On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE) - The Obidi Actions and the Obidi Field Equations

On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE)

The Generalized Obidi Action of the Theory of Entropicity (ToE)

A Representative Equation for the Entropy Field Equations of the Theory of Entropicity (ToE)

On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE)

The Mathematical Landscape of ToE

The Theory of Entropicity (ToE) is built on the radical idea that entropy is the fundamental field of reality. To capture this, ToE introduces a suite of unprecedented constructs — the Obidi Action, the Master Entropic Equation (MEE), the Vuli–Ndlela Integral, the Entropy Potential Equation, and the No-Rush Theorem — all designed to unify thermodynamics, relativity, and quantum mechanics within a single entropic continuum. These tools recast the familiar laws of physics as consequences of entropy field dynamics.

The Obidi Field Equation (OFE)

The Obidi Field Equation (generally called the Master Entropic Equation — MEE) is a key component of the Theory of Entropicity (ToE), which redefines entropy as the fundamental field of reality. It is derived from the Obidi Action and is used to govern the evolution of the entropy field in spacetime. The equation is nonlinear and nonlocal, reflecting the probabilistic nature of entropy. It is solved iteratively and is used to describe the dynamics of the entropic field, which is believed to be responsible for all physical phenomena.

Information Geometries

At the foundation of ToE lies a manifold of states equipped with multiple information geometries. The Fisher–Rao geometry measures statistical curvature, the Fubini–Study geometry encodes quantum coherence, and the α-geometry introduces asymmetry and irreversibility into information transport. Together, these geometries provide the scaffolding on which entropy flows, ensuring that both classical and quantum domains are represented within a single entropic continuum.

Entropy Sector

ToE incorporates a wide family of entropy measures, each adapted to different physical regimes. The Shannon entropy is the classical measure of uncertainty in probability distributions, forming the bedrock of information theory. The von Neumann entropy extends this to quantum states, capturing the informational content of density matrices. Alongside these, ToE includes the Rényi entropy for scale sensitivity, the Tsallis entropy for nonextensive systems, the Kullback–Leibler divergence for directional information change, and the Araki relative entropy for quantum comparisons. By weaving all of these into its framework, ToE ensures that entropy is not treated as a single formula but as a universal family of measures that adapt to classical, quantum, and statistical contexts.

Local Obidi Action

The Local Obidi Action is the geometric sector of ToE. It integrates curvature, asymmetric transport, and entropy gradients into a single variational principle. This action describes how entropy interacts with the underlying geometry of the manifold, ensuring that entropic flow is inseparable from the curvature and structure of space itself. It is here that ToE begins to unify thermodynamics and relativity, showing that spacetime curvature can be understood as a manifestation of entropy dynamics.

Spectral Obidi Action and Operator Geometry

Complementing the local action is the Spectral Obidi Action, which introduces a Dirac-type entropy operator. The spectrum of this operator regulates coherence, scale, and regularity. This spectral perspective allows ToE to encode quantum features directly into its mathematics. The spectral operator geometry emerges from the asymptotics of this operator, linking spectral data to geometric structure. In this way, coherence and irreversibility are not external assumptions but built into the entropic field itself.

Coupling Terms

ToE includes explicit coupling terms that link geometry, entropy, and spectral structure. These couplings ensure that no sector evolves in isolation: geometry modulates entropy flow, entropy interacts with spectral coherence, and spectral properties feed back into geometric curvature. This interdependence is what makes ToE a unified framework rather than a patchwork of separate theories.

Master Entropic Equation (MEE)

From the unified Obidi Action arises the Master Entropic Equation (MEE). This is the governing equation of the entropy field, balancing geometric diffusion, entropy production, spectral coherence, and causal correction. It is highly nonlinear and nonlocal, reflecting the complexity of reality itself. The MEE is the mathematical heart of ToE, the place where all sectors converge into a single dynamical law.

Vuli–Ndlela Integral

A distinctive innovation of ToE is the Vuli–Ndlela Integral, which reformulates quantum path integrals to include irreversibility. Unlike traditional formulations that treat time symmetrically, the Vuli–Ndlela Integral weights paths by entropic cost, embedding the arrow of time directly into quantum mechanics. This construct explains temporal asymmetry not as an emergent phenomenon but as a fundamental feature of entropic dynamics.

