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.

Thursday, 20 November 2025

Entropy as a Field: Can the Spectral Obidi Action Go Beyond Araki Relative Entropy? A Most Radical Conceptualization of a Unified Field in the Theory of Entropicity (ToE)

Introduction: The Measure That Became a Candidate for a Field

Entropy has always been one of the most enigmatic concepts in science. From the early days of thermodynamics, where it was introduced as a measure of disorder, to the information age, where Shannon reframed it as a measure of uncertainty, entropy has been treated as a tool rather than a thing. It is something we calculate, something we use to compare states, something we invoke to explain irreversibility. But it has never been considered a field in the same way that electromagnetism or gravity are fields.

The Theory of Entropicity (ToE) challenges this long-standing assumption. It proposes that entropy is not merely a statistical measure but the fundamental continuum of reality itself. At the heart of this proposal lies the Spectral Obidi Action (SOA), an action principle that looks strikingly similar to Araki relative entropy but is claimed to serve a radically different purpose.

This raises a provocative question: is ToE simply repeating Araki’s work under a new name, or does the SOA genuinely open a new path by treating entropy as a field?

Araki Relative Entropy: The Established Framework

To appreciate the novelty of SOA, we need to understand what Araki relative entropy already does. In operator algebra and quantum field theory, Araki relative entropy is defined as:

This expression compares two quantum states, ρ\rho and σ\sigma, through the modular operator:

Its meaning is precise: it quantifies how distinguishable one state is from another. It is a measure of relative information, deeply tied to the structure of von Neumann algebras and modular theory.

Araki entropy has been deployed extensively. It appears in studies of entanglement entropy, in modular Hamiltonians, and in the algebraic formulation of quantum field theory. It is mathematically rigorous, physically interpretable, and widely accepted. But it is always used as a measure. It does not evolve. It does not generate equations of motion. It does not act as a field.

The Spectral Obidi Action: A Radical Reinterpretation

The Spectral Obidi Action (SOA) proposed in the Theory of Entropicity (ToE) is written as:

**Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE)**

At first glance, this looks like Araki entropy stripped of its dependence on specific states. But ToE interprets it differently. Instead of being a measure of distinguishability, SOA is framed as an action principle — something to be varied, something that generates dynamics.

This is a profound shift in theoretical physics. In physics, an action principle is not just a mathematical curiosity. It is the foundation of dynamics. The Einstein–Hilbert action generates Einstein’s equations. The Yang–Mills action generates the equations of gauge fields. By proposing SOA as an action, ToE is suggesting that entropy itself can be treated as a field variable, with its own equations of motion.

In this framework, entropy is no longer emergent. It is fundamental. And SOA is not isolated — it is coupled with other terms: geometric actions, generalized entropies (Shannon, von Neumann, Rényi, Tsallis, KL, Araki), spectral operator geometry, and causal constraints. Together, these form the Generalized Obidi Action, from which the Master Entropic Equation (MEE) emerges.

Why No One Else Has Tried This

The absence of prior work in this direction is not accidental. Researchers have avoided treating entropy as a field for several reasons.

First, entropy has always been understood as emergent. It arises from coarse-graining, from statistical descriptions, from the loss of information about microstates. To elevate it to a fundamental field risks stripping it of its meaning.

Second, entropy in its traditional forms does not generate dynamics. Araki relative entropy, for example, is relational. It compares states but does not evolve them. It is not designed to produce equations of motion.

Third, the physics community is cautious. Without clear predictions or experimental consequences, entropy-as-field risks being mathematically elegant but physically empty. Researchers prefer frameworks that yield testable results, and entropy has always been seen as a derived quantity rather than a fundamental one.

Is SOA Valid or Useful?

This brings us to the crux of the matter: is the SOA action principle valid, and is it useful?

Validity here means more than mathematical consistency. It requires that varying SOA yields well-defined field equations. It requires that those equations integrate coherently with the rest of physics. And it requires that the framework does not collapse into redundancy with existing measures like Araki entropy.

Usefulness, meanwhile, demands predictive novelty. If SOA can lead to testable predictions — finite-rate entanglement formation, corrections to general relativity, causal bounds — then it offers something new. If it can integrate entropy measures into a unified field theory that explains time’s arrow, spacetime curvature, and quantum coherence, then it is more than a restatement.

But if it cannot, then it risks being a formal repackaging of known entropy measures, elegant but empty.

