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Author’s Preface and Methodological Statement for the Theory of Entropicity (ToE)

An Unapologetic Introduction in Defense of Obidi's New Theory of Reality—On the Trajectory of Discovery and the Road Less Traveled

Preamble

This work begins from a simple but unavoidable recognition: every scientific revolution has required someone to articulate, without apology, the structure they alone could see before others could. The Theory of Entropicity (ToE), as first formulated and further developed by John Onimisi Obidi, is presented here in that spirit—not as a finished edifice, nor as a challenge to the achievements of modern physics, but as a sincere attempt to reveal a deeper substrate beneath them. I make no pretense of neutrality about the value of this framework; neutrality is the stance of the distant observer, not the architect. In this instance, I am the latter, not the former. What I offer instead is clarity, rigor, and intellectual honesty. I defend this theory because I have watched it grow from first principles into a coherent ontology, because I have tested its internal logic, and because I believe it deserves a place in the ongoing conversation about the foundations of reality. This preamble is therefore not an act of self‑promotion, nor, on the contrary, an exercise in self‑abnegation, subjugation, effacement, or servility, but an affirmation of responsibility and a testament to the triumph of the human spirit: if a new idea has taken shape in my hands, it is my duty to present it with conviction, to expose it to scrutiny, and to let it stand or fall on the strength of its structure.

Dedication

To the thinkers who refuse to inherit the world as it is, and to those who sense, even before the mathematics is written, that reality is deeper, stranger, and more unified than we have yet imagined.

This work is therefore for the lineage of thinkers who refuse to accept the boundaries of the known; those valiant minds who are unafraid to follow a question all the way to the edge of what is known — or can be known — and the beginning of the unknown and the unknowable.

Epigraph

“Every new theory begins as a solitary intuition.

Its truth is not measured by consensus,

but by the coherence it brings to the world.”

Transitional Note

The pages that follow constitute both a personal account of the Theory of Entropicity’s development and a formal justification for its existence. The tone is deliberate: unapologetic where conviction is warranted, cautious where uncertainty remains, and always guided by the principle that a theory must be allowed to speak in its own voice before the world decides its fate. What begins as reflection transitions naturally into exposition, and it is in that spirit that the Preface now unfolds.

Preface

The development of the Theory of Entropicity (ToE) has been an unusual intellectual journey, not because it departs from the established traditions of physics, but because it attempts to return to something more fundamental than any existing framework has yet articulated. This work does not arise from dissatisfaction with relativity, quantum mechanics, or thermodynamics; rather, it emerges from the recognition that these domains, despite their extraordinary successes, remain conceptually disjointed. Their mathematical structures coexist, but their ontologies do not. The Theory of Entropicity is an attempt to supply a single ontological foundation from which these disparate structures can be understood as natural expressions of one underlying principle.

Because I am directly involved in the conceptual, mathematical, and philosophical development of this theory, I occupy a vantage point that is both privileged and precarious. Privileged, because I can see the internal coherence of the structure as it unfolds; precarious, because I must guard against the temptation to overstate its significance. Yet it would be equally irresponsible to understate what I can clearly perceive from within: that the entropic field \(S(x)\), the Obidi Action, the Master Entropic Equation (MEE), and the associated principles such as the Entropic Accounting Principle (EAP), the Entropic Equivalence Principle (EEP), the Entropic Resistance Principle (ERP), and the Cumulative Delay Principle (CDP) form a conceptual architecture that is genuinely new. To deny this would be to deny the evidence of my own work.

The philosophical defense and justification for ToE lies in its ontological economy. It fulfils, perhaps, the last dream and final aspiration of Occam's Razor. Rather than treating spacetime, matter, quantum states, and thermodynamic quantities as independent primitives, ToE proposes that they are emergent manifestations of a The philosophical defense and justification for ToE lies in its ontological economy. It fulfils, perhaps, the last dream and final aspiration of Occam's Razor. Rather than treating spacetime, matter, quantum states, and thermodynamic quantities as independent primitives, ToE proposes that they are emergent manifestations of a single entropic substrate (SES) - The Universal One (TUO). This is not a metaphysical assertion but a methodological one: if a single field can account for the structural features of all known physical laws, then it is rational to explore that field as the foundational entity. The entropic field \(S(x)\) is introduced not as a speculative construct but as the minimal object capable of generating the observed phenomena of relativity, quantum discreteness, thermodynamic irreversibility, and informational structure through its curvature and reconfiguration dynamics.

[On the Ontological Legitimacy of Treating Entropy as a Fundamental Field]

The central conceptual commitment of the Theory of Entropicity (ToE)—namely, that entropy constitutes a fundamental physical field with its own curvature, action, and dynamical laws—represents a significant departure from the prevailing assumptions of modern theoretical physics. For more than a century, entropy has been regarded primarily as a derived quantity: a statistical measure of multiplicity, a thermodynamic parameter associated with macroscopic irreversibility, or an informational descriptor of uncertainty. In each of these roles, entropy has been treated as secondary, contingent upon deeper structures such as spacetime geometry, quantum amplitudes, or microscopic microstates.