Entropy Potential Equation

The Entropy Potential Equation defines the effective energy landscape of the entropy field. It describes how entropy gradients shape the evolution of the field, guiding flow and interaction. This equation provides the structure within which iterative solutions of the MEE are carried out, ensuring that entropic evolution follows a coherent trajectory.

No-Rush Theorem

The No-Rush Theorem imposes a universal temporal bound on interactions. It formalizes the principle that entropy cannot redistribute instantaneously but is constrained by a finite rate. This bound corresponds to the constancy of light, making Einstein’s second postulate a consequence of entropic dynamics. In ToE, causality is not imposed externally but arises naturally from the finite-rate redistribution of entropy.

Unified Vision

Taken together, these constructs form a tightly interwoven system. The Local Obidi Action governs geometric aspects, the Spectral Obidi Action encodes coherence, the entropy sector provides multiple measures of information including Shannon and von Neumann, the coupling terms bind everything together, the MEE governs dynamics, the Vuli–Ndlela Integral introduces irreversibility, the Entropy Potential Equation defines structure, and the No-Rush Theorem enforces causality.

Interpretive Summary

The mathematics of ToE is complex because its ambition is vast: to unify thermodynamics, relativity, and quantum theory within a single entropy-driven continuum. By elevating entropy to the status of a universal field, ToE provides a rigorous architecture in which time’s arrow, the constancy of light, quantum coherence, and spacetime curvature are all explained as consequences of entropic dynamics. In this vision, entropy shapes geometry, governs motion, and creates spacetime.

References

1. Obidi, J. O. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
2. John Onimisi Obidi. (6th November, 2025). Comparative analysis between john onimisi obidi’s theory of entropicity (toe) and waldemar marek feldt’s feldt–higgs universal bridge (f–hub) theory. International Journal of Current Science Research and Review, 8(11), pp. 5642–5657, 19th November 2025. URL: https: //doi.org/10.47191/ijcsrr/V8-i11–21.
3. Obidi, John Onimisi. 2025. On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
4. Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
5. Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
6. Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
7. Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).

Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE — https://theoryofentropicity.blogspot.com

2. LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true

3. Notion: Theory of Entropicity (ToE)

4. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack

5. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium

6. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

7. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

8. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).

9. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

10. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

11. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

12. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia

13. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate

14. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare

15. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea

16. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN

17. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).

18. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar

19. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory

The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory

                               The Obidi Action of the Theory of Entropicity (ToE)

