A Brief Overview of the Conceptual and Mathematical Evolution of the Theory of Entropicity (ToE)

By John Onimisi Obidi

Abstract

The Theory of Entropicity (ToE),

The Theory of Entropicity (ToE),

^[1]

as first formulated and further developed by John Onimisi Obidi,

^{[2][3][4][5][6][7][8]}  redefines entropy from a measure of disorder into the fundamental field of existence — the invisible curvature that gives rise to motion, time, gravitation, and consciousness.

By extending information geometry and uniting the α-connection of Amari–Čencov with the physical reality of entropy flow, ToE introduces the Entropic Metric Equation, where entropy becomes the true source of curvature, irreversibility, and evolution.

It absorbs Tsallis’ and Rényi’s formulations as natural consequences of the same underlying α-curvature and connects thermodynamics, quantum mechanics, and relativity into a single, entropic law of being.

This work summarizes the mathematical, physical, and philosophical implications of ToE, showing that entropy is not the shadow of physics — it is the light by which physics is seen.

redefines entropy from a measure of disorder into the fundamental field of existence — the invisible curvature that gives rise to motion, time, gravitation, and consciousness.

1. From Mathematics to Reality: The Entropic Metric Equation

At first glance, the ToE entropic metric equation (EME)

gᵢⱼ^(α) = ∂²ψ(θ)/∂θᵢ∂θⱼ + α Tᵢⱼₖ(θ)

appears similar to the Amari–Čencov α-connection known from information geometry.

But the similarity is purely formal — the meaning of each term has been transformed.

In Amari’s framework, the metric describes how information about probability distributions curves; in ToE, it describes how reality itself curves under entropy flow.

The potential ψ(θ) is no longer a mathematical function of data — it is the entropy potential field that drives time and structure.

The tensor Tᵢⱼₖ(θ) is no longer an abstract measure of statistical skewness — it is the irreversibility tensor, encoding the arrow of time.

And the constant α is not a deformation parameter — it is a physical curvature constant of entropy, the slope of irreversibility that propels the universe forward.

Thus, this single equation becomes the Einstein equation of entropy: while Einstein’s (pure) geometry is determined by stress–energy, ToE’s geometry is created and driven by the continuous flow of entropy.

The manifold it describes is alive, breathing through the expansion and contraction of entropy itself.

2. The Entropic Metric as the New Fabric of Reality

Einstein’s relativity taught that energy curves spacetime.

The Theory of Entropicity teaches that entropy curves existence itself.

The term α·Tᵢⱼₖ is not a small correction — it is the seed of becoming.

It is the asymmetry that ensures the future differs from the past.

If α were zero, there would be no irreversible processes, no aging, no evolution — only a frozen, reversible symmetry.

But α is not zero.

Reality, therefore, is the manifestation of entropy’s will to change.

In this sense, the entropic metric is not about static structure but about motion, time, and emergence.

It is the mathematical expression of the fact that to exist is to evolve.

3. The Equation as a Bridge Between Domains

Each great theory of physics has its own geometry:

• Einstein used the Riemannian metric to describe spacetime curvature.
• Schrödinger used the Fubini–Study metric to describe quantum phase space.
• Amari and Čencov used the Fisher–Rao metric to describe information curvature.

ToE unites them.

It shows that all these geometries are shadows of the same deeper manifold — the entropic manifold, whose curvature determines both the flow of time and the structure of information.

Einstein’s gravity, quantum amplitudes, and statistical inference all become facets of one universal geometry of entropy.

4. The Shift from Epistemology to Ontology

The Amari–Čencov metric described how we know;

the Theory of Entropicity describes how we exist.

This is the difference between a map and the landscape itself.

Information geometry is epistemic — a theory of knowledge.

ToE is ontological — a theory of being.

It turns entropy from a bookkeeping tool into a living field — one that shapes the trajectories of matter, thought, and time.

In this sense, the Entropic Metric is not a metaphorical geometry of information, but a physical geometry of reality.

5. The Unification of the Three α’'s

In different sciences, the symbol α has appeared in three guises:

as Tsallis’ nonadditivity constant, Rényi’s information scaling parameter, and Amari’s geometric connection constant.

ToE reveals that all three are manifestations of one universal α — the curvature of irreversibility in the entropy field.

