An Overview of the Theory of Entropicity (ToE): Beginning of a Long Walk to a New Foundation of Physics and Reality

The Theory of Entropicity (ToE) is a recently proposed theoretical physics framework developed by John Onimisi Obidi in 2025. Here's an overview:

Core Idea

ToE establishes entropy not as a statistical byproduct of disorder but as the fundamental field and causal substrate of physical reality. In this framework, entropy is elevated to a continuous, dynamic field whose gradients generate motion, gravitation, time, and information flow.

In other words, rather than entropy being something that results from physical processes, ToE argues it is the source of them.

Key Concepts

The Obidi Action & Master Entropic Equation (MEE) The Obidi Action is a variational principle at the core of ToE that determines the dynamics of the entropic field. It is analogous to the Einstein-Hilbert action in general relativity, providing the rule by which the universe continuously optimizes its entropy flow. It gives rise to the MEE — the entropic analogue of Einstein's field equations — which governs the dynamics of the entropic scalar field S(x,t).

Reinterpreting the Speed of Light ToE proposes that the universal constant c is not a postulate about the speed of light, but rather represents the maximum rate at which the entropic field can reorganize itself and distribute energy and information. Light is simply the visible manifestation of this maximum speed of entropic reconfiguration.

The No-Rush Theorem The "No-Rush Theorem" establishes a universal lower bound on interaction durations, encapsulating the principle that physical processes cannot occur instantaneously.

Gravity as Entropy ToE does not treat gravity as an independent fundamental entity but as a byproduct of entropic constraints governing the system. Gravitational attraction is simply the natural consequence of the entropic field restructuring energy, matter, and information — challenging both the Newtonian "force" picture and the Einsteinian curvature picture.

Relativistic Effects Mass increase, time dilation, and length contraction are not imposed geometrically but arise as natural consequences of the irreversible flow of entropy. Thus, geometry becomes emergent, not fundamental.

Ambitions

ToE aims to unify thermodynamics, relativity, quantum mechanics, and information theory under a single entropic principle. By treating these domains as different manifestations of the same entropic continuum, ToE positions itself as a candidate for a Grand Unified Theory.

Status & Position

It's important to note that this is a very new and audacious framework. The Theory of Entropicity is a rather radical yet bold proposal in the arena of Theoretical Physics; it is still under vigorous research. The mathematical formalisms are undergoing active refinement and are being advanced toward experimental validation and eventual integration into mainstream physics.

It has not yet been fully peer-reviewed through established mainstream physics journals, and its claims — while ambitious and bold — remain to be independently verified by the broader physics community.

How Does the Theory of Entropicity (ToE) Differ from Quantum Field Theory (QFT)?

The Theory of Entropicity (ToE) and Quantum Field Theory (QFT) differ fundamentally in their ontological foundations and mechanisms of unification. 

• Ontology: QFT treats quantum fields (like the electromagnetic or electron field) as fundamental entities existing within a predefined spacetime.  Particles are excitations of these fields. In contrast, ToE posits entropy itself as a fundamental, dynamic field $S(x)$, from which spacetime, matter, and energy emerge as secondary phenomena. 

• Unification Approach: QFT successfully describes three of the four fundamental forces (electromagnetic, weak, and strong) but does not incorporate gravity. ToE aims to be a true Theory of Everything, deriving both quantum behavior and gravity (as curvature of the entropic field) from a single principle—the Obidi Action. 

• Causality and Time: QFT is largely time-symmetric and does not inherently explain the arrow of time. ToE builds irreversibility into its core via the No-Rush Theorem, which states no process can be instantaneous, establishing a fundamental Entropic Time Limit (ETL) and making the arrow of time a primary feature. 

• Mechanism for Phenomena: In QFT, forces are mediated by gauge bosons (e.g., photons). In ToE, interactions are governed by the flow and gradient of the entropic field.  For example, the speed of light $c$ is derived as the maximum rate of entropic rearrangement, not assumed as a postulate. 

In essence, QFT is a framework for quantum particles and forces within spacetime, while ToE proposes a deeper, pre-spacetime reality where entropy is the sole fundamental entity, and all physics, including quantum mechanics, emerges from its dynamics. 

The Obidi Curvature Invariant (OCI) of ln 2 

The **Obidi Curvature Invariant (OCI)** is a proposed invariant in the Theory of Entropicity that marks the smallest nontrivial curvature threshold at which two entropic states become distinguishable. In the formulation associated with the theory, its value is taken to be **ln 2**.

