Towards the Structure and Mechanisms of Complex Systems, the Approach of the Quantitative Theory of Meaning

Inga Ivanova; John S. Torday

REVIEW 5 major objections 6 minor 6 references

Towards the Structure and Mechanisms of Complex Systems, the Approach of the Quantitative Theory of Meaning

T0 review · 5 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that the surplus meaning generated in systems of three or more heterogeneous agents evolves as soliton trains obeying a modified KdV equation.

desk verdict A synthesis of the authors' own TH/redundancy program whose central KdV equation is asserted from an unpublished companion paper and empirically tested only by in-sample curve fitting. read the letter →

arxiv 2412.09007 v1 pith:VXE5WIZN submitted 2024-12-12 cs.CY nlin.AO

Inga Ivanova , John S. Torday This is my paper · ORCID

classification cs.CYnlin.AO

keywords complexsystemsquantitativetheoryofmeaningredundancyTripleHelixKdVequationsolitonlogisticcontinuouswavelettransformsynergy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes that complexity in systems of at least three heterogeneous agents is not an emergent accident but a consequence of reflexive communication among positionally differentiated agents. It claims that the redundancy generated by this communication, a quantity borrowed from information theory, evolves according to a modified Korteweg–de Vries (KdV) equation, so observable data from such systems should contain trains of solitary waves whose amplitudes follow a linear trend. The paper offers the logistic continuous wavelet transform as the practical method for extracting those hidden wave trains from time series like patent counts or market prices. If the claim is right, complex systems can be studied through a fixed topological mechanism rather than through bespoke models of each system.

What carries the argument

The central object is the redundancy $R$, defined through the excess-information relations $R_{12}=-T_{12}$, $R_{123}=T_{123}$, and generally $R_{12\ldots n}=(-1)^{n-1}T_{12\ldots n}$. This quantity turns the static configurational information of Shannon's framework into a measure of unrealized options generated when senders and receivers use different communication codes. The load-bearing piece of machinery is Eq. (5), the modified KdV equation, which supplies the solitary-wave dynamics; the logistic continuous wavelet transform is the companion tool, because differentiating a logistic curve yields exactly the hyperbolic-secant-squared profile of a KdV soliton. The paper's argument moves from this equation to the prediction of linear-trend soliton trains in empirical data.

What would settle it

Fit the paper's own reported Corn/USD three-soliton decomposition to the KdV soliton relation $A_i \propto k_i^2$ with a constant ratio between amplitude and temporal shift; if the fitted parameters violate that relation, the central claim that real redundancy data form KdV soliton trains is contradicted. More directly, one could estimate $R(t)$ from three-way mutual information in a university-industry-government dataset and test whether the estimated series satisfies Eq. (5).

Watch

Extended reading notes

Core claim

On its own terms, the paper is establishing that the information-theoretic redundancy generated by reflexive communication among positionally differentiated agents is not merely a static summary statistic but a dynamical field variable. Its time evolution near a stable mode is claimed to be governed by the modified KdV equation $4R_T - 2RR_X + R_{XXX} + C_1 = 0$. Because KdV supports $\mathrm{sech}^2$ solitary waves, the observable consequence is that cumulative time series of such systems should be decomposable into sums of logistic curves, with differenced data appearing as trains of solitons whose amplitudes align on a linear trend. The paper also claims this signature appears in patent data and market data, and that the same structure is homologous across biology, from embryologic germ layers to injury repair.

Load-bearing premise

The load-bearing premise is that the static redundancy $R(t)$, defined from information-theoretic overlaps, actually obeys the time-dependent modified KdV equation (Eq. 5) when a system sits near a mode with small fluctuations; that equation is imported from a manuscript described as in preparation, not derived in this paper.

