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Does the introduction explain the objective of the research presented in the preprint?

Yes

Are the methods well-suited for this research?

Somewhat appropriate

The mathematical methods employed in Section 3 are rigorous and well-executed. The proof of Theorem 1.1 is sound, the use of periodic reduction to a single interval is elegant, and the construction of generalized harmonic functions is creative and mathematically valid. The corollaries follow correctly from the main theorem. However, the methods become less rigorous in Section 4, where the transition to physical applications is made without sufficient justification. The assumptions regarding Planck's constant and the amplitude profile are introduced without derivation, and the connection between the mathematical model and the double-slit experiment is asserted rather than demonstrated. While the mathematical core is solid, the physical modeling would benefit from more rigorous development. Overall, the methods are appropriate for the mathematical portion of the research but need strengthening for the physical claims to be fully supported

Are the conclusions supported by the data?

Somewhat unsupported

The mathematical conclusions in Section 3 are well-supported by the proofs provided. Theorem 1.1 and its corollaries follow logically from the defined functions, and the uniform convergence is correctly demonstrated. However, the physical conclusions in Section 4 are not adequately supported by the data or derivations presented. The claim that the model 'reproduces the experimentally observed data' of the double-slit experiment is stated without any quantitative comparison, without any derived interference pattern, and without any equation linking the mathematical framework to the predicted intensity distribution. The assumptions regarding Planck's constant ($h = cA_0^2$) and the amplitude profile are introduced arbitrarily and are not derived from the mathematical model developed in Section 3. The disconnect between the mathematical framework and the physical conclusions means that while the mathematics is internally consistent, the physical interpretation overreaches what the data and derivations can support. The conclusions would be strengthened by either providing rigorous derivations connecting the model to physical observables, or by limiting the conclusions to the mathematical results alone.

Are the data presentations, including visualizations, well-suited to represent the data?

Neither appropriate and clear nor inappropriate and unclear

This manuscript is primarily a mathematical exposition and does not contain data presentations or visualizations in the traditional sense. There are no graphs, tables of numerical results, or comparisons with experimental data. The mathematical derivations are presented textually with equations, which is appropriate for the type of content in Section 3. However, given that Section 4 makes claims about reproducing experimental results (the double-slit experiment), the absence of any visual comparison—such as an interference pattern plot, a graph of intensity versus position, or a table comparing predicted and measured values—is a significant omission. For a paper that asserts physical applicability, the lack of data visualization to support these claims makes it difficult to evaluate the validity of the physical conclusions. The presentation of the mathematical content is clear and well-structured. The presentation of the physical claims would benefit substantially from the inclusion of appropriate visualizations and quantitative comparisons.

How clearly do the authors discuss, explain, and interpret their findings and potential next steps for the research?

Neither clearly nor unclearly

The authors present their mathematical findings clearly in Section 3, with well-structured proofs and logical derivations. Theorem 1.1 and its corollaries are stated explicitly and the proof is accessible. However, the discussion and interpretation of these findings are limited. The manuscript lacks a dedicated discussion or conclusion section that explains the significance of the mathematical results, their limitations, or their potential applications. The reader is left to infer the importance of the work without guidance from the authors. Furthermore, Section 4 introduces physical interpretations that are not clearly connected to the mathematical framework. The claims regarding the double-slit experiment and the photon heuristic are stated without explaining how they follow from the earlier derivations. This creates confusion rather than clarity. Most importantly, there is no discussion of potential next steps, future research directions, or open questions. The manuscript ends abruptly after Section 4 without any concluding remarks. A clear discussion of findings, limitations, and future work would substantially strengthen the manuscript.

Is the preprint likely to advance academic knowledge?

Moderately likely

This preprint offers a modest but genuine contribution to academic knowledge. The mathematical framework developed in Section 3—generalized harmonic functions that converge uniformly to linearity—is original and mathematically sound. Theorem 1.1 and its corollaries represent a valid addition to the literature on harmonic analysis and function approximation. However, the potential for broader impact is currently limited by two factors. First, the physical applications proposed in Section 4 are not adequately developed or supported. The claims regarding the double-slit experiment and photon behavior, while intriguing, lack the rigorous derivations and experimental comparisons necessary to constitute a meaningful contribution to physics. Second, the manuscript lacks a discussion of the mathematical results' significance, limitations, and potential applications, which would help readers understand the value and context of the work. With further development—particularly by deepening the mathematical framework and either providing rigorous physical derivations or focusing solely on the mathematics—this line of research could advance to a more substantial contribution. In its current form, the preprint offers a promising foundation rather than a completed advance.

Would it benefit from language editing?

Yes

The manuscript would benefit from language editing. While the mathematical sections are generally clear and the proofs are logically structured, the physical discussion in Section 4 contains vague phrasing and undefined terminology that hinder comprehension. Expressions such as 'the fundamental effect of time in Universe at each scale is kind of the above example' and 'this model might be more appropriate at those scales' are imprecise and weaken the scientific rigor of the arguments. Additionally, some sentences are awkwardly constructed, and there are minor grammatical errors throughout. The introduction and abstract could be rewritten for greater clarity and impact. Professional language editing would help ensure that the mathematical contributions are communicated with the precision they deserve, and that the physical interpretations are stated unambiguously.

Would you recommend this preprint to others?

Yes, but it needs to be improved

I would recommend this preprint to others, particularly mathematicians and theoretical physicists interested in harmonic analysis and its potential applications, but with the caveat that it needs improvement before it can be considered a fully developed contribution. The strengths of the preprint lie in its mathematical core: Theorem 1.1 and the construction of generalized harmonic functions are original, properly proven, and worthy of attention. Researchers working on function approximation, harmonic analysis, or alternative mathematical frameworks for quantum mechanics may find valuable insights here. However, readers should be aware of the preprint's limitations. The physical claims in Section 4 are not adequately supported by the mathematical framework or by experimental data. The manuscript lacks a conclusion, a discussion of limitations, and proper references to the existing literature. These shortcomings should be addressed before the work can be considered complete. With revisions—particularly by either strengthening the physical derivations or focusing the manuscript on its mathematical contributions—this work could become a solid addition to the literature. In its current form, it represents a promising but incomplete piece of research.

Is it ready for attention from an editor, publisher or broader audience?

No, it needs a major revision

Competing interests

The author declares that they have no competing interests.

Use of Artificial Intelligence (AI)

The author declares that they did not use generative AI to come up with new ideas for their review.