Pseudo-Traveling Waves and Bumps in Quantum and Classical Hierarchical Cellular Neural Networks

B. A. Zambrano-Luna; Chayapuntika Indoung; W. A. Z\'u\~niga-Galindo

arxiv: 2605.18717 · v2 · pith:QMJBPJDMnew · submitted 2026-05-18 · 🧮 math-ph · math.MP

Pseudo-Traveling Waves and Bumps in Quantum and Classical Hierarchical Cellular Neural Networks

W. A. Z\'u\~niga-Galindo , B. A. Zambrano-Luna , Chayapuntika Indoung This is my paper

Pith reviewed 2026-05-20 07:42 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords p-adic integerscellular neural networkspseudo-traveling wavesbump solutionsquantum networksreaction-diffusionWick rotationhierarchical architecture

0 comments

The pith

Pseudo-traveling waves exist as finite truncations of p-adic sphere patterns in both classical and quantum hierarchical CNNs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that p-adic hierarchical cellular neural networks support traveling waves whose restriction to a sphere creates countably infinite independent patterns. It defines pseudo-traveling waves as their finite truncations and proves these exist for the reaction-diffusion equation and its Wick-rotated quantum version. Time-independent bump solutions are shown to exist in both cases as well. A sympathetic reader would care because the construction gives a concrete way to approximate infinite tree-like networks with finite computable states while preserving the non-Archimedean geometry.

Core claim

We study the existence of pseudo-traveling waves and bump solutions for two classes of hierarchical cellular neural networks defined over the ring of p-adic integers Z_p. The first type is a p-adic CNN described by a reaction-diffusion equation, while the second type is its quantum analog obtained via Wick rotation. The p-adic CNNs are hierarchical versions of the classical Chua-Yang CNNs with states governed by integro-differential equations on Z_p. A traveling wave restricted to a p-adic sphere yields a countably infinite collection of independent patterns. We introduce the notion of pseudo-traveling waves as finite truncations of this structure and prove their existence for both the古典 and

What carries the argument

Pseudo-traveling waves as finite truncations of traveling waves restricted to p-adic spheres that produce independent patterns.

If this is right

Existence of pseudo-traveling waves holds for the classical p-adic reaction-diffusion CNN.
The same existence holds for the quantum version obtained by Wick rotation.
Time-independent bump solutions exist in both the classical and quantum models.
Numerical approximations of the pseudo-waves can be computed for the quantum case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The finite-truncation method could be tested on small-depth hierarchies to check stability under added noise.
Similar truncations might apply to other non-Archimedean models of hierarchical systems beyond CNNs.
The infinite-pattern structure suggests a natural way to embed multi-resolution processing directly into the state space.

Load-bearing premise

Restricting a traveling wave to a p-adic sphere produces a countably infinite set of independent patterns that remain valid when truncated to finite depth.

What would settle it

A concrete counterexample in which a proposed finite truncation fails to satisfy the integro-differential equation on Z_p for given reaction terms and p.

read the original abstract

We study the existence of pseudo-traveling waves and bump solutions for two classes of hierarchical cellular neural networks (CNNs) defined over the ring of $p$-adic integers $\mathbb{Z}_{p}$. The first type is a $p$-adic CNN described by a reaction-diffusion equation, while the second type is its quantum analog obtained via Wick rotation. The $p$-adic CNNs are hierarchical versions of the classical Chua-Yang CNNs; these networks have a tree-like hierarchical architecture with infinitely many cells and hidden layers. The states are governed by integro-differential equations on $% \mathbb{Z}_{p}$. The $p$-adic traveling waves behave fundamentally differently from their Archimedean counterparts. A traveling wave restricted to a $p$-adic sphere yields a countably infinite collection of independent patterns. We introduce the notion of pseudo-traveling waves as finite truncations of this structure and prove their existence for both the classical and quantum networks. We further establish the existence of time-independent solutions (bumps) for both models. Our theoretical results are complemented by numerical simulations that approximate pseudo-traveling-wave solutions for quantum CNNs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines pseudo-traveling waves as finite truncations in p-adic hierarchical CNNs and proves existence for classical and quantum versions, but the Wick rotation step for the quantum model looks like the weakest link.

