On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics

Conceptual and Mathematical Foundations of Theory of Entropicity(ToE)

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This entry is adapted from: 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. https://doi.org/10.6084/m9.figshare.30337396.v1

On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics

The Theory of Entropicity (ToE) reconceives entropy not as a statistical by‑product of disorder but as the fundamental field and causal substrate of physical reality. In this framework, entropy 𝑆(𝑥) is elevated to a continuous, dynamical field whose gradients generate motion, gravitation, time, and information flow. The Obidi Action, a variational principle for the entropy field, yields the Master Entropic Equation, Entropic Geodesics, and the Entropy Potential Equation, establishing a unified dynamical foundation. By integrating Fisher–Rao’s classical information metric and the Fubini–Study quantum metric through the Amari–Čencov α‑connection formalism, ToE provides a rigorous geometric and probabilistic structure for physical evolution within an entropic manifold.

At its core, the theory reformulates the speed of light as the maximum rate of entropic rearrangement, deriving relativistic and quantum constraints from finite entropy propagation. The No‑Rush Theorem imposes a universal temporal bound on interactions, while the Vuli‑Ndlela Integral, an entropy‑weighted reformulation of Feynman’s path integral, introduces irreversibility and temporal asymmetry into quantum mechanics. Together, these constructs unify thermodynamics, relativity, and quantum theory within a single entropy‑driven continuum.

Extending its foundations, ToE incorporates Rényi and Tsallis entropies, establishing a correspondence between generalized entropy measures and geometric structures. The entropic order parameter α emerges as a universal deformation index linking geometry, information, and entropy flow. In this synthesis, ToE reproduces Einstein’s field equations as a limiting case and subsumes Bianconi’s “Gravity from Entropy” as a special instance. Beyond physics, ToE suggests that mass, energy, spacetime, and consciousness arise as emergent constraints of a single entropic reality.

In this work, the Theory of Entropicity(ToE) confirms the G-Field and small positive cosmological constant predicted by the Theory of Ginestra Bianconi in her momentous work “Gravity from Entropy.”

Theory of Entropicity ToE Vuli-Ndlela Integral Ginestra Bianconi Unification of Physics Relativity Quantum Mechanics Fisher-Rao Information Geometry Fubini-Study Quantum Information Geometry Entropic Gravity Erik Verlinde

On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics

The Theory of Entropicity (ToE), as first formulated and developed by John Onimisi Obidi,^[1][2] reconceives entropy not as a statistical by‑product of disorder but as the fundamental field and causal substrate of physical reality. In this framework, entropy 𝑆(𝑥) is elevated to a continuous, dynamical field whose gradients generate motion, gravitation, time, and information flow. The Obidi Action, a variational principle for the entropy field, yields the Master Entropic Equation, Entropic Geodesics, and the Entropy Potential Equation, establishing a unified dynamical foundation. By integrating Fisher–Rao’s classical information metric and the Fubini–Study quantum metric through the Amari–Čencov α‑connection formalism, ToE provides a rigorous geometric and probabilistic structure for physical evolution within an entropic manifold.

At its core, the theory reformulates the speed of light as the maximum rate of entropic rearrangement, deriving relativistic and quantum constraints from finite entropy propagation. The No‑Rush Theorem imposes a universal temporal bound on interactions, while the Vuli‑Ndlela Integral, an entropy‑weighted reformulation of Feynman’s path integral, introduces irreversibility and temporal asymmetry into quantum mechanics. Together, these constructs unify thermodynamics, relativity, and quantum theory within a single entropy‑driven continuum.

Extending its foundations, ToE incorporates Rényi and Tsallis entropies, establishing a correspondence between generalized entropy measures and geometric structures. The entropic order parameter α emerges as a universal deformation index linking geometry, information, and entropy flow. In this synthesis, ToE reproduces Einstein’s field equations as a limiting case and subsumes Bianconi’s “Gravity from Entropy” as a special instance. Beyond physics, ToE suggests that mass, energy, spacetime, and consciousness arise as emergent constraints of a single entropic reality. In this work, the Theory of Entropicity(ToE) confirms the G-Field and small positive cosmological constant predicted by the Theory of Ginestra Bianconi in her momentous work “Gravity from Entropy.”

Physics: Theory of Entropicity (ToE) on Quantum Gravity and the Unification of Physics

By John Onimisi Obidi (Independent Research)
7 October 2025

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Abstract

The Theory of Entropicity (ToE) redefines entropy as the fundamental generative field of the universe. Where classical thermodynamics and information theory regard entropy as a statistical measure of disorder or uncertainty, ToE elevates it to a primary ontological status—the underlying reality from which motion, time, geometry, and matter emerge.

