In the Theory of Entropicity (ToE), information has temperature

In ToE, this statement is not a metaphor. It is a literal physical principle.

ToE teaches that information is not abstract—it is a physical configuration of the entropic field. And because every configuration of entropy carries an energetic and dynamical cost, information necessarily possesses a temperature.

Let us state put this in the cleanest possible way.

1. Information = Entropy Configuration

In the Theory of Entropicity:

- Information is a pattern in the entropy field \( S(x) \).  

- Patterns require energy to maintain.  

- Energy and entropy are inseparable.  

Thus, information is not “cold mathematics.”  

It is a thermodynamic structure.

2. Temperature = Rate of Entropic Reconfiguration

In ToE, temperature is defined as:

> the rate at which entropy can reorganize within a system.

If information is a configuration of entropy, then:

- Changing information requires entropic reconfiguration.  

- Entropic reconfiguration has a finite rate.  

- A finite rate implies a temperature.

Therefore:

\[

\textbf{Information has temperature because information is entropy.}

\]

3. Landauer’s Principle as a Shadow of ToE

Classical physics already hints at this:

- Landauer’s principle says erasing one bit of information costs  

  \[

  k_B T \ln 2.

  \]

This is a thermodynamic statement:  

information processing requires heat.

ToE generalizes this:

> Not only does information processing require heat—information itself is a thermal object.

4. Information Temperature in ToE

In the Theory of Entropicity:

- Every bit of information corresponds to a microstate distribution.  

- Every microstate distribution has an entropic curvature.  

- Every entropic curvature has a temperature.  

Thus:

\[

\textbf{Information carries temperature because it is a local excitation of the entropy field.}

\]

This is why:

- Information cannot be transmitted infinitely fast.  

- Information cannot be copied without cost.  

- Information cannot be observed simultaneously by multiple observers.  

All of these are consequences of the finite entropic temperature of information.

5. Why This Matters

This principle allows ToE to unify:

- thermodynamics,  

- information theory,  

- quantum measurement,  

- relativity,  

- and entanglement dynamics  

under a single entropic framework.

It also explains:

- why computation generates heat,  

- why observation requires entropic collapse,  

- why information flow has a speed limit,  

- why entanglement has a formation time,  

- why measurement is irreversible.

Because information has temperature, and temperature is the rate of entropic change.

6. The ToE Statement in One Line

> Information has temperature because information is entropy, and entropy cannot exist without thermodynamic structure.

From the Temperature of Information to the Temperature of Geometry: The Foundations of the Theory of Entropicity (ToE) and the Unification of Quantum and Entropic Reality - A Unified Framework for the Thermodynamic, Quantum, and Geometric Foundations of Physical Reality

From the Temperature of Information to the Temperature of Geometry: The Foundations of the Theory of Entropicity (ToE) and the Unification of Quantum and Entropic Reality - 

A Unified Framework for the Thermodynamic, Quantum, and Geometric Foundations of Physical Reality

Dedication

Joseph Polchinski’s work reshaped how we think about the universe. Even in his final days, when illness clouded his physical strength, his clarity of thought remained untouched. His commitment to understanding the deepest structures of reality continues to inspire new generations of thinkers.

This work is dedicated to him. Where he saw strings vibrating in spacetime, the Theory of Entropicity (ToE) extends that vision: information itself vibrates, reorganizes, and shapes the geometry of the universe. His legacy lives on in every attempt to understand the hidden architecture of reality.

Prologue

Modern physics stands on three towering pillars—quantum mechanics, general relativity, and thermodynamics. Each describes the universe with extraordinary precision, yet they speak different conceptual languages. Quantum theory deals in probabilities and discrete events. Relativity describes smooth curvature and the geometry of spacetime. Thermodynamics explains energy, entropy, and irreversible processes.

What has been missing is a framework that explains how information animates all three.

The Theory of Entropicity (ToE) proposes that information is not a secondary description of physical systems—it is the foundation from which physical systems arise. In this view, spacetime, matter, and energy are the visible shadows of a deeper informational field. Energy becomes the motion of information. Geometry becomes the structure information creates. Temperature becomes the rate at which information reorganizes itself.

A central idea of ToE is the existence of a “temperature of information,” a measure of how quickly the informational field changes. Regions where information reorganizes rapidly behave like “hot” geometries—dynamic, curved, and energetic. Regions where information changes slowly behave like “cold” geometries—stable, smooth, and quiet.

This leads to a unifying principle: quantum behavior, thermodynamic behavior, and geometric behavior are all different expressions of how information flows.