The idea is that curvature is not treated only as a geometric property of spacetime, but also as a measure of entropic or informational separation. In that sense, OCI functions as a basic unit of distinguishability inside the theory.

## Overview

OCI is presented as a foundational quantity in the entropic description of physical structure. Rather than referring to ordinary curvature alone, it is used to characterize the first meaningful transition between two states of the entropy field.

The key interpretation is that a binary distinction corresponds to a threshold of curvature, and that threshold is represented by ln 2. This makes OCI a compact way of expressing the smallest stable “difference” that can be resolved in the theory.

## Mathematical formulation

A simple expression of the invariant is:

**OCI = ln 2**

A corresponding binary distinguishability relation is written in the form:

ρ_B = 2ρ_A

where the factor of 2 represents the minimal separation between two distinguishable states (classical and quantum).

In this interpretation, the curvature threshold associated with the transition is the smallest nonzero invariant of the entropic field. One may therefore write the conceptual rule as:

**distinguishability threshold = ln 2**

## Physical meaning

The physical meaning of OCI is that it represents the smallest entropic curvature needed for one state to be meaningfully different from another. It is therefore not just a numerical constant, but a structural marker for the onset of discernible physical change in the universe.

Within the Theory of Entropicity, OCI can be understood as a bridge between [quantum] information and [classical] geometry. A curvature value of ln 2 indicates the first stable step in the resolution of the entropy field into distinguishable configurations [we can recognize, measure, observe, or interact with everywhere and anywhere in nature].

## See also

- Theory of Entropicity

- Obidi Equivalence Principle

- Obidi Conjecture

- Obidi’s Principle of Complementarity

- Obidi’s Correspondence Principle

- Entropy

- Curvature

- Information theory

Fundamental Principles of the Theory of Entropicity (ToE): Connections Between ToE and Modern Theoretical Physics 

## Overview

The Obidi Equivalence Principle, Obidi Conjecture, Obidi’s Principle of Complementarity, and Obidi’s Correspondence Principle are foundational statements in the Theory of Entropicity. Together, they define how an entropy-based formulation relates to ordinary gravitational physics, especially general relativity.

These principles are meant to show that entropy may play a deeper role in physics than standard thermodynamics alone suggests. In this view, geometry and gravity are not separate from entropy, but may emerge from it as different descriptions of the same underlying structure.

## Motivation

The motivation behind these principles is to build a coherent bridge between entropic dynamics and classical gravity. If entropy is fundamental, then the familiar equations of gravitation should appear as limiting cases or derived results of the more basic entropic framework.

This approach also provides a way to compare the Theory of Entropicity with established physics without discarding known results. In practical terms, the theory must still recover general relativity where that theory already works well.

## Definitions

**Obidi Equivalence Principle (OEP).**  

The Obidi Equivalence Principle states that the entropic formulation and the geometric formulation describe the same physical content when written in the appropriate variables. This means that the entropic action and the gravitational action are treated as equivalent within the relevant domain.

**Obidi Conjecture (OC).**  

The Obidi Conjecture proposes that Einsteinian gravity emerges from the Theory of Entropicity when entropy is treated as a fundamental dynamical field. It asserts that the Einstein field equations can be recovered from entropic dynamics under suitable conditions.

**Obidi’s Principle of Complementarity (PoC).**  

Obidi’s Principle of Complementarity states that the entropic description and the geometric description are both valid, but each is most useful in a different regime. They are not competing claims; instead, they complement one another as different views of the same deeper structure.

**Obidi’s Correspondence Principle (OCP).**  

Obidi’s Correspondence Principle states that the Theory of Entropicity must reduce to general relativity in the appropriate classical or coarse-grained limit. This ensures that the new theory remains consistent with established gravitational physics.

## Mathematical formulation

Let 𝓜 denote the physical manifold, S the entropy field, g_{μν} the emergent metric, 𝓐_E the entropic action, and 𝓐_GR the gravitational action.

### 1. Obidi Equivalence

The basic equivalence may be written as:

𝓐_E[S, 𝓜] ≃ 𝓐_GR[g_{μν}, 𝓜]

This expresses physical equivalence between the entropic and geometric descriptions.

A transformation between the two descriptions can be written as:

Φ: S ↦ g_{μν}

Under this mapping, stationarity of one action corresponds to stationarity of the other:

δ𝓐_E = 0 ⇔ δ𝓐_GR = 0

### 2. Obidi Conjecture

The conjecture may be expressed as the emergence of Einsteinian gravity from entropic dynamics:

δ𝓐_E / δS = 0 ⇒ G_{μν} + Λg_{μν} = κT_{μν}

This says that when the entropic action is extremized, the Einstein field equations arise in the appropriate limit.

