On the Conceptual and Mathematical Beauty of Obidi’s Theory of Entropicity (ToE)
From Geometric Relativity to Geometric Entropicity
JOHN ONIMISI OBIDI
October 20, 2025
On the Conceptual and Mathematical Beauty of Obidi's Theory of Entropicity (ToE)
From Geometric Relativity to Geometric Entropicity
When Lev Davidovich Landau, one of the most brilliant physicists of the twentieth century, first studied Einstein's General Theory of Relativity, he is said to have exclaimed that it was "so beautiful that it must be true."
He was not merely admiring the equations; he was recognizing a kind of inner perfection — a harmony between mathematical necessity and natural truth. To Landau, as to many of his generation, beauty was not an ornament in physics but its highest proof.
In the same spirit, the Theory of Entropicity (ToE) emerges in our time as a work of comparable aesthetic inevitability.
It proposes that entropy — long considered a measure of disorder, uncertainty, or information loss — is in fact the very foundation of physical existence.
Everything we call matter, energy, space, and time is not built upon entropy but from it. Entropy is the invisible current that gives rise to geometry, to motion, to matter and even to the passage of time itself.
1. From Order to Origin
For two centuries, physics has treated entropy as a secondary concept: a measure of how far systems have drifted from order, a bookkeeping device for thermodynamic processes.
In statistical mechanics, entropy was a way of counting microstates. In information theory, it became a measure of uncertainty. And in cosmology, it was invoked to describe the arrow of time — the universe's relentless drift toward equilibrium.
Even when Stephen Hawking made his momentous discovery of black hole radiation — now known as Hawking Radiation — as a profound manifestation of entropy on a cosmic scale, it remained fundamentally described in terms of quantum information and particle–antiparticle processes near the event horizon.
Yet, in all these formulations, entropy remained passive. It described change but never caused it. It was a bystander to the dynamics of the universe.
The Theory of Entropicity (ToE) — as first formulated and further developed by John Onimisi Obidi — changes that forever. It proposes that entropy is not an effect but the cause, not an outcome but the origin. It is the active field that drives the universe toward unfolding complexity and, in doing so, creates the phenomena we observe as energy, space, and time.
Entropy is the universal generator, and everything else — geometry, gravity, quantum uncertainty, even consciousness — are the shadows it casts upon reality.
2. The New Foundation
Einstein's General Relativity replaced the Newtonian notion of force with the curvature of spacetime.
The Theory of Entropicity (ToE) goes one step deeper: it replaces spacetime itself with entropy as the true substrate of existence.
In ToE, space and time are not pre-existing arenas in which events occur; they are emergent manifestations of entropy's dynamic structure.
Just as the ripples on a pond are not separate from the water that carries them, so space and time are not separate from entropy — they are its motion, its geometry, its rhythm.
This reordering of ontology is both radical and simple. It restores unity where modern physics had split reality into incompatible domains. Thermodynamics, quantum theory, and relativity — once treated as separate pillars of science — are revealed as different expressions of the same entropic principle.
The Theory of Entropicity (ToE) reveals that energy is the quantitative measure of entropy in motion, gravity emerges as the curvature of entropy, and quantum probability arises from entropy's intrinsic irreversibility at microscopic scales.
3. The Entropic Field of ToE
Central to the Theory of Entropicity (ToE) is the idea that entropy is not a number, a function, nor a measure of ignorance, but a field — a real, continuous entity that permeates all existence.
Every point in the universe is filled with entropy, and the dynamics of that field determine everything we experience as physical reality.
Objects move, not because they are pulled by forces or guided by geometric geodesics, but because the entropic field rearranges itself to minimize constraint and maximize flow. ToE says: “spacetime is the macroscopic expression of an entropic function minimizing constraint.”
4. Artificial Intelligence and Autonomous Vehicles as a Symbolism for the Entropic Field of ToE
In the Theory of Entropicity, entropy is an autonomous physical field whose dynamics are governed by a self-consistent variational principle. This field continually reorganizes the configuration of space, time, and energy to minimize internal constraint and maximize entropic flow. Matter and motion arise as local consequences of this universal optimization, much as intelligent systems adjust their states autonomously to sustain functionality.
4.1 What “autonomous” means in ToE
When ToE says entropy is an autonomous physical field, it does not mean that entropy “thinks” or “decides” like a mind — but that it acts according to intrinsic laws of self-regulation, independent of external instruction or external force.
- In classical physics: forces act on bodies; geometry constrains motion; energy is conserved within a given framework.
- In ToE: the entropic field itself is the framework — it both generates and evolves the conditions that define motion, curvature, and even conservation.