From this perspective:

• String theory becomes a theory of informational vibrations.

• Quantum field theory becomes a theory of entropic curvature.

• The Casimir effect becomes a pressure created by constrained informational flow.

ToE reframes temperature as a measure of informational activity, not molecular agitation. Entropy becomes the active substance of reality, not a passive measure of disorder. The laws of physics emerge from the invisible geometry of information itself.

This work lays out the conceptual foundations of that idea.

Abstract

The Theory of Entropicity (ToE) proposes that information and entropy form the true foundation of physical reality. In this framework, information is not something we use to describe matter and energy—matter and energy emerge from information.

Spacetime, particles, and forces arise as thermodynamic projections of an underlying informational manifold. A key insight of ToE is that information has its own intrinsic temperature, which determines how quickly the informational field reorganizes. This temperature shapes geometry, energy, and the behavior of physical systems.

From this foundation, ToE constructs an informational action principle and derives a field equation that generalizes Einstein’s equation into the informational domain. Phenomena such as the Casimir effect, inertial mass, and gravity are reinterpreted as consequences of entropic curvature rather than quantum vacuum fluctuations.

In this view, the universe is a thermodynamic image of an informational continuum. The laws of physics are expressions of its self‑organizing flow.

Keywords: Casimir Effect, Entropic Gravity, Entropy, Information Geometry, Informational Field Theory, Quantum Thermodynamics, Spacetime Emergence, Temperature of Geometry, Temperature of Information, Theory of Entropicity (ToE)

1. Introduction: The Temperature of Information and Geometry

Physics today is built on three major frameworks: quantum mechanics, general relativity, and thermodynamics. Each is powerful, but none explains why the universe behaves the way it does at the deepest level. They coexist, but they do not fully connect.

The Theory of Entropicity (ToE) proposes a new unifying idea: information is the fundamental substance of the universe. Everything else—matter, energy, spacetime—emerges from how information flows and organizes itself.

In this view:

• Energy is the motion of information.

• Geometry is the structure information creates.

• Temperature is the rate at which information reorganizes.

This leads to the idea of a temperature of information, a measure of how quickly the informational field changes. When information changes rapidly, geometry becomes dynamic and curved. When information changes slowly, geometry becomes stable and smooth.

This idea builds on earlier insights from black hole thermodynamics, entropic gravity, and information geometry, but extends them into a unified framework where information itself carries temperature and generates curvature.

1.1 Motivation and Background

The connection between thermodynamics and geometry has been hinted at for decades. Black hole physics revealed that horizons have entropy and temperature. Later work showed that gravitational equations can be derived from thermodynamic principles.

Information geometry demonstrated that statistical systems form curved manifolds, suggesting that information has geometric structure.

ToE brings these ideas together by proposing that information itself is a physical field. Once this is accepted, temperature becomes a natural descriptor of how information evolves—and geometry becomes the visible expression of that evolution.

1.2 The Conceptual Innovation

ToE introduces the idea that information has its own intrinsic temperature. This temperature does not measure heat. It measures how quickly the informational field changes.

When the informational field changes rapidly, geometry becomes “hot”—dynamic, curved, and energetic. When the field changes slowly, geometry becomes “cold”—smooth and stable.

This leads to a unifying principle: quantum behavior and geometric behavior are two sides of the same informational process.

1.3 Goals and Structure of the Paper

This work aims to establish the conceptual and mathematical foundations of ToE. It introduces the temperature of information, develops an informational uncertainty principle, constructs an informational action, and derives a field equation that generalizes Einstein’s equation.

The later sections explore how classical gravity, cosmology, and physical phenomena emerge from informational geometry.

1.4 Why the Temperature of Information Matters

If correct, ToE would unify quantum mechanics, thermodynamics, and gravity by showing that all three arise from the same informational substrate. It suggests that the constants of nature are not independent—they are expressions of one underlying informational process.

In this view, the universe is a thermodynamic hologram of an informational continuum.

2. Temperature as a Property of Information and Geometry

In everyday physics, temperature is tied to matter. It measures how fast particles move, how much energy they share, and how chaotic their motion becomes. But in the Theory of Entropicity (ToE), temperature takes on a deeper meaning. It becomes a property of the informational structure of the universe itself.

If information is the true foundation of physical reality, then temperature must describe how actively that information is changing. Instead of measuring molecular agitation, temperature becomes a measure of informational agitation — the intensity with which the informational field reorganizes.