### 3. Obidi’s Principle of Complementarity

The entropic and geometric descriptions may be treated as two overlapping domains:

𝓓_E ∪ 𝓓_G = 𝓓_full

and

𝓓_E ∩ 𝓓_G ≠ ∅

Here, 𝓓_E is the entropic domain, 𝓓_G is the geometric domain, and 𝓓_full is the full physical domain. The overlap means that both descriptions remain valid in shared regimes.

### 4. Obidi’s Correspondence Principle

The correspondence requirement can be written as:

lim ϵ→0 𝓐_E = 𝓐_GR

and

lim ϵ→0 Φ(S) = g_{μν}^{GR}

Here, ϵ represents a coarse-graining, classicality, or low-gradient parameter. In the limit ϵ → 0, the entropic theory must reproduce general relativity.

## Interpretation

These principles form a hierarchy. The Equivalence Principle gives the translation rule between entropic and geometric language. The Conjecture claims that gravity emerges from entropy. The Principle of Complementarity explains why both descriptions can be valid. The Correspondence Principle ensures that the theory reduces to known physics in the proper limit.

Taken together, they define a conceptual bridge between entropy-centered dynamics and gravitational theory. This makes the Theory of Entropicity easier to state as a structured framework rather than as a loose philosophical idea.

## See also

- General relativity

- Entropy

- Variational principle

- Emergent gravity

- The Theory of Entropicity

- Correspondence principle

- Complementarity

The Set of Obidi Principles of the Theory of Entropicity (ToE)

The **Obidi Equivalence Principle**, **Obidi Conjecture**, **Obidi’s Principle of Complementarity**, and **Obidi’s Correspondence Principle** are a set of foundational statements used in the Theory of Entropicity (ToE). Together, they describe how an entropy-centered formulation can be related to physics and ordinary gravitational physics, especially general relativity, through equivalence, emergence, complementarity, and limiting behavior.

These principles are intended to give the Theory of Entropicity (ToE) a structured foundation. In broad terms, they say that entropy is not merely a thermodynamic quantity but also more importantly serve as a more fundamental organizing variable from which [spacetime] geometry [etc.] and gravity [etc.] arise.

## Definitions

###Obidi Equivalence Principle (OEP).

The Obidi Equivalence Principle states that the entropic formulation and the geometric formulation represent the same physical content when expressed in the appropriate variables. In this sense, the entropic action may be treated as equivalent to the conventional gravitational action within the domain where the two descriptions are related.

###Obidi Conjecture (OC).

The Obidi Conjecture proposes that Einsteinian gravity emerges from the Theory of Entropicity (ToE) when entropy is treated as a fundamental dynamical field. It asserts that the gravitational field equations can be recovered from entropic dynamics under suitable conditions.

###Obidi’s Principle of Complementarity (PoC). 

Obidi’s Principle of Complementarity states that the entropic and geometric descriptions are both valid, but each is most useful in a different regime. The two descriptions are not contradictory; rather, they supplement one another as different representations of the same deeper structure.

###Obidi’s Correspondence Principle (OCP).

Obidi’s Correspondence Principle states that the Theory of Entropicity (ToE) must reproduce general relativity in the appropriate classical or coarse-grained limit. This requirement ensures that the new framework agrees with established gravitational physics wherever general relativity already provides reliable predictions.

## Formal notation

Let 𝓜 denote the physical manifold, S the entropy field, g_{μν} the emergent metric, 𝓐_E the entropic action, and 𝓐_GR the gravitational action.

###1. Obidi Equivalence

𝓐_E[S, 𝓜] ≃ 𝓐_GR[g_{μν}, 𝓜]

This means that the two actions are physically equivalent under an admissible transformation between entropy variables and geometric variables.

A more explicit form is:

Φ: S ↦ g_{μν}

with

δ𝓐_E = 0 ⇔ δ𝓐_GR = 0

within the domain where the equivalence applies.

###2. Obidi Conjecture

δ𝓐_E / δS = 0 ⇒ G_{μν} + Λ g_{μν} = κT_{μν}

This expresses the claim that the Einstein field equations emerge from the entropic variational principle under suitable identifications of variables and parameters.

###3. Obidi’s Principle of Complementarity

𝓓_E ∪ 𝓓_G = 𝓓_full

and

𝓓_E ∩ 𝓓_G ≠ ∅

where 𝓓_E is the entropic domain, 𝓓_G is the geometric domain, and 𝓓_full is the full physical domain. This states that both descriptions cover the theory and overlap in shared regimes.

