Obidi’s Revolutionary Vision for Modern Theoretical Physics in His Audacious Theory of Entropicity (ToE)

General Introduction

The Theory of Entropicity (ToE) reconceptualizes entropy not as a secondary statistical measure of disorder, but as the fundamental physical field from which all structure, dynamics, and causation arise. In this framework, entropy is ontic rather than epistemic: it is not a descriptor of ignorance, but the generative substrate of reality itself. Matter, geometry, temporal flow, and interaction are understood as emergent manifestations of the behavior of this entropic field.

At the core of ToE lies the Obidi Action, a universal variational principle that governs the dynamics of entropy directly. Unlike conventional actions, which treat entropy as a derived quantity or impose thermodynamic constraints after the fact, the Obidi Action elevates entropy to the status of the primary dynamical variable. Physical laws emerge as extremal conditions on entropic evolution, rather than as independent postulates imposed on spacetime or matter fields.

ToE is grounded in a rigorous information-geometric structure. By unifying the Fisher–Rao metric of classical information geometry with the Fubini–Study metric of quantum state space, and embedding both within the Amari–Čencov α-connection formalism, the theory provides a continuous geometric bridge between classical, quantum, and gravitational regimes. This integration allows entropy to function simultaneously as a measure of informational divergence, a generator of curvature, and a driver of irreversible dynamics.

Within this framework, spacetime geometry is not fundamental but emerges from gradients and flows of the entropic field. Curvature, inertia, and gravitational attraction arise as entropic responses to informational imbalance, while time itself is identified with the irreversible sequencing of entropic updates. The familiar equations of physics—Einstein’s field equations, Yang–Mills dynamics, scalar field evolution, and fermionic motion—are recovered as effective limits of Obidi's entropic variational principle, rather than as independently assumed laws.

Crucially, ToE transforms spectral and operator-based formulations from optional mathematical reformulations into foundational physical principles. Both bosonic and fermionic dynamics are unified within a single entropic–spectral framework, eliminating the traditional asymmetry between local field actions and operator-based quantum descriptions.

In this way, the Theory of Entropicity (ToE) offers a coherent and unified foundation for physics, in which entropy is not merely conserved or maximized, but actively governs the emergence, evolution, and consistency of the physical universe itself.

Author's Preamble

The Theory of Entropicity (ToE), conceived by John Onimisi Obidi, represents one of the most daring conceptual restructurings of modern physics. It is not merely an extension of existing frameworks, nor a reinterpretation of known principles. Instead, it proposes a radical inversion of the traditional hierarchy of physical concepts: entropy is not a derived quantity, but the fundamental field from which all physical structures and laws emerge. This shift demands a new mathematical architecture, one that draws from the deepest regions of quantum field theory, operator algebras, spectral analysis, and variational principles. As the theory evolves, its mathematics naturally grows in complexity, reflecting the profound ambition of its vision.

This paper presents a comprehensive exposition of Obidi’s conceptual and mathematical innovations, clarifying how the Theory of Entropicity reimagines the role of Araki Relative Entropy, how it constructs a new entropic action principle, and how it positions itself relative to other entropy‑based approaches in contemporary physics.

The Conceptual Core: Entropy as the Substrate of Reality

At the heart of Obidi’s theory lies a simple but transformative idea: entropy is not a secondary descriptor of physical systems, nor a measure of ignorance, nor a thermodynamic bookkeeping device. Instead, entropy is the primary field of the universe. It is the ontological substrate from which geometry, matter, quantum behavior, and thermodynamic phenomena arise.

This conceptual inversion mirrors the historical pattern of major theoretical revolutions. Just as Einstein elevated spacetime geometry from a passive stage to an active participant in gravitational dynamics, Obidi elevates entropy from a statistical summary to the generative engine of physical law. In this view, the universe is not governed by geometric or energetic principles that later give rise to entropy; rather, the universe evolves by following the dynamics of an entropic field, and geometry and energy are emergent expressions of that evolution.

Araki Relative Entropy: From Diagnostic Tool to Generative Principle

Araki Relative Entropy is one of the most sophisticated constructs in quantum field theory and quantum information theory. Traditionally, it serves as a measure of distinguishability between quantum states, defined in the context of von Neumann algebras and modular theory. It is a scalar functional, not an operator, and is used to study entanglement, energy conditions, and information flow in quantum systems.