The Stakes for ToE

The stakes are high. If entropy can be treated as a field, physics could be recast in entropic terms. Spacetime curvature would be understood as entropic geometry. Quantum coherence would emerge from spectral entropy dynamics. Causality would be enforced by finite-rate entropy redistribution. And time’s arrow would be explained as entropic asymmetry.

This would be a profound unification, bringing together thermodynamics, relativity, and quantum mechanics under a single entropic continuum. But it requires ToE to demonstrate that SOA is not just Araki entropy in disguise, but a genuine action principle with predictive power.

Conclusion: A Bold but Risky Leap

The Spectral Obidi Action is bold. It risks redundancy, but it also opens the door to a new way of thinking. No other researchers have tried to recast Araki relative entropy into a field-theoretic action because, in its traditional form, it does not yield physical meaning or dynamics. ToE is unusual in attempting it.

Whether SOA is valid depends on whether ToE can show that entropy-as-field yields new dynamics and testable predictions. If it can, this could mark a turning point in physics. If not, it will remain an elegant but empty reformulation.

Either way, the question is captivating: Can entropy itself be the field that unifies physics?

References

Obidi, J. O. (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).

Further Resources on the Theory of Entropicity (ToE):

Litmus Test of the Theory of Entropicity (ToE): If Einstein’s Relativity is Emergent from Entropy, then the Observer is Dethroned, and Physics Gains a New Foundation in the Theory of Entropicity (ToE)

Einstein’s relativity says length contraction is only a kinematic effect, but ToE says it is more than kinematic and that it is physical due to entropic field constraints

In Einstein’s Relativity

In special relativity, length contraction is treated as a purely kinematic effect. It arises because observers in relative motion disagree about simultaneity. When you measure the length of a moving rod, you must define the endpoints at the same time in your frame. Due to relativity of simultaneity, those “same‑time” slices differ between frames, and the rod appears contracted.

Importantly, in Einstein’s view, nothing physically happens to the rod itself. In its own rest frame, it is unchanged. The contraction is a matter of perspective, not a physical compression.

In the Theory of Entropicity (ToE)

ToE reinterprets this phenomenon. It argues that length contraction is not merely perspectival but physically real, because it is constrained by the entropic field.

Here’s the reasoning:

Entropy is treated as a fundamental field in ToE, not just a measure.
Motion through spacetime involves redistribution of entropy.
The No‑Rush Theorem in ToE enforces finite‑rate bounds on entropic redistribution, analogous to the constancy of light speed.
As a result, when an object moves, its geometry is not just “seen differently” but is physically constrained by entropic gradients.

Thus, contraction is interpreted as a field‑driven adjustment of the object’s spatial extension, enforced by entropic dynamics. In this view, the rod’s length genuinely changes because the entropic field does not permit unconstrained extension at high velocities.

Why This Matters

Einstein’s relativity: contraction is a coordinate effect, a matter of how different observers slice spacetime.
ToE: contraction is a physical effect, rooted in entropic field constraints. It is not just kinematic but dynamical.

This difference is profound. If ToE is correct, then relativistic effects like time dilation and length contraction are not merely observational artifacts but physical consequences of entropy as a field. That would mean relativity itself is emergent from entropic dynamics.

The Challenge

The burden of proof lies with ToE. To establish that length contraction is physically real, ToE must:

Show mathematically how entropic field equations enforce contraction.
Demonstrate predictions that differ from Einstein’s purely kinematic interpretation.
Provide experimental or observational evidence that contraction has physical consequences beyond relativity’s perspective effects.

Without this, the mainstream view — that contraction is kinematic — remains dominant.

💡So, the distinction is this: Einstein says contraction is a matter of how you look; ToE says contraction is a matter of how reality itself is constrained by entropy.

To compare Einstein’s kinematic interpretation with ToE’s entropic reinterpretation, and show how each frames the meaning of contraction, that would give the Theory of Entropicity (ToE) a strong piece to share with both technical and general audiences.

This is critical, because ToE puts the observer as fundamentally dependent on entropy, and that what we see or measure or observe is also equally fundamentally constrained by the entropic field itself, so that observer input becomes naturally secondary; hence relativity is emergent from the entropic field. This kinematic exposition as being actually deeply a physical effect is a crucial test of the Theory of Entropicity (ToE).

Thus, we have identified the crucial pivot point of the Theory of Entropicity (ToE): it reframes relativity not as a framework built on observer kinematics, but as something emergent from the entropic field itself. Let us distill why this point and observation matters, and why it becomes a decisive test for ToE.