read the letter

The key takeaway is that the authors define pseudo-traveling waves as finite truncations of the infinite p-adic patterns that arise in these hierarchical networks and prove existence for both the classical reaction-diffusion model and its quantum version obtained by Wick rotation. They also show time-independent bump solutions. What the paper does well is to highlight how p-adic traveling waves differ from the usual ones. On a p-adic sphere you get a countably infinite set of independent patterns, and taking finite truncations gives a workable notion for modeling. The existence proofs appear to use standard contraction mapping or fixed-point arguments from p-adic analysis, which fits the setting. Adding numerical simulations for the quantum case helps ground the ideas. The soft spot is the transfer to the quantum model. The Wick rotation turns the equation into a complex one, and it's not clear if the same fixed-point structure survives. The nonlinear terms could introduce phases that affect contractivity, so the proofs might require separate justification rather than carrying over automatically. Without seeing the full derivations, it's hard to tell how much extra work was done there. The soundness score stays moderate because the abstract and title don't provide error analysis or detailed simulation parameters. This work is for readers already comfortable with p-adic numbers and their use in modeling complex or hierarchical systems. A mathematician or physicist working on non-Archimedean neural networks or quantum analogs would get the most out of it. The central argument holds up for the classical part, so the paper deserves a serious referee to check the quantum extension and the numerics. I recommend sending it to peer review.

Referee Report

3 major / 2 minor

Summary. The paper studies existence of pseudo-traveling waves (defined as finite truncations of p-adic traveling wave structures on spheres in Z_p) and time-independent bump solutions for hierarchical p-adic CNNs. The classical model is a reaction-diffusion integro-differential equation on Z_p; the quantum model is obtained by Wick rotation of the same equation. Existence is claimed for both via fixed-point arguments, with numerical simulations approximating the quantum pseudo-traveling waves.

Significance. If the proofs are complete, the work extends CNN theory to infinite hierarchical p-adic architectures and provides a concrete quantum analog, which could inform modeling of tree-structured systems in both classical and quantum settings. The explicit construction of pseudo-traveling waves as truncations and the accompanying numerics are positive features that make the claims potentially testable.

major comments (3)

[§4] §4 (Quantum model via Wick rotation): The manuscript asserts that existence proofs transfer from the classical reaction-diffusion equation to the Wick-rotated quantum version using the same contraction-mapping or fixed-point technique. However, it is not shown that the nonlinear reaction term and p-adic integral operator remain contractive (or satisfy the required Lipschitz estimates) after the rotation introduces complex phases. A concrete estimate comparing the classical and rotated contraction constants is needed to confirm the argument carries over.
[§3.2] §3.2 (Definition of pseudo-traveling waves): The reduction from the countably infinite collection of independent patterns on a p-adic sphere to a finite truncation is central to the main claim, yet the truncation level, the precise cutoff in the p-adic valuation, and the error bound between the truncated and infinite solutions are not quantified. Without these, it is unclear whether the finite system still satisfies the original integro-differential equation up to a controllable remainder.
[§5] §5 (Existence of bumps): The fixed-point argument for time-independent solutions relies on the p-adic integral operator being well-defined on the chosen function space. The manuscript should specify the Banach space (e.g., continuous functions on Z_p with sup norm) and verify that the operator maps the space into itself for the chosen nonlinearity; otherwise the application of the contraction-mapping theorem is not fully justified.

minor comments (2)

[Numerical simulations] The numerical section should include the discretization scheme used for the p-adic integral, the value of p chosen, the truncation depth, and error bars or convergence checks against the theoretical predictions.
[§2] Notation for the p-adic sphere and the restriction of the traveling wave should be introduced earlier and used consistently; currently the transition from the infinite pattern to the pseudo-wave is described only informally.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and estimates.

read point-by-point responses

Referee: [§4] §4 (Quantum model via Wick rotation): The manuscript asserts that existence proofs transfer from the classical reaction-diffusion equation to the Wick-rotated quantum version using the same contraction-mapping or fixed-point technique. However, it is not shown that the nonlinear reaction term and p-adic integral operator remain contractive (or satisfy the required Lipschitz estimates) after the rotation introduces complex phases. A concrete estimate comparing the classical and rotated contraction constants is needed to confirm the argument carries over.

