⭐ The Obidi Equivalence Principle (OEP) of the Theory of Entropicity (ToE)

Spacetime is the macroscopic projection of an underlying information‑geometric manifold, and every geometric property of physical spacetime corresponds to an entropic property of that information manifold.

More formally:

> The curvature, geodesics, and metric structure of physical spacetime arise from, and are isomorphic to, the curvature, geodesics, and Fisher‑information metric of the underlying information‑geometric manifold after coarse‑graining.

The Obidi Equivalence Principle proposes a global isomorphism between an underlying information-geometric manifold endowed with a Fisher metric and emergent physical spacetime, asserting that all geometric and dynamical properties of spacetime correspond to entropic and informational structures under a coarse-graining map.

🧠 Collective Historical Insight Summary

What ToE has formulated as the Obidi Equivalence Principle (OEP) is essentially an attempt to do for entropy/information what Einstein did for gravity:

Establish a strict equivalence (isomorphism) between two domains:

1) information geometry

2) physical spacetime geometry

This idea does have precedents in fragments across physics:

holography (geometry ↔ entanglement)

AdS/CFT (bulk ↔ boundary)

information geometry (Fisher metric ↔ statistical structure)

But:

👉 No mainstream framework currently enforces a full, global, invertible isomorphism of the kind which Obidi's Theory of Entropicity (ToE) has demanded.

So ToE's OEP is:

not baseless

but much stronger than anything currently accepted in traditional physics.

✅ Legitimacy of the Use of the Fisher Information Metric in the Theory of Entropicity (ToE)

From established literature:

Amari (2016) shows Fisher information defines a Riemannian metric

Anza & Crutchfield (2022) connect entropy and geometric structure

Franzosi et al. (2016) define geometric entropy via curvature

👉 So ToE's starting point:

“information → geometry”

is fully grounded in existing research

✅ ToE is Attempting a True Equivalence Principle

This is both important and audacious at once.

Compare:

Einstein EP

ToE's OEP

gravity ≡ geometry (Einstein)

spacetime ≡ information geometry (Obidi)

inertial = gravitational mass  (Einstein)

physical = entropic geometry (Obidi)

local equivalence  (Einstein)

global mapping (Obidi)

👉 Structurally, this is the right kind of move for a foundational theory.

This is the core demand of the Theory of Entropicity (ToE):

\[

(\mathcal{M}{info}, g{F}) \;\xrightarrow{\text{emergence}}\; (\mathcal{M}{spacetime}, g{\mu\nu})

\]

with the requirement that:

\[

\Phi: \mathcal{M}{info} \to \mathcal{M}{spacetime}

\]

is a smooth, invertible mapping preserving curvature, geodesics, and entropy production.

That is:

Emergence relation:

(𝓜₍ᵢₙfₒ₎, gF) ⟶₍ₑₘₑᵣgₑₙcₑ₎ (𝓜₍ₛₚₐcₑₜᵢₘₑ₎, gμν)

Correspondence map:

Φ : 𝓜₍ᵢₙfₒ₎ → 𝓜₍ₛₚₐcₑₜᵢₘₑ₎

⭐ What the principle asserts (in plain language)

1. Information geometry is the substrate.  

   It exists before spacetime.

2. Spacetime emerges from information geometry.  

   Not as a metaphor — as a coarse‑grained projection.

3. Every physical geometric quantity has an entropic counterpart.  

   - Spacetime curvature ↔ Entropic curvature  

   - Geodesics ↔ Paths of extremal entropy flow  

   - Mass ↔ Information‑curvature density  

   - Energy ↔ Rate of information change  

4. Gravity is not a force but an entropic gradient.  

   This is the entropic analogue of Einstein’s “gravity = geometry.”

5. The arrow of time is the monotonic increase of Fisher information.

⭐ Why this principle is necessary for the Theory of Entropicity (ToE)

Without the OEP:

- spacetime and information geometry become dualistic  

- entropy cannot be the fundamental invariant  

- gravity cannot be entropic  

- quantum mechanics cannot be geometric  

- the ToE collapses into two incompatible layers  

With the Obidi Equivalence Principle (OEP):

- quantum → statistical geometry  

- gravity → entropic curvature  

- time → information ordering  

- energy → information flow  

- spacetime → emergent macro‑geometry  

Everything becomes one coherent structure.

