On the Heretical Foundation and Yet Logical and Mathematical Consistency of the Theory of Entropicity (ToE) in Modern Theoretical Physics: Obidi's Heresy! 

In the Theory of Entropicity (ToE), John Onimisi Obidi has been able to invent the Local Obidi Action (LOA) and incorporated such diverse components like generalized entropies of Rényi and Tsallis, the information geometry of Fisher-Rao and Fubini-Study metrics, and  the Amari-Čencov alpha connections. After inventing this local action, Obidi went on further to develop the Spectral Obidi Action (SOA), thus incorporating spectral operator formalism into the Theory of Entropicity ToE as an overarching Action that subtly incorporates the Araki Relative Entropy and Spectral Geometry. 

As far as we know based on all current literature accessible to us, no other researcher and investigator has attempted such a generalization and feat of unification.

Thus, what John Onimisi Obidi has achieved is unusual; and—in the literal, technical sense—nobody else has built a single local variational action that explicitly weaves together:

• generalized entropies (Tsallis, Rényi, Araki, etc.),
• Fisher–Rao, Fubini–Study, and Amari–Čencov geometry,
• and gravity/field dynamics

in the way the Local Obidi Action (LOA) and its pairing with SOA do in ToE. At an initial hasty glance, this rather unusual integration could be deemed heretical. Only after some sorber and humble reflection can one realize that this feat is both original and groundbreaking in the annals of Science.

Let us now investigate how Obidi has managed to do it, and why the literature hasn’t gone there before Obidi did.

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1. What the Local Obidi Action actually is, structurally

If you strip away the Entropicity branding and look at the bare structure, the LOA is essentially built from three intertwined pieces:

1. A scalar field sector – the entropy field with a kinetic term and a potential.
2. A geometric sector – a metric (or metric family) that is not just the spacetime metric , but an entropic metric that already carries Fisher–Rao / Fubini–Study / Amari structure.
3. A coupling to gravity – Ricci scalar and curvature built from the entropically deformed metric.

So instead of:

• “Here is a scalar field + GR as two separate things,”

Obidi declares that:

• “The scalar field is entropy,
• and its geometry is information geometry,
• and its coupling to curvature is gravity.”

Once we accept that axiom, it becomes natural (for us) to pull in Fisher–Rao, Fubini–Study, and Amari–Čencov, because in accordance to Obidi's mental picture:

“Wherever there is entropy, there is an information metric; wherever there is an information metric, there is geometry; wherever there is geometry, there can be a variational action.”

Most people in the literature stop at one of those arrows and never compose all three [in the way Obidi has done].

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2. Why LOA can host Fisher–Rao, Fubini–Study and Amari–Čencov at once

There are three existing “worlds” that Obidi noticed all share the same backbone:

• Fisher–Rao – the metric on probability distributions (classical information geometry).
• Fubini–Study – the metric on pure quantum states (projective Hilbert space).
• Amari–Čencov α–connections – the family of affine connections that live on statistical manifolds, encoding duality and irreversibility.

In mainstream [physics] work, these are usually treated as:

• mathematical curiosities in statistics or quantum foundations,
• tools for machine learning or information geometry,
• or at most, hints about “information in gravity.”

But Obidi did something slightly heretical but logically and mathematically clean — Obidi boldly declares:

1. Promote entropy from a number to a field .
2. Promote its information geometry to the actual physical geometry of the “entropic manifold.”
3. Explicitly choose Fisher–Rao + Fubini–Study + Amari α–connections as the components of that geometry, and then
4. Feed that composite geometry directly into an action principle (LOA) that also talks to gravity.

So, the Local Obidi Action (LOA) is not “randomly mixing” things; it is taking seriously the idea that:

“The right geometry for an entropy field is information geometry, and the right language for dynamics on a geometry is a variational action.”

Once we write down that action, all three structures appear almost inevitably. 

This is Obidi's [First] Heresy!

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3. Why no one else has done exactly this (so far)

There are several historical and sociological reasons for this, not just “they missed it”:

a) Different starting questions

• Jacobson, Padmanabhan, Verlinde:
  They started from: “Can gravity be derived from thermodynamics?”
  So entropy is a constraint or diagnostic on an already-existing spacetime; the metric is primary.

• Amari, Čencov, information geometers:
  They started from: “What is the natural geometry of probability distributions?”
  Gravity and spacetime are not their target; physics is at best a side application.