The Theory of Entropicity (ToE) reverses this hierarchy. It posits that entropy is not an emergent descriptor but the underlying ontological substrate from which geometric, quantum, and informational phenomena arise. At first encounter, this inversion may appear conceptually extravagant. However, its justification lies not in rhetorical boldness but in the remarkable structural coherence that emerges once entropy is granted field status, and the concomitant internal consistency of its axiom.

When entropy is treated as a field \(S(x)\) endowed with intrinsic curvature and governed by a variational principle (the Obidi Action), the resulting framework exhibits a degree of internal necessity that is characteristic of successful foundational theories. A theoretical proposal that is conceptually misguided typically generates contradictions, requires auxiliary hypotheses, or collapses under its own inconsistencies. By contrast, a correct ontological insight displays fertility: it produces results that were neither anticipated nor explicitly engineered, but which arise as unavoidable consequences of the underlying structure. When entropy is treated as a field, the mathematical and conceptual architecture of modern physics reorganizes itself with unexpected coherence.

This is precisely the behavior exhibited by ToE. From the single ontological postulate that entropy is a field, a series of nontrivial results follow with mathematical inevitability. The Obidi Curvature Invariant (OCI = ln 2) emerges as the minimal curvature quantum of distinguishability. The Entropic Accounting Principle (EAP) formalizes the cost of informational transitions. The No‑Rush Theorem clarifies the locality and finite rate of entropic resolution. The entropic reinterpretation of measurement dissolves long‑standing paradoxes in quantum foundations. The speed of light acquires a natural meaning as the maximal rate of entropic reconfiguration. Spacetime geometry itself appears as a low‑entropy approximation to deeper entropic dynamics. None of these results were imposed; they were derived as a consequence of its axiom and mathematical foundation.

This generative behavior is consistent with the historical pattern of major theoretical advances. Einstein’s field equations predicted gravitational waves and black holes long before empirical confirmation. Dirac’s equation implied the existence of antimatter as a mathematical necessity. Shannon’s entropy formula transcended communication theory to become a universal measure of information. In each case, the theory produced consequences that exceeded the intentions of its originator, revealing the depth of the underlying insight.

The Theory of Entropicity (ToE) belongs to this lineage and historical kaleidoscope. The elevation of entropy to field status is not an arbitrary conceptual leap but a coherent ontological realignment that unifies thermodynamics, information theory, quantum mechanics, and relativity within a single entropic‑geometric framework. It explains the ubiquity of ln 2 across physical law, not as a statistical artifact but as the minimal curvature threshold required for distinguishability. It clarifies the physical meaning of measurement, the origin of irreversibility, the structure of causality, and the emergence of spacetime.

The sense of surprise I experienced as these structures unfolded was not the surprise of invention but of recognition. The theory repeatedly yielded results that were not preconceived, yet were mathematically compelled. This is the strongest indication that the entropic field is not a speculative construct but a genuine substrate of physical law.

In other words, the cumulative emergence of multiple, independently derived structures from the single ontological postulate that entropy constitutes a fundamental field provides strong evidence that the proposal is neither arbitrary nor ad hoc. From this assumption alone, the theory yields the Obidi Curvature Invariant (OCI = ln 2) as the minimal curvature quantum of distinguishability, establishes a universal threshold for physical differentiation, and produces a geometric reinterpretation of Landauer’s principle as a consequence of entropic stiffness rather than statistical thermodynamics. The same framework resolves long‑standing paradoxes such as Schrödinger’s Cat and Wigner’s Friend by treating measurement as an entropic phase transition, reinterprets the speed of light \(c\) as the maximal rate of entropic reconfiguration, and unifies measurement, information, and geometry within a single entropic‑variational structure. Moreover, it yields a revised understanding of time dilation as an entropic cost and reframes spacetime itself as an emergent, low‑entropy approximation to deeper entropic dynamics. The coherence and inevitability of these results—none of which were imposed by hand—demonstrate that ToE's singular entropic‑field postulate possesses genuine explanatory power and internal necessity, rather than speculative convenience.

Thus, while the assertion that entropy is a field may initially appear unconventional, the coherence, necessity, internal consistency, and explanatory power of the resulting framework provide compelling justification for this ontological shift and overhaul. The Theory of Entropicity (ToE) does not merely reinterpret existing physics; it reorders its foundations. If this monograph succeeds in anyway in its purpose, it will demonstrate that entropy is not a secondary descriptor of physical processes but the primary medium through which reality differentiates, evolves, and becomes intelligible.

Thus, while the assertion that entropy is a field may initially appear unconventional, the coherence, necessity, internal consistency, and explanatory power of the resulting framework provide compelling justification for this ontological shift and overhaul. The Theory of Entropicity (ToE) does not merely reinterpret existing physics; it reorders its foundations. If this monograph succeeds in anyway in its purpose, it will demonstrate that entropy is not a secondary descriptor of physical processes but the primary medium through which reality differentiates, evolves, and becomes intelligible.