Einstein’s Relativity: Observer-Centric

In Einstein’s special relativity, the observer is central. Length contraction, time dilation, and simultaneity shifts are all explained as consequences of how different observers slice spacetime. The rod doesn’t physically shrink; rather, its measured length depends on the observer’s frame. The observer’s input — choice of frame, simultaneity convention — is fundamental.

This makes relativity a kinematic theory: it describes how measurements transform between observers, but it doesn’t claim that the underlying physical object is altered.

ToE’s Reinterpretation: Entropy-Centric

ToE turns this logic inside out. It argues that the observer is secondary, because all observation is constrained by the entropic field. What we see, measure, or record is not simply a matter of perspective — it is fundamentally shaped by entropy’s dynamics.

In this view:

Length contraction is not just a perspectival artifact. It is a physical effect, enforced by entropic field constraints.
The No-Rush Theorem ensures that entropy redistribution cannot occur instantaneously, embedding finite-rate bounds into reality.
The observer’s frame is not the cause of contraction; it is merely a reflection of deeper entropic dynamics.

Thus, relativity itself is emergent from the entropic field. The observer’s role is demoted: they do not impose relativity, they inherit it from entropy.

Why This Is a Crucial Test

If ToE is correct, then relativistic phenomena — length contraction, time dilation, causality — must be demonstrable as physical consequences of entropy dynamics, not just coordinate effects. This is a bold claim, and it sets up a clear test:

Mathematical demonstration: ToE must show that varying the entropic action (SOA + couplings) yields contraction and dilation as solutions of the field equations.
Physical interpretation: These effects must be explained as genuine changes in geometry due to entropy constraints, not just observer-dependent slicing.
Experimental distinction: ToE must predict subtle differences between “observer-only” relativity and “entropy-driven” relativity — differences that could, in principle, be tested.

If ToE can pass this test, it elevates entropy from a statistical measure to the fundamental field of reality. If it cannot, then relativity remains kinematic, and entropy-as-field risks being a philosophical overlay.

The Stakes

This is why the above point is so critical: ToE’s claim that relativity is emergent from entropy is not a minor reinterpretation — it is the litmus test of whether ToE is a genuine physical theory or a formal restatement.

If relativity is emergent from entropy, then the observer is dethroned, and physics gains a new foundation in the Theory of Entropicity (ToE).
If relativity remains purely kinematic, then ToE’s entropic field risks redundancy.

💡In summary: The observer’s dependence on entropy, and the claim that relativity emerges from entropic constraints, is the decisive test of ToE. It is here that ToE either proves itself as a new physical theory or collapses into repetition of Einstein’s framework.

Hence, “The Observer Dethroned: Why Relativity Emerges from Entropy in ToE” speaks to this challenge. We are thus at the center stage of our work on the Theory of Entropicity (ToE), where we must present this pivotal idea in a way that captures both technical depth and philosophical drama.

Further Expository Insights

We can see a crucial point in this: The No-Rush Theorem ensures that entropy redistribution cannot occur instantaneously, thus embedding finite-rate bounds into reality. If that be so, is it not true and logically sound then that what the observer observes is also constrained by the entropic field, including his/her coordinates; so that it is the entropic field that guarantees that observation. So, if the No-Rush Theorem posits the above, then mere observation or mere geometric transformations [alone] do not effect a change independent of the entropic field, hence what the observer sees or measures must have been so computed before the observer does, and hence [Einstein’s relativistic] kinematic effects are not a priori. This has great philosophical and physical implications also.

We shall hereunder provide further inputs pertaining the above, to give the reader more ground for understanding and rationality for the radical claims of the Theory of Entropicity (ToE).

The No‑Rush Theorem as a Constraint on Reality

The No‑Rush Theorem in ToE states that entropy redistribution cannot occur instantaneously. This is not just a technical condition — it is a fundamental bound on how reality evolves. Just as the speed of light in relativity sets a maximum rate for causal influence, the No‑Rush Theorem sets a maximum rate for entropic change, and hence a limit on how reality can evolve and compute.

This means that every physical process, every redistribution of information, every adjustment of geometry is constrained by entropy’s finite‑rate dynamics. Nothing “jumps” outside of entropy’s bounds.