In this framework, regions where information changes rapidly behave like “hot” zones. Their geometry is dynamic, curved, and responsive. Regions where information changes slowly behave like “cold” zones, where geometry is smooth, stable, and nearly static.

This leads to a profound equivalence: the temperature of information and the temperature of geometry are two sides of the same phenomenon. When information becomes more active, geometry becomes more dynamic. When information settles, geometry cools and smooths out.

In ToE, temperature is not something matter possesses. It is something information does.

3. Relation to Landauer’s Principle

Landauer’s Principle is one of the most important links between information and physics. It states that erasing information requires energy. In other words, information is physical — it cannot be changed or destroyed without a cost.

ToE takes this idea much further.

Instead of treating information as something that merely influences physical systems, ToE treats information as the system itself. The temperature of information becomes a universal property of the informational field, not just a measure of energy dissipation in computation.

Where Landauer describes the cost of erasing information, ToE describes the geometric consequences of possessing it.

In this view, energy, entropy, and geometry are inseparable. They are all expressions of how information flows. Landauer’s insight becomes a local manifestation of a deeper principle: the universe is driven by the irreversible flow of information, and temperature is the measure of that flow.

4. Understanding the Temperature of Information

In traditional physics, we don’t talk about information having a temperature. Temperature is something matter has. But ToE reframes this idea by asking a simple question: what does temperature really measure?

Temperature measures how quickly energy changes when entropy changes. Since entropy is fundamentally an informational quantity, temperature already has an informational meaning built into it.

ToE simply makes this explicit.

The temperature of information describes how rapidly the informational structure of the universe reorganizes. It is not about heat. It is about activity — the pace at which information reshapes its own geometry.

This makes the concept surprisingly natural. Information theory and thermodynamics already mirror each other. Entropy measures uncertainty in both. Free energy measures usable information. Temperature measures responsiveness.

ToE extends this parallel by giving information its own intrinsic temperature — a measure of how energetically it evolves.

4.1 What Temperature Really Measures

Temperature is often described as “how hot something is,” but at its core, it measures how sensitive energy is to changes in entropy. It is a measure of responsiveness. When entropy increases, temperature tells us how much energy must change in response.

Since entropy is an informational quantity, temperature already has an informational interpretation. ToE simply shifts the focus from matter to information itself.

4.2 The Step Taken by ToE

ToE proposes that wherever information is being reorganized, there is a flow of entropy. And wherever entropy flows, there is a temperature associated with that flow.

This temperature does not describe heat. It describes informational activity.

The temperature of information is the rate at which the informational field changes. It is the intensity of informational motion. And because geometry emerges from information, this temperature naturally becomes the temperature of geometry as well.

4.3 Why This Is Conceptually Natural

Thermodynamics and information theory have been intertwined for decades. They share the same mathematical structures. They describe the same kinds of uncertainty and organization.

If entropy and information are the same quantity, then temperature — which measures how systems respond to entropy — should also have an informational meaning.

In ToE, the temperature of information measures how rapidly the informational field reorganizes. When information changes quickly, geometry becomes more dynamic. When information changes slowly, geometry becomes more stable.

This is why ToE speaks of a “temperature of geometry.” Geometry is simply the visible expression of informational activity.

4.4 How It Differs from Ordinary Temperature

The temperature of information is not heat. It is not kinetic agitation. It is not something you can feel with your hand.

Instead, it is a field property that describes how energetically information reshapes its own geometry.

Ordinary temperature measures the motion of particles. Informational temperature measures the motion of information.

This distinction is subtle but transformative. It shifts temperature from being a property of matter to being a property of the universe’s informational foundation.

4.5 Why This Concept Is New

No existing physical theory assigns temperature to information itself. Thermodynamics ties temperature to energy. Information theory treats information as abstract. Even modern theories of gravity that use entropy still tie temperature to energy flow, not informational flow.

ToE is the first framework to propose that information itself has temperature — and that geometry inherits this temperature because it emerges from information.

This is a conceptual leap, but one that follows naturally from decades of hints in thermodynamics, quantum theory, and gravitational physics.

4.6 Summary: What the Temperature of Information Really Means

The temperature of information is not a metaphor. It is a measure of how rapidly information changes its own geometry. It is the pulse of the informational universe.

In ToE, temperature is not what matter feels. It is what information does.

The universe is constantly reorganizing itself, computing its own structure. The temperature of information measures how energetically that computation unfolds.

5. The Kinetic Analogy: Why Information Has a Temperature

In classical physics, temperature is tied to motion. When molecules move quickly, the temperature is high. When they move slowly, the temperature is low. This idea is so deeply embedded in our thinking that it’s easy to forget what temperature really represents: the average intensity of microscopic activity.