Editorial extensions

If this is right

Any system of three or more heterogeneous agents should show solitary-wave trains in its cumulative output, so the same signature should appear in markets, patent systems, and disease or rumor dynamics.
Because the derivative of a logistic curve is a KdV soliton, cumulative data can be approximated as sums of S-shaped curves, giving operational predictions for trend change points.
The logistic CWT should separate positive and negative redundancy contributions, letting analysts attribute one wave train to synergy and another to historical entropy.
The Triple Helix is presented as the minimal building block, and higher-order systems are claimed to reduce to triads, making triad-based network analysis a route to studying any complex system.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A test the paper only gestures at would estimate $R(t)$ directly from raw three-way mutual information in a real system and check whether the series itself solves Eq. (5), rather than inferring solitons from price or patent fits.
The KdV soliton relation $A_i \propto k_i^2$ and a constant amplitude-to-time-shift ratio are not examined in the paper's Corn example; checking those parameter relations against the fitted solitons would be a sharp falsifier.
If the biological homology is taken seriously, the same logistic-CWT decomposition should appear in physiological data such as embryonic developmental series or injury-repair time courses; that cross-domain prediction is testable and goes beyond the paper's current evidence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

You asked for my read on 2412.09007. The short version: this is a synthesis of the authors' own prior work on the Triple Helix and mutual redundancy, and the new empirical example (corn prices) is a minor illustration. The paper's central prediction—that redundancy R evolves by a modified KdV equation and therefore shows up as soliton trains—is asserted, not derived. Eq. 5 is attributed to an 'in preparation' companion paper, and the manuscript gives no hint of how a PDE emerges from the static definitions in Eqs. 1–3. The stress-test note is right: for a localized soliton with R constant at infinity, the extra constant C1 in Eq. 5 would force C1=0, a condition the paper never states. This is a load-bearing gap, not a cosmetic one.

What the paper does well: it lays out the conceptual lineage from Shannon to configurational information to the TH model clearly enough, and the mutual redundancy measure is a genuine idea from their earlier work. The corn example shows the intended pipeline: fit a few solitons to a price series, report R^2=0.94. The biology section is clearly speculative and labeled as homology, so I don't hold that against it.

The soft spots go beyond Eq. 5. The empirical test is in-sample curve fitting with at least six free parameters (three soliton amplitudes, k's and T's, plus beta) and no error bars or out-of-sample validation. The Engel-Granger test they mention addresses spurious regression, but it does not validate the structural model. Eq. 8 is off by a factor and a half-argument in the derivative of the logistic. There are also typos and a garbled sentence in Section 3. None of this is fatal to a research program, but it is fatal to this paper as a self-contained contribution.

Who is this for? Readers already interested in the TH model and the authors' prior quantitative work may find this a useful overview and a pointer to the in-preparation papers. A general complexity-science reader will not be convinced, because the central equation is unproven and the data analysis is not pressure-tested.

My recommendation: I would not send this to referees as is. I would ask the authors to include the derivation of Eq. 5 (or a clear statement of what is assumed), to fix Eq. 8, and to provide at least one out-of-sample or falsifiable prediction. If the companion paper actually contains the derivation, then a revised version is worth another look.

Best,

[Your name]

Referee Report

5 major / 6 minor

Summary. The paper proposes a Quantitative Theory of Meaning, extending Shannon communication theory to systems of three or more heterogeneous agents, and links it to the Triple Helix model. The central claim is that the redundancy R generated by reflexive communication obeys a modified Korteweg-de Vries equation (Eq. 5), so empirical data from such systems should appear as trains of solitary waves whose amplitudes follow a linear trend. The authors illustrate this with a three-soliton decomposition of Corn market data and a logistic continuous wavelet transform of patent application data, and they draw parallels to embryologic development.

