, providing a comparative quantitative overview.

In summary:

We expanded and critically deepened the literature review, added comparative tables and figures, clarified the unique contribution of the paper, merged and sharpened the future directions section with concrete modeling proposals, removed redundancies, and revised the conclusion. We believe these changes address the reviewer’s concerns and significantly strengthen the manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors
  1. Line  196: ....."Because fractal structures can be generated using both recursive and iterative procedures....."do you have any supporting article with this statement? If so Please cite.
  2.  In review. Its must that the authors have at least two published works related to this field. if so cite and explain.
  3. Line 269: Its better to add some images/figures with supporting of this "Neuroimaging Applications"
  4. Overall the review drafted well and explain clearly.
Comments on the Quality of English Language

Minor editing needed 

Author Response

  1. Line 196: “Because fractal structures can be generated using both recursive and iterative procedures…” do you have any supporting article?

Thank you for this suggestion. We revised the sentence, and added both references and alluded to the example in Figure 1. “Because recursive and iterative rules can produce perceptually similar outputs [7,31] (Figure 1i)...”

 

  1. It is a must that the authors have at least two published works related to this field. If so, cite and explain.

Thank you for the suggestion. We clarified our input to the field by adding Table 2 and Figure 3, which explicitly cite our work.

 

  1. Line 269: It is better to add some images/figures supporting the “Neuroimaging Applications.”

Thank you for the suggestion. We have now added Figure 3, which summarizes the neuroimaging findings.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

My comments on the initial version of the manuscript have been sufficiently addressed by the authors in this revised version. I have no further comments on the technical aspects. The manuscript may be considered for publication after a proofreading.