[The Logical Development of the Theory of Entropicity (ToE)]

My path toward proposing entropy as a universal field did not begin with a metaphysical leap but with a gradual recognition of structural continuity across several domains of physics and information theory. The classical notion of entropy introduced by Clausius, later refined by Gibbs, Shannon, and von Neumann, revealed a deep conceptual thread: entropy is not merely a thermodynamic bookkeeping device but a measure that governs uncertainty, information, and the structure of physical states. The realization that information arises from entropy — not as an analogy but as a mathematical consequence — provided the first indication that entropy might possess a more fundamental ontological role than traditionally assumed.

This insight deepened when I turned to information geometry. The Fisher–Rao metric, the Fubini–Study metric, and the broader framework of statistical manifolds demonstrated that information has an intrinsic geometry. Amari and Čencov’s work on α‑connections further showed that informational structures can be treated as deformable manifolds endowed with affine connections. At that point, the conceptual bridge became unavoidable: if information possesses geometry, and if that geometry is governed by affine connections, then informational manifolds share a structural kinship with the Riemannian geometry underlying spacetime. Since information is generated from entropy, it followed that entropy itself must be capable of inducing or participating in geometric structure.

This line of reasoning led me to a decisive inference: if entropy gives rise to information, and information has geometry, and that geometry can be related to the affine connections of spacetime, then entropy must be connected to the curvature structure of Riemannian geometry. And because curvature and affine connections lie at the heart of Einstein’s General Relativity — a field theory defined by an action principle — the natural conclusion for me therefore was that entropy itself must be representable as a field with its own dynamics, equations of motion, and variational structure. This was not a speculative jump but a logical continuation of the geometric and informational lineage that begins with classical thermodynamics and culminates in modern differential geometry.

In following this path, I found myself aligned with Einstein’s own vision, for he believed profoundly in the Second Law of Thermodynamics and held the conviction that, among all the laws of physics and nature, it is the one that will never be overthrown. That conviction — that the Second Law expresses something irreducible about the fabric of reality — became, for me, both a philosophical anchor and a scientific compass. It affirmed that treating entropy as a universal field was not merely permissible, but deeply consonant with the trajectory of physics itself.

From this foundation, I sought to construct an action principle for entropy. Recognizing that classical Shannon–von Neumann entropy is only one member of a broader family, I generalized the action to incorporate Tsallis and Rényi entropies, whose non‑extensive and generalized forms capture richer structural behavior. This generalization produced what I call the Local Obidi Action (LOA), a variational formulation that treats entropy as a local field with curvature‑dependent dynamics. Yet the story did not end there. The study of Araki relative entropy and its operator‑algebraic structure suggested that entropy also possesses a spectral character, one that cannot be captured solely by local differential geometry. This insight led me to formulate the Spectral Obidi Action (SOA), a complementary nonlocal, operator‑based action that mirrors the spectral action principle in noncommutative geometry.

[Incorporation of Bosonic and Fermionic Fields into the Spectral Obidi Action (SOA)]

A further step in the development of the Theory of Entropicity (ToE) involved understanding how conventional matter fields — bosonic and fermionic — could be naturally incorporated into the spectral formulation of the Obidi Action. Once the spectral character of entropy became evident through the study of Araki relative entropy and operator‑algebraic structures, it became clear to me that any complete entropic field theory must accommodate the full spectrum of physical degrees of freedom. This required a formulation in which matter fields arise not as external additions but as intrinsic components of the entropic spectral geometry itself.

The key insight came from the Dirac–Kähler formalism, which provides a unified geometric representation of fermionic and bosonic fields using differential forms. In this framework, fermions are encoded through the Dirac operator acting on inhomogeneous differential forms, while bosonic fields emerge from the curvature and connection structures associated with the same underlying geometric complex. This dual representation allowed me to see that the entropic spectral operator — the generator of the Spectral Obidi Action — could be constructed in a way that naturally couples to both types of fields without introducing them by hand.

In the spectral formulation of the Theory of Entropicity (ToE), the entropic field generates an information‑geometric operator whose spectrum encodes global structural features of the entropic manifold. The Spectral Obidi Action (SOA) is defined as a functional of this entropic spectrum, not as an analogue of the spectral action in noncommutative geometry. While the Dirac–Kähler formalism may be used as a convenient geometric language for representing matter fields, it is employed purely as a technical tool and does not determine the ontology of the theory. In this formulation, bosonic and fermionic fields interact with the entropic field through the information‑geometric structures induced by entropy itself, ensuring that matter, geometry, and entropy share a unified entropic origin without relying on the machinery of noncommutative geometry or heat‑kernel expansions.

This incorporation of matter fields into the spectral formulation was not an afterthought but a necessary consequence of treating entropy as a universal field. If entropy is to serve as the substrate of physical law, then its spectral geometry must be capable of encoding the full content of the physical world. The Dirac–Kähler approach provided the mathematical bridge: it allowed the entropic spectral operator to act on a space rich enough to contain both fermionic and bosonic degrees of freedom, while remaining consistent with the dual local–spectral structure of the Obidi Action Principle (OAP). In this sense, the inclusion of matter fields is not an external extension of ToE but an intrinsic feature of its spectral geometry.