• Quantum geometers (Fubini–Study, Bures, etc.):
  They started from: “What is the geometry of quantum state space?”
  Again: no attempt to build a full gravitational field theory from it.

But, again, Obidi started from a very different question:

“What if entropy is the fundamental field of nature, and its information geometry is the geometry that all physics lives on?”

Once we ask that question and insist on a true field theory, the Local Obidi Action (LOA) becomes the natural language to express it.

b) Disciplinary silos

The people who deeply understand:

• Araki entropy and modular operators,
• spectral geometry and Connes–Chamseddine actions,
• Fisher–Rao / Amari α–connections,
• Fubini–Study in QFT/QI,
• and entropic gravity

are usually not the same person or even in the same research community.

Obidi's work is thus unusual because we see that Obidi has indeed:

• studied entropic gravity (Verlinde, Bianconi, Jacobson),
• studied information geometry (Amari–Nagaoka, Čencov),
• studied quantum geometry and spectral stuff,
• and then refused to treat them as separate hobbies or disciplines.

That cross-silo synthesis is rare—even more so for someone working independently.

c) Risk profile

It is much safer in academia to:

• add a small correction to GR,
• or apply Fisher–Rao to machine learning,
• or talk about “entropic gravity” near black holes,

than to say:

“I am writing a new action principle where entropy is the fundamental field and all of Fisher–Rao, Fubini–Study, Amari α, Tsallis, Rényi, and Araki are unified as one geometric-variational structure.”

That sounds “too big” and will scare most grant reviewers. But Obidi has not been constrained by that incentive structure, so Obidi can afford to be bold.

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4. Is it really true that no one has done what Obidi has done?

We are able to note the following in an objective fashion:

• There are works that connect Fisher information and gravity;
• there are works that connect entropy and gravity (Verlinde, Bianconi, etc.);
• there are works that connect Fisher–Rao and Fubini–Study;

But:

• We do not find any pre–ToE work in which a single local variational action:
  • takes an entropy field as fundamental,
  • uses information geometry (Fisher–Rao, Fubini–Study, Amari α) as its metric content, and
  • couples that to gravity as the dynamical backbone for all of physics.

The only place that precise combination appears is in Obidi's own Theory of Entropicity (ToE) papers, blog, and encyclopedia entry.

So, we can state as follows:

The ingredients existed in the literature, but the specific synthesis and the Local Obidi Action as a unifying entropic action are original to ToE, as far as current published research shows.

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5. Closure on Obidi's originality 

What John Onimisi Obidi has done is not:

• invent Fisher–Rao,
• invent Fubini–Study,
• invent Amari α–connections,
• or invent entropy itself.

What is new is Obidi's conceptual foundation and integration:

• treating entropy as a field,
• insisting that its natural geometry is information geometry,
• lifting that geometry directly into a gravitationally coupled action,
• and then using that as the backbone of a unified theory (ToE).

This is Obidi's First Heresy!

That kind of “conceptual recombination” is where a lot of real originality lives in theoretical physics. 

Einstein did not invent Riemannian geometry; he saw what it meant physically. John Onimisi Obidi has therefore done something similar at the level of entropy + information geometry + gravity.

The Theory of Entropicity (ToE) App: Homepage of the Theory of Entropicity (ToE) App - https://replit.com/@jonimisiobidi/Entropy-Weaver

 The Theory of Entropicity (ToE) App

Entropy Weaver - Replit

https://replit.com/@jonimisiobidi/Entropy-Weaver

Here is an interactive web application for exploring and understanding the Theory of Entropicity (ToE) with visualization tools, mathematical frameworks, and collaborative research features

The Theory of Entropicity (ToE) application includes the following features:

1.) Interactive concept explorer with hierarchical visualization of ToE's core concepts (entropy as fundamental field, emergent gravity, ETL, speed of light reinterpretation)

2.) Mathematical framework viewer displaying key equations (Obidi Action, Master Entropic Equation,

3.) Vuli-Ndlela Integral) with LaTeX rendering and interactive parameters

4.) Visual entropy field simulator showing 2D/3D representations of the entropic field S(x,t) with adjustable parameters

5.) Comparative analysis tool to explore relationships between ToE and other theories (General Relativity,

Quantum Mechanics, Verlinde's entropic gravity)

6.) Research notebook feature for documenting insights, hypotheses, and extensions with rich text editing and equation support

7.) Knowledge base with comprehensive ToE documentation, key papers, and concept definitions organized by topic

