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

Dedication

This work is respectfully and wholeheartedly dedicated to Professor Tadashi Takayanagi, Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Japan. 

His groundbreaking contributions to holography, quantum entanglement, and the profound interplay between geometry and information have transformed the landscape of modern theoretical physics. 

From the AdS/BCFT correspondence to the pioneering formulation of pseudo-entropy and the illumination of entanglement structures in de Sitter holography, his ideas have continually opened new conceptual horizons for understanding spacetime, duality, and the quantum foundations of geometry. 

The Theory of Entropicity (ToE) is offered here in admiration and profound appreciation of his influence and in recognition of the intellectual foundations he has laid for future generations of researchers in quantum gravity, holography, and the physics of information.

Prologue

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 conceptual innovations embedded in the mathematical structure of the Theory of Entropicity (ToE) introduce a framework that resonates with, and extends beyond, the established formalisms of modern physics. These formulations yield results with potential implications for the Standard Model and, in particular, for the interpretation of the Higgs field. They demonstrate how comparable mathematical architectures—such as those that govern mass generation and field curvature—may naturally emerge within an alternative informational or thermodynamic context.

Within this perspective, the appearance of an effective mass term derived from the curvature of an entropy potential mirrors the function of the Higgs potential, which generates particle masses through spontaneous symmetry breaking. This parallel does not supplant the Higgs mechanism but suggests that similar structures can arise from purely entropic or information-geometric principles.

In this broader interpretation, the Higgs field and its associated boson exemplify a particular mani festation of a deeper and more general phenomenon: the emergence of inertial mass from curvature in an underlying scalar potential—be it physical, geometric, or entropic in origin. Consequently, the ToE framework expands the conceptual boundaries of field theory, proposing that mass, symmetry, and curvature can all be understood as distinct expressions of entropy-driven geometry.

More broadly still, this view intimates that the fundamental dynamical laws of the Standard Model may share profound structural affinities with thermodynamic and informational principles. It implies that the long-standing division between energy-based and entropy-based descriptions of nature may be less absolute than once believed, hinting at a deeper unifying language underlying both.

The current work further embeds pseudo–entropy into a broader landscape of entropic approaches— Jacobson’s thermodynamic derivation of Einstein equations, Padmanabhan’s emergent spacetime, Verlinde’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 dynamics from an underlying entropic field. This paper 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.

App Deployment on the Theory of Entropicity (ToE):

App on the Theory of Entropicity (ToE): Click or Open on web browser (a GitHub Deployment - WIP): Theory of Entropicity (ToE)

https://phjob7.github.io/JOO_1PUBLIC/index.html

 

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References

1. Obidi, John Onimisi (27th December, 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.30958670
2. Obidi, John Onimisi. (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
3. 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
4. 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
5. 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
6. 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
7. 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
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9. Obidi, John Onimisi. (4 November, 2025). The Theory of Entropicity (ToE) Derives Einstein’s Relativistic Speed of Light © as a Function of the Entropic Field: ToE Applies Logical Entropic Concepts and Principles to Derive Einstein’s Second Postulate. Cambridge University. https://doi.org/10.33774/coe-2025-f5qw8-v2
10. Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
11. HandWiki contributors, “Physics:Theory of Entropicity (ToE) Derives Einstein’s Special Relativity,” HandWiki, https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity_(ToE)_Derives_Einstein%27s_Special_Relativity&oldid=3845936

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-1: Theory of Entropicity (ToE)
4. Notion-2: Theory of Entropicity (ToE)
5. Notion-3: Theory of Entropicity (ToE)
6. Notion-4: Theory of Entropicity (ToE)
7. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack
8. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium
9. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
10. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
11. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).
12. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
13. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
14. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
15. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia
16. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate
17. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare
18. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea
19. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN
20. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).
21. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar
22. IJCSRR: International Journal of Current Science Research and Review - Theory of Entropicity (ToE) - John Onimisi Obidi | IJCSRR
23. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)