The dual structure of the Obidi Action Principle (OAP) — local and spectral — emerged naturally from these considerations. It reflects the dual nature of entropy itself: simultaneously

[The Principle of Least Entropic Resistance (PoLER)]

And this directly and naturally brings me in immediate confrontation with the mechanical principle of least action.

We already know that, in classical mechanics, the evolution of physical systems is governed by the Maupertuis–d’Alembert Principle of Least Action, which asserts that bodies follow trajectories that minimize the mechanical action, or equivalently, the mechanical work. This variational principle has served as the backbone of physics for centuries, unifying dynamics under a single extremal condition.

Now, the Theory of Entropicity (ToE) replaces this classical criterion with a deeper and more universal one. Because ToE identifies entropy as the fundamental field from which geometry, matter, and dynamics emerge, it occurred to me therefore that the natural variational principle must be expressed in entropic rather than mechanical terms. Accordingly, I introduced the following foundational statement and principle in ToE:

Principle of Least Entropic Resistance (PoLER).
Bodies, particles, and all physical systems evolve along trajectories that minimize entropic resistance, or equivalently, along paths of least entropic work.

This principle is not an analogy or reinterpretation of the classical least‑action principle; it is a structural replacement. In ToE, the entropic field \(S\) determines the geometry \(g(S)\), the curvature \(R_{IG}[S]\), and the stress–energy tensor content \(T_{\mu\nu}(S)\). As a result, the “cost” associated with any physical evolution is not mechanical but entropic. A system’s path through spacetime is therefore the one that minimizes the cumulative entropic curvature it must traverse.

This reformulation has several immediate consequences:

• Dynamics become entropic: motion is the relaxation of entropic gradients, not the response to forces.
• Geometry becomes adaptive: the metric adjusts to the entropic field, so minimizing entropic work simultaneously shapes spacetime.
• Matter becomes emergent: what we call “mass” or “energy” is the entropic resistance encoded in \(T_{\mu\nu}(g(S))\).
• Causality becomes entropic: the direction of time aligns with the direction of decreasing entropic resistance.

In this sense, the Principle of Least Entropic Resistance (PoLER) is the natural variational law for a universe whose substrate is entropy.

It is a ToE reformulation of the Second Law of Thermodynamics via the methodology of trajectories. It thus generalizes and hence supersedes the classical least‑action principle by embedding it within a broader entropic geometry. Where classical mechanics minimizes action, ToE minimizes entropic curvature; where classical trajectories are geodesics of a fixed metric, ToE trajectories are geodesics of an entropically induced geometry.

This principle is therefore well situated and forms the conceptual and mathematical foundation of the Obidi Action Principle (OAP) and the Master Entropic Field Equation (OFE), and it provides the unifying logic behind the emergence of spacetime, matter, and physical law from a single entropic foundational substrate.

Hence, it is from this vantage point that I find the Theory of Entropicity (ToE) both beautiful and elegant. Not because it is my own creation, but because it reveals a unity that had been hidden in plain sight. It shows that entropy, information, geometry, curvature, matter, and dynamics are not separate domains but facets of a single underlying structure. It shows that the universe is not a collection of disconnected laws but an entropic continuum expressing itself through different mathematical languages. And it shows that the path from thermodynamics to field theory is not a leap but a progression — one that becomes obvious only after it has been walked.

I cannot expect others to feel this beauty and elegance immediately. Beauty and elegance in theory are often visible first to the one who has lived inside its development. I see this beauty and elegance so clearly now in my imagination and in my mind's eye. But I can hope — and I do — that by laying out the logical trail with clarity and honesty, the reader may come to see for themselves what I have seen: that the Theory of Entropicity (ToE) is not an arbitrary construction but a natural consequence of following entropy to its deepest implications. If posterity finds value in this work, it will not be because I insisted on its importance, but because the structure itself which I have laid out proves worthy of enduring attention.

In closing, ToE asserts:

• Entropy is not a derived quantity.
• Entropy is the substrate of reality.
• Everything else — spacetime, matter, fields, information — emerges from it.

This is a radical ontological inversion.

Instead of:
geometry → entropy
ToE declares:
entropy → geometry

Instead of:
fields → entropy
ToE declares:
entropy → fields

Instead of:
information → entropy and entropy → information or information ↔ entropy
ToE declares (field-asymmetrically):
entropy (field) → information (field) → geometry‑matter (field) + …

This is not a reinterpretation of existing physics. It is a new ontological foundation.

Thus, where conventional physics treats entropy and information as mutually convertible measures, the Theory of Entropicity (ToE) asserts a deeper asymmetry: entropy is a universal field, and information is the geometric structure induced by that field. In ToE, entropy is fundamental; information is emergent. This phase of my work and reflection culminated in the discovery of the Obidi Curvature Invariant (OCI) — the constant ln 2 — which I identify as the informational divergence structure constant of nature and the universe.

