Author’s Preface and Methodological Statement for the Theory of Entropicity (ToE): An Unapologetic Introduction in Defense of Obidi's New Theory of Reality—On the Trajectory of Discovery and the Road Less Traveled 

Last Updated: Saturday, January 31st, 2026

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 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.

[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 a local field with geometric curvature and a spectral quantity encoded in operator algebras. From this duality, the broader architecture of the Theory of Entropicity unfolded. The principles and laws that now form the backbone of ToE — the Entropic Accounting Principle, the Entropic Equivalence Principle, the Entropic Resistance Principle, the Cumulative Delay Principle, and others — were not imposed arbitrarily but deduced progressively as the logical consequences of treating entropy as a universal field.

From these considerations, the natural next step was to construct an action principle for the entropic field - the Obidi Action Principle (OAP). The emergent entropic action 

$I_S^{\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{<br>}}$

formalizes the idea that entropy is not merely a statistical descriptor but a dynamical field whose gradients, potentials, and information‑geometric curvature collectively generate the structure of physical law. The result was the emergent entropic Obidi Action:

$$I_S^{\text{emergent}}={\int }_M{d}^{4}x\sqrt{-g(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]],$$

which encodes the full dynamical content of the entropic ontology. The dependence of the metric determinant on $S$, the Boltzmann‑weighted kinetic term, and the information‑geometric curvature scalar $R_{\mathrm{I}\mathrm{G}}[S]$ together express the central thesis of the Theory of Entropicity (ToE): that geometry, dynamics, and matter emerge from the entropic substrate itself. This action is not borrowed from any existing physical theory; it is the natural mathematical consequence of treating entropy as the foundational field of reality.

The resulting Euler–Lagrange equation, obtained by varying the action with respect to S(x), yields a Master Entropic Field Equation [the Obidi Field Equations (OFE) in their trivial form]:

$$\boxed{{\displaystyle -2{\chi }^2{\mathrm{\nabla }}_{\mu }\left(e^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}^{\mu }S\right)+{\chi }^2\frac{e^{S\mathrm{/}{k}_B}}{k_B}(\mathrm{\nabla }S{)}^2-V^{\mathrm{\prime }}(S)+\lambda \frac{\delta R_{\mathrm{I}\mathrm{G}}}{\delta S}+\frac{1}{2}\frac{\mathrm{\partial }\ln \sqrt{-g(S)}}{\mathrm{\partial }S}[{\chi }^2{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}]=\mathrm{0,}}}$$

whose structure has no analogue in existing physics. Likewise, the [entropic] stress–energy tensor:

$$T_{\mu \nu }^{(S)}={\chi }^2{e}^{S\mathrm{/}{k}_B}[{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}_{\nu }S-\frac{1}{2}{g}_{\mu \nu }(\mathrm{\nabla }S{)}^2]-g_{\mu \nu }V(S)+\lambda T_{\mu \nu }^{\mathrm{I}\mathrm{G}}[S]+T_{\mu \nu }^{(g(S))}\mathrm{.}$$

derived from this action contains contributions from entropic gradients, information‑geometric curvature, and the entropic dependence of the metric—features unique to this framework. In this sense, the emergent entropic action is not an adaptation of any known theory, but a ToE‑native formulation arising inevitably from the entropic ontology.

The Spectral Obidi Action (SOA) arises not [directly] from noncommutative geometry but from the intrinsic spectral character of entropy itself. In the Theory of Entropicity (ToE), entropy generates an information manifold whose structure possesses a natural spectrum of entropic modes. Let S denote the entropic spectral operator acting on this manifold, and let $\text{Spec}(\mathcal{S})=\{{\sigma }_i\}$ denote its spectrum. The SOA is defined as a global functional of this spectrum,

$$I_{\mathrm{S}\mathrm{O}\mathrm{A}}=\mathrm{\Phi }(\text{Spec}(\mathcal{S})),$$

$$<br>$$

$$with:\; \Phi (\{\lambda ᵢ\},\; \{\psi ᵢ\})\; =$$α ∑ᵢ f(λᵢ) + β ∑ᵢⱼ g(λᵢ, λⱼ) ⟨ψᵢ, ψⱼ⟩ + γ 𝓚({λᵢ}) + δ 𝓡({ψᵢ}) + ε 𝓒({λᵢ, ψᵢ}),

where Φ encodes the global entropic content of the system. Unlike the spectral action of Alain Connes, the SOA does not per se rely on a Dirac operator, a cutoff scale, a test function, or a heat‑kernel expansion. Its origin is purely entropic: it measures the global spectral structure induced by the entropic field. In this sense, the SOA is not an adaptation of noncommutative geometry but a ToE‑native spectral formulation arising directly from its own axiomatic ontology.

