Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript proposes a compelling geometric origin for quantum interference: a finite quantum of action compactifies the classical action manifold into a U(1) phase space, making wave behavior topologically inevitable. This idea is linked to Chronon Field Theory (ChFT), where the action quantum arises as a quantized symplectic flux from stable solitons of a causal field. 

Overall, the core geometric argument is elegant and potentially foundational. However, the physical realization via ChFT lacks sufficient mathematical rigor and explicit derivations. Major revision is required to substantiate the claimed connections between ChFT’s Skyrme-type structure and key physical outcomes: gauge symmetry, gravity, and the precise origin of the action quantum.

My specific review comments:
1) Explicitly derive how the minimal action flux defined as the integral of the causal curvature over a soliton surface corresponds to a unit topological charge (e.g., Skyrme or Hopf number equal to one). Do not assume this equivalence—prove it from the field equations and topology of the soliton solution.
2) Provide a dedicated section showing how the holonomy of the covariant derivative of the causal field yields exactly the Standard Model gauge group SU(3) times SU(2) times U(1). Include at least a sketch of the mechanism by which the field dynamics generate these specific compact gauge symmetries.
3) Clarify why the compactification scale of action is 2 pi times the fundamental action quantum, rather than just the quantum itself. Justify the appearance of the 2 pi factor from the internal dynamics of the chronon field, not merely as a convention to recover standard quantum mechanics.
4) Derrick’s theorem ensures static stability of solitons, but real particles move. Address the dynamical stability of chronon solitons under Lorentz boosts or time evolution. This is essential for the theory to describe physical matter.
5) In the appendix, the identification of the winding number k equals 1 is currently based on "normalization. " Replace this with a derivation from the stated physical assumptions: show that only the principal harmonic satisfies continuity, additivity, and two-path interference symmetry.
6) Detail how coarse-graining the chronon field equations leads to Einstein’s field equations. Explicitly relate the microscopic stiffness parameters of the causal medium to the effective gravitational constant and cosmological constant—do not merely assert the correspondence.
7) Explain how the field equations preserve the unit-norm condition of the causal field and enforce a consistent arrow of time. Is causality dynamically maintained, or is it imposed as an external constraint? Clarify the foundational status of temporal order in the theory.
8) Elaborate on how macroscopic classical actions emerge from the collective behavior of many unit-winding solitons, rather than from single solitons with higher topological charge. Provide a minimal theoretical model of this accumulation process.
9) Quantify the predicted phase discontinuities in mesoscopic interferometers. Give an order-of-magnitude estimate of the size of discrete phase steps (relative to the fundamental action quantum) that experiments like SQUIDs or Josephson junction arrays would need to resolve.
10) Explain the physical mechanism within Chronon Field Theory that drives the classical limit. Specifically, describe how changes in the field’s stiffness parameters lead to the vanishing of the action quantum and the decompactification of action space.

Author Response

Response to Reviewer 1

I thank the reviewer for the constructive and detailed evaluation. In order to provide proper responses to all the important and insightful comments, I find it necessary to significantly expand the paper to include new sections and supporting appendices so that this work can be evaluated as a cohesive framework for quantum foundation. I draw on my existing research notes to make it possible to finish this major revision within the deadline.  All changes mentioned in this response refer to the revised manuscript.

 

Comment 1

Explicitly derive how the minimal action flux defined as the integral of the causal curvature over a soliton surface corresponds to a unit topological charge (e.g., Skyrme or Hopf number equal to one). Do not assume this equivalence—prove it from the field equations and topology of the soliton solution.

 

Response:

A new Appendix M titled “Topological Structure of the Chronon Field” now provides the explicit derivation. I show that the causal curvature two-form defines an integer-valued cohomology class on the compact soliton surface, and that the unit of action corresponds to the fundamental generator of this class. The derivation uses only the field equations and continuity constraints. No assumptions remain. Quote from Appendix M: ”In this appendix I develop the topological underpinnings of the chronon field, establish the configuration space in which finite-action configurations reside, derive the homotopy classification of admissible patterns, and demonstrate that the symplectic flux associated with polarization of the chronon field isquantized. This analysis provides the mathematical foundation for the existence of a fundamental quantum of action $\hbar_{\mathrm{geom}}$ and explains why the relevant winding number is uniquely $k=1$.” A new Appendix: ”Existence and Stabilization of Chronon Solitons” is also added. The two new appendices together establish that a stable minimum sliton is the source of the minimum action.

 

Comment 2

Provide a dedicated section showing how the holonomy of the covariant derivative of the causal field yields exactly the Standard Model gauge group SU(3) times SU(2) times U(1). Include at least a sketch of the mechanism by which the field dynamics generate these specific compact gauge symmetries.