Obidi’s innovation is to promote this quantity from a static measure to a dynamic generator. In the Theory of Entropicity, Araki Relative Entropy is not merely a tool applied to a pre‑existing physical theory; it becomes a foundational ingredient of the theory itself. The modular structures underlying Araki’s definition—particularly the relative modular operator and its spectral properties—are reinterpreted as the mathematical backbone of an entropic field. The entropy between states associated with local regions of spacetime becomes the local expression of this field, and the evolution of the field is governed by an action principle built from these entropic quantities.

This transformation is unprecedented. In standard quantum field theory, Araki Relative Entropy never generates dynamics, never curves spacetime, and never unifies physical domains. In ToE, it does all three.

The Obidi Action: A New Entropic Variational Principle

The introduction of the Obidi Action marks the structural heart of the Theory of Entropicity (ToE). In classical and quantum physics, actions are typically geometric or energetic in nature. They encode the dynamics of fields, particles, and spacetime through variational principles. Obidi’s action departs from this tradition by making entropy—not geometry, not energy—the primary variational quantity.

The Obidi Action is constructed from entropic functionals derived from Araki-type structures. It is designed to be local, meaning that each region of the entropic manifold contributes a finite entropic density. This requires a careful assignment of local algebras, local states, and local entropic contributions, ensuring that the action remains well-defined and variationally differentiable. When the action is extremized, it yields the Master Entropic Equation (MEE), a new field equation that replaces Einstein’s equations as the generator of spacetime dynamics.

The MEE is not a modification of general relativity, nor a correction to quantum field theory. It is a fundamentally new equation, entropic in origin and universal in scope. Through it, relativistic effects such as time dilation and mass increase emerge from the behavior of the entropic field, rather than from geometric postulates. Quantum correlations, entanglement, and thermodynamic irreversibility likewise become expressions of the same underlying entropic dynamics.

Mathematical Legitimacy: Can Araki Entropy Be Combined with a Local Action?

A natural question arises: is it mathematically legitimate to combine Araki Relative Entropy—a scalar functional defined on pairs of states—with a local action principle? The answer is yes, provided the construction is handled with the appropriate operator‑algebraic rigor.

Araki Relative Entropy itself is a number, but it is built from modular operators that possess rich spectral structures. These operators can be localized by associating them with local algebras of observables, allowing the entropic functional to be decomposed into contributions from local regions. The resulting entropic density can then be integrated or summed to form a local action. For this to be mathematically sound, the theory must specify the class of algebras, the nature of the states, the locality structure, and the variational framework. When these elements are clearly defined, the combination becomes not only legitimate but natural.

This is analogous to how Einstein transformed the Ricci scalar—a geometric quantity—into the Einstein-Hilbert action. The Ricci scalar was not originally an action density; it became one through conceptual innovation and mathematical precision. Obidi’s use of Araki entropy follows the same pattern.

Comparison with Other Entropy-Based Approaches in Physics

The Theory of Entropicity (ToE) stands in contrast to several influential entropy‑related frameworks in modern physics, including holography, Jacobson’s thermodynamic gravity, and Verlinde’s entropic gravity. While these approaches use entropy in profound ways, none of them elevate entropy to the status of a fundamental field.

In holography, entropy—particularly entanglement entropy—plays a central role in reconstructing bulk geometry from boundary quantum states. However, the underlying theory remains a quantum field theory or string theory defined on a boundary. Entropy is a diagnostic of deeper structures, not the substrate of reality.

Jacobson’s thermodynamic gravity derives Einstein’s equations from thermodynamic relations applied to local horizons. This is a reinterpretation of general relativity, not a replacement. Entropy is tied to coarse-graining and horizon structure, not to a universal entropic field.

Verlinde’s entropic gravity proposes that gravity is an emergent entropic force, but the underlying microscopic theory is unspecified, and no entropic field equation is provided. The approach is phenomenological rather than foundational.

Ginestra Bianconi’s use of Relative Entropy in her momentous paper on Gravity from Entropy 

A more technically accurate comparison emerges when examining Ginestra Bianconi’s “Gravity from Entropy,” where she employs relative entropy as a geometric comparator between two metrics: the metric induced by matter distributions and the background spacetime metric. In her formulation, entropy is not merely a count of microstates but a measure of deviation between the geometry generated by matter and the reference geometry of the underlying manifold. Bianconi constructs an entropic functional based on the relative entropy between probability distributions associated with these two metrics, effectively quantifying how much the presence of matter “distorts” the background geometry. This distortion, expressed through relative entropy, becomes the driver of gravitational dynamics. In her framework, gravity emerges because the system evolves toward configurations that minimize the relative entropy between the matter‑induced metric and the background metric, thereby aligning geometric structure with entropic optimality. Her approach thus treats spacetime geometry as a statistical equilibrium configuration determined by the entropic comparison of two metric structures.