⭐ The 3 Axioms of the Theory of Entropicity (ToE):

- Axiom 1: Entropic Primacy  

- Axiom 2: Information‑Geometric Substrate  

- Axiom 3: Obidi Equivalence Principle (OEP)  

References 

1)

https://theoryofentropicity.blogspot.com/2026/04/obidi-equivalence-principle-oep.html

2)

http://youtube.com/post/UgkxMndtJGNXut5AISg1-2nRtwCBDUmhkxYM?si=xnYZNWea-SgUSb9M

3)

https://medium.com/@jonimisiobidi/the-obidi-equivalence-principle-oep-of-the-theory-of-entropicity-toe-8ff4c199a3d7

How Complicated is the Mathematical Foundation of the Theory of Entropicity (ToE)?

The mathematical foundation of the Theory of Entropicity (ToE), pioneered by researcher John Onimisi Obidi, is highly complex because it seeks to replace the geometric foundations of traditional physics (like spacetime) with an "entropic field". Unlike standard physics where entropy is a secondary statistical measure, ToE promotes it to a primary "ontic" field that generates reality. [1, 2, 3, 4]

The complexity stems from its integration of several advanced mathematical and conceptual frameworks: [5, 6, 7]

1. Information Geometry & Metric Transformation

ToE builds upon information geometry, which links statistical and geometric concepts. [2]

• Statistical Metrics: It utilizes the Fisher-Rao metric (classical distinguishability) and the Fubini-Study metric (quantum distinguishability).
• $\alpha$-Connections: It employs Amari–Čencov $\alpha$-connections as a "deformation index" to transform these informational metrics into physical metric-affine geometries that resemble spacetime curvature. [1, 3, 8, 9]

2. Core Field Equations

The theory replaces Einstein's geometric equations with its own entropic counterparts: [10, 11]

• The Obidi Action: A foundational variational principle that determines the dynamics of the entropic field, generalizing classical and quantum actions.
• Master Entropic Equation (MEE): Derived from the Obidi Action, this is the entropic analogue of Einstein's field equations. It is described as highly nonlinear and nonlocal, reflecting a universe that "computes" its own state.
• Vuli–Ndlela Integral: A reformulation of Feynman's path integrals that weights paths by their "entropic cost," embedding the arrow of time directly into quantum mechanics. [1, 8, 12, 13, 14]

3. Emergent Physical Principles

The mathematics must derive existing physical constants and laws as emergent effects rather than starting postulates: [15]

• The No-Rush Theorem: Formalizes a universal temporal bound on interactions, stating that entropy cannot redistribute instantaneously.
• Speed of Light ($c$): Reinterpreted as the maximum propagation rate of the entropic field, rather than a fundamental axiom of relativity.
• Entropic Resistance Principle (ERP): Used to explain inertia and relativistic mass increase as the field's resistance to reconfiguration. [1, 2, 3, 8, 16, 17]

Current Status

While the foundational structure is established, ToE is an emerging and audacious proposal. Critics and reviewers note that further explicit mathematical construction is needed for its full development, including full quantization of the entropy field and detailed coupling to Standard Model fields. [7, 15, 18]

Would you like to dive deeper into how information geometry specifically bridges the gap between entropy and gravity in this theory?

[1] https://medium.com

[2] https://medium.com

[3] https://medium.com

[4] https://medium.com

[5] https://medium.com

[6] https://www.researchgate.net

[7] https://www.cambridge.org

[8] https://medium.com

[9] https://figshare.com

[10] https://www.preprints.org

[11] https://www.authorea.com

[12] https://medium.com

[13] https://ggnpreprints.authorea.com

[14] https://client.prod.orp.cambridge.org

[15] https://encyclopedia.pub

[16] https://www.authorea.com

[17] https://client.prod.orp.cambridge.org

[18] https://www.cambridge.org

How Obidi Has Developed a New Foundation of Physics: Basic Concepts of the Formulation of the Theory of Entropicity (ToE)

The Central Ontological Shift

ToE posits entropy — S(x) — as the fundamental, continuous, and dynamic field from which matter, gravity, space, and time emerge. It treats entropy not as a measure of disorder, but as the active, primary cause of physical reality. [Medium](https://medium.com/@jonimisiobidi/the-beauty-of-obidis-theory-of-entropicity-toe-the-universe-as-an-accounting-mechanism-with-7bfdc225832c) The philosophical ambition is sweeping: just as Einstein replaced Newtonian gravity with the geometry of curved spacetime, Obidi proposes replacing Einstein's curved spacetime with the dynamics of an entropic field. [Medium](https://medium.com/@jonimisiobidi/a-brief-historical-and-conceptual-introduction-to-the-foundations-of-the-theory-of-entropicity-1c72bc612765)