• In Tsallis and Rényi theories, α modifies how entropy accumulates — describing non-extensive behavior in complex systems.
• In Amari’s framework, α modifies how curvature measures information divergence.
• In ToE, α modifies the curvature of existence itself, connecting thermodynamic nonadditivity with geometric asymmetry.

Thus, the “statistical α,” “informational α,” and “geometric α” are not separate — they are different shadows cast by the same physical α.

6. How Tsallis and Rényi Appear in ToE

Tsallis’ and Rényi’s formulations are encoded in the potential ψ(θ) of the Entropic Metric.

When ψ(θ) adopts logarithmic, power-law, or nonextensive forms, it naturally reproduces the behavior of these entropies within ToE’s geometry.

They are not added to the theory — they emerge from it as different curvature regimes of the entropic manifold.

The third-order tensor Tᵢⱼₖ(θ) contains the same nonadditive structure that Tsallis and Rényi described statistically, but ToE embeds it geometrically.

Entropy’s nonlinear accumulation becomes the curvature of space and time themselves.

Thus, even though the Tsallis and Rényi forms aren’t written explicitly in your metric, their mathematical signatures live inside the Tᵢⱼₖ(θ) term.

That tensor is responsible for third-order nonlinearities — precisely the kind of deviation from additivity that Tsallis and Rényi entropies describe.

In ToE, we have:

• Tᵢⱼₖ(θ) = the “nonextensive curvature tensor” — capturing how local entropy interactions deviate from equilibrium.
• α multiplies this tensor, physically scaling how strongly these nonadditive effects influence geometry.

So, Tsallis and Rényi are not “added into” ToE —

they are geometrically encoded in Tᵢⱼₖ(θ), just as the Schwarzschild curvature tensor encodes gravity in general relativity.

That is why you “don’t see” them explicitly —

because ToE has absorbed them into its geometric structure.

6.1 Level 1 — Statistical Foundations: Tsallis and Rényi α’s

6.1. Level 1 — Statistical Foundations: Tsallis and Rényi α's

At the base of the landscape lie the statistical fields of Tsallis and Rényi.

Here, α (often written as q) modifies how probabilities combine and how entropy scales.

When α = 1, systems behave normally — additive, linear, and extensive.

But as α drifts away from 1, strange things happen: correlations form, interactions couple, systems remember their past.

In this region, α is the thermodynamic artist shaping how disorder accumulates.

It governs how entropy departs from classical equilibrium, allowing galaxies, cells, and markets alike to self-organize.

Tsallis’ and Rényi’s equations are mathematical portraits of this behavior — paintings of entropy when it ceases to be simple.

They tell us how entropy behaves when the world becomes complex, but they do not tell us why.

6.2 Level 2 — Informational Structure: The Amari–Čencov α-Connection

6.2. Level 2 — Informational Structure: The Amari–Čencov α-Connection

Climbing higher, we reach the terrain of information geometry.

Here, the same symbol α reappears, but now it bends manifolds instead of probabilities.

It defines how informational distances curve, how dual coordinate systems diverge, and how inference becomes asymmetric.

It is the geometer’s α, describing how knowledge itself warps when systems exchange information.

In this zone, the world is abstract — built from probability manifolds and statistical connections —

but it hints at a deeper reality: curvature and irreversibility are not accidents of thought; they may be the universe’s native language.

Yet even here, α was treated as a symbol of convenience, not a measurable constant of nature.

It encoded perspective, not physical cause.

6.3 Level 3 — Physical Reality: The Entropic α of ToE

6.3. Level 3 — Physical Reality: The Entropic α of ToE

Now we reach the summit — the Theory of Entropicity.

At this altitude, α is no longer statistical or symbolic; it becomes a universal curvature constant of irreversibility, woven into the very fabric of existence.

From this viewpoint, Tsallis’ and Rényi’s α describe how entropy accumulates in microscopic configurations,

while Amari’s α describes how information curves across abstract manifolds.

Both, says ToE, are local projections of the same underlying curvature: the entropic field α.

This α is the universal slope of time — the reason events unfold, the measure of how the universe leans forward into its own future.

When α = 0, time stands still and reality freezes in reversible symmetry.

When α > 0, entropy flows forward, producing motion, growth, and evolution.

When α < 0, entropy contracts, creating order, coherence, and the seeds of consciousness.

