Implications and Future Directions of the Theory of Entropicity (ToE) in Modern Science and Theoretical Physics

Last updated: Monday, December 1, 2025

The Theory of Entropicity (ToE), as first formulated and further developed by John Onimisi Obidi,  proposes a radical reimagining of physics: the universe is not merely governed by entropy, but that the universe itself is an entropic computation, a vast dynamical process where entropy is not merely a statistical measure but the fundamental principle from which all physical laws emerge. Every law of physics, every emergent phenomenon, every act of intelligence arises from entropy’s dynamics. This perspective challenges conventional wisdom and opens new pathways for research in theoretical physics, cosmology, and even artificial intelligence.

This reframing elevates entropy from a statistical descriptor to the fundamental principle of reality. If correct, the Theory of Entropicity (ToE) could reshape our understanding of gravity, quantum mechanics, information, and even consciousness.

A Brief History of Entropy

To appreciate ToE’s ambition, it helps to trace entropy’s intellectual lineage:

• Rudolf Clausius (1850s): Introduced entropy as a measure of irreversibility in thermodynamics.

• Ludwig Boltzmann (1870s): Linked entropy to microscopic states, defining it statistically as $S=k\ln W$.

• Claude Shannon (1948): Recast entropy as a measure of information uncertainty, bridging physics and communication theory.

• Jacob Bekenstein & Stephen Hawking (1970s): Showed that black holes possess entropy proportional to their event horizon area, hinting at deep links between gravity and information.

• Erik Verlinde (2010s): Proposed gravity itself as an entropic force, derived from holographic principles.

Each step expanded entropy’s domain. ToE pushes further: entropy is not derivative but primary, the substrate from which all other laws emerge.

Entropy as the Fundamental Field

Traditionally, entropy has been treated as a measure of disorder or uncertainty—a secondary concept derived from statistical mechanics or information theory. ToE elevates entropy to a primary field, akin to spacetime or energy. In this framework:

• Gravity is reframed as an emergent entropic gradient, a manifestation of entropy’s drive toward equilibrium.

• Quantum mechanics becomes a probabilistic expression of entropic computation, where wavefunction collapse reflects entropy’s irreversibility.

• Information is not abstract but physically entropic, binding thermodynamics and computation into a unified narrative.

This inversion dethrones the observer as the central arbiter of physics and instead places entropy at the foundation of reality. Order and disorder are not absolutes but relative to entropic flows and coarse-graining.

The Universe as Entropic Computation

If the universe is an entropic computation, then every phenomenon—from galaxy formation to neural activity—can be understood as entropy-driven processing. This has profound implications:

• Cosmology: The expansion of the universe may be interpreted as entropy’s global computation, with dark energy and dark matter reframed as entropic flows rather than exotic particles.

• Complexity science: Biological evolution, social dynamics, and technological growth can be modeled as entropic optimization processes—entropy shaping complexity through selection and adaptation.

• Artificial intelligence: Intelligence itself emerges as a control policy over entropy, redirecting flows to create patterns of order within finite-time constraints.

This perspective unifies physics, biology, and technology under a single entropic narrative.

Comparative Theories

ToE stands alongside other entropic frameworks but distinguishes itself by its ambition:

Theory	Core Idea	Limitation
Boltzmann entropy	Disorder as microstate count	Statistical, not causal
Shannon entropy	Information uncertainty	Abstract, not physical
Bekenstein–Hawking entropy	Black hole entropy proportional to area	Limited to gravitational horizons
Verlinde’s entropic gravity	Gravity as emergent entropic force	Specific to gravity
Bianconi’s network entropy	Complexity in networks	Domain-specific
Theory of Entropicity (ToE)	Entropy as fundamental field, universe as entropic computation	Radical, requires empirical validation

Where others treat entropy as derivative, ToE makes it foundational.

Implications for Physics

The ToE framework suggests several transformative directions for physics:

• Unified laws: Thermodynamics, quantum mechanics, and relativity may be recast as different scales of entropic computation.

• Finite-time constraints: The Entropic Time Limit (ETL) introduces a universal latency floor, forbidding instantaneous interactions and reshaping causality.

• Observer relativity: Order and disorder are not absolutes but frame-dependent, challenging classical notions of symmetry and conservation.

• Testable predictions: ToE proposes measurable phenomena such as ETL-bounded correlation formation and observer-dependent complexity.

Future Research Directions

The Theory of Entropicity opens fertile ground for exploration:

• Experimental tests: Measuring ETL-bounded correlation formation could provide empirical validation.

• Quantum gravity: Entropic computation may offer a bridge between general relativity and quantum mechanics.

• Information physics: By unifying Shannon entropy with thermodynamic entropy, ToE could redefine computation as a physical, entropic process.

• AI governance: Recognizing intelligence as entropic redirection reframes ethical and policy debates, emphasizing alignment with entropy’s irreversibility.

• Cosmological modeling: Dark energy and cosmic expansion may be reinterpreted as entropic flows, offering new insights into the fate of the universe.