In this framework, entropy is not a passive outcome of physical interactions but an active, continuous field that governs all processes of transformation and perception. The Obidi Action, a variational principle central to ToE, expresses this concept mathematically by unifying thermodynamics, relativity, and quantum mechanics under one entropic field equation. Through it, the physical universe is described as a vast dynamic network of entropic flows seeking equilibrium, with curvature, inertia, and mass appearing as emergent features of that ongoing reorganization.

The theory further reinterprets the speed of light as the maximum permissible rate of entropic rearrangement, establishing an absolute limit on causal propagation encapsulated in the No-Rush Theorem. It reformulates the Feynman Path Integral into a new construct—the Vuli Ndlela Integral—where probability amplitudes are modified by entropy-dependent weighting terms that enforce time irreversibility and the arrow of time.

Mathematically, ToE synthesizes the Fisher–Rao information metric (the basis of statistical distinguishability), the Fubini–Study quantum metric (which defines distances between quantum states), and the Amari–Čencov α-connections (which capture dual geometric structures of information flow). Together, these are fused into an entropy-weighted geometric manifold that describes the structure of physical reality. Within this triadic framework, the Rényi and Tsallis entropies extend Shannon’s foundational measure, introducing deformation parameters that represent the nonlinearity of real-world systems and their tendency toward irreversibility.

The outcome is a coherent, mathematically rigorous, and conceptually profound model of the cosmos, offering an alternative path toward quantum gravity and the unification of physics—a path grounded not in quantization of spacetime, but in the self-organizing logic of entropy itself.

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The Entropic Foundation of Existence

From Newton’s deterministic mechanics to Einstein’s relativistic geometry, physics has evolved by refining our understanding of motion, force, and curvature. Yet even Einstein’s elegant theory left unanswered the question of why space and time curve or what drives their emergence. The Theory of Entropicity ventures beyond these boundaries by identifying entropy as the primal field from which both geometry and motion originate.

In traditional thermodynamics, entropy quantifies the number of microscopic configurations compatible with a macroscopic state. In statistical mechanics, it reflects the degree of uncertainty. In ToE, however, entropy becomes a local dynamical field—a continuum that permeates existence, whose gradients generate both the flow of time and the curvature of space.

This insight transforms the famous Einstein–Wheeler dictum—

“Matter tells spacetime how to curve, and spacetime tells matter how to move.”

into the Entropic Dictum:

“Entropy tells information how to flow, and information flow tells reality how to exist.”

Under this reinterpretation, space, time, inertia, and curvature are no longer fundamental building blocks of reality. They are emergent consequences of entropy’s drive to redistribute itself. The universe is not a static stage upon which entropy unfolds—it is the unfolding itself.

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Mathematical Framework: From Information Geometry to Entropy Geometry

ToE extends and transforms information geometry, the mathematical study of probability spaces endowed with geometric structure. Traditionally, information geometry employs three key formalisms: the Fisher–Rao metric, the Fubini–Study metric, and the Amari–Čencov α-connections. ToE shows that these are not isolated tools but facets of a deeper entropic manifold, unified by a single scalar field, S, and parameterized by a universal entropic order α.

1. The Fisher–Rao Metric
   In classical statistics, the Fisher–Rao metric quantifies the “distance” between probability distributions by measuring how distinguishable they are. ToE interprets this metric as the curvature of the macroscopic entropy field, where high entropy gradients correspond to strong curvature. The exponential weighting term, derived from the Obidi Action, embeds entropy directly into the geometry of the manifold, allowing gravitational-like curvature to arise naturally from entropic density.

2. The Fubini–Study Metric
   In quantum mechanics, the Fubini–Study metric defines the distance between quantum states in Hilbert space. In ToE, this metric acquires an entropic deformation: it is weighted by the local entropy density, making coherence and decoherence geometric consequences of entropy flow. In other words, quantum entanglement and collapse become expressions of entropic curvature in the complex projective space of states.

3. The Amari–Čencov α-Connections
   The Amari–Čencov formalism introduces a continuum of affine connections—mathematical structures that describe how vectors and information change along the manifold. The parameter α labels these dual connections, which ToE reinterprets as physical manifestations of entropy’s irreversibility. The deviation of α from zero (the Levi-Civita limit) quantifies the degree of time asymmetry in the system—the same asymmetry that gives rise to the arrow of time.

By incorporating the Rényi and Tsallis entropies, ToE further extends this geometry. These generalized entropies modify how probability distributions are weighted, leading to curvature deformations consistent with complex, non-equilibrium systems. Thus, the α-parameter of Amari–Čencov geometry becomes physically identified with the α of Rényi/Tsallis statistics: both control the degree of deviation from equilibrium and symmetry.

In this synthesis, Einstein’s field equations emerge as the α → 1 limit of the broader entropic field equations derived from the Obidi Action. What Einstein described as curvature of spacetime is reinterpreted in ToE as curvature of the entropic manifold—a more fundamental substratum from which geometry itself is born.

