Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper tries to start from a very simple and standard probabilistic setting，and then build up “chronon layers” ChN​, finite-dimensional “slot” Hilbert spaces, and permutation representations. From this, the author claims to see the appearance of SU(3), SU(2), U(1) and Standard-Model–like structures.

Overall I think the idea is original and interesting. The construction looks new to me, and the way it connects probability theory, representation theory and particle-physics–type structures is quite unusual. Some physical interpretation is still speculative, but I think the work is worth publishing after some clarification. So my recommendation is: accept after minor–to–moderate revision.

Below some more concrete comments.

1. The basic probabilistic setup should be written more clearly. It is relatively new  to ones who are major in physics. For example, the definition of the Ch_N is neccessary.
2. Explain better what the finite-dimensional "slot spaces" really are.  I think one simple toy example would help a lot here, even in a short appendix.
3. Separate clearly: what is theorem, what is modeling choice, what is physical guess.

Author Response

Comment 1: The basic probabilistic setup should be written more clearly. It is relatively new  to ones who are major in physics. For example, the definition of the Ch_N is neccessary.
Response.
We agree that accessibility for a physics readership can be improved. The rigorous definition of the layers is already provided in Appendix B via the factor map pi_N​, induced (B_N, mu_N), and the stroboscopic dynamics (Appendix B, “Temporal layers”). However, we accept that the reader should not have to “hunt” for the definition. We will add a short boxed/inline definition of Ch_N​ early in Sec. II, with a clear pointer to Appendix B for the formal measure-theoretic details.

Comment 2: Explain better what the finite-dimensional "slot spaces" really are.  I think one simple toy example would help a lot here, even in a short appendix.
Response.
We agree a short illustrative example will help substantially. We emphasize that H_N \simeq \mathbb C^N is an internal slot space for one Ch_N​ cell, used to represent slot relabelings and to constrain the effective stroboscopic operator U_N​; it is not the full QFT Hilbert space. This is already stated at the level of definitions in Appendix B, but is easy to misread without an explicit worked illustration. We add a half-page toy example at the end of Appendix B showing slot basis {∣1⟩,…,∣N⟩}, permutation representation, commutant constraint on H_{\mathrm{eff}}^{(N)}​, continuous symmetry emerging as unitary freedom on degenerate subspaces.

Comment 3: Separate clearly: what is theorem, what is modeling choice, what is physical guess.
Response.
We agree, and we can do this without restructuring the manuscript. At present, definitions and algebraic results are mixed with interpretive comments (especially in later sections). We will add a short “Scope/status” paragraph in the Introduction and insert light labels in key places (“Assumption,” “Proposition,” “Interpretation”), keeping the text otherwise unchanged.

Reviewer 2 Report

Comments and Suggestions for Authors

 In this paper, authors propose an interesting physical model related to the emergence of "discrete" layers of time in theories where macroscopic time is a "continuous" parameter. In this model, the symmetry groups U(1), SU(2), SU(3), and electroweak interactions naturally arise in the corresponding discrete layers of time.

In addition, a number of other interesting results have been obtained within this model, including the emergence of Hypercharge, Weak Isospin, Electric Charge, Anomaly Cancellation, Higgs, and Neutrino Sectors.

This model is mathematically correct, but I have two significant questions that need to be answered.

1) The phrase "quantum gravity" appears in the introduction and keywords. However, the content of the paper has nothing to do with quantum gravity, except for its mention in the last paragraph on page 5.

This needs to be clarified.

2) What new testable predictions does this model provide compared to existing results?

 

This  paper can be published once  these questions are answered.

Author Response

Question 1: The phrase "quantum gravity" appears in the introduction and keywords. However, the content of the paper has nothing to do with quantum gravity, except for its mention in the last paragraph on page 5.

This needs to be clarified.

Response.
We agree the current framing can be read as claiming a quantum-gravity model in the narrow sense. This is not intended: the manuscript is primarily a discrete-time/chronon kinematical construction, with only brief qualitative remarks on gravity (Sec. IX) and no UV-complete quantization of GR. We will add one explicit sentence in the Introduction clarifying scope, and we will replace or qualify the keyword “quantum gravity” (unless the editor requests otherwise) with more precise terms (discrete time / chronon layering / emergent gauge symmetry).

Question 2: What new testable predictions does this model provide compared to existing results?