The core dictum encapsulates this: ToE redefines the Einstein-Wheeler dictum ("matter curves spacetime") to assert that "entropy curves existence itself." [Medium](https://medium.com/@jonimisiobidi/foundations-of-obidis-theory-of-entropicity-toe-conceptual-mathematical-and-physical-pillars-929690e65c55)

The Mathematical Architecture

The theory is built on several interlocking constructs:

- The Obidi Action — a universal variational principle that governs the dynamics of the entropic field, from which all physical laws can be derived. [Medium](https://medium.com/@jonimisiobidi/the-theory-of-entropicity-toe-a-new-framework-for-understanding-reality-d6d1e038c53e)

- The Master Entropic Equation (MEE) — the fundamental dynamical equation of the entropy field, analogous to Einstein's Field Equations but derived entirely from entropic principles. [Medium](https://medium.com/@jonimisiobidi/the-theory-of-entropicity-toe-lays-down-the-prolegomenon-to-the-foundation-of-modern-theoretical-15d6a93b018a)

- Triadic Information Geometry — the theory synthesizes three geometric formalisms: the Fisher-Rao Metric (encoding classical entropy curvature), the Fubini-Study Metric (representing quantum entropy curvature), and the Amari-Čencov α-Connection (introducing the asymmetric, irreversible flow of entropy — establishing the arrow of time). [Medium](https://medium.com/@jonimisiobidi/foundations-of-obidis-theory-of-entropicity-toe-conceptual-mathematical-and-physical-pillars-929690e65c55)

- The Vuli-Ndlela Integral — an entropy-weighted reformulation of the Feynman path integral that directly embeds irreversibility into quantum dynamics. [Medium](https://medium.com/@jonimisiobidi/foundations-of-obidis-theory-of-entropicity-toe-conceptual-mathematical-and-physical-pillars-929690e65c55)

- The Obidi Curvature Invariant (OCI) — identified as ln(2), this is considered the fundamental unit of entropic cost or distinguishability. Reality only "acknowledges" a state once entropic curvature exceeds this threshold — explaining why quantum measurements produce discrete outcomes. [Medium](https://medium.com/@jonimisiobidi/foundations-of-obidis-theory-of-entropicity-toe-conceptual-mathematical-and-physical-pillars-929690e65c55)

Reinterpreting Known Physics

Rather than discarding established results, ToE seeks to derive them from entropic principles:

- Relativistic effects — mass increase, time dilation, and length contraction — are derived directly from entropic principles, viewing them as inevitable results of "entropic resistance" rather than independent geometrical postulates. [Medium](https://medium.com/@jonimisiobidi/the-beauty-of-obidis-theory-of-entropicity-toe-the-universe-as-an-accounting-mechanism-with-7bfdc225832c)

- Mercury's perihelion precession is reproduced (43 arcseconds/century) via entropic gradients, without curved spacetime. Gravitational light deflection (1.75 arcseconds) is reproduced via entropic field coupling. [Medium](https://medium.com/@jonimisiobidi/a-brief-historical-and-conceptual-introduction-to-the-foundations-of-the-theory-of-entropicity-1c72bc612765)

- The cosmological constant emerges naturally as a feature of the entropic field's geometry, not an arbitrary parameter, and the accelerating expansion of the universe is explained as a manifestation of large-scale entropic flow. [Medium](https://medium.com/@jonimisiobidi/a-brief-historical-and-conceptual-introduction-to-the-foundations-of-the-theory-of-entropicity-1c72bc612765)

- Dark matter and dark energy are no longer mysteries requiring unseen particles. The non-equilibrated spectral modes of the entropy field generate contributions to effective energy density that behave like cold dark matter, while slight deviations from global entropic equilibrium produce a small, positive entropic pressure acting as dark energy. [Medium](https://medium.com/@jonimisiobidi/the-theory-of-entropicity-toe-lays-down-the-prolegomenon-to-the-foundation-of-modern-theoretical-15d6a93b018a)

The Speed of Light Reinterpreted

The universal constant *c* is reinterpreted — not as a postulate about photons, but as the maximum rate at which the entropic field can reorganize energy and information. Light is simply the visible manifestation of this maximum entropic reconfiguration speed. [Medium](https://medium.com/@jonimisiobidi/a-brief-historical-and-conceptual-introduction-to-the-foundations-of-the-theory-of-entropicity-1c72bc612765)

The No-Rush Theorem

A key derived principle states that nature cannot be rushed — all physical interactions must occur within a finite, non-zero time interval governed by the entropic field. [Medium](https://medium.com/@jonimisiobidi/foundations-of-obidis-theory-of-entropicity-toe-conceptual-mathematical-and-physical-pillars-929690e65c55) This grounds causality and the arrow of time in the physics of entropic flow rather than geometric postulate.