This paper presents a systematic comparison between the recently developed pseudo entropy framework of Takayanagi, Kusuki, and Tamaoka and the Theory of Entropicity (ToE). While pseudo–entropy has revealed a remarkable boundary route to the linearized Einstein equation in dS3, the Theory of Entropicity proposes a far more fundamental idea: that entropy is not a boundary diagnostic of geometry, but the underlying field from which geometry, matter, motion, and time themselves emerge. The discussion that follows demonstrates how the pseudo–entropy program fits naturally within the broader structure of ToE, and how the ToE framework generalizes, extends, and ultimately surpasses it. The pseudo–entropy construction shows that a non–Hermitian generalization of entanglement entropy in a two–dimensional CFT satisfies a first law whose bulk dual reproduces the perturbative Einstein equation in dS3. Moreover, infinitesimal variations of pseudo–entropy obey a Klein–Gordon equation on a kinematic dS2 space, suggesting the emergence of time from Euclidean CFT data. In this paper, we reinterpret these results within the Theory of Entropicity by showing that the same Klein–Gordon structure appears as the boundary–projected, linearized limit of the Master Entropic Equation derived from the Local Obidi Action. Thus, what pseudo–entropy identifies kinematically from the boundary, ToE generates dynamically in the bulk through the entropic field S(x). The manuscript further embeds pseudo–entropy into a broader landscape of entropic approaches — Jacobson’s thermodynamic derivation of Einstein equations, Padmanabhan’s emergent spacetime, Ver linde’s entropic gravity, Caticha’s entropic inference, and Bianconi’s metric relative entropy. Where these earlier programs emphasize information, thermodynamics, or emergence, ToE provides a uni fying ontological principle: entropy itself is the fundamental field of the universe. By promoting the modular–like operator ∆ to a dynamical object through the Spectral Obidi Action, ToE offers a natural explanation of dark matter, dark energy, and vacuum entropic pressure — domains entirely absent from the pseudo–entropy framework. This paper shows explicitly how Bianconi’s relative–entropy action and the Takayanagi–Kusuki–Tamaoka pseudo–entropy construction both appear as limiting cases of the Obidi Actions. Finally, we demonstrate that ToE provides a unified entropic–spectral variational principle in which bosons and fermions arise from the same foundational structure. The spectral interpretation of bosonic actions, the Dirac–based fermionic bilinears, and geometric actions such as Einstein–Hilbert and Yang–Mills all emerge as projections of the Local and Spectral Obidi Actions. This paper therefore positions pseudo–entropy not as an alternative to ToE, but as a special holographic shadow of a deeper entropic field theory. In this sense, the present work does not merely compare two independent approaches. Rather, it establishes a hierarchical synthesis: pseudo–entropy reconstructs gravity from boundary information, while the Theory of Entropicity constructs gravity, geometry, quantum structure, and temporal dynam ics from an underlying entropic field. This manuscript argues that pseudo–entropy is best understood not as a standalone gravitational principle, but as a boundary manifestation of the universal entropic dynamics formulated by the Theory of Entropicity (ToE).

Abstract

The recent work of Takayanagi, Kusuki, and Tamaoka has introduced the concept of holographic pseudo-entropy in non-unitary CFT2 and demonstrated a striking equivalence: the first law of pseudo entropy is precisely dual to the linearized Einstein equation in three-dimensional de Sitter space (dS3) once one allows complexified extremal surfaces in the bulk. Moreover, variations of pseudo-entropy obey a Klein–Gordon equation on the kinematical space dS2, offering an emergent time structure arising from an Euclidean boundary theory. In this paper we show that while the holographic pseudo-entropy program represents an important boundary diagnostic of gravitational dynamics, it remains a restricted kinematical construction tied to holography, non-unitary conformal field theories, and perturbative de Sitter gravity. By contrast, the Theory of Entropicity (ToE) treats entropy S(x) as the fundamental physical field of nature, endowed with a local variational principle (the Local Obidi Action) and a spectral variational principle (the Spectral Obidi Action). From these actions one derives the Master Entropic Equation, entropic geodesics, irreversible dynamics, and a unified description of gravity, time, quantum processes, and information geometry. The goal of this work is threefold. First, we present a precise and self-contained exposition of the Takayanagi–Kusuki–Tamaoka framework. Second, we develop the Theory of Entropicity as a universal entropic field theory whose dynamics extend far beyond the holographic pseudo-entropy correspondence. Third, we provide a systematic comparison showing how ToE absorbs pseudo-entropy as a special boundary manifestation of a deeper entropic field, thereby revealing why pseudo-entropy reproduces only the linearized sector of gravitational physics while ToE yields a fully nonlinear, time-asymmetric, and information-geometric unification of physical law.

Keywords

Amari–Čencov α–Connections; Araki Relative Entropy; Atiyah–Singer Index Theorem;

Bekenstein–Hawking Entropy; Bosons; Canonical Quantization; Complex Geodesics; Dark Matter; Dark

Energy; dS/CFT Correspondence; Dirac–Kähler Fermions; Dirac Spinors; Einstein–Hilbert Action; Emer

gent Geometry; Entropic Field; Entropic Geodesics; Entropy Geometry; Entropy as Ontic Field; Fermions;

Fisher–Rao Metric; Fubini–Study Metric; G-Field (Bianconi); Ginestra Bianconi; Holographic Pseudo

Entropy; Information Geometry; Jacobson Thermodynamics; Kinematic Space (dS2); Klein–Gordon

Equation (Pseudo-Entropy); Local Obidi Action (LOA); Master Entropic Equation (MEE); Modular Op

erator ∆; Nonlinear Entropic Dynamics; Obidi Actions; Padmanabhan Entropic Gravity; Pseudo-Entropy