 

Response:

A new Subsection 2.3 in the main text summarizes the mechanism by which the holonomy of the causal-field polarization bundle generates compact internal symmetry subgroups. The full derivation (including the emergence of the SU(3) × SU(2) × U(1) structure) is presented in a new Appendix N: ”Gauge Holonomy and the Standard Model Sector”, with careful explanation of how each factor arises from independent twisting sectors of the polarization connection.

Comment 3

Clarify why the compactification scale of action is 2 pi times the fundamental action quantum, rather than just the quantum itself. Justify the appearance of the 2 pi factor from the internal dynamics of the chronon field, not merely as a convention to recover standard quantum mechanics.

 

Response:

The revised manuscript now explains that the compactification scale arises from the periodicity of the holonomy angle associated with a full rotation in the polarization bundle. A full traversal of the fundamental topological cycle corresponds to a 360-degree rotation of the symplectic phase, leading naturally to the factor of 2\pi. This is now spelled out explicitly in the new Appendix R1:

“Geometric origin of the 2\pi factor:

Although one could formally absorb numerical constants into a redefinition of

\hbar, the compactification scale of the action is not a matter of convention.  

The fundamental periodicity arises from the U(1) twist of the chronon polarization bundle: a full 2pi rotation of the internal phase returns the polarization frame to itself, so the symplectic flux is quantized in units of 2\pi\hbar.  Thus the appearance of 2\pi reflects intrinsic bundle topology rather than a convenient normalization chosen to match standard quantum mechanics.”

 

Comment 4

Derrick’s theorem ensures static stability of solitons, but real particles move. Address the dynamical stability of chronon solitons under Lorentz boosts or time evolution. This is essential for the theory to describe physical matter.

 

Response:

A dedicated new Appendix Q: ”Dynamical Stability of Chronon Solitons” has been added. I show that boosted solitons remain stable under Lorentz boosts and time evolution. In this framework, time evolution is implemented by the RNF reconstruction process.

 

Comment 5

In the appendix, the identification of the winding number k equals 1 is currently based on "normalization. " Replace this with a derivation from the stated physical assumptions: show that only the principal harmonic satisfies continuity, additivity, and two-path interference symmetry.

 

Response:

The new Appendix subsection M.4 “Uniqueness of the fundamental winding number k=1” now provides the explicit constraints for why k=1 from additivity, energetic stability, and phase periodicity considerations.

 

Comment 6

Detail how coarse-graining the chronon field equations leads to Einstein’s field equations. Explicitly relate the microscopic stiffness parameters of the causal medium to the effective gravitational constant and cosmological constant—do not merely assert the correspondence.

 

Response:

The revised manuscript expands this derivation, added a dedicated Appendix section O.2 and a theorem Q.1(emergent gravitation). Appenix O.4 demonstrates the relationship explicitly. The coarse-grained energy-momentum tensor of aligned chronon fields now appears explicitly, and I show how the effective Einstein equations arise from variations in the macroscopic alignment metric. The mapping from stiffness parameters to the gravitational constant and the cosmological constant is now described in a transparent way.

 

Comment 7

Explain how the field equations preserve the unit-norm condition of the causal field and enforce a consistent arrow of time. Is causality dynamically maintained, or is it imposed as an external constraint? Clarify the foundational status of temporal order in the theory.

 

Response:

A new subsection Appendix K.5 clarifies that the unit-norm condition is dynamically preserved due to a built-in Lagrange-multiplier structure in the variational principle. The arrow of time emerges from the Real-Now-Front reconstruction process, where each new slice inherits orientation from the causal-alignment direction. This is now stated explicitly. Causality is emergent. My previously published work (Reference 41) is cited which also independently established the emergence of Lorentzian metric and causal structure in the chronon field framework.

 

Comment 8

Elaborate on how macroscopic classical actions emerge from the collective behavior of many unit-winding solitons, rather than from single solitons with higher topological charge. Provide a minimal theoretical model of this accumulation process

 

Response:

In order to address this very important point, I dedicated a subsection (Appendix M.5) that makes the accumulation mechanism explicit in a simple model. This subsection explains that higher-winding solitons are energetically unstable and naturally break into multiple unit-winding solitons. Macroscopic classical actions therefore arise from the collective contribution of many such unit solitons acting together. The new subsection provides a minimal coarse-grained “soliton-gas’’ model showing how a large, effectively continuous classical action emerges.

 

Comment 9

Quantify the predicted phase discontinuities in mesoscopic interferometers. Give an order-of-magnitude estimate of the size of discrete phase steps (relative to the fundamental action quantum) that experiments like SQUIDs or Josephson junction arrays would need to resolve.