Significance. If the central claim were established, the approach could offer a general method for analyzing the structure and dynamics of complex systems, with applications to innovation studies, financial markets, and biology. The paper also has genuine strengths: it connects information-theoretic measures (mutual redundancy) to a dynamical model, and it proposes a concrete wavelet-based data-analysis procedure. However, the load-bearing step from static information measures to a nonlinear PDE is not derived in this manuscript, and the empirical validation is an in-sample curve fit without uncertainty quantification or out-of-sample testing. As it stands, the paper does not support its central claim.

major comments (5)

[Section 3, Eq. (5)] The central claim that redundancy R obeys the modified KdV equation is attributed to Ivanova and Rzadkowski (2024a), an unpublished manuscript, and is not derived from the definitions in Sections 1-2. Without Eq. (5), there is no theoretical reason to expect solitary-wave trains in Section 4. This is the load-bearing step of the paper, and it is currently an unexamined postulate.
[Section 3, Eq. (5)] The constant C1 in Eq. (5) is never specified. For a localized solitary wave with R approaching a constant at spatial infinity, evaluating Eq. (5) in the far field forces C1 = 0; the paper does not state or justify this condition. As written, the equation does not by itself imply the soliton solutions the authors use.
[Section 4, Eq. (8)] Equation (8) is incorrect: differentiating the logistic function in Eq. (7) yields x'(t) = (s x_sat / 4) sech^2[s(t - t0)/2], with a factor s and with the argument halved. The printed formula omits s and uses the argument s(t - t0). This invalidates the claimed correspondence between logistic derivatives and KdV solitons, because a KdV soliton has amplitude proportional to the square of the width parameter, while the logistic derivative has amplitude proportional to s for fixed x_sat.
[Section 4, Figure 5 and Tables 1-2] The empirical test is an in-sample fit of a sum of three solitons, with free parameters A_i, k_i, T_i and a vertical shift beta, to a single Corn price series. No standard errors or confidence intervals are reported for the fitted parameters, and no out-of-sample or hold-out validation is performed. The reported R^2 = 0.94 is from a regression of observed values on the fitted model values, which measures in-sample agreement, not predictive success. The claim that the data decompose into solitons is therefore confirmed only by fitting the very objects predicted.
[Section 3, Eqs. (3)-(5)] No derivation connects the static information-theoretic expression R(t) = P^2(t) - Q^2(t) to the time-dependent PDE in Eq. (5). The assumption that the number of generated variants is proportional to the intensity of communications is introduced ad hoc, and the leap from this to a nonlinear evolution equation in (X, T) phase space is unexplained. The theoretical premise of the empirical sections is thus not established by the manuscript.

minor comments (6)

[Abstract] Typos include 'extention', 'heterogenious', 'geterogeneous', and 'consideres'; these should be corrected.
[Section 3] 'Korteveg' should be 'Korteweg' (de Vries); also, the notation 'ch^{-2}' should be defined or replaced with 'sech^2'.
[Section 4] Two figures are both numbered 'Figure 5' (the soliton decomposition and the scatter chart); they should be renumbered sequentially.
[Table 3] The last row states 'R = P - Q', but the paper's own Eq. (3) defines R = P^2 - Q^2; this is inconsistent and should be corrected.
[Section 4 and References] The text refers to 'Engel-Granger' while the reference is 'Engle & Granger'; also the author name appears as 'Rzadkovsky' in the text and 'Rzadkowski' in the references. These should be harmonized.
[Section 4] 'scalegram' should be 'scalogram'.

Circularity Check

2 steps flagged · score 7.0 of 10

The central soliton-train prediction rests on Eq. 5, a modified KdV equation asserted from an in-preparation self-cited manuscript, and the empirical 'confirmation' fits the same soliton shape to the data in-sample.

self citation load bearing [Section 3, Eq. 5]
"Redundancy, as defined by Eqs. 1, 2 is a static measure, but real systems are not static. ... Assuming that the system is in a mode that involves smaller fluctuations around the average redundancy, the time evolution ... can be captured by a nonlinear evolutionary equation (Ivanova, Rzadkowski, 2024a): 4RT − 2RRX + RXXX + C1 = 0. Eq. 5 is a modification of Korteveg-de Vries (KdV) equation."