The Theory of Entropicity (ToE) extends this idea to information itself.

If information is a dynamic field — something that can change, propagate, oscillate, and reorganize — then it must also have a measure of how energetically it behaves. In ToE, the informational field is not static. It fluctuates, spreads, and reshapes itself across the fabric of reality. These fluctuations are not random; they define the structure of geometry and the behavior of physical systems.

The temperature of information is the measure of this activity. When the informational field fluctuates rapidly, the temperature is high. When it changes slowly, the temperature is low. This mirrors the kinetic interpretation of temperature in ordinary physics, but applied to the deeper substrate of information rather than matter.

In this sense, the universe is constantly “computing” its own structure, and the temperature of information tells us how energetically that computation is taking place.

6. How the Temperature of Information Could Be Measured

Although the temperature of information is introduced as a theoretical concept, it is not beyond the reach of measurement. Just as ordinary temperature can be inferred from the behavior of particles, the temperature of information can be inferred from the behavior of systems where information and entropy play a central role.

Several physical contexts offer clues:

Quantum information systems

In quantum computers and simulators, entanglement spreads through a system in measurable ways. The rate at which this entanglement grows can serve as a proxy for informational temperature. Faster growth means a “hotter” informational environment.

Black hole physics

Black holes provide one of the clearest windows into the relationship between entropy and geometry. Their temperature and entropy are directly tied to the structure of spacetime. In ToE, these quantities reflect the temperature of the underlying informational field.

Cosmology

The universe as a whole evolves through massive changes in entropy. As the cosmos expands and cools, its informational geometry changes as well. Large‑scale entropy production can be interpreted as a shift in the temperature of information across cosmic history.

These examples show that the temperature of information is not an abstract idea. It is a measurable feature of systems where information, entropy, and geometry interact. We may not yet have a direct “informational thermometer,” but the physical world already provides indirect ways to observe the phenomenon.

7. Why the Temperature of Information Is a New Concept

The idea that information has temperature is not found in traditional physics. Historically, temperature has always been tied to matter — to particles, energy, and heat. Even when entropy entered the world of information theory, temperature remained firmly rooted in thermodynamics.

Only recently have theories of gravity and spacetime begun to treat geometry as a thermodynamic system. But even in these approaches, temperature is still tied to energy flow, not informational flow.

The Theory of Entropicity (ToE) takes the next step. It proposes that information itself is a physical field with its own dynamics. Once information is treated as something that moves, interacts, and evolves, it naturally acquires a temperature. This temperature is not metaphorical. It is the thermodynamic descriptor of informational activity.

This is why the temperature of information is a genuinely new insight. It emerges only when information is elevated from a descriptive tool to the fundamental substance of reality.

8. Why the Temperature of Information Matters

Throughout the history of physics, temperature has been tied to matter. It measures how particles move and how energy is distributed. But as our understanding of entropy deepened, it became clear that entropy is not just a property of matter — it is a property of information.

Shannon showed that entropy measures uncertainty. Jaynes showed that statistical mechanics can be derived from principles of information inference. Black hole physics revealed that geometry itself carries entropy and temperature. Modern theories of gravity suggest that spacetime may be a thermodynamic system.

ToE brings all these threads together.

If information is the foundation of reality, then temperature must describe the activity of information itself. The temperature of information becomes the missing link that unifies thermodynamics, quantum mechanics, and geometry.

In this view:

• Temperature measures how energetically information reorganizes.

• Geometry is the visible expression of informational activity.

• The universe evolves through the flow of information, not the motion of matter.

The temperature of information is the pulse of the informational universe — the rate at which reality computes itself.

App Deployment on the Theory of Entropicity (ToE):

App on the Theory of Entropicity (ToE): Click or Open on web browser (a GitHub Deployment - WIP): Theory of Entropicity (ToE)

https://phjob7.github.io/JOO_1PUBLIC/index.html

 

Sources — help

1. ijcsrr.org
2. researchgate.net
3. encyclopedia.pub
4. medium.com
5. medium.com
6. medium.com
7. medium.com
8. encyclopedia.pub
9. figshare.com
10. researchgate.net
11. medium.com
12. researchgate.net
13. cambridge.org