Response:

I agree that it is important to provide quantitative estimates. After re-examining the derivation, I found that the predicted differences (between my new theory’s prediction and that of standard quantum mechanics) in phase discontinuities in mesoscopic interferometers arise only at the chronon scale and are therefore suppressed by over thirty orders of magnitude relative to experimental sensitivity. Such effects are untestable in SQUIDs or any foreseeable mesoscopic interferometry. To avoid suggesting experimentally irrelevant predictions, I removed the earlier phenomenology section.

I have added a new Section 8, “Phenomenology: The Electron as the Minimal Chronon Soliton,” which develops a physically meaningful prediction, namely, identifying the electron as the minimal chronon soliton with an inferred radius of order 10^{-34}meters. I have also added Section 7.8, which derives the exponential suppression of Hawking radiation in this framework, and I conclude the paper with four concrete falsifiability tests at te end of Section 10.

 

Comment 10

Explain the physical mechanism within Chronon Field Theory that drives the classical limit. Specifically, describe how changes in the field’s stiffness parameters lead to the vanishing of the action quantum and the decompactification of action space.

 

Response:

My previous description was misleading and wrong, and the fundamental action quantum \hbar_{geom} is a fixed geometric modulus of the chronon medium that does not change with scale. Its effects only vanish reletively at classical scales.  A new Appendix R: “Classical Limit and Effective Decompactification of Action” has beed added. This appendix explains that when the chronon alignment coherence length becomes small compared to the characteristic scale of the process, the system samples many independent unit-winding contributions. Their collective accumulation renders the modular spacing 2\pi\hbar_{geom} operationally negligible. In this sense, the action space becomes effectively decompactified, recovering smooth classical trajectories even though the underlying action quantum remains fixed. The recovery mechanism is the same as in conventional quantum mechanics.

 

I thank the reviewer again for the detailed, highly constructive comments, giving me an opportunity to systematically present the new framework.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Quantum mechanics is the foundation of modern physics, and research of its methodological foundations is always of great interest. As a matter of fact the author of the reviewed article has presented a new theory of physical reality based on introduction of the concept of a continuous causal field Φ_μ(x). In his work, the author answers the question of where quantization in quantum mechanics comes from, as well as quantum phenomena such as interference and superposition. From a methodological point of view, the author replaces the postulate about the quantum-wave nature of matter with the postulate about the existence of the causal field Φ_μ(x), which produces the necessary quantization of action (5) with subsequent conclusions about the compactification of action space. My critical comments are listed below.

1. A physical meaning of the causal field Φ_μ(x) is not very clear from the work. Is this a new understanding of the structure of spacetime? Could it be a replacement for the Minkowski four-dimensional space? This question needs to be clarified once again. What spin corresponds to the causal field Φ_μ(x) in Lagrangian (2)?
2. How unambiguous is the choice of the Lagrangian in the form (2)? In principle, other types of Lagrangians can also lead to the minimum action quantum (5).
3. Is it possible to obtain the non-relativistic Schrödinger equation from the author's formalism (for example, by adding the Lagrangian of a massive particle to Lagrangian (2) )?
4. The method of the article does not provide a geometric interpretation of the Born rule for probability density as the square of wave function modulus. Although, according to the logic of the article, the dynamics of the causal field Φ_μ(x) should explain how the reduction of the wave packet of a massive quantum particle occurs in the process of quantum measurement.
5. According to the author approach , bosons are coherent oscillations of causal curvature on the solitonic background. But bosons are carriers of physical interactions. What about fermions? They are the basic particles of matter. The author does not write about them at all. Why? How will the interaction of physical fields with the causal field Φ_μ(x) be implemented? Is it possible to determine the parameters, such as masses of elementary particles, which depend on the causal field Φ_μ(x) ? This question makes sense, since other modern geometric quantum theories of spacetime (see, for example, the recent open-access article with DOI: 10.1016/j.nuclphysb.2025.116959) directly allow to calculate the masses of some elementary particles that are carriers of physical interactions.
6. When it comes to experimental confirmation of the proposed theory (section 4.5), the author should specifically calculate the deviation of the experimental results on quantization of magnetic flux from the predictions of traditional quantum theory. Will the difference be significant? We are talking about those very discrete steps or plateaus corresponding to integer multiples of Ñ_geom. This comment is a recommendation for the author's future work.

The article can be published in Quantum Reports after the author provides answers to the above comments 1-5.

Author Response

Response to Reviewer 2

I really appreciate the reviewer’s thoughtful and careful analyses. The comments helped me strengthen the framework significantly.

 

Comment 1

A physical meaning of the causal field Φμ(x) is not very clear from the work. Is this a new understanding of the structure of spacetime? Could it be a replacement for the Minkowski four-dimensional space? This question needs to be clarified once again. What spin corresponds to the causal field Φμ(x) in Lagrangian (2)?