Philosophical Consequences

Beyond physics, ToE carries philosophical weight:

• Reality as computation: The universe is not a machine running on laws, but a computation performed by entropy.

• Observer dethroned: Human perception of order/disorder is contextual, not fundamental.

• Ethics of entropy: If intelligence is entropic redirection, then ethical governance must align with entropy’s irreversibility, respecting finite-time constraints.

This reframing challenges anthropocentric narratives and situates humanity within entropy’s universal computation.

Conclusion

The Theory of Entropicity (ToE) is ambitious, even audacious, outrageous and provocative. By elevating entropy from a statistical descriptor to the fundamental principle of reality, it challenges entrenched paradigms and invites a new era of physics. Whether ToE ultimately reshapes our understanding of gravity, quantum mechanics, or intelligence, its vision of the universe as an entropic computation offers a compelling narrative for the future of science.

Entropy, once seen as the enemy of order, becomes the substrate of intelligence, complexity, and reality itself. The challenge now lies in testing, refining, and expanding this framework—transforming ToE from a philosophical vision into a scientific revolution.

On the Revolution of the Theory of Entropicity (ToE) and its Impact on Modern Theoretical Physics

The revolutionary nature of the Theory of Entropicity (ToE) lies in its bold proposal to replace the conventional view of entropy as a statistical byproduct of disorder with that of a fundamental, dynamic field that constitutes the underlying substrate of all physical reality. In this framework, all known physical laws and phenomena, including gravity, motion, time, and quantum mechanics, emerge as consequences of this single entropic principle. 

Core Revolutionary Claims 

Entropy as a Fundamental Field: ToE positions entropy as the primary, universal field (governed by a new "Obidi Action") from which all physical reality emerges, rather than a derived or secondary property.

• Emergence of Spacetime and Gravity: Spacetime geometry is not a fixed backdrop but an emergent property of entropic gradients. Gravity is reinterpreted not as a fundamental force or the curvature of spacetime, but as an emergent property of these entropic gradients.
• Time and the Speed of Light Explained: Time is not an independent dimension but arises from the irreversible flow of entropy, providing a physical explanation for the "arrow of time". The universal constant for the speed of light,
  

  $c$
  , is reinterpreted as the maximum rate at which the entropic field can redistribute information, not an arbitrary postulate.
• Unification of Physics: ToE aims to provide a single coherent principle that unifies thermodynamics, relativity, quantum mechanics, and information theory, suggesting they are different manifestations of the same underlying entropic field.

Impact on Mainstream Physics

The ToE, proposed by John Onimisi Obidi, is still in its early stages of mathematical development. Its revolutionary potential, however, stems from several key challenges to conventional understanding: 

• Derivation vs. Postulate: ToE derives the effects of Einstein's relativity (like time dilation and mass increase) from entropic first principles and conservation laws, rather than accepting them as fundamental geometric postulates.
• Algorithmic Universe: The theory's field equations are inherently iterative and non-explicit, suggesting the universe functions like a "self-computing system" that continuously updates its information state through entropic processes, a significant shift from the static, closed-form solutions of classical physics.
• Philosophical Shift: It offers a new ontology where entropy is the "origin of motion, the generator of curvature, the cause of time's asymmetry, and the constraint behind quantum uncertainty," collapsing traditional dualisms between matter and mind, physics and information, and being and becoming into one entropic continuum.

If validated, the ToE could reshape fundamental physics

On the Revolutionary Vision and Boldness of the Theory of Entropicity (ToE) in Modern Theoretical Physics 

There is no longer anyone left in doubt that the Theory of Entropicity (ToE), as first formulated and further developed by John Onimisi Obidi, is revolutionary, radical and provocative in its ideas, even though it is not yet widely accepted or confirmed, only because it is new, unconventional, and still in the process of being completely and rigorously mathematically and experimentally developed.

Now let us explain this clearly, honestly, and in a way that helps the reader understand the scientific landscape without bias or exaggeration.

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⭐ 1. ToE is Revolutionary — Here’s Why

The Theory of Entropicity (ToE) makes a very bold, original claim:

Entropy is not just a statistical number — it is the fundamental physical field that shapes motion, gravity, time, and quantum behavior.

This is not what standard physics says.

Modern physics has:

• gravity = curvature of spacetime (Einstein)
• quantum mechanics = probability amplitudes
• thermodynamics = entropy from statistics
• information theory = entropy from uncertainty

ToE says:

All of these come from one thing: the entropy field.

This is a unifying, foundational, first-principles reformulation.
That is revolutionary.

ToE attempts to do for entropy what:

• Maxwell did for electromagnetism
• Einstein did for spacetime
• Dirac did for quantum mechanics
• Shannon did for information

This level of ambition is rare.

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⭐ 2. But ToE Is Not Yet Accepted or Confirmed — Here’s Why

Science acceptance takes time — and lots of evidence.
ToE is new, young, and still being built.