Response.
We emphasize that the robust output of the present paper is structural: the internal symmetry algebra and charge/anomaly pattern emerge from the Ch12Ch_{12}Ch12​ slot organization. Quantitative predictions (masses, mixings, relic abundances, cosmological evolution) require specifying the microscopic update rule and mixing terms - explicitly listed by us as open problems. That said, the framework suggests in-principle falsifiable directions, contingent on dynamics: possible Floquet-type periodic signatures tied to tau_{12}​ (qualitative in this paper), existence of SM-neutral gravitating excitations from misaligned layers (qualitative), robust structural “selection rule”: the minimal Ch_{12}​ mechanism yields exactly three inequivalent generation classes (a fourth SM-like chiral generation would falsify this minimal Ch_{12}​ picture). We will add a short “Phenomenological outlook / falsifiability” paragraph in the Discussion clarifying what is predicted structurally now vs what requires dynamical completion.

Reviewer 3 Report

Comments and Suggestions for Authors

File is being attached 

Comments for author File: Comments.pdf

Author Response

Question 1: Regarding the two descriptions of time how precisely mathematical relationship between the 
two could be established that leads to the twelve-layered ChN? 

Response.
The relationship is implemented in two steps: 1) Layering (rigorous): Ch_N​ is defined as a factor description obtained by the block map pi_N​ and induced (B_N, mu_N, T^N) (Appendix B); 2) Matching (interpretive/compatibility): chronon updates are embedded into macroscopic spacetime with endpoints localized near t≈tau_1​ (Sec. II / Appendix B), ensuring consistency of discrete update order with macroscopic time ordering. 
We will add 1–2 sentences in Sec. II explicitly connecting “factor definition” to “matching” to prevent scope confusion.

Question 2: What exactly the temporal layers ChN stand for? Do they specify Hilbert subspaces, 
dynamical orbits or representation sectors etc. and how do algebraic operators apply? 

Response.
Formally, Ch_N​ is the coarse-grained factor system of the base dynamics. The associated H_N≃\mathbb C^N​ is an internal slot space for one layer cell, used for representation/commutant analysis of the effective stroboscopic operator. We add an explicit sentence stating “Ch_N​ is a factor description; H_N​ is an internal slot representation space, not the full QFT Hilbert space.”

Question 3: The authors state that temporal layers ChN naturally support the symmetries U(1), SU(2) and 
SU(3). Are these symmetries located by matching commutant structure and dimensionality or 
layer structure and choron dynamics, etc. Could the authors explicate the mechanism in more 
details?

Response.
They arise from slot-permutation invariances of the coarse-grained statistics and the resulting commutant constraints on H_{\mathrm{eff}}^{(N)}​. Continuous unitary symmetry appears as the symmetry of degenerate/locked sectors allowed by these constraints (and, in maximally isotropic cases, by symmetry enhancement). We add the Appendix B toy example explicitly illustrating the mechanism.

Question 4: It is demonstrated that ChN12 gives rise to three triads and three quartets leading to $\mathbb{C}^3 \otimes \mathbb{C}^4$. How are these embeddings of three triads and three quartets enforced? Additionally, is $\mathbb{C}^{12} \cong \mathbb{C}^3 \otimes \mathbb{C}^4$ unique or chosen comparatively?

Answer: Please see the attachment.

 

Question 5: Does the regaining of $SU(3)_c \times SU(2)_L \times U(1)_Y$ via $\mathbb{C}^3 \otimes \mathbb{C}^4$ unique or gives rise to rather larger symmetry group such as $SU(4)$ etc.?

Answer: Please see the attachment.

Question 6: Does the pattern of hypercharges and anomaly cancellation stem from ChN themselves or the 
standard model constraints afterward? Is the anomaly cancellation explicitly considered in the 
model?

Response.
In our construction, Y and T_3​ are defined as the linear invariants compatible with the triad/quartet structure on Ch_{12}​; electric charge follows as Q=T_3+Y. Anomaly cancellation is then checked as an explicit identity in this assignment (Sec. VII). We will add one sentence emphasizing that anomaly sums are computed from the model-defined assignments, not imposed externally.

 

Question 7: Authors interpret three fermion generations in terms of inequivalent embeddings of a triad into 
dodecad which preserve the quartet structure. What is the basis of this interpretation Could there 
be any other predictions? 

Response.
The basis is combinatorial: inequivalent triad embeddings in the 12-cycle that preserve quartet structure form exactly three equivalence classes (Sec. VII). Other predictions are currently structural rather than numerical; quantitative predictions require a dynamical completion, explicitly listed as open problems, approaches for which wil be presented in next papers. We add a short “what is robust now vs what is future work” sentence in the Discussion.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Review report on the paper titled "Complementary Continuous-Discrete Time, Chronon Layering and Temporal Folding" by M. E. Abishev and D. Z. Berkimbayev.

In this paper the authors have considered a discrete-time chronon model's approach, including a dual-time and a hierarchy of temporal layers denoted as ChN. The paper  studies some specific layers, as Ch2, Ch3 and Ch4, observing a compatibility with U(1), SU(3) and pair-locked SU(2) symmetry. In the case of Ch12 the intersection of the  commutants are compatible with the standard model algebra. 