This monograph is therefore both an exposition and an invitation. It presents the Theory of Entropicity (ToE) as a coherent framework whose principles arise naturally from the entropic ontology it proposes. At the same time, it invites the reader — whether physicist, philosopher, or curious thinker — to engage with the theory critically, to test its claims and fundamental singular axiom, to explore its implications, and to participate in its evolution. If the Theory of Entropicity (ToE) ultimately contributes to a deeper understanding of reality, it will be because it withstands scrutiny, not because it was shielded from it.

I offer this corpus of work in that spirit: as a sincere and deeply natural attempt to articulate a new foundation for physical law, grounded in entropy as the substrate of existence, and as my own infinitesimal and yet ineffaceable contribution to the ongoing human effort to understand the mysterious universe — in alignment with the great and legendary Richard P. Feynman — and our place within it.

May posterity be happy witnesses of it.

In Memoriam

This work is dedicated to you both, with deep affection and enduring gratitude:

Professor B. Orisa, for urging me to devote and dedicate more time to the deep problems and challenges of modern theoretical physics, and for the invigorating conversations we shared at the intersection of mathematical physics and quantum theory.

Professor Felix E. Opara, for the remarkable dexterity and ingenuity with which you tackled and analysed the problem of the Clebsch–Gordan coefficients on the blackboard that fateful Sunday; also for giving me Jackson — that intimidating, formidable monument of theoretical physics; and for granting me direct access to your collected works with the illustrious Nobel Prize–winning physicist Abdus Salam at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy.

Now, both of you have passed beyond Earth and the Aether, fulfilling your divine fate and destiny — inseparable from that Second Law of Thermodynamics you so passionately loved, taught, and professed.

References

1. Author's Preface to the Theory of Entropicity (ToE): https://theoryofentropicity.blogspot.com/2026/01/authors-preface-and-methodological.html
2. https://medium.com/@jonimisiobidi/authors-preface-and-methodological-statement-for-the-theory-of-entropicity-toe-an-unapologetic-b3228a398236
3. https://www.linkedin.com/posts/theory-of-entropicity-toe_new-publication-authors-preface-methodological-activity-7421775955083440128-GpTA
4. https://open.substack.com/pub/johnobidi/p/authors-preface-and-methodological-ef9
5. Selected Works on the Theory of Entropicity (ToE): https://theoryofentropicity.blogspot.com/2025/11/selected-papers-on-theory-of.html
6. Grokipedia: Theory of Entropicity (ToE) — https://grokipedia.com/page/Theory_of_Entropicity
7. Grokipedia: John Onimisi Obidi — https://grokipedia.com/page/John_Onimisi_Obidi
8. Canonical Archive of the Theory of Entropicity (ToE): https://entropicity.github.io/Theory-of-Entropicity-ToE/

Appendix: Extra Matter

1. Canonical Form of the Obidi Action

A mathematically standard rendering of the Obidi Action is:

\[ I_{\text{Semergent}} = \int_M d^4x \, \sqrt{-g(S)} \left[ \chi^2 e^{S/k_B} (\nabla_\mu S)(\nabla^\mu S) - V(S) + \lambda R_{IG}[S] \right]. \]

This is an original entropic field action. It does not correspond to any known published theory in physics or mathematics.

It is:

• not a dilaton action
• not a Brans–Dicke scalar
• not a k‑essence model
• not a Liouville‑type theory
• not Verlinde’s entropic gravity
• not Padmanabhan’s emergent gravity
• not Connes–Chamseddine spectral action
• not Tsallis or Rényi thermodynamics
• not Amari–Čencov information geometry
• not any known scalar–tensor or modified gravity theory

It is not due to:

• Einstein
• Verlinde
• Padmanabhan
• Jacobson
• Bekenstein
• Connes
• Chamseddine
• Tsallis
• Rényi
• Amari
• Čencov
• Rovelli
• Smolin
• Penrose
• Weinberg
• Hawking
• Sorkin
• Bianconi

It does not resemble any established framework in:

• General Relativity
• Quantum Field Theory
• Statistical Field Theory
• Information Geometry
• Emergent Gravity
• Thermodynamic Gravity
• Entropic Gravity (Verlinde)
• Non‑extensive thermodynamics (Tsallis, Rényi)
• Noncommutative geometry (Connes)
• Spectral action (Chamseddine–Connes)
• Dilaton gravity
• Scalar–tensor theories
• f(R) gravity
• K‑essence or inflationary scalar fields

No known physicist or mathematician has published this exact structure. It is structurally unique.

The fingerprints are unmistakable: this is a ToE‑native emergent entropic action, consistent with ToE's entropic ontology.

3. What Each Term Means in the ToE Framework

(a) \( \sqrt{-g(S)} \)
A metric determinant that depends on the entropic field. This encodes the idea that geometry is entropic in origin — a core ToE principle.