4.2 How entropy becomes a dynamic field in ToE
In the ToE framework, the Obidi Action (simplistic form) defines the Lagrangian density for the entropic field:
$$
A_{ToE} = \int \left( \nabla_i S \, \nabla^i S \;-\; V(S) \;+\; J(x,S) \right) \, \sqrt{-g} \, d^4x
$$
Here:
- \(S(x,t)\) is the entropic potential field,
- \(V(S)\) is its self-interaction potential,
- \(J(x,S)\) represents coupling with matter and information.
By varying this action, you obtain the Master Entropic Equation (MEE):
$$
\Box S + \mathcal{A}(S, \nabla S, g) = 0
$$
This shows that the field evolves according to its own gradient structure, not due to external imposition. The term \(\mathcal{A}\) encodes irreversibility and entropy flow, ensuring that the field’s rearrangement always tends toward minimal constraint and maximal flow.
4.3 “Reorganizing reality dynamically”
- Matter = localized entropic condensation (regions of high entropic density)
- Motion = redistribution of entropic gradients
- Gravity = curvature induced by those gradients
- Time = sequential update of the field’s configuration as entropy flows irreversibly
Thus, when we say the entropic field reorganizes reality dynamically, we mean that the field is both substrate and law — it does not operate within space and time but generates them through its ongoing evolution.
In other words, reality is the field thinking itself through entropy flow — not in the
cognitive sense, but in the physical sense of continuous self-adjustment to maintain
the directionality of existence.
4.4 The analogy with autonomous vehicles and AI
This is a powerful and very apt analogy. Think of it this way:
| Analogy |
Autonomous Vehicle / AI |
Entropic Field (ToE) |
| System identity |
The AI system that perceives, decides, and acts |
The entropic field that generates, adjusts, and evolves all physical states |
| Internal rule set |
Neural architecture + algorithms (learning laws) |
Entropic field equations (Global Action, MEE, etc.) |
| Sensors / inputs |
Environment data, feedback loops |
Gradient information from within itself (\(\nabla S, g\)) |
| Goal function |
Optimize path / minimize loss |
Minimize constraint / maximize entropy flow |
| Output / behavior |
Adjusts trajectory autonomously |
Reorganizes spacetime and energy distributions autonomously |
So, just as an autonomous AI navigates its environment by continually adjusting its trajectory to new data in order to maintain optimal performance, the entropic field adjusts its own geometry and entropy rules to maintain optimal entropy flow through the environment.
In other words, the entropic system can self‑distribute itself‑regulating intelligence — not mental intelligence, but field intelligence encoded in the laws of entropic evolution.
4.5 Why this is a radical shift from previous physics
| Classical Physics |
Theory of Entropicity (ToE) |
| Forces act on matter |
Entropy reorganizes all matter and geometry |
| Space-time is background |
Space-time is emergent from entropy field |
| Motion follows geodesics |
Motion is field self-rearrangement |
| Energy is conserved in static geometry |
Energy is redistributed dynamically through entropy flow |
| Entropy is a measure |
Entropy is the generator |
Thus, the ToE transforms the universe from a passive system of forces and geometry
into an autonomous field of continual self-organization — a living, evolving system
whose every process is a manifestation of entropic intelligence.
5. The Bridge from Information to Geometry
One of the most striking achievements of modern physics has been the recognition
that information and physical law are intimately connected.
The work of Claude Shannon, Rolf Landauer, and later thinkers showed that
information is not abstract — it is physical. Every bit stored, every computation performed,
has an energy cost and an entropic footprint.
The Theory of Entropicity extends this connection to its ultimate form. It asserts that
information is born from entropy itself. When entropy differentiates, information arises
as the local ordering of its field.
And since information can be measured geometrically — by the Fisher–Rao metric in
probability space or by the Fubini–Study distance in quantum state space — it follows
that geometry itself is an expression of entropy.
The ToE thus explains why geometry, information, and energy are inseparably linked.
Where traditional physics starts with geometry and ends with entropy as a derived
quantity, ToE reverses the chain: geometry is the effect, not the cause.
6. The Role of Irreversibility
Time, in classical physics, was a parameter. In relativity, it became a coordinate.
In ToE, it becomes a process — a manifestation of entropy’s irreversible flow.
The universe’s arrow of time is not an imposed boundary condition; it is a natural
property of the entropic field. Because entropy flows only one way — from constraint
toward expansion — the evolution of the universe has an intrinsic direction.