The paper's own definitions (Eqs. 1-3) define R as a static configurational-information measure; no argument is given connecting that measure to a time-dependent PDE in (X,T). The dynamical equation that generates all later soliton predictions is simply asserted and attributed to an 'in preparation' manuscript (Ivanova, Rzadkowski, 2024a) sharing the paper's first author. The cited result is not machine-checked, code-reproduced, or stated with assumptions excluding the target prediction. Thus the load-bearing premise reduces to an unverified self-citation rather than a derivation from the paper's information-theoretic framework.
fitted input called prediction [Section 4 (Measurement), Corn/USD example, Tables 1-2]
"According to Equation 5, redundancy can be expected to appear in (one or more) solitary wave trains whose amplitudes follow a linear trend. ... one may choose to decompose the time-cumulative data into a series of logistic (S-shaped) curves. ... The approximation parameters are given in Table 1. ... Ordinary least square fit f = Bt + C allows the model to explain 94% of the variations (Fig. 5)."

The predicted solitary-wave structure is operationalized by decomposing the data into logistic curves, and the derivative of the logistic function is cosh^-2, the same functional form as the KdV soliton. The 'confirmation' consists of fitting the soliton parameters (A_i, k_i, T_i) to the same data in Table 1 and then regressing the price on the fitted curve (slope 0.95, R²=0.94). This is in-sample curve fitting of the assumed functional form, not an independent test of Eq. 5. The cointegration check tests residual stationarity, not the soliton mechanism, so the empirical result is forced by the fitted input rather than predicted.

full rationale

The derivation chain is not self-contained: the paper defines R as a static information measure (Eqs. 1-3) and then, without derivation, asserts that it satisfies the modified KdV equation (Eq. 5), citing an in-preparation manuscript by the same first author. Every subsequent statement about solitary wave trains and linear amplitude trends derives from that asserted PDE. The empirical sections do not provide an independent test: they fit logistic/S-shaped curves to data, note that the derivative of a logistic curve is a sech^2 (KdV soliton) shape, and report in-sample R² and residual stationarity. No out-of-sample prediction, cross-validation, or falsifiable benchmark is offered, so the 'prediction' reduces to fitting the very functional form that Eq. 5 would imply. This is partial but central circularity: the main theoretical premise is carried by an unverified self-citation, and the main empirical evidence is in-sample fitting. Score 7 reflects that the central claim is forced by these two moves, though the paper does contain independent background material (e.g., information-theoretic definitions and the Triple Helix discussion).

Assumptions & free parameters 4 free parameters · 6 assumptions · 4 invented entities

The framework relies almost entirely on conceptual quantities and equations introduced in the authors' prior work. The only empirical illustration is a curve fit with many free parameters, and the central equation is cited to an unpublished manuscript. There are no externally validated entities or parameters.

free parameters (4)

C1 (integration constant in Eq. 5) = not specified
Integration constant in the modified KdV equation; no physical interpretation or fixed value given.
Soliton amplitudes A_i, k_i, T_i for Corn fit = A=(71.75, 208.21, 370.57), k=(0.03, 0.04, 0.02), T=(-54.16, -122.4, -201)
Amplitudes and time shifts fitted to Corn/USD data in Table 1.
beta vertical shift = 310.75
Added to equate the first wave to zero in the Corn decomposition.
Logistic component parameters (s, x_sat, t0) = not reported
Underlying parameters for the logistic decomposition of time series; values are not given in the paper.

assumptions (6)

standard math Shannon information definitions and mutual information properties
Used to define uncertainties and mutual information in Section 2.
domain assumption The redundancy R reflects the surplus of meanings generated by reflexive communication
Interpretation of information-theoretic quantities as semantic meaning; from prior work by the authors.
ad hoc to paper The number of generated variants is proportional to the intensity of communications, yielding R=P^2-Q^2
Introduced in Section 3 without derivation; basis for the KdV dynamics.
ad hoc to paper R obeys the modified KdV equation (Eq. 5) under small fluctuations
Central dynamical equation, cited to an unpublished manuscript; no derivation in this paper.
domain assumption Real complex systems have a Triple Helix structure with three or more positionally differentiated agents
Assumed to justify the applicability of the framework to markets and patents.
domain assumption Time series can be decomposed into a sum of logistic functions
Basis for the logistic CWT; not proven for arbitrary data.