Where It Stands

The Theory of Entropicity is very new — it began taking shape in 2025 and is still having its mathematical structure developed and refined. It has been published in article, preprint, and working-paper forms across various platforms, including Cambridge Open Engage, Academia, ResearchGate, SSRN, and others. Critics note it must still undergo rigorous independent peer review, experimental prediction, and falsifiability testing before being accepted into mainstream physics. [Medium](https://medium.com/@jonimisiobidi/a-brief-historical-and-conceptual-introduction-to-the-foundations-of-the-theory-of-entropicity-1c72bc612765)

It is an ambitious, still-evolving framework — but one with a clear and radical thesis: entropy is not a consequence of physical laws; it *is* the fundamental law.

Progress of the Theory of Entropicity (ToE): Foundations of a New Physics of Reality and Nature 

The Theory of Entropicity (ToE) is a radical, emerging framework in theoretical physics that reinterprets entropy as the fundamental, dynamic field of reality rather than a mere statistical byproduct. Proposed by John Onimisi Obidi in 2025, the theory is currently in its early stages of mathematical development and is undergoing vigorous research and active refinement for eventual integration into mainstream physics. [1, 2, 3]

Core Conceptual Developments

The theory posits a "monistic" foundation where all known physical phenomena—including space, time, and gravity—emerge from a single entropic field, denoted as $S(x,t)$. [4, 5]

• Entropy as an Ontic Field: Unlike traditional physics, ToE treats entropy as a primary physical field that permeates existence and acts as the causal engine of the universe.
• The "No-Rush Theorem": A central principle asserting that no physical interaction can occur instantaneously. It establishes a finite, non-zero duration for all processes, providing a mechanistic basis for causality.
• Reinterpretation of the Speed of Light ($c$): The theory suggests $c$ is not a separate postulate but the maximum rate at which the entropic field can reorganize energy and information.
• Emergent Gravity and Spacetime: Gravity is reinterpreted as the curvature of the entropic field, while space and time are seen as emergent maps of entropic gradients and flow. [1, 3, 4, 5, 6, 7]

Mathematical and Theoretical Progress

As of 2025–2026, several key mathematical structures have been established to formalize the theory:

• The Obidi Action: A universal variational principle that governs entropic dynamics. It includes the Local Obidi Action (LOA) for local field dynamics and the Spectral Obidi Action (SOA) for global geometric constraints.
• Master Entropic Equation (MEE): The entropic analogue to Einstein's field equations, governing the dynamics of the scalar field $S(x,t)$.
• Vuli–Ndlela Integral: An entropy-weighted reformulation of Feynman's path integral, used to embed the "arrow of time" and irreversibility directly into quantum mechanics.
• Unification of Formalisms: Recent papers demonstrate how ToE serves as a "superset" of other theories, such as Ginestra Bianconi's "Gravity from Entropy," and incorporates generalized entropies like Tsallis and Rényi. [1, 4, 8, 9, 10]

Current Status and Outlook

While ToE offers a potential "Grand Unified Theory," it remains a speculative and non-mainstream proposal that has not yet gained unanimous acceptance. [2, 6]

• Publications: Foundational writings have been traces through various platforms, including preprints on SSRN, ResearchGate, and Cambridge Open Engage.
• Experimental Verification: Efforts are underway to identify falsifiable predictions, such as searching for attosecond-scale delays in quantum entanglement formation and refining observations of Mercury's perihelion precession through entropic gradients. [5, 9, 11]

Would you like to explore the mathematical specifics of the Obidi Action or its proposed applications in fields like AI and biology?

[1] https://medium.com

[2] https://medium.com

[3] https://medium.com

[4] https://encyclopedia.pub

[5] https://medium.com

[6] https://encyclopedia.pub

[7] https://www.researchgate.net

[8] https://www.researchgate.net

[9] https://medium.com

[10] https://figshare.com

[11] https://papers.ssrn.com

A Brief History of the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE), formulated by John Onimisi Obidi, posits entropy not as a mere statistical byproduct, but as the fundamental, foundational field of reality. It builds on the classical thermodynamics of Clausius and Boltzmann, the "arrow of time" concept, and Verlinde's entropic gravity,, elevating entropy to be the underlying substrate of time, space, and physical interaction. 