This directionality, which gives rise to causality itself, emerges directly from entropy
dynamics. The passage of time, the aging of stars, and the unfolding of thought are all
expressions of one cosmic asymmetry — entropy’s irreversible will to transform.
7. Rényi–Tsallis Entropies and Amari–Čencov’s α-Connection
Among the most graceful achievements of the Theory of Entropicity (ToE) is its ability
to weave together two of the most influential generalizations of entropy in modern
science — the Rényi and Tsallis formulations — into a single physical narrative.
For decades, these entropies were regarded as mathematical extensions of the
Boltzmann–Gibbs definition, useful mainly in statistics and complex-systems theory.
They quantified how systems depart from ordinary additivity, capturing correlations,
long-range interactions, and multifractal behavior. Yet they remained abstract tools,
powerful but isolated from the geometry of the physical world.
The Theory of Entropicity (ToE) changes that completely. It interprets Rényi and
Tsallis entropies not as detached statistical curiosities but as genuine signatures of the
geometry of nature itself. Within ToE, the parameter that measures non-additivity
becomes a physical indicator of how the universe’s entropic field bends, connects,
and evolves.
In this view, every departure from perfect additivity corresponds to a subtle deformation
of the underlying informational fabric of reality. The very same measure that describes
correlated probabilities in statistical mechanics now tells us how spacetime, matter,
and information are entropically curved.
ToE goes further by linking these statistical deformations to the geometric language
of Amari and Čencov’s information connections. Whereas Rényi and Tsallis describe
how probabilities combine, the Amari–Čencov framework describes how information
flows through a curved manifold of possibilities. ToE reveals that these two languages
— one statistical, one geometric — are in fact two views of a single entropic phenomenon.
When entropy ceases to be perfectly additive, the manifold of information cannot
remain perfectly flat. It acquires curvature, torsion, and asymmetry, and these
geometric distortions manifest as the gravitational and dynamical structures we call
spacetime.
This is where the beauty of ToE becomes evident. The theory identifies a deep unity
between the measure of complexity in probability space and the structure of geometry in physical space. The same principle that governs how probabilities blend in a turbulent plasma or a living cell also governs how stars curve light and how time
unfolds. No other framework (Jacobson’s Thermodynamic Gravity, Caticha’s Entro-Dynamics, Verlinde’s Entropic Gravity, Bianconi’s G-Field, etc.) has so elegantly
fused the mathematical generalizations of entropy with the physical geometry of th
universe. None has produced a self-contained field theory where entropy itself is t
dynamical field generating both probabilistic and physical geometry.
This expresses ToE’s idea of universality of entropy flow — that the same entropic field equations apply across scales:
- From microscopic thermodynamics (biological or plasma systems),
- To macroscopic spacetime curvature (gravitational or cosmological systems).
This echoes Toe’s strong claim that entropy is scale-independent and causal,
producing order and motion across all domains of reality.
In ToE, the same entropic functional Λ(x) defines both:
- The curvature of probability manifolds (statistical complexity), and
- The curvature of spacetime geometry (physical structure).
Thus, this statement expresses the ToE postulate that information geometry and
physical geometry are isomorphic through entropy.
Hence, the beauty of the Theory of Entropicity (ToE) becomes evident, as all of the
above reveals a profound unity between the measure of complexity in probability space and the structure of geometry in physical space. The same entropic principle that
governs how probabilities evolve in a turbulent plasma or a living cell also dictates
how stars curve light and how time itself unfolds. In ToE, the mathematical
generalizations of entropy—spanning the Fisher–Rao, Fubini–Study, and Rényi–Tsallis frameworks—are seamlessly fused with the physical geometry of the universe.
No other framework has so elegantly unified these domains under a single variational
principle.
In the Rényi and Tsallis formulations, the parameter that quantifies non-extensivity
varies across systems: in astrophysical plasmas it may capture collective interaction
in quantum entanglement it measures non-local correlations; in cosmology it encodes
the deviation from equilibrium of the cosmic horizon. The Theory of Entropicity (T
unifies all these instances under one law: they are diverse expressions of the same
entropic field behaving under different boundary conditions. By embedding the
generalized entropies directly within its variational principle—the Obidi Action—T
transforms what were once empirical fitting parameters into physically meaningful
constants of nature.
What makes this connection so extraordinary is its symmetry and economy. Where
classical thermodynamics used energy as the universal currency, and relativity used
geometry, ToE employs entropy itself as the unifying medium. Rényi and Tsallis
provided the mathematical vocabulary for complexity; ToE provides the physical
grammar that lets the universe speak that language. The non-extensive index that o
belonged to abstract probability now dictates how curvature arises, how systems
exchange information, and how the arrow of time becomes irreversible.