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Implications and Unification

The implications of ToE are profound and far-reaching. The theory establishes a unifying language for phenomena traditionally separated by scale and domain.

• Mass and Inertia as Entropic Phenomena
  Mass arises as a concentration of internal entropy—an entropic “knot” in the manifold that resists reconfiguration. Inertia, likewise, becomes resistance to changes in the local entropy distribution. Thus, Newton’s first law and Einstein’s mass-energy equivalence both find their origin in the stability properties of the entropic field.

• Gravity as Entropic Curvature
  Gravity is not a force but a manifestation of entropy gradients. Massive bodies deform the surrounding entropy field, and that deformation manifests as curvature—exactly as in general relativity, but derived from a more primitive principle.

• Quantum Behavior as Entropic Fluctuation
  Quantum probabilities emerge from micro-scale fluctuations in the entropic field. The Vuli Ndlela Integral replaces Feynman’s formulation by embedding entropy directly into the weighting of each path, thereby incorporating irreversibility and explaining the collapse of the wavefunction as a natural entropic process.

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Relation to Bianconi’s “Gravity from Entropy”

Recent research by Ginestra Bianconi has advanced an entropic view of gravity, proposing that spacetime geometry emerges from the relative entropy between two metrics—a matter-induced metric and a background metric. This leads to modified Einstein equations containing an auxiliary G-field and a small positive cosmological constant.

The Theory of Entropicity reveals that Bianconi’s framework is a special case of its own. When the ToE’s α-parameter approaches unity (the Shannon–Kullback–Leibler limit), the ToE field equations reduce precisely to Bianconi’s modified Einstein equations. In this limit, the entropic curvature tensor simplifies, and the auxiliary G-field emerges naturally as a Lagrange multiplier enforcing entropic consistency. The small positive cosmological constant identified by Bianconi corresponds directly to the entropic potential term in the Obidi Action, confirming that her theory is nested within ToE’s general structure.

This unification not only validates Bianconi’s approach but also shows that her cosmological constant is a geometric manifestation of the entropic field’s residual curvature—a universal background entropy density that drives cosmic acceleration.

Thus, Einstein’s geometric gravity and Bianconi’s entropic gravity are both sub-theories—limiting cases within the broader and more general Theory of Entropicity.

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Philosophical and Foundational Insight

Beyond mathematics, ToE transforms the philosophical understanding of physics. It restores unity to domains that have long seemed disjointed—thermodynamics, relativity, and quantum theory—by recognizing that each is a projection of one universal principle: entropy’s continuous drive toward equilibrium.

Time, in this view, is not an independent dimension but the ordering consequence of entropy flow. The passage of time reflects the sequential redistribution of entropy across the manifold. Stillness does not imply timelessness; it merely indicates local entropic equilibrium. Eternity, therefore, is not the absence of time but the complete uniformity of entropy—a timeless state of perfect equilibrium.

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Conclusion

The Theory of Entropicity (ToE) reveals that entropy is not disorder, but the source of order, structure, and geometry itself. Its field equations provide a universal bridge between the macroscopic and microscopic worlds, uniting gravity, quantum mechanics, and thermodynamics through a shared entropic geometry.

By embedding Rényi–Tsallis entropy structures within the Fisher–Rao, Fubini–Study, and Amari–Čencov frameworks, ToE pushes physics to its ultimate frontier—the entropic origin of reality. Through the Obidi Action, the No-Rush Theorem, and the Vuli Ndlela Integral, it constructs a coherent and testable foundation for quantum gravity and the unification of physics.

Entropy, in this grand synthesis, is not merely a measure of uncertainty—it is the language of existence, the field that writes the geometry of the universe, and the invisible current that drives all that is, all that becomes, and all that endures.

References

1. 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. https://doi.org/10.6084/m9.figshare.30337396.v1

2. Physics:HandWiki Master Index of Source Papers on Theory of Entropicity(ToE). (2025, September 9). HandWiki, . Retrieved 17:33, September 9, 2025 from https://handwiki.org/wiki/index.php?title=Physics:HandWiki_Master_Index_of_Source_Papers_on_Theory_of_Entropicity%28ToE%29&oldid=3743060

3. Hoffman DD. Observer Mechanics: A Formal Theory of Perception. Oxford (UK): Oxford University Press; 1989. Available from: https://global.oup.com/academic/product/observer-mechanics-9780198539767

4. Hoffman DD. Visual Intelligence: How We Create What We See. New York (NY): W. W. Norton & Company; 1998. Available from: https://wwnorton.com/books/9780393319675

5. Hoffman DD, Massa R. Automotive Lighting and Human Vision. Boca Raton (FL): CRC Press; 2005. Available from: https://www.crcpress.com/Automotive-Lighting-and-Human-Vision/Hoffman-Massa/p/book/9780849323071