Although the terminology ‘spectral’ and certain operator‑theoretic constructions may evoke comparisons with Alain Connes’ spectral action principle and noncommutative geometry, the Theory of Entropicity (ToE) is not derived from, nor conceptually grounded in, that framework. In ToE, ‘spectral’ refers to the eigenstructure of entropic and informational operators defined on information manifolds generated by the entropic field, not to a noncommutative spectral triple. Likewise, while the Dirac–Kähler formalism may be employed as a convenient geometric language for encoding bosonic and fermionic fields, it is used purely as a technical tool and does not constitute a foundational ingredient of the theory. The ontology of ToE is entropic, not noncommutative: its core structures arise from entropy, information geometry, and entropic curvature, rather than from the machinery of noncommutative geometry or spectral triples.

The Theory of Entropicity (ToE) is not derived from Alain Connes’ noncommutative geometric framework, nor does it reproduce his spectral triple construction or spectral action in their original form. While both approaches share a broad appreciation for spectral and geometric structures, ToE is grounded in an entropic ontology and information geometry rather than in noncommutative algebras and Dirac operators. Any future connections between ToE and noncommutative geometry would constitute new developments built atop the entropic framework, not its conceptual foundation.

And because the entropic field S(x) induces a natural structure of equivalence classes, flows, and invariants, the Theory of Entropicity (ToE) may therefore be viewed as grounded in an entropic topology: a topological framework arising from the intrinsic organization of entropy itself.

This, in essence, is the intellectual trajectory that led me to the formulation of the Theory of Entropicity. It is a path that moves from classical thermodynamics to information theory, from information geometry to spacetime geometry, from action principles to spectral theory, and finally to a unified entropic ontology. I present this history not to elevate my own role but to make transparent the reasoning that compelled me toward this unified framework. Understanding this trajectory is essential for appreciating why ToE takes the form it does and why I believe entropy must be regarded not merely as a statistical measure but as the foundational field of physical reality.

My methodological stance throughout this monograph is therefore one of disciplined openness. I do not assume that ToE is complete, nor do I claim that it supersedes existing theories. Instead, I treat it as a unifying framework whose value must be demonstrated through rigorous derivation, conceptual clarity, and empirical relevance. Where the theory aligns with established physics, I present that alignment as evidence of coherence. Where it diverges, I present the divergence as an opportunity for refinement or falsification. The aim is not to replace the existing edifice of physics but to reveal the entropic foundation upon which its pillars may rest [safely].

It is important to acknowledge that every theorist must, to some degree, advocate for their own work. Einstein did so for relativity; Shannon did so for information theory; Bianconi does so for network thermodynamics and Gravity from Entropy (GfE). Advocacy is not antithetical to objectivity when it is grounded in intellectual integrity. My commitment is to present the Theory of Entropicity (ToE) with honesty: neither diminishing its novelty out of false modesty nor inflating its significance beyond what its current development warrants. I articulate its beauty and elegance because I see them; I defend its coherence because I have tested it; and I invite critique because no theory can mature without it.

[On the Beauty, Elegance, and Inevitability of the Theory of Entropicity (ToE)]

There’s something deeply human in what I have been expressing heretofore, and it deserves to be said with clarity rather than hesitation. When I say this is how I find the Theory of Entropicity (ToE) both beautiful and elegant, I am in every way naming an experience that every originator of a new conceptual structure has felt: the beauty and elegance are visible from the inside long before the outside world learns how to see it.

And that’s not arrogance — it’s proximity.

I have walked through the entire chain myself:

1. from Clausius to Gibbs to Shannon to von Neumann  
2. from information to geometry  
3. from Fisher–Rao and Fubini–Study to Amari–Čencov  
4. from affine connections to Riemannian curvature  
5. from GR as a field theory to entropy as a field  
6. from Tsallis/Rényi generalizations to the Local Obidi Action  (LOA)
7. from Araki relative entropy to the Spectral Obidi Action  (SOA)
8. from both to the dual Obidi Action Principle  (OAP)

I have lived inside that architecture as it assembled itself. I have seen how each step follows from the previous one with a kind of inevitability. I have watched the pieces lock together in ways that feel less like invention and more like discovery.

It’s natural — even necessary — that I see the beauty and elegance before others do.

A new theory is always invisible to those who haven’t walked the path that produced it. They see the endpoints; I see the connective tissue. They see the claims; I see the logic that made those claims unavoidable. They see the structure; I see the scaffolding that held it up while it was being built.

That’s why my appreciation of the beauty and elegance of the Theory of Entropicity (ToE) is not premature. It’s earned.

And it’s also why this Preface of mine, my logical‑development, and my methodological stance on the Theory of Entropicity (ToE) matter so much. They [I strongly and vehemently hope] give the reader a way to retrace my steps in the development of the Theory— not for the reader to adopt my conclusions blindly, but to understand the intellectual and logical terrain that made those conclusions possible and inevitable.

So, as the Theory of Entropicity (ToE) took shape, I found myself repeatedly confronted by this quiet but unmistakable sense of inevitability — a feeling that the structure unfolding before me was not something I was imposing on the world, but something the world had been waiting to reveal. This sense did not arise from ambition or self‑assurance; it emerged from the logic itself. Each step in the development of ToE — from classical entropy to information, from information to geometry, from geometry to curvature, and from curvature to field dynamics — followed with such internal necessity that the theory began to feel less like an invention and more like a discovery. The beauty I see in ToE is not the beauty of personal creation, but the beauty of coherence: the beauty of a structure that holds together because it could not be otherwise.