Thus, all three α’s — Tsallis’, Rényi’s, and Amari’s — are revealed as different cross-sections of one multidimensional entropic curvature.

All three are unified by one constant: α — the curvature of irreversibility.

It is the reason time moves forward, systems evolve, and consciousness arises.

8. The Entropic Metric in the Universe

The implications are vast:

• In cosmology, the entropic curvature explains the universe’s accelerated expansion as a natural result of residual entropy flow.
• In quantum physics, the finite speed of entanglement formation is the time it takes for entropy to equalize between systems.
• In thermodynamics, irreversibility is no longer an approximation — it is a fundamental law of geometry.
• In cognition, entropy defines the flow of awareness and memory — the geometry of thought itself.

Entropy thus ceases to be a shadow cast by energy and becomes the light illuminating all domains.

9. Empirical Predictions and Experimental Tests

ToE is testable.

It predicts measurable phenomena across scales:

1. Starlight bends not because of spacetime curvature alone, but because of entropy gradients.

1. The 232-attosecond delay in quantum entanglement is the fundamental entropic handshake time.

1. The small positive cosmological constant is the residual curvature of entropy at cosmic scale.

6.5. In Summary — The Synthesis

We can now describe the relationship like this:

The Rényi and Tsallis entropies capture statistical manifestations of nonadditive entropy behavior.

The Amari–Čencov α–connection captures the geometric manifestation of the same asymmetry.

The Theory of Entropicity reveals that both are projections of one underlying entropic geometry, governed by the universal constant α — the curvature of irreversibility in the entropy field.

Thus, the following holds:

• Tsallis/Rényi α ↔ information scaling of entropy,
• Amari α ↔ geometric scaling of entropy,
• ToE α ↔ physical scaling of reality’s curvature itself.

And that is why, even though the Tsallis and Rényi equations are not visible in ToE’s metric’s outer form, their essence is fully contained in it —

they are embedded as the internal nonlinear structure of the entropy potential ψ(θ) and its curvature tensor Tᵢⱼₖ(θ) in the field equations of the Theory of Entropicity (ToE).

7. The Entropic Geometry Diagram of Correspondence

Visualize three mountains:

• At the base, Tsallis and Rényi describe how entropy behaves statistically when systems interact or correlate.
• In the middle, Amari–Čencov describe how information bends and diverges geometrically.
• At the summit, ToE reveals that both are merely different faces of one entropic mountain — the curvature of existence itself.

From this higher perspective:

1. A measurable entropic pressure should exist just outside event horizons.

1. Neural activity patterns follow entropic balancing laws identical to thermodynamic systems.

• Tsallis and Rényi express

• how entropy accumulates

• ;

• Amari and Čencov
• express how entropy bends;
• ToE expresses why entropy exists and flows.

Through such tests, entropy transitions from abstraction to observable force.

10. Deriving Bianconi’s G-Field and Cosmological Constant

Ginestra Bianconi proposed the G-Field to explain why vacuum energy yields a small, positive cosmological constant.

ToE derives this naturally: the G-Field is the long-wavelength harmonic mode of the entropic field — the universe’s slow, gentle curvature toward equilibrium.

The cosmological constant is not fine-tuned; the Theory of Entropicity (ToE) shows us that Bianconi’s small, positive cosmological constant is the residual entropic pressure that remains as the universe asymptotically seeks balance.

Bianconi’s field becomes a special case of ToE’s entropy field — not an addition, but a rediscovery from deeper principles.

11. The Philosophical Horizon

Where earlier physics asked what the universe is made of, ToE asks why it must exist at all.

Entropy, in this theory, is not decay but creation — not chaos but coherence.

It is the process by which the universe writes itself into being.

Dualities dissolve:

• Mind and matter become entropic reflections of each other.
• Time and space become gradients of entropy.
• Energy and information become conjugate expressions of one entropic potential.

ToE turns science back toward unity, revealing that every law of nature is an echo of a deeper truth: the flow of entropy is the act of existence itself.

12. Postscript — The Whisper of Entropy

Entropy is no longer the destroyer of order; it is the composer of the cosmos.

Every transformation, every thought, every heartbeat is a curvature of entropy seeking equilibrium.

Where others saw decay, ToE sees direction.

Where others saw randomness, ToE sees rhythm.

We are not apart from entropy — we are how it perceives itself.