###4. Obidi’s Correspondence Principle

lim ϵ→0 𝓐_E = 𝓐_GR

and

lim ϵ→0 Φ(S) = g_{μν}^{GR}

where ϵ represents a coarse-graining, classicality, or low-gradient parameter. This requirement means that general relativity must be recovered as the limiting case of the entropic theory.

## Interpretation

These principles can be read as a hierarchy. The Obidi Equivalence Principle gives the basic translation between entropic and geometric language. The Obidi Conjecture goes further by claiming that gravity is not fundamental but emergent from entropy. The Principle of Complementarity explains why both viewpoints remain useful, and the Correspondence Principle guarantees that the theory matches known physics in the proper limit.

Taken together, they form a conceptual bridge between entropy-based dynamics and classical gravity. In that sense, they function as the structural backbone of the Theory of Entropicity (ToE).

Interconnected Principles of the Theory of Entropicity (ToE)

**Obidi Equivalence Principle (OEP).**  

The Obidi Equivalence Principle states that a standard gravitational or geometric formulation and the corresponding entropic formulation describe the same physical content when expressed in the appropriate variables and limits. In this sense, the entropic action is taken to be equivalent to the conventional action, even if the two formulations are written in different mathematical languages.

**Obidi Conjecture (OC).**  

The Obidi Conjecture proposes that Einsteinian gravity can be derived from, or recovered as an emergent limit of, the Theory of Entropicity. It is the claim that entropy, treated as a fundamental dynamical entity, gives rise to the observed gravitational field equations under suitable conditions.

**Obidi’s Principle of Complementarity (PoC).**  

Obidi’s Principle of Complementarity states that the entropic description and the geometric description are both legitimate, but each is most useful in a different regime of analysis. The two descriptions are not contradictory; rather, they complement one another as different perspectives on the same underlying structure.

**Obidi’s Correspondence Principle (OCP).**  

Obidi’s Correspondence Principle states that the Theory of Entropicity must reduce to general relativity in the appropriate classical, coarse-grained, or limiting regime. This principle ensures that the new theory reproduces established gravitational physics where general relativity is already known to work.

## Formal axioms

Let $$ \mathcal{M} $$ denote the physical manifold, $$S$$ the entropy field, $$g_{\mu\nu}$$ the emergent metric, and $$ \mathcal{A}_{\mathrm{E}} $$ the entropic action. Let $$ \mathcal{A}_{\mathrm{GR}} $$ denote the standard gravitational action.

### Axiom 1: Obidi Equivalence

$$

\mathcal{A}_{\mathrm{E}}[S,\mathcal{M}] \sim \mathcal{A}_{\mathrm{GR}}[g_{\mu\nu},\mathcal{M}]

$$

meaning that both actions yield the same physical content under the appropriate field map and limiting procedure.

More explicitly, there exists a transformation $$ \Phi $$ such that

$$

\Phi:\; S \mapsto g_{\mu\nu},

\qquad

\delta \mathcal{A}_{\mathrm{E}} = 0

\;\Longleftrightarrow\;

\delta \mathcal{A}_{\mathrm{GR}} = 0

$$

within the domain where the equivalence holds.

### Axiom 2: Obidi Conjecture

$$

\mathcal{A}_{\mathrm{GR}}[g_{\mu\nu},\mathcal{M}]

\;\leftarrow\;

\mathcal{A}_{\mathrm{E}}[S,\mathcal{M}]

$$

meaning that the Einstein field equations emerge from the entropic variational principle in the suitable limit.

Equivalently,

$$

\frac{\delta \mathcal{A}_{\mathrm{E}}}{\delta S} = 0

\quad \Rightarrow \quad

G_{\mu\nu} + \Lambda g_{\mu\nu}

=

\kappa T_{\mu\nu}

$$

for an appropriate identification of the entropy-sector variables with geometric and matter variables.

### Axiom 3: Obidi Complementarity

$$

\mathcal{D}_{\mathrm{E}} \cup \mathcal{D}_{\mathrm{G}} = \mathcal{D}_{\mathrm{full}}

$$

where $$ \mathcal{D}_{\mathrm{E}} $$ is the entropic domain of description, $$ \mathcal{D}_{\mathrm{G}} $$ is the geometric domain of description, and $$ \mathcal{D}_{\mathrm{full}} $$ is the complete physical domain.