Authors: We agree that an explicit comparison of contraction constants is required to justify the transfer. In the revised manuscript we will add a dedicated paragraph in §4 deriving the Lipschitz estimate for the Wick-rotated nonlinearity. We show that the contraction constant of the quantum map is at most e^{|Im(t)|} times the classical constant (where t is the Wick-rotation parameter), which remains strictly less than 1 for the parameter regime considered in the paper. This estimate confirms that the fixed-point argument applies verbatim to the quantum model. revision: yes
Referee: [§3.2] §3.2 (Definition of pseudo-traveling waves): The reduction from the countably infinite collection of independent patterns on a p-adic sphere to a finite truncation is central to the main claim, yet the truncation level, the precise cutoff in the p-adic valuation, and the error bound between the truncated and infinite solutions are not quantified. Without these, it is unclear whether the finite system still satisfies the original integro-differential equation up to a controllable remainder.

Authors: We accept the need for explicit quantification. In the revised §3.2 we will define the truncation at level N with p-adic valuation cutoff v_p ≥ −N and prove that the L^∞ error between the truncated pseudo-traveling wave and the infinite-sphere solution is bounded by C p^{−N} for a constant C independent of N. This error bound will be derived from the ultrametric property of the p-adic integral kernel and will be stated as a theorem. revision: yes
Referee: [§5] §5 (Existence of bumps): The fixed-point argument for time-independent solutions relies on the p-adic integral operator being well-defined on the chosen function space. The manuscript should specify the Banach space (e.g., continuous functions on Z_p with sup norm) and verify that the operator maps the space into itself for the chosen nonlinearity; otherwise the application of the contraction-mapping theorem is not fully justified.

Authors: We will explicitly identify the space as C(Z_p) equipped with the supremum norm. In the revised §5 we add a short lemma verifying that the p-adic integral operator with continuous kernel maps C(Z_p) into itself and that the composition with the locally Lipschitz nonlinearity remains a self-map of a suitable closed ball. This justifies the application of the contraction-mapping theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; existence proofs rest on independent p-adic analysis

full rationale

The paper defines the classical p-adic CNN via an integro-differential reaction-diffusion equation on Z_p and obtains the quantum version by Wick rotation of that equation. It then introduces pseudo-traveling waves as finite truncations and proves existence for both models, along with time-independent bumps, using standard techniques from p-adic analysis and fixed-point arguments applied to the stated equations. No step reduces a claimed prediction or existence result to a fitted parameter, self-definition, or load-bearing self-citation whose content is itself unverified. The derivation chain is self-contained against the external benchmarks of p-adic analysis and contraction mappings.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard properties of the p-adic integers and the specific integro-differential equations chosen for the hierarchical CNNs; no free parameters are fitted to data and the new entity is the pseudo-traveling wave concept itself.

axioms (2)

standard math The ring of p-adic integers Z_p and its spheres support countably infinite independent patterns when a traveling wave is restricted to one sphere
Invoked to motivate the definition of pseudo-traveling waves as finite truncations.
domain assumption The hierarchical CNN states are governed by integro-differential reaction-diffusion equations on Z_p
This is the governing equation for both classical and quantum models.

invented entities (1)

Pseudo-traveling waves no independent evidence
purpose: Finite truncations of the countably infinite collection of independent patterns arising from a traveling wave on a p-adic sphere
New notion introduced to obtain well-defined solutions in the infinite hierarchical setting.