8.) Clean, academic interface with dark mode option, professional typography optimized for reading complex physics content

Homepage of the Theory of Entropicity (ToE) App

Comparison Between the Theory of Entropicity (ToE) and Einstein's General Relativity (GR)

Theory of Entropicity

General Relativity

Aspect	Theory of Entropicity	General Relativity
Fundamental Entity	Entropy field S(x,t)	Spacetime metric g_μν
Gravity Source	Entropy gradients	Mass-energy curvature
Speed of Light	Maximum entropy reorganization rate	Postulated constant
Time	Emergent from entropy dynamics	Coordinate in spacetime manifold
Field Equations	Master Entropic Equation	Einstein Field Equations
Variational Principle	Obidi Action	Einstein-Hilbert Action
Mercury Precession	Entropy corrections	Spacetime curvature

Understanding the Comparison

Key Differences

ToE fundamentally differs from traditional physics by treating entropy as the primary ontological entity. While other theories treat spacetime, fields, or particles as fundamental, ToE proposes that these all emerge from the entropic field.

Unification Approach

Rather than trying to quantize gravity or geometrize quantum mechanics, ToE takes a third path: deriving both from a single entropic principle. This is analogous to how thermodynamics unified heat and mechanics.

Understanding the Comparison

Key Differences

ToE fundamentally differs from traditional physics by treating entropy as the primary ontological entity. While other theories treat spacetime, fields, or particles as fundamental, ToE proposes that these all emerge from the entropic field.

Unification Approach

Rather than trying to quantize gravity or geometrize quantum mechanics, ToE takes a third path: deriving both from a single entropic principle. This is analogous to how thermodynamics unified heat and mechanics.

How Entropy Generates Gravity in the Theory of Entropicity (ToE)

The statement “entropy generates gravity” is meant in the Theory of Entropicity (ToE) framework that gravity is sourced from the entropy field itself. At first glance it looks circular: you write down an action that already contains the Einstein–Hilbert term (gravity), then add entropy terms, and afterwards claim gravity comes from entropy. But the subtlety is in how the coupling is interpreted.

🔍 Why it’s not circular

1. Einstein–Hilbert term as a geometric scaffold

   • The $\frac{1}{16\pi G}\int \sqrt{-g}R\backslash frac\{1\}\{16\backslash pi\; G\}\backslash int\; \backslash sqrt\{-g\}\backslash ,R$ term is the standard geometric action.

   • In ToE, this is not taken as an independent “given” but as a background structure that is ultimately determined by the entropy field $S(x)S(x)$.

   • The metric $g_{\mu \nu }g\_\{\backslash mu\backslash nu\}$ is varied alongside $SS$, and its dynamics are sourced entirely by the entropy stress-energy tensor.

2. Entropy as the source of curvature

   • The entropy Lagrangian contributes a stress-energy tensor $T_{\mu \nu }^{(S)}T\_\{\backslash mu\backslash nu\}^\{(S)\}$.

   • The Einstein tensor $G_{\mu \nu }G\_\{\backslash mu\backslash nu\}$ is equated to this $T_{\mu \nu }^{(S)}T\_\{\backslash mu\backslash nu\}^\{(S)\}$.

   • In other words, the geometry (gravity) is not free-standing—it is driven by entropy gradients and potentials. Without entropy, the right-hand side vanishes, and spacetime is flat [and (hence) non-existent].

3. Coupling vs. generation

   • Saying “entropy generates gravity” does not mean the Einstein–Hilbert term disappears. It means that the content of the gravitational field equations is entirely determined by entropy dynamics.

   • The EH term ensures the correct geometric structure (curvature, diffeomorphism invariance), while entropy provides the source and modulation.

4. Analogy with scalar-tensor theories

   • In Brans–Dicke theory, you also have both a scalar field and the Einstein–Hilbert term. Gravity is still “generated” by the scalar field because the scalar controls the effective coupling and dynamics.

   • Similarly, in ToE, entropy is not just an extra matter field—it is the organizing principle that determines how geometry behaves.

🧠 Conceptual Resolution

So it’s not a logical circle, but a hierarchy of roles:

• The Einstein–Hilbert term provides the mathematical language of curvature.

• The entropy field provides the physical content that sources curvature.

• Together, varying the action shows that spacetime curvature is nothing but the manifestation of entropy gradients and interactions.

Thus, “gravity is generated by entropy” means: the reason spacetime curves is because entropy fields deform it, not because curvature exists independently. The EH term is the vessel; entropy is