jmp Journal of Modern Physics 2153-120X 2153-1196 Scientific Research Publishing 10.4236/jmp.2026.173016 jmp-150072 Article Physics Mathematics Derivation of Complex Phenomena from a Unified Theory Based on the Holographic Principle 0000-0002-1354-8476 Xiu Rulin 1 Hawaii Theoretical Physics Research Center, Pahoa, HI, USA

The author declares no conflicts of interest regarding the publication of this paper.

 12 03 2026 03 2026 17 03 283 294 13 01 2026 09 03 2026 12 03 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/jmp.2026.173016

Complex systems across a wide range of scientific domains consistently exhibit universal features, including fractal geometries, 1/f noise, and powerlaw distributions such as Zipf’s law and Gutenberg-Richter Law. The ubiquity of these empirical regularities strongly suggests the presence of a common underlying principle governing their emergence. This paper shows that these phenomena can be rigorously derived from a unified theoretical framework grounded in the holographic principle. In our previous work, we introduced a holographic action derived directly from the holographic principle—a generalized action that encompasses quantum physics, string theory, general relativity, and thermodynamics. We demonstrated that all elementary particles, fundamental forces, dark matter, dark energy, the observed value of the cosmological constant, the matter-antimatter asymmetry, and CP violation in weak interaction emerge from this single mathematical structure. In the present paper, we extend this framework to complex systems. We show that the observed power-law behaviors—fractal scaling, 1/f noise, Zipf’s law, and Gutenberg-Richter Law—arise naturally as consequences of the holographic action. Deriving these diverse phenomena from one cohesive theoretical foundation offers a new perspective on the origins of complexity and self-organization in the universe. This work proposes a candidate for the fundamental mathematical structure underlying these ubiquitous and otherwise disparate patterns, marking a step toward a deeper understanding of self-organized criticality. It also provides an additional demonstration of the predictive power of the unified theory based on the holographic principle.

 Holographic Principle Complex Systems Fractals 1/<i>f</i> Noise Zipf’s Law Gutenberg-Richter Law Scaling Invariance Power Laws Self-Organized Criticality Unified Theory Quantum Gravity 1. Introduction

Complex systems, characterized by their vast range of length scales and intricate dynamics, are a frontier of modern science. A remarkable feature of such systems is their tendency to evolve into a poised, “critical” state, where minor disturbances can trigger events of all sizes, often called avalanches [1]. This emergent scale-invariance is a hallmark of complexity.

Striking empirical regularities are observed across seemingly unrelated complex systems. These include:

Fractals: Geometric shapes with self-similar patterns at increasingly smaller scales are ubiquitous in nature, from coastlines and snowflakes to the large-scale structure of the universe [2][3].1/fNoise: Also known as pink or fractal noise, it is a signal whose power spectral density is inversely proportional to its frequency. It is commonly observed in physical and biological systems [4][5].Zipf’sLaw: An empirical law stating that the frequency of an item in a ranked list is inversely proportional to its rank. This is famously observed in the frequency of words in natural languages [6].CatastrophicEvents: Phenomena like earthquakes, described by the Gutenberg-Richter law, exhibit a power-law relationship between the magnitude and frequency of events [7].

These diverse phenomena share a common mathematical characteristic: a power-law relationship between the magnitude of an event and its frequency. The universality of these patterns suggests they arise from the intrinsic dynamics of the system itself, without external tuning or design. This raises a fundamental question: Is there a universal physical principle or mathematical framework that can explain the emergence of such organized complexity from simple physical laws? This remains one of the most profound challenges in modern science.

The theory of self-organized criticality (SOC), introduced by Bak, Tang, and Wiesenfeld, was proposed as a unifying principle [8][9]. SOC suggests that complex systems naturally evolve to a critical state, giving rise to scale-invariant behaviors. However, a consensus on its abstract mathematical formulation and a fundamental underlying mechanism remains elusive.

In our previous work [10], we proposed a unified theoretical framework grounded in the holographic principle that integrates quantum physics, string theory, general relativity, and thermodynamics. Central to this framework is the concept of elementaryinformation as the fundamental constituent underlying all physical phenomena. From this idea, we derived a holographic action that characterizes the maximum amount of elementary information encoded on a holographic surface. This action generalizes the standard actions of string theory, quantum mechanics, general relativity, and thermodynamics into a single mathematical structure.