References

1. Obidi, John Onimisi (30th December, 2025). From the Temperature of Information to the Temperature of Geometry: The Foundations of the Theory of Entropicity (ToE) and the Unification of Quantum and Entropic Reality - A Unified Framework for the Thermodynamic, Quantum, and Geometric Foundations of Physical Reality. Figshare. https://doi.org/10.6084/m9.figshare.30976342
2. Obidi, John Onimisi (27th December, 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.30958670
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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
9. Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).
10. 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
11. 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
12. 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%27s_Special_Relativity&oldid=3845936

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-1: Theory of Entropicity (ToE)
4. Notion-2: Theory of Entropicity (ToE)
5. Notion-3: Theory of Entropicity (ToE)
6. Notion-4: Theory of Entropicity (ToE)
7. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack
8. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium
9. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
10. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
11. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).
12. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
13. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
14. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
15. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia
16. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate
17. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare
18. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea
19. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN
20. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).
21. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar
22. IJCSRR: International Journal of Current Science Research and Review - Theory of Entropicity (ToE) - John Onimisi Obidi | IJCSRR
23. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)

Entropicity, Neutrino Mixing, and the PMNS Matrix:  A New Perspective on Neutrino Oscillations and Symmetries Based on New Insights from the Theory of Entropicity(ToE)

When Entropy Enters the Neutrino: Rethinking the PMNS Matrix Through Obidi’s Theory of Entropicity (ToE)

Abstract

This paper develops a novel interpretation of neutrino mixing and oscillation phenomena by embedding them within the Theory of Entropicity (ToE), a framework in which entropy is promoted from a statistical descriptor to a fundamental dynamical entity. Within this paradigm, entropy is treated as an active field capable of coupling to quantum systems, inducing irreversible dynamics, modifying symmetry principles, and reshaping conservation laws. Applying ToE to neutrino physics, the work proposes that entropy-driven mechanisms can naturally generate or influence the structure of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix, offering a new physical rationale for large leptonic mixing angles and CP violation.

The paper introduces a reinterpretation of CPT symmetry in the presence of intrinsic entropy flow, formulates an Entropic Noether’s theorem linking symmetry breaking to entropy production, and proposes a thermodynamic uncertainty principle that establishes a fundamental entropic time limit for quantum processes. These concepts are applied to neutrino oscillations, mass hierarchy, CP violation, and the Dirac–Majorana question. The analysis further explores how entropic effects may induce decoherence, modify oscillation probabilities, generate environment-dependent CP phases, and lead to subtle violations of conventional conservation laws. Finally, the paper discusses phenomenological consequences for current and future experiments, including T2K, NOνA, DUNE, and JUNO, outlining potential observational signatures that could distinguish entropic dynamics from Standard Model expectations.

Long Overview: Entropy, Neutrinos, and the Foundations of Mixing

1. Motivation: Why Revisit Neutrino Physics Through Entropy?

Neutrinos occupy a unique position in fundamental physics. They are extraordinarily abundant, weakly interacting, and deeply connected to unresolved questions about mass generation, CP violation, and the matter–antimatter asymmetry of the universe. While the PMNS matrix successfully parametrizes neutrino mixing and oscillations, it does not explain why the mixing angles take their observed values, nor why CP violation appears to be potentially large in the lepton sector.

This paper argues that these unresolved features may be signaling physics that lies beyond conventional quantum field theory—specifically, physics associated with irreversibility, information flow, and entropy. The Theory of Entropicity (ToE) provides the conceptual and mathematical setting for this investigation by treating entropy as a real, dynamical participant in fundamental processes rather than as a passive statistical measure.

---

2. Core Idea of the Theory of Entropicity

At the heart of ToE is the assertion that entropy possesses field-like properties: it can flow, couple to matter, break symmetries, and influence dynamics at the most fundamental level. In this view, entropy is associated with a field $𝑆(𝑥)$ and an entropy current ${𝑆}^{𝜇}$, allowing irreversibility to be encoded directly into the equations of motion.

This has profound implications. Time-reversal symmetry is no longer exact; probability conservation becomes approximate within subsystems; and information loss is treated as a physical transfer into an entropy sector rather than as a mere epistemic limitation. The paper builds on these ideas to show how neutrinos—because of their weak interactions and long propagation distances—are exceptionally sensitive probes of entropic dynamics.

---

3. Reinterpreting Neutrino Mixing and the PMNS Matrix

In the Standard Model, neutrino oscillations arise because flavor eigenstates are superpositions of mass eigenstates. The PMNS matrix encodes this mismatch, but its structure remains unexplained. The paper proposes that entropy-driven mixing mechanisms may provide a natural explanation.