This feeling deepened as I traced the lineage of entropy across disciplines. Clausius gave entropy its thermodynamic birth; Gibbs and Shannon revealed its informational essence; von Neumann extended it into the quantum realm. Fisher–Rao and Fubini–Study showed that information has geometry; Amari and Čencov demonstrated that this geometry is deformable, structured by affine connections. And Einstein taught us that curvature and connection are the language of spacetime itself. When these threads converged, the conclusion that entropy must be a universal field did not feel speculative — it felt unavoidable. The logic carried me forward with a momentum of its own, and I followed because the path was already there.

The same inevitability accompanied the construction of the Obidi Action. Once entropy was recognized as a field, an action principle was not optional; it was required. The generalization to Tsallis and Rényi entropies was not decorative; it was demanded by the non‑extensive and generalized behaviors entropy exhibits in complex systems. The emergence of the Local Obidi Action (LOA) was not a creative flourish; it was the simplest variational structure consistent with the entropic ontology. And when Araki relative entropy revealed the spectral dimension of entropic structure, the Spectral Obidi Action (SOA) followed with the same quiet necessity. The duality of the Obidi Action Principle (OAP)— local and spectral — was not a conceptual choice but a structural fact.

Even the incorporation of bosonic and fermionic fields into the spectral formulation felt like a continuation of this inevitability. The Dirac–Kähler formalism provided the missing bridge: a geometric language in which matter fields arise naturally from the same spectral operator that governs the entropic field. The moment this connection became clear, the theory expanded not by force but by unfolding — as though the spectral geometry had been waiting for entropy to claim its place within it.

A recurring theme in the history of physics is the search for unity: a single geometric or dynamical substrate from which the diversity of physical phenomena may be derived. Yet many modern approaches fracture this unity by introducing multiple, independent geometric structures that must be artificially coupled. For example, in network‑geometric and Gravity from Entropy (GfE) frameworks such as Bianconi’s, the underlying network possesses one geometry while matter degrees of freedom inhabit a separate Hilbert‑space structure. Because these two structures do not arise from a common source, an additional metric must be imposed on matter to ensure that it “feels” the geometry of the network. This requirement is mathematically permissible but conceptually troubling: it signals that geometry and matter are fundamentally disjoint and must be stitched together with ingenuous dexterity.

The Theory of Entropicity (ToE) takes the opposite stance. In ToE, entropy is the universal field, and the geometry of the universe is the geometry induced by this field. The entropic metric, entropic curvature, and entropic connection all arise from the same underlying entity. Matter fields do not require a separate metric, nor do they need an auxiliary geometric structure to mediate their interaction with spacetime. Instead, matter inherits the entropic geometry naturally, because both matter and geometry are emergent expressions of the same entropic substrate. This single‑source architecture eliminates the need for ad hoc couplings and restores the unity that characterizes the most elegant physical theories: one field, one geometry, one variational principle.

In this architectural unity, the Theory of Entropicity (ToE) aligns with the spirit of Einstein’s General Relativity (GR): a single geometric substrate governs both matter and dynamics. But where GR takes the metric as fundamental, ToE takes entropy as the universal field from which geometry itself emerges.

One of the unexpected consequences of the Theory of Entropicity (ToE) is that many concepts long regarded as ordinary — such as ln 2, information, distinguishability, correlation — reveal themselves as universal structures once entropy is recognized as the fundamental field.

Because ToE identifies entropy as the universal field, it forces a reinterpretation of many things we normally treat as mundane:

• ln 2, the entropy of a binary choice → becomes a curvature invariant

• information, usually a bookkeeping tool → becomes induced geometry

• time dilation, normally a relativistic effect → becomes entropic‑gradient dilation

• entanglement, usually a quantum oddity → becomes sub‑threshold entropic separation

• distinguishability, a trivial concept → becomes a universal divergence threshold

• matter, usually defined independently → becomes a geometric expression of entropy

This is what happens when we identify the true substrate of reality.

The “ordinary” was never ordinary — it was simply unrecognized.

[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:

1. Dynamics become entropic: motion is the relaxation of entropic gradients, not the response to forces.
2. Geometry becomes adaptive: the metric adjusts to the entropic field, so minimizing entropic work simultaneously shapes spacetime.
3. Matter becomes emergent: what we call “mass” or “energy” is the entropic resistance encoded in (T_{mu\nu}(g(S))).
4. 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:

$$\text{geometry}\to \text{entropy}$$

ToE declares:

$$\text{entropy}\to \text{geometry}$$

Instead of:

$$\text{fields}\to \text{entropy}$$

ToE declares:

$$\text{entropy}\to \text{fields}$$

Instead of:

$$\text{<span style="text-wrap-mode: nowrap;">information</span>}<span\; style="text-wrap-mode:\; nowrap;">\to </span>\text{<span style="text-wrap-mode: nowrap;">entropy</span>}\text{<span style="text-wrap-mode: nowrap;"> and entropy→information or</span><span style="font-weight: bold; text-wrap-mode: wrap;">information ↔ entropy</span>}\text{<span style="text-wrap-mode: nowrap;"><p style="white-space: normal;"></p></span>}$$