We demonstrated that a wide range of physical laws and phenomena emerge naturally from this holographic formulation. Elementary particles arise from Poincaré symmetry, gravitational and gauge interactions follow from diffeomorphic symmetry, and classical equations of motion emerge from Weyl symmetry. Moreover, dark matter and dark energy appear as vibrational modes on the horizon scale of the universe, enabling us to derive a value for the cosmological constant consistent with observational data. In subsequent work [11], we further showed that the observed matter-antimatter asymmetry and the CP violation in weak interactions also emerge naturally from the holographic action. Together, these results outline a pathway toward a grand unified theory derived from first principles, offering a coherent framework with concrete, testable predictions.

Building on this foundation, the present paper demonstrates that fractals, 1/f noise, Zipf’s law, and the Gutenberg-Richter law can also be derived from this holographic quantum theory. We propose that these complex phenomena emerge naturally from the unified theory, thereby providing a candidate for the mathematical formulation and underlying principle for understanding complex systems.

We will first review the holographic quantum theory derived from the holographic principle. We will then show how to derive the aforementioned complex phenomena by analyzing the simplest case of the holographic action in a negligible background field. Finally, we will discuss the implications and predictions of these results and suggest directions for future work.

 2. Review of a Unified Theory Based on the Holographic Principle

We begin by positing the holographic principle as fundamental [10], stated as follows:

Allphysicalphenomenaemergefromahologramthatencodestheinformationofasystem.

This principle implies that information is the basic ingredient determining all physical reality. To mathematically formulate the holographic principle, we must identify the form of the basic information underlying all physics, which we refer to as elementary information (EI). We propose that elementary information is encoded by spacetime, which we term elementary information spacetime (EI spacetime).

Based on general considerations from quantum physics and general relativity, we find that the holographic action A calculates the maximum amount of observable information that can be encoded in EI spacetime (τ, σ) [10]. It takes the form:

A = α ∫ Δ τ   Δ σ

where the integral symbol ∫ represents the summation over EI space and time (τ, σ), and α is a constant derived to be:

α = 1 / ( l p t p )

Here l p is the Planck length, t p is the Planck time, and l p = c t p = ( ℏ G / c 3 ) 1 / 2 .

To derive observable physical phenomena from the holographic action (2), it is necessary to view the physical spacetime X μ as a projection from the EI spacetime (τ, σ):

X μ : ( τ , σ ) → X μ ( τ , σ ) .

In physical spacetime, the holographic action becomes:

A ′ h = α ∫ 0 T d τ ∫ 0 L d σ   ∂ τ X μ ∂ σ X μ

The more general version of this holographic action is:

A ″ h = α ∫ 0 T d τ ∫ 0 L d σ   γ a b ∂ a X μ ∂ b X μ

Here a and b correspond to τ or σ. One can define the holographic function Ψh as:

Ψ h ( T , L ) = exp ( i A h ) = ∑ sum   over   possible   X μ exp ( i A ′ h )

Notice that the holographic action (3) and the function (4) share a structural similarity with the Polyakov action in bosonic string theory. However, there are two important differences. First, in string theory the worldsheet time coordinate is integrated from 0 to infinity, whereas in the holographic action the integration runs only from 0 to (T), where (T) is the characteristic time scale associated with the system under consideration. Second, the physical interpretation of the string action remains an open question, while the holographic function Ψ h ( T , L ) has a clear meaning: it is directly related to the elementary information content of the system, specifically the number of possible states ( E i ) in the system, through the relation:

E i = A h = i ln ψ h ψ h ( T , L ) = e i A h = ∑ sum   over   all   possible   X μ e i A ′ h

In the presence of a background field G μ ν and B μ ν in physical spacetime, the holographic action becomes

A h = i α ∫ 0 T d τ ∫ 0 L d σ ( G μ ν + B μ ν ) γ a b ∂ a X μ ∂ b X ν

The holographic function Ψh now becomes:

Ψ h ( X μ ( L , T ) , G μ ν ( X μ ( L , T ) ) ) = ∑ sum   over   possible   X μ   and   G μ ν exp [ i α ∫ 0 T d τ ∫ 0 L d σ ( G μ ν + B μ ν ) γ a b ∂ a X μ ∂ b X ν ]

As demonstrated in previous work [10], this framework can serve as a single mathematical basis from which all known fundamental forces and elementary particles, as well as phenomena like dark matter and dark energy, can be derived. It provides a unified theory integrating quantum physics, general relativity, string theory, and thermodynamics. In the following sections, we will show how the universal laws of complex systems also emerge from this theory.

 3. Derivation of Universal Complex Phenomena 3.1. Fractals and Scaling Invariance

According to the holographic theory, all physical phenomena are described by the holographic action and function. A key feature of the action in Equation (6) is its scaling invariance. Consider the following rescaling of the EI spacetime coordinates and metric:

X ′ μ ( λ τ , λ σ ) = X μ ( τ , σ )

At the same time, one can rescale the EI spacetime metrics

γ ′ a b = λ − 2 γ a b

This transformation leaves the holographic action (3) or (7) invariant. This fundamental symmetry implies that the physical phenomena described by the theory should exhibit scaling invariance. Fractals are geometric objects defined by their self-similarity across different scales, which is a direct physical manifestation of scaling invariance. The holographic theory thus provides a fundamental principle for the ubiquity of fractals and other scale-invariant phenomena in nature.

 3.2. 1/ <italic>f</italic> Noise

1/f noise, or pink noise, is a process whose power spectral density S ( f ) is inversely proportional to its frequency f, i.e., S ( f ) ∝ 1 / f . This implies that each octave (a doubling of frequency) contains an equal amount of noise energy.

In the simplest case of a flat physical spacetime ( G μ ν = η μ ν , B μ ν = 0 ), the holographic action (Equation (3)):

A ″ h = α ∫ 0 T d τ ∫ 0 L d σ   γ a b ∂ a X μ ∂ b X μ

describes a quantum system similar to that in bosonic string theory [12][13]. To calculate the quantum spectrum of the action A ″ h , one chooses the light cone gauge [12][13],

X ± = 2 − 1 2 ( X 0 ± X 1 ) , X i , i = 2 , ⋯ , D − 1.

Choose X + = τ and rewrite the Lagrangian A ″ h in the Hamiltonian form:

H = α ′ ∫ 0 L d σ ( 2 π α   Π i Π i + 1 2 π α ∂ σ X i ∂ σ X i ) ,

with

Π i = δ A ″ h δ ( ∂ τ X i )

The Hamiltonian (9) indicates the wave equation:

∂ τ ′ 2 X i = c 2 ∂ σ ′ 2 X i

Here c is a constant related to the velocity of the vibrations inside the system. The value of the c depends on the EI specatime metric γ a b , the background field G μ ν and B μ ν , as well as the space and time scale L and T.

To change (9) and (3) into a scale invariant format, introduce:

τ ′ = τ T ,     σ ′ = σ L , z = τ ′ − i σ ′ / c , z ¯ = τ ′ + i σ ′ / c .

The Equation (10) also indicates:

∂ z X i ∂ z ¯ X i = 0

The Equations (10) and (11) reveals that the physical spacetime X μ is composed of a series of vibrational modes in the form:

X i = x i + p L i z + p R i z ¯ + i ( 1 π α ) 1 / 2 ∑ n = − ∞ n ≠ 0 ∞ 1 n [ α n i exp ( − i π n z ) + α ˜ n i exp ( − i π n z ¯ ) ] .

Equation (12) shows that the system is composed of two series of vibrations. One series vibrating in the z direction, the other series vibrating in the z ¯ direction. Here α n i and α ˜ n i are the generators of the vibrations in the z and z ¯ directions respectively. The oscillators are labeled by the direction of oscillation in the physical spacetime i and the harmonic number n.