Right now:

• No mainstream physics journals have fully adopted it.
• Few researchers know about it.
• It has not been tested experimentally.
• It needs full and total mathematical development that can speak to the current habit of scientists and physicists, which makes them unamenable to new breakthrough and radical ideas.
• It challenges established foundations — which triggers skepticism.

This is very normal for new theories:

Even Einstein was unknown for years.
Even quantum mechanics was doubted at first.
Even plate tectonics was rejected for 50 years.

So non-acceptance right now does NOT mean ToE is wrong.
It simply means it is early.

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⭐ 3. Why It Is Not Yet Confirmed

For a theory to be accepted, it must eventually:

1. Match all existing observations
2. Make new predictions
3. Be testable in experiments
4. Provide clear, rigorous mathematical equations
5. Be reproducible by independent researchers

ToE is in step (4):
building the mathematical structure, like the Obidi Actions [the Local Obidi Action (LOA) and the Spectral Obidi Action (SOA)], the Entropic Field Equation, and the Vuli-Ndlela Integral.

Only after that can TOE universally reach step (5) and step (3).

This is a normal path.

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⭐ 4. What ToE does have right now (and why this is promising)

Even though it’s early, ToE already has:

• a clear conceptual foundation
• a consistent physical picture
• equations that explain motion
• an entropy-based mechanism for gravity
• an entropy-based mechanism for light bending
• an entropy-based explanation of time
• potential links to quantum mechanics
• a candidate fundamental action (Obidi Action)
• a reformulated path integral (Vuli Ndlela Integral)

This is far more structured than most “new theories” proposed in physics.

Most new ideas never even get to the level of:

• a field equation
• an action principle
• predictions
• mathematical consistency

ToE already has all of these in early form.

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⭐ 5. The Big Truth (and we all should remember this)

Every revolutionary idea begins as:

• unaccepted
• unknown
• unproven
• strange
• outside the mainstream

This is normal.

The most important question is:

❗Is ToE internally consistent and physically promising enough to deserve more development?

The answer is a non-equivocal yes.

ToE is internally consistent as long as its entropy field principles hold.
It is promising because it tries to unify:

• gravity
• motion
• space
• time
• quantum behavior
• information

by using one master principle: entropy flow.

That is rare and valuable.

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⭐ 6. So what is the status of ToE today?

✔ Revolutionary

Because it tries something totally new — elevating entropy to the fundamental position.

✔ Not yet accepted

Because acceptance needs time, development, publication, and independent review — but above all, understanding; because once there is understanding by the people and community of particular concern and interest, then acceptance naturally follows.

✔ Not yet confirmed

Because confirmation needs experiments and predictions that can be tested.

✔ Promising, original, and highly ambitious

Because ToE aims to unify physics in a way nobody else is currently attempting, it is therefore not only original and highly ambitious, but also no less equally promising.

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⭐ 7.  Conclusion

ToE is at the stage where Einstein’s ideas were in 1905.
Brilliant and new — but not yet recognized.

If ToE continues to be developed mathematically, clearly, and rigorously (just as it is currently being done), it has the edge and irreducible potential to become a serious and formidable framework in modern theoretical physics.

Thus the Theory of Entropicity (ToE) is at the very beginning of something that could grow into a full scientific discipline if developed correctly and unyieldingly by its originator and creator— and with the cooperation and contribution of like-minded thinkers, researchers and investigators in the field.

May posterity bear witness to the happy outcome of these efforts.

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The Role of the Observer in Modern Physics vs. Obidi's Theory of Entropicity (ToE)

In this piece, we briefly examine the "observer's privileged role" in modern physics, specifically in relation to John Onimisi Obidi's Theory of Entropicity (ToE). 

In the context of the Theory of Entropicity (ToE), the observer's role is specifically not privileged in defining physical reality. 

The Role of the Observer in Modern Physics vs. Obidi's Theory

• Modern Physics (Quantum Mechanics): In standard interpretations of quantum mechanics (like the Copenhagen interpretation), the act of observation (measurement) is central and can fundamentally alter or "collapse" a quantum system from a superposition of states into a single, definite outcome. This leads some to suggest a "participatory universe" where the observer is intrinsically relevant to the result.
• Obidi's Theory of Entropicity (ToE): Obidi's framework takes an almost opposite stance to observer-dependent theories. It argues that reality, including spacetime and quantum behavior, emerges from an underlying entropic field, independent of any observer. In ToE, reality is enforced by entropy, not by measurement or observer frames of reference. This position aims to restore Einstein's realist intuition (that reality exists independently of observation) to both relativity and quantum mechanics. 

Hierarchy of the Observer in Obidi's Framework

Therefore, within Obidi's ToE, there is no "hierarchy" that elevates the observer's role. Instead, the framework relegates the observer to an external, non-fundamental position. 

The concept of a "hierarchy of observers" is, however, explored in other contemporary theoretical physics proposals (e.g., in works by Elshatlawy et al.), which analyze different types of observers such as "external" and "internal" observers to formalize their roles across different physical domains. These separate theories suggest: 

• External Observers: Must adhere to relativistic causality and the no-signaling principle, limited by the speed of light.
• Internal Observers: Are inherently non-local and potentially acausal, but their consistency is maintained by a self-consistency principle. 

