Space-Time from the Perspective of Feynman Graphon Models
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsI have attached the results from my reviewer.
Comments for author File: 
Comments.pdf
Author Response
Responses to Reviewer 1
Dear Reviewer,
I would like to thank because of your constructive comments. In general, Introduction and Section 3 are revised / reorganized and a new section is added. Here I address a point-by-point changes in the paper according to your comments in the report.
1- In Response to Clarifying foundational definitions, Definitions of qft-state, entanglement dept, spin foam of graphons, the notion of universality and the construction process of spin foams, New paragraphs are added and Section 3 is reorganized. In this regard, please check: Page 6--7: Lines 241–290, Page 8: Lines 336--346, Page 19: Lines 759 – 777, Page 22: Lines 849 – 866, Pages 25--30: Lines 942–1063, Page 34: Corollary 4, Page 39: Corollary 6.
2- In Response to Strengthen Mathematical Rigor in Spin Foam Construction, The concepts of spin networks, spin foams of Dyson--Schwinger equations, and explanation of how dynamic works at the level of spin foams of stretched Feynman graphons are step by step are defined and discussed in the language of cells. In this regard, new paragraphs are added to Pages 25–30: Lines 942–1063. In addition, transition amplitudes at the level of stretched Feynman graphons is governed by their corresponding amplitudes of Feynman diagrams and Green’s functions. Convergence at the level of these amplitudes have already been discussed in my previous published paper “subsystems via quantum motions” and “renormalization bi-Heyting algebra”. These papers are in the Reference List.
3- In Response to Improve Connection to Established Physics, some new paragraphs are added to the paper to address some connections to the established physics, algebraic QFT, General Relativity, Standard Model of particles, UV finiteness and challenges of this new platform. In this regard, please check Pages 5–7: Lines 197–290, Pages 40--42: Lines 1346—1431.
4- In Response to Refine the Universal Model Claim, the suggestion about the phrase universal model has been revised in the text.
5- In Response to Enhance Logical Flow and Motivation, Introduction and Section 3 are reorganized according to the suggested steps to address better the construction process. In addition, a new section 5 is added to compare this new framework with other theories.
6- New papers 61,62,63,64,65,66,67,68,69,71,72 are added to the Reference List in the support of the above changes.
Reviewer 2 Report
Comments and Suggestions for AuthorsSuggestions for Authors in attached file!
Comments for author File: Comments.pdf
Author Response
Responses to Reviewer 2
Dear Reviewer,
I would like to thank for spending time to write a report about my paper. It is important to me that you, as an expert, acknowledged the novelty of the idea used in the paper, and the correctness of the mathematical skeleton of the paper. However, I disagree with your opinion about physical motivation / background / technical details of the paper, and I think many of observations / descriptions that you mentioned in the report are irrelevant to the topic of the paper and some of them are not correct. Here I try to explain these cases point-by-point which I have found incorrect / irrelevant in your report and also, I have made some clarification in the paper.
1- The conceptual gap, which is mentioned in the report, is not correct. Indeed, this new framework of spin foams of DSEs / Feynman graphons constructively build space-time via non-trivial correlations encoded by solutions of combinatorial Dyson—Schwinger equations in a particular class of combinatorial Hopf algebras which are represented by a completely abstract topological Hopf algebra on some pure combinatorial objects. Here I address the procedure. (i) “space-time=correlations” is asserted. This new theory of spin networks and spin foams asserts the structure of space-time when there is a given background depended physical theory, such as a gauge field theory, and we have the opportunity to analyze/describe space-time from the perspective of this physical theory. At this level, theoretically, space-time has already been fixed to define this physical theory and therefore we need to apply a version of our spin foam models decorated by combinatorial Dyson--Schwinger equations of the physical theory and their Feynman graphon representations. These decorated spin foams provide a discrete model for the Banach manifold of quantum motions of the physical theory which leads us to achieve a dynamical model of space-time. This procedure is explained with full details in Section 3. (ii) “space-time=correlations” is derived. Space-time is derived from this new theory of spin networks and spin foams when we lift our construction program onto a universal setting formulated on the basis of non-decorated spin foams of (stretched) graphons governed by an abstract combinatorial topological Hopf algebra. At this level, no physical theory is pre-assumed and space-time is built in terms of spin networks and spin foams of a certain class of non-trivial correlations extracted from (stretched) graphon representations of solutions of a particular class of recursive Hochschild equations. This procedure is explained with full details in Section 4. So the paper addresses also the opposite of the direction that you addressed as the conceptual gap.