To quantize, impose the equal time canonical commutation relations:

[ X i ( z , z ¯ ) , Π j ( z ′ , z ¯ ′ ) ] = i δ i j δ ( z − z ′ ) δ ( z ¯ − z ¯ ′ )

with all other commutators between the independent variables vanishing. From the constraints of (13), one obtains:

[ x i , p j ] = i δ i j , p j = p L j + p R j [ α m i , α n j ] = m δ i j δ m , − n [ α ˜ m i , α ˜ n j ] = m δ i j δ m , − n

For each vibration mode α m i or α ˜ m j satisfy a harmonic oscillator algebra with nonstandard normalization, with

α m μ ~ m 1 / 2 a , α − m μ ~ m 1 / 2 a † α ˜ m μ ~ m 1 / 2 a ˜ , α ˜ − m μ ~ m 1 / 2 a ˜ †

where [ a , a † ] = 1 and [ a ˜ , a ˜ † ] = 1 . Here a † and a ˜ † are generators of a vibrational state; a and a ˜ are the annihilator of a vibrational state. With this normalization, the Equation (12) becomes:

X μ ~ x μ + p L μ z + p R μ z ¯ + i ( 1 π α ) 1 / 2 ∑ n = − ∞ n ≠ 0 ∞ 1 n [ a exp ( − i π n z ) + a ˜ exp ( − i π n z ¯ ) ]

The frequency of the nth vibrations in each series is:

f n = n / T

The probability of the occurrence of the n-th vibration is:

P n = 1 n P 1

This reveals that the probability of the occurrence of a vibration is inversely proportional to its rank or frequency. The energy of the nth vibration is:

E n = n E 1

This indicates that the energy of each oscillation is proportional to its rank or frequency. The contribution from the n-th vibration to the total energy over the time is:

E ′ n = P n E n = P 1 E 1

This demonstrates that the energy contribution from each vibrational mode is the same for all frequencies.

Equation (12) reveals that the holographic action, in the presence of a negligible background field, describes a system in which physical spacetime consists of a series of vibrations. The amplitude of each vibration is inversely proportional to the square root of its frequency, while the energy contribution from each frequency remains constant. These characteristics are hallmarks of 1/f noise. Therefore, 1/f noise can be interpreted as arising from spacetime excitations within the framework of holographic quantum theory under negligible background conditions.

DerivationofZipf’sLawfromHolographicQuantumTheory

Zipf’s law is an empirical law that states that for many types of data, the frequency of any item is inversely proportional to its rank in the frequency table. The most famous example is in natural language, where the frequency of any word is proportional to its rank in the frequency table.

In the context of holographic quantum theory, the derivation of pink noise—outlined in Equations (12) to (17)—reveals that the probability P n associated with the occurrence of a discrete state is inversely proportional to the integer n. This integer can be naturally interpreted as the rank of the discrete state. If one models the occurrence of a word as a discrete state within this framework, then the derived expression implies that the probability of a word’s occurrence (analogous to a vibration mode) scales inversely with its rank.

This correspondence suggests that Zipf’s law emerges intrinsically from the statistical structure of spacetime excitations governed by holographic quantum dynamics in the limit of a negligible background field. The result provides a theoretical foundation for Zipf’s law, linking them to holographic principles and the unified theory derived from it.

TheGutenberg-RichterLawandHolographicQuantumTheoryinSeismology

In seismology, the Gutenberg-Richter (GR) law describes the statistical relationship between the magnitude M of an earthquake and the total number N of earthquakes exceeding that magnitude:

log 10 N = a − b M

where a and b are empirical constants.

We can model Earth’s seismic activity within the holographic framework. As a firstorder approximation, we treat the Earth’s interior as uniform, which renders the background field effectively constant. Under this assumption, the Earth supports internal vibrational modes analogous to those described by Equations (10)-(17). These modes correspond to macroscopic standing waves, with characteristic wavelengths on the order of L/n, where L is the Earth’s radius and n is an integer ranging from 1 to very large values.