(Takayanagi–Kusuki–Tamaoka); Quantum Entanglement; Quantum Gravity; Rényi Entropy; Relative

Entropy; Shannon Information; Small Positive Cosmological Constant; Spectral Action; Spectral Dynam

ics; Spectral Geometry; Spectral Obidi Action (SOA); Spectral Theories; Takayanagi–Kusuki–Tamaoka

Pseudo-Entropy; Theory of Entropicity (ToE); Thermodynamic Gravity; Tsallis Entropy; Vuli–Ndlela

Integral; Yang–Mills Theory.

References

Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1

Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE —

https://theoryofentropicity.blogspot.com

2. LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true

3. Notion: Theory of Entropicity (ToE)

4. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack

5. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium

6. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

7. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

8. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).

9. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

10. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

11. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

12. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia

13. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate

14. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare

15. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea

16. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN

17. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).

18. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar

19. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)

On the Union of Ginestra Bianconi and John Onimisi Obidi: Progress of the Theory of Entropicity (ToE) as a Field Theory in Theoretical and Mathematical Physics

On the Union of Ginestra Bianconi and John Onimisi Obidi: Progress of the Theory of Entropicity (ToE) as a Field Theory in Theoretical and Mathematical Physics

                     The Obidi Action of the Theory of Entropicity (ToE)

The evolution of theoretical physics has often been marked by unexpected unions — moments when distinct intellectual traditions converge to produce new frameworks of understanding. The collaboration between Ginestra Bianconi, known for her pioneering work in network theory and statistical mechanics, and John Onimisi Obidi, whose contributions to entropic formulations and mathematical physics have been steadily gaining recognition, represents one such union. Their combined perspectives have given rise to a more formal articulation of the Theory of Entropicity (ToE), positioning it as a candidate for a field theory in theoretical and mathematical physics.

                            The Obidi Action of the Theory of Entropicity (ToE)

Entropicity as a Conceptual Foundation

Entropy has long been a cornerstone of physics, from thermodynamics to information theory. Yet, the notion of entropicity extends beyond entropy as a measure of disorder. It suggests a principle of generative structure, where systems evolve not merely toward equilibrium but toward complex configurations that balance order and randomness.

Bianconi’s work on multilayer networks and phase transitions provides the mathematical scaffolding for this idea, while Obidi’s formulations emphasize entropicity as a field property — a quantity that can be distributed, conserved, and transformed across domains of physics.

Toward a Field Theory of ToE

The ambition of ToE is not simply to describe entropy but to elevate entropicity into a unifying field. This involves:

• Defining entropicity as a tensorial field, capable of interacting with matter and energy.
• Exploring its role in phase transitions, where entropicity mediates between microstates and macrostates.
• Extending its reach into quantum information, where entropicity may serve as a bridge between classical thermodynamics and quantum coherence.

Such a formulation aligns with the broader tradition of field theories in physics, from electromagnetism to quantum chromodynamics, but introduces a novel axis: the dynamics of complexity itself.

Implications for Theoretical and Mathematical Physics

The union of Bianconi and Obidi’s approaches suggests several promising directions:

• Network Cosmology: Viewing the universe as a multilayer network, where entropicity governs connectivity and evolution.
• Information Geometry: Embedding entropicity within geometric frameworks, linking statistical manifolds to physical fields.
• Complex Systems Physics: Providing a rigorous field-theoretic language for phenomena ranging from biological evolution to social dynamics.

In each case, entropicity functions not as a metaphor but as a quantifiable field variable, opening pathways for predictive modeling and experimental validation.

Conclusion: A New Horizon

The Theory of Entropicity, as advanced through the union of Bianconi and Obidi’s insights, represents a bold step toward reconceptualizing entropy not as a passive measure but as an active field of physics. If successful, ToE could unify disparate domains — thermodynamics, information theory, quantum mechanics — under a single entropic framework.

For theoretical and mathematical physics, this is more than an incremental advance; it is a reorientation toward complexity as a fundamental property of nature.