The current paper is problematic due to the following aspects:

• the  existence of a dual-time scheme as the one used in the present paper; i don't see any reason to include a dual-time;
• the theory possess a scale-dependent Newton constant which is quite problematic;
• the frequency dependent corrections to gravitational wave propagation are not analyzed and discussed;
• although for the standard model the approach is compatible, some specific extensions of the standard model are not discussed;
• i don't see how the misaligned layers can play the role of dark sectors in the cosmological theory, and how this would lead to an accelerated expansion; also, how this would be related to inflation;
• the ultraviolet completion of the theory is not present which is a crucial aspect to be taken into consideration;
• also, how the "misaligned" layers would be related to dark matter particles? what candidates for such particles are viable in this scenario? 

Author Response

Please see the attachment for expanded responses.

Comment 1: the  existence of a dual-time scheme as the one used in the present paper; i don't see any reason to include a dual-time;

Response.
We do not introduce two physical clock times. The “dual-time” language distinguishes macroscopic continuous EFT time t from the microscopic discrete update counter n∈Z used to define stroboscopic layers. This is needed to define Ch_N​ without forcing a rigid lattice on macroscopic time. Add a short clarifying sentence in Sec. II.A.

Comment 2: the theory possess a scale-dependent Newton constant which is quite problematic;

Response.
We agree the current wording can be read too strongly. The gravitational discussion is not used anywhere in deriving the Standard-Model–like algebra/charges; it is purely outlook. We do not claim a derived running G, nor do we compute amplitudes/bounds. We will rephrase the paragraph after Eq. (10) to remove any strong “prediction” tone (including “scale-dependent Newton constant”) and present it as a speculative direction subject to strong constraints.

Comment 3: the frequency dependent corrections to gravitational wave propagation are not analyzed and discussed;

Response.
Correct: we do not perform a quantitative GW propagation analysis. This would require specifying the microdynamics and estimating the modulation amplitude. Add one sentence noting what would need to be computed and that it is beyond the scope of this kinematical paper.

Comment 4: although for the standard model the approach is compatible, some specific extensions of the standard model are not discussed;

Response.
The goal is minimal: show how the SM-like algebra/charges arise from Ch_{12}​ structure. Extensions require additional assumptions (other layers, additional symmetry breaking, extra sectors), which are not fixed by the current kinematics. Add one short paragraph in the Discussion stating that BSM extensions are outside scope and indicating (briefly) what extra input would be required in this framework.

Comment 5: i don't see how the misaligned layers can play the role of dark sectors in the cosmological theory, and how this would lead to an accelerated expansion; also, how this would be related to inflation;

Response.
We do not claim inflation or accelerated expansion. The misaligned-layer remark is a qualitative idea: excitations can be SM-neutral under the Ch_{12}​ algebra but still gravitate. Add one explicit sentence: “This work does not address inflation or dark energy.”

Comment 6: the ultraviolet completion of the theory is not present which is a crucial aspect to be taken into consideration;

Response.
We agree a UV completion is crucial for a full theory; we explicitly list it as an open problem. The main algebraic result does not depend on the UV completion, but a full dynamical account does. Strengthen one sentence in the Discussion making this explicit as a limitation and future task.

Comment 7:  how the "misaligned" layers would be related to dark matter particles? what candidates for such particles are viable in this scenario?  

Response.
At the present stage we do not propose a specific particle candidate with computed mass/couplings. The framework suggests a class of effective SM-singlet (with respect to Ch_{12}​) excitations carrying energy-momentum. Viability (stability, relic abundance) requires a specified microdynamics. Add one sentence clarifying that “misaligned-layer dark matter” is currently an EFT-level class of SM-neutral excitations, not yet a concrete particle model.

Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors

The manuscript "Complementary Continuous-Discrete Time..." presents a chronon model, in which the Authors recreate the gauge group structure of the Standard Model as well as discuss the charge stability, neutrino and Higgs sector, and the impact on the metric of space-time. The idea of an underlying mechanism, which explains the very foundation of physics is attractive and deserves proper attention.

I have some critical remarks, though, which should be clarified before considering this work for publication. The construction of the ChN layer is first described as the introduction of (line 41) "N internal temporal slots". In Ch.3 you construct the layers as "N quanta of action", which transforms into "N-state system"; from the doublets in Ch2 and triplets in Ch3, and pairs in Ch4 you find symmetry groups U(1), SU(2), and SU(3), which you connect with the Standard Model. The question is: what are the quanta of action, what do they represent in terms of physical interpretation? I have a problem linking this construction to the gauge group of the SM, where the gauge group describes the symmetries of the bosonic sector. It is true that we don't know how this symmetry emerged, the GUT theory with SU(5) symmetry breaking being an option, but your observation and conclusions are difficult to understand. Not to mention, that the "N temporal slots" are proven in the Appendix (A20, A31) to have the same sign as the spatial coordinates in the metric, i.e., opposite to the sign of dt_0. 