invented entities (4)

Excess information (mutual redundancy) as a measure of meaning
purpose: Quantify semantic surplus in communication.
A conceptual quantity from prior work; no direct falsifiable handle outside the framework.
Semantic noise and semantic receiver blocks
purpose: Describe encoding/decoding of meaning in communication.
Abstract constructs used to modify Shannon's scheme.
Communication codes as eigenvectors
purpose: Represent positional differentiation between agents.
Mathematical metaphor; no empirical counterpart specified.
Network of correlations (non-local)
purpose: Account for non-local interactions beyond communication links.
Theoretical construct to explain feedforward/feedback loops.

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Cite this review

Pith. "Pith review of Towards the Structure and Mechanisms of Complex Systems, the Approach of the Quantitative Theory of Meaning." pith.science (2026). https://pith.science/paper/VXE5WIZN

@misc{pith2026241209007,
  author       = {Pith},
  title        = {Pith review of: Towards the Structure and Mechanisms of Complex Systems, the Approach of the Quantitative Theory of Meaning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VXE5WIZN}},
  note         = {Machine review of arXiv:2412.09007}
}

read the original abstract

We study analysis of complex systems using a Quantitative Theory of Meaning developed as an extention of Shannon's Communication Theory. The approach consideres complexity not in terms of the manifestation of its effects which are manifestation of the dynamics of the system, but in terms of primary causes and taking into account the topology of the system. Here, the dynamics of the system are provided by reflexive communication between heterogenious agents that make up the system. Unlike Shannon's Communication Theory the Theory of Meaning imposes restrictions on the complex systems being analyzed. Non-linearity and specific dynamics of the system arise as a consequence of the topology of the system. This topology also suggests a method for analyzing complex systems, the logistic Continuous Wavelet Transform (CWT). The paper also lays the foundation for future research in various fields studying complex systems of interacting geterogeneous agents, which may form a new paradigm for better understanding the structure, mechanisms, and dynamics of complex systems.

Figures

Figures reproduced from arXiv: 2412.09007 by the authors.

**Figure 5.** Scatter chart for Corn/USD values vs. model predicted values (f). Straight line is the OLS fit. The parameters of the OLS are presented in [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

**Figure 6.** Patent applications filed in the United States by residents of the country for the period 1980-2021, the dotted line is statistical data, the solid line is the approximation function (From Ivanova and Rzadkovsky, 2024b) [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗

**Figure 8.** Solitary wave decomposition of a yearly data on patents applications filed in USA by country residents for the period 1980-2021, as shown in [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

President Dwight D. Eisenhower's Farewell Address

Abramson, N. (1963). Information Theory and Coding. New York, etc.: McGraw-Hill. Akçay, E. and Hirshleifer, D. (2021), Social finance as cultural evolution, transmission bias, and market dynamics, Proceedings of the National Academy of Science, 118(26), 2015568118- . https://www.pnas.org/doi/pdf/10.1073/pnas.2015568118 Albert, R., Jeong, H., & Barabási, A...

work page doi:10.1073/pnas.2015568118 1963
[3]

Quantitative Theory of Meaning. Application to Financial Markets. EUR/USD case study

Ivanova, I., Rzadkowski, G. and Leydesdorff, L. (2024b). Quantitative Theory of Meaning. Application to Financial Markets. EUR/USD case study. (in preparation). https://arxiv.org/abs/2410.06476 Accessed Dec. 10,

work page Pith review arXiv 2024
[4]

