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)