Theory of Entropicity (ToE) by John Onimisi Obidi

The Theory of Entropicity (ToE) establishes entropy not as a statistical byproduct of disorder but as the fundamental field and causal substrate of physical reality. Central to this formulation is the Obidi Action, a variational principle. By integrating the Fisher–Rao and Fubini–Study metrics through the Amari–Čencov alpha-connection formalism, ToE provides a rigorous information-geometric foundation for entropy-driven dynamics. The Obidi Action comprises the Local and Spectral Obidi Actions.

Sunday, 23 November 2025

On the Historical Lineage of the Theory of Entropicity (ToE) - The Long Walk to a Field Theory of Entropy in Modern Theoretical Physics

The long walk to a Field Theory of Entropy by the Theory of Entropicity (ToE) is both mind-blowing and mind boggling — and it should feel that way. This is because ToE has just crossed the boundary between:

• classical thermodynamics,

• statistical mechanics,

• information theory,

• relativity,

• quantum theory,

• spectral geometry,

• and field theory …

to arrive at a single principle that binds them all. There are only a few moments in the history of physics where something like this happens:

• when Newton unified terrestrial and celestial motion,

• when Maxwell unified electricity, magnetism, and light,

• when Einstein unified inertia, gravity, and spacetime geometry,

• when Dirac unified quantum mechanics and special relativity,

• when Feynman unified electromagnetism and quantization through the path integral,

• when Bekenstein and Hawking connected gravity, thermodynamics, and quantum theory,

• when Connes connected geometry to operator theory.

The Theory of Entropicity (ToE) is on that level — and in some ways, beyond it. The following tells us why what we are seeing feels so powerful.

🔥 1. ToE discovered something everyone missed for 120 years

All these giants:

• Einstein

• Hawking

• Jacobson

• Verlinde

• Padmanabhan

• Bianconi

• Connes

• Caticha

• Penrose

• Witten …

touched entropy or information in one domain, but no one connected the dots that entropy is the actual physical field of reality. ToE saw it. ToE linked:

• entropy

• geometry

• time

• matter

• energy

• information

• computation

• causality …

through a single object: S(x). This is the leap no one before ToE made.

🔥 2. ToE unified local physics (Lagrangians) and global physics (spectral operators) This is something even Connes did not do. ToE created the Local Obidi Action (LOA) and the Spectral Obidi Action (SOA) …and showed they are dual descriptions of the same entropic field. This is almost too elegant to be accidental. It is the same type of unity that:

• Yang–Mills got from gauge symmetry

• String theory got from dualities

• Conformal field theory got from modular invariance

But ToE derived it from entropy alone.

🔥 3. ToE showed the speed of light is not geometric — it is entropic

This one insight overturns the entire conceptual foundation of relativity:

The speed of light is the maximum speed at which the entropic field can update reality. This is such a powerful idea that it practically rewrites physics:

• time dilation = entropic accounting

• length contraction = entropic reallocation

• mass increase = frozen entropic capacity

• light cones = entropic cones

• Minkowski spacetime = entropic equilibrium metric

• GR curvature = Hessian of S(x)

No one else has said this before. Not even remotely.

🔥 4. ToE introduces the first-ever dynamical entropy field with propagation, inertia, and causality Entropy was always:

• descriptive

• statistical

• informational

• epistemic

Never ontological.

Never dynamical.

Never propagating.

ToE gave entropy:

• a field equation,

• a potential,

• a source term,

• a propagation limit,

• a spectral curvature,

• and a variational origin.

This is unprecedented.

🔥 5. ToE solved major open problems that the community has failed to solve Bianconi explicitly ended her paper with:

• Challenge 1: gravity needs a canonical quantization

• Challenge 2: the G-field interpretation is incomplete

ToE solved both. This alone puts ToE on the level of note-worthy breakthroughs.

🔥 6. ToE unified bosons and fermions Physicists have tried for decades to unify:

• Einstein–Hilbert action

• Yang–Mills theories

• Dirac/fermionic actions

ToE derives all of them from the same Obidi Action. No other theory has ever done this using entropy. Even Connes’ spectral action did not unify entropy with physics. ToE did.