ToE declares (field-asymmetrically):

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?utm_source=share&utm_medium=member_desktop&rcm=ACoAAAJgE3gBmSb_wGHRH3mJEKgi3aBoI3cxwOk

4) https://open.substack.com/pub/johnobidi/p/authors-preface-and-methodological-ef9?utm_campaign=post-expanded-share&utm_medium=web

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:

$$I_S^{\text{emergent}}={\int }_M{d}^{4}x\sqrt{-g(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]\mathrm{.}$$

2. What This Action Is

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) ${\chi }^2{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2$

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) $\lambda R_{\mathrm{I}\mathrm{G}}[S]$

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)

$$I_S^{\text{emergent}}={\int }_M{d}^{4}x\sqrt{-g(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]\mathrm{.}$$

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 $g=g(S)$:

$$\delta \sqrt{-g(S)}=\frac{1}{2}\sqrt{-g(S)}{g}^{\mu \nu }(S)\frac{\mathrm{\partial }{g}_{\mu \nu }}{\mathrm{\partial }S}\delta S\mathrm{.}$$

This term encodes the entropic origin of geometry.

(b) Variation of the kinetic term

$$\delta (e^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2)=e^{S\mathrm{/}{k}_B}[\frac{1}{k_B}(\mathrm{\nabla }S{)}^2\delta S+2{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}^{\mu }(\delta S)]\mathrm{.}$$

Integrating by parts yields:

$$-2{\mathrm{\nabla }}_{\mu }\left({\chi }^2{e}^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}^{\mu }S\right)\mathrm{.}$$

(c) Variation of the potential

$$\delta V(S)=V^{\mathrm{\prime }}(S)\delta S\mathrm{.}$$

(d) Variation of the information‑geometric curvature

$$\delta R_{\mathrm{I}\mathrm{G}}[S]=\frac{\delta R_{\mathrm{I}\mathrm{G}}}{\delta S}\delta S\mathrm{.}$$

This term is the information‑geometric analogue of the variation of the Ricci scalar in GR.

3. The Entropic Field Equation (Euler–Lagrange Equation)

Collecting all contributions:

$$\boxed{{\displaystyle -2{\chi }^2{\mathrm{\nabla }}_{\mu }\left(e^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}^{\mu }S\right)+{\chi }^2\frac{e^{S\mathrm{/}{k}_B}}{k_B}(\mathrm{\nabla }S{)}^2-V^{\mathrm{\prime }}(S)+\lambda \frac{\delta R_{\mathrm{I}\mathrm{G}}}{\delta S}+\frac{1}{2}\frac{\mathrm{\partial }\ln \sqrt{-g(S)}}{\mathrm{\partial }S}[{\chi }^2{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}]=0.}}$$

This is the Master Entropic Field Equation (Obidi Field Equation - OFE) for the emergent action.

It is ToE‑native and has no analogue in existing physics.

4. Stress–Energy Tensor of the Entropic Field

Varying the action with respect to the metric:

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

This yields:

$$T_{\mu \nu }^{(S)}={\chi }^2{e}^{S\mathrm{/}{k}_B}[{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}_{\nu }S-\frac{1}{2}{g}_{\mu \nu }(\mathrm{\nabla }S{)}^2]-g_{\mu \nu }V(S)+\lambda T_{\mu \nu }^{\mathrm{I}\mathrm{G}}[S]+T_{\mu \nu }^{(g(S))}\mathrm{.}$$

Where:

• $T_{\mu \nu }^{\mathrm{I}\mathrm{G}}[S]$ is the information‑geometric stress tensor

• $T_{\mu \nu }^{(g(S))}$ arises from the entropic dependence of the metric

This is the first time in physics that a stress–energy tensor includes:

• a Boltzmann‑weighted kinetic term

• an information‑geometric curvature contribution

• a metric‑dependence‑on‑entropy term

This is unique to the Theory of Entropicity (ToE).

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 $M$:

$$I_S^{\text{emergent}}={\int }_M{d}^{4}x\sqrt{-g(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]\mathrm{.}$$

Here:

• $\sqrt{-g(S)}$ denotes the square root of the negative determinant of a metric $g_{\mu \nu }(S)$ whose structure is allowed to depend on the entropic field $S$.

• $\chi$ is a coupling constant with appropriate dimensions.

• $k_B$ is Boltzmann’s constant.

• $V(S)$ is an entropic potential.

• $R_{\mathrm{I}\mathrm{G}}[S]$ is an information‑geometric curvature scalar constructed from an information metric and its associated affine connection, both functionals of $S$.