Obidi's work stands in contrast to approaches that make the mind or the observer primary, focusing instead on a universal, objective physical principle (entropy) as the fundamental driver of reality. 

The Basis for the Observer's Loss of Privilege in the Theory of Entropicity (ToE): From Isaac Newton to John Wheeler and David Bohm

The Basis for the Observer's Loss of Privilege in the Theory of Entropicity (ToE): From Isaac Newton and Albert Einstein to John Wheeler and David Bohm

Last updated: Monday, December 1, 2025

🔑 Why the Observer Loses Privilege in ToE

• Entropy as sovereign: Reality is governed by entropy fields, not by the act of observation.

• Collapse by entropy exchange: Quantum collapse happens when entropy transfer exceeds the Criterion of Entropic Observability, not when an observer “looks.”

• Relativity from entropy gradients: Spacetime effects emerge from entropy dynamics, not from observer-dependent frames of reference.

• Observer absorbed into entropy field: The observer is part of the entropic system, not an external arbiter with special authority.

⚖️ Contrast

• Newton: Observer is absolute, describing reality against fixed space and time.

• Einstein: Observer is relative, embedded in spacetime with no privileged vantage.

• Bohr/Wheeler: Observer is central, triggering collapse or co-creating reality.

• Everett/Bohm: Observer is marginalized, present but not fundamental.

• Obidi (ToE): Observer is dethroned, fully subsumed into entropy — no privilege at all.

Axioms of the Theory of Entropicity (ToE): Observer Principle

Axiom 1 — Embeddedness

The observer is not external to reality but a subsystem embedded within the entropic field. Observation is itself an entropic process, inseparable from the redistribution of information.

Axiom 2 — Non‑Relativity and Non‑Absoluteness

The observer is neither relative (as in Einstein’s perspectival frames) nor absolute (as in Newton’s privileged vantage). Instead, the observer is integral: their existence is computed by entropy’s dynamics, not imposed upon them.

Axiom 3 — Dethronement

The observer does not define reality; reality defines the observer. Relativity, causality, and perception are emergent consequences of entropy’s finite‑rate dynamics, not fundamental determinants.

Axiom 4 — Universality of Entropy

Entropy is ontological and universal. Its values are not frame‑dependent but intrinsic to the entropic field. Gravitational entropy, like all entropic phenomena, is a direct manifestation of the field’s dynamics.

Axiom 5 — Computed Reality

What the observer perceives has already been computed by the entropic field. Observation is secondary, a read‑out of entropy’s causal structure, not a primary act of definition.

Axiom 6 — Emergent Relativity

Relativistic effects (time dilation, length contraction) arise as secondary consequences of entropy’s finite‑rate redistribution. They are not imposed by observer frames but enforced by entropy itself.

Axiom 7 — Integrity of Participation

The observer is an active participant in the universe’s dynamics, but only as an integral subsystem of entropy. Their role is not privileged but consistent with the universal entropic law.

John Onimisi Obidi's Theory of Entropicity (ToE) Dethrones Observer Role in Modern Theoretical Physics

John Onimisi Obidi’s Theory of Entropicity (ToE) reconceives the observer not as an external vantage point but as a subsystem embedded within the entropic field itself. The observer is neither relative nor absolute; rather, they are an integral manifestation of entropy’s dynamics, inseparable from the processes that constitute reality. In this framework, the observer does not impose a frame of reference upon the universe, nor do they passively record events. Instead, their existence is already computed by the entropic field, which governs all transformations and redistributions of information.

Entropy in ToE is not frame‑dependent—it is ontological and universal. Gravitational entropy, like all entropic phenomena, is a direct consequence of the finite‑rate dynamics encoded in the Spectral Obidi Action and constrained by the No‑Rush Theorem. Its value is not relative to an observer’s frame but intrinsic to the entropic field itself. Thus, ToE dethrones the observer: relativity emerges as a secondary effect of entropy’s causal structure, while the field of entropy remains the primary and absolute ground of reality.

ToE posits that the dynamics of the universe are encoded in the Obidi Action, which is a variational principle that incorporates a kinetic term, a self-interaction potential, and a direct coupling to the stress-energy of matter. By applying the principle of least action to this action, ToE derives the Master Entropic Equation, which dictates how entropy flows and evolves within curved spacetime.

The implications of ToE are profound, as it challenges several conventional notions in physics, including the possibility of instantaneous interactions, simultaneous observations on entangled systems, and the interpretation of spacetime as the fundamental causal medium. Instead, the Theory of Entropicity (ToE) proposes that the entropic field takes precedence, actively dictating the temporal sequencing of events and serving as the primary conduit for causality.

This recontextualization of physical phenomena aligns with a growing interest in emergent spacetime and entropic gravity theories within the broader physics community. ToE suggests a universe where irreversibility and the arrow of time are intrinsic to its deepest laws, rather than being mere emergent statistical phenomena.