6. Hoffman DD. The Case Against Reality: How Evolution Hid the Truth from Our Eyes. New York (NY): W. W. Norton & Company; 2019. Available from: https://wwnorton.com/books/9780393607002

7. Obidi, John Onimisi. Master Equation of the Theory of Entropicity (ToE). Encyclopedia; 2025. https://encyclopedia.pub/entry/58596

8. 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.(2025-06-30). https://doi.org/10.33774/coe-2025-hmk6nI

9. Obidi, John Onimisi . “On the Discovery of New Laws of Conservation and Uncertainty, Probability and CPT-Theorem Symmetry-Breaking in the Standard Model of Particle Physics: More Revolutionary Insights from the Theory of Entropicity (ToE)”. Cambridge University. (14 June 2025). https://doi.org/10.33774/coe-2025-n4n45

10. 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

11. Obidi, John Onimisi. Review and Analysis of the Theory of Entropicity (ToE) in Light of the Attosecond Entanglement Formation Experiment: Toward a Unified Entropic Framework for Quantum Measurement, Non-Instantaneous Wave-Function Collapse, and Spacetime Emergence. Cambridge University; 29 March 2025. https://doi.org/10.33774/coe-2025-7lvwh

12. Obidi, John Onimisi. Attosecond Constraints on Quantum Entanglement Formation as Empirical Evidence for the Theory of Entropicity (ToE). Cambridge University; 25 March 2025. https://doi.org/10.33774/coe-2025-30swc

13. Obidi, John Onimisi. ‘’The Theory of Entropicity (ToE) Validates Einstein’s General Relativity (GR) Prediction for Solar Starlight Deflection via an Entropic Coupling Constant η’‘. Cambridge University. (23 March 2025). https://doi.org/10.33774/coe-2025-1cs81

14. Obidi, John Onimisi. ‘’The Theory of Entropicity (ToE): An Entropy-Driven Derivation of Mercury’s Perihelion Precession Beyond Einstein’s Curved Spacetime in General Relativity (GR)’‘. Cambridge University. (16 March 2025). https://doi.org/10.33774/coe-2025-g55m9

15. Obidi, John Onimisi. ‘’How the Generalized Entropic Expansion Equation (GEEE) Describes the Deceleration and Acceleration of the Universe in the Absence of Dark Energy’‘. Cambridge University. (12 March 2025). https://doi.org/10.33774/coe-2025-6d843

16. Obidi, John Onimisi. ‘’Corrections to the Classical Shapiro Time Delay in General Relativity (GR) from the Entropic Force-Field Hypothesis (EFFH)’‘. Cambridge University. (11 March 2025). https://doi.org/10.33774/coe-2025-v7m6c

17. Obidi, John Onimisi. The Theory of Entropicity (ToE) Validates Einstein’s General Relativity (GR) Prediction for Solar Starlight Deflection via an Entropic Coupling Constant η. Cambridge University; 23 March 2025. https://doi.org/10.33774/coe-2025-1cs81

18. Obidi, John Onimisi. The Theory of Entropicity (ToE): An Entropy-Driven Derivation of Mercury’s Perihelion Precession Beyond Einstein’s Curved Spacetime in General Relativity (GR). Cambridge University; 16 March 2025. https://doi.org/10.33774/coe-2025-g55m9

19. Obidi, John Onimisi. How the Generalized Entropic Expansion Equation (GEEE) Describes the Deceleration and Acceleration of the Universe in the Absence of Dark Energy. Cambridge University; 12 March 2025. https://doi.org/10.33774/coe-2025-6d843

20. Obidi, John Onimisi. Corrections to the Classical Shapiro Time Delay in General Relativity (GR) from the Entropic Force-Field Hypothesis (EFFH). Cambridge University; 11 March 2025. https://doi.org/10.33774/coe-2025-v7m6c

21. Obidi, John Onimisi. Exploring the Entropic Force-Field Hypothesis (EFFH): New Insights and Investigations. Cambridge University; 20 February 2025. https://doi.org/10.33774/coe-2025-3zc2w

22. Obidi, John Onimisi. The Entropic Force-Field Hypothesis: A Unified Framework for Quantum Gravity. Cambridge University; 18 February 2025. https://doi.org/10.33774/coe-2025-fhhmf

23. Obidi, John Onimisi. Exploring the Entropic Force-Field Hypothesis (EFFH): New Insights and Investigations. Cambridge University; 20 February 2025. https://doi.org/10.33774/coe-2025-3zc2w

24. Obidi, John Onimisi. The Entropic Force-Field Hypothesis: A Unified Framework for Quantum Gravity. Cambridge University; 18 February 2025. https://doi.org/10.33774/coe-2025-fhhmf