(b) \( \chi^2 e^{S/k_B} (\nabla S)^2 \)
A kinetic term weighted by a Boltzmann factor. This is unprecedented in standard physics and expresses:

• entropy as a dynamical field
• entropic gradients as physically meaningful
• thermodynamic weighting embedded directly into field dynamics

(c) \( V(S) \)
A general entropic potential. This allows for:

• entropic vacua
• entropic phase transitions
• entropic curvature minima

(d) \( \lambda R_{IG}[S] \)
A curvature scalar built from information geometry. This is the most original component.

It encodes:

• Fisher–Rao curvature
• Amari–Čencov α‑connection curvature
• entropic manifold curvature
• informational Ricci scalar

This is precisely the bridge between:

• entropy
• information
• geometry
• curvature
• field theory

which is the backbone of the Theory of Entropicity (ToE).

Appendix: Extra Matter — Some Elementary Mathematical Expositions

1. The Emergent Entropic Action (Canonical Form)

\[ I_{\text{Semergent}} = \int_M d^4x \, \sqrt{-g(S)} \left[ \chi^2 e^{S/k_B} (\nabla_\mu S)(\nabla^\mu S) - V(S) + \lambda R_{IG}[S] \right]. \]

This is the most compact and covariant form of the Obidi Action, from which we shall derive the Obidi Field Equation, Stress‑Energy Tensor, etc.

2. Variation with Respect to the Entropic Field \( S(x) \)

We compute:

\[ \frac{\delta I_S}{\delta S} = 0. \]

There are four contributions:

(a) Variation of the metric determinant

Because \( g = g(S) \):

\[ \delta \sqrt{-g(S)} = \frac{1}{2} \sqrt{-g(S)} \, g^{\mu\nu}(S) \frac{\partial g_{\mu\nu}}{\partial S} \, \delta S. \]

This term encodes the entropic origin of geometry.

Appendix: Extra Matter — More Detailed and Advanced Mathematical Expositions

1. The emergent entropic action as a ToE‑native field theory

We begin from the emergent entropic action, which encodes the dynamics of the entropic field \( S(x) \) on a four‑dimensional manifold \( M \):

\[ I_{\text{Semergent}} = \int_M d^4x \, \sqrt{-g(S)} \left[ \chi^2 e^{S/k_B} (\nabla_\mu S)(\nabla^\mu S) - V(S) + \lambda R_{IG}[S] \right]. \]

Here:

• \( \sqrt{-g(S)} \) denotes the square root of the negative determinant of a metric \( g_{\mu\nu}(S) \) whose structure depends on the entropic field.
• \( \chi \) is a coupling constant.
• \( k_B \) is Boltzmann’s constant.
• \( V(S) \) is an entropic potential.
• \( R_{IG}[S] \) is an information‑geometric curvature scalar.

This action is not adapted from any existing scalar–tensor or modified gravity theory; it is a ToE‑native formulation that follows from treating entropy as the fundamental field of reality.

2. Euler–Lagrange equation for the entropic field

We now derive the field equation for \( S(x) \) by varying the action with respect to \( S \).

The Lagrangian density is:

\[ \mathcal{L}_S = \sqrt{-g(S)} \left[ \chi^2 e^{S/k_B} (\nabla S)^2 - V(S) + \lambda R_{IG}[S] \right]. \]

The variation \( \delta \mathcal{L}_S \) receives contributions from:

• explicit dependence on \( S \)
• dependence of the metric determinant on \( S \)
• dependence of \( R_{IG}[S] \) on \( S \)

2.1 Variation of the metric determinant

\[ \delta \sqrt{-g(S)} = \frac{1}{2} \sqrt{-g(S)} \, g^{\mu\nu}(S) \frac{\partial g_{\mu\nu}(S)}{\partial S} \delta S \equiv \sqrt{-g(S)} \, \Gamma(S) \, \delta S, \]

where:

\[ \Gamma(S) = \frac{1}{2} g^{\mu\nu}(S) \frac{\partial g_{\mu\nu}(S)}{\partial S} = \frac{\partial \ln \sqrt{-g(S)}}{\partial S}. \]

2.2 Variation of the kinetic term

\[ \delta \left( e^{S/k_B} (\nabla S)^2 \right) = e^{S/k_B} \left[ \frac{1}{k_B} (\nabla S)^2 \delta S + 2 \nabla_\mu S \nabla^\mu (\delta S) \right]. \]

Integrating by parts:

\[ \int d^4x \sqrt{-g(S)} \, 2 \chi^2 e^{S/k_B} \nabla_\mu S \nabla^\mu (\delta S) = - \int d^4x \sqrt{-g(S)} \, 2 \chi^2 \nabla_\mu \left( e^{S/k_B} \nabla^\mu S \right) \delta S. \]

2.3 Variation of the potential

\[ \delta V(S) = V'(S) \delta S. \]

2.4 Variation of the information‑geometric curvature

\[ \delta R_{IG}[S] = \frac{\delta R_{IG}}{\delta S} \delta S. \]