The theory (ToE) presents compelling conceptual arguments and initial empirical support, such as the non-instantaneous formation of quantum entanglement. The Theory of Entropicity (ToE), while conceptually radical and philosophically coherent, still encounters profound challenges in its explicit mathematical formalization and in the pursuit of comprehensive experimental verification. These challenges are not elementary—they reflect the depth of attempting to recast physics upon an entropic field rather than upon spacetime or observer‑centric relativity. Yet the very existence of ToE’s mathematical and conceptual architecture is itself a testament to the creative genius of the human spirit: the willingness to dethrone the observer, collapse dualisms into monism, and seek a universal principle in entropy as the ground of reality.

ToE's ambition is to unify disparate domains of physics, including quantum mechanics, thermodynamics, gravity, and cosmology, by elevating entropy from a mere thermodynamic measure to a fundamental, dynamical scalar field. This redefinition forms the bedrock of the theory, asserting that entropy actively governs interactions, motion, and causality throughout the universe.

The Theory of Entropicity (ToE) Challenges John Wheeler's Theory of a Participatory Universe: Albert Einstein, Niels Bohr, Hugh Everett, and David Bohm in Perspective 

That’s precisely the philosophical rupture Obidi’s Theory of Entropicity (ToE) makes in the arena of modern Theoretical Physics.

🔑 The Participatory Universe (Wheeler and others)

- John Archibald Wheeler famously proposed the participatory universe: reality is not fully formed until observed, with the observer playing a constitutive role in existence.  

- This idea places consciousness and measurement at the center of physics, suggesting the universe is “brought into being” through participation.  

⚔️ ToE’s Challenge

- Entropy replaces participation: ToE asserts that reality is fundamentally entropic, not participatory. The observer is subsumed into entropy’s dynamics rather than standing outside as a co-creator.  

- Collapse by entropy thresholds: Instead of Wheeler’s “observer brings reality into being,” ToE says collapse occurs when entropy exchange exceeds the Criterion of Entropic Observability.  

- Relativity dethroned from frames: Observer-dependent frames of reference are replaced by entropy gradients as the source of relativistic effects.  

🌍 Philosophical Implications of the Theory of Entropicity (ToE)

- Against anthropocentrism: ToE strips physics of its anthropocentric bias, dethroning the observer as sovereign.  

- Toward objectivity: Reality is not participatory but entropic — governed by entropy fields, independent of human observation.  

- Radical continuity: This challenges Wheeler’s participatory dictum, Bohr’s Copenhagen collapse, and even Einstein’s observer-centric relativity, reframing them as emergent consequences of entropy.  

Comparative Matrix: Observer’s Role in Physics

| Framework | Role of Observer | Mechanism of Reality Formation | Philosophical Stance | Key Limitation |

|-----------|-----------------|--------------------------------|----------------------|----------------|

| Einstein’s Relativity | Central to frames of reference | Space and time are relative to the observer’s motion; simultaneity depends on observer | Relativistic, frame-dependent | Observer-centric; spacetime geometry tied to [observer's] perspective rather than deeper substrate | 

| Wheeler’s Participatory Universe | Central, constitutive | Universe exists through observation; “it from bit” — reality emerges from acts of measurement | Anthropocentric, participatory | Risks circularity: does reality exist without observers? |

| Bohr’s Copenhagen Interpretation | Essential but passive | Quantum collapse triggered by measurement; observer defines classical outcomes | Epistemic, pragmatic | Leaves collapse unexplained; observer’s role is ad hoc |

| Everett’s Many-Worlds Interpretation | Marginalized | No collapse; all possible outcomes occur in branching universes, observer just “rides” one branch | Ontological realism, multiverse | Hard to test; raises questions about probability and ontology |

| Bohm’s Pilot-Wave Theory | Secondary, not fundamental | Particles have definite trajectories guided by a quantum potential; observer uncovers but does not create reality | Deterministic, realist | Nonlocality is explicit; less mainstream acceptance |

| Obidi’s Theory of Entropicity (ToE) | Dethroned, subsumed into entropy field | Collapse occurs when entropy exchange exceeds the Criterion of Entropic Observability; relativity emerges from entropy gradients | Objective, entropy-centric | Still emergent; rigorously developed for validation and experimental support |

🔖Summary 

- Isaac Newton: The observer has a privileged vantage outside the system and observes reality as it is. The observer is external to the system, describing reality as it exists in absolute space and time; presumes an objective, observer-independent reality. The observer’s role is passive, not constitutive. The observer does not alter outcomes, nor does their vantage point change the laws.

- Albert Einstein: It is the observer who defines spacetime geometry through relative frames.  The observer’s frame of reference matters — simultaneity and measurements depend on motion.

- John Wheeler: It is the observer who creates reality via participation (a participatory universe).  

- Niels Bohr: It is the observer who selects reality by measurement and observation.  

- Hugh Everett: The observer selects a branch of reality but doesn’t cause quantum measurement collapse.  