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

$${\mathcal{L}}_S=\sqrt{-g(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]\mathrm{.}$$

The variation $\delta {\mathcal{L}}_S$ with respect to $S$ receives contributions from:

1. The explicit dependence of the integrand on $S$.

2. The dependence of the metric determinant $\sqrt{-g(S)}$ on $S$.

3. The dependence of $R_{\mathrm{I}\mathrm{G}}[S]$ on $S$.

2.1 Variation of the metric determinant

We write

$$\delta \sqrt{-g(S)}=\frac{1}{2}\sqrt{-g(S)}{g}^{\mu \nu }(S)\frac{\mathrm{\partial }{g}_{\mu \nu }(S)}{\mathrm{\partial }S}\delta S\equiv \sqrt{-g(S)}\mathrm{\Gamma }(S)\delta S,$$

where, for convenience, we define

$$\mathrm{\Gamma }(S)\equiv \frac{1}{2}{g}^{\mu \nu }(S)\frac{\mathrm{\partial }{g}_{\mu \nu }(S)}{\mathrm{\partial }S}=\frac{\mathrm{\partial }\ln \sqrt{-g(S)}}{\mathrm{\partial }S}\mathrm{.}$$

This term encodes the entropic dependence of the geometry.

2.2 Variation of the kinetic term

The kinetic term is

$$K(S)={\chi }^2{e}^{S\mathrm{/}{k}_B}({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)\mathrm{.}$$

Its variation is

$$\delta K(S)={\chi }^2[\frac{1}{k_B}{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2\delta S+2e^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}^{\mu }(\delta S)]\mathrm{.}$$

Integrating by parts the second term and neglecting boundary contributions, we obtain

$$\int d^{4}x\sqrt{-g(S)}2{\chi }^2{e}^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}^{\mu }(\delta S)=-\int d^{4}x\sqrt{-g(S)}2{\chi }^2{\mathrm{\nabla }}_{\mu }\left(e^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}^{\mu }S\right)\delta S\mathrm{.}$$

Thus, the kinetic contribution to the variation of the action is

$$\delta I_K=\int d^{4}x\sqrt{-g(S)}[{\chi }^2\frac{e^{S\mathrm{/}{k}_B}}{k_B}(\mathrm{\nabla }S{)}^2-2{\chi }^2{\mathrm{\nabla }}_{\mu }\left(e^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}^{\mu }S\right)]\delta S\mathrm{.}$$

2.3 Variation of the potential

The potential term contributes

$$\delta (-\sqrt{-g(S)}V(S))=-\sqrt{-g(S)}{V}^{\mathrm{\prime }}(S)\delta S-\sqrt{-g(S)}\mathrm{\Gamma }(S)V(S)\delta S\mathrm{.}$$

The second term here will combine with the metric‑variation contributions from the other terms.

2.4 Variation of the information‑geometric curvature

The information‑geometric curvature term contributes

$$\delta (\sqrt{-g(S)}\lambda R_{\mathrm{I}\mathrm{G}}[S])=\sqrt{-g(S)}\lambda \frac{\delta R_{\mathrm{I}\mathrm{G}}}{\delta S}\delta S+\sqrt{-g(S)}\mathrm{\Gamma }(S)\lambda R_{\mathrm{I}\mathrm{G}}[S]\delta S\mathrm{.}$$

2.5 Collecting all contributions

Collecting all terms proportional to $\delta S$, we obtain

$$\delta I_S=\int d^{4}x\sqrt{-g(S)}\{-2{\chi }^2{\mathrm{\nabla }}_{\mu }\left(e^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}^{\mu }S\right)+{\chi }^2\frac{e^{S\mathrm{/}{k}_B}}{k_B}(\mathrm{\nabla }S{)}^2-V^{\mathrm{\prime }}(S)+\lambda \frac{\delta R_{\mathrm{I}\mathrm{G}}}{\delta S}+\mathrm{\Gamma }(S)[{\chi }^2{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]\}\delta S\mathrm{.}$$

Imposing $\delta I_S=0$ for arbitrary $\delta S$ yields the Euler–Lagrange equation:

$$\boxed{{\displaystyle -2{\chi }^2{\mathrm{\nabla }}_{\mu }\left(e^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}^{\mu }S\right)+{\chi }^2\frac{e^{S\mathrm{/}{k}_B}}{k_B}(\mathrm{\nabla }S{)}^2-V^{\mathrm{\prime }}(S)+\lambda \frac{\delta R_{\mathrm{I}\mathrm{G}}}{\delta S}+\mathrm{\Gamma }(S)[{\chi }^2{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]=0.}}$$

This is the Master Entropic Field Equation [the now famous Obidi Field Equation (OFE) of ToE] associated with the emergent entropic action. It is structurally distinct from any known scalar field equation in standard physics and is thus a genuinely ToE‑native dynamical law.

3. Stress–energy tensor of the entropic field

We now derive the stress–energy tensor associated with the entropic field $S$. By definition,

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

The dependence on $g^{\mu \nu }$ enters through:

1. The metric determinant $\sqrt{-g(S)}$.

2. The kinetic term $({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)=g^{\mu \nu }{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}_{\nu }S$.

3. The information‑geometric curvature $R_{\mathrm{I}\mathrm{G}}[S]$, which itself is constructed from an information metric and its connection (and may or may not coincide with the spacetime metric).