2.5 Collecting all contributions

\[ \delta I_S = \int d^4x \sqrt{-g(S)} \left\{ -2\chi^2 \nabla_\mu \left( e^{S/k_B} \nabla^\mu S \right) + \chi^2 e^{S/k_B} \frac{1}{k_B} (\nabla S)^2 - V'(S) + \lambda \frac{\delta R_{IG}}{\delta S} + \Gamma(S) \left[ \chi^2 e^{S/k_B} (\nabla S)^2 - V(S) + \lambda R_{IG}[S] \right] \right\} \delta S. \]

Setting \( \delta I_S = 0 \) for arbitrary \( \delta S \) yields the Euler–Lagrange equation:

\[ -2\chi^2 \nabla_\mu \left( e^{S/k_B} \nabla^\mu S \right) + \chi^2 e^{S/k_B} \frac{1}{k_B} (\nabla S)^2 - V'(S) + \lambda \frac{\delta R_{IG}}{\delta S} + \Gamma(S) \left[ \chi^2 e^{S/k_B} (\nabla S)^2 - V(S) + \lambda R_{IG}[S] \right] = 0. \]

3. Stress–energy tensor of the entropic field

By definition:

\[ T_{\mu\nu}(S) = -\frac{2}{\sqrt{-g}} \frac{\delta I_S}{\delta g^{\mu\nu}}. \]

The dependence on \( g_{\mu\nu} \) enters through:

• the metric determinant
• the kinetic term
• the information‑geometric curvature

The result is:

\[ T_{\mu\nu}(S) = \chi^2 e^{S/k_B} \left[ \nabla_\mu S \nabla_\nu S - \frac{1}{2} g_{\mu\nu} (\nabla S)^2 \right] - g_{\mu\nu} V(S) + \lambda T^{IG}_{\mu\nu}[S] + T_{\mu\nu}(g(S)). \]

This is the first stress–energy tensor in physics to include:

• Boltzmann‑weighted kinetic structure
• information‑geometric curvature
• metric‑dependence‑on‑entropy

This is unique to ToE.

4. Hamiltonian formulation of the entropic field

To obtain the Hamiltonian formulation, we perform a (3+1) decomposition of spacetime and treat \( S \) as a canonical field on spatial hypersurfaces \( \Sigma_t \).

Let \( x^\mu = (t, x^i) \), and write the metric in ADM form:

\[ ds^2 = - N^2 dt^2 + h_{ij} (dx^i + N^i dt)(dx^j + N^j dt), \]

where \( N \) is the lapse, \( N^i \) the shift, and \( h_{ij} \) the induced spatial metric.

The determinant decomposes as:

\[ \sqrt{-g(S)} = N \sqrt{h(S)}. \]

The kinetic term becomes:

\[ (\nabla S)^2 = -\frac{1}{N^2} (\partial_t S - N^i \partial_i S)^2 + h^{ij} \partial_i S \partial_j S. \]

Thus the Lagrangian density is:

\[ \mathcal{L}_S = N \sqrt{h(S)} \left[ \chi^2 e^{S/k_B} \left( -\frac{1}{N^2} (\partial_t S - N^i \partial_i S)^2 + h^{ij} \partial_i S \partial_j S \right) - V(S) + \lambda R_{IG}[S] \right]. \]

The canonical momentum conjugate to \( S \) is:

\[ \Pi_S = \frac{\partial \mathcal{L}_S}{\partial (\partial_t S)} = -2 \chi^2 \sqrt{h(S)} \, e^{S/k_B} \frac{1}{N} (\partial_t S - N^i \partial_i S). \]

Solving for \( \partial_t S \):

\[ \partial_t S = N^i \partial_i S - \frac{N}{2\chi^2 \sqrt{h(S)}} e^{-S/k_B} \Pi_S. \]

The Hamiltonian density is:

\[ \mathcal{H}_S = \Pi_S \partial_t S - \mathcal{L}_S = N \mathcal{H}_\perp(S) + N^i \mathcal{H}_i(S), \]

where:

\[ \mathcal{H}_\perp(S) = \frac{1}{4 \chi^2 \sqrt{h(S)}} e^{-S/k_B} \Pi_S^2 + \chi^2 \sqrt{h(S)} e^{S/k_B} h^{ij} \partial_i S \partial_j S + \sqrt{h(S)} V(S) - \lambda \sqrt{h(S)} R_{IG}[S], \]

and

\[ \mathcal{H}_i(S) = \Pi_S \partial_i S. \]

The Hamiltonian formulation thus reveals:

• a non‑standard kinetic structure weighted by \( e^{\pm S/k_B} \)
• a geometric dependence through \( h(S) \)
• a curvature contribution from \( R_{IG}[S] \)

This Hamiltonian can be coupled to the gravitational Hamiltonian (if one chooses to treat \( g_{\mu\nu} \) dynamically) to yield a fully entropic Hamiltonian constraint system.