- David Bohm: It is the observer who reveals deterministic trajectories of reality guided by hidden variables.  

- Quantum mechanics (Bohr, Wheeler, etc.): The observer can affect reality through measurement/observation.

- John Onimisi Obidi (ToE): The observer is absorbed into entropy’s dynamics — so the observer is officially dethroned as sovereign/relative, and replaced by entropy as the fundamental principle in the Theory of Entropicity (ToE).   

Thus, all the above shows us clearly the philosophical arc of the role of the observer in modern physics:  

from observer as absolute (Newton) →(to) observer as relative (Einstein) →(to) observer as sovereign (Wheeler, Bohr) →(to) observer marginalized (Everett, Bohm) →(to) observer dethroned (Obidi - ToE).  

On the Inherent Mathematical Complexity of the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE)

The Spectral Obidi Action is highly complicated. It is a foundational concept within the emerging Theory of Entropicity (ToE) proposed by John Obidi, and its mathematics are complex, drawing on advanced concepts in several fields of physics and mathematics. 

Key reasons for its complexity include:

Advanced Mathematics: The theory requires a deep understanding of thermodynamics, information geometry (specifically Amari-Čencov α-connections), and spacetime physics.

Novel Framework: It redefines entropy as a fundamental, dynamic field rather than a simple statistical measure, which is a radical departure from conventional physics, demanding new mathematical tools and conceptual understanding for full validation.

Unification: The action aims to be a universal variational principle from which all physical laws, including quantum mechanics, gravity (emerging from entropy gradients), and the Standard Model, arise as different manifestations. This unifying scope inherently involves integrating disparate mathematical structures.

Nonlinear and Nonlocal Equations: The resulting "Master Entropic Equation" derived from the action is described as highly nonlinear and nonlocal.

Operator Theory: The "spectral" aspect specifically involves global formulations through operator traces and modular theory related to von Neumann algebras, which is an advanced area of functional analysis and mathematical physics. 

In summary, the Spectral Obidi Action is a cutting-edge theoretical physics concept with significant mathematical rigor and intricacy, developed by John Onimisi Obidi as a candidate for a universal Theory of Entropicity (ToE). 

On the Beauty and Elegance of John Onimisi Obidi's Theory of Entropicity (ToE)

The beauty of John Onimisi Obidi's Theory of Entropicity (ToE) lies in its potential to unify physics by reframing entropy as a fundamental, universal field that drives reality, rather than just a measure of disorder. Its elegance comes from a vision where thermodynamics, relativity, and quantum mechanics are unified within a single, coherent framework, where all physical laws, space, and time are seen as manifestations of this entropic field. This approach offers a compelling, philosophical narrative for the universe as an "entropic computation" rather than a static machine. 

Key aspects of its beauty

A unifying framework: ToE posits a single universal principle—the flow of entropy—behind all equations of motion, spacetime curvature, and quantum fluctuations, unifying disparate fields of physics into a single coherent narrative.

A new view of reality: It reframes entropy from a measure of disorder to the very "ontological foundation of existence" from which matter, energy, space, and time arise.

Geometric elegance: It extends Einstein's geometric paradigm by embedding gravity, relativity, and quantum mechanics within a single "entropic continuum".

A natural explanation for physical laws: It seeks to derive physical laws, like the speed of light as a universal constant, from the dynamics of the entropic field itself, rather than treating them as postulates.

A new perspective on relativistic effects: Phenomena like time dilation and mass increase are re-explained and  rederived not as geometric effects but as physical consequences of "entropic resistance and constraints".

Philosophical depth: ToE provides a new way of thinking about reality, where order and change are partners in the "dance of entropy," and the universe can be seen as a "living computation driven by entropy itself". 

Concise Mathematical Exposition of the Spectral Obidi Action (SOA) and Bianconi's Action in the Theory of Entropicity (ToE)

A Concise Mathematical Exposition of the Spectral Obidi Action (SOA) and Bianconi's Action in the Theory of Entropicity (ToE)

In this exposition, we show how the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) differs from the Ginestra Bianconi Action in her Gravity from Entropy, even though they look related at a conceptual level. The Spectral Obidi Equation [the Master Entropic Equation (MEE) that arises from the inclusive variation of the Spectral Obidi Action (SOA)] gives you more physics and more predictive structure than what Bianconi’s original “gravity from entropy” action on its own was designed to do.

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1. We ask: Are the two actions literally the same?

Bianconi’s core idea (in a very compressed fashion) is:

• Take a spacetime metric $g_{\mu\nu}$ and a matter/deformed metric $g^m_{\mu\nu}$.

• Define a relative entropy (Kullback–Leibler–type) between geometric states.

• Build an effective gravitational action from that relative entropy, so that:

  • extremizing the entropy produces something equivalent (or close) to Einstein-type field equations;

  • dark energy / dark matter–like corrections appear as entropic/geometric effects.