🔥 7. ToE introduced an idea no one else dared to suggest

That the laws of physics themselves evolve because the entropic field evolves. This is a radical philosophical shift. It is the next step after:

• Aristotle’s teleology

• Newton’s determinism

• Einstein’s covariance

• Wheeler’s “it from bit”

• Rovelli’s relativized information

• Bekenstein’s informational bounds

But ToE didn’t stop at philosophy — ToE built the [integral] mathematics.

🌟 Closing Words

People's reaction — “this is mind-blowing and mind boggling” — is exactly what Einstein, Dirac, and Feynman felt when they first uncovered their deepest insights. ToE has produced a new unifying architecture of reality.

Thus, the Theory of Entropicity (ToE) is not just a paper or a model. It is a new language for physics. And ToE is its architect.

Sources — help

References

Obidi, John Onimisi. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
John Onimisi Obidi. (6th November, 2025). Comparative analysis between john onimisi obidi’s theory of entropicity (toe) and waldemar marek feldt’s feldt–higgs universal bridge (f–hub) theory. International Journal of Current Science Research and Review, 8(11), pp. 5642–5657, 19th November 2025. URL:https: //doi.org/10.47191/ijcsrr/V8-i11–21.
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. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).
Obidi, John Onimisi. (4 November, 2025). The Theory of Entropicity (ToE) Derives Einstein’s Relativistic Speed of Light © as a Function of the Entropic Field: ToE Applies Logical Entropic Concepts and Principles to Derive Einstein’s Second Postulate. Cambridge University. https://doi.org/10.33774/coe-2025-f5qw8-v2
Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
HandWiki contributors, “Physics:Theory of Entropicity (ToE) Derives Einstein’s Special Relativity,” HandWiki, https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity_(ToE)_Derives_Einstein’s_Special_Relativity&oldid=3845936

Further Resources on the Theory of Entropicity (ToE):

Beyond Comparison: Why the Theory of Entropicity (ToE) Reconstructs Reality from Entropy

Introduction:

Two Roads into Entropy

Entropy has always been a measure—of disorder, of uncertainty, of information. But in recent years, physicists have begun to ask whether entropy might be more than a tool. Could it be the foundation of reality itself?

Two striking approaches have emerged. Ginestra Bianconi’s Gravity from Entropy (2025) reframes gravity as arising from quantum relative entropy between spacetime and matter metrics. John Onimisi Obidi’s Theory of Entropicity (ToE) goes further, proposing that entropy is not a comparison but the field of reality itself.
Both are bold. Both are entropic. But they are not the same.

Bianconi’s Entropic Gravity:

Geometry as Information

In Bianconi’s framework, gravity is derived from the quantum relative entropy between two metrics:

\\mathcal{L} = \\mathbf{Tr}

\tilde{G}\, the metric of spacetime
\tilde{g}\, the metric induced by matter fields

The Bianconi action is then written as:

$\mathcal{L} = \mathbf{Tr} \, \tilde{g} \ln \tilde{G}^{-1} = - \mathbf{Tr}_F \ln \tilde{G} \tilde{g}^{-1}$

This formulation treats gravity as the entropic mismatch between geometry and matter. By introducing a G-field as a Lagrange multiplier, Bianconi derives modified Einstein equations with a small positive cosmological constant.

Her approach is elegant:

Gravity is not imposed but emerges from information geometry. Entropy becomes the language of spacetime curvature.

ToE’s Entropic Field:

Reality as Entropy

John Onimisi Obidi’s Theory of Entropicity (ToE) takes a more radical stance. It argues that entropy is not a measure of reality—it is reality.

In ToE:

The Spectral Obidi Action (SOA) reframes Araki relative entropy as a dynamical action principle, not a static measure.

$\mathcal{S}_{\mathrm{SOA}} = - \mathrm{Tr} \ln \Delta$

The No-Rush Theorem enforces finite-rate bounds on entropy redistribution, making causality and relativistic effects physical consequences of entropy.
Length contraction and time dilation are not kinematic artifacts but field-driven realities, enforced by entropy’s constraints.
Observers are secondary: what they see has already been computed by the entropic field. Relativity itself is emergent.

ToE integrates Shannon, von Neumann, Rényi, Tsallis, KL, and Araki entropies into a unified spectral-geometric framework, coupling them with Fisher–Rao and Fubini–Study metrics via Amari–Čencov connections. It is not just a theory of gravity—it is a theory of everything entropic.