References

Obidi, John Onimisi (12 Nov. 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Figshare. https://doi.org/10.6084/m9.figshare.30596129.v1

Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE —

https://theoryofentropicity.blogspot.com

2. LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true

3. Notion: Theory of Entropicity (ToE)

4. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack

5. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium

6. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

7. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

8. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).

9. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

10. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

11. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

12. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia

13. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate

14. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare

15. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea

16. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN

17. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).

18. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar

19. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE) 

The Theory of Entropicity (ToE) Explains the Constancy of the Speed of Light c in Albert Einstein’s Theory of Relativity (ToR): The Entropic Origin of the Speed of Light c from the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE) Explains the Constancy of the Speed of Light c in Albert Einstein’s Theory of Relativity (ToR)

The Entropic Origin of the Speed of Light c from the Theory of Entropicity (ToE)

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The Theory of Entropicity (ToE) Explains Why the Speed of Light c is a Constant in Einstein’s Theory of Relativity (ToR)

We have previously dwelt on how the Theory of Entropicity (ToE) explains why the speed of light is a constant (c) according to Einstein’s beautiful and revolutionary Theory of Relativity (ToR). Here, we dive deeper into giving the core mechanism for this action in the Theory of Entropicity (ToE) [first formulated by John Onimisi Obidi — Researcher, Investigator, Thinker, Physicist, Philosopher, and Humanist — different from the unrelated social media personality of a similar name] that explains Einstein’s constancy of the speed of light in his Theory of Relativity (ToR) - that is, we are providing a deeper version of ToE’s explanation of the "why" behind the "what."

Further Exploratory and Revolutionary Insights of the Theory of Entropicity (ToE) Into Explaining the Constancy of the Speed of Light (c) in Einstein’s Theory of Relativity (ToR).

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Table of Contents

1. What "You Are Made of the Field" Actually Means
Let's use a better analogy than a movie screen which we have used elsewhere:[1] A whirlpool in a river.

The River: This represents the fundamental Entropic Field. It has an intrinsic property: the maximum speed its current can flow.

The Whirlpool: This represents a stable object—a particle, a clock, an observer. A whirlpool is not a separate "thing" floating in the water; it is a stable, organized pattern of the water itself. Its existence is defined by its structure and its interaction with the surrounding flow.

When we say "you are made of the field,"[1] it means you (and your clock and ruler) are like that whirlpool. You are not an independent entity moving through a passive, empty space. You are a complex, stable pattern of activity within the universe's fundamental entropic field.

Your physical being, the forces holding your atoms together, and the regular processes inside your clock are all manifestations of the rules of this field. Therefore, your properties are not absolute; they are relational and dependent on your interaction with the rest of the field (the river).

2. The ToE Mechanism for Time Dilation and Length Contraction
To explain why time slows and length contracts, we must use the core principles of ToE, which are all based on entropy.

Here is the proposed mechanism, step-by-step.

The Foundational Principle: Existence is a Battle Against Entropy
According to ToE (and standard physics), any organized system—a clock, a ruler, a living cell—is a low-entropy state. To exist, it must constantly work to maintain its structure and order against the universe's natural tendency to dissolve into chaos (the Second Law of Thermodynamics). It does this by processing information and managing entropy flow. Think of it as having an "entropic budget"[2][3][4][5][6] just to remain stable and functional.

Step 1: Explaining Time Dilation (Why Clocks Slow Down)
What is a Clock? A clock is a system designed to perform a highly regular, repeating internal process (a "tick"). This tick could be the swing of a pendulum, the vibration of a crystal, or the transition of a cesium atom.

The "Entropic Cost" of Ticking: For this tick to be regular, the clock must use part of its "entropic budget" to ensure the process repeats identically, fighting off microscopic disorder. This is its normal operating cost while stationary.

The "Entropic Cost" of Motion: According to ToE, "motion" is not passive. Moving through the entropic field means a system is constantly interacting with new regions of the field. This creates an additional and continuous entropic "load" or "stress" on the system. It's like a swimmer not only having to manage their own body heat and energy (internal entropy) but also having to fight the current and drag of the water (external entropic interaction).