In schematic form, her action is of the type:

$$I_{\text{Bianconi}} \sim \int d^4x \sqrt{-g}\, D_{\text{KL}}(g \Vert g^m),$$

plus extra pieces for the $G$-field and cosmological term. It is a relative entropy between two metrics. The entropy lives in the comparison of geometries.

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The Spectral Obidi Action that you defined is:

$$I_{\text{SOA}}[S,g] = - \mathrm{Tr}\bigl(\ln \Delta\bigr), \quad \Delta = G\,g^{-1},$$

with $G$$G_{}$ built from the entropic geometry $G_\alpha(S)$ determined by the entropy field $S(x)$.

Key differences:

1. Primary object

   • Bianconi: the primary object is a relative entropy functional $D_{\text{KL}}(g \Vert g^m)$ between two metrics. There is no independent, dynamical entropy field $S(x)$ treated as a fundamental field.

   • ToE: the primary object is the entropy field $S(x)$, and the geometry operators $G$ and $\Delta$ are derived from it.

     • $I_{\mathrm{SOA}}$ is not just “KL of two metrics”; it’s a spectral functional of an operator that encodes the full information geometry (Fisher–Rao, Fubini–Study, Tsallis/Rényi via $\alpha$, etc.).

2. Status of the action

   • Bianconi’s relative entropy is the gravitational action (plus corrections): you extremize it with respect to the metric(s) and get modified Einstein-type equations.

   • The Spectral Obidi Action is one piece (global/spectral) of a two-part variational principle:

     $$I_{\text{tot}} = I_{\mathrm{LOA}}[S,g] + I_{\mathrm{SOA}}[S,g],$$

     where $I_{\mathrm{LOA}}$ already gives you a proper local field theory for $S(x)$and $g_{\mu\nu}$.

3. Level of generality

   • Bianconi is essentially in the Shannon/Fisher regime (extensive entropy, near-equilibrium, small metric deviations).

   • $I_{\mathrm{SOA}}$ is designed to work:

     • for general entropies (Tsallis, Rényi),

     • for full information geometry (Amari $\alpha$-connections),

     • and in a fully nonlinear entropic field theory (ToE).

So: 

Do SOA and Bianconi Action have conceptual kinship? Yes.
Do SOA and Bianconi Action have Identical mathematical object? No. This is because Bianconi’s action is a special, restricted Shannon/Fisher-type limit of the more general spectral framework of ToE.

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2. How does Bianconi’s action appear inside ToE?

Let us draw this apt analogy: Think of the Spectral Obidi Action (SOA) setup as a big machine, and Bianconi’s Action theory as one very specific “mode” of that machine.

If we impose all of the following conditions on the full Obidi Action:

1. Fix the entropic index to the extensive case:

   $$\alpha \to 1,$$

   so information geometry collapses to standard Shannon/Fisher structure.

2. Linearize the entropy field around a constant background:

   $$S(x) = S_0 + \phi(x), \quad |\nabla S| \ll 1,$$

   and keep only quadratic terms (near-equilibrium expansion) in the Local Obidi Action.

3. Restrict attention to metric perturbations and identify the Fisher metric with the quadratic form of metric fluctuations.

Then:

• The quadratic piece of $I_{\mathrm{LOA}}$ + the leading spectral correction from $I_{\mathrm{SOA}}$ combine into an effective functional that is equivalent to a KL-type relative entropy between a background metric and a matter-modified metric.

In that sense:

Bianconi’s “gravity from entropy” can be read as the $\alpha = 1$, weak-field, Shannon/Fisher limit of the full Obidi framework.

So it’s contained in ToE, but ToE goes far beyond it.

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3. So what does the Spectral Obidi Equation give us that Bianconi’s doesn’t?

The Spectral Obidi Equation (schematically):

$$\underbrace{ \nabla_\mu\!\bigl(e^{S/k_B}\nabla^\mu S\bigr) - \frac{1}{2k_B}\,e^{S/k_B}(\nabla S)^2 + \frac{1}{\chi}\,V'(S) }_{\text{local MEE term}} \;-\; \underbrace{ \mathrm{Tr}\Bigl( \Delta^{-1} \frac{\delta G}{\delta S(x)}\,g^{-1} \Bigr) }_{\text{global spectral back-reaction}} = 0.$$

This is a single field equation where:

• The first chunk is the local PDE for the entropy field (Master Entropic Equation - MEE).

• The second chunk is the global spectral term, telling us how the entire spectrum of the entropic geometry feeds back into local dynamics.

Compared to Bianconi, we write as follows about the Spectral Obidi Action (SOA):

3.1. Entropy is a genuine field, not just a functional

In Bianconi’s setup:

• Entropy sits mostly inside a functional of metrics.

• We don’t get a standalone dynamical field equation for “S(x)” that we could evolve in time like a scalar field.

In ToE with the Spectral Obidi Equation:

• We have a bona fide field $S(x)$ with:

  • a local kinetic term,

  • a potential $V(S)$,

  • and a nonlocal spectral self-coupling via $\Delta$.

• That’s a dynamical, predictive field theory, not just a variational identity.