Observation as Entropy‑Dependent

If entropy governs redistribution at finite rates, then observation itself is constrained by the entropic field. An observer’s coordinates, measurements, and perceptions are not free-floating — they are guaranteed by entropy’s structure.

In other words:

What the observer sees is not an independent act of perception.
It is the entropic field that computes reality first, and the observer inherits that computation.
The observer’s coordinates are secondary, because they are already embedded in the entropic continuum.

This reverses the usual logic of relativity. In Einstein’s framework, kinematic effects arise from the observer’s frame. In ToE, kinematic effects are not a priori — they are consequences of entropy’s finite‑rate constraints, which the observer merely reflects.

Philosophical Implications

This has profound consequences:

Observer dethroned: The observer is no longer the primary agent of relativity. They are a derivative phenomenon, constrained by entropy.
Relativity emergent: Relativistic effects like length contraction and time dilation are not just perspectival — they are physical consequences of entropic dynamics.
Reality pre‑computed: What the observer measures has already been determined by the entropic field before observation occurs. Measurement is not creative; it is receptive.

This shifts the philosophy of physics from an observer‑centric model to an entropy‑centric one. It suggests that reality is not shaped by how we look at it, but by how entropy itself evolves.

Hence, what we take to be reality, and how we see reality, changes forever due to the Principles of the Theory of Entropicity (ToE).

Physical Implications

If this is true, then ToE makes testable claims:

Relativistic effects should be derivable directly from entropic field equations, not just from Lorentz transformations.
There may be subtle differences between “observer‑only” relativity and “entropy‑driven” relativity — differences that could, in principle, be measured.
The entropic field becomes the guarantor of causality, geometry, and observation itself.

Once again, that is why this is a crucial test of ToE — If ToE can demonstrate that relativity is emergent from entropy, then it has succeeded in re‑founding physics on entropic grounds. If not, then entropy remains a measure, and relativity remains kinematic.

⚙ Closure Highlight: Thus, the No‑Rush Theorem implies that observation is fundamentally constrained by entropy. What the observer sees is already computed by the entropic field, making kinematic effects secondary rather than primary. This is both a philosophical revolution — dethroning the observer — and a physical test that will determine whether ToE is genuinely novel or merely a reinterpretation.

References

Obidi, J. O. (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).

Further Resources on the Theory of Entropicity (ToE):

Wednesday, 19 November 2025

On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE) - The Obidi Actions and the Obidi Field Equations

On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE)

The Generalized Obidi Action of the Theory of Entropicity (ToE)

The Generalized Obidi Action of the Theory of Entropicity (ToE)

A Representative Equation for the Entropy Field Equation of the Theory of Entropicity (ToE) — **A Representative Equation for the Entropy Field Equations of the Theory of Entropicity (ToE)**

On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE)

The Mathematical Landscape of ToE

The Theory of Entropicity (ToE) is built on the radical idea that entropy is the fundamental field of reality. To capture this, ToE introduces a suite of unprecedented constructs — the Obidi Action, the Master Entropic Equation (MEE), the Vuli–Ndlela Integral, the Entropy Potential Equation, and the No-Rush Theorem — all designed to unify thermodynamics, relativity, and quantum mechanics within a single entropic continuum. These tools recast the familiar laws of physics as consequences of entropy field dynamics.

The Obidi Field Equation (OFE)

The Obidi Field Equation (generally called the Master Entropic Equation — MEE) is a key component of the Theory of Entropicity (ToE), which redefines entropy as the fundamental field of reality. It is derived from the Obidi Action and is used to govern the evolution of the entropy field in spacetime. The equation is nonlinear and nonlocal, reflecting the probabilistic nature of entropy. It is solved iteratively and is used to describe the dynamics of the entropic field, which is believed to be responsible for all physical phenomena.

Information Geometries

At the foundation of ToE lies a manifold of states equipped with multiple information geometries. The Fisher–Rao geometry measures statistical curvature, the Fubini–Study geometry encodes quantum coherence, and the α-geometry introduces asymmetry and irreversibility into information transport. Together, these geometries provide the scaffolding on which entropy flows, ensuring that both classical and quantum domains are represented within a single entropic continuum.