Assuming an earthquake is triggered when a region of the Earth resonates with and absorbs the energy of one of these vibrational modes, the earthquake’s magnitude M can be related to the energy En of the n-th mode. Use a logarithmic scale for magnitude of the earthquake:

M ~ log 10 E n = log 10 ( n E 1 ) = log 10 n + const

Furthermore, the frequency of occurrence N of earthquakes of a given magnitude is proportional to the probability Pn of the corresponding vibrational mode being excited:

N ~ P n = 1 n P 1

From Equation (19), we have log 10 N = − log 10 n + const . Substituting n from Equation (20) ( M ~ log 10 n ), we get:

log 10 N = − M + const

This is precisely the form of the Gutenberg-Richter law with a prediction that constant b appearing in the expression of Gutenberg-Richter (GR) law (15) being 1 in the approximation of a uniform earth.

In the derivation of the Gutenberg-Richter (GR) law presented above, we assume that earthquakes result from the absorption of the Earth’s internal vibrations. This hypothesis requires to be validated by the experiment.

4. Discussion

The derivation of fractals, 1/f noise, Zipf’s law, and the Gutenberg-Richter law from a unified theory based on the holographic principle carries significant implications. Although self-organized criticality (SOC) has long served as a popular framework for explaining these phenomena, it has lacked a fundamental mathematical formulation grounded in first principles. The derivation presented here provides a more foundational and comprehensive explanation for the emergence of complexity. It demonstrates that the unified theory based on the holographic principle not only integrates quantum physics and general relativity, but also possesses the predictive power to account for and unify the behavior of complex systems.

Our work suggests that the observed universality of power-law behaviors in nature is not a coincidence but a direct consequence of the holographic nature of reality. The scaling invariance inherent in the holographic action provides a natural explanation for the self-similarity observed in fractals and the scale-free nature of 1/f noise, Zipf’s law, and Gutenberg-Richter (GR) law. This provides a deeper and more unified understanding than phenomenological models like the sandpile models of SOC, which are often system-specific.

The holographic framework also provides a new perspective on the relationship between information, spacetime, and physical phenomena. By treating information as the fundamental entity, we can derive the properties of spacetime and the laws of physics, rather than assuming them as a starting point. This approach has the potential to resolve some of the long-standing problems in physics, such as the unification of gravity and quantum mechanics, derivation of the cosmological constant, matter-anitmatter asymmetry, and CP violation in weak interaction from a fundamental theory [10][11].

4.1. Comparison with Self-Organized Criticality

The theory of self-organized criticality (SOC), introduced by Bak, Tang, and Wiesenfeld [8][9], has been a popular and influential framework for explaining the emergence of power-law behaviors in complex systems. SOC proposes that complex systems naturally evolve to a critical state without the need for external tuning, giving rise to the observed scale-invariant behaviors. However, SOC has largely remained a phenomenological model. While it provides valuable insights through specific examples like the sandpile model, it lacks a general, abstract mathematical formulation and a fundamental underlying mechanism rooted in the basic laws of physics.

The holographic approach presented in this paper offers a more fundamental and comprehensive explanation. By grounding the emergence of complexity in the established principles of quantum physics and general relativity, we provide a theoretical foundation that goes beyond system-specific models. The scaling invariance inherent in the holographic action is not an emergent property of a particular system but a fundamental symmetry of the underlying theory. This provides a natural and elegant explanation for the self-similarity observed in fractals and the scale-free nature of 1/f noise, Zipf’s law, and Gutenberg-Richter (GR) law. In essence, our work suggests that the observed universality of power-law behaviors in nature is not a mere coincidence but a direct and inevitable consequence of the holographic nature of reality.

 4.2. Information, Spacetime, and Physical Phenomena

The holographic framework provides a new and powerful perspective on the intricate relationship between information, spacetime, and physical phenomena. By treating information as the fundamental entity, we are able to derive the properties of spacetime and the laws of physics, rather than assuming them as a starting point. This approach represents a paradigm shift in our understanding of the physical world. It suggests that spacetime is not a pre-existing stage upon which physical events unfold, but rather an emergent structure that arises from the underlying information content of the universe.

This information-centric view has the potential to resolve some of the most long-standing and challenging problems in modern physics, such as the unification of gravity and quantum mechanics. By providing a common mathematical framework that encompasses both quantum field theory and general relativity and explain the universal phenomena in complex system, the holographic principle offers a promising pathway towards a theory of everything.