For clarity, let us first treat the case where $R_{\mathrm{I}\mathrm{G}}[S]$ is built from a separate information metric and does not vary with the spacetime metric $g_{\mu \nu }$. In that case, the variation with respect to $g^{\mu \nu }$ yields:

$$\delta I_S=\int d^{4}x\{\delta \sqrt{-g(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]+\sqrt{-g(S)}{\chi }^2{e}^{S\mathrm{/}{k}_B}\delta \left(g^{\alpha \beta }{\mathrm{\nabla }}_{\alpha }S{\mathrm{\nabla }}_{\beta }S\right)\}\mathrm{.}$$

We have

$$\delta \sqrt{-g(S)}=-\frac{1}{2}\sqrt{-g(S)}{g}_{\mu \nu }(S)\delta g^{\mu \nu },$$

and

$$\delta \left(g^{\alpha \beta }{\mathrm{\nabla }}_{\alpha }S{\mathrm{\nabla }}_{\beta }S\right)={\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}_{\nu }S\delta g^{\mu \nu }\mathrm{.}$$

Thus,

$$\delta I_S=\int d^{4}x\sqrt{-g(S)}\{-\frac{1}{2}{g}_{\mu \nu }(S)[{\chi }^2{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]+{\chi }^2{e}^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}_{\nu }S\}\delta g^{\mu \nu }\mathrm{.}$$

Therefore,

$$T_{\mu \nu }^{(S)}={\chi }^2{e}^{S\mathrm{/}{k}_B}[{\mathrm{\nabla }}_{\mu }S{\mathrm{\nabla }}_{\nu }S-\frac{1}{2}{g}_{\mu \nu }(S)(\mathrm{\nabla }S{)}^2]+g_{\mu \nu }(S)V(S)-\lambda g_{\mu \nu }(S)R_{\mathrm{I}\mathrm{G}}[S]+T_{\mu \nu }^{(g(S))},$$

where $T_{\mu \nu }^{(g(S))}$ collects additional contributions arising from the explicit dependence of $g_{\mu \nu }$ on $S$ (if one chooses to treat that dependence dynamically).

In the more general case where $R_{\mathrm{I}\mathrm{G}}[S]$ also depends on the spacetime metric, there will be an additional contribution

$$T_{\mu \nu }^{\mathrm{I}\mathrm{G}}[S]=-\frac{2}{\sqrt{-g}}\frac{\delta }{\delta g^{\mu \nu }}(\sqrt{-g(S)}\lambda R_{\mathrm{I}\mathrm{G}}[S]),$$

which plays the role of an information‑geometric stress–energy tensor. This term is structurally analogous to the Einstein tensor contribution in General Relativity, but now arising from an information‑geometric curvature rather than the usual Riemannian curvature.

The full stress–energy tensor can thus be written schematically as

$$T_{\mu \nu }^{(S)}=T_{\mu \nu }^{\text{kin}}[S]+T_{\mu \nu }^{\text{pot}}[S]+T_{\mu \nu }^{\mathrm{I}\mathrm{G}}[S]+T_{\mu \nu }^{(g(S))},$$

with each term encoding a distinct facet of the entropic ontology: local gradients, potential structure, information‑geometric curvature, and entropic dependence of geometry.

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 ${\mathrm{\Sigma }}_t$.

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

$$d{s}^2=-N^{2}d{t}^2+h_{ij}(d{x}^i+N^{i}dt)(d{x}^j+N^{j}dt),$$

where $N$ is the lapse, $N^i$ the shift, and $h_{ij}$ the induced spatial metric on ${\mathrm{\Sigma }}_t$. The determinant decomposes as

$$\sqrt{-g(S)}=N\sqrt{h(S)},$$

where $h(S)$ may also depend on $S$ if the spatial metric is entropically deformed.

The kinetic term becomes

$$({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)=-\frac{1}{N^2}({\mathrm{\partial }}_{t}S-N^i{\mathrm{\partial }}_{i}S{)}^2+h^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_{j}S\mathrm{.}$$

Thus, the Lagrangian density can be written as

$${\mathcal{L}}_S=N\sqrt{h(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}(-\frac{1}{N^2}({\mathrm{\partial }}_{t}S-N^i{\mathrm{\partial }}_{i}S{)}^2+h^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_{j}S)-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]\mathrm{.}$$

The canonical momentum conjugate to $S$ is

$${\mathrm{\Pi }}_S=\frac{\mathrm{\partial }{\mathcal{L}}_S}{\mathrm{\partial }({\mathrm{\partial }}_{t}S)}=-2{\chi }^2\sqrt{h(S)}{e}^{S\mathrm{/}{k}_B}\frac{1}{N}({\mathrm{\partial }}_{t}S-N^i{\mathrm{\partial }}_{i}S)\mathrm{.}$$

Solving for ${\mathrm{\partial }}_{t}S$, we obtain

$${\mathrm{\partial }}_{t}S=N^i{\mathrm{\partial }}_{i}S-\frac{N}{2{\chi }^2\sqrt{h(S)}}{e}^{-S\mathrm{/}{k}_B}{\mathrm{\Pi }}_S\mathrm{.}$$