Seen through this lens, the Rényi and Tsallis entropies cease to be mere statistical
inventions; they are windows into the entropic architecture of reality. ToE shows that when the universe departs from perfect equilibrium, it does so along directions
defined by these entropies. Their parameters record the memory of correlations, the
tension between order and freedom, and the gradient along which the universe
evolves. Each value corresponds to a different geometric temperament of nature—a
different way the entropic field sculpts space, time, and matter.
This unification has profound implications. It implies that the same mathematics that describes information propagation in neural networks (Artificial Intelligence -AI),
energy distribution in galaxies, and coherence loss in quantum systems stems from one underlying entropic principle. By linking generalized entropies with the geometry of spacetime, ToE gives physicists and mathematicians a common language that
bridges complexity theory, quantum mechanics, thermodynamics, and gravitation. It opens the possibility that by measuring non-extensive statistical behavior in one
domain, we can infer geometric properties in another—a predictive power absent fr
previous theories.
The beauty of ToE, therefore, lies not only in its ambition but in its coherence. It
unites the abstract and the tangible, the statistical and the geometric, the micro and
the cosmic, within a single entropic continuum. Where earlier theories saw separate
realms—information versus space, statistics versus gravity—ToE perceives a seamless
f
low governed by entropy’s universal logic. In doing so, it transforms the Rényi and
Tsallis entropies from mathematical curiosities into the living fingerprints of the
universe’s most fundamental law: that everything evolves, curves, and connects
through entropy itself.
8. The beauty of unification
What makes ToE beautiful is not only what it explains but how it explains. It does not rely on arbitrary postulates or patchwork equations. It begins with one concept — entropy — and allows all else to follow logically from it. Each phenomenon becomes a manifestation of a single underlying principle, expressed differently at different scales.
This kind of unification is the highest form of beauty in science. It is the kind of beauty Einstein recognized in the curvature of spacetime and that Maxwell found in the symmetry of his electromagnetic equations. It is the beauty that comes from economy: the ability of one idea to illuminate a hundred phenomena. Occam’s razor is thus utilized at its best.
ToE achieves this by treating entropy not as an effect of processes, but as the cause of processes. Where General Relativity describes how mass curves spacetime, ToE describes why spacetime exists at all. Where quantum mechanics describes probabilities, ToE explains why probabilities arise — as the measurable expression of entropy’s irreversibility. Where thermodynamics sets limits on efficiency, ToE reveals those limits as laws of nature’s entropic architecture.
In this view, every physical law becomes an emergent rule of entropy’s game. From the smallest particle to the largest galaxy, the same field plays out its dynamics — harmoniously, relentlessly, beautifully.
9. The human connection
Beyond its physics, ToE also offers a profound philosophical reflection on existence. If entropy is the substrate of reality, then human consciousness — with its capacity for memory, imagination, and choice — is part of the same entropic continuum.
Our thoughts are not exceptions to the universe’s laws; they are extensions of them. Every act of perception is an act of entropic ordering — the mind’s attempt to reduce uncertainty, to carve clarity out of possibility. In this light, the human quest for knowledge is itself an entropic process.
We, as observers, are not outside the system but participants in entropy’s unfolding. Every experiment, every theory, every equation we write is part of the universe’s self‑discovery. To understand entropy, therefore, is to understand not only the cosmos but ourselves.
10. The elegance of inevitability
What makes the Theory of Entropicity (ToE) so strikingly beautiful is its inevitability. Once the principle is stated — that entropy is the fundamental field — everything follows with logical precision.
- Geometry must arise, because differences in entropy define structure.
- Time must flow, because entropy’s transformation is irreversible.
- Quantum uncertainty must exist, because entropy governs probability.
- Even gravity must emerge, because the entropic field shapes the motion of all things toward states of maximal equilibrium.
There is nothing arbitrary about any of this. It is the natural unfolding of one principle through many forms. And that inevitability, that inner necessity, is what gives the theory its aesthetic power.
When Einstein first wrote down his field equations, he believed he had discovered not only a law of physics but a law of beauty. The same can now be said of ToE. It does not simply add to the existing framework of science — it reorders it, placing entropy where geometry once stood, and geometry where effects once seemed primary.
11. The future horizon
The Theory of Entropicity (ToE) is still young. Its mathematical structure is being refined, its predictions explored, its implications tested. But already, it points toward a vast landscape of research — from the nature of black holes to the origin of time, from quantum entanglement to cosmological expansion.