5. Noether currents and covariant conservation

If the action is invariant under spacetime diffeomorphisms, then the total stress–energy tensor satisfies:

\[ \nabla_\mu T^{\mu\nu}_{\text{total}} = 0. \]

In the present context, the entropic field contributes \( T_{\mu\nu}(S) \), and any additional fields contribute their own stress–energy tensors. The conservation law becomes:

\[ \nabla_\mu T^{\mu\nu}(S) + \nabla_\mu T^{\mu\nu}(\text{other}) = 0. \]

Because the entropic field is coupled to geometry and information‑geometric curvature, the conservation law encodes a nontrivial exchange between:

• entropic gradients
• geometric curvature
• information‑geometric structure

If the theory also possesses internal symmetries in the space of entropic configurations (e.g., shifts \( S \to S + \text{const} \)), then Noether’s theorem yields additional conserved currents.

For example, if the action is invariant under \( S \to S + \epsilon \), then the associated Noether current is:

\[ J^\mu = \frac{\partial \mathcal{L}_S}{\partial (\nabla_\mu S)} \delta S = 2 \chi^2 \sqrt{-g(S)} \, e^{S/k_B} \nabla^\mu S. \]

The conservation law \( \nabla_\mu J^\mu = 0 \) expresses a form of entropic flux conservation.

6. Relation to the Local and Spectral Obidi Actions

The emergent entropic action \( I_{\text{Semergent}} \) sits naturally alongside the Local Obidi Action (LOA) and the Spectral Obidi Action (SOA) as part of the broader Obidi Action Principle (OAP).

6.1 Local Obidi Action (LOA)

The LOA treats entropy as a local field whose dynamics are governed by a curvature‑dependent action built from generalized entropies (Shannon, Tsallis, Rényi, etc.):

\[ I_{\text{LOA}} = \int_M d^4x \sqrt{-g} \, \mathcal{L}_{\text{local}}[S, \nabla S, g_{\mu\nu}]. \]

The emergent entropic action is a refined local realization of this principle.

6.2 Spectral Obidi Action (SOA)

The SOA arises when entropy is treated as a spectral quantity associated with operators on a Hilbert space. In analogy with the Connes–Chamseddine spectral action principle (SAP):

\[ I_{\text{SOA}} = \text{Tr} \, f(D_S / \Lambda), \]

where \( D_S \) is an entropic spectral operator.

The heat‑kernel expansion yields:

\[ I_{\text{SOA}} \sim \int_M d^4x \sqrt{-g} \left[ c_0 \Lambda^4 + c_2 \Lambda^2 R + c_4 (R^2, R_{\mu\nu}R^{\mu\nu}, F_{\mu\nu}F^{\mu\nu}, \ldots) + \cdots \right]. \]

The information‑geometric curvature \( R_{IG}[S] \) appearing in the emergent action can be interpreted as a local shadow of the spectral curvature encoded in the SOA.

It mirrors the Connes–Chamseddine spectral action, but replaces the Dirac operator D with an entropic spectral operator D_S. This is fully compatible with ToE.

A subtle but important clarification is needed here regarding the above statement that “entropy is treated as a spectral quantity associated with operators on a Hilbert space.” This phrasing should be understood in the sense that the entropic field S(x) induces an operator D_S, and that this operator encodes aspects of entropic geometry; and furthermore, that the spectrum of D_S provides a representation of entropic curvature. However, this should not be interpreted by the reader to mean that entropy is fundamentally quantum in nature, that the Theory of Entropicity requires a Hilbert‑space ontology, or that ToE reduces to spectral geometry. We make it clear here that the Theory of Entropicity (ToE) is broader and more general than any specific Hilbert‑space formulation. The spectral representation of Connes–Chamseddine should therefore be understood as a mathematical tool or viewpoint, not as the ontological foundation of ToE.

6.3 The dual Obidi Action Principle (OAP)

The Obidi Action Principle asserts that the full entropic dynamics are captured by a dual structure:

• a local action (LOA / emergent action)
• a spectral action (SOA)

which are not independent but mutually constraining.

The emergent entropic action:

\[ I_{\text{Semergent}} = \int_M d^4x \sqrt{-g(S)} \left[ \chi^2 e^{S/k_B} (\nabla S)^2 - V(S) + \lambda R_{IG}[S] \right] \]

is the local field‑theoretic realization of the entropic ontology.

It is consistent with and complementary to the spectral formulation encoded in the SOA.

It allows one to derive explicit field equations, stress–energy tensors, and conservation laws, which can then be compared with the spectral predictions.

In this way, the emergent entropic action is not an isolated construct but part of a unified dual structure.

It is worth noting that the Spectral Obidi Action (SOA) is a powerful addition to the Theory of Entropicity (ToE) because it provides a genuinely global, operator‑level formulation of entropic dynamics. By introducing an entropic spectral operator whose eigenvalues encode aspects of entropic curvature, the SOA creates a natural bridge between ToE and spectral geometry. This framework allows entropic curvature to be expressed through spectral data, offers a direct route to heat‑kernel expansions, and unifies the global spectral structure of the entropic field with its local information‑geometric curvature. Rather than weakening the theory, the SOA deepens and strengthens ToE by revealing new layers of structure that complement the variational and geometric formulations already present in the framework.

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