Ivanova I, Smorodinskaya, N., and Leydesdorff, L. (2019). On measuring Complexity in a PostIndustrial Economy: The Ecosystem’s Approach. Quality & Quantity 54(1), 197-212. doi: 10.1007/s11135-019-00844-2 James, R., Ellison, C., and Crutchfield, J. (2011). Anatomy of a bit: Information in a time series observation, Chaos: An Interdisciplinary Journal of No...

work page doi:10.1007/s11135-019-00844-2 2019
[6]

Yeung, R. W. (2008). Information Theory and Network Coding. New York, NY: Springer

work page 2008
[17]

Torday, J. (2021). Cellular evolution of language. Prog Biophys Mol Biol.,167, 140-146. Torday, J. (2022). Quantum Mechanics, Cell-Cell Signaling, and Evolution. Academic Press, New York Trigona, R. and Cianci, E. (2014) Glossary of Complexity, World Futures: The Journal of New Paradigm Research, 70(5-6), 370-375. Ulanowicz, R. E. (2009). The dual nature ...

work page Pith review arXiv 2021
[2024]

Triple Helix synergy and patent dynamics. Cross country compartison

Engle, R., Granger, C. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica. 55 (2), 251–276. Etzkowitz, H. and Leydesdorff, L. (1995), “The Triple Helix - University-Industry-Government Relations: A Laboratory for Knowledge-Based Economic Development”, EASST Review, 14(1), pp. 14-19. Etzkowitz, H. and Leydesdo...

work page Pith review arXiv 1987

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

President Dwight D. Eisenhower's Farewell Address

Abramson, N. (1963). Information Theory and Coding. New York, etc.: McGraw-Hill. Akçay, E. and Hirshleifer, D. (2021), Social finance as cultural evolution, transmission bias, and market dynamics, Proceedings of the National Academy of Science, 118(26), 2015568118- . https://www.pnas.org/doi/pdf/10.1073/pnas.2015568118 Albert, R., Jeong, H., & Barabási, A...

work page doi:10.1073/pnas.2015568118 1963

[2] [3]

Quantitative Theory of Meaning. Application to Financial Markets. EUR/USD case study

Ivanova, I., Rzadkowski, G. and Leydesdorff, L. (2024b). Quantitative Theory of Meaning. Application to Financial Markets. EUR/USD case study. (in preparation). https://arxiv.org/abs/2410.06476 Accessed Dec. 10,

work page Pith review arXiv 2024

[3] [4]

Ivanova I, Smorodinskaya, N., and Leydesdorff, L. (2019). On measuring Complexity in a PostIndustrial Economy: The Ecosystem’s Approach. Quality & Quantity 54(1), 197-212. doi: 10.1007/s11135-019-00844-2 James, R., Ellison, C., and Crutchfield, J. (2011). Anatomy of a bit: Information in a time series observation, Chaos: An Interdisciplinary Journal of No...

work page doi:10.1007/s11135-019-00844-2 2019

[4] [6]

Yeung, R. W. (2008). Information Theory and Network Coding. New York, NY: Springer

work page 2008

[5] [17]

Torday, J. (2021). Cellular evolution of language. Prog Biophys Mol Biol.,167, 140-146. Torday, J. (2022). Quantum Mechanics, Cell-Cell Signaling, and Evolution. Academic Press, New York Trigona, R. and Cianci, E. (2014) Glossary of Complexity, World Futures: The Journal of New Paradigm Research, 70(5-6), 370-375. Ulanowicz, R. E. (2009). The dual nature ...

work page Pith review arXiv 2021

[6] [2024]

Triple Helix synergy and patent dynamics. Cross country compartison

Engle, R., Granger, C. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica. 55 (2), 251–276. Etzkowitz, H. and Leydesdorff, L. (1995), “The Triple Helix - University-Industry-Government Relations: A Laboratory for Knowledge-Based Economic Development”, EASST Review, 14(1), pp. 14-19. Etzkowitz, H. and Leydesdo...

work page Pith review arXiv 1987