The Crucial Difference

The difference between Bianconi and ToE is subtle but decisive:

Bianconi uses entropy to compare metrics. Gravity emerges from mismatch.
ToE uses entropy to generate metrics, motion, and observation. Reality emerges from entropy itself.

In Bianconi’s world, entropy is a tool. In ToE’s world, entropy is the field.

Why ToE Is Original and Useful

ToE’s originality lies in its ontological inversion. It dethrones the observer, making relativity emergent from entropy rather than fundamental. It reframes spacetime, motion, and causality as consequences of entropy’s finite-rate dynamics.

Its usefulness lies in its unification power:

A field-theoretic origin of relativity, quantum coherence, and spacetime geometry
A natural explanation for dark matter and dark energy via entropic vacuum pressure
A new lens on time’s arrow, causality, and the emergence of physical law
Where Bianconi reformulates gravity, ToE reconstructs physics itself.

Conclusion:

Entropy as the Fabric of Reality

Both Bianconi and Obidi are pioneers in the entropic turn of physics. Bianconi shows that gravity can be understood through entropy. Obidi shows that reality itself can be understood as entropy.

The difference is not just technical—it is philosophical. In ToE, entropy is not a measure of reality. It is the fabric of reality.

If ToE succeeds, it will mark a turning point: relativity, quantum mechanics, and spacetime itself will be seen not as fundamental, but as emergent from the entropic field.
And that is why ToE is more than a comparison. It is a reconstruction.

Sources — help

References

Obidi, John Onimisi. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
John Onimisi Obidi. (6th November, 2025). Comparative analysis between john onimisi obidi’s theory of entropicity (toe) and waldemar marek feldt’s feldt–higgs universal bridge (f–hub) theory. International Journal of Current Science Research and Review, 8(11), pp. 5642–5657, 19th November 2025. URL:https: //doi.org/10.47191/ijcsrr/V8-i11–21.
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. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).
Obidi, John Onimisi. (4 November, 2025). The Theory of Entropicity (ToE) Derives Einstein’s Relativistic Speed of Light © as a Function of the Entropic Field: ToE Applies Logical Entropic Concepts and Principles to Derive Einstein’s Second Postulate. Cambridge University. https://doi.org/10.33774/coe-2025-f5qw8-v2
Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
HandWiki contributors, “Physics:Theory of Entropicity (ToE) Derives Einstein’s Special Relativity,” HandWiki, https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity_(ToE)_Derives_Einstein’s_Special_Relativity&oldid=3845936

Further Resources on the Theory of Entropicity (ToE):

The Mathematical Ingenuity, Sophistication and Complexity of the Theory of Entropicity (ToE)

The mathematical complexity of the Theory of Entropicity (ToE) stems from its ambition to unify physics by framing entropy as a fundamental field, which requires a sophisticated and unconventional mathematical framework that combines information geometry and non-equilibrium thermodynamics. Its complexity is characterized by the use of concepts like α-connections, non-extensive entropy formalisms (Rényi and Tsallis), and an iterative, self-correcting computational logic instead of classical differential geometry. This requires developing the Obidi Action and the Master Entropic Equation (MEE) to describe how entropy deforms and creates spacetime and gravity.

Key areas of mathematical complexity

Unifying diverse frameworks: ToE aims to provide a single framework that integrates the distinct mathematical languages of thermodynamics, general relativity, and quantum mechanics by treating entropy as the central, unifying element.

Information geometry and gravity: It builds upon concepts from information geometry, such as the Fisher-Rao metric, and transforms them into physical metrics of spacetime curvature. This involves using non-extensive formalisms (Rényi-Tsallis α-q) to mathematically link information flow to geometric structures.

Entropy as an ontological field:

Rather than a statistical byproduct, entropy is the fundamental field 𝑆(𝑥,𝑡) whose dynamics are governed by the Obidi Action and described by the Master Entropic Equation (MEE), which is the entropic analogue of Einstein's field equations.

Computational and iterative logic:

ToE’s mathematical model is described as more akin to computation and artificial intelligence than classical calculus because the field doesn't just evolve within spacetime; it evolves the geometry itself through an iterative process.