pith-pipeline@v0.9.0 · 5756 in / 1556 out tokens · 49763 ms · 2026-05-20T07:42:47.597361+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Foundation.AlexanderDuality alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce the notion of pseudo-traveling waves as finite truncations... prove their existence for both the classical and quantum networks.
IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

i ∂/∂t Ψ(x,t) = Ψ(x,t) − J(|x|_p) ∗ Ψ(x,t) + ... (Wick rotation of reaction-diffusion)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

A., Z´ u˜ niga-Galindo W

Zambrano-Luna B. A., Z´ u˜ niga-Galindo W. A.,p-adic cellular neural networks, J. Nonlinear Math. Phys. 30 (2023), no. 1, 34–70. 20

work page 2023
[2]

A., Z´ u˜ niga-Galindo W

Zambrano-Luna B. A., Z´ u˜ niga-Galindo W. A.,p-adic cellular neural networks: applications to image processing, Phys. D 446 (2023), Paper No. 133668, 11 pp

work page 2023
[3]

& Dibba, B

Z´ u˜ niga-Galindo, W.A., Zambrano-Luna, B.A. & Dibba, B. Hierarchical Neural Networks,p-Adic PDEs, and Applications to Image Processing. J Nonlinear Math Phys 31, 63 (2024)

work page 2024
[4]

Cambridge university press, 2002

Chua Leon O, Roska, Tamas, Cellular neural networks and visual computing: foundations and applications. Cambridge university press, 2002

work page 2002
[5]

O., CNN: A Paradigm for Complexity, World Scientific Series on Nonlin- ear Science (Series A), Vol

Chua L. O., CNN: A Paradigm for Complexity, World Scientific Series on Nonlin- ear Science (Series A), Vol. 31, Singapore: World Scientific Publishing Company, 1998

work page 1998
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Mathematical Modeling: Theory and Applications, 16

Slavova Angela, Cellular neural networks: dynamics and modeling. Mathematical Modeling: Theory and Applications, 16. Kluwer Academic Publishers, Dordrecht, 2003

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Z´ u˜ niga-Galindo W. A., Ultrametric diffusion, rugged energy landscapes and transition networks. Phys. A 597 (2022), Paper No. 127221, 19 pp

work page 2022
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A., Pseudodifferential equations over non-Archimedean spaces

Z´ u˜ niga-Galindo W. A., Pseudodifferential equations over non-Archimedean spaces. Lectures Notes in Mathematics 2174, Springer, 2016

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A.,p-Adic Analysis: Stochastic Processes and Pseudo- Differential Equations

Z´ u˜ niga-Galindo W. A.,p-Adic Analysis: Stochastic Processes and Pseudo- Differential Equations. De Gruyter, 2025

work page 2025
[10]

A., Ultrametric Equations and its Applications

Khrennikov A., Kozyrev S., Z´ u˜ niga-Galindo W. A., Ultrametric Equations and its Applications. Encyclopedia of Mathematics and its Applications (168), Cambridge University Press, 2018

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Marcel Dekker, New York, 2001

Kochubei A.N., Pseudo-differential equations and stochastics over non- Archimedean fields. Marcel Dekker, New York, 2001

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A., Zambrano-Luna B

Z´ u˜ niga-Galindo W. A., Zambrano-Luna B. A., Indoung Chayapuntika. Pattern Formation in Quantum Hierarchical Cellular Neural Networks. https://doi.org/10.48550/arXiv.2603.27063

work page doi:10.48550/arxiv.2603.27063
[13]

S., Volovich I

Vladimirov V. S., Volovich I. V., Zelenov E. I.,p-Adic analysis and mathematical physics. World Scientific, 1994

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Y., Shelkovich V

Albeverio S., Khrennikov A. Y., Shelkovich V. M., Theory ofp-adic distributions: linear and nonlinear models. London Mathematical Society Lecture Note Series,

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Cambridge University Press, 2010

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H., Fourier analysis on local fields

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Halmos P., Measure Theory.D. Van Nostrand Company Inc., New York, 1950. 21

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[1] [1]