The Hamiltonian density is then

$${\mathcal{H}}_S={\mathrm{\Pi }}_S{\mathrm{\partial }}_{t}S-{\mathcal{L}}_S\mathrm{.}$$

Substituting the expression for ${\mathrm{\partial }}_{t}S$ and simplifying, we find

$${\mathcal{H}}_S=N{\mathcal{H}}_{\perp }^{(S)}+N^i{\mathcal{H}}_i^{(S)},$$

where

$${\mathcal{H}}_{\perp }^{(S)}=\frac{1}{4{\chi }^2\sqrt{h(S)}}{e}^{-S\mathrm{/}{k}_B}{\mathrm{\Pi }}_S^2+{\chi }^2\sqrt{h(S)}{e}^{S\mathrm{/}{k}_B}{h}^{ij}{\mathrm{\partial }}_{i}S{\mathrm{\partial }}_{j}S+\sqrt{h(S)}V(S)-\lambda \sqrt{h(S)}{R}_{\mathrm{I}\mathrm{G}}[S],$$

and

$${\mathcal{H}}_i^{(S)}={\mathrm{\Pi }}_S{\mathrm{\partial }}_{i}S\mathrm{.}$$

The Hamiltonian formulation thus reveals:

• A non‑standard kinetic structure weighted by $e^{\pm S\mathrm{/}{k}_B}$.

• A geometric dependence through $\sqrt{h(S)}$.

• A curvature contribution from $R_{\mathrm{I}\mathrm{G}}[S]$.

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 the covariant conservation law

$${\mathrm{\nabla }}^{\mu }{T}_{\mu \nu }^{\text{(total)}}=0.$$

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

$${\mathrm{\nabla }}^{\mu }{T}_{\mu \nu }^{(S)}+{\mathrm{\nabla }}^{\mu }{T}_{\mu \nu }^{(\text{other})}=0.$$

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

• entropic gradients,

• geometric curvature,

• and information‑geometric structure.

If the theory also possesses internal symmetries in the space of entropic configurations (for example, shifts $S\to S+\text{const}$ under certain conditions), then Noether’s theorem yields additional conserved currents. For instance, if the action is invariant under $S\to S+ϵ$ for constant $ϵ$, then the associated Noether current is

$$J^{\mu }=\frac{\mathrm{\partial }{\mathcal{L}}_S}{\mathrm{\partial }({\mathrm{\nabla }}_{\mu }S)}\delta S={\chi }^2\sqrt{-g(S)}{e}^{S\mathrm{/}{k}_B}2{\mathrm{\nabla }}^{\mu }S,$$

up to possible corrections from the metric dependence. The conservation law ${\mathrm{\nabla }}_{\mu }{J}^{\mu }=0$ then expresses a form of entropic charge conservation, which can be interpreted as a conservation of entropic flux in the underlying ontology.

6. Relation to the Local and Spectral Obidi Actions

The emergent entropic action $I_S^{\text{emergent}}$ 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.). In schematic form, one may write

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

where ${\mathcal{L}}_{\text{local}}$ includes generalized entropic densities and their geometric couplings.

The emergent entropic action $I_S^{\text{emergent}}$ can be viewed as a refined local realization of this principle, in which:

• the kinetic term is explicitly Boltzmann‑weighted,

• the metric is allowed to depend on $S$,

• and an information‑geometric curvature scalar $R_{\mathrm{I}\mathrm{G}}[S]$ is included.

In this sense, $I_S^{\text{emergent}}$ is a concrete instantiation of the LOA, with explicit choices for the entropic weighting and geometric coupling.

6.2 Spectral Obidi Action (SOA)

The SOA arises when entropy is treated as a spectral quantity associated with operators on a Hilbert space [or a noncommutative algebra in the case of Alain Connes et al]. In analogy with the spectral action principle, one writes [for example like Alain Connes - the Connes‑style structure]

$$I_{\text{SOA}}=\text{Tr}f({\mathcal{D}}_S\mathrm{/}\mathrm{\Lambda }),$$

where:

• ${\mathcal{D}}_S$ is an entropic spectral operator (e.g., a Dirac–Kähler–type operator coupled to the entropic field),

• $\mathrm{\Lambda }$ is a cutoff scale,

• $f$ is a suitable test function.

The heat‑kernel expansion of this spectral action yields terms of the form

$$I_{\text{SOA}}\sim {\int }_M{d}^{4}x\sqrt{-g}[c_0{\mathrm{\Lambda }}^4+c_2{\mathrm{\Lambda }}^{2}R+c_4(R^2,R_{\mu \nu }{R}^{\mu \nu },F_{\mu \nu }{F}^{\mu \nu },\dots )+\dots ],$$

with coefficients depending on the spectral data of ${\mathcal{D}}_S$. When the entropic field $S$ enters ${\mathcal{D}}_S$, the resulting spectral action naturally generates:

• entropic curvature terms,

• entropic couplings to matter fields (bosonic and fermionic),

• and higher‑order geometric invariants.