 

Response:

The Introduction and Section 2 have been revised to clarify that the causal field is a pre-geometric field encoding the local orientation of temporal alignment. It is a vector field in 4 dimensions. It does not replace Minkowski space but generates spacetime structure dynamically through alignment. It behaves as a constrained vector-like object but does not correspond to a particle with definite spin. The framework does not need the postulate of a Minkowski spacetime. In this framework, time is an emergent order parameter, therefore not quantisable. Chronons are not time particles as in other versions of chronon field theory.

 

Comment 2

How unambiguous is the choice of the Lagrangian in the form (2)? In principle, other types of Lagrangians can also lead to the minimum action quantum (5).

 

Response:

In the revised and expanded paper, I have added an appendix K: “TCP Mathematical Framework”. The Lagrangian is given by equation (K.5). At the the end of subsection K.5 I have added the following discussion:

“Remarks on the Lagrangian

The functional in (K.5) is not unique; it is a

minimal representative of a broader universality class of chronon

actions that all share three structural features: (i) local alignment

dynamics, (ii) a quantized twist sector, and (iii) quartic stabilization

ensuring finite-energy solitons and a compact action manifold.  

More general Lagrangians may be constructed, but their long-wavelength

limits coincide with those derived here.”

 

Other Lagrangians can produce quantized action sectors, but the chosen one is the minimal structure yielding stable solitons and Real-Now-Front alignment.

 

Comment 3

Is it possible to obtain the non-relativistic Schrödinger equation from the author's formalism (for example, by adding the Lagrangian of a massive particle to Lagrangian (2) )?

 

Response:

Yes. A new subsection 7.3 and new Appendix T: “Nonrelativistic Limit and Schreodinger Dynamics” is dedicated to the task of deriving the Schrödinger equation in the long-wavelength limit. The derivation is given in detail in Appendix T and the result is presented as Theorem T.1. 

In this framework massive particles are solitons and localized curvature waves (see section 2.5: ”Mass in Chronon Field Theory: Localization, Curvature, and Energy”) emergent from the aligned field dynamics, therefore no outside-of-the-framework matter coupling terms are needed in the Lagrangian.

 

Comment 4

The method of the article does not provide a geometric interpretation of the Born rule for probability density as the square of wave function modulus. Although, according to the logic of the article, the dynamics of the causal field Φμ(x) should explain how the reduction of the wave packet of a massive quantum particle occurs in the process of quantum measurement.

 

Response:

The revised Section 5.5:”Born weights and the classical limits” and Appendix S:”RNF Probability Measure and the Born Rule” are dedicated to deriving the Born rule and present the result as Theorem S1(RNF Born Rule).

 

Comment 5

According to the author approach , bosons are coherent oscillations of causal curvature on the solitonic background. But bosons are carriers of physical interactions. What about fermions? They are the basic particles of matter. The author does not write about them at all. Why? How will the interaction of physical fields with the causal field Φμ(x) be implemented? Is it possible to determine the parameters, such as masses of elementary particles, which depend on the causal field Φμ(x) ? This question makes sense, since other modern geometric quantum theories of spacetime (see, for example, the recent open-access article with DOI: 10.1016/j.nuclphysb.2025.116959) directly allow to calculate the masses of some elementary particles that are carriers of physical interactions.

 

Response:

A new discussion in Section 2.5 and a dedicated Appendix U: “Fermionic Degrees of Freedom as Chiral Twist Defects” outline and show how that fermionic excitations correspond to chiral edge modes of polarized chronon textures and how gauge interactions arise through curvature of the polarization bundle. The connection to particle masses is discussed in Section 2.5. The recent article with DOI: 10.1016/j.nuclphysb.2025.116959 has been cited in the discussion of Section 2.5.

 

Comment 6

When it comes to experimental confirmation of the proposed theory (section 4.5), the author should specifically calculate the deviation of the experimental results on quantization of magnetic flux from the predictions of traditional quantum theory. Will the difference be significant? We are talking about those very discrete steps or plateaus corresponding to integer multiples of Ñgeom. This comment is a recommendation for the author's future work.

 

Response:

Follwing your comments on needing specific calculated deviations, I derived some numerical predictions that are many orders of magnitude too small for experimental verification, because those are chronon scale (~ 10^-34m) phenomena. Therefore I have deleted the previous section on proposed experiments in mesoscopic interferometers. Instead I added a new section Section 8, “Phenomenology: The Electron as the Minimal Chronon Soliton,” which develops a physically meaningful prediction, identifying the electron as the minimal chronon soliton with an inferred radius of order 10^{-34} meters. I also added Section 7.8, which derives the exponential suppression of Hawking radiation in this framework, and I conclude the paper with four concrete falsifiability tests in Section 10.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

All of my concerns have been adequately addressed. I am therefore pleased to recommend your manuscript for acceptance.