Historical Foundation and Development:

• Classical Thermodynamics (19th Century): The roots lie in Rudolf Clausius’s 1865 definition of entropy as energy dispersal and Ludwig Boltzmann’s 1872 statistical formulation (
  
  
  
  ), which established entropy as the reason for the arrow of time and the second law of thermodynamics.
• Modern Reinterpretation (2011): Erik Verlinde’s work, which argued that gravity is an entropic force rather than a fundamental one, paved the way for treating entropy as a foundational physical driver.
• The Theory of Entropicity (2020s): John Onimisi Obidi expands these concepts by arguing that entropy is the "heartbeat of existence". This theory reinterprets Einstein's relativistic speed of light (
  

  $$
  ) not as a postulate, but as a necessary result of the maximum rate of entropic reconfiguration (the "[Master Entropic Equation]" or "No-Rush Theorem").
• Core Tenets: The Theory of Entropicity posits that spacetime itself is not a pre-existing arena but an emergent structure created by the distribution of entropic gradients. 

This theory aims to unify thermodynamics, quantum mechanics, and relativity by revealing them as different expressions of a single, governing entropic principle. 

📚 Key Supporting / Related Works on Entropic Gravity and the Theory of Entropicity (ToE)

Verlinde, E. (2011). On the origin of gravity and the laws of Newton. JHEP.

https://link.springer.com/content/pdf/10.1007/JHEP04(2011)029.pdf⁠�

Carroll, S. M., & Remmen, G. N. (2016). What is the entropy in entropic gravity? Phys. Rev. D.

https://link.aps.org/accepted/10.1103/PhysRevD.93.124052⁠�

Bianconi, G. (2025). Gravity from entropy. Phys. Rev. D.

https://link.aps.org/pdf/10.1103/PhysRevD.111.066001⁠�

Plastino, A., & Rocca, M. (2020). Quantum field theory of entropic gravity. Annals of Physics.

https://www.researchgate.net/publication/337010441⁠�

Kowalski-Glikman, J. (2010). Gravity and BF topological field theory. Phys. Rev. D.

https://arxiv.org/pdf/1002.1035⁠�

Lee, J. W. (2012). On the origin of entropic gravity and inertia. Foundations of Physics.

https://arxiv.org/pdf/1003.4464⁠�

Gao, S. (2011). Is gravity an entropic force? Entropy.

https://www.mdpi.com/1099-4300/13/5/936⁠�

Vopson, M. (2025). Is gravity evidence of a computational universe? AIP Advances.

https://pubs.aip.org/aip/adv/article/15/4/045035/3345217⁠�

Chen, W. X. (2025). Geometry as thermodynamics.

https://hal.science/hal-05223730v1/file/SSS.pdf⁠�

Obidi, J. O. (2025). Foundations of the Theory of Entropicity.

http://www.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c⁠�

 📚 Key Academic Sources and Publications on the Theory of Entropicity (ToE)

(Most are part of the ToE corpus)

Obidi, J. O. (2025). On the conceptual and mathematical foundations of the theory of entropicity (ToE). Cambridge Open Engage.

http://www.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c⁠�

Obidi, J. O. (2025). Collected Works on the Evolution of the Foundations of ToE. ResearchGate.

https://www.researchgate.net/profile/John-Obidi-4/publication/399267338⁠�

Obidi, J. O. (2025). Review and analysis of ToE in light of entanglement experiments. Cambridge Open Engage.

https://www.cambridge.org/engage/coe/article-details/67e999596dde43c908bf7dda⁠�

Obidi, J. O. (2025). Entropy-driven derivation of Mercury’s perihelion precession.

https://www.cambridge.org/engage/coe/article-details/67e63abe6dde43c9086de9e0⁠�

Obidi, J. O. (2025). ToE and the unification of relativity effects.

https://www.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a⁠�

Obidi, J. O. (2025). A critical review of ToE and its mathematical structure.

https://www.cambridge.org/engage/coe/article-details/68630f541a8f9bdab5e1939d⁠�

Obidi, J. O. (2026). Further expositions on ToE and entropy-based gravity.

https://www.cambridge.org/engage/coe/article-details/69513828083c11e4a170b0b2⁠�

Obidi, J. O. (2025). ToE and string theory reinterpretation. SSRN.

https://papers.ssrn.com/sol3/Delivery.cfm?abstractid=5986134⁠�

Obidi, J. O. (2025). Unifying mathematical architecture of ToE.

https://www.cambridge.org/engage/coe/article-details/67fcb510fa469535b91e7b7d⁠�

Obidi, J. O. (2026). From information temperature to geometry.

https://www.cambridge.org/engage/coe/article-details/69543375098cdc781fdccf9e⁠