While Bianconi’s work shares with Obidi’s Theory of Entropicity (ToE) the conviction that entropy can generate gravitational behavior, the two theories diverge sharply in their foundational commitments. Bianconi’s model remains rooted in metric comparison and statistical equilibrium, where relative entropy quantifies the mismatch between matter geometry and background geometry. The gravitational field is emergent from this entropic reconciliation. Obidi’s ToE, by contrast, does not rely on comparing metrics, nor does it assume a background geometry at all. Instead, it posits entropy itself as the fundamental field, with Araki-type entropic functionals forming the variational core of the Obidi Action. In Bianconi’s picture, gravity emerges from the entropic alignment of two geometric structures; in Obidi’s, gravity is one manifestation of a universal entropic field whose dynamics are primary, continuous, and not reducible to metric comparison. This distinction underscores the conceptual boldness of ToE: it does not derive gravity from entropy acting on geometry, but derives geometry itself from the dynamics of entropy. Obidi seeks to derive all physical law from the dynamics of his singular postulate of entropy itself.

Thus, Obidi’s ToE differs fundamentally. It does not treat entropy as a property of something else, nor as a tool for reconstructing geometry, nor as a thermodynamic by-product. Instead, entropy is the primary field, the action is entropic, and the field equations are entropic. Geometry, quantum behavior, and thermodynamics all emerge from the same entropic substrate.

The Growing Mathematical Complexity: A Sign of Theoretical Maturity

As the Theory of Entropicity (ToE) develops, its mathematics naturally becomes more intricate. This is not a flaw but a hallmark of genuine theoretical progress. Every major physical theory began with a simple conceptual insight that later required sophisticated mathematics to express rigorously. Newton needed calculus, Einstein needed differential geometry, quantum mechanics needed operator theory, and quantum field theory needed renormalization and algebraic structures.

Obidi’s theory is following the same trajectory. The conceptual core remains elegant and simple, but the mathematical scaffolding required to express it—modular theory, spectral analysis, entropic manifolds, and variational calculus—reflects the depth of the idea. Complexity is the price of rigor, not a sign of conceptual failure.

Conclusion: A Bold New Architecture for Fundamental Physics

Obidi’s Theory of Entropicity (ToE) is an audacious attempt to reframe the foundations of physics. By elevating entropy to the status of a fundamental field and constructing a new entropic action principle grounded in Araki Relative Entropy, the theory proposes a unified framework in which quantum mechanics, gravity, and thermodynamics emerge from a single entropic substrate. Its mathematical demands are substantial, but they are the natural consequence of its ambition.

Whether ToE ultimately becomes a new pillar of physics will depend on the continued development of its mathematical foundations and its ability to make contact with empirical phenomena. But as a vision, Obidi's Theory of Entropicity (ToE) stands as one of the most original and conceptually daring proposals in modern theoretical physics.

References 

1. https://theoryofentropicity.blogspot.com/2026/01/what-are-applications-and-differences.html

2. https://theoryofentropicity.blogspot.com/2026/01/on-john-onimisi-obidis-ingenious.html

Theory of Entropicity (ToE) — The Yuletide Papers 

1. Obidi, John Onimisi. Collected Works on the Evolution of the Foundations of the Theory of Entropicity(ToE): Establishing Entropy as the Fundamental Field that Underlies and Governs All Observations, Measurements, and Interactions - Volume I: The Conceptual and Philosophical Expositions (Version 1.0). (December 31, 2025). The Yuletide Volume. Available at SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5996415
2. 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
3. Obidi, John Onimisi (29th December, 2025). The Theory of Entropicity (ToE) Sheds Light on String Theory, Quantum Field Theory, and the Casimir Effect: Strings and Branes are Vibrations of Information [Geometry] in the Entropic Field of ToE. Figshare. https://doi.org/10.6084/m9.figshare.30968344
4. Obidi, John Onimisi (28th December, 2025). 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). Figshare. https://doi.org/10.6084/m9.figshare.30964483
5. Obidi, John Onimisi (28th December, 2025). Further Expositions on the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: How the Theory of Entropicity (ToE) Unifies Spectral and Araki Entropies with Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov Formalisms. Figshare. https://doi.org/10.6084/m9.figshare.30959819
6. 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