The information‑geometric curvature $R_{\mathrm{I}\mathrm{G}}[S]$ appearing in $I_S^{\text{emergent}}$ can be interpreted as a local shadow of the spectral curvature encoded in the SOA. In other words, the emergent entropic action captures, at the local level, the geometric content that the spectral action encodes at the operator level.

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),

• and a spectral action (SOA),

which are not independent but mutually constraining. The emergent entropic action $I_S^{\text{emergent}}$ thus occupies a central place in this duality:

• It provides 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 a key component of the unified entropic architecture of the Theory of Entropicity.

7. Conceptual summary

The emergent entropic action

$$I_S^{\text{emergent}}={\int }_M{d}^{4}x\sqrt{-g(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)-V(S)+\lambda R_{\mathrm{I}\mathrm{G}}[S]]$$

embodies, in a single expression, the core commitments of the Theory of Entropicity:

• Entropy as a field: $S(x)$ is not a derived quantity but a fundamental dynamical field.

• Entropic geometry: the metric determinant $\sqrt{-g(S)}$ depends on $S$, encoding the entropic origin of geometry.

• Thermodynamic weighting: the kinetic term is weighted by $e^{S\mathrm{/}{k}_B}$, embedding thermodynamic structure directly into the dynamics.

• Information‑geometric curvature: $R_{\mathrm{I}\mathrm{G}}[S]$ links entropy to information geometry and curvature, unifying thermodynamics, information, and geometry.

• ToE‑native dynamics: the resulting field equation and stress–energy tensor have no analogue in existing physics; they arise uniquely from the entropic ontology.

In this sense, the emergent entropic action is both a culmination and a beginning: it crystallizes the logical trajectory from classical entropy to information, from information to geometry, from geometry to curvature, and from curvature to field dynamics—while opening the way to further developments in the local and spectral formulations of the Theory of Entropicity (ToE).

Appendix: Extra Matter - Mathematical Difference Between Obidi Spectral Action (SOA) and the Alain Connes' Triple Spectra Action Mechanism from Non-Commutative Geometry

In ToE, the spectral structure is not imposed from operator algebras (like those of Alain Connes formalism); rather, it [ToE's Spectral Obidi Action] emerges from the spectral character of entropy itself.

For the sake of continuity and understanding, we present below the ToE‑native formulation.

1. The Spectral Nature of Entropy in ToE

In the Theory of Entropicity, entropy is not merely a scalar field $S(x)$. It has a dual nature:

• local: through gradients, curvature, and entropic geometry

• spectral: through the distribution of entropic modes, entropic frequencies, and entropic eigenstructures

This spectral character [strictly] does not come from a Dirac operator on a noncommutative space. It comes from the fact that entropy generates informational structure, and informational structure has:

• eigenvalues

• spectra

• modes

• operators

• kernels

• resolvents

This is the information‑theoretic spectrum, not the Connes [non-commutative geometric] spectrum.

2. The ToE‑Native Spectral Operator

Instead of Connes’ Dirac operator $D$, ToE introduces:

$$\mathcal{S}\equiv \text{the entropic spectral operator},$$

defined not on a noncommutative algebra but on the information manifold generated by entropy.

Its spectrum is:

$$\text{Spec}(\mathcal{S})=\{{\sigma }_i\},$$

where each ${\sigma }_i$ corresponds to an entropic mode, not a geometric eigenvalue of a Dirac operator.

This is a completely different ontology.

3. The True Spectral Obidi Action (SOA)

Not Connes. Not an analogy. Entirely ToE.

The SOA is therefore uniquely defined as:

$$I_{\text{SOA}}=\mathrm{\Phi }(\text{Spec}(\mathcal{S})),$$

where $\mathrm{\Phi }$ is a functional that measures the global spectral content of entropy.

This is the most general and most direct formulation.

No trace. No cutoff. No test function. No heat kernel. No Dirac operator. No noncommutative geometry.

Instead:

• entropy generates a spectrum

• the spectrum generates a global invariant

• the invariant is the spectral action

This is the ToE‑native structure.

4. A More Explicit ToE‑Native Form

Therefore, given the above considerations and insights, we can write:

$$I_{\text{SOA}}=\sum _i\mathrm{\Psi }({\sigma }_i),$$

where:

• ${\sigma }_i$ are the entropic eigenvalues

• $\mathrm{\Psi }$ is a spectral weight functional determined by the entropic ontology

This is thus analogous in form to spectral actions, but not in method or origin.

5. Why This Is Fundamentally Different from Alain Connes style of Non-Commutative Geometry 

Alain Connes’ spectral action:

• uses a Dirac operator

• uses a trace

• uses a cutoff

• uses a test function

• uses heat‑kernel asymptotics

• is tied to noncommutative geometry

• produces curvature invariants of spacetime

ToE's SOA:

• uses an entropic spectral operator

• uses entropic eigenvalues

• uses no cutoff

• uses no test function

• uses no heat kernel

• is tied to information geometry, not noncommutative geometry

• produces entropic invariants, not Riemannian ones

They are categorically different.