Formalization and new equations:

The theory requires the development of new equations, such as the Obidi Action and the Master Entropic Equation, and further formalization of concepts like the full quantization of the entropy field.

Ongoing development:

While a foundational structure exists, ToE is still in a state of active development, with ongoing efforts to provide more explicit and detailed mathematical constructions, especially concerning the full quantization of the entropy field and its coupling to standard model fields, as noted in critical reviews on various online publication repositories, such as OSF, Authorea, ResearchGate, SSRN, International Journal of Current Science Research and Review (IJCSRR), Cambridge University Open Engage (CoE).

Sources — help

References

Obidi, John Onimisi. (12th November, 2025). On the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: A Rigorous Derivation of Bianconi’s Results from the Entropic Obidi Actions of the Theory of Entropicity (ToE). Cambridge University. https//doi.org/10.33774/coe-2025-g7ztq
John Onimisi Obidi. (6th November, 2025). Comparative analysis between john onimisi obidi’s theory of entropicity (toe) and waldemar marek feldt’s feldt–higgs universal bridge (f–hub) theory. International Journal of Current Science Research and Review, 8(11), pp. 5642–5657, 19th November 2025. URL:https: //doi.org/10.47191/ijcsrr/V8-i11–21.
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. Cambridge University. Published October 17, 2025. https://doi.org/10.33774/coe-2025-1dsrv
Obidi, John Onimisi (17th October 2025). On the Conceptual and Mathematical Foundations of the Theory of Entropicity (ToE): An Alternative Path toward Quantum Gravity and the Unification of Physics. Figshare. https://doi.org/10.6084/m9.figshare.30337396.v2
Obidi, John Onimisi. 2025. A Simple Explanation of the Unifying Mathematical Architecture of the Theory of Entropicity (ToE): Crucial Elements of ToE as a Field Theory. Cambridge University. Published October 20, 2025. https://doi.org/10.33774/coe-2025-bpvf3
Obidi, John Onimisi (15 November 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30627200.v1
Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).
Obidi, John Onimisi. (4 November, 2025). The Theory of Entropicity (ToE) Derives Einstein’s Relativistic Speed of Light © as a Function of the Entropic Field: ToE Applies Logical Entropic Concepts and Principles to Derive Einstein’s Second Postulate. Cambridge University. https://doi.org/10.33774/coe-2025-f5qw8-v2
Obidi, John Onimisi. (28 October, 2025). The Theory of Entropicity (ToE) Derives and Explains Mass Increase, Time Dilation and Length Contraction in Einstein’s Theory of Relativity (ToR): ToE Applies Logical Entropic Concepts and Principles to Verify Einstein’s Relativity. Cambridge University. https://doi.org/10.33774/coe-2025-6wrkm
HandWiki contributors, “Physics:Theory of Entropicity (ToE) Derives Einstein’s Special Relativity,” HandWiki, https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity_(ToE)_Derives_Einstein’s_Special_Relativity&oldid=3845936

Further Resources on the Theory of Entropicity (ToE):

Theory of Entropicity (ToE) by John Onimisi Obidi

Wikipedia

Sunday, 23 November 2025

On the Historical Lineage of the Theory of Entropicity (ToE) - The Long Walk to a Field Theory of Entropy in Modern Theoretical Physics

On the Historical Lineage of the Theory of Entropicity (ToE) - The Long Walk to a Field Theory of Entropy in Modern Theoretical Physics

ToE gave entropy:

References

Beyond Comparison: Why the Theory of Entropicity (ToE) Reconstructs Reality from Entropy

Beyond Comparison: Why the Theory of Entropicity (ToE) Reconstructs Reality from Entropy

Bianconi’s Entropic Gravity:

Geometry as Information

ToE’s Entropic Field:

Reality as Entropy

References

The Mathematical Ingenuity, Sophistication and Complexity of the Theory of Entropicity (ToE)

The Mathematical Ingenuity, Sophistication and Complexity of the Theory of Entropicity (ToE)

References

Author’s Preface and Methodological Statement for the Theory of Entropicity (ToE): An Unapologetic Introduction in Defense of Obidi's New Theory of Reality—On the Trajectory of Discovery and the Road Less Traveled

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