In this regard, new paragraphs are added in Pages 5–7, Lines 197–290, Pages 40--42: Lines 1346–1431 to clarify the above misunderstanding.
2- It is addressed in the report that “No explicit reconstruction theorem is provided … the connection between the combinatorial constructions and physical spacetime remains philosophical rather than demonstrably structural”.
They are not correct observations and here I explain why. First of all, sections 3 and 4 present a step by step construction program for space-time via non-trivial correlations extracted from Feynman graphon models of DSEs. Secondly, Theorem 5 delivers a new dynamical model for space-time. Corollary 4 reconstructs “zero path length problem” and offers a solution for this issue by linking it to my recent work “zero charge problem via graphon processes”. Thirdly, the Connes--Kreimer Hopf algebraic renormalization has already been developed to abelian and non-abelian gauge field theories where quantum gauge symmetries are encoded by a certain class of Hopf ideals. This Hopf algebraic framework coherently works for Standard Model of particles minimally coupled with Einstein--Hilbert action. In other words, the Connes--Kreimer renormalization Hopf algebra and its topological enriched version are tied to the physical space-time. Therefore decorated versions of our new spin foam models, formulated in Section 3, are tied to physical space-time.
In this regard, some new paragraphs are added in Pages 5--8: Lines 197—345 and also, the section 3 is reorganized with additional parts to clarify the construction process and relating physical space-time to this new framework. New Section 5 is added to compare this new framework with other cases.
3- It is addressed in the report that “The manuscript advances strong interpretive claims such as associating strong coupling with continuous micro-spacetime, weak coupling with discrete micro-spacetime, and identifying the Landau pole as the boundary between them without supporting derivations, explicit toy-model calculations, or connections to established frameworks (e.g., lattice gauge theory, asymptotic safety, holography), rendering these assertions largely speculative.”
They are not correct observations and here I explain why. First of all, interpretation is a way of linking formal objectivity, formulated by mathematical structures and models, to real objectivity, achieved by empirical knowledge under stances of realism, anti-realism and neo-realism. Fundamental elements of mathematical and theoretical physics, such as wave functions, space-time, Hilbert space formalism, creation and annihilation operators, etc. …, are all based on interpretations. They are not real physical entities. String Theory and Quantum Gravity are strongly depended on interpretations without any experimental data. So your objection about interpretation has no point at this level of formal objectivity. Secondly, there is no global accepted definition or golden rule to recognize which abstract objects correspond to physical entity, and which mathematical model correspond to real physics. At quantum scales, the only confirmed mathematical models that are supported by experiments are Quantum Mechanics, perturbative QED and perturbative QCD. The rest of models are knowledge at the level of formal objectivity. On the contrary of your opinion, the strength of this paper is to bring a new link from this new formal understanding of the fabric of space-time to real objectivity of space-time hidden in the concept of “zero charge problem” in physics of high energy. In my recent published paper about “zero charge problem”, I developed a mathematical machinery to discuss triviality of QED in terms of a new class of discrete Markov chains and this is indeed the conceptual motivation of the present paper. In short, relating landau pole to discrete-continuous nature of the fabric of space-time is the result of resolving zero-charge-problem in terms of stochastic models and this result is applied in this work to analyze the fabric of space-time.
In this regard, new paragraphs are added in Pages 40–42: Lines 1346--1431
4- It is addressed in the report that “key claims (such as defining spin-foam volume, formulating a path integral over spin-foam trajectories, and interpreting mass/energy as deformations of spin foams) remain mathematically underdeveloped, since no explicit measure, action functional, or amplitude assignment is provided to support these physics-loaded statements.”
They are not correct observations. Proofs of Theorem 5 and Corollaries 4 and 6, and also Remark 6 provide answers to many of those concepts. It is important to pay attention that everything is lifted onto the Banach manifold structure on the space of quantum motions underlying the space of built spin foams and path integral is explained in the text at this level. The process of amplitude assignment to Feynman graphon models has already been discussed in my previous published works which I addressed them in the Reference List and actually, it is completely out of the scope of the present work.
In this regard, the section 3 is reorganized with additional parts to clarify the construction process. Furthermore, new paragraphs are added in Page 33: Lines 1125–1127, Page 34: Corollary 4, Page 39: Corollary 6, Pages 40--42: Lines 1346--1431.
5- It is addressed in the report that “The manuscript does not modify this solution theory. It does not: • alter the fixed-point structure of cDSEs, • introduce a new renormalization scheme, • provide a new combinatorial equation, or • change the uniqueness properties of perturbative solutions.”
Actually, almost all these points have been considered in my previous published works which I addressed some of them in Reference List. In addition, these topics are out of the scope of the present paper.
6- It is addressed in the report that “Rather, the novelty consists in reinterpreting Hopf subalgebras generated by cDSE solutions as vertices of combinatorial graphs decorated by Lie group representations. The “new theory” of spin networks and spin foams (introduced via Definitions 4–7 in Section 3.2) amounts to additional combinatorial constructions layered on top of the standard Connes–Kreimer framework. No new dynamical principle or amplitude prescription is supplied. In this sense, the claim of introducing a “new theory” appears overstated.
First of all Section 3 is reorganized to present better the mathematics behind this new theory of spin foams. Secondly, Theorems 5 and 6 together with Corollaries 4 and 6 explain this dynamical model. Actually, this new theory of spin networks and foams is not only formulated by additional combinatorial data on top of the standard CK framework. Topological enrichment of CK framework + formulating Banach manifold structure on the space of quantum motions are building blocks of this new theory.
7- It is addressed in the report that “The phrase “construction program” is essentially rhetorical: it refers to a sequence of definitions and embeddings, not to a constructive derivation yielding a unique spacetime geometry with measurable physical predictions.”
They are not correct observations. First of all, the logical background of Feynman graphon approach to quantum field theory is provided by Heyting algebras which are not Boolean. Therefore the mathematical foundation of quantum field theory based on topological hopf algebra of renormalization is constructive. This topic has already been explained in my previous publications and I addressed them in Reference List. The structure of this new class of spin networks and foams does not need axiom of choice in general. If we work on Feynman graphons defined on the Lebesgue measure space $\mathbb{R}$, we need axiom of choice but it is easily possible to replace Lebesgue measure space $\mathbb{R}$ with another $\sigma$-finite algebra to proceed the construction process without applying full version of Axiom of choice. I used stretched Feynman graphons in the whole text to address this possibility. However for the simplicity of explanation to readers I made a choice to write the platform by using real numbers. Secondly, there are only few number of definitions in this 40+ pages paper and almost all elements used in those definitions have already been constructed in the text or in the previous works.
In this regard, new paragraphs are added in Pages 5–8: Lines 197–289, and the section 3 is reorganized with additional parts to clarify the construction process
8- It is addressed in the report that “However, no quantitative entanglement measure is derived, and no reconstruction of causal or metric structure from such correlations is demonstrated; thus the role of “entanglement” remains interpretive rather than physically established.”
They are not correct observations. First of all, quantum entanglement, in the language of towers of combinatorial DSEs provide evolution of spin networks in this new model and generating spin foams. Please check details of the construction program given in Section 3. Secondly, the metric structure that you are addressing belongs to formal objectivity of General Relativity and here everything is lifted on the Banach manifold structure on the space of all quantum motions of the physical theory. Figure 14 summarizes this machinery. In addition, the Connes—Kreimer Hopf algebraic renormalization is valid for the standard model of particles minimally coupled with gravity (Einstein—Hilbert action). Therefore the spin foam model built on the basis of topological Hopf algebra of renormalization and related combinatorial constructions are linked to physical space-time. Thirdly, the objection about interpretive rather than physically established has no point here, because quantum entanglement at the level of quantum mechanics and Standard Model of particles, as perturbative QFT, are only real objectivities about physics at high energy derived from our formal objectivity knowledge. Quantum entanglement at non-perturbative regimes are still at the level of formal objectivity which report the possibility of the existence of some real objectivity such as confinement and triviality of physical theories.
In this regard, Section 3 is reorganized / revised and new paragraphs are added in Pages 40--42: Lines 1346--1432.
9- It is addressed in the report that “it lacks a substantive comparison with major emergent-spacetime frameworks (tensor networks/MERA, AdS/CFT entanglement wedge reconstruction, causal sets, group field theory, and random graph/network quantum geometry), which is essential given its claim of a “universal graphon model” of spacetime.”
The most important conceptual point that this new framework brings rather than other frameworks is linking “zero charge problem”, which encodes triviality of QED, to discrete-continuous behavior of the fabric of space-time. This case has already been addressed in the text. In addition, section 3 is about space-time from the perspective of this new spin foam model and its consequences. Section 4 is about an abstraction of this spin foam model.
In this regard, new paragraphs are added in Pages 5–8: Lines 197–346, Page 33: Lines 1125–1127, Page 39: Corollary 6. In addition, new Section 5 is added for more clarification and comparison with other models. However, full details of comparison is out of the scope of the paper.
10- It is addressed in your report that “The manuscript relies heavily on the author’s prior work for several central results, which may raise concerns of fragility and reproducibility unless the key arguments and proofs are restated in sufficient detail within the present paper”
Yes, I am working more than 15 years to prepare the building blocks of this new model and this is a theoretical research task. These issues of fragility and reproducibility exist also for several other models such as quantum gravity and string theory which are developed without any rigorous experimental data. They proceed because Mathematics works to build advanced formal objectivities. Taking time for new ideas to be introduced, developed and popularized in theoretical science is a usual situation.
11- It is addressed in your report about “publication-ready contribution to mathematical physics”.
As I tried to explain in my responses, three distinct concepts “objective reality”, “real objectivity” and “formal objectivity” have been mixed in your comments. Physical theories beyond quantum mechanics, such as quantum field theories, are not realistic theories and neo-realistic approaches are needed to apply these theories for the description of the physics at high energy scales. Mathematical physics is not physics. When you are asking about giving physical evidence / investigation in the support of a new candidate model for the fabric of space-time, it means that objective reality and real objectivity are fully overlapped in your question which is not technically and mathematically correct.
12- New papers 61,62,63,64,65,66,67,68,69,71,72 are added to the Reference List in the support of the above changes.
Reviewer 3 Report
Comments and Suggestions for AuthorsPaper Summary:
The paper develops a formal framework in which space-time is modeled using renormalization Hopf algebras and Feynman graphons. Non-perturbative structures in quantum field theory are represented as infinite sums of “stretched” Feynman graphons, organized by a topologically enriched Connes–Kreimer Hopf algebra and solutions of combinatorial Dyson–Schwinger equations.
On this basis, the author proposes to replace local von Neumann algebras A(O) by suitable topological Hopf subalgebras attached to regions of space-time, and builds a new version of spin networks and spin foams in the space of DSE solutions. Finally, the construction is lifted to a universal setting of real-valued stretched graphons on [0,∞), yielding a “graphon–Hopf” model of the fabric of space-time in which the micro-scale appears discrete in the perturbative regime and continuous in the non-perturbative regime, with the QCD Landau pole acting as the interface between these two behaviors.
Results:
Section 2
This section reviews the path-integral/QFT setup, BPHZ renormalization, the Connes–Kreimer Hopf algebra of Feynman graphs, Dyson–Schwinger equations, and basic graphon theory. The material appears standard and is technically consistent, but the exposition is quite dense and could likely be streamlined with more explicit pointers to the literature. The main new ingredient seems to be the use of "stretched" Feynman graphons on a σ-finite measure space, which allows the author to accommodate infinite combinatorics and non-perturbative Dyson–Schwinger towers within a single Banach/graphon framework. This is a reasonable mathematical choice, but in later sections most constructions rely primarily on having a suitable graphon Banach space with a cut metric, rather than on any particularly distinctive property of the “stretched” setting. It may help the reader if the specific conceptual or technical advantages of stretching, beyond infinite containment, are stated more explicitly in one place.
Section 3
This section builds on this framework to introduce qft-states, a notion of “entanglement depth,” and a family of spin-network and spin-foam–type structures, including their localization to bounded regions of space-time. These constructions are mathematically coherent, but many of the key statements feel primarily definitional once the underlying machinery is fixed. For example, the invariance of entanglement depth under graphon equivalence, and the existence of localized spin foams on , follow quite directly from how these quantities and subspaces are defined. The idea of replacing local von Neumann algebras by appropriate topological Hopf subalgebras is interesting, but at present it is more stated than developed into a precise structural theorem or worked example. Clarifying a single central payoff in this section, for instance, one concrete result that cannot be formulated as cleanly in the standard language, would help the reader see more clearly what is gained by this reformulation.
Section 4
Section 4 hoists the preceding, theory-dependent constructions into a universal shuffle Hopf algebra of stretched graphon “words” and reconstructs the associated spin-foam/metric structure in that setting. This provides a natural unifying framework: different quantum field theories, with their own renormalization Hopf algebras and Dyson–Schwinger solutions, can in principle be represented within a single ambient graphon/foam space. I view this as the main mathematical contribution of the final section.
However, the manuscript does not yet fully demonstrate how this universal framework is used in practice. The discrete/continuous fabric and zero–path-length considerations appear to be restatements of earlier, theory-specific results at a higher level of abstraction, and there are no explicit examples comparing two distinct theories or deriving a new classification or constraint that genuinely requires the universal space. While I find the idea very interesting, it would strengthen this section to either (a) illustrate at least one concrete application of this projection/unification mechanism, or (b) more clearly position the universal construction as an organizational device, with specific expectations about how it might be used in future work.
While the prose is demanding to read, I do see potential in this universal space: it could become a useful “common ground” for translations and correlations between different quantum field theoretic models.
Author Response
Responses to Reviewer 3
Dear Reviewer,
I would like to thank because of your pointed suggestions regarding improving the mathematical background studies and addressing advantages of this working platforms. In general, Introduction and Section 3 are revised / reorganized and a new section is added. Here I address a point-by-point changes in the paper according to your comments in the report.
1- In Response to “It may help the reader if the specific conceptual or technical advantages of stretching, beyond infinite containment, are stated more explicitly in one place”, New paragraphs are added to clarify the nature of stretching and its importance. In this regard, please check Pages 11--12: Lines 469—505, Page 14: Lines 577—590.
2- In Response to “Clarifying a single central payoff in this section, for instance, one concrete result that cannot be formulated as cleanly in the standard language, would help the reader see more clearly what is gained by this reformulation”, New paragraphs are added to clarify advantages of this new setting rather than standard languages. In this regard, please check Pages 5—7: Lines 197—290, Pages 40–42: Lines 1345–1431.
3- In Response to “the manuscript does not yet fully demonstrate how this universal framework is used in practice”, New paragraphs are added to show the mechanism of applying universal framework. In this regard, please check Pages 5--8: Lines 197—346, Pages 39--40: Corollary 6 and Remark 7, Pages 40--42: Lines 1345–1431.
4- In Response to “there are no explicit examples comparing two distinct theories or deriving a new classification or constraint that genuinely requires the universal space”, New paragraphs are added to compare this new theory of spin foams with mainstream theories. In this regard, please check Pages 5--8: Lines 197—346, Pages 39--40: Corollary 6 and Remark 7, Pages 40--42: Lines 1345–1431.
5- In Response to “it would strengthen this section to either (a) illustrate at least one concrete application of this projection/unification mechanism, or (b) more clearly position the universal construction as an organizational device, with specific expectations about how it might be used in future work”, in this regard conceptual advantages / applications of this new spin foam model, comparing it with other theories and its universal level are addressed by adding new paragraphs. In this regard, new paragraphs are added in Pages 5--8: Lines 197—346, Pages 39--40: Corollary 6 and Remark 7, Pages 40--42: Lines 1345–1431.
6- New papers 61,62,63,64,65,66,67,68,69,71,72 are added to the Reference List in the support of the above changes.
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsFor corrections to this article, please refer to the attachment.
Comments for author File: Comments.pdf
Author Response
Dear Reviewer,
Thank you for your suggestion. Here is a short list of point-by-point responses.
1- In Response to Verify Mathematical Rigor of Hopf Operations under Topological Completion: This topic is fully discussed in my previous publications whose addressed in the Reference List. They are open access work available easily. In short, Hopf algebraic operators are bounded linear operators between Banach spaces which is equivalent to their continuity. This case together with the compatibility of the algebraic structure with topological enrichment is fully presented in the open access papers:
[36] A. Shojaei-Fard, The dynamics of non-perturbative phases via Banach bundles, Nuclear Phys. B 969(2021), paper no. 115478, 39 pp.
[69] A. Shojaei-Fard, Renormalization of unbounded stretched graphons, Open J. Math. Analys.9(2), 193–213, 2025.
Since repeating these cases increase similarity rate of the paper, therefore in my opinion no more clarification / modification is needed in the text.
2- In Response to Explicit Mapping to Physical Observables, these new path integrals are defined on the space of graphon representations of 1PI Green's functions or their fixed point equations. This approach replaces working on a single Green's function by working on a geometric space of Green's functions of the physical theory. The required measure for this new path integral is provided by Borel measure derived from the metric structure introduced in Theorem 5. Section 2 starts with these N-point correlation functions to clarify the procedure of passing from standard formalism of QFT to this Graphon approach to QFT.
Therefore in my opinion no more clarification / modification is needed in this regard.
3- In Response to Preservation of Gauge Symmetries in the Graphon Limit: This topic is fully discussed in my previous publications whose addressed in Reference List. One of them is open access and the preprint of other is available in my ResearchgGate profile. Please check
[34] A. Shojaei-Fard, Non-perturbative graph languages, halting problem and complexity, Forum Math. 34(5), 1159–1185, 2022.
[36] A. Shojaei-Fard, The dynamics of non-perturbative phases via Banach bundles, Nuclear Phys. B 969(2021), paper no. 115478, 39 pp.
Since repeating these cases increase similarity rate of the paper, therefore in my opinion no more clarification / modification is needed in this regard.
4- In Response to Computational Tractability and Approximation Schemes: The dimension of Banach space $\mathcal{S}_{\approax}^{\Phi}$ is determined by Feynman graphons $W_{G^{r}}$ for all amplitudes $r$ of the physical theory. In addition, the geometry of this particular Banach space and approximation capabilities of its geodesics are discussed in the following publication
[36] A. Shojaei-Fard, The dynamics of non-perturbative phases via Banach bundles, Nuclear Phys. B 969(2021), paper no. 115478, 39 pp.
Since repeating these cases increase similarity rate of the paper, therefore in my opinion no more clarification / modification is needed in this regard.
5- In Response to Quantitative Benchmarking with Mainstream Approaches, Proof of Corollary 5, Remark 6 and Remark 7 provide quantitative comparisons or specific test cases (behavior of coupling constants near the Landau pole) against established mainstream frameworks. In addition, Corollary 4 and Corollary 6 address new path integral setting at the level of lengths shorter than the Planck scale independent of T-duality which is another quantitative comparison against established mainstream frameworks.
Therefore in my opinion no more clarification / modification is needed in this regard.
Reviewer 2 Report
Comments and Suggestions for Authorssee the comments attached
Comments for author File: Comments.pdf
Author Response
Dear Reviewer,
Thank you for your comments, however, your points about rigorous reconstruction of the metric has fully explained by proof of Lemma 2, proof of Theorem 3, proof of Theorem 5, proof of Corollary 4 and proof of Corollary 5. In addition, one of the most important advantages of this new framework than other mainstream frameworks is offering a new solution to the zero path length problem independent of T-duality, please check Corollary 4 and Corollary 6.
Therefore in my opinion no more clarification / modification is needed in this regard.