Two complementary ideas are developed. First, entropy maximization arguments suggest that large mixing angles can emerge as equilibrium configurations in the early universe or other high-entropy environments. Second, explicit coupling between neutrinos and the entropy field can induce flavor-changing effects analogous to open quantum system dynamics. In this picture, neutrino flavor transitions are not purely unitary oscillations but may involve subtle exchanges of entropy with an underlying reservoir.

These mechanisms offer new intuition for why two mixing angles are large, one is smaller, and why neutrinos differ qualitatively from quarks.

---

4. CPT Symmetry and Entropy-Induced Time Asymmetry

A central conceptual advance of the paper is its treatment of CPT symmetry. Conventional quantum field theory guarantees CPT invariance under broad assumptions, including unitarity and time-reversal symmetry. ToE challenges these assumptions by introducing a fundamental arrow of time through entropy flow.

Rather than abandoning CPT entirely, the paper proposes a generalized CPT* symmetry in which CPT invariance is restored only when entropy conjugation is included. This leads to the possibility that neutrinos and antineutrinos experience slightly different effective dynamics due to asymmetric coupling to entropy. Such effects could manifest as tiny differences in oscillation parameters, decoherence rates, or CP-violating phases—effects that next-generation experiments may be able to probe. 

---

5. Entropic Noether’s Theorem and Modified Conservation Laws

The introduction of entropy as a dynamical field necessitates a generalization of Noether’s theorem. The paper formulates an Entropic Noether’s principle, according to which symmetries broken by entropy coupling lead to modified continuity equations with entropy-dependent source terms.

Applied to neutrino physics, this framework opens the door to controlled violations of lepton number and lepton flavor conservation. Such violations may appear as non-unitarity, exotic decay channels, or apparent anomalies in oscillation data. Importantly, these effects are not arbitrary but are quantitatively linked to entropy production, preserving a deeper generalized conservation structure.

---

6. Thermodynamic Uncertainty and the Entropic Time Limit

Another key contribution of the paper is the extension of quantum uncertainty principles to include entropy. By placing energy and entropy on equal footing, the theory introduces an entropic time limit—a fundamental minimum duration for physical processes.

When applied to neutrino oscillations, this principle suggests new bounds on coherence lengths, oscillation speeds, and decoherence mechanisms. While current experiments are unlikely to be directly sensitive to such effects, extreme environments such as supernovae, the early universe, or ultra-long-baseline propagation may reveal signatures of entropy-limited dynamics.

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7. Phenomenology and Experimental Outlook

The final sections translate the theoretical framework into phenomenological consequences. The paper discusses how entropy-induced decoherence could slightly damp oscillation probabilities, how CP violation might acquire environment-dependent corrections, and how the neutrino mass hierarchy and Dirac–Majorana nature could be reinterpreted within an entropic framework.

Crucially, the work outlines potential tests using current and upcoming experiments such as T2K, NOνA, DUNE, JUNO, and Hyper-Kamiokande. While ToE effects are expected to be subtle, their cumulative impact over long baselines or high-entropy conditions may eventually distinguish them from conventional new-physics scenarios. 

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8. Conceptual Significance

Beyond specific predictions, the paper makes a broader conceptual claim: neutrino physics may be one of the first domains where the dynamical role of entropy becomes experimentally relevant. By linking mixing, CP violation, irreversibility, and information flow within a single framework, the Theory of Entropicity offers a unifying perspective that challenges the traditional separation between thermodynamics and fundamental particle physics.

In this sense, the work positions neutrinos not merely as particles with tiny masses, but as windows into the deep entropic structure of physical law itself.

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Further Notes on Application of the Theory of Entropicity (ToE) to the Physics of Neutrinos

For decades, neutrinos have been quietly unsettling our understanding of the universe.

They pass through Earth in astronomical numbers, barely interacting, almost ghostlike. And yet, these elusive particles do something profoundly strange: they change identity as they travel. A neutrino created as one “flavor” can later be detected as another. This phenomenon — neutrino oscillation — is now well established and mathematically encoded in the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix.

But here is the uncomfortable truth: while the PMNS matrix works astonishingly well, it explains nothing about why it is the way it is.

Why are the mixing angles so large compared to those of quarks?
 Why does the lepton sector seem predisposed to strong CP violation?
 Why does neutrino physics feel so different from the rest of the Standard Model?

The Theory of Entropicity (ToE) begins from a radical but increasingly unavoidable suspicion: perhaps the missing ingredient is entropy — not as a statistical afterthought, but as a fundamental physical actor.

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The Uneasy Status of Entropy in Fundamental Physics

Entropy is everywhere in physics, yet nowhere at its foundations.