 4.3. Experimental Predictions and Testability

A significant challenge for any fundamental theory is to provide predictions that are experimentally testable. The holographic framework presented here offers several avenues for verification and experimental exploration:

1) Deviations from Ideal Power Laws: The derivations of 1/f noise, Zipf’s law, and the Gutenberg-Richter law were performed in the simplified limit of a negligible background field (i.e., flat spacetime). The theory predicts that in the presence of significant background fields Gμν or Bμν (as described in Equation (6)), deviations from the ideal 1/f or 1/r power laws should occur. This is a strong, testable prediction. For instance, one could study condensed matter systems where effective background fields can be controlled (e.g., via electromagnetic fields or strain) and look for systematic deviations in the noise spectrum of electron transport. Similarly, in astrophysics, the 1/f noise observed in the light curves of quasars could be analyzed for deviations that correlate with the surrounding spacetime environment.

2) Cosmological Signatures: The theory proposes that the large-scale structure of the universe is fundamentally holographic and fractal. This could lead to specific, non-Gaussian signatures in the Cosmic Microwave Background (CMB) radiation or in the statistical distribution of galaxies. While current observations are consistent with the standard ΛCDM model, a re-analysis of cosmological data through the lens of this holographic theory could reveal subtle correlations or anisotropies not predicted by standard models.

3) Universality of the Gutenberg-Richter “b” value: The theory predicts that the “b” value in the Gutenberg-Richter law (Equation (10)) should be approximately 1 in the idealized case. However, the model also predicts that this value should be modulated by the local geological properties of a region, which act as an effective background field. Therefore, the theory predicts a relationship between the measured “b” value in a specific seismic zone and its underlying physical characteristics (e.g., stress, temperature, composition). This provides a framework for moving beyond empirical observation to a predictive model of seismicity.

4) Interdisciplinary Applications: The framework can be used to model complex systems outside of physics. For example, in neuroscience, brain activity often exhibits 1/f noise. The holographic model would predict that changes in cognitive state or external stimuli (acting as background fields) should lead to predictable changes in the spectral properties of EEG or fMRI signals. This provides a new theoretical tool for analyzing data in these fields.

5) The derivation of the Gutenberg-Richter (GR) law presented here posits that earthquakes arise from the absorption of internal vibrational energy within the Earth’s structure. If this hypothesis is correct, at the similar geographic setting, such as a series of earthquakes at the same places close to each other in time, one should observe the energy of each of the earthquake should be of the form n E 0 . One can test this whether this is true with current seismic data.

The above predictions can be investigated with current and future experimental capabilities. We will refer this to the future work. These testable predictions demonstrate our unified theory based on the holographic principle is not merely a philosophical construct but has tangible consequences and is able to make useful predictions.

 4.4. Limitations and Range of Validity

It is crucial to acknowledge the approximations and limitations inherent in the derivations presented. These limitations do not diminish the potential of the holographic framework. Rather, they highlight the path forward for developing it from a foundational principle into a fully predictive, quantitative theory of complexity.

 5. Conclusions

In this paper, we have shown that the universal phenomena of complex systems, including fractals, 1/f noise, and Zipf’s law, can be derived from a unified theoretical framework based on the holographic principle. By introducing a holographic action that unifies quantum physics, general relativity, and thermodynamics, we have provided a mathematical foundation for the emergence of complexity from simple physical laws. Our results suggest that the holographic principle is not only a key to understanding the quantum nature of gravity, all fundamental forces, elementary particles, dark matter, dark energy, large structure of our universe, but also a fundamental principle governing the organization of complex systems throughout the universe.

The unified holographic quantum theory offers a new and powerful tool for studying complexity, providing a more fundamental and unified perspective than previous approaches such as self-organized criticality. This work opens up new avenues for research into the nature of complexity and the fundamental laws of the universe. By continuing to explore the implications of the holographic principle, we may one day be able to answer some of the deepest questions in science, from the origin of life to the nature of consciousness.

 Acknowledgements

The author thanks Kunquan Lu at the Institute of Physics, Chinese Academy of Sciences, for his insightful research, discussions, and advice which inspired this work. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.

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