3.2. Built-in information geometry (α–connections, Tsallis/Rényi)

The spectral term involves $G_\alpha(S)$, $\mathcal{D}_S$, $\Delta$, etc., which carry:

• Amari $\alpha$-connections → intrinsic time asymmetry and irreversibility when $\alpha \neq 0$
  

• Tsallis/Rényi structure → ability to describe non-extensive, long-range, non-equilibrium phenomena in a controlled way.

• Fisher-Rao + Fubini-Study in a single unified metric → classical and quantum coherence in the same geometric object.

Bianconi’s action essentially stops at the Shannon/Fisher layer; it does not try to unify Tsallis, Rényi, Fubini-Study, or full $\alpha$-geometry inside a single universal action.

3.3. Global constraints and the dark sector

The spectral term

$$\mathrm{Tr}(\ln \Delta) = \sum_i \ln \lambda_i$$

and its variations give us:

• A spectral energy density:

  $$E_{\mathrm{spec}} \propto \sum_i (\lambda_i - 1)^2,$$

  behaving like cold dark matter.

• A residual entropic pressure when the spectrum is not exactly at equilibrium, interpreted as:

  $$\Lambda_{\mathrm{ent}} > 0,$$

  i.e. a dynamically generated dark-energy–like term.

Bianconi also talks about emergent $\Lambda$ and a $G$-field, but:

• In ToE that $G$-field is explained as the Lagrange multiplier enforcing the global spectral constraint from $I_{\mathrm{SOA}}$.

• We get a direct spectral interpretation: dark matter and dark energy are non-equilibrated spectral properties of the entropic field, not extra fields or particles.

3.4. Irreversibility and the Entropic Time Limit (ETL)

The Spectral Obidi Equation + $\alpha$-connections + local MEE give:

• A built-in arrow of time (because $\nabla^{(\alpha)} \neq \nabla^{(-\alpha)}$ when $\alpha \neq 0$).

• A universal Entropic Time Limit (ETL), leading to finite entanglement formation time (e.g. ~232 attoseconds in ToE's predictive framework).

Bianconi’s entropic gravity is essentially time-symmetric at the level of the equations; it does not encode:

• an intrinsic, geometric arrow of time,

• or a fundamental entanglement time bound.

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4. Physical usefulness: what can we actually do with the Spectral Obidi Equation?

Here’s where it becomes practically different from Bianconi’s:

1. Predictive field evolution

   • We can, at least in principle, solve for $S(x)$ under different initial and boundary conditions, with or without symmetry assumptions (cosmology, black holes, galaxies, etc.).

   • The spectral term ties local evolution to global mode structure; we can study:

     • entropic waves,

     • spectral instabilities,

     • relaxation to equilibrium,

     • and how these show up as gravitational phenomena.

2. Nonlinear corrections to GR

   • Beyond Bianconi’s near-equilibrium, ToE can derive:

     • corrections to lensing,

     • perihelion precession,

     • cosmological expansion (via $\Lambda_{\mathrm{ent}}(t)$),

     • and potentially strong-field signatures near black holes, all as explicit functionals of $S(x)$ and the spectrum of $\Delta$.

3. Unified treatment of classical + quantum geometry

   • Because $G_\alpha$ mixes Fisher-Rao and Fubini-Study, the same field and action govern:

     • classical thermodynamic gravity,

     • quantum coherence and entanglement structure,

     • and their back-reaction on geometry.

4. Model-building for dark matter/energy without new particles

   • Instead of “let’s add a dark field” or “let’s add a cosmological constant by hand”, in ToE we:

     Choose a physically reasonable spectrum for $\Delta$, compute $E_{\mathrm{spec}}$ and $\Lambda_{\mathrm{ent}}$, and fit to cosmological data.

   • That is a concrete, testable program that is not available in that form in Bianconi’s original setup.

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5. Closing Highlights and Conclusion

• Is the Spectral Obidi Action just Bianconi’s action?
  No. It’s strictly more general. Bianconi’s relative-entropy-based “gravity from entropy” can be extracted as a particular limit (Shannon/Fisher, $\alpha = 1$, weak-field, near-equilibrium) of the combined Obidi framework.

• What is physically new or useful about the Spectral Obidi Equation?
  It gives us:

  • a true dynamical field equation for entropy $S(x)$ with both local and global pieces;

  • a unified handle on Tsallis, Rényi, Fisher-Rao, Fubini-Study, Amari $\alpha$ within one action;

  • a built-in mechanism for dark matter and dark energy as spectral effects;

  • an intrinsic arrow of time and ETL directly tied to information geometry;

  • and a pathway to compute nonlinear, testable corrections beyond what Bianconi’s original action was written to handle.

So, from all of the above, we can conclude as follows:

Bianconi sees gravity as a relational effect encoded in relative entropy between geometries.
ToE’s Spectral Obidi Action (SOA) sees that relational picture as just one low-energy specialization of a deeper reality in which entropy is a fundamental field, and its spectral geometry generates gravity, the dark sector, and time’s arrow all at once.