It governs the arrow of time, limits computation, shapes black holes, and dictates thermodynamic behavior. But in particle physics and quantum field theory, entropy is usually treated as a bookkeeping tool — something we calculate after the fact, not something that acts.

This uneasy separation has always been philosophically troubling. The laws of motion are time-reversible, yet the universe is not. Quantum mechanics is unitary, yet measurement is irreversible. Field theories are symmetric, yet physical processes are directional.

The Theory of Entropicity takes these tensions seriously and proposes a bold resolution: entropy is not merely a measure of disorder — it is a dynamical field that participates in physical law.

Once this idea is accepted, neutrinos emerge as natural messengers of entropic physics.

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Why Neutrinos Are the Perfect Entropic Probes

Neutrinos are unlike any other known particles. They interact weakly, travel immense distances, and retain quantum coherence far longer than most systems. If entropy subtly influences quantum evolution, neutrinos are among the few particles capable of revealing it.

In conventional treatments, neutrino oscillations are purely unitary phenomena. A neutrino’s flavor changes because mass eigenstates accumulate different phases during propagation. Entropy plays no direct role.

ToE challenges this picture by asking a deeper question: what if neutrino oscillations occur within an entropic background that is itself evolving?

In such a setting, flavor change is no longer just a geometric rotation in Hilbert space. It becomes a process influenced by entropy flow, information redistribution, and irreversible coupling to an entropic field.

The PMNS matrix, from this perspective, is not merely a mixing matrix. It is a thermodynamic fingerprint.

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Entropy as a Driver of Mixing, Not Just a Spectator

One of the most striking features of neutrino physics is the size of its mixing angles. Two of them are large — almost maximal — while the quark sector shows only small mixings. The Standard Model offers no explanation for this contrast.

Within the Theory of Entropicity, this disparity becomes less mysterious.

Entropy tends to favor configurations that maximize accessible states and information flow. Large mixing angles correspond to states that are more entropically connected, more delocalized in flavor space, and less constrained by symmetry. In high-entropy environments — such as the early universe — such configurations are not accidental; they are favored.

Neutrino mixing, in this view, is not arbitrary. It is entropically natural.

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CP Violation, Time’s Arrow, and the Entropic Asymmetry

Perhaps the most profound implication of introducing entropy as a fundamental field lies in its relationship to time.

Entropy is inherently directional. It distinguishes past from future. By embedding entropy directly into fundamental dynamics, ToE introduces a subtle but unavoidable time asymmetry into particle physics.

This has direct consequences for CP and CPT symmetries.

Standard quantum field theory guarantees CPT invariance under broad assumptions, including exact unitarity and time reversibility. ToE relaxes these assumptions — not by discarding them recklessly, but by extending them. It proposes that CPT symmetry must be generalized to include entropy conjugation.

In this framework, neutrinos and antineutrinos may experience slightly different entropic environments, leading to effective asymmetries that resemble CP violation. Crucially, these effects arise not from arbitrary symmetry breaking, but from the same entropic mechanisms that define the arrow of time itself.

Suddenly, CP violation in the lepton sector is no longer an isolated mystery. It becomes part of a deeper story about irreversibility.

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Conservation Laws Revisited: An Entropic Noether Principle

One of the most unsettling ideas in the Theory of Entropicity is that traditional conservation laws may not be absolute within subsystems.

This does not mean energy or lepton number simply disappear. Rather, ToE proposes that conservation laws must be generalized to include entropy flow.

When entropy couples dynamically to matter, symmetries give rise not to strict conservation equations, but to balance laws with entropic source terms. This is formalized through an Entropic Noether principle.

Applied to neutrinos, this opens the door to small, controlled deviations from perfect unitarity — effects that could appear as decoherence, apparent non-conservation, or anomalous oscillation behavior.

These deviations are not arbitrary. They are tightly constrained by entropy production itself.

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The Entropic Time Limit and Quantum Evolution

Another striking idea introduced in the paper is the existence of a fundamental entropic time limit.

Just as quantum mechanics limits how precisely energy and time can be simultaneously defined, ToE suggests that entropy imposes a minimum timescale on physical processes. No interaction can occur faster than the rate at which entropy can reorganize.

For neutrinos, this implies subtle bounds on coherence lengths, oscillation speeds, and flavor transition rates. While such effects are likely tiny, they may become relevant over cosmic distances or in extreme astrophysical environments.

Once again, neutrinos appear as ideal laboratories for probing the deep structure of physical law.

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What Experiments Might See

The Theory of Entropicity does not reject existing neutrino experiments; it invites them to look closer.