Entropy Sector

ToE incorporates a wide family of entropy measures, each adapted to different physical regimes. The Shannon entropy is the classical measure of uncertainty in probability distributions, forming the bedrock of information theory. The von Neumann entropy extends this to quantum states, capturing the informational content of density matrices. Alongside these, ToE includes the Rényi entropy for scale sensitivity, the Tsallis entropy for nonextensive systems, the Kullback–Leibler divergence for directional information change, and the Araki relative entropy for quantum comparisons. By weaving all of these into its framework, ToE ensures that entropy is not treated as a single formula but as a universal family of measures that adapt to classical, quantum, and statistical contexts.

Local Obidi Action

The Local Obidi Action is the geometric sector of ToE. It integrates curvature, asymmetric transport, and entropy gradients into a single variational principle. This action describes how entropy interacts with the underlying geometry of the manifold, ensuring that entropic flow is inseparable from the curvature and structure of space itself. It is here that ToE begins to unify thermodynamics and relativity, showing that spacetime curvature can be understood as a manifestation of entropy dynamics.

Spectral Obidi Action and Operator Geometry

Complementing the local action is the Spectral Obidi Action, which introduces a Dirac-type entropy operator. The spectrum of this operator regulates coherence, scale, and regularity. This spectral perspective allows ToE to encode quantum features directly into its mathematics. The spectral operator geometry emerges from the asymptotics of this operator, linking spectral data to geometric structure. In this way, coherence and irreversibility are not external assumptions but built into the entropic field itself.

Coupling Terms

ToE includes explicit coupling terms that link geometry, entropy, and spectral structure. These couplings ensure that no sector evolves in isolation: geometry modulates entropy flow, entropy interacts with spectral coherence, and spectral properties feed back into geometric curvature. This interdependence is what makes ToE a unified framework rather than a patchwork of separate theories.

Master Entropic Equation (MEE)

From the unified Obidi Action arises the Master Entropic Equation (MEE). This is the governing equation of the entropy field, balancing geometric diffusion, entropy production, spectral coherence, and causal correction. It is highly nonlinear and nonlocal, reflecting the complexity of reality itself. The MEE is the mathematical heart of ToE, the place where all sectors converge into a single dynamical law.

Vuli–Ndlela Integral

A distinctive innovation of ToE is the Vuli–Ndlela Integral, which reformulates quantum path integrals to include irreversibility. Unlike traditional formulations that treat time symmetrically, the Vuli–Ndlela Integral weights paths by entropic cost, embedding the arrow of time directly into quantum mechanics. This construct explains temporal asymmetry not as an emergent phenomenon but as a fundamental feature of entropic dynamics.

Entropy Potential Equation

The Entropy Potential Equation defines the effective energy landscape of the entropy field. It describes how entropy gradients shape the evolution of the field, guiding flow and interaction. This equation provides the structure within which iterative solutions of the MEE are carried out, ensuring that entropic evolution follows a coherent trajectory.

No-Rush Theorem

The No-Rush Theorem imposes a universal temporal bound on interactions. It formalizes the principle that entropy cannot redistribute instantaneously but is constrained by a finite rate. This bound corresponds to the constancy of light, making Einstein’s second postulate a consequence of entropic dynamics. In ToE, causality is not imposed externally but arises naturally from the finite-rate redistribution of entropy.

Unified Vision

Taken together, these constructs form a tightly interwoven system. The Local Obidi Action governs geometric aspects, the Spectral Obidi Action encodes coherence, the entropy sector provides multiple measures of information including Shannon and von Neumann, the coupling terms bind everything together, the MEE governs dynamics, the Vuli–Ndlela Integral introduces irreversibility, the Entropy Potential Equation defines structure, and the No-Rush Theorem enforces causality.

Interpretive Summary

The mathematics of ToE is complex because its ambition is vast: to unify thermodynamics, relativity, and quantum theory within a single entropy-driven continuum. By elevating entropy to the status of a universal field, ToE provides a rigorous architecture in which time’s arrow, the constancy of light, quantum coherence, and spacetime curvature are all explained as consequences of entropic dynamics. In this vision, entropy shapes geometry, governs motion, and creates spacetime.

References

Obidi, J. O. (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).

Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE — https://theoryofentropicity.blogspot.com

2. LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true

3. Notion: Theory of Entropicity (ToE)

4. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack

5. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium

6. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

7. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

8. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).

9. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

10. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

11. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

12. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia

13. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate

14. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare

15. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea

16. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN

17. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).

18. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar

19. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)

Saturday, 15 November 2025

The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory

**The Obidi Action of the Theory of Entropicity (ToE)**

This paper presents a systematic comparison between the recently developed pseudo entropy framework of Takayanagi, Kusuki, and Tamaoka and the Theory of Entropicity (ToE). While pseudo–entropy has revealed a remarkable boundary route to the linearized Einstein equation in dS3, the Theory of Entropicity proposes a far more fundamental idea: that entropy is not a boundary diagnostic of geometry, but the underlying field from which geometry, matter, motion, and time themselves emerge. The discussion that follows demonstrates how the pseudo–entropy program fits naturally within the broader structure of ToE, and how the ToE framework generalizes, extends, and ultimately surpasses it. The pseudo–entropy construction shows that a non–Hermitian generalization of entanglement entropy in a two–dimensional CFT satisfies a first law whose bulk dual reproduces the perturbative Einstein equation in dS3. Moreover, infinitesimal variations of pseudo–entropy obey a Klein–Gordon equation on a kinematic dS2 space, suggesting the emergence of time from Euclidean CFT data. In this paper, we reinterpret these results within the Theory of Entropicity by showing that the same Klein–Gordon structure appears as the boundary–projected, linearized limit of the Master Entropic Equation derived from the Local Obidi Action. Thus, what pseudo–entropy identifies kinematically from the boundary, ToE generates dynamically in the bulk through the entropic field S(x). The manuscript further embeds pseudo–entropy into a broader landscape of entropic approaches — Jacobson’s thermodynamic derivation of Einstein equations, Padmanabhan’s emergent spacetime, Ver linde’s entropic gravity, Caticha’s entropic inference, and Bianconi’s metric relative entropy. Where these earlier programs emphasize information, thermodynamics, or emergence, ToE provides a uni fying ontological principle: entropy itself is the fundamental field of the universe. By promoting the modular–like operator ∆ to a dynamical object through the Spectral Obidi Action, ToE offers a natural explanation of dark matter, dark energy, and vacuum entropic pressure — domains entirely absent from the pseudo–entropy framework. This paper shows explicitly how Bianconi’s relative–entropy action and the Takayanagi–Kusuki–Tamaoka pseudo–entropy construction both appear as limiting cases of the Obidi Actions. Finally, we demonstrate that ToE provides a unified entropic–spectral variational principle in which bosons and fermions arise from the same foundational structure. The spectral interpretation of bosonic actions, the Dirac–based fermionic bilinears, and geometric actions such as Einstein–Hilbert and Yang–Mills all emerge as projections of the Local and Spectral Obidi Actions. This paper therefore positions pseudo–entropy not as an alternative to ToE, but as a special holographic shadow of a deeper entropic field theory. In this sense, the present work does not merely compare two independent approaches. Rather, it establishes a hierarchical synthesis: pseudo–entropy reconstructs gravity from boundary information, while the Theory of Entropicity constructs gravity, geometry, quantum structure, and temporal dynam ics from an underlying entropic field. This manuscript argues that pseudo–entropy is best understood not as a standalone gravitational principle, but as a boundary manifestation of the universal entropic dynamics formulated by the Theory of Entropicity (ToE).

Abstract

The recent work of Takayanagi, Kusuki, and Tamaoka has introduced the concept of holographic pseudo-entropy in non-unitary CFT2 and demonstrated a striking equivalence: the first law of pseudo entropy is precisely dual to the linearized Einstein equation in three-dimensional de Sitter space (dS3) once one allows complexified extremal surfaces in the bulk. Moreover, variations of pseudo-entropy obey a Klein–Gordon equation on the kinematical space dS2, offering an emergent time structure arising from an Euclidean boundary theory. In this paper we show that while the holographic pseudo-entropy program represents an important boundary diagnostic of gravitational dynamics, it remains a restricted kinematical construction tied to holography, non-unitary conformal field theories, and perturbative de Sitter gravity. By contrast, the Theory of Entropicity (ToE) treats entropy S(x) as the fundamental physical field of nature, endowed with a local variational principle (the Local Obidi Action) and a spectral variational principle (the Spectral Obidi Action). From these actions one derives the Master Entropic Equation, entropic geodesics, irreversible dynamics, and a unified description of gravity, time, quantum processes, and information geometry. The goal of this work is threefold. First, we present a precise and self-contained exposition of the Takayanagi–Kusuki–Tamaoka framework. Second, we develop the Theory of Entropicity as a universal entropic field theory whose dynamics extend far beyond the holographic pseudo-entropy correspondence. Third, we provide a systematic comparison showing how ToE absorbs pseudo-entropy as a special boundary manifestation of a deeper entropic field, thereby revealing why pseudo-entropy reproduces only the linearized sector of gravitational physics while ToE yields a fully nonlinear, time-asymmetric, and information-geometric unification of physical law.