The Result of the Theory of Entropicity (ToE): Reallocation of the Budget. The clock still has the same fundamental priority: it must maintain its structural integrity. Faced with this new external entropic load from motion, it must divert resources from its "entropic budget" to deal with it. The work of simply staying intact in the face of this motion takes precedence. Consequently, there is less "budget" available for its primary function—the regular "ticking."

Conclusion of the Theory of Entropicity (ToE): The internal process of the clock—its tick—is forced to run slower. Time dilation is the observable consequence of a system prioritizing its structural integrity over its internal functions when under the entropic stress of motion. Ordinarily, this is not obvious and not readily observable; but it becomes crucial as the speed increases; as the speed increases, it reaches a limit at which it wants to cross the entropic bound, then it is halted, because it cannot go faster than entropy! This is the No-Rush Theorem and also why Einstein's Relativity of the speed of light is constant and why it is the maximum speed of all propagations.

Entropy thus ensures that no interaction can occur faster or slower than Entropy allows, and no propagation can go faster than Entropy permits. This is ToE's No-Rush Theorem[7] in its most encompassing and celebrated form. Thus, Entropy is what actually dictates how fast or slow any motion should be. This conclusion is only natural, unavoidable and inescapable, because since Entropy is what dictates and constrains motion according to the Theory of Entropicity (ToE), then the same Entropy must constrain the speed of motion itself; that is, how fast or slow interactions and propagations and objects can move in the Entropic Field itself.

Step 2: Explaining Length Contraction (Why Rulers Shrink)
What is a Ruler? A ruler is a rigid object. Its length is a stable property defined by the equilibrium distance between its atoms. This equilibrium is a delicate balance of electromagnetic forces, which ToE recasts as constraints within the entropic field.

The "Entropic Headwind": When the ruler moves, it experiences the same entropic stress, but this time it's directional. The front of the ruler is continuously interacting with "new" parts of the field before the back does. This creates a kind of "entropic pressure" or "headwind" that pushes against the front of the ruler.

The Result: A New Equilibrium. The system (the ruler) must find a new stable state to cope with this constant directional pressure. The internal forces readjust to a new, slightly compressed equilibrium in the direction of motion. The atoms are pushed closer together until their repulsive forces are strong enough to balance the new external "entropic headwind."

Conclusion: The ruler physically becomes shorter in its direction of motion. Length contraction is the physical deformation of an object as it re-establishes structural equilibrium under the directional entropic pressure of moving through the field.

Table

This explanation attempts to ground the strange effects of relativity in a physical, causal mechanism rooted in entropy management, rather than leaving them as abstract geometric consequences of a postulate. It is a bold and unproven claim, but it is the kind of deeper explanation that the Theory of Entropicity aims to provide.

References

Obidi, John Onimisi. The Theory of Entropicity (ToE) Simply Explained Qualitatively. Encyclopedia. Available online: https://encyclopedia.pub/entry/58652 (accessed on 20 July 2025).
Obidi, John Onimisi. A Critical Review of the Theory of Entropicity (ToE) on Original Contributions, Conceptual Innovations, and Pathways towards Enhanced Mathematical Rigor: An Addendum to the Discovery of New Laws of Conservation and Uncertainty. Cambridge University; 30 June 2025. https://doi.org/10.33774/coe-2025-hmk6n
Obidi, John Onimisi. Einstein and Bohr Finally Reconciled on Quantum Theory: The Theory of Entropicity (ToE) as the Unifying Resolution to the Problem of Quantum Measurement and Wave Function Collapse. Cambridge University; 14 April 2025. https://doi.org/10.33774/coe-2025-vrfrx
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References

Obidi, John Onimisi (12 Nov. 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Figshare. https://doi.org/10.6084/m9.figshare.30596129.v1

Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE —  https://theoryofentropicity.blogspot.com

2. LinkedIn: Theory of Entropicity ToE —  https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true

3. Notion: Theory of Entropicity (ToE)

4. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack

5. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium

6. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

7. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

8. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).

9. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

10. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

11. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

12. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia

13. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate

14. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare

15. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea

16. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN

17. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).

18. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar

19. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)

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