Long-baseline experiments such as T2K, NOνA, DUNE, and JUNO already test neutrino oscillations with extraordinary precision. ToE predicts that, beyond the Standard Model parameters, there may exist faint signatures of entropic coupling: environment-dependent CP phases, tiny decoherence effects, or deviations from perfect unitarity that accumulate over long distances.

None of these effects would overthrow existing data. They would refine it.

In this sense, ToE does not compete with the Standard Model — it extends its interpretive reach.

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A Broader Shift in Perspective

At its deepest level, this work is not only about neutrinos.

It is about the role of entropy in fundamental physics.

Recent developments in quantum information theory, black hole physics, and thermodynamics increasingly suggest that entropy and information are not peripheral concepts. They are structural. The Theory of Entropicity embraces this trend and pushes it further, proposing that entropy is the substrate from which geometry, quantum behavior, and symmetry itself emerge.

Neutrinos, with their quiet strangeness, may be among the first particles to reveal this hidden layer of reality.

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Final Thought: From Ghost Particles to Entropic Messengers

Neutrinos were once thought to be massless. Then almost irrelevant. Now they sit at the center of some of the deepest questions in physics.

The Theory of Entropicity (ToE) suggests that their true role may be even more profound. They may be messengers — not just of physics beyond the Standard Model, but of a deeper entropic architecture underlying all physical law.

If that is the case, then the PMNS matrix is not merely a numerical artifact.

It is a window into how entropy shapes the universe at its most fundamental level.

App Deployment on the Theory of Entropicity (ToE):

App on the Theory of Entropicity (ToE): Click or Open on web browser (a GitHub Deployment - WIP): Theory of Entropicity (ToE)

https://phjob7.github.io/JOO_1PUBLIC/index.html

 

Sources — help

1. ijcsrr.org
2. researchgate.net
3. encyclopedia.pub
4. medium.com
5. medium.com
6. medium.com
7. medium.com
8. encyclopedia.pub
9. figshare.com
10. researchgate.net
11. medium.com
12. researchgate.net
13. cambridge.org

References

1. Obidi, John Onimisi (27th December, 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.30958670
2. 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
3. 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
4. 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
5. 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
6. 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
7. 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
8. Obidi, John Onimisi. Unified Field Architecture of Theory of Entropicity (ToE). Encyclopedia. Available online: https://encyclopedia.pub/entry/59276 (accessed on 19 November 2025).
9. 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
10. 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
11. 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%27s_Special_Relativity&oldid=3845936

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-1: Theory of Entropicity (ToE)
4. Notion-2: Theory of Entropicity (ToE)
5. Notion-3: Theory of Entropicity (ToE)
6. Notion-4: Theory of Entropicity (ToE)
7. Substack: Theory of Entropicity (ToE) — John Onimisi Obidi | Substack
8. Medium: Theory of Entropicity (ToE) — John Onimisi Obidi — Medium
9. SciProfiles: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
10. Encyclopedia.pub: Theory of Entropicity (ToE) — John Onimisi Obidi | Author
11. HandWiki contributors, “Biography: John Onimisi Obidi,” HandWiki, https://handwiki.org/wiki/index.php?title=Biography:John_Onimisi_Obidi&oldid=2743427 (accessed October 31, 2025).
12. HandWiki Contributions: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
13. HandWiki Home: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
14. HandWiki Homepage-User Page: Theory of Entropicity (ToE) — John Onimisi Obidi | HandWiki
15. Academia: Theory of Entropicity (ToE) — John Onimisi Obidi | Academia
16. ResearchGate: Theory of Entropicity (ToE) — John Onimisi Obidi | ResearchGate
17. Figshare: Theory of Entropicity (ToE) — John Onimisi Obidi | Figshare
18. Authoria: Theory of Entropicity (ToE) — John Onimisi Obidi | Authorea
19. Social Science Research Network (SSRN): Theory of Entropicity (ToE) — John Onimisi Obidi | SSRN
20. Wikidata contributors, Biography: John Onimisi Obidi “Q136673971,” Wikidata, https://www.wikidata.org/w/index.php?title=Q136673971&oldid=2423782576 (accessed November 13, 2025).
21. Google Scholar: ‪John Onimisi Obidi — ‪Google Scholar
22. IJCSRR: International Journal of Current Science Research and Review - Theory of Entropicity (ToE) - John Onimisi Obidi | IJCSRR
23. Cambridge University Open Engage (CoE): Collected Papers on the Theory of Entropicity (ToE)