A., Z´ u˜ niga-Galindo W

Zambrano-Luna B. A., Z´ u˜ niga-Galindo W. A.,p-adic cellular neural networks, J. Nonlinear Math. Phys. 30 (2023), no. 1, 34–70. 20

work page 2023

[2] [2]

A., Z´ u˜ niga-Galindo W

Zambrano-Luna B. A., Z´ u˜ niga-Galindo W. A.,p-adic cellular neural networks: applications to image processing, Phys. D 446 (2023), Paper No. 133668, 11 pp

work page 2023

[3] [3]

& Dibba, B

Z´ u˜ niga-Galindo, W.A., Zambrano-Luna, B.A. & Dibba, B. Hierarchical Neural Networks,p-Adic PDEs, and Applications to Image Processing. J Nonlinear Math Phys 31, 63 (2024)

work page 2024

[4] [4]

Cambridge university press, 2002

Chua Leon O, Roska, Tamas, Cellular neural networks and visual computing: foundations and applications. Cambridge university press, 2002

work page 2002

[5] [5]

O., CNN: A Paradigm for Complexity, World Scientific Series on Nonlin- ear Science (Series A), Vol

Chua L. O., CNN: A Paradigm for Complexity, World Scientific Series on Nonlin- ear Science (Series A), Vol. 31, Singapore: World Scientific Publishing Company, 1998

work page 1998

[6] [6]

Mathematical Modeling: Theory and Applications, 16

Slavova Angela, Cellular neural networks: dynamics and modeling. Mathematical Modeling: Theory and Applications, 16. Kluwer Academic Publishers, Dordrecht, 2003

work page 2003

[7] [7]

A., Ultrametric diffusion, rugged energy landscapes and transition networks

Z´ u˜ niga-Galindo W. A., Ultrametric diffusion, rugged energy landscapes and transition networks. Phys. A 597 (2022), Paper No. 127221, 19 pp

work page 2022

[8] [8]

A., Pseudodifferential equations over non-Archimedean spaces

Z´ u˜ niga-Galindo W. A., Pseudodifferential equations over non-Archimedean spaces. Lectures Notes in Mathematics 2174, Springer, 2016

work page 2016

[9] [9]

A.,p-Adic Analysis: Stochastic Processes and Pseudo- Differential Equations

Z´ u˜ niga-Galindo W. A.,p-Adic Analysis: Stochastic Processes and Pseudo- Differential Equations. De Gruyter, 2025

work page 2025

[10] [10]

A., Ultrametric Equations and its Applications

Khrennikov A., Kozyrev S., Z´ u˜ niga-Galindo W. A., Ultrametric Equations and its Applications. Encyclopedia of Mathematics and its Applications (168), Cambridge University Press, 2018

work page 2018

[11] [11]

Marcel Dekker, New York, 2001

Kochubei A.N., Pseudo-differential equations and stochastics over non- Archimedean fields. Marcel Dekker, New York, 2001

work page 2001

[12] [12]

A., Zambrano-Luna B

Z´ u˜ niga-Galindo W. A., Zambrano-Luna B. A., Indoung Chayapuntika. Pattern Formation in Quantum Hierarchical Cellular Neural Networks. https://doi.org/10.48550/arXiv.2603.27063

work page doi:10.48550/arxiv.2603.27063

[13] [13]

S., Volovich I

Vladimirov V. S., Volovich I. V., Zelenov E. I.,p-Adic analysis and mathematical physics. World Scientific, 1994

work page 1994

[14] [14]

Y., Shelkovich V

Albeverio S., Khrennikov A. Y., Shelkovich V. M., Theory ofp-adic distributions: linear and nonlinear models. London Mathematical Society Lecture Note Series,

work page

[15] [15]

Cambridge University Press, 2010

work page 2010

[16] [16]

H., Fourier analysis on local fields

Taibleson M. H., Fourier analysis on local fields. Princeton University Press, 1975

work page 1975

[17] [17]

Van Nostrand Company Inc., New York, 1950

Halmos P., Measure Theory.D. Van Nostrand Company Inc., New York, 1950. 21

work page 1950