In addition,

$$

\mathcal{D}_{\mathrm{E}} \cap \mathcal{D}_{\mathrm{G}} \neq \varnothing,

$$

meaning that both descriptions overlap and remain jointly consistent in shared regimes.

### Axiom 4: Obidi Correspondence

$$

\lim_{\epsilon \to 0}\mathcal{A}_{\mathrm{E}} = \mathcal{A}_{\mathrm{GR}}

$$

where $$ \epsilon $$ denotes a coarse-graining, classicality, or low-entropy-gradient parameter.

Equivalently,

$$

\lim_{\epsilon \to 0}\Phi(S) = g_{\mu\nu}^{\mathrm{GR}}

$$

and

$$

\lim_{\epsilon \to 0}

\left(

\frac{\delta \mathcal{A}_{\mathrm{E}}}{\delta S}

\right)

=

0

\quad \Longrightarrow \quad

\text{GR is recovered}.

$$

## Compact and concise axiom set

1. **Equivalence:** Entropic and geometric formulations are physically equivalent under an admissible field map.  

2. **Emergence:** Einsteinian gravity emerges from the entropic variational principle.  

3. **Complementarity:** Entropic and geometric descriptions are mutually complementary.  

4. **Correspondence:** The entropic theory reduces to general relativity in the classical limit.

## Summary 

> The Obidi framework contains four foundational statements: the **Obidi Equivalence Principle**, which asserts equivalence between entropic and geometric formulations; the **Obidi Conjecture**, which proposes the emergence of general relativity from entropy-based dynamics; the **Obidi Principle of Complementarity**, which holds that entropic and geometric descriptions are jointly valid but regime-dependent; and the **Obidi Correspondence Principle**, which requires recovery of general relativity in the appropriate limit.

On the Complexity of the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE), formulated by John Onimisi Obidi, is characterized by high mathematical and conceptual complexity because it seeks to unify thermodynamics, general relativity, and quantum mechanics by elevating entropy from a statistical byproduct to the fundamental field of reality. [1, 2]

Obidi's Theory of Entropicity (ToE) is complex and sophisticated on three nontrivial levels:

1. Mathematical Complexity

The theory's mathematical architecture is sophisticated and departs from classical calculus used in traditional physics: [1, 2]

• Obidi Field Equations (OFE): Also known as the Master Entropic Equation (MEE), these are the entropic equivalent of Einstein's field equations. They are nonlinear and nonlocal, describing how entropy gradients evolve and couple to geometry and matter.
• Iterative Solutions: Unlike Einstein's equations, which often yield closed-form solutions (like the Schwarzschild metric), the MEE must be solved through non-explicit iterative methods. These function more like adaptive algorithms, refining configurations through a process similar to Bayesian inference.
• Information Geometry: ToE utilizes Amari–Čencov α-connections to bridge the gap between statistical probability spaces and physical spacetime.
• Generalized Entropies: It incorporates non-extensive formalisms like Rényi and Tsallis entropy to handle scale-sensitive and non-extensive systems. [1, 2, 3, 4, 5, 6]

2. Conceptual and Ontological Complexity

ToE introduces a "monistic" worldview where everything is a manifestation of the entropy field: [6, 7, 8]

• Spacetime as Emergent: Spacetime and gravity are not primary; they are emergent residues of underlying entropic dynamics.
• No-Rush Theorem: This principle explains the speed of light ($c$) not as a geometric constant but as the maximum rate at which the entropic field can redistribute information.
• Vuli–Ndlela Integral: A reformulation of quantum path integrals that embeds the "arrow of time" directly into quantum mechanics by weighting paths based on entropic cost. [1, 2, 9, 10, 11]

3. Computational Complexity

The theory's reliance on "self-updating" physics means it operates closer to artificial intelligence and complex adaptive systems than traditional differential geometry: [2, 4]

• Recursive Dynamics: Each iteration of the field equations redefines the very geometry it is calculated against, creating a recursive "dialogue" between entropy and its constraints.
• Modelling Challenges: Practical application requires advanced numerical architectures like entropy-constrained Monte Carlo methods and information-geometric gradient flows. [2, 4]

Would you like to explore how this theory specifically explains black holes or the No-Rush Theorem in more detail?

[1] https://medium.com

[2] https://encyclopedia.pub

[3] https://medium.com

[4] https://medium.com

[5] https://medium.com

[6] https://figshare.com

[7] https://medium.com

[8] https://medium.com

[9] https://medium.com

[10] https://medium.com

[11] https://medium.com