Our awareness is entropy’s self-reflection, its way of knowing that it flows.

To understand entropy is to understand the very reason why time has meaning, why change is inevitable, and why consciousness is possible.

The Theory of Entropicity does not merely add a new equation to physics.

It completes the sentence that began with Einstein and continued through Shannon, Schrödinger, and Bianconi:

Entropy is the foundation of everything that exists.

📘 Reader’s Summary: One Alpha, One Universe

For decades, the symbol α (alpha) appeared across multiple disciplines — in Tsallis and Rényi’s entropies, in Amari–Čencov geometry — always hinting at something universal but unseen. The Theory of Entropicity (ToE) changes everything.

It shows that all these α’s are not coincidences, but reflections of a single, universal constant — the curvature of entropy flow through the universe.

The Theory of Entropicity reveals that these α’s are one and the same: the curvature of entropy flow, the slope of irreversibility that defines time and evolution.

• Tsallis and Rényi describe how entropy behaves.
• Amari describes how entropy bends.
• The Theory of Entropicity (ToE) explains why entropy exists.

This single constant unites thermodynamics, quantum mechanics, information theory, and consciousness under one principle:

everything that exists flows along the curvature of entropy.

When entropy flows freely, α is positive and time moves forward; when systems reach equilibrium, α vanishes; when coherence or self-organization occurs, α becomes negative, giving rise to order and awareness.

Thus, the same constant that shapes heat, light, gravity, quantum behavior, and even thought is one and the same:

α — the curvature of the entropic field.

✍️ Author’s Note — John Onimisi Obidi

When I began studying entropy, I did not expect to find a new physics — only deeper understanding.

But as I followed its traces through thermodynamics, geometry, and information theory, I saw something profound: entropy is not an afterthought of energy — it is the origin of existence.

The Theory of Entropicity (ToE) is my attempt to give that realization structure and voice.

It unifies what science has long kept apart: the physical, the informational, and the conscious.

It tells us that the same flow that shapes stars also shapes thought — that the universe is one continuous act of entropic creation.

If these ideas resonate with you, I invite you to journey further.

In the next parts of this series, we will explore the mathematics, the predictions, and the experimental frontiers of this theory — from entropic cosmology and light deflection to entropy’s role in perception and time.

Thank you for walking this new path with me.

Entropy has always been speaking —

the Theory of Entropicity (ToE) simply teaches us to listen.

— John Onimisi Obidi

Originator & Developer, The Theory of Entropicity (ToE)

© 2025

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.

[2] Physics:HandWiki Master Index of Source Papers on Theory of Entropicity(ToE). (2025, September 9). HandWiki. Retrieved 17:33, September 9, 2025

[3] Obidi, John Onimisi. Conceptual and Mathematical Foundations of Theory of Entropicity(ToE). Encyclopedia. Available online: (accessed on 13 October 2025)

[4] Wissner-Gross, A. D., & Freer, C. E. (2013). Causal Entropic Forces. Physical Review Letters.

[5] Amari, S. (2016). Information Geometry and Its Applications. Springer.

[6] Jaynes, E. T. (1957). Information Theory and Statistical Mechanics. Physical Review.

[7] Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

[8] Vidal, G. (2008). Class of Quantum Many-Body States That Can Be Efficiently Simulated. Physical Review Letters.

[9] Methods of Information Geometry — Shun-ichi Amari & Hiroshi Nagaoka (American Mathematical Society, 2000)

[10] The Vuli-Ndlela Integral in the Theory of Entropicity (ToE) — John Onimisi Obidi (2025)

[11] The Obidi Action and the Foundation of the Entropy Field Equation — John Onimisi Obidi (2025)

[12] The Master Entropic Equation (MEE) — John Onimisi Obidi (2025)

[13] Psych entropy and the Entropy of the Mind — John Onimisi Obidi (2025)

[14] Bianconi, G. (2009). Entropy of network ensembles. Physical Review E.

[15] Bianconi, G., & Barabási, A.-L. (2001). Competition and multiscaling in evolving networks. Europhysics Letters.

[16] Bianconi G. Gravity from entropy. Phys Rev D. 2025 Mar 3;111(6):066001. doi:10.1103/PhysRevD.111.066001.

[17] 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.

[18] 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).

[19] 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).

[20] 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).

Entropy is the foundation of everything that exists.