It opens new questions in mathematics, new tools for computation, and new metaphors for philosophy. And like all great theories, it does more than explain; it invites participation. It offers to the next generation of physicists, mathematicians, and philosophers a new field of exploration — a chance to discover how the most abstract quantity in physics, entropy, is in fact the most real.
12. A return to beauty
The Theory of Entropicity (ToE) returns physics to the aesthetic ideal that guided its greatest discoveries: the conviction that truth and beauty are inseparable. It reminds us that the universe is not a machine but a melody — a pattern of flows, gradients, and balances that resonate with mathematical harmony.
Entropy, in this view, is the rhythm that moves everything, from galaxies to minds, from the birth of stars to the birth of ideas. And so, when one contemplates the Theory of Entropicity (ToE), one cannot help but echo Landau’s feeling about Einstein’s work: it is so beautiful that it must be true.
Not because we wish it so, but because its beauty lies in its inevitability — in the way it makes sense of everything that was once fragmented, and in the way it restores unity to the cosmos and to our understanding of it.
The Theory of Entropicity (ToE) is not the end of physics; it is the beginning of a new kind of simplicity — the simplicity that lies beyond complexity, where all the diverse patterns of the universe emerge from one inexhaustible source: the living field of entropy itself.
May posterity bear witness to it.
John Onimisi Obidi
References
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- Obidi, John Onimisi (2025). A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE). Link
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References
- Obidi, John Onimisi (2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and Unification of Physics. Link
- Physics: HandWiki Master Index of Source Papers on Theory of Entropicity (ToE) (2025, September 9). HandWiki. Retrieved 17:33, September 9, 2025. Link
- Obidi, John Onimisi. Conceptual and Mathematical Foundations of Theory of Entropicity (ToE). Encyclopedia. Available online (accessed 13 October 2025). Link
- Wissner-Gross, A. D., & Freer, C. E. (2013). Causal Entropic Forces. Physical Review Letters. Link
- Amari, S. (2016). Information Geometry and Its Applications. Springer. Link
- Jaynes, E. T. (1957). Information Theory and Statistical Mechanics. Physical Review. Link
- Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. Link
- Vidal, G. (2008). Class of Quantum Many-Body States That Can Be Efficiently Simulated. Physical Review Letters. Link
- Amari, S., & Nagaoka, H. (2000). Methods of Information Geometry. American Mathematical Society. Link
- Obidi, John Onimisi (2025). The Vuli-Ndlela Integral in the Theory of Entropicity (ToE). Link
- Obidi, John Onimisi (2025). The Obidi Action and the Foundation of the Entropy Field Equation. Link
- Obidi, John Onimisi (2025). The Master Entropic Equation (MEE). Link
- Obidi, John Onimisi (2025). Psych Entropy and the Entropy of the Mind. Link
- Bianconi, G. (2009). Entropy of network ensembles. Physical Review E. Link
- Bianconi, G., & Barabási, A.-L. (2001). Competition and multiscaling in evolving networks. Europhysics Letters. Link
- Bianconi, G. (2025). Gravity from entropy. Physical Review D, 111(6):066001. doi:10.1103/PhysRevD.111.066001. Link
- Obidi, John Onimisi (2025). The Theory of Entropicity (ToE): An Entropy-Driven Derivation of Mercury's Perihelion Precession Beyond Einstein's Curved Spacetime General Relativity (GR). Cambridge University. Link
- Obidi, John Onimisi (2025). 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. Link
- Obidi, John Onimisi (2025). 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. Link
- Obidi, John Onimisi (2025). A Critical Review of the Theory of Entropicity (ToE) on Original Contributions, Conceptual Innovations, and Pathways towards Enhanced Mathematical Rigor. Cambridge University. Link
- A Brief Critical Review of John Onimisi Obidi's Recent Paper: On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. (2025). Link
- The Theory of Entropicity (ToE) On the Geometry of Existence and the Curvature of Space-Time. (2025). Link
- On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. (2025). Link
- The Discovery of the Entropic α-Connection: How the Theory of Entropicity (ToE) Transformed Information Geometry into a Physical Law. (2025). Link
- An Introduction to the Theory of Entropicity (ToE): On the Evolution of its Conceptual and Mathematical Foundations (Part I). (2025). Link
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- Obidi, John Onimisi (2025). On the Conceptual and Mathematical Beauty of Obidi's Theory of Entropicity (ToE): From Geometric Relativity to Geometric Entropicity. Link
- Collected Works on the Theory of Entropicity (ToE). Medium Publications. (2025). Link
- Collected Works on the Theory of Entropicity (ToE). Substack Publications. (2025). Link