Keywords

Amari–Čencov α–Connections; Araki Relative Entropy; Atiyah–Singer Index Theorem;

Bekenstein–Hawking Entropy; Bosons; Canonical Quantization; Complex Geodesics; Dark Matter; Dark

Energy; dS/CFT Correspondence; Dirac–Kähler Fermions; Dirac Spinors; Einstein–Hilbert Action; Emer

gent Geometry; Entropic Field; Entropic Geodesics; Entropy Geometry; Entropy as Ontic Field; Fermions;

Fisher–Rao Metric; Fubini–Study Metric; G-Field (Bianconi); Ginestra Bianconi; Holographic Pseudo

Entropy; Information Geometry; Jacobson Thermodynamics; Kinematic Space (dS2); Klein–Gordon

Equation (Pseudo-Entropy); Local Obidi Action (LOA); Master Entropic Equation (MEE); Modular Op

erator ∆; Nonlinear Entropic Dynamics; Obidi Actions; Padmanabhan Entropic Gravity; Pseudo-Entropy

(Takayanagi–Kusuki–Tamaoka); Quantum Entanglement; Quantum Gravity; Rényi Entropy; Relative

Entropy; Shannon Information; Small Positive Cosmological Constant; Spectral Action; Spectral Dynam

ics; Spectral Geometry; Spectral Obidi Action (SOA); Spectral Theories; Takayanagi–Kusuki–Tamaoka

Pseudo-Entropy; Theory of Entropicity (ToE); Thermodynamic Gravity; Tsallis Entropy; Vuli–Ndlela

Integral; Yang–Mills Theory.

References

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

Further Resources on the Theory of Entropicity (ToE):

1. Website: Theory of Entropicity ToE —

https://theoryofentropicity.blogspot.com

2. LinkedIn: Theory of Entropicity ToE — https://www.linkedin.com/company/theory-of-entropicity-toe/about/?viewAsMember=true

3. Notion: Theory of Entropicity (ToE)

4. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack

5. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium

6. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

7. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author

8. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).

9. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

10. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

11. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki

12. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia

13. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate

14. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare

15. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea

16. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN

17. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).

18. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar

19. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)

Wikipedia

Thursday, 20 November 2025

Entropy as a Field: Can the Spectral Obidi Action Go Beyond Araki Relative Entropy? A Most Radical Conceptualization of a Unified Field in the Theory of Entropicity (ToE)

Entropy as a Field: Can the Spectral Obidi Action Go Beyond Araki Relative Entropy? A Most Radical Conceptualization of a Unified Field in the Theory of Entropicity (ToE)

Introduction: The Measure That Became a Candidate for a Field

Araki Relative Entropy: The Established Framework

The Spectral Obidi Action: A Radical Reinterpretation

Why No One Else Has Tried This

Is SOA Valid or Useful?

The Stakes for ToE

Conclusion: A Bold but Risky Leap

References

Further Resources on the Theory of Entropicity (ToE):

Litmus Test of the Theory of Entropicity (ToE): If Einstein’s Relativity is Emergent from Entropy, then the Observer is Dethroned, and Physics Gains a New Foundation in the Theory of Entropicity (ToE)

Litmus Test of the Theory of Entropicity (ToE): If Einstein’s Relativity is Emergent from Entropy, then the Observer is Dethroned, and Physics Gains a New Foundation in the Theory of Entropicity (ToE)

In Einstein’s Relativity

In the Theory of Entropicity (ToE)

Why This Matters

The Challenge

Einstein’s Relativity: Observer-Centric

ToE’s Reinterpretation: Entropy-Centric

Why This Is a Crucial Test

The Stakes

Further Expository Insights

The No‑Rush Theorem as a Constraint on Reality

Observation as Entropy‑Dependent

Philosophical Implications

Physical Implications

References

Wednesday, 19 November 2025

On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE) - The Obidi Actions and the Obidi Field Equations

On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE)

On the Complexity and Intricacy of the Mathematical Foundations of the Theory of Entropicity (ToE)

References

Saturday, 15 November 2025

The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory

The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory

Abstract

Keywords

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 (Last Updated: Friday, June19th, 2026)