Please consider and clarify the following points:

• a clear description of the ChN layers with sound physical interpretation
• what do the actions in ChN represent?
• why can it be linked to the gauge structure of the SM?
• are you sure that at each `slot' of ChN new temporal parameters are introduced?
• if new temporal coordinates appear, which time is measured by external macroscopic clocks and which time is the coordinate used for Lorentz transformations?
• is the measured time continuous or discrete and how do you define the continuity limit in the case of discrete time steps?

There is also "causality" mentioned in the paper; you have to be careful with this if you discuss quantum systems, where the time-energy uncertainty principle works, and time localization of the system is made via a measurement. 

The paper, as the Authors admit, is more of an observation than derivation, and without addressing the points above it is difficult to accept the conclusions. I do not recommend publication in its present form.

Author Response

Please see the attachment for expanded responses.

Point 1: Provide a clear description of ChNCh_NChN​ with sound physical interpretation.

Response.
Ch_N​ is rigorously defined as the coarse-grained (stroboscopic) factor description obtained by sampling the base update process every N steps via \pi_N​. Physically, a Ch_N​ “cell” contains N microscopic update slots between two coarse instants. We agree this needs to be stated earlier and more plainly. We add a short inline definition of Ch_N​ in Sec. II and refer to Appendix B for the full measure-theoretic construction.

Point 2: What do the actions in Ch_N​ represent?

Response.
The “quantum of action” postulate is a modeling choice to label discrete microscopic updates by integer action/phase increments (introducing \tau_1​ without a macroscopic time lattice). It is not a claim that this object is a Standard-Model particle or a directly measurable low-energy action quantum.  We add a short paragraph in Sec. II.A clarifying this interpretation.

Point 3: Why can it be linked to the gauge structure of the SM (bosonic sector)?

Response.
We agree the text must separate clearly what is derived: the internal symmetry algebra of the effective theory on Ch_{12}​ (as an intersection of commutants in H12≃C3⊗C4, and the associated charge/anomaly structure; what is not derived here: the full dynamical emergence of gauge bosons and local gauge interactions. Our identification is with the global internal symmetry algebra; the step to local gauge fields is the standard EFT gauging principle and is outside the present paper’s dynamical scope. We add one sentence in the Introduction/Discussion explicitly stating this scope boundary.

Point 4: Are you sure each ‘slot’ introduces new temporal parameters?

Response.
No: we do not introduce multiple physical clock times. Slots label sub-steps within one coarse cell, and H_N\simeq\mathbb C^N is an internal kinematic space. The auxiliary coordinates in Appendix A are not additional physical times; the Signature Theorem makes them space-like, and the text itself rescales them to spatial coordinates x_i​. We add a “Notation” sentence in Appendix A: internal directions are space-like and may be written as x_i​.

 

Point 5: Which time is measured by clocks and which time is used for Lorentz transformations?

Response.
Macroscopic clocks measure the single EFT time (denoted t or t_0​). The Lorentz transformations refer to the emergent effective geometry with one time-like coordinate t_0​ and N space-like coordinates x_i​ (derived in Appendix A). We clarify in Appendix A by renaming/reinterpreting internal directions as x_i​.

 

Point 6: Is measured time continuous or discrete? How is the continuity limit defined?

Response.
Macroscopic time t is continuous by construction (EFT). The microscopic update index nnn is discrete and used only to define stroboscopic coarse-graining. The “continuity limit” is the regime where \tau_1​ is far below experimental resolution, so the stroboscopic dynamics is approximated by an effective Hamiltonian.  We add 1–2 sentences in Sec. II emphasizing this.

 

Point 7: Causality in quantum systems; be careful with time-energy uncertainty.

Response.
Our use of “causality” refers to compatibility between discrete update ordering and macroscopic causal ordering in the Lorentzian embedding/emergent causal cones; it is not a statement about arbitrarily sharp time localization of quantum states or evasion of time–energy uncertainty. We adjust one sentence to avoid any wording that could be interpreted as making claims about measurement-time localization.

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

Comments and Suggestions for Authors

The revised version does not contain any new  computations which can address the previous problems.

Reviewer 5 Report

Comments and Suggestions for Authors

Thank you for the clarifications, they make the presentation much more sound